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https://math.stackexchange.com/questions/3378740/whats-the-difference-between-glpp-and-lpp | # What's the difference between GLPP and LPP
$$a)$$ Express the following optimization problem as a linear programming problem(LPP): $$\text{maximize }3x+3y-30$$$$\text{subject to }|x-2|+|y|\le5$$
Hint: you will need to express the inequality in another way so there is no absolute signs. For example the inequality $$|x|\le3$$ is equivalent to the pair of inequilities $$x\le3\wedge x\ge-3$$
$$b)$$ Explain why the following optimization problem is not a LPP: $$\text{maximize }3x+3y-30$$$$\text{subject to }|x-2|-|y|\le5$$
$$a)$$ Consider $$|x-2|+|y|\le5$$
$$\Leftrightarrow x-2+|y|\le5\wedge 2-x+|y|\le5$$
$$\Leftrightarrow |y|\le7-x\wedge |y|\le3+x$$
$$\Leftrightarrow (y\le-x+7\wedge y\ge x-7)\wedge(y\le x+3\wedge y\ge-x-3)$$
Basicly we have
maximize
$$3x+3y-30$$
subject to
$$y+x\le7$$$$x-y\le 7$$$$-x+y\le 3$$$$-x-y\le3$$
Which is in the form of general linear programming problem(GLPP)
$$b)$$
Definition(GLPP)
The general linear programming problem can be stated as follows:
Find values of $$x_1,x_2,\dots,x_n$$ that will
maximize or minimize $$z=c_1x_1+c_2x_2+\dots+c_nx_n\tag*{(1)}$$
subject to the restrictions $$\left.\begin{array}{r}a_{11}x_1+a_{12}x_2+\dots+a_{1n}x_n\le(\ge)(=)b_1\\a_{21}x_1+a_{22}x_2+\dots+a_{2n}x_n\le(\ge)(=)b_2\\\color{lightgrey}{\text{by the way, can anyone help me eidt those dots to\color{darkwhite}{\text{ (about here)}} center, thx}\rightarrow}\vdots\\a_{m1}x_1+a_{m2}x_2+\dots+a_{mn}x_n\le(\ge)(=)b_m\end{array}\right\}\tag*{(2)}$$
where in each inequality in $$(2)$$ one and only one of the symbols, $$\le,\ge,=$$occurs.
(from "Elementary Linear Programming With Application" by Bernard Kolman $$\cdot$$ Robert E. Beck)
I think there are two reasons that make it isn't a general linear programming problem
First it has costant on left side of inequality which isn't in the form of GLPP
Second it has absolute value apears in the inequality, that also not allowed in GLPP
Does this answers $$b)?$$
I only found definition for GLPP from the book, is it same as LPP $$?$$
Thanks for your help.
• A GLP(P) is an LP(P). I think a GLP(P) should be distinguished from an LP(P) in standard form. GLP(P) is typically the abbreviation for a generalized LP (which is not an LP). – LinAlg Oct 3 '19 at 1:09 | 2020-01-22 14:33:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 27, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8440231084823608, "perplexity": 742.8204566640071}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607118.51/warc/CC-MAIN-20200122131612-20200122160612-00371.warc.gz"} |
http://mathhelpforum.com/calculus/203696-applications-integration-probability.html | # Math Help - applications of integration to probability
1. ## applications of integration to probability
how are constraints used to modify a joint density function?
2. ## Re: applications of integration to probability
Hey Kvandesterren.
Basically if you have an unconstrained PDF, you can select a subset and use that as your new probability space (you may have to normalize it if it's not normalized yet).
If the PDF is defined, then usually what happens is you restrict the domain. So for example you might have a normal distribution but put a constraint that X >= 0 (But this still behaves the same way probabilistically as the un-constrained). So if this is the case, you mask out the negative values, re-normalize and this becomes your distribution.
You also have what I call "conditional slicing". Basically all this means is in a joint distribution, you fix one parameter and then you get the distribution for the other given whatever fixed parameter. Mathematically it would look something like P(Y|X = x) or P(Y|X < x) or something similar. You don't have to fix one value though: you can do something like P(Y|a < X < b) or something along those lines.
So in short, you start with some unconstrained density function (possibly many variables) and then you select a subset of that probability space and typically if you want to treat this as a PDF, then you normalize it and it becomes its own distribution and you can do all the things you do with a normal distribution like expectation, variance, cumulative probability, whatever. | 2014-09-23 03:13:12 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8189361095428467, "perplexity": 467.61106064179637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657137906.42/warc/CC-MAIN-20140914011217-00215-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://slawekk.wordpress.com/ | ## IsarMathLib 1.20.0: Ordered loops and loop valued metrics
April 28, 2021
Just to be clear, the loops in the title are algebraic structures and are not related to continuous images of circles or flow control constructs in programming languages.
Some time ago I got curious about a minimal setup in which a (pseudo) metric defined on a set would result in making this set a topological space. It turned out that if the metric is defined as a function with values in an partially ordered group we can use the standard definition of an open disk and the collection of unions of such open disks is a topology. However, I noticed then that this should be possible to be pushed a step farther and one should be able to define a metric as a function valued in an ordered loop and still get a topological space. In this release I added some material on ordered loops and rewrote the metric spaces section to be based on metrics valued in ordered loops.
Loops (the algebraic structures) are similar to groups, except that their binary operation does not have to be associative (I wrote more about loops in the previous post). An ordered loop (see for example Lattice-ordered loops and quasigroups by T. Evans) is a loop with a partial order such that $x\leq y$ if and only if $x+z\leq y+z$ and $x\leq y$ if and only if $z+x\leq z+y$. The “if and only if” is worth noting here as this is different from the ordered group definition where one-sided implications are sufficient (I lost a couple of hours working with a wrong definition).
## IsarMathLib 1.19.0: Loops
April 6, 2021
This release adds a theory file about loops. Loops (the kind we talk about here, not to be confused with for example loop algebras) are algebraic structures that are almost like groups, except that we don’t require the operation to be associative. Wikipedia has a nice picture illustrating dependencies between various algebraic structures between magmas and groups which I copy below:
As the picture shows we get a loop when we take a quasigroup and require that an identity (a neutral element of the operation) exists. It can be shown that that identity is unique in the loop. If we denote the loop identity as $e$, then we can define the left and right inverses of a loop element $a$ as the only solutions to the equations $x\cdot a = e$ and $a\cdot x = e$. In a loop these inverses do not have to be the same in general. If the loop operation is associative then we get a group and the left and right inverses unify to become the group inverse.
## IsarMathLib 1.18.0: Quasigroups
March 24, 2021
A quasigroup is a set with a binary operation (usually written with the multiplicative notation) that supports a certain kind of “divisibility”: every equation of the form $a\cdot x = b$ or $x\cdot a = b$ has a unique solution. So, a quasigroup is like a group except we do not require associativity or existence of a neutral element. If we denote those solution as $a\backslash b$ and $b/a$ respectively we get the identities $a\cdot (a\backslash b) = b$ and $(b/a)\cdot a = b$ by definition and, with the uniqueness, we get $a\backslash (a\cdot b) = b$ and $(b\cdot a)/a =b$, which may serve as an alternative definition of quasigroup.
According to this paper “One of the mathematical subfields where automated theorem provers are heavily used is the field of quasigroup and loop theory”.
## IsarMathLib 1.17.0: Real numbers as a metric and topological space
March 7, 2021
In this release I added a new theory about real numbers. It’s mostly about setting up notation and validating other contexts (related to ordered fields, rings, groups, metric spaces etc.) so that theorems formulated in those contexts can be used when considering real numbers. At the end there are two theorems about the distance function $(x,y) \mapsto |x-y|$ being a metric and one stating that the collection of unions of open disks is a topology on reals.
IsarMathLib had been upgraded to Isabelle2021 in the previous release 1.16.1 (not announced on this blog) but at that time I didn’t notice that the proof documents were not generated properly. The pdf files generated by Isabelle didn’t get the formal text right, some symbols (like $\{$ and $\}$, very important in mathematics based on set theory) were replaced by some placeholder symbol. Errors related to LaTeX are difficult to debug in Isabelle (at least for me), but this time I was lucky. The Isabelle2021 release notes contained information that the “The standard LaTeX engine is now lualatex, according to settings
variable ISABELLE_PDFLATEX.” . Indeed changing the ISABELLE_PDFLATEX="lualatex --file-line-error" line in the settings file back to ISABELLE_PDFLATEX="pdflatex -file-line-error" like it was in Isabelle2020 fixed the problem.
Read the rest of this entry »
## IsarMathLib version 1.16.0: metric spaces
February 1, 2021
In this release I added a new theory on (pseudo-)metric spaces. Daniel de la Concepción Sáez contributed a theorem stating that in topological groups the group inverse is uniformly continuous with respect to the Roelcke uniformity.
A pseudo-metric is typically defined as a function $d: X\times X \rightarrow [0,\infty)$ that satisfies the conditions $d(x,x) = 0$, $d(x,y) = 0 \Rightarrow x=y$ and $d(x,z) \leq d(x,y)+d(y,z)$ for all $x,y,z \in X$. This is sufficient to prove that the collection of subsets of $X$ that are unions of disks (defined as $\text{disk}(c,r) = \{x\in X: d(c,x)) is a topology, making this metric space a topological space. If we replace the interval $[0,\infty)$ in the definition of pseudo-metric by the nonnegative set of a linearly ordered, abelian, Archimedean group then we are not changing anything as such group is isomorphic to a subgroup of the additive group of real numbers. I was curious: what is really needed to get a topological space from a pseudo-metric space this way? Do we really need the group to be Archimedean? Does the group operation have to be commutative? Do we need the order to be total? Amusingly, the answer to these questions is “no”. The group does not have to be abelian and the Archimedean property is not needed. The linear order condition can be relaxed by assuming that the positive set is a meet semi-lattice (every two-element subset has a greatest lower bound) under the group order relation. I don’t know if this is useful to know, but surely my personal curiosity got satisfied.
## IsarMathLib version 1.15.0: comparison with Mizar
December 13, 2020
Some time ago Victor Makarov asked on the Mizar forum about two theorems in group theory. The first theorem states that the union of two subgroups is a subgroup iff one of the subgroups is a subset of the other subgroup. The other one gives a criterion for when for two subgroups $H,K$ the set $H\cdot K = \{x\cdot y | x\in H,y\in K \}$ is also a subgroup. These theorems turned out not to be in the Mizar Mathematical Library, but Roland Coghetto posted formalization of those two theorems in the Mizar proof language. In this release I added the formalization of those two theorems to IsarMathLib so that one can easily compare the syntax and style of Isabelle’s Isar proof language (as it is used in IsarMathLib) and the Mizar proof language.
The result can be viewed as theorems named union_subgroups and prod_subgr_subgr. For the proof of the first one I followed the informal source on Stack Exchange provided by Roland, so it can be directly compared to its Mizar version. The proof of the second one is probably different than the Mizar version, but I think they are still useful as examples of the same theorem in different proof assistants.
## IsarMathLib version 1.14.0: lattices
September 29, 2020
I released the next version of IsarMathLib. This release adds a short theory file about lattices with definitions, notation and a proof that join and and meet-semilattices form idempotent ($x\cdot x = x$) semigroups.
## IsarMathLib version 1.13.0: Roelcke uniformity
August 3, 2020
I have released version 1.13.0 of IsarMathLib. This release adds the definition and basic properties of the Roelcke uniformity, contributed by Daniel de la Concepción Sáez.
So, what is Roelcke uniformity? As we all know each topological group is a uniform space with two natural uniformities: the left uniformity and right uniformity. The collection of uniformities on a set forms a lattice (in fact, a complete one) so it is natural to ask what are the meet and join of the left and right uniformities. The meet is called the Roelcke uniformity. More details with references can be found in answers to this StackExchange question and the formalization is available in the IsarMathLib’s TopologicalGroup_Uniformity_ZF theory.
## IsarMathLib version 1.12.1: update to Isabelle2020
May 12, 2020
This version updates IsarMathLib to Isabelle2020. It was the easiest update ever – nothing had to be changed, so I just changed the abstract in the proof document with the project description.
## IsarMathLib version 1.12.0: topological groups and uniformities
May 5, 2020
This release adds three theory files contributed by Daniel de la Concepción Sáez. Two of them, Topology_ZF_properties_2 and Topology_ZF_properties_3 have been in IsarMathLib for a long time, but I forgot to add them to the file that lists theories to be presented, so they were not showing up in the Isabelle generated HTML presentation or the proof document. Now they do.
The third theory TopologicalGroup_Uniformity_ZF (see also the section Topological groups – uniformity in the proof document or the outline) is brand new in this release and contains a proof that every topological group is a uniform space. | 2021-05-09 02:03:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 30, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7511842846870422, "perplexity": 540.5059072188077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988953.13/warc/CC-MAIN-20210509002206-20210509032206-00011.warc.gz"} |
https://python-advanced.quantecon.org/black_litterman.html | # Two Modifications of Mean-Variance Portfolio Theory¶
## Overview¶
### Remarks About Estimating Means and Variances¶
The famous Black-Litterman (1992) [BL92] portfolio choice model that we describe in this lecture is motivated by the finding that with high or moderate frequency data, means are more difficult to estimate than variances.
A model of robust portfolio choice that we’ll describe also begins from the same starting point.
To begin, we’ll take for granted that means are more difficult to estimate that covariances and will focus on how Black and Litterman, on the one hand, an robust control theorists, on the other, would recommend modifying the mean-variance portfolio choice model to take that into account.
At the end of this lecture, we shall use some rates of convergence results and some simulations to verify how means are more difficult to estimate than variances.
Among the ideas in play in this lecture will be
• Mean-variance portfolio theory
• Bayesian approaches to estimating linear regressions
• A risk-sensitivity operator and its connection to robust control theory
In [1]:
import numpy as np
import scipy as sp
import scipy.stats as stat
import matplotlib.pyplot as plt
%matplotlib inline
from ipywidgets import interact, FloatSlider
### Adjusting Mean-variance Portfolio Choice Theory for Distrust of Mean Excess Returns¶
This lecture describes two lines of thought that modify the classic mean-variance portfolio choice model in ways designed to make its recommendations more plausible.
As we mentioned above, the two approaches build on a common and widespread hunch – that because it is much easier statistically to estimate covariances of excess returns than it is to estimate their means, it makes sense to contemplated the consequences of adjusting investors’ subjective beliefs about mean returns in order to render more sensible decisions.
Both of the adjustments that we describe are designed to confront a widely recognized embarrassment to mean-variance portfolio theory, namely, that it usually implies taking very extreme long-short portfolio positions.
### Mean-variance Portfolio Choice¶
A risk-free security earns one-period net return $r_f$.
An $n \times 1$ vector of risky securities earns an $n \times 1$ vector $\vec r - r_f {\bf 1}$ of excess returns, where ${\bf 1}$ is an $n \times 1$ vector of ones.
The excess return vector is multivariate normal with mean $\mu$ and covariance matrix $\Sigma$, which we express either as
$$\vec r - r_f {\bf 1} \sim {\mathcal N}(\mu, \Sigma)$$
or
$$\vec r - r_f {\bf 1} = \mu + C \epsilon$$
where $\epsilon \sim {\mathcal N}(0, I)$ is an $n \times 1$ random vector.
Let $w$ be an $n \times 1$ vector of portfolio weights.
A portfolio consisting $w$ earns returns
$$w' (\vec r - r_f {\bf 1}) \sim {\mathcal N}(w' \mu, w' \Sigma w )$$
The mean-variance portfolio choice problem is to choose $w$ to maximize
$$U(\mu,\Sigma;w) = w'\mu - \frac{\delta}{2} w' \Sigma w \tag{1}$$
where $\delta > 0$ is a risk-aversion parameter. The first-order condition for maximizing (1) with respect to the vector $w$ is
$$\mu = \delta \Sigma w$$
which implies the following design of a risky portfolio:
$$w = (\delta \Sigma)^{-1} \mu \tag{2}$$
### Estimating the Mean and Variance¶
The key inputs into the portfolio choice model (2) are
• estimates of the parameters $\mu, \Sigma$ of the random excess return vector$(\vec r - r_f {\bf 1})$
• the risk-aversion parameter $\delta$
A standard way of estimating $\mu$ is maximum-likelihood or least squares; that amounts to estimating $\mu$ by a sample mean of excess returns and estimating $\Sigma$ by a sample covariance matrix.
### The Black-Litterman Starting Point¶
When estimates of $\mu$ and $\Sigma$ from historical sample means and covariances have been combined with reasonable values of the risk-aversion parameter $\delta$ to compute an optimal portfolio from formula (2), a typical outcome has been $w$’s with extreme long and short positions.
A common reaction to these outcomes is that they are so unreasonable that a portfolio manager cannot recommend them to a customer.
In [2]:
np.random.seed(12)
N = 10 # Number of assets
T = 200 # Sample size
# random market portfolio (sum is normalized to 1)
w_m = np.random.rand(N)
w_m = w_m / (w_m.sum())
# True risk premia and variance of excess return (constructed
# so that the Sharpe ratio is 1)
μ = (np.random.randn(N) + 5) /100 # Mean excess return (risk premium)
S = np.random.randn(N, N) # Random matrix for the covariance matrix
V = S @ S.T # Turn the random matrix into symmetric psd
# Make sure that the Sharpe ratio is one
Σ = V * (w_m @ μ)**2 / (w_m @ V @ w_m)
# Risk aversion of market portfolio holder
δ = 1 / np.sqrt(w_m @ Σ @ w_m)
# Generate a sample of excess returns
excess_return = stat.multivariate_normal(μ, Σ)
sample = excess_return.rvs(T)
# Estimate μ and Σ
μ_est = sample.mean(0).reshape(N, 1)
Σ_est = np.cov(sample.T)
w = np.linalg.solve(δ * Σ_est, μ_est)
fig, ax = plt.subplots(figsize=(8, 5))
ax.set_title('Mean-variance portfolio weights recommendation \
and the market portfolio')
ax.plot(np.arange(N)+1, w, 'o', c='k', label='$w$ (mean-variance)')
ax.plot(np.arange(N)+1, w_m, 'o', c='r', label='$w_m$ (market portfolio)')
ax.vlines(np.arange(N)+1, 0, w, lw=1)
ax.vlines(np.arange(N)+1, 0, w_m, lw=1)
ax.axhline(0, c='k')
ax.axhline(-1, c='k', ls='--')
ax.axhline(1, c='k', ls='--')
ax.set_xlabel('Assets')
ax.xaxis.set_ticks(np.arange(1, N+1, 1))
plt.legend(numpoints=1, fontsize=11)
plt.show()
Black and Litterman’s responded to this situation in the following way:
• They continue to accept (2) as a good model for choosing an optimal portfolio $w$.
• They want to continue to allow the customer to express his or her risk tolerance by setting $\delta$.
• Leaving $\Sigma$ at its maximum-likelihood value, they push $\mu$ away from its maximum value in a way designed to make portfolio choices that are more plausible in terms of conforming to what most people actually do.
In particular, given $\Sigma$ and a reasonable value of $\delta$, Black and Litterman reverse engineered a vector $\mu_{BL}$ of mean excess returns that makes the $w$ implied by formula (2) equal the actual market portfolio $w_m$, so that
$$w_m = (\delta \Sigma)^{-1} \mu_{BL}$$
### Details¶
Let’s define
$$w_m' \mu \equiv ( r_m - r_f)$$
as the (scalar) excess return on the market portfolio $w_m$.
Define
$$\sigma^2 = w_m' \Sigma w_m$$
as the variance of the excess return on the market portfolio $w_m$.
Define
$${\bf SR}_m = \frac{ r_m - r_f}{\sigma}$$
as the Sharpe-ratio on the market portfolio $w_m$.
Let $\delta_m$ be the value of the risk aversion parameter that induces an investor to hold the market portfolio in light of the optimal portfolio choice rule (2).
Evidently, portfolio rule (2) then implies that $r_m - r_f = \delta_m \sigma^2$ or
$$\delta_m = \frac{r_m - r_f}{\sigma^2}$$
or
$$\delta_m = \frac{{\bf SR}_m}{\sigma}$$
Following the Black-Litterman philosophy, our first step will be to back a value of $\delta_m$ from
• an estimate of the Sharpe-ratio, and
• our maximum likelihood estimate of $\sigma$ drawn from our estimates or $w_m$ and $\Sigma$
The second key Black-Litterman step is then to use this value of $\delta$ together with the maximum likelihood estimate of $\Sigma$ to deduce a $\mu_{\bf BL}$ that verifies portfolio rule (2) at the market portfolio $w = w_m$
$$\mu_m = \delta_m \Sigma w_m$$
The starting point of the Black-Litterman portfolio choice model is thus a pair $(\delta_m, \mu_m)$ that tells the customer to hold the market portfolio.
In [3]:
# Observed mean excess market return
r_m = w_m @ μ_est
# Estimated variance of the market portfolio
σ_m = w_m @ Σ_est @ w_m
# Sharpe-ratio
sr_m = r_m / np.sqrt(σ_m)
# Risk aversion of market portfolio holder
d_m = r_m / σ_m
# Derive "view" which would induce the market portfolio
μ_m = (d_m * Σ_est @ w_m).reshape(N, 1)
fig, ax = plt.subplots(figsize=(8, 5))
ax.set_title(r'Difference between $\hat{\mu}$ (estimate) and \
$\mu_{BL}$ (market implied)')
ax.plot(np.arange(N)+1, μ_est, 'o', c='k', label='$\hat{\mu}$')
ax.plot(np.arange(N)+1, μ_m, 'o', c='r', label='$\mu_{BL}$')
ax.vlines(np.arange(N) + 1, μ_m, μ_est, lw=1)
ax.axhline(0, c='k', ls='--')
ax.set_xlabel('Assets')
ax.xaxis.set_ticks(np.arange(1, N+1, 1))
plt.legend(numpoints=1)
plt.show()
Black and Litterman start with a baseline customer who asserts that he or she shares the market’s views, which means that he or she believes that excess returns are governed by
$$\vec r - r_f {\bf 1} \sim {\mathcal N}( \mu_{BL}, \Sigma) \tag{3}$$
Black and Litterman would advise that customer to hold the market portfolio of risky securities.
Black and Litterman then imagine a consumer who would like to express a view that differs from the market’s.
The consumer wants appropriately to mix his view with the market’s before using (2) to choose a portfolio.
Suppose that the customer’s view is expressed by a hunch that rather than (3), excess returns are governed by
$$\vec r - r_f {\bf 1} \sim {\mathcal N}( \hat \mu, \tau \Sigma)$$
where $\tau > 0$ is a scalar parameter that determines how the decision maker wants to mix his view $\hat \mu$ with the market’s view $\mu_{\bf BL}$.
Black and Litterman would then use a formula like the following one to mix the views $\hat \mu$ and $\mu_{\bf BL}$
$$\tilde \mu = (\Sigma^{-1} + (\tau \Sigma)^{-1})^{-1} (\Sigma^{-1} \mu_{BL} + (\tau \Sigma)^{-1} \hat \mu) \tag{4}$$
Black and Litterman would then advise the customer to hold the portfolio associated with these views implied by rule (2):
$$\tilde w = (\delta \Sigma)^{-1} \tilde \mu$$
This portfolio $\tilde w$ will deviate from the portfolio $w_{BL}$ in amounts that depend on the mixing parameter $\tau$.
If $\hat \mu$ is the maximum likelihood estimator and $\tau$ is chosen heavily to weight this view, then the customer’s portfolio will involve big short-long positions.
In [4]:
def black_litterman(λ, μ1, μ2, Σ1, Σ2):
"""
This function calculates the Black-Litterman mixture
mean excess return and covariance matrix
"""
Σ1_inv = np.linalg.inv(Σ1)
Σ2_inv = np.linalg.inv(Σ2)
μ_tilde = np.linalg.solve(Σ1_inv + λ * Σ2_inv,
Σ1_inv @ μ1 + λ * Σ2_inv @ μ2)
return μ_tilde
τ = 1
μ_tilde = black_litterman(1, μ_m, μ_est, Σ_est, τ * Σ_est)
# The Black-Litterman recommendation for the portfolio weights
w_tilde = np.linalg.solve(δ * Σ_est, μ_tilde)
τ_slider = FloatSlider(min=0.05, max=10, step=0.5, value=τ)
@interact(τ=τ_slider)
def BL_plot(τ):
μ_tilde = black_litterman(1, μ_m, μ_est, Σ_est, τ * Σ_est)
w_tilde = np.linalg.solve(δ * Σ_est, μ_tilde)
fig, ax = plt.subplots(1, 2, figsize=(16, 6))
ax[0].plot(np.arange(N)+1, μ_est, 'o', c='k',
label=r'$\hat{\mu}$ (subj view)')
ax[0].plot(np.arange(N)+1, μ_m, 'o', c='r',
label=r'$\mu_{BL}$ (market)')
ax[0].plot(np.arange(N)+1, μ_tilde, 'o', c='y',
label=r'$\tilde{\mu}$ (mixture)')
ax[0].vlines(np.arange(N)+1, μ_m, μ_est, lw=1)
ax[0].axhline(0, c='k', ls='--')
ax[0].set(xlim=(0, N+1), xlabel='Assets',
title=r'Relationship between $\hat{\mu}$, \
$\mu_{BL}$and$\tilde{\mu}$')
ax[0].xaxis.set_ticks(np.arange(1, N+1, 1))
ax[0].legend(numpoints=1)
ax[1].set_title('Black-Litterman portfolio weight recommendation')
ax[1].plot(np.arange(N)+1, w, 'o', c='k', label=r'$w$ (mean-variance)')
ax[1].plot(np.arange(N)+1, w_m, 'o', c='r', label=r'$w_{m}$ (market, BL)')
ax[1].plot(np.arange(N)+1, w_tilde, 'o', c='y',
label=r'$\tilde{w}$ (mixture)')
ax[1].vlines(np.arange(N)+1, 0, w, lw=1)
ax[1].vlines(np.arange(N)+1, 0, w_m, lw=1)
ax[1].axhline(0, c='k')
ax[1].axhline(-1, c='k', ls='--')
ax[1].axhline(1, c='k', ls='--')
ax[1].set(xlim=(0, N+1), xlabel='Assets',
title='Black-Litterman portfolio weight recommendation')
ax[1].xaxis.set_ticks(np.arange(1, N+1, 1))
ax[1].legend(numpoints=1)
plt.show()
### Bayes Interpretation of the Black-Litterman Recommendation¶
Consider the following Bayesian interpretation of the Black-Litterman recommendation.
The prior belief over the mean excess returns is consistent with the market portfolio and is given by
$$\mu \sim \mathcal{N}(\mu_{BL}, \Sigma)$$
Given a particular realization of the mean excess returns $\mu$ one observes the average excess returns $\hat \mu$ on the market according to the distribution
$$\hat \mu \mid \mu, \Sigma \sim \mathcal{N}(\mu, \tau\Sigma)$$
where $\tau$ is typically small capturing the idea that the variation in the mean is smaller than the variation of the individual random variable.
Given the realized excess returns one should then update the prior over the mean excess returns according to Bayes rule.
The corresponding posterior over mean excess returns is normally distributed with mean
$$(\Sigma^{-1} + (\tau \Sigma)^{-1})^{-1} (\Sigma^{-1}\mu_{BL} + (\tau \Sigma)^{-1} \hat \mu)$$
The covariance matrix is
$$(\Sigma^{-1} + (\tau \Sigma)^{-1})^{-1}$$
Hence, the Black-Litterman recommendation is consistent with the Bayes update of the prior over the mean excess returns in light of the realized average excess returns on the market.
### Curve Decolletage¶
Consider two independent “competing” views on the excess market returns
$$\vec r_e \sim {\mathcal N}( \mu_{BL}, \Sigma)$$
and
$$\vec r_e \sim {\mathcal N}( \hat{\mu}, \tau\Sigma)$$
A special feature of the multivariate normal random variable $Z$ is that its density function depends only on the (Euclidiean) length of its realization $z$.
Formally, let the $k$-dimensional random vector be
$$Z\sim \mathcal{N}(\mu, \Sigma)$$
then
$$\bar{Z} \equiv \Sigma(Z-\mu)\sim \mathcal{N}(\mathbf{0}, I)$$
and so the points where the density takes the same value can be described by the ellipse
$$\bar z \cdot \bar z = (z - \mu)'\Sigma^{-1}(z - \mu) = \bar d \tag{5}$$
where $\bar d\in\mathbb{R}_+$ denotes the (transformation) of a particular density value.
The curves defined by equation (5) can be labeled as iso-likelihood ellipses
Remark: More generally there is a class of density functions that possesses this feature, i.e.
$$\exists g: \mathbb{R}_+ \mapsto \mathbb{R}_+ \ \ \text{ and } \ \ c \geq 0, \ \ \text{s.t. the density } \ \ f \ \ \text{of} \ \ Z \ \ \text{ has the form } \quad f(z) = c g(z\cdot z)$$
This property is called spherical symmetry (see p 81. in Leamer (1978) [Lea78]).
In our specific example, we can use the pair $(\bar d_1, \bar d_2)$ as being two “likelihood” values for which the corresponding iso-likelihood ellipses in the excess return space are given by
\begin{aligned} (\vec r_e - \mu_{BL})'\Sigma^{-1}(\vec r_e - \mu_{BL}) &= \bar d_1 \\ (\vec r_e - \hat \mu)'\left(\tau \Sigma\right)^{-1}(\vec r_e - \hat \mu) &= \bar d_2 \end{aligned}
Notice that for particular $\bar d_1$ and $\bar d_2$ values the two ellipses have a tangency point.
These tangency points, indexed by the pairs $(\bar d_1, \bar d_2)$, characterize points $\vec r_e$ from which there exists no deviation where one can increase the likelihood of one view without decreasing the likelihood of the other view.
The pairs $(\bar d_1, \bar d_2)$ for which there is such a point outlines a curve in the excess return space. This curve is reminiscent of the Pareto curve in an Edgeworth-box setting.
Dickey (1975) [Dic75] calls it a curve decolletage.
Leamer (1978) [Lea78] calls it an information contract curve and describes it by the following program: maximize the likelihood of one view, say the Black-Litterman recommendation while keeping the likelihood of the other view at least at a prespecified constant $\bar d_2$
\begin{aligned} \bar d_1(\bar d_2) &\equiv \max_{\vec r_e} \ \ (\vec r_e - \mu_{BL})'\Sigma^{-1}(\vec r_e - \mu_{BL}) \\ \text{subject to } \quad &(\vec r_e - \hat\mu)'(\tau\Sigma)^{-1}(\vec r_e - \hat \mu) \geq \bar d_2 \end{aligned}
Denoting the multiplier on the constraint by $\lambda$, the first-order condition is
$$2(\vec r_e - \mu_{BL} )'\Sigma^{-1} + \lambda 2(\vec r_e - \hat\mu)'(\tau\Sigma)^{-1} = \mathbf{0}$$
which defines the information contract curve between $\mu_{BL}$ and $\hat \mu$
$$\vec r_e = (\Sigma^{-1} + \lambda (\tau \Sigma)^{-1})^{-1} (\Sigma^{-1} \mu_{BL} + \lambda (\tau \Sigma)^{-1}\hat \mu ) \tag{6}$$
Note that if $\lambda = 1$, (6) is equivalent with (4) and it identifies one point on the information contract curve.
Furthermore, because $\lambda$ is a function of the minimum likelihood $\bar d_2$ on the RHS of the constraint, by varying $\bar d_2$ (or $\lambda$ ), we can trace out the whole curve as the figure below illustrates.
In [5]:
np.random.seed(1987102)
N = 2 # Number of assets
T = 200 # Sample size
τ = 0.8
# Random market portfolio (sum is normalized to 1)
w_m = np.random.rand(N)
w_m = w_m / (w_m.sum())
μ = (np.random.randn(N) + 5) / 100
S = np.random.randn(N, N)
V = S @ S.T
Σ = V * (w_m @ μ)**2 / (w_m @ V @ w_m)
excess_return = stat.multivariate_normal(μ, Σ)
sample = excess_return.rvs(T)
μ_est = sample.mean(0).reshape(N, 1)
Σ_est = np.cov(sample.T)
σ_m = w_m @ Σ_est @ w_m
d_m = (w_m @ μ_est) / σ_m
μ_m = (d_m * Σ_est @ w_m).reshape(N, 1)
N_r1, N_r2 = 100, 100
r1 = np.linspace(-0.04, .1, N_r1)
r2 = np.linspace(-0.02, .15, N_r2)
λ_grid = np.linspace(.001, 20, 100)
curve = np.asarray([black_litterman(λ, μ_m, μ_est, Σ_est,
τ * Σ_est).flatten() for λ in λ_grid])
λ_slider = FloatSlider(min=.1, max=7, step=.5, value=1)
@interact(λ=λ_slider)
def decolletage(λ):
dist_r_BL = stat.multivariate_normal(μ_m.squeeze(), Σ_est)
dist_r_hat = stat.multivariate_normal(μ_est.squeeze(), τ * Σ_est)
X, Y = np.meshgrid(r1, r2)
Z_BL = np.zeros((N_r1, N_r2))
Z_hat = np.zeros((N_r1, N_r2))
for i in range(N_r1):
for j in range(N_r2):
Z_BL[i, j] = dist_r_BL.pdf(np.hstack([X[i, j], Y[i, j]]))
Z_hat[i, j] = dist_r_hat.pdf(np.hstack([X[i, j], Y[i, j]]))
μ_tilde = black_litterman(λ, μ_m, μ_est, Σ_est, τ * Σ_est).flatten()
fig, ax = plt.subplots(figsize=(10, 6))
ax.contourf(X, Y, Z_hat, cmap='viridis', alpha =.4)
ax.contourf(X, Y, Z_BL, cmap='viridis', alpha =.4)
ax.contour(X, Y, Z_BL, [dist_r_BL.pdf(μ_tilde)], cmap='viridis', alpha=.9)
ax.contour(X, Y, Z_hat, [dist_r_hat.pdf(μ_tilde)], cmap='viridis', alpha=.9)
ax.scatter(μ_est[0], μ_est[1])
ax.scatter(μ_m[0], μ_m[1])
ax.scatter(μ_tilde[0], μ_tilde[1], c='k', s=20*3)
ax.plot(curve[:, 0], curve[:, 1], c='k')
ax.axhline(0, c='k', alpha=.8)
ax.axvline(0, c='k', alpha=.8)
ax.set_xlabel(r'Excess return on the first asset, $r_{e, 1}$')
ax.set_ylabel(r'Excess return on the second asset, $r_{e, 2}$')
ax.text(μ_est[0] + 0.003, μ_est[1], r'$\hat{\mu}$')
ax.text(μ_m[0] + 0.003, μ_m[1] + 0.005, r'$\mu_{BL}$')
plt.show()
Note that the line that connects the two points $\hat \mu$ and $\mu_{BL}$ is linear, which comes from the fact that the covariance matrices of the two competing distributions (views) are proportional to each other.
To illustrate the fact that this is not necessarily the case, consider another example using the same parameter values, except that the “second view” constituting the constraint has covariance matrix $\tau I$ instead of $\tau \Sigma$.
This leads to the following figure, on which the curve connecting $\hat \mu$ and $\mu_{BL}$ are bending
In [6]:
λ_grid = np.linspace(.001, 20000, 1000)
curve = np.asarray([black_litterman(λ, μ_m, μ_est, Σ_est,
τ * np.eye(N)).flatten() for λ in λ_grid])
λ_slider = FloatSlider(min=5, max=1500, step=100, value=200)
@interact(λ=λ_slider)
def decolletage(λ):
dist_r_BL = stat.multivariate_normal(μ_m.squeeze(), Σ_est)
dist_r_hat = stat.multivariate_normal(μ_est.squeeze(), τ * np.eye(N))
X, Y = np.meshgrid(r1, r2)
Z_BL = np.zeros((N_r1, N_r2))
Z_hat = np.zeros((N_r1, N_r2))
for i in range(N_r1):
for j in range(N_r2):
Z_BL[i, j] = dist_r_BL.pdf(np.hstack([X[i, j], Y[i, j]]))
Z_hat[i, j] = dist_r_hat.pdf(np.hstack([X[i, j], Y[i, j]]))
μ_tilde = black_litterman(λ, μ_m, μ_est, Σ_est, τ * np.eye(N)).flatten()
fig, ax = plt.subplots(figsize=(10, 6))
ax.contourf(X, Y, Z_hat, cmap='viridis', alpha=.4)
ax.contourf(X, Y, Z_BL, cmap='viridis', alpha=.4)
ax.contour(X, Y, Z_BL, [dist_r_BL.pdf(μ_tilde)], cmap='viridis', alpha=.9)
ax.contour(X, Y, Z_hat, [dist_r_hat.pdf(μ_tilde)], cmap='viridis', alpha=.9)
ax.scatter(μ_est[0], μ_est[1])
ax.scatter(μ_m[0], μ_m[1])
ax.scatter(μ_tilde[0], μ_tilde[1], c='k', s=20*3)
ax.plot(curve[:, 0], curve[:, 1], c='k')
ax.axhline(0, c='k', alpha=.8)
ax.axvline(0, c='k', alpha=.8)
ax.set_xlabel(r'Excess return on the first asset, $r_{e, 1}$')
ax.set_ylabel(r'Excess return on the second asset, $r_{e, 2}$')
ax.text(μ_est[0] + 0.003, μ_est[1], r'$\hat{\mu}$')
ax.text(μ_m[0] + 0.003, μ_m[1] + 0.005, r'$\mu_{BL}$')
plt.show()
### Black-Litterman Recommendation as Regularization¶
First, consider the OLS regression
$$\min_{\beta} \Vert X\beta - y \Vert^2$$
which yields the solution
$$\hat{\beta}_{OLS} = (X'X)^{-1}X'y$$
A common performance measure of estimators is the mean squared error (MSE).
An estimator is “good” if its MSE is relatively small. Suppose that $\beta_0$ is the “true” value of the coefficient, then the MSE of the OLS estimator is
$$\text{mse}(\hat \beta_{OLS}, \beta_0) := \mathbb E \Vert \hat \beta_{OLS} - \beta_0\Vert^2 = \underbrace{\mathbb E \Vert \hat \beta_{OLS} - \mathbb E \beta_{OLS}\Vert^2}_{\text{variance}} + \underbrace{\Vert \mathbb E \hat\beta_{OLS} - \beta_0\Vert^2}_{\text{bias}}$$
From this decomposition, one can see that in order for the MSE to be small, both the bias and the variance terms must be small.
For example, consider the case when $X$ is a $T$-vector of ones (where $T$ is the sample size), so $\hat\beta_{OLS}$ is simply the sample average, while $\beta_0\in \mathbb{R}$ is defined by the true mean of $y$.
In this example the MSE is
$$\text{mse}(\hat \beta_{OLS}, \beta_0) = \underbrace{\frac{1}{T^2} \mathbb E \left(\sum_{t=1}^{T} (y_{t}- \beta_0)\right)^2 }_{\text{variance}} + \underbrace{0}_{\text{bias}}$$
However, because there is a trade-off between the estimator’s bias and variance, there are cases when by permitting a small bias we can substantially reduce the variance so overall the MSE gets smaller.
A typical scenario when this proves to be useful is when the number of coefficients to be estimated is large relative to the sample size.
In these cases, one approach to handle the bias-variance trade-off is the so called Tikhonov regularization.
A general form with regularization matrix $\Gamma$ can be written as
$$\min_{\beta} \Big\{ \Vert X\beta - y \Vert^2 + \Vert \Gamma (\beta - \tilde \beta) \Vert^2 \Big\}$$
which yields the solution
$$\hat{\beta}_{Reg} = (X'X + \Gamma'\Gamma)^{-1}(X'y + \Gamma'\Gamma\tilde \beta)$$
Substituting the value of $\hat{\beta}_{OLS}$ yields
$$\hat{\beta}_{Reg} = (X'X + \Gamma'\Gamma)^{-1}(X'X\hat{\beta}_{OLS} + \Gamma'\Gamma\tilde \beta)$$
Often, the regularization matrix takes the form $\Gamma = \lambda I$ with $\lambda>0$ and $\tilde \beta = \mathbf{0}$.
Then the Tikhonov regularization is equivalent to what is called ridge regression in statistics.
To illustrate how this estimator addresses the bias-variance trade-off, we compute the MSE of the ridge estimator
$$\text{mse}(\hat \beta_{\text{ridge}}, \beta_0) = \underbrace{\frac{1}{(T+\lambda)^2} \mathbb E \left(\sum_{t=1}^{T} (y_{t}- \beta_0)\right)^2 }_{\text{variance}} + \underbrace{\left(\frac{\lambda}{T+\lambda}\right)^2 \beta_0^2}_{\text{bias}}$$
The ridge regression shrinks the coefficients of the estimated vector towards zero relative to the OLS estimates thus reducing the variance term at the cost of introducing a “small” bias.
However, there is nothing special about the zero vector.
When $\tilde \beta \neq \mathbf{0}$ shrinkage occurs in the direction of $\tilde \beta$.
Now, we can give a regularization interpretation of the Black-Litterman portfolio recommendation.
To this end, simplify first the equation (4) characterizing the Black-Litterman recommendation
\begin{aligned} \tilde \mu &= (\Sigma^{-1} + (\tau \Sigma)^{-1})^{-1} (\Sigma^{-1}\mu_{BL} + (\tau \Sigma)^{-1}\hat \mu) \\ &= (1 + \tau^{-1})^{-1}\Sigma \Sigma^{-1} (\mu_{BL} + \tau ^{-1}\hat \mu) \\ &= (1 + \tau^{-1})^{-1} ( \mu_{BL} + \tau ^{-1}\hat \mu) \end{aligned}
In our case, $\hat \mu$ is the estimated mean excess returns of securities. This could be written as a vector autoregression where
• $y$ is the stacked vector of observed excess returns of size $(N T\times 1)$ – $N$ securities and $T$ observations.
• $X = \sqrt{T^{-1}}(I_{N} \otimes \iota_T)$ where $I_N$ is the identity matrix and $\iota_T$ is a column vector of ones.
Correspondingly, the OLS regression of $y$ on $X$ would yield the mean excess returns as coefficients.
With $\Gamma = \sqrt{\tau T^{-1}}(I_{N} \otimes \iota_T)$ we can write the regularized version of the mean excess return estimation
\begin{aligned} \hat{\beta}_{Reg} &= (X'X + \Gamma'\Gamma)^{-1}(X'X\hat{\beta}_{OLS} + \Gamma'\Gamma\tilde \beta) \\ &= (1 + \tau)^{-1}X'X (X'X)^{-1} (\hat \beta_{OLS} + \tau \tilde \beta) \\ &= (1 + \tau)^{-1} (\hat \beta_{OLS} + \tau \tilde \beta) \\ &= (1 + \tau^{-1})^{-1} ( \tau^{-1}\hat \beta_{OLS} + \tilde \beta) \end{aligned}
Given that $\hat \beta_{OLS} = \hat \mu$ and $\tilde \beta = \mu_{BL}$ in the Black-Litterman model, we have the following interpretation of the model’s recommendation.
The estimated (personal) view of the mean excess returns, $\hat{\mu}$ that would lead to extreme short-long positions are “shrunk” towards the conservative market view, $\mu_{BL}$, that leads to the more conservative market portfolio.
So the Black-Litterman procedure results in a recommendation that is a compromise between the conservative market portfolio and the more extreme portfolio that is implied by estimated “personal” views.
### Digression on A Robust Control Operator¶
The Black-Litterman approach is partly inspired by the econometric insight that it is easier to estimate covariances of excess returns than the means.
That is what gave Black and Litterman license to adjust investors’ perception of mean excess returns while not tampering with the covariance matrix of excess returns.
The robust control theory is another approach that also hinges on adjusting mean excess returns but not covariances.
Associated with a robust control problem is what Hansen and Sargent [HS01], [HS08a] call a ${\sf T}$ operator.
Let’s define the ${\sf T}$ operator as it applies to the problem at hand.
Let $x$ be an $n \times 1$ Gaussian random vector with mean vector $\mu$ and covariance matrix $\Sigma = C C'$. This means that $x$ can be represented as
$$x = \mu + C \epsilon$$
where $\epsilon \sim {\mathcal N}(0,I)$.
Let $\phi(\epsilon)$ denote the associated standardized Gaussian density.
Let $m(\epsilon,\mu)$ be a likelihood ratio, meaning that it satisfies
• $m(\epsilon, \mu) > 0$
• $\int m(\epsilon,\mu) \phi(\epsilon) d \epsilon =1$
That is, $m(\epsilon, \mu)$ is a non-negative random variable with mean 1.
Multiplying $\phi(\epsilon)$ by the likelihood ratio $m(\epsilon, \mu)$ produces a distorted distribution for $\epsilon$, namely
$$\tilde \phi(\epsilon) = m(\epsilon,\mu) \phi(\epsilon)$$
The next concept that we need is the entropy of the distorted distribution $\tilde \phi$ with respect to $\phi$.
Entropy is defined as
$${\rm ent} = \int \log m(\epsilon,\mu) m(\epsilon,\mu) \phi(\epsilon) d \epsilon$$
or
$${\rm ent} = \int \log m(\epsilon,\mu) \tilde \phi(\epsilon) d \epsilon$$
That is, relative entropy is the expected value of the likelihood ratio $m$ where the expectation is taken with respect to the twisted density $\tilde \phi$.
Relative entropy is non-negative. It is a measure of the discrepancy between two probability distributions.
As such, it plays an important role in governing the behavior of statistical tests designed to discriminate one probability distribution from another.
We are ready to define the ${\sf T}$ operator.
Let $V(x)$ be a value function.
Define
\begin{aligned} {\sf T}\left(V(x)\right) & = \min_{m(\epsilon,\mu)} \int m(\epsilon,\mu)[V(\mu + C \epsilon) + \theta \log m(\epsilon,\mu) ] \phi(\epsilon) d \epsilon \cr & = - \log \theta \int \exp \left( \frac{- V(\mu + C \epsilon)}{\theta} \right) \phi(\epsilon) d \epsilon \end{aligned}
This asserts that ${\sf T}$ is an indirect utility function for a minimization problem in which an evil agent chooses a distorted probability distribution $\tilde \phi$ to lower expected utility, subject to a penalty term that gets bigger the larger is relative entropy.
Here the penalty parameter
$$\theta \in [\underline \theta, +\infty]$$
is a robustness parameter when it is $+\infty$, there is no scope for the minimizing agent to distort the distribution, so no robustness to alternative distributions is acquired As $\theta$ is lowered, more robustness is achieved.
Note: The ${\sf T}$ operator is sometimes called a risk-sensitivity operator.
We shall apply ${\sf T}$to the special case of a linear value function $w'(\vec r - r_f 1)$ where $\vec r - r_f 1 \sim {\mathcal N}(\mu,\Sigma)$ or $\vec r - r_f {\bf 1} = \mu + C \epsilon$and $\epsilon \sim {\mathcal N}(0,I)$.
The associated worst-case distribution of $\epsilon$ is Gaussian with mean $v =-\theta^{-1} C' w$ and covariance matrix $I$ (When the value function is affine, the worst-case distribution distorts the mean vector of $\epsilon$ but not the covariance matrix of $\epsilon$).
For utility function argument $w'(\vec r - r_f 1)$
$${\sf T} ( \vec r - r_f {\bf 1}) = w' \mu + \zeta - \frac{1}{2 \theta} w' \Sigma w$$
and entropy is
$$\frac{v'v}{2} = \frac{1}{2\theta^2} w' C C' w$$
### A Robust Mean-variance Portfolio Model¶
According to criterion (1), the mean-variance portfolio choice problem chooses $w$ to maximize
$$E [w ( \vec r - r_f {\bf 1})]] - {\rm var} [ w ( \vec r - r_f {\bf 1}) ]$$
which equals
$$w'\mu - \frac{\delta}{2} w' \Sigma w$$
A robust decision maker can be modeled as replacing the mean return $E [w ( \vec r - r_f {\bf 1})]$ with the risk-sensitive
$${\sf T} [w ( \vec r - r_f {\bf 1})] = w' \mu - \frac{1}{2 \theta} w' \Sigma w$$
that comes from replacing the mean $\mu$ of $\vec r - r\_f {\bf 1}$ with the worst-case mean
$$\mu - \theta^{-1} \Sigma w$$
Notice how the worst-case mean vector depends on the portfolio $w$.
The operator ${\sf T}$ is the indirect utility function that emerges from solving a problem in which an agent who chooses probabilities does so in order to minimize the expected utility of a maximizing agent (in our case, the maximizing agent chooses portfolio weights $w$).
The robust version of the mean-variance portfolio choice problem is then to choose a portfolio $w$ that maximizes
$${\sf T} [w ( \vec r - r_f {\bf 1})] - \frac{\delta}{2} w' \Sigma w$$
or
$$w' (\mu - \theta^{-1} \Sigma w ) - \frac{\delta}{2} w' \Sigma w \tag{7}$$
The minimizer of (7) is
$$w_{\rm rob} = \frac{1}{\delta + \gamma } \Sigma^{-1} \mu$$
where $\gamma \equiv \theta^{-1}$ is sometimes called the risk-sensitivity parameter.
An increase in the risk-sensitivity parameter $\gamma$ shrinks the portfolio weights toward zero in the same way that an increase in risk aversion does.
## Appendix¶
We want to illustrate the “folk theorem” that with high or moderate frequency data, it is more difficult to estimate means than variances.
In order to operationalize this statement, we take two analog estimators:
• sample average: $\bar X_N = \frac{1}{N}\sum_{i=1}^{N} X_i$
• sample variance: $S_N = \frac{1}{N-1}\sum_{t=1}^{N} (X_i - \bar X_N)^2$
to estimate the unconditional mean and unconditional variance of the random variable $X$, respectively.
To measure the “difficulty of estimation”, we use mean squared error (MSE), that is the average squared difference between the estimator and the true value.
Assuming that the process $\{X_i\}$is ergodic, both analog estimators are known to converge to their true values as the sample size $N$ goes to infinity.
More precisely for all $\varepsilon > 0$
$$\lim_{N\to \infty} \ \ P\left\{ \left |\bar X_N - \mathbb E X \right| > \varepsilon \right\} = 0 \quad \quad$$
and
$$\lim_{N\to \infty} \ \ P \left\{ \left| S_N - \mathbb V X \right| > \varepsilon \right\} = 0$$
A necessary condition for these convergence results is that the associated MSEs vanish as $N$ goes to infinity, or in other words,
$$\text{MSE}(\bar X_N, \mathbb E X) = o(1) \quad \quad \text{and} \quad \quad \text{MSE}(S_N, \mathbb V X) = o(1)$$
Even if the MSEs converge to zero, the associated rates might be different. Looking at the limit of the relative MSE (as the sample size grows to infinity)
$$\frac{\text{MSE}(S_N, \mathbb V X)}{\text{MSE}(\bar X_N, \mathbb E X)} = \frac{o(1)}{o(1)} \underset{N \to \infty}{\to} B$$
can inform us about the relative (asymptotic) rates.
We will show that in general, with dependent data, the limit $B$ depends on the sampling frequency.
In particular, we find that the rate of convergence of the variance estimator is less sensitive to increased sampling frequency than the rate of convergence of the mean estimator.
Hence, we can expect the relative asymptotic rate, $B$, to get smaller with higher frequency data, illustrating that “it is more difficult to estimate means than variances”.
That is, we need significantly more data to obtain a given precision of the mean estimate than for our variance estimate.
### A Special Case – IID Sample¶
We start our analysis with the benchmark case of IID data. Consider a sample of size $N$ generated by the following IID process,
$$X_i \sim \mathcal{N}(\mu, \sigma^2)$$
Taking $\bar X_N$ to estimate the mean, the MSE is
$$\text{MSE}(\bar X_N, \mu) = \frac{\sigma^2}{N}$$
Taking $S_N$ to estimate the variance, the MSE is
$$\text{MSE}(S_N, \sigma^2) = \frac{2\sigma^4}{N-1}$$
Both estimators are unbiased and hence the MSEs reflect the corresponding variances of the estimators.
Furthermore, both MSEs are $o(1)$ with a (multiplicative) factor of difference in their rates of convergence:
$$\frac{\text{MSE}(S_N, \sigma^2)}{\text{MSE}(\bar X_N, \mu)} = \frac{N2\sigma^2}{N-1} \quad \underset{N \to \infty}{\to} \quad 2\sigma^2$$
We are interested in how this (asymptotic) relative rate of convergence changes as increasing sampling frequency puts dependence into the data.
### Dependence and Sampling Frequency¶
To investigate how sampling frequency affects relative rates of convergence, we assume that the data are generated by a mean-reverting continuous time process of the form
$$dX_t = -\kappa (X_t -\mu)dt + \sigma dW_t\quad\quad$$
where $\mu$is the unconditional mean, $\kappa > 0$ is a persistence parameter, and $\{W_t\}$ is a standardized Brownian motion.
Observations arising from this system in particular discrete periods $\mathcal T(h) \equiv \{nh : n \in \mathbb Z \}$with$h>0$ can be described by the following process
$$X_{t+1} = (1 - \exp(-\kappa h))\mu + \exp(-\kappa h)X_t + \epsilon_{t, h}$$
where
$$\epsilon_{t, h} \sim \mathcal{N}(0, \Sigma_h) \quad \text{with}\quad \Sigma_h = \frac{\sigma^2(1-\exp(-2\kappa h))}{2\kappa}$$
We call $h$ the frequency parameter, whereas $n$ represents the number of lags between observations.
Hence, the effective distance between two observations $X_t$ and $X_{t+n}$ in the discrete time notation is equal to $h\cdot n$ in terms of the underlying continuous time process.
Straightforward calculations show that the autocorrelation function for the stochastic process $\{X_{t}\}_{t\in \mathcal T(h)}$ is
$$\Gamma_h(n) \equiv \text{corr}(X_{t + h n}, X_t) = \exp(-\kappa h n)$$
and the auto-covariance function is
$$\gamma_h(n) \equiv \text{cov}(X_{t + h n}, X_t) = \frac{\exp(-\kappa h n)\sigma^2}{2\kappa} .$$
It follows that if $n=0$, the unconditional variance is given by $\gamma_h(0) = \frac{\sigma^2}{2\kappa}$ irrespective of the sampling frequency.
The following figure illustrates how the dependence between the observations is related to the sampling frequency
• For any given $h$, the autocorrelation converges to zero as we increase the distance – $n$– between the observations. This represents the “weak dependence” of the $X$ process.
• Moreover, for a fixed lag length, $n$, the dependence vanishes as the sampling frequency goes to infinity. In fact, letting $h$ go to $\infty$ gives back the case of IID data.
In [7]:
μ = .0
κ = .1
σ = .5
var_uncond = σ**2 / (2 * κ)
n_grid = np.linspace(0, 40, 100)
autocorr_h1 = np.exp(-κ * n_grid * 1)
autocorr_h2 = np.exp(-κ * n_grid * 2)
autocorr_h5 = np.exp(-κ * n_grid * 5)
autocorr_h1000 = np.exp(-κ * n_grid * 1e8)
fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(n_grid, autocorr_h1, label=r'$h=1$', c='darkblue', lw=2)
ax.plot(n_grid, autocorr_h2, label=r'$h=2$', c='darkred', lw=2)
ax.plot(n_grid, autocorr_h5, label=r'$h=5$', c='orange', lw=2)
ax.plot(n_grid, autocorr_h1000, label=r'"$h=\infty$"', c='darkgreen', lw=2)
ax.legend()
ax.grid()
ax.set(title=r'Autocorrelation functions, $\Gamma_h(n)$',
xlabel=r'Lags between observations, $n$')
plt.show()
### Frequency and the Mean Estimator¶
Consider again the AR(1) process generated by discrete sampling with frequency $h$. Assume that we have a sample of size $N$ and we would like to estimate the unconditional mean – in our case the true mean is $\mu$.
Again, the sample average is an unbiased estimator of the unconditional mean
$$\mathbb{E}[\bar X_N] = \frac{1}{N}\sum_{i = 1}^N \mathbb{E}[X_i] = \mathbb{E}[X_0] = \mu$$
The variance of the sample mean is given by
\begin{aligned} \mathbb{V}\left(\bar X_N\right) &= \mathbb{V}\left(\frac{1}{N}\sum_{i = 1}^N X_i\right) \\ &= \frac{1}{N^2} \left(\sum_{i = 1}^N \mathbb{V}(X_i) + 2 \sum_{i = 1}^{N-1} \sum_{s = i+1}^N \text{cov}(X_i, X_s) \right) \\ &= \frac{1}{N^2} \left( N \gamma(0) + 2 \sum_{i=1}^{N-1} i \cdot \gamma\left(h\cdot (N - i)\right) \right) \\ &= \frac{1}{N^2} \left( N \frac{\sigma^2}{2\kappa} + 2 \sum_{i=1}^{N-1} i \cdot \exp(-\kappa h (N - i)) \frac{\sigma^2}{2\kappa} \right) \end{aligned}
It is explicit in the above equation that time dependence in the data inflates the variance of the mean estimator through the covariance terms. Moreover, as we can see, a higher sampling frequency—smaller $h$—makes all the covariance terms larger, everything else being fixed. This implies a relatively slower rate of convergence of the sample average for high-frequency data.
Intuitively, the stronger dependence across observations for high-frequency data reduces the “information content” of each observation relative to the IID case.
We can upper bound the variance term in the following way
\begin{aligned} \mathbb{V}(\bar X_N) &= \frac{1}{N^2} \left( N \sigma^2 + 2 \sum_{i=1}^{N-1} i \cdot \exp(-\kappa h (N - i)) \sigma^2 \right) \\ &\leq \frac{\sigma^2}{2\kappa N} \left(1 + 2 \sum_{i=1}^{N-1} \cdot \exp(-\kappa h (i)) \right) \\ &= \underbrace{\frac{\sigma^2}{2\kappa N}}_{\text{IID case}} \left(1 + 2 \frac{1 - \exp(-\kappa h)^{N-1}}{1 - \exp(-\kappa h)} \right) \end{aligned}
Asymptotically the $\exp(-\kappa h)^{N-1}$ vanishes and the dependence in the data inflates the benchmark IID variance by a factor of
$$\left(1 + 2 \frac{1}{1 - \exp(-\kappa h)} \right)$$
This long run factor is larger the higher is the frequency (the smaller is $h$).
Therefore, we expect the asymptotic relative MSEs, $B$, to change with time-dependent data. We just saw that the mean estimator’s rate is roughly changing by a factor of
$$\left(1 + 2 \frac{1}{1 - \exp(-\kappa h)} \right)$$
Unfortunately, the variance estimator’s MSE is harder to derive.
Nonetheless, we can approximate it by using (large sample) simulations, thus getting an idea about how the asymptotic relative MSEs changes in the sampling frequency $h$ relative to the IID case that we compute in closed form.
In [8]:
def sample_generator(h, N, M):
ϕ = (1 - np.exp(-κ * h)) * μ
ρ = np.exp(-κ * h)
s = σ**2 * (1 - np.exp(-2 * κ * h)) / (2 * κ)
mean_uncond = μ
std_uncond = np.sqrt(σ**2 / (2 * κ))
ε_path = stat.norm(0, np.sqrt(s)).rvs((M, N))
y_path = np.zeros((M, N + 1))
y_path[:, 0] = stat.norm(mean_uncond, std_uncond).rvs(M)
for i in range(N):
y_path[:, i + 1] = ϕ + ρ * y_path[:, i] + ε_path[:, i]
return y_path
In [9]:
# Generate large sample for different frequencies
N_app, M_app = 1000, 30000 # Sample size, number of simulations
h_grid = np.linspace(.1, 80, 30)
var_est_store = []
mean_est_store = []
labels = []
for h in h_grid:
labels.append(h)
sample = sample_generator(h, N_app, M_app)
mean_est_store.append(np.mean(sample, 1))
var_est_store.append(np.var(sample, 1))
var_est_store = np.array(var_est_store)
mean_est_store = np.array(mean_est_store)
# Save mse of estimators
mse_mean = np.var(mean_est_store, 1) + (np.mean(mean_est_store, 1) - μ)**2
mse_var = np.var(var_est_store, 1) \
+ (np.mean(var_est_store, 1) - var_uncond)**2
benchmark_rate = 2 * var_uncond # IID case
# Relative MSE for large samples
rate_h = mse_var / mse_mean
fig, ax = plt.subplots(figsize=(8, 5))
ax.plot(h_grid, rate_h, c='darkblue', lw=2,
label=r'large sample relative MSE, $B(h)$')
ax.axhline(benchmark_rate, c='k', ls='--', label=r'IID benchmark')
ax.set_title('Relative MSE for large samples as a function of sampling \
frequency \n MSE($S_N$) relative to MSE($\\bar X_N$)')
ax.set_xlabel('Sampling frequency, $h$')
ax.legend()
plt.show()
The above figure illustrates the relationship between the asymptotic relative MSEs and the sampling frequency
• We can see that with low-frequency data – large values of $h$ – the ratio of asymptotic rates approaches the IID case.
• As $h$ gets smaller – the higher the frequency – the relative performance of the variance estimator is better in the sense that the ratio of asymptotic rates gets smaller. That is, as the time dependence gets more pronounced, the rate of convergence of the mean estimator’s MSE deteriorates more than that of the variance estimator.
• Share page | 2020-12-01 08:18:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9689967036247253, "perplexity": 2221.7236885745515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141672314.55/warc/CC-MAIN-20201201074047-20201201104047-00637.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-quotient-b-6divb-3 | # How do you find the quotient b^6divb^3?
Feb 19, 2017
See the entire solution process below:
#### Explanation:
First, rewrite the expression as:
${b}^{6} / {b}^{3}$
Now, use this rule of exponents to find the quotient:
${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} - \textcolor{b l u e}{b}}$
${b}^{\textcolor{red}{6}} / {b}^{\textcolor{b l u e}{3}} = {b}^{\textcolor{red}{6} - \textcolor{b l u e}{3}} = {b}^{3}$ | 2021-12-05 21:03:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9029816389083862, "perplexity": 2929.755234135344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363216.90/warc/CC-MAIN-20211205191620-20211205221620-00560.warc.gz"} |
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# Livestock host composition rather than land use or climate explains spatial patterns in bluetongue disease in South India
## Abstract
Culicoides-borne arboviruses of livestock impair animal health, livestock production and livelihoods worldwide. As these arboviruses are multi-host, multi-vector systems, predictions to improve targeting of disease control measures require frameworks that quantify the relative impacts of multiple abiotic and biotic factors on disease patterns. We develop such a framework to predict long term (1992–2009) average patterns in bluetongue (BT), caused by bluetongue virus (BTV), in sheep in southern India, where annual BT outbreaks constrain the livelihoods and production of small-holder farmers. In Bayesian spatial general linear mixed models, host factors outperformed landscape and climate factors as predictors of disease patterns, with more BT outbreaks occurring on average in districts with higher densities of susceptible sheep breeds and buffalo. Since buffalo are resistant to clinical signs of BT, this finding suggests they are a source of infection for sympatric susceptible sheep populations. Sero-monitoring is required to understand the role of buffalo in maintaining BTV transmission and whether they must be included in vaccination programs to protect sheep adequately. Landscape factors, namely the coverage of post-flooding, irrigated and rain-fed croplands, had weak positive effects on outbreaks. The intimate links between livestock host, vector composition and agricultural practices in India require further investigation at the landscape scale.
## Introduction
Culicoides (Diptera: Ceratopogonidae) are tiny biting flies that transmit arboviruses of livestock, wildlife and humans. These viruses cause diseases that have a significant impact on animal health and welfare, livestock production, trade and livelihoods1,2,3. In the past 20 years, Culicoides-borne arboviruses have undergone global changes in epidemiology that have been linked to changes in climate, land use, globalisation of trade and changes in animal husbandry4. Culicoides are also involved in transmitting human diseases with Culicoides paraensis (Goeldi) being the major vector of Oropouche virus (Orthobunyavirus) that causes a febrile illness of people in the neo-tropics5. Understanding and predicting how the impacts of Culicoides-borne arboviruses vary between geographical areas to inform targeting of control measures6 is hampered by their ecological complexity. Culicoides, require semi-aquatic developmental sites for egg, larval and pupal development and usually rely on mammalian hosts as a source of blood to produce eggs. Many aspects of this life cycle are affected by variations in temperature, moisture and host and habitat availability4,7. In most regions, several Culicoides vector species and both wild deer and domestic mammals are involved concurrently in transmission4,8 and each of these possesses different climate sensitivities and associations with natural and managed ecosystems9.
The emergence of bluetongue virus (BTV: Reoviridae; Orbivirus) and Schmallenberg virus (Bunyaviridae: Orthobunyavirus) in Europe has focussed research effort upon understanding and predicting spatial variability in Culicoides-borne arboviruses in relation to environmental drivers4,10,11,12,13. Research activity has been far less intense in poor tropical and sub-tropical arbovirus endemic zones where selective resistance in livestock tends to be greater, and where vaccines may be used routinely to reduce the impact of what clinical disease occurs4. Prior studies of Culicoides-borne disease patterns in tropical endemic areas focussed largely on climatic factors, primarily temperature and rainfall14. More recent studies have incorporated land use15, vegetation16 and hosts17 as potential explanatory factors, although the relative contribution of these factors to BT impacts are rarely quantified together.
In India between 1997 and 2005 endemic circulation of over 20 different BTV serotypes resulted in over 2000 outbreaks in sheep, involving 0.4 million cases and around 64 000 deaths, making it the top viral cause of disease in this host18. In addition to mortality (with local case fatality rates of up to 30%19), clinical impacts include weight loss, reductions in wool quality, infertility and lameness. Economic costs include those for veterinary treatment, vaccination, surveillance and trade restrictions20. Although there is limited information on breed susceptibility to bluetongue in India21, indigenous sheep breeds tend to be asymptomatic despite being widely infected with BTV (high antibody prevalence). Outbreaks were detected largely in exotic (introduced for improvement), or cross-breeds of sheep until the 1980s when cases began to be detected in indigenous sheep20,22. These latter cases are thought to have arisen from exotic strains of BTV introduced during breed improvement exercises. Bluetongue mitigation is currently achieved primarily through the use of inactivated pentavalent vaccines23, supplied by the state governments of the affected states to small-holder farmers through village veterinary officers.
In common with the transmission of many other arboviral diseases14,24,25, rainfall variability during monsoon events is hypothesised to affect the size and timing of BT epidemics in India20 and to restrict its spatial distribution primarily to South India which receives high rainfall in both the south west and north east monsoon seasons26. However, BT severity varies substantially between districts even within zones subject to similar monsoon conditions, suggesting that factors such as landscape, host and husbandry factors may modulate climate effects on transmission27,28 and should be accounted for in an ideal predictive framework. Furthermore, research in Europe has linked vector seasonality and abundance and spread of bluetongue to landscape and host factors alongside climate29,30,31.
The bluetongue system in India is epidemiologically complex. Most of the 26 global BTV serotypes have been detected in India19,26,32, and circulate alongside a diverse Culicoides fauna that includes at least seven species that have been implicated in arbovirus transmission in other countries (Culicoides actoni Smith; Culicoides brevitarsis Kieffer; Culicoides fulvus Sen and Das Gupta; Culicoides wadai Kitaoka; Culicoides imicola Kieffer; Culicoides oxystoma Kieffer)33,34,35,36,37. The immature stages of these species develop in a wide range of semi-aquatic habitats, ranging from animal dung (e.g. buffalo dung: C. oxystoma) to organically enriched moist soil (e.g. C. imicola, C. schultzei, C. peregrinus). Systematic studies of vectorial capacity and distribution are lacking33,38. In the mixed farming systems, practised by the small and marginal farmers of South India, mixed sheep and goat flocks are kept for meat production and indigenous cattle and buffalo maintained for milk production and for draught purposes. Although cattle, goats and buffalo are susceptible to BTV infection and in India are widely infected, displaying high BTV antibody prevalence, they do not usually show clinical disease20. Despite this, bovines and some breeds of goats39 develop levels of viraemia equivalent to sheep40 and can thus theoretically infect Culicoides and be involved in the onward transmission of BTV making them potential reservoir hosts. Furthermore, the 14 different sheep breeds present in affected states in South India vary in susceptibility to disease effects21. Thus, a high diversity of both susceptible sheep and disease resistant, potential reservoir species for bluetongue virus, such as cattle and buffalo, are kept in the same landscape in South India. Breed and species composition of the livestock population, alongside landscape and climate factors are expected to contribute to patterns in bluetongue disease.
Utilising a substantial dataset of clinical BT outbreaks collected since 1992 by the Indian Council of Agricultural Research’s National Institute for Veterinary Epidemiology and Disease Informatics (NVIEDI), this paper investigates the relative roles of climate, land-use and availability of livestock hosts in driving long-term spatial variation in the severity of BT outbreaks in sheep across districts in South India.
This paper uses spatial Bayesian Generalized Linear Mixed Modelling approaches41 with the aim of enhancing current disease management systems in the region. Understanding how multiple environmental factors interact to produce variation in disease severity across districts requires quantitative methods that can deal with collinearity between potential risk factors and spatial dependency in errors. The latter may arise due to intrinsic processes (such as disease spread between districts, represented by the spatial autocorrelation) and extrinsic processes (such as trends or drifts in the response which can be partly or totally explained by environmental covariates)42. Bayesian generalized linear mixed models overcome these problems by modelling the spatial dependence as random effects, through a prior distribution43. We use a modified version of the Besag-York-Mollie (BYM) model44 that quantifies the compromise between spatially structured and spatially unstructured random error variation (to account for overdispersion) as well as fixed effects of environmental predictors41.
Our specific aims for South India were to test for associations between the number of disease outbreaks in farmed sheep and:
1. (a)
coverage of rain-fed cropland and irrigated cropland45, that may encompass more abundant semi-aquatic habitat for immature Culicoides46.
2. (b)
coverage of open forest types and grassland or shrubland, used by small-holder farmers for grazing
3. (c)
coverage of closed forest types that are avoided by small-holders for grazing.
4. (d)
sheep densities, particular of sheep of susceptible exotic and cross-breeds in which cases have been more frequently recorded historically22.
5. (e)
densities of disease-resistant hosts such as cattle and buffalo numbers in which infection may circulate and be sustained silently.
The dependent variable is the average number of villages reporting bluetongue outbreaks in sheep per district from 1992 to 2009 (offset by the total number of villages per district). An outbreak is defined as a village with more than one disease affected sheep in a given transmission year. The village is the epidemiological unit for disease reporting in South India because villages are assumed to be relatively homogenous in animal husbandry and environmental conditions. The analysis was restricted to data from three states of South India in which outbreaks are regularly reported, namely Karnataka (n = 27 districts), Andhra Pradesh (n = 22 districts) and Tamil Nadu (n = 29 districts), with 61 out of the total of 78 constituent districts reporting outbreaks of BT over the 18 year study period (see Fig. S1 for state boundaries and location of the study region within India). We used a hierarchical multi-model approach to infer the relative importance of climate, host and landscape effects in explaining disease patterns (see Methods for full details). The predictors were divided into climate, landscape and host categories (Table 1, Figs S2 to S5). Within each of these categories, all possible predictor combinations were fitted and ranked by Deviance Information Criteria (DIC47) to identify a best model with the lowest DIC and a top model set that had DIC values within 2 DIC units of the best model. Predictors that were present (in >70% of top models) and significant (in >50%) of top models were offered to a second stage of model fitting. In this stage, all possible model combinations of the top predictors across climate, host and landscape categories were fitted and ranked by DIC. In the second stage, the frequency with which predictors were present and significant in top set of models was again used to infer their importance in explaining disease patterns.
## Results
The average annual number of outbreaks ranged from 0 to 33.8 and was highly variable between districts (mean ± s.d. = 4.93 ± 8.10) (Fig. 1a). The average annual number of outbreaks was highest across Andhra Pradesh (particularly in the south of the state) and at the southern tip of India, in Tamil Nadu. Here we first contrast the overall performance and important predictors of the top ranking models in each of the landscape, climate and host categories. Secondly we examine the performance and important predictors of top ranking models that combined predictors across these categories. We also analyse the potential impact of temporal mismatches in our predictor data versus the bluetongue outbreak data on our results.
### Landscape-driven models of BT outbreaks
For the landscape suite of predictors, there were 32 models with similar support in the data that were within 2 DIC units of the best model with the lowest DIC (Table S1, DIC = 202.01, Fig. 2a). The best model contained three predictors – cover of post-flooding/irrigated croplands (irrig. crop), cover of rain-fed croplands (Rain crop), and cover of broad-leaved deciduous forest (open decid.). Cover of post-flooding/irrigated croplands and rain-fed croplands had significant (i.e. credible interval of posterior distribution of parameter estimate did not bridge zero) positive effects on mean number of BT outbreaks in all of the top models and produced large increases in DIC when dropped from the best model – 2.36 and 5.44 DIC units respectively. By contrast, cover of broad-leaved deciduous forest was selected in only half of top models, was never significant and caused an increase of only 0.09 DIC units when dropped from the best model, suggesting that this predictor has a weak effect, if any, on BT outbreaks. The remaining landscape predictors never had a significant effect on BT outbreaks (despite each being added to half of the top models, Fig. 2a). Cover of post-flooding/irrigated croplands and rain-fed croplands were the only LANDSCAPE variables offered to the second stage of the environmental variable selection. The best landscape model outperformed the null model substantially (DIC = 210.52, ∆DIC = 8.51).
### Climate-driven models of BT outbreaks
For the climate suite of predictors, there were 13 models with similar support in the data that were within 2 DIC units of the best model with the lowest DIC (Table S2, DIC = 207.08, Fig. 2b). The best performing model contained average North East Monsoon rainfall amount (ne_rain) and average annual rainfall (ann_rain) amount whilst the second best model containing average annual rainfall amount, average annual temperature (ann_temp) and the interaction between these variables had very similar support in the data (∆DIC = 0.31). The best climate model outperformed the null model only marginally (∆DIC = 3.44). None of the climate predictors fulfilled the criteria for being added to the combined environmental model. Although annual rainfall appeared in 10 of the 13 top models and caused an increase of 2.75 DIC units if dropped from the best model, it had a significant (negative) effect on BT outbreaks in only four top models (Fig. 2b). North East Monsoon rainfall was added to seven models but had a significant (negative) effect on BT outbreaks in only six models and produced an increase of only 0.47 DIC units when dropped from the top model. South West Monsoon rainfall was added to seven of the top models and had a significant (negative) effect on BT outbreaks in only five models. Annual temperature was also selected in seven of the top models but had a significant (negative) effect on BT outbreaks in only four models. The annual temperature: annual rainfall interaction was added to four of the top models but was never significant.
### Host-driven models of BT incidence
For the host suite of predictors, there were 10 models with similar support in the data that were within 2 DIC units of the best model with the lowest DIC (Table 2, DIC = 197.63, Fig. 2c). All top models contained significant positive effects of densities of buffalo and non-descript sheep on BT outbreaks and 80% of them contained a significant positive effect of the density of exotic and cross-bred sheep (Fig. 2c). The best model contained the density of buffalo, non-descript sheep, exotic and cross-bred sheep and indigenous cattle and had much better support in the data than the null model (∆DIC = 12.89). Buffalo and non-descript sheep were the most important effects, leading to increases in DIC of 4.18 and 3.92, if dropped from the best model, followed by exotic and cross-bred sheep (∆DIC = 1.96 if dropped from top model). The indigenous cattle variable was added to six of the top models and had a significant negative impact on BT outbreaks in four of these. The effect of indigenous cattle was weak overall however, leading to an increase of only 0.91 DIC units if dropped from the top model. Thus density of buffalos, non-descript sheep and exotic sheep were the only HOST variables offered to the next step of the environmental variable selection.
Comparing between best models based on a single suite of predictors, the HOST-driven model (DIC = 197.63) outperformed the LANDSCAPE-driven model substantially (DIC = 202.01, ∆DIC = 4.37), and vastly outperformed the CLIMATE-driven model (DIC = 207.08, ∆DIC = 9.45).
### Combined host- and landscape -driven models of BT outbreaks
All possible combinations of five landscape and host predictors were fitted (irrig. crop, rain crop, buffalo, nsheep, esheep). Of the 33 models fitted, 10 models had similar support in the data, being within 2 DIC units of the best landscape and host model (Table 3, Fig. 2d). The best landscape and host model had a DIC of 198.54, and contained only buffalos, density of non-descript sheep and density of exotic and cross-bred sheep and had slightly less support in the data than the best host model (DIC = 197.63, Table S1, that had also included density of indigenous cattle). Thus combining landscape and host predictors did not improve the ability of models to explain patterns in BT outbreaks compared to models with HOST variables possibly because of the substantial collinearity between landscape and host predictors. Both buffalo (r = 0.650, p < 0.001) and non-descript sheep (r = 0.573, p < 0.001) are positively correlated with post-flooding or irrigated croplands whilst exotic and cross-bred sheep are not (r = −0.170, p = 0.132). Since these combined models perform less well than host only models, we infer the importance of the host predictors from the top host models, namely that densities of buffalos, non-descript sheep and exotic and cross-bred sheep all have consistent and important positive effects on BT outbreaks (Fig. 2c).
The overall accuracy of host models was good with low RMSE values in comparison to the observed range of variability in average BT outbreaks, ranging from 0.63 to 0.66 across top models (Table 2, see match between Fig. 1a,b). Out-of-fit model performance was also good, with little evidence that CPO values clustered around 0 for any of the top host models (Fig. S6), and a low logarithmic score for all top host models (Table 2).
In the models based only on hosts, the proportion of marginal variance explained by the spatially structured effect (ϕ) ranged from 0.64 to 0.78 (where 1 represents a purely spatial model), indicating a prevalent effect of the spatially structured random effect on average BT outbreak numbers (Table 2). Of the overall variance explained by the top host model for example, the unstructured random effect makes up 18.8%, the spatially-structured random effect 69.4% and the fixed effects 11.8%. Although relatively weak effects, the host covariates are statistically significant in the model, as the credible intervals for corresponding coefficients do not cross zero (Table 4) and the DIC for the null intercept-only model (without fixed effects) is 12.9 DIC units higher than the best model with fixed host and random effects. This is also confirmed by the high Pearson correlation values obtained between observations and predictions when top host models are fitted only with fixed effects but no random effects (Table S3). Thus, these host covariates are a good approximation of the general spatial trend in the data, despite additional noise, with the majority of the departures from this trend due to spatial autocorrelation (probably dependent on the disease transmission process).
In addition to the landscape-host associations mentioned above, across South India densities of non-descript sheep are positively correlated with those of buffalo (r = 0.669, p < 0.001), indigenous cattle (r = 0.545, p < 0.001) and goats (r = 0.447, p < 0.001). Densities of exotic sheep are by contrast negatively correlated with buffalo (r = −0.377, p < 0.001) and indigenous cattle (r = −0.363, p < 0.001), but positively correlated with cross-bred cattle (r = 0.371, p < 0.001). Examining the geographical coincidence of the key environmental predictors of BT outbreaks (Figs 1a and 3), the Andhra Pradesh hot spot of outbreaks is characterised by high densities of buffalo, non-descript sheep and post-flooding or irrigated croplands. In contrast, in Tamil Nadu at the tip of India, the areas of higher BT incidence are characterised by high densities of exotic and crossbred sheep together with high densities of buffalo and non-descript sheep (all exceeding 100,000 head per district), but with very little post-flooding or irrigated cropland. The district values of the spatial random effects are depicted in Fig. S7 are particularly substantial in this southern region of Tamil Nadu, indicating that alternative environmental effects may be constraining outbreaks there.
Considering the potential impacts on our analysis of temporal mismatches in climate predictors and the bluetongue data, very little change in median or range in district-level values of climate predictors was observed between snapshots at the start, middle and end of the study period. Districts have become a fraction warmer (<0.5 °C) on average from the start to the end of the period, and the middle period was drier on average than the start and end of the period (Fig. S8). Most importantly, values of climate predictors were highly correlated between the start, middle and end of the periods at the district level, meaning that districts that started out as warmer or wetter on average, remained warmer or wetter respectively (Table S5). For host predictors, there were substantial changes between livestock census periods (Fig. S9), with the total numbers of sheep and goats per district increasing over the study period, the number of cattle per district decreasing before increasing again to values comparable to the start of the study period, and the number of buffalos per district remaining fairly stable throughout the study period. However, the values of host predictors were highly correlated between livestock census periods at the district level, meaning that districts with more of a particular livestock type at the start of the study period tend to have more of the same livestock type at the end of the study period.
## Discussion
This study advances understanding of the geographical determinants of BT in South India by quantifying the roles of climate, landscape and host factors across a wide geographical area within the same analysis. The resulting models predict average spatial patterns in BT with a high degree of accuracy. We found that host factors, primarily the availability of both susceptible (exotic and cross-bred sheep breeds) and disease-resistant reservoir hosts (buffalo), are more important than land use and climate factors as predictors of variability in BT outbreaks in sheep between districts in South India.
At the national scale, BT is restricted to those parts of India that are heavily affected by monsoons and is linked anecdotally to the timing and intensity of the monsoon seasons20. However, our models suggest that within this affected area, spatial variability between districts in long term outbreak numbers is driven by host and landscape heterogeneity rather than by climate. In line with known effects of temperature and moisture on Culicoides life cycle parameters (reviewed in4) and on the basic reproduction number of bluetongue48, many prior studies in tropical and sub-tropical areas have found significant impacts of climatic factors on sub-national patterns of Culicoides-borne disease or sero-conversion rates49. The direction and magnitude of inferred climate effects, however, varies between regions. For example, in South Africa Baylis et al.24 found that African horse sickness virus epizootics were associated with the sequence of drought and flood brought by the warm phase of the El Nino Southern Oscillation. In tropical Australia14, heavy rainfall during the cyclonic season was fond to be unfavourable for BT transmission, possibly reducing vector population size by destroying breeding sites.
The lack of a detectable climate effect in our study may be because we modelled average patterns in BT outbreaks over a long period of time, potentially blurring important climate-driven inter-annual dynamics (cf. Purse et al.50) or due to the applied spatial scale units. By averaging patterns across villages within a district we may also miss the local scale effects of landscape factors on the BTV system. We also modelled patterns in outbreak numbers rather than seroconversion rates. Seroconversion rates are directly related to recent infection events and, if related to concurrent environmental conditions, should provide a more accurate picture of the conditions under which transmission occurs. But in India, seroconversion data are collected much less often, and very few places, compared to BT outbreak data22, and are thus likely currently to represent a narrow range of the environments in which transmission can occur.
The finding that more outbreaks occur in sheep on average in areas with high buffalo densities is consistent with the high antibody prevalence against BTV observed in this species, without clinical disease, in both India51 and China52 and indicates this species’ potential importance as a reservoir host. Although the duration and level of BTV viraemia has not been measured in buffalo, isolations have been made from aborted buffalos in India, suggesting a transmissible viraemia53. Moreover adult Culicoides vectors from Europe are observed to feed preferentially on larger animals such as cattle54 or horses55. This is explained in terms of their greater body surface but also their greater metabolic weight (calculated as body weight0.75) and emission of semio-chemicals including carbon dioxide, that are used by Culicoides in host-location. Whether buffalos are preferred over small ruminants as biting hosts for adult Culicoides in the Indian setting requires empirical confirmation, because such host preferences have direct consequences for species’ roles in transmission56.
Densities of exotic and cross-bred sheep also had a positive effect on outbreak numbers. Such breeds are known to be more susceptible to clinical signs of BT in India57 and Nepal17 than indigenous local breeds and so infection is more likely to result in a recorded outbreak. Indigenous local sheep breeds in Asia also show antibody prevalence against BT with few disease effects58,59, although sporadic clinical cases have been observed in these breeds since the 1980’s20,60. The only metric of indigenous sheep density available to our analysis were densities from breeds categorised as non-descript, defined as not having more than 50% similarity to any recognized local breed. These non-descript sheep were found to have a consistent positive association with BT cases. Whether this effect arises because these breeds are disease-resistant and maintain transmission silently or because they are susceptible to disease effects and contribute to recorded cases is difficult to separate given the current lack of empirical studies in India of breed susceptibility. Susceptibility of indigenous breeds in BTV transmission is extremely important to understand given that non-descript sheep constitute a huge proportion of the total sheep population in South India and often co-occur with buffalo and known susceptible sheep breeds (Figs 3, S5). Routine recording and analysis of breed composition for disease cases may also give some insight into their role in transmission.
Although landscape predictors did not improve the description of bluetongue patterns substantially when added to models containing host factors, the landscape-driven models identified important land use types that favour Culicoides populations, namely post-flooding/irrigated croplands (irrig. crop) and rain-fed croplands (Rainfed crop), and that are intimately linked to particularly communities of susceptible hosts for bluetongue within agricultural systems in India. The distribution and vectorial capacity of candidate vector species has not been well studied in India33 and no quantitative links between BTV-infection or incidence rates and vector community composition have been made. Anecdotally, three key species, namely C. imicola, C. peregrinus, and C. oxystoma are abundant in BT-affected areas36,61. Populations of C. imicola and C. peregrinus, which both breed in moist and organically enriched soil, have been significantly associated with irrigated areas or other areas of high soil moisture availability (C. imicola62; C. peregrinus63). C. oxystoma, which develops in a range of habitats including buffalo wallows64, has also been found in both active and abandoned rice paddy fields (encompassed by the irrigated cropland class in this study) elsewhere in Asia46). It is possible that the extensive rice belt found in Andhra Pradesh (the state most severely affected by BT) makes a substantial contribution to maintaining BT transmission by supporting high, proximate populations of buffalo (Fig. 1c) and C. oxystoma.
Landscape or habitat configuration may provide a potential explanation for why buffalos play a stronger role in driving variability in bluetongue outbreaks in sheep compared to cattle in this context. Indigenous cattle were found to have a minor effect and cross-bred cattle no effect on outbreaks compared to buffalos despite the high sero-prevalence of infection detected in cattle in India51 and the comparable variation in densities of these species compared to buffalos across districts in the region (Table S4). Whether the body size or herd sizes of buffalos makes them more attractive to host-seeking midges than cattle requires empirical confirmation but buffalos seem to overlap in habitat to a greater extent with susceptible sheep and C. oxystoma midges within rice paddies than do cattle. This illustrates the importance of taking a resource or habitat-based approach to predicting the interactions between key hosts and vectors in a disease system9.
The inference that buffalos (and potentially non-descript) indigenous breeds of sheep may be playing a key role in BTV transmission has considerable implications for BT control through vaccination in India. Inactivated pentavalent vaccine doses were supplied in 2015 by the respective state governments to small-holder farmers through village veterinary officers. These were targeted only at the susceptible sheep population in endemic districts and cattle and buffalo were not included as vaccination targets. Buffalo outnumber exotic and cross-bred sheep in all but five of the 80 districts in the focal states of Karnataka, Tamil Nadu and Andhra Pradesh, usually by more than two orders of magnitude. This means that partial coverage of the susceptible sheep population is unlikely to achieve herd immunity where-ever these species co-occur. BTV strains could likely attain high levels of transmission in the buffalo population which could act as a source for infection and re-infection of the sheep population. Transmission models from Europe show that vaccinating bovine reservoirs can be an effective strategy for reducing BTV transmission and disease impacts in the sheep population56. It is advisable to conduct sero-monitoring of buffalo populations (alongside resistant sheep breeds26) in different parts of the country, to modify the composition of the multi-valent inactivated vaccine administered to susceptible sheep accordingly, and to quantify with mathematical models whether and where vaccination of buffalo is needed to reduce transmission of BTV to sheep.
Our models showed good accuracy in both within-sample and out-of-fit tests but a high proportion of the overall variation was attributed to spatially structured random effects (this was particularly true for the districts at the southern tip of India). This indicates that the model could be improved by integration of other unmeasured, spatially structured environmental factors. These could include soil-related parameters (e.g. soil type and water retention capacity), or animal husbandry factors such as dung management and use as fertiliser, local drainage and flooding that influence vector populations or pathogen-related factors. In addition, the land use availability at the start of the study period may have been different due to the conversion of grassland and shrubland to cropland seen across the focal states65. Overall, we expect the temporal mismatch between the bluetongue outbreak data (1992–2009) and the rainfall (2004–2010) and host data (2007) to have had limited impact on our results because these conditions, when drawn from coarser resolution data, were highly correlated at district level between the start, middle and end of the study period.
Our approach was to fit a very robust spatial model that carefully analyses the importance of different types of spatial structure alongside climate, land use and hosts in determining bluetongue outbreak patterns as a baseline to inform future space-time model approaches. A properly specified space-time model would have to account for the greater sparsity and lower reliability of disease patterns observed at the monthly level as well as autocorrelation in outbreaks and covariates over time and the possibility that spatial correlation may be structured over time. More-over only the climate covariates are observed on the same monthly time scale as the outbreak data whilst the most epidemiologically relevant host and land use data are available for only a single snapshot in the latter part of the study period. The resulting difficulty in identifying all parameters using a space-time approach would trade-off against determining the relative roles of average climate, land use and host variability in making some areas more susceptible to outbreaks than others. Our framework is highly complementary to the NADRES (National Animal Disease Referral System), the existing early warning system for BT in India, developed and maintained by NIVEDI. NADRES predicts presence rather than outbreak number, quantifies the importance of wide ranging host, landscape and climate drivers but does not account for spatial dependence. To improve targeting of disease control and risk communication, future modelling frameworks for India should investigate the scale-dependent contribution of climate, host and landscape variability to outbreak patterns, over seasonal and inter-annual time-scales, from districts to village level. To optimise control measures that are often taken at the landscape scale, links between particular agricultural ecosystems (like rice paddy cultivation), reservoir and vector community composition and dynamics, and BT impacts should be quantified.
## Methods
### Epidemiological data
District level (Admin-2) annual BT outbreak data (1992–2009) were provided by NIVEDI of the Indian Council of Agricultural Research (ICAR). NIVEDI maintains the livestock diseases database for India, collating outbreak data at district level, based on clinical symptoms observed and reported each month by village-level veterinary officers. Since most goats, cattle and buffalo are disease resistant, reported BT outbreaks are from sheep and thus disease patterns in sheep are analysed here. The village is considered as the epidemiological unit for reporting animal disease outbreaks in India. An outbreak is defined as a village with more than one disease affected sheep in a given transmission year. The analysis was restricted to data from three states of South India in which outbreaks are regularly reported, namely Karnataka (n = 27 districts), Andhra Pradesh (n = 22 districts) and Tamil Nadu (n = 29 districts), with 61 out of the total of 78 constituent districts reporting outbreaks of BT over the 18 year study period (see Fig. S1 for state boundaries and location of the study region within India). The boundaries taken for Andhra Pradesh were those applicable before Telangana state was separated to become an independent state in 2014. District boundaries have changed over the time period for some states, but NIVEDI allocate the past outbreaks to districts according to the most recent district boundaries, prior to providing the data. The dependent variable is the average number of villages reporting bluetongue outbreaks in sheep per district from 1992 to 2009 with total number of villages per district used as the offset (see Modelling Approach) to account for the fact that districts with more villages and more village veterinary officers are likely to report more outbreaks. The city administrative districts of Chennai and Hyderabad were omitted from the analysis as urban districts with little farming they recorded village outbreak data extremely rarely and inconsistently over the study period.
### Environmental data
Landscape: Land use types vary in their suitability for grazing for susceptible and reservoir domestic hosts and in the likely extent of semi-aquatic breeding habitat available to Culicoides BTV vectors. The absolute areal coverage in km2 of each of seven land use types per district in 2009 were extracted from the Globcover 2009 land-cover map66 that has an original pixel resolution of 300 m using Zonal statistics in ArcMap 10.1 (ESRI, Inc., Redlands, CA, U.S.A.) (see Figs S3 and S4). Although the Glob-cover dataset includes additional forest and water-body land use types, these made up less than 3% of the area of districts on average and were therefore unlikely to have been major drivers of overall outbreak numbers.
Host species composition: Densities of the following six host taxa were extracted from the database of National Livestock census data 2007 (http://www.dahd.nic.in/ accessed on 5th May 2014): (1) nondescript indigenous sheep; (2) exotic and cross-bred sheep; (3) goats; (4) crossbred cattle; (5) indigenous cattle and (6) buffaloes. These host density data are collected at village level during the census and then provided as summed district level densities. These predictors were log-transformed (see Fig. S5). Non-descript sheep breeds are indigenous breeds that do not have more than 50% similarity (phenotypically) to any recognized local breed.
Climate variables: Culicoides life cycle and BTV transmission cycle parameters are highly sensitive to temperature and moisture availability (reviewed in4). The availability of moist soil breeding sites is highly dependent on seasonal rainfall patterns interacting with management factors like dung storage practices, irrigation and drainage. High temperatures increase rates of viral replication, blood digestion, immature midge development and adult biting, but decrease adult survival rates and dry up moist soil breeding sites. Monthly Rainfall Estimates (RFE) were obtained for seven years (2004–2010) from the NOAA/Climate Prediction centre RFE 2.067 at a pixel resolution of 0.1° latitude and longitude and were extracted for districts using zonal statistics ArcMap 10.1 (ESRI, Inc., Redlands, CA, U.S.A.). The average annual total rainfall, south-west monsoon rainfall (months of June to September) and north-east monsoon rainfall (months of October to December) were calculated. The annual mean temperature layer was downloaded from Worldclim68 (1950–2000) at a spatial resolution of around 1 km2 and extracted and averaged across districts using zonal statistics ArcMap 10.1 (Fig. S2). In addition to these four main effect climate predictors, the interaction between annual rainfall and annual temperature was also considered.
### Modelling approach
All environmental predictors were centred and standardised prior to model-fitting. To take account of spatial autocorrelation, the relationship between the average number of BT outbreaks in sheep per year and environmental predictors was quantified using spatial generalised linear mixed models, implemented in a Bayesian framework.
Let Ei denote the number of villages at risk of BT outbreaks in district i(i = 1, ….. n) used as the offset. The response variable, yi, the average number of BT outbreaks per year (as an integer) over the study period in district i, is assumed to follow a Poisson distribution:
$${y}_{i}|{\theta }_{i} \sim Poisson({E}_{i}{\theta }_{i})$$
where θi denotes the underlying true area-specific relative risk.
The log risk ηi = log(θi) was assumed to have the decomposition:
$${\eta }_{i}=\mu +{{z}_{i}}^{{\rm{{\rm T}}}}\beta +{b}_{i}$$
Here μ denotes the overall risk level, $${{z}_{i}}^{{\rm{{\rm T}}}}={({z}_{i1},\mathrm{....}.{z}_{ip})}^{{\rm{{\rm T}}}}$$ a set of p covariates with corresponding regression parameters $$\beta ={({\beta }_{1},\mathrm{....}.{\beta }_{p})}^{{\rm{{\rm T}}}}$$, and bi a random effect41. The random effects, $$b=({b}_{1},\,\,\ldots {b}_{n})$$, are used to account for extra-Poisson variation, or spatial dependence between districts due to intrinsic factors or unmeasured abiotic risk factors. Areas that are close in space are assumed to be more similar than areas that are not close, and here districts i and j were defined as neighbours if they shared a common border, denoted as i~j.
The area-specific random effect b, was modelled considering a modified Besag-York-Mollie model with a parameterisation suggested by Dean et al.69
$${\boldsymbol{b}}=\frac{1}{\sqrt{\tau }}(\sqrt{1-\varphi }{\boldsymbol{v}}+\sqrt{\varphi }{{\boldsymbol{u}}}_{\ast }),$$
having covariance matrix
$$\mathrm{Var}({\boldsymbol{b}}|{\tau }_{b})={\tau }_{b}^{-1}((1-\varphi ){\bf{I}}+\varphi {{\bf{Q}}}_{\ast }^{-}).$$
where u* is the scaled spatially structured effect, governed by a Gaussian Markov random field (GMRF) with regions conditionally independent unless classed as neighbours i~j, Q* is the precision matrix of the Besag model with $${{\bf{Q}}}_{\ast }^{-}$$ denoting the generalised inverse and vi is the unstructured effect modified by Riebler et al.41 to make the model more intuitive and interpretable. In this parameterisation both ui and vi are standardised to have (generalised) variance equal to one. The marginal precision is then τ and the proportion of the marginal variance explained by the spatial effect (u) is ϕ. Employing the hyper-parameterization structure proposed by41 allows these hyper-parameters to be mathematically interpretable and not confounded (as in the BYM model). The advantage of this formulation is that the compromise between pure over-dispersion and spatially structured correlation is reflected by the mixing parameter ϕ. When ϕ = 0, the model reduces to pure over-dispersion, whilst when ϕ = 1, the model reduces to the Besag model, i.e. only spatially structured correlation. The model was fitted by integrated nested Laplace approximation using the INLA R package (www.r-inla.org)70 and “bym2” model specified inside the model formula. Weak prior distributions were used for μ and β given by Gaussian distributions with zero mean and precision of 1 × 106. The prior distributions on the standard deviation $$1/\sqrt{\tau }$$ and ϕ are both based on transformed exponential distributions following the penalised complexity framework as described in Simpson et al.71. Using a transformed exponential distribution has the property that greatest density is at the lowest values for both the precision parameter (τ) and the mixing parameter (ϕ). This implies that the penalised complexity prior will tend to shrink towards a model of no spatial smoothing in the absence of sufficient support for spatial complexity.
### Model building and selection of environmental predictors
It was not computationally possible to fit all possible combinations (>130,000 combinations) of the 17 main effect predictors and one interaction term. Instead, the main effect predictors were divided into three categories, landscape predictors, climate predictors and host predictors and within each individual category all possible model combinations were first fitted to identify the best predictor variables from that category. Prior to this step, pairs of highly correlated predictors within each category were identified using Pairwise Pearson correlation analyses (r > 0.7, p < 0.001). These pairs were precluded from appearing together in model combinations. Once all models were fitted within a category, the model with the lowest Deviance Information Criteria (DIC)47 was identified as the best model. DIC is a generalisation of the Akaike Information Criterion (AIC), and is derived as the mean deviance adjusted for the estimated number of parameters in the model, compromising between model fit and complexity, and providing a measure of out-of-sample predictive error72. All models within 2 DIC units of the best model in a category were defined as the top model set, having similar support in the data as the best model, based on the often used rule of thumb of Burnham and Anderson73. The number of times that each predictor appeared and had a significant effect on BT outbreaks across the top model set was calculated. Predictors were only passed to the second stage of model selection if they had appeared in over 70% of within-category models, and had significant effects on BT outbreaks in over half of these models.
In the second stage of model selection, all possible combinations of the predictors selected in the first stage were fitted, and the top model set was again identified using DIC. The number of times that each predictor appeared and had a significant effect on BT outbreaks in the top model set was again calculated and mapped against the constituent predictors in the model. This approach reveals whether the effects of individual environmental predictors on BT outbreaks is robust to presence or absence of other predictors in the model and was an attempt to understand the importance of predictors whilst avoiding the drawbacks of model averaging pointed out by Cade74.
Considering metrics of model accuracy, we calculated the Root Mean Square Error between the predicted posterior mean values and the corresponding observed annual mean number of BT outbreaks per district. To test the out-of-fit predictive performance of the model, leave-one-out cross validation statistics, namely Conditional Predictive Ordinates (CPOs) were calculated75. The CPO expresses the posterior probability of observing the value (or set of values) of yi when the model is fitted to all data except yi, with a larger value implying a better fit of the model to yi, and very low CPO values suggest that yi is an outlier and an influential observation. When many CPO values cluster near zero, the model demonstrates poor out-of-fit performance. When many CPO values cluster near one, the model demonstrates good out-of-fit performance76. We then calculated the cross-validated logarithmic score77, by taking the negative sum of logged CPO values across districts, a lower value of which indicates a better prediction quality of the model.
In the model specification, the generalised variance is rescaled to 141, hence the summed variance across the components was set to be equal to 1. This allowed the contribution of each of the model components (spatially structured and unstructured random effects and the fixed effects) to the overall variance explained to be calculated. This follows a similar approach to that taken by Riebler et al.41.
However, the default posteriors for the spatial random effect parameters in INLA can result in a bias away from variance being attributed to fixed effects.
To avoid any bias in attributing variance to specific model components, the goodness of fit statistics and the proportion of variance explained by each component was calculated for a set of nested models. For each model in the top models, a full model, including fixed, random and spatial components, was compared with a set of sub-models in which one or two of these components were removed. Thus identifying any bias in the variance attribution by comparison amongst the model set.
### Temporal mismatches in environmental predictors and outbreak data
A potential drawback in our analysis is that while the temperature data reflect the whole period from which outbreak data are drawn, the rainfall, land cover data (from 2009) and the host data (from 2007) are drawn only from the latter half of the period. Datasets that spanned the total period were available for climate and land use. The Indian Meteorological Department New High Spatial Resolution (0.25 × 0.25 degree) Long Period (1901–2013) Daily Gridded Rainfall Data Set Over India78, and the ERA Interim daily 2 m temperature (0.25 × 0.25 degree)79 were too coarse in resolution compared to the size of a smaller districts (mean district size ± s.d. in km2 = 7452 ± 12, range in km2 = 178–19223). The alternative longitudinal dataset for land use, the ESA: Land Cover CCI Product80, divides the landscape into land use categories that are less explicitly related to midge and host habitats than those in the Globcover products. Similarly, only the more recent livestock census data divide livestock into indigenous and exotic breeds which are known to have different clinical responses to bluetongue (see above).
These temporal mismatches in data will only have a large impact on our analyses of which factors make some districts more susceptible to bluetongue outbreaks in sheep on average, if the late period conditions of climate, hosts and land use are poorly correlated with the equivalent conditions in a district during the earlier parts of the study period. We tested this by performing Pearson’s correlation analysis, paired by district, between the values of climate and host predictors in different snapshots across the study periods.
## Data Availability
The bluetongue data that were analysed during the current study are available from Indian Council of Agricultural Research-National Institute of Veterinary Epidemiology and Disease Informatics (ICAR-NIVEDI) and were obtained from the Director of ICAR-NIVEDI. The livestock data were downloaded from the website of Department of Animal Husbandry, Dairying and Fisheries and were used under licence for this study. The environmental data were compiled from third party sources as referenced in the methods. The bluetongue data are available from the authors on reasonable request, contingent on permission of the Director of ICAR-NIVEDI.
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## Acknowledgements
Authors would like to sincerely acknowledge Dr. H. Rahman, (prior Project Director of PD_ADMAS) for sharing the BT outbreak data and Drs Y.K. Reddy, Y.N. Reddy, S.M. Byregowda and M. Prasad for useful discussions of preliminary models. The authors would like to thank Andrew Lawson, Andrea Riebler and Havard Rue for invaluable advice on setting up spatial models in INLA. M.C., S.C. and B.V.P. were supported by IBVNet project (http://www.ibvnet.com/) grant no. BB/H009167/1 under the Combating Infectious Diseases of Livestock for International Development (CIDLID) program jointly funded by BBSRC, DFID, and the Scottish government. Additional support was provided to MC by the Indian Council of Agriculture Research and to BP from NERC under the SUNRISE project (Grant no. NE/R000131/1).
## Author information
Authors
### Contributions
B.P. and M.C. conceived and conducted the analysis and wrote the main manuscript. S.C. and G.P. provided expertise in Culicoides biology and bluetongue epidemiology whilst MRG provided outbreak data and interpretation. L.S. and P.H. provided statistical support. All authors reviewed and edited the manuscript.
### Corresponding author
Correspondence to B. V. Purse.
## Ethics declarations
### Competing Interests
The authors declare no competing interests.
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Chanda, M., Carpenter, S., Prasad, G. et al. Livestock host composition rather than land use or climate explains spatial patterns in bluetongue disease in South India. Sci Rep 9, 4229 (2019). https://doi.org/10.1038/s41598-019-40450-8
• Accepted:
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• DOI: https://doi.org/10.1038/s41598-019-40450-8
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EcoHealth (2021) | 2022-06-28 13:46:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5505523681640625, "perplexity": 7640.762159523408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103516990.28/warc/CC-MAIN-20220628111602-20220628141602-00356.warc.gz"} |
https://acikerisim.iku.edu.tr/entities/publication/7303eb34-01eb-4a57-9bb8-0b6b4c7e6302 | ## Yayın: The radius of starlikeness of the certain classes of p-valent functions defined by multiplier transformations
Yükleniyor...
2008-12
Acu, Mugur
Polatoğlu, Yaşar
Yavuz, Emel
##### Yayımcı
Springer International Publishing
##### Özet
The aim of this paper is to give the radius of starlikeness of the certain classes of Open image in new window-valent functions defined by multiplier transformations. The results are obtained by using techniques of Robertson (1953,1963) which was used by Bernardi (1970), Libera (1971), Livingstone (1966), and Goel (1972). | 2023-03-23 18:46:24 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9118430614471436, "perplexity": 2492.4371302941527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945182.12/warc/CC-MAIN-20230323163125-20230323193125-00283.warc.gz"} |
https://cs.stackexchange.com/questions/100496/graph-with-two-classes-of-nodes-a-and-b-densest-subgraph-with-same-node-classes | # Graph with two classes of nodes A and B: densest subgraph with same node classes cardinality
I'm searching (without results) a problem that can be reduced to finding the densest subgraph with the same cardinality between two classes of nodes.
Consider a graph with nodes of class A and nodes of class $$B$$ (not bipartite and $$|A|$$ it is not necessarily the same as $$|B|$$). The problem is to find the densest subgraph that respects the costraint $$|A\cap D|=|B\cap D|$$ where $$D$$ is the set of the nodes of the desired subgraph.
Before asking I tried with the partition problem, but the transformation between partition in this problem is pseudo polynomial and not polynomial (for each number N in the input set i build a clique of N nodes so the transformation depends on the value of the numbers in the set and this is not ok).
Thanks for the help :)
Edit: the definition of densest is the subgraph that maximizes #edges/#nodes. This subgraph can be computed in polynomial time with an algorithm by Goldberg.
• Is $|A \cap D| = 1 = |B \cap D|$ an optimal solution? What do you mean by densest? Did you also consider taking the complement of the graph? Then the problem might be related to finding the largest (balanced) induced bipartite subgraph. – Pål GD Nov 24 '18 at 18:07 | 2019-05-23 05:38:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7881072163581848, "perplexity": 263.28262015512644}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257100.22/warc/CC-MAIN-20190523043611-20190523065611-00176.warc.gz"} |
http://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-8th-edition/chapter-2-section-2-7-derivatives-and-rates-of-change-2-7-exercises-page-150/49 | ## Calculus: Early Transcendentals 8th Edition
(a) The average rate of change from 1990 to 2005 is $1169.6th/da.yr$ (b) The instantaneous rate of change in 2000 is $1397.8th/da.yr$
(a) The amount of oil in unit of thousands of barrels per day in 1990 is $66,533$ and in 2005 is $84,077$. So, the average rate of change from 1990 to 2005 is $$r=\frac{84,077-66,533}{2005-1990}=1,169.6\hspace{0.1cm}(th/da.yr)$$ The unit of the numerator is $th/da$ (thousands of barrels per day) and that of the denominator is $yr$ (year). Therefore, the unit of the average rate of change is $th/da.yr$ (b) The amount of oil in unit of thousands of barrels per day in 1995 is $70,099$, in 2000 is $76,784$ and in 2005 is $84,077$. So, the average rate of change (i) from 1995 to 2000: $\frac{76,784-70,099}{2000-1995}=1337(th/da.yr)$ (ii) from 2000 to 2005: $\frac{84,077-76,784}{2005-2000}=1458.6(th/da.yr)$ The instantaneous rate of change in 2000 would be the average of the average rates of change from 1995 to 2000 and from 2000 to 2005: $$\frac{1337+1458.6}{2}=1397.8(th/da.yr)$$ Taking the average does not change the unit. That means the unit here is still $th/da.yr$ | 2017-05-30 01:47:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.772658109664917, "perplexity": 178.87878958742684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613738.67/warc/CC-MAIN-20170530011338-20170530031338-00238.warc.gz"} |
https://www.physicsforums.com/threads/qm-problem-ii.62384/ | # Qm problem II
1. Feb 3, 2005
### broegger
I am trying to prove that
$$\frac{d\langle p \rangle}{dt} = \langle -\frac{\partial V}{\partial x} \rangle$$
I am done if I can just prove that
$$\left[ \Psi^*\frac{\partial^2 \Psi}{\partial x^2} \right]_{-\infty}^{\infty} = 0$$
$$\left[ \frac{\partial \Psi}{\partial x} \frac{\partial \Psi^*}{\partial x} \right]_{-\infty}^{\infty} = 0$$
My suggestion is that since $$\Psi$$ is a wavefunction, it is normalizable and must approach 0 as $$x \rightarrow \pm\infty$$, and so must its derivatives. I don't know if this argument holds?
Last edited: Feb 3, 2005
2. Feb 3, 2005
### vanesch
Staff Emeritus
I would think that's a valid reason, no ?
cheers,
Patrick.
3. Feb 3, 2005
### da_willem
Good luck proving something incorrect! :tongue2:
4. Feb 3, 2005
### broegger
I see your point :) I mean the time derivative, of course.
5. Feb 3, 2005
### dextercioby
That's something totally different.It is simply CORRECT...
Daniel.
6. Feb 3, 2005
### vanesch
Staff Emeritus
No, it also excaped me, but of course what is correct is:
d/dt <p> = <- dV/dx >
cheers,
Patrick. | 2017-11-24 11:21:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7553308606147766, "perplexity": 6687.859108919933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934807650.44/warc/CC-MAIN-20171124104142-20171124124142-00027.warc.gz"} |
https://mathhelpboards.com/threads/question-on-proof-of-completness-of-space-of-all-complex-valued-functions.2304/ | # question on proof of completness of space of all complex valued functions
#### oblixps
##### Member
in the proof of showing that the vector space of all complex valued functions with the norm $$|f|_u = sup(|f(x)|)$$ over all x in the domain is complete, there was a step that was confusing:
let $${f_n}$$ be a Cauchy sequence in the normed space Z. We know that $$|f_n(x) - f_m(x)| \leq |f_n - f_m|_u$$. So $${f_{n}(x)}$$ is a Cauchy sequence in $$\mathbb{C}$$ which is complete so f_n(x) converges to f(x) for every x. Letting n approach infinite on both sides of the inequality, we get $$|f(x) - f_n(x)| \leq lim \inf |f_n - f_m|_u$$.
my question is where did that lim inf come from?
#### Opalg
##### MHB Oldtimer
Staff member
in the proof of showing that the vector space of all complex valued functions with the norm $$|f|_u = sup(|f(x)|)$$ over all x in the domain is complete, there was a step that was confusing:
let $${f_n}$$ be a Cauchy sequence in the normed space Z. We know that $$|f_n(x) - f_m(x)| \leq |f_n - f_m|_u$$. So $${f_{n}(x)}$$ is a Cauchy sequence in $$\mathbb{C}$$ which is complete so f_n(x) converges to f(x) for every x. Letting n approach infinite on both sides of the inequality, we get $$|f(x) - f_n(x)| \leq lim \inf |f_n - f_m|_u$$.
my question is where did that lim inf come from?
In general, if $a_n\leqslant b_n$ and $a_n\to a$, then $a\leqslant \liminf b_n$. In fact, given $\varepsilon>0$ there exists $N$ such that $|a_n-a|<\varepsilon$ whenever $n\geqslant N.$ But there exists $m>N$ such that $b_m < \liminf b_n + \varepsilon$. For that value of $m$, $$a < a_m+\varepsilon \leqslant b_m+\varepsilon < \liminf b_n +2\varepsilon.$$ Since $\varepsilon$ is arbitrary, it follows that $a\leqslant \liminf b_n$. | 2021-03-03 21:58:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9703740477561951, "perplexity": 97.66445534775166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178367790.67/warc/CC-MAIN-20210303200206-20210303230206-00435.warc.gz"} |
https://chem.libretexts.org/Courses/UWMilwaukee/CHE_125%3A_GOB_Introductory_Chemistry/11%3A_Energy_and_Chemical_Processes/11.04%3A_Phase_Changes | # 11.4: Phase Changes
Learning Objectives
• Determine the heat associated with a phase change.
Matter can exist in one of several different states, including a gas, liquid, or solid state. The amount of energy in molecules of matter determines the state of matter.
• A gas is a state of matter in which atoms or molecules have enough energy to move freely. The molecules come into contact with one another only when they randomly collide.
• A liquid is a state of matter in which atoms or molecules are constantly in contact but have enough energy to keep changing positions relative to one another.
• A solid is a state of matter in which atoms or molecules do not have enough energy to move. They are constantly in contact and in fixed positions relative to one another.
The following are the changes of state:
Solid → Liquid Melting or fusion Liquid → Gas Vaporization Liquid → Solid Freezing Gas → Liquid Condensation Solid → Gas Sublimation
• If heat is added to a substance, such as in melting, vaporization, and sublimation, the process is endothermic. In this instance, heat is increasing the speed of the molecules causing them move faster (examples: solid to liquid; liquid to gas; solid to gas).
• If heat is removed from a substance, such as in freezing and condensation, then the process is exothermic. In this instance, heat is decreasing the speed of the molecules causing them move slower (examples: liquid to solid; gas to liquid). These changes release heat to the surroundings.
• The amount of heat needed to change a sample from solid to liquid would be the same to reverse from liquid to solid. The only difference is the direction of heat transfer.
Example $$\PageIndex{1}$$
Label each of the following processes as endothermic or exothermic.
1. water boiling
2. ice forming on a pond
Solution
1. endothermic - you must put a pan of water on the stove and give it heat in order to get water to boil. Because you are adding heat/energy, the reaction is endothermic.
2. exothermic - think of ice forming in your freezer instead. You put water into the freezer, which takes heat out of the water, to get it to freeze. Because heat is being pulled out of the water, it is exothermic. Heat is leaving.
Exercise $$\PageIndex{1}$$
Label each of the following processes as endothermic or exothermic.
1. water vapor condensing
2. gold melting
a. exothermic
b. endothermic
A phase change is a physical process in which a substance goes from one phase to another. Usually the change occurs when adding or removing heat at a particular temperature, known as the melting point or the boiling point of the substance. The melting point is the temperature at which the substance goes from a solid to a liquid (or from a liquid to a solid). The boiling point is the temperature at which a substance goes from a liquid to a gas (or from a gas to a liquid). The nature of the phase change depends on the direction of the heat transfer. Heat going into a substance changes it from a solid to a liquid or a liquid to a gas. Removing heat from a substance changes a gas to a liquid or a liquid to a solid.
Two key points are worth emphasizing. First, at a substance’s melting point or boiling point, two phases can exist simultaneously. Take water (H2O) as an example. On the Celsius scale, H2O has a melting point of 0°C and a boiling point of 100°C. At 0°C, both the solid and liquid phases of H2O can coexist. However, if heat is added, some of the solid H2O will melt and turn into liquid H2O. If heat is removed, the opposite happens: some of the liquid H2O turns into solid H2O. A similar process can occur at 100°C: adding heat increases the amount of gaseous H2O, while removing heat increases the amount of liquid H2O (Figure $$\PageIndex{1}$$).
Water is a good substance to use as an example because many people are already familiar with it. Other substances have melting points and boiling points as well.
Second, as shown in Figure $$\PageIndex{1}$$, the temperature of a substance does not change as the substance goes from one phase to another. In other words, phase changes are isothermal (isothermal means “constant temperature”). Again, consider H2O as an example. Solid water (ice) can exist at 0°C. If heat is added to ice at 0°C, some of the solid changes phase to make liquid, which is also at 0°C. Remember, the solid and liquid phases of H2O can coexist at 0°C. Only after all of the solid has melted into liquid does the addition of heat change the temperature of the substance.
For each phase change of a substance, there is a characteristic quantity of heat needed to perform the phase change per gram (or per mole) of material. The heat of fusion (ΔHfus) is the amount of heat per gram (or per mole) required for a phase change that occurs at the melting point. The heat of vaporization (ΔHvap) is the amount of heat per gram (or per mole) required for a phase change that occurs at the boiling point. If you know the total number of grams or moles of material, you can use the ΔHfus or the ΔHvap to determine the total heat being transferred for melting or solidification using these expressions:
$\text{heat} = n \times ΔH_{fus} \label{Eq1a}$
where $$n$$ is the number of moles and $$ΔH_{fus}$$ is expressed in energy/mole or
$\text{heat} = m \times ΔH_{fus} \label{Eq1b}$
where $$m$$ is the mass in grams and $$ΔH_{fus}$$ is expressed in energy/gram.
For the boiling or condensation, use these expressions:
$\text{heat} = n \times ΔH_{vap} \label{Eq2a}$
where $$n$$ is the number of moles) and $$ΔH_{vap}$$ is expressed in energy/mole or
$\text{heat} = m \times ΔH_{vap} \label{Eq2b}$
where $$m$$ is the mass in grams and $$ΔH_{vap}$$ is expressed in energy/gram.
Remember that a phase change depends on the direction of the heat transfer. If heat transfers in, solids become liquids, and liquids become solids at the melting and boiling points, respectively. If heat transfers out, liquids solidify, and gases condense into liquids. At these points, there are no changes in temperature as reflected in the above equations.
Example $$\PageIndex{2}$$
How much heat is necessary to melt 55.8 g of ice (solid H2O) at 0°C? The heat of fusion of H2O is 79.9 cal/g.
Solution
We can use the relationship between heat and the heat of fusion (Equation $$\PageIndex{1}$$) to determine how many cal of heat are needed to melt this ice:
\begin{align*} \ce{heat} &= \ce{m \times ΔH_{fus}} \\[4pt] \mathrm{heat} &= \mathrm{(55.8\: \cancel{g})\left(\dfrac{79.9\: cal}{\cancel{g}}\right)=4,460\: cal} \end{align*}
Exercise $$\PageIndex{2}$$
How much heat is necessary to vaporize 685 g of H2O at 100°C? The heat of vaporization of H2O is 540 cal/g.
\begin{align*} \ce{heat} &= \ce{m \times ΔH_{vap}} \\[4pt] \mathrm{heat} &= \mathrm{(685\: \cancel{g})\left(\dfrac{540\: cal}{\cancel{g}}\right)=370,000\: cal} \end{align*}
Table $$\PageIndex{1}$$ lists the heats of fusion and vaporization for some common substances. Note the units on these quantities; when you use these values in problem solving, make sure that the other variables in your calculation are expressed in units consistent with the units in the specific heats or the heats of fusion and vaporization.
Table $$\PageIndex{1}$$: Heats of Fusion and Vaporization for Selected Substances
Substance ΔHfus (cal/g) ΔHvap (cal/g)
aluminum (Al) 94.0 2,602
gold (Au) 15.3 409
iron (Fe) 63.2 1,504
water (H2O) 79.9 540
sodium chloride (NaCl) 123.5 691
ethanol (C2H5OH) 45.2 200.3
benzene (C6H6) 30.4 94.1
Sublimation
There is also a phase change where a solid goes directly to a gas:
$\text{solid} \rightarrow \text{gas} \label{Eq3}$
This phase change is called sublimation. Each substance has a characteristic heat of sublimation associated with this process. For example, the heat of sublimation (ΔHsub) of H2O is 620 cal/g.
We encounter sublimation in several ways. You may already be familiar with dry ice, which is simply solid carbon dioxide (CO2). At −78.5°C (−109°F), solid carbon dioxide sublimes, changing directly from the solid phase to the gas phase:
$\mathrm{CO_2(s) \xrightarrow{-78.5^\circ C} CO_2(g)} \label{Eq4}$
Solid carbon dioxide is called dry ice because it does not pass through the liquid phase. Instead, it does directly to the gas phase. (Carbon dioxide can exist as liquid but only under high pressure.) Dry ice has many practical uses, including the long-term preservation of medical samples.
Even at temperatures below 0°C, solid H2O will slowly sublime. For example, a thin layer of snow or frost on the ground may slowly disappear as the solid H2O sublimes, even though the outside temperature may be below the freezing point of water. Similarly, ice cubes in a freezer may get smaller over time. Although frozen, the solid water slowly sublimes, redepositing on the colder cooling elements of the freezer, which necessitates periodic defrosting (frost-free freezers minimize this redeposition). Lowering the temperature in a freezer will reduce the need to defrost as often.
Under similar circumstances, water will also sublime from frozen foods (e.g., meats or vegetables), giving them an unattractive, mottled appearance called freezer burn. It is not really a “burn,” and the food has not necessarily gone bad, although it looks unappetizing. Freezer burn can be minimized by lowering a freezer’s temperature and by wrapping foods tightly so water does not have any space to sublime into.
## Concept Review Exercises
1. Explain what happens when heat flows into or out of a substance at its melting point or boiling point.
2. How does the amount of heat required for a phase change relate to the mass of the substance?
3. What is the direction of heat transfer in boiling water?
4. What is the direction of heat transfer in freezing water?
5. What is the direction of heat transfer in sweating?
1. The energy goes into changing the phase, not the temperature.
2. The amount of heat is a constant per gram of substance.
3. Boiling. Heat is being added to the water to get it from the liquid state to the gas state.
4. Freezing. Heat is exiting the system in order to go from liquid to solid. Another way to look at it is to consider the opposite process of melting. Energy is consumed (endothermic) to melt ice (solid to liquid) so the opposite process (liquid to solid) must be exothermic.
5. Sweating. Heat is consumed to evaporate the moisture on your skin which lowers your temperature.
## Key Takeaway
• There is an energy change associated with any phase change.
## Exercises
1. How much energy is needed to melt 43.8 g of Au at its melting point of 1,064°C?
2. How much energy is given off when 563.8 g of NaCl solidifies at its freezing point of 801°C?
3. What mass of ice can be melted by 558 cal of energy?
4. How much ethanol (C2H5OH) in grams can freeze at its freezing point if 1,225 cal of heat are removed?
5. What is the heat of vaporization of a substance if 10,776 cal are required to vaporize 5.05 g? Express your final answer in joules per gram.
6. If 1,650 cal of heat are required to vaporize a sample that has a heat of vaporization of 137 cal/g, what is the mass of the sample?
7. What is the heat of fusion of water in calories per mole?
8. What is the heat of vaporization of benzene (C6H6) in calories per mole?
9. What is the heat of vaporization of gold in calories per mole?
10. What is the heat of fusion of iron in calories per mole?
1. 670 cal
2. 69,630 cal
3. 6.98 g
4. 27.10 g
1. 8,930 J/g
6. 12.0 g
1. 1,440 cal/mol
8. 7,350 cal/mol
9. 80,600 cal/mol
10. 3,530 cal/mol | 2021-10-22 20:36:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7184187173843384, "perplexity": 1135.2502681083704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585518.54/warc/CC-MAIN-20211022181017-20211022211017-00155.warc.gz"} |
http://www.theinfolist.com/php/SummaryGet.php?FindGo=dimension_of_a_vector_space | dimension of a vector space
TheInfoList
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say $V$ is if the dimension of $V$ is wiktionary:finite, finite, and if its dimension is infinity, infinite. The dimension of the vector space $V$ over the field $F$ can be written as $\dim_F\left(V\right)$ or as $\left[V : F\right],$ read "dimension of $V$ over $F$". When $F$ can be inferred from context, $\dim\left(V\right)$ is typically written.
# Examples
The vector space $\R^3$ has $\left\$ as a standard basis, and therefore $\dim_\left(\R^3\right) = 3.$ More generally, $\dim_\left(\R^n\right) = n,$ and even more generally, $\dim_\left(F^n\right) = n$ for any Field (mathematics), field $F.$ The complex numbers $\Complex$ are both a real and complex vector space; we have $\dim_\left(\Complex\right) = 2$ and $\dim_\left(\Complex\right) = 1.$ So the dimension depends on the base field. The only vector space with dimension $0$ is $\,$ the vector space consisting only of its zero element.
# Properties
If $W$ is a linear subspace of $V$ then $\dim \left(W\right) \leq \dim \left(V\right).$ To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if $V$ is a finite-dimensional vector space and $W$ is a linear subspace of $V$ with $\dim \left(W\right) = \dim \left(V\right),$ then $W = V.$ The space $\R^n$ has the standard basis $\left\,$ where $e_i$ is the $i$-th column of the corresponding identity matrix. Therefore, $\R^n$ has dimension $n.$ Any two finite dimensional vector spaces over $F$ with the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If $B$ is some set, a vector space with dimension $, B,$ over $F$ can be constructed as follows: take the set $F^\left(B\right)$ of all functions $f : B \to F$ such that $f\left(b\right) = 0$ for all but finitely many $b$ in $B.$ These functions can be added and multiplied with elements of $F$ to obtain the desired $F$-vector space. An important result about dimensions is given by the rank–nullity theorem for linear maps. If $F / K$ is a field extension, then $F$ is in particular a vector space over $K.$ Furthermore, every $F$-vector space $V$ is also a $K$-vector space. The dimensions are related by the formula $\dim_K(V) = \dim_K(F) \dim_F(V).$ In particular, every complex vector space of dimension $n$ is a real vector space of dimension $2n.$ Some formulae relate the dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If $V$ is a vector space over a field $F$ then and if the dimension of $V$ is denoted by $\dim V,$ then: :If dim $V$ is finite then $, V, = , F, ^.$ :If dim $V$ is infinite then $, V, = \max \left(, F, , \dim V\right).$
# Generalizations
A vector space can be seen as a particular case of a matroid, and in the latter there is a well-defined notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces. The Krull dimension of a commutative Ring (algebra), ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
## Trace
The dimension of a vector space may alternatively be characterized as the Trace (linear algebra), trace of the identity operator. For instance, $\operatorname\ \operatorname_ = \operatorname \left\left(\begin 1 & 0 \\ 0 & 1 \end\right\right) = 1 + 1 = 2.$ This appears to be a circular definition, but it allows useful generalizations. Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an Algebra over a field, algebra $A$ with maps $\eta : K \to A$ (the inclusion of scalars, called the ''unit'') and a map $\epsilon : A \to K$ (corresponding to trace, called the ''counit''). The composition $\epsilon \circ \eta : K \to K$ is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in bialgebras, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension ($\epsilon := \textstyle \operatorname$), so in these cases the normalizing constant corresponds to dimension. Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "trace class operators" on a Hilbert space, or more generally nuclear operators on a Banach space. A subtler generalization is to consider the trace of a ''family'' of operators as a kind of "twisted" dimension. This occurs significantly in representation theory, where the Character (mathematics), character of a representation is the trace of the representation, hence a scalar-valued function on a Group (mathematics), group $\chi : G \to K,$ whose value on the identity $1 \in G$ is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: $\chi\left(1_G\right) = \operatorname\ I_V = \dim V.$ The other values $\chi\left(g\right)$ of the character can be viewed as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant, $j$-invariant is the graded dimension of an infinite-dimensional graded representation of the monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.
* * * * * , also called Lebesgue covering dimension
* | 2022-01-26 23:41:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 74, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9367984533309937, "perplexity": 128.1646695689957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305006.68/warc/CC-MAIN-20220126222652-20220127012652-00578.warc.gz"} |
http://accessanesthesiology.mhmedical.com/content.aspx?bookid=1471§ionid=85199818 | Pharmacology
### Pharmacology
Which of the following is true regarding seizures as one of the multiple side effects from the use of opioids?
(A) Morphine and related opioids can cause seizure activity when moderate doses are given
(B) Seizure activity is more likely with meperidine, especially in the elderly and with renal dysfunction
(C) Seizure activity is mediated through stimulation of N-methyl-d-aspartate (NMDA) receptors
(D) Naloxone is very effective in treating seizures produced by morphine and related drugs including meperidine
(E) Seizure activity is most likely related with the fact that opioids stimulate the production of γ-aminobutyric acid (GABA)
(B) Extremely high doses of morphine and related opioids can produce seizures, presumably by inhibiting the release of GABA (at synaptic level).
Normeperidine a metabolite of meperidine is prone to produce seizures and tends to accumulate in patients with renal dysfunction and in the elderly.
Naloxone may not effectively treat seizures produced by meperidine.
Which of the following is true regarding respiratory depression related to the use of opioids?
(A) Opioid agonists, partial agonists, and agonist/antagonists produce the same degree of respiratory depression
(B) Opioids produce a leftward shift of the CO2 response curve
(C) Depression of respiration is produced by a decrease in respiratory rate, with a constant minute volume
(D) Naloxone partially reverses the opioid-induced respiratory depression
(E) The apneic threshold is decreased
(E) Opioids produce a dose-dependant respiratory depression by acting directly on the respiratory centers on the brainstem. Partial agonist and agonist-antagonist opioids are less likely to cause severe respiratory depression, as are the selective K-agonist.
Therapeutic doses of morphine decrease minute ventilation by decreasing respiratory rate (as oppose to tidal volume).
Opioids depress the ventilatory response to carbon dioxide; the carbon dioxide-response curve shows a decrease slope and rightward shift.
The apneic threshold is decreased and also the increase in ventilatory response to hypoxemia is blunted by opioids.
Naloxone can effectively and fully reverse the respiratory depression from opioids.
The use of which of the following opioids would produce the greatest incidence of delayed respiratory depression?
(A) 25 μg intravenous (IV) fentanyl (bolus)
(B) 4 mg IV morphine (bolus)
...
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Connect to the full suite of AccessAnesthesiology content and resources including procedural videos, interactive self-assessment, real-life cases, 20+ textbooks, and more | 2017-03-31 00:35:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28238362073898315, "perplexity": 8172.735031080308}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218205046.28/warc/CC-MAIN-20170322213005-00059-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://cstheory.stackexchange.com/questions/31097/johnson-lindenstrauss-for-random-variables | # Johnson Lindenstrauss for Random variables?
Does the Johnson-Lindenstrauss Lemma apply to any finite-dimensional Hilbert Space? In particular, I am interested in the space of random variables $X = (X_1,...,X_N)$ over $N$ uncertain states. If $\pi_i$ is the probability of state $i$, then this space has an inner product $\langle X, Y \rangle = E(XY) = \sum_{i=1}^n \pi_i X_i Y_i$ and a norm $\|X\|^2_{\pi} = \langle X, X \rangle = \sum_{i=1}^N \pi_i X_i^2$.
The standard JL lemma says that, if $S$ is a set of $m$ points in $\mathbb{R}^n$ and $n > C \frac{ln(m)}{\varepsilon^2}$ then there a (suitably scaled) random orthogonal projection $f:\mathbb{R}^N \to \mathbb{R}^n$ will satisfy
$$(1-\varepsilon)\|u-v\|^2_{N} \leq \|f(u) - f(v)\|_{n}^2 \leq (1+\varepsilon)\|u-v\|_{N}^2$$ where I have used $\|\cdot\|_{n},\|\cdot\|_{N}$ to denote the standard euclidean norms in $\mathbb{R}^n$ and $\mathbb{R}^N.$
Does there exist a version of the lemma with the weighted norm $\|X\|^2_{\pi}$? What would the corresponding norm in the lower $n$-dimensional space look like?
• It is not clear from your question if $\pi$ is fixed or not. If it is not fixed, then this is impossible, as your can expose every coordinate in the original input. If $\pi$ is fixed, then it can be interepeted as a linear scaling of space ($\sqrt(\pi_i)$ in each coordinate, naturally. Of course, you need that JL works for dot products, but that is well known. – Sariel Har-Peled Apr 11 '15 at 22:49
• Yes, JL does work with any dot product you define, because all Hilbert spaces of the same dimension are linearly isometric. – Sasho Nikolov Apr 12 '15 at 1:24 | 2019-07-20 18:42:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9666110277175903, "perplexity": 171.0622418686965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526560.40/warc/CC-MAIN-20190720173623-20190720195623-00246.warc.gz"} |
https://www.gamedev.net/forums/topic/341162-2d-collision-detection-with-triangles/ | # 2d collision detection with triangles
This topic is 4898 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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I'm programing a physics engine and have run into a problem. I don't know how to separate two objects and find the point of collision. My objects are stored as triangles and the triangles can be arranged in any way (not connected, overlapping etc.). I have SAT implemented for detecting a collision but I don't know what to do next. If I have to I will implement a different method but I don't know what other algorithms there are. Could someone help me or tell me where I can find a tutorial?
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Check here: http://uk.geocities.com/olivier_rebellion/
And specifically this demo (includes lots of usefull documentation): http://uk.geocities.com/olivier_rebellion/Polycolly.zip
Of course, do not forget to google for it. I'm sure you'll find planty of other tutorials over the net.
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Ive read Olis tutorials (I have the polycolly docs open in my browser right now).
Unfortunately he only uses polygons and doesn't describe how to use groups of triangles which is what I need.
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that problem is difficult. The SAT works well for convex geometry, but for polygon soups, it's a lot more complicated usually, due to the number of contacts it can generate (if you do a per-poly calculation).
The thing with per-triangle tests, there is no notion of volume. It's just a bunch of triangles colliding.
You need to analyse the intersections and turn them into meaningful data.
I never really tried to solve those problems, maybe subconsciously trying to avoid the nightmares they generate, but here's a thought.
...I don't think there is a full-proof design on how to extract informations from intersections, but if you store the intersections between triangles (will gives segments), those segments should give you closed contour lines.
Now what to do with those lines... Well, they should roughly follow a plane, and then you can guess where this is going. You need to extract orientation information from those contours (like, use a covariance matrix, and eigen vectors to form a basis, and take the smallest of the vector deviation as your plane normal).
With non-convex geometry, you will get several contours. Each of them will produce a contact point (the 'centre' of the countour) and a normal (the normal of the plane that best fits the contour points).
That's for intersections.
For collisions, where triangles naturally collide after a given time, then you just need a bit of sorting and merging the contact points that are close together to avoid redundancies (and over the top responses).
That, or order your model into convex pieces and use the SAT / GJK / whatever.
my 2 cents.
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Quote:
Original post by oliiiconfusing stuffThat, or order your model into convex pieces and use the SAT / GJK / whatever.
I think Ill do the latter.
But theres still the problem of how to represent the static geometry of the world (which will have concave shapes). How does one handle this?
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triangles, that you collide with simple shapes, like boxes and sphere, and some fudge to make it work.
Here is an example of a method I used.
It's not particularly good nor clever, and the collision system it was used for was only dealing with intersecting geometry. My main problem was hitting edges of triangles, which will provide a hard collision, even if the surface appears smooth. This was mainly a problem with bxes, spheres would bump a bit, but nothing major, but it was still a problem.
A way I dealt with was to generate normals at vertices. then when I got a edge-edge collision between a triangle and a box, I'd find teh point on the triangle edge that was in contact. On that point, I'd interpolate the normal from the two edge vertices. That normal was then used for the collision impulse, not the intersection direction. That way, for flat surfaces, the normal on vertices would point up, and so was the interpolated normal on the edge.
That produced a smooth collision impulse.
There was also the matter of cleaning up the contacts (avoid duplications and redundancies), sorting them (along the velocity), and the small matter of dealing with reducing intersections, ect... It worked OK in the end, but not perfect by any means. Again, one way to reduce those problems is deal with swept tests.
So, lots of fudge.
demos.
- keys : F1->F8 for various demos, AWSD, mouse.
- docs included for the swept SAT.
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I cant get it to work. [sad] Somethings wrong with the impulse or collision point but I cant figure it out. I dont know what to do.
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if you don't tell us what you are doing, we can't help you.
BTW, I missed the "2d" part in the OP. but anyway, show us what you are doing.
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I know that the SAT is working so it must be something in this code.
Sorry for the bad variable names. This is in Python but I'm not too worried about performance yet.
# finds the collision pointdef __findCP(self,ta,tb,n): # ta is a triangle from object a and tb is from object b. n is the collision normal ma = 1000000.0 mb = -1000000.0 # find the minimum and maximum points on the triangles for p in ta: t = p.dot(n) if t < ma: ma = t for p in tb: t = p.dot(n) if t > mb: mb = t na = 0 nb = 0 cpa = None cpb = None # count the points on the axis for p in ta: t = p.dot(n) if math.fabs(t-ma) <= 0.001: na += 1 cpa = p for p in tb: t = p.dot(n) if math.fabs(t-mb) <= 0.001: nb += 1 cpb = p if na > 1: return cpb return cpa############################### this is after a collision has been foundd = self.pos - body.pos # get the difference between bodysif d.dot(p) < 0.0: # p is the normal of collision p = Vector([0,0])-p # i didnt make a negative operatorfa = 1.0 # this code seperates the bodys (whats the correct way to do this?)fb = 1.0if self.invmass == 0: fa = 0.0if body.invmass == 0: fb = 0.0self.pos += p*fabody.pos -= p*fbta = Triangle(self.tris[cp[0]]) # make a copy of the colliding triangle on this bodytb = Triangle(body.tris[cp[1]]) # make a copy of the colliding triangle on the other bodyif math.fabs(self.r) > 0.0001: # rotate ta.rotate(self.r)if math.fabs(body.r) > 0.0001: tb.rotate(body.r)ta += self.pos # translatetb += body.poscp = self.__findCP(ta,tb,p) # finds the colliding pointpa = self.pos-cp # difference of cp and center of this objectpa = Vector([-pa.y,pa.x]) # make it perpendicularpb = body.pos-cp # same as above but for other objectpb = Vector([-pb.y,pb.x])pa1 = pa.dot(p) # used in impulse calculationpb1 = pb.dot(p)# impulsej = -(1.0+1.0)*((self.dpos-body.dpos).dot(p)) / (p.dot(p)*(self.invmass+body.invmass) + pa1*pa1*self.invinertia + pb1*pb1*body.invinertia)# change the velocity and rotational velocityself.dpos = self.dpos + j*self.invmass*pbody.dpos = body.dpos - j*body.invmass*pself.dr = self.dr + pa.dot(-j*p)*self.invinertiabody.dr = body.dr + pb.dot(j*p)*body.invinertia
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If your problem is: search for collisions in a set of triangles the algorithm is really simple!
First you need a data structure to store your triangles.
In 2D you can use quadtree or, best, an hash matrix (voxel).
The idea is to divide your space in a grid of cells.
In each cell you store a list of the triangles intersecting the cell.
As insertion key you can use, conservatively, the bounding rect of the triangle.
The hash map (quadtree, matrix, ...) is very important if you have a lot of triangles!
In fact if you compare each triangles with the entire set you will get O(N^2); by using an hash matrix you can reduce considerably the work.
Obviously a trick is to compare
T1 with T2,T3,...,Tn then
T2 with T3,..,Tn (T1 and T2 already compared)
Ti with Ti+1,...,Tn
So you have the half of comparisons.
Instead with an hash map, given a triangle, you can find a set of possible intersections.
Given two triangles the algorithm is the edge-edge test. If you find an edge of triangle A that intersects an edge of triangle B you have found an intersection.
So in the worst case you have to perform 6 tests.
Tip: you can avoid the intersection test if the bounding rectangles have no intersection. This test is fast.
In 3D it's more complicate but not so much; in this case you can use a frustum cull.
This solution applied to 2D is interesting because if the clipping is not empty you have also the intersection area (in the form of a convex polygon).
So your problem is reduced to implement an hash map and to compute segment-segment intersection.
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× | 2019-01-24 08:57:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27389442920684814, "perplexity": 2197.7613947609243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584519757.94/warc/CC-MAIN-20190124080411-20190124102411-00452.warc.gz"} |
https://tahitianhoney.com/napier/euclidean-geometry-in-mathematical-olympiads-egmo-pdf.php | # Euclidean geometry in mathematical olympiads egmo pdf
## গণিত অলিম্পিয়াডের প্রস্তুতি – SONP ARENA
MAA Press Books American Mathematical Society Home. Evan is also the author of a popular textbook Euclidean Geometry in Math Olympiads used by students preparing for mathematical olympiads. As a high school student, Evan was himself an IMO gold medalist and winner of the 2014 USAMO (which he took from 12:30am to 5am in Taiwan). Outside of math and teaching, Evan enjoys StarCraft and Korean pop, 9/19/2018 · There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of ….
### Euclidean Geometry in Mathematical Olympiads Evan Chen
Evan Chen & Geo Book (EGMO). we remind the reader that angle chasing is only a small part of olympiad geometry, and not to overuse it. Problem for this Section Problem 2.2. Find an example of two triangles ABC and XYZ such that AB: XY = BC: YZ, BCA= YZX,but ABC and XYZare not similar. 2.2 Power of a Point Cyclic quadrilaterals have many equal angles, so it should come as, Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads..
Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads. 10/4/2017 · EGMO is quite useful, but has already been mentioned. I would recommend having a look at Yufei Zhao’s handouts on his website (just search up Yufei Zhao on google and you’ll find it). He has some handouts on configurations that come up in geometry...
You can also purchase a PDF. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or … A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian
2/1/2018 · Euclidean Geometry in Mathematical Olympiads MAA Problem Book Series March 25, 2016 Textbook for students preparing for national/international olympiads such as USAMO and IMO. Euclidean Geometry In Mathematical Olympiads Study group for Euclidean Geometry in Mathematical Olympiads by Evan Chen. UlTiMaTe Poll Forum! Collections of forums, tags, topics, and posts from the AoPS community. 3. V New Topic k Locked. M. Nothing matches your input.
While Euclidean geometry was codified almost 2,500 years ago, the field has been anything but stagnant since then. This book is an existence proof of how dynamic it is. The problems are original, challenging and have that achievable level of difficulty that makes them worthy Mathematical Olympiad A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian
The booklets in the series, A Taste of Mathematics, are published by the Canadian Mathematical Society (CMS).They are designed as enrichment materials for high school students with an interest in and aptitude for mathematics. 10/4/2017 · EGMO is quite useful, but has already been mentioned. I would recommend having a look at Yufei Zhao’s handouts on his website (just search up Yufei Zhao on google and you’ll find it). He has some handouts on configurations that come up in geometry...
List of mathematics competitions. Language Watch Edit Mathematics competitions or mathematical olympiads are competitive events where participants sit a mathematics test. These tests may require multiple choice or numeric answers, or a detailed written solution or proof. European Girls' Mathematical Olympiad (EGMO) — since April 2012; A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian
Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. Our Team. Our History. Jobs. Site Info. Terms. Euclidean Geometry In Mathematical Olympiads Study group for Euclidean … Evan Chen & Geo Book (EGMO) email: evan [at] evanchen•cc. evan chen (陳誼廷) euclidean geometry in mathematical olympiads (egmo) you can get a hard copy from amazon or the ams. you can also purchase a pdf. euclidean geometry in mathematical olympiads (often abbreviated egmo, despite an olympiad having the same name) is a comprehensive problem-solving book in
Euclidean Geometry in Mathematical Olympiads By Evan Chen This is a challenging problem solving book in Euclidean Geometry, assuming nothing of the reader other than a good deal of courage. Topics covered include cyclic quadtrilaterals, power of a point, homothety, and triangle centers. The emphasis of the book is placed squarely on the Euclidean Geometry in Mathematical Olympiads By Evan Chen This is a challenging problem solving book in Euclidean Geometry, assuming nothing of the reader other than a good deal of courage. Topics covered include cyclic quadtrilaterals, power of a point, homothety, and triangle centers. The emphasis of the book is placed squarely on the
See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. If you notice any errors, please let me know! LaTeX notes: I provided the LaTeX source for most of these files as an example here. See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. If you notice any errors, please let me know! LaTeX notes: I provided the LaTeX source for most of these files as an example here.
File Format: PDF/Adobe Acrobat - Quick View Areal Co-ordinate Methods in Euclidean Geometry. Tom Lovering. April 11, 2008. Introduction. In this article I aim to briefly develop the theory of areal (or File Format: PDF/Adobe Acrobat - Quick View Areal Co-ordinate Methods in Euclidean Geometry. Tom Lovering. April 11, 2008. Introduction. In this article I aim to briefly develop the theory of areal (or
### Further reading Complex Projective 4-Space
গণিত অলিম্পিয়াডের প্রস্তুতি – SONP ARENA. Not a single complaint of any sort from anyone, anywhere in the world. I was just trying to give away - for free - PDFs of public domain texts on mathematics and mathematical astronomy - PDFs that I had either made myself, or found on other sites and "repaired" by replacing missing pages with scans from copies I had found at research libraries., UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries. The Romanian Master of Mathematics; The European Girls' Mathematical Olympiad (EGMO). The Benelux Mathematical Olympiad. The Pan African Mathematical Olympiad. Here is specific advice about learning Euclidean Geometry. Contact Information:.
### Egmo.co.il site-stats.org
গণিত অলিম্পিয়াডের প্রস্তুতি – SONP ARENA. Euclidean Geometry in Mathematical Olympiads With 248 Illustrations c 2016 by The Mathematical Association of America each with four problems to be solved in four hours EGMO The European Girls’ Mathematical Olympiad, "Analysis of Mixed Natural and Symbolic Language Input in … https://en.wikipedia.org/wiki/Euclid_axioms Evan Chen & Geo Book (EGMO) email: evan [at] evanchen•cc. evan chen (陳誼廷) euclidean geometry in mathematical olympiads (egmo) you can get a hard copy from amazon or the ams. you can also purchase a pdf. euclidean geometry in mathematical olympiads (often abbreviated egmo, despite an olympiad having the same name) is a comprehensive problem-solving book in.
• geometry Ill-known/original/interesting investigations
• MAA Press Books American Mathematical Society Home
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• problem collections with solutions from various International Mathematical Olympiads olympiad geometry notes, books, magazines, articles, links Here I am gonna add the full issues from free magazines of Euclidean Geometry as an online backup. geometry problems in pdf with aops links in Greek (GR) Balkan MO Geometry 1984 - 2017 GR This challenging problem-solving book on Euclidean geometry is suitable for students preparing for national or international mathematical Olympiads. In addition to offering a guided tour through the classical results of Euler, Pascal, and others, it provides several carefully chosen worked examples and over 300 practice problems from contests
Geometry G1. Let ABC be an acute triangle with D,E,F the feet of the altitudes lying on BC,CA,AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that AP AQ. (United Kingdom) Solution 1. The line EF intersects the circumcircle at two points. Depending on the choice File Format: PDF/Adobe Acrobat - Quick View Areal Co-ordinate Methods in Euclidean Geometry. Tom Lovering. April 11, 2008. Introduction. In this article I aim to briefly develop the theory of areal (or
10/4/2017 · EGMO is quite useful, but has already been mentioned. I would recommend having a look at Yufei Zhao’s handouts on his website (just search up Yufei Zhao on google and you’ll find it). He has some handouts on configurations that come up in geometry... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. Our Team. Our History. Jobs. Site Info. Terms. Euclidean Geometry In Mathematical Olympiads Study group for Euclidean …
9/19/2018 · There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of … For a beginner, especially to master Geometry ‘Pre college Mathematics’ is the way to start. For insights and approach on problem solving ‘Solving Problems in Geometry:Insights and strategies’ by Kim Hoo Hang published by world scientific is a sup...
Asia International Mathematical Olympiad Past Papers. Asia International Mathematical Olympiad Past Papers Yuka has been added to the squad of eleven from which the team of six for the IMO in Bath will be selected at the Leeds camp. Our thanks to Man Group for their support of the EGMO team.. European Girls’ Mathematical Olympiad Day 2 (10 April 2019). The problems from the second paper at the eighth European Girls’ Mathematical Olympiad, sat today in Kyiv, are now available.
152 8. Inversion let X be the point on closest to O (so OX⊥ ).Then X∗ is the point on γ farthest from O, so that OX∗ is a diameter of γ.Since O, X, X∗ are collinear by definition, this implies the result. In a completely analogous fashion one can derive the converse—the image of a circle passing through O is a line. Also, notice how the points on ω are fixed during the whole The booklets in the series, A Taste of Mathematics, are published by the Canadian Mathematical Society (CMS).They are designed as enrichment materials for high school students with an interest in and aptitude for mathematics.
4/5/2013 · File Format: PDF/Adobe Acrobat - Quick View BMO Marking. Each question on the British Mathematical Olympiad is marked out of a total of. 10 marks. Evan is also the author of a popular textbook Euclidean Geometry in Math Olympiads used by students preparing for mathematical olympiads. As a high school student, Evan was himself an IMO gold medalist and winner of the 2014 USAMO (which he took from 12:30am to 5am in Taiwan). Outside of math and teaching, Evan enjoys StarCraft and Korean pop
9/19/2018 · There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of … Euclidean Geometry In Mathematical Olympiads Study group for Euclidean Geometry in Mathematical Olympiads by Evan Chen. UlTiMaTe Poll Forum! Collections of forums, tags, topics, and posts from the AoPS community. 3. V New Topic k Locked. M. Nothing matches your input.
In fact, there exists a $4 \times 4$ linear representation of the co-called conformal geometry including another category of non-linear transforms, translations. Reference: the very good book "Riemannian Geometry" by S. Gallot, D. Hullin, J. Lafontaine, 2nd edition 1993. Universitext, Springer, pages 175-176. Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. Our Team. Our History. Jobs. Site Info. Terms. Euclidean Geometry In Mathematical Olympiads Study group for Euclidean …
This challenging problem-solving book on Euclidean geometry is suitable for students preparing for national or international mathematical Olympiads. In addition to offering a guided tour through the classical results of Euler, Pascal, and others, it provides several carefully chosen worked examples and over 300 practice problems from contests UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries. The Romanian Master of Mathematics; The European Girls' Mathematical Olympiad (EGMO). The Benelux Mathematical Olympiad. The Pan African Mathematical Olympiad. Here is specific advice about learning Euclidean Geometry. Contact Information:
Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads. UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries. The Romanian Master of Mathematics; The European Girls' Mathematical Olympiad (EGMO). The Benelux Mathematical Olympiad. The Pan African Mathematical Olympiad. Here is specific advice about learning Euclidean Geometry. Contact Information:
## List of mathematics competitions Wikipedia
Euclidean geometry in mathematical olympiads. 2/1/2018 · Euclidean Geometry in Mathematical Olympiads MAA Problem Book Series March 25, 2016 Textbook for students preparing for national/international olympiads such as USAMO and IMO., Yuka has been added to the squad of eleven from which the team of six for the IMO in Bath will be selected at the Leeds camp. Our thanks to Man Group for their support of the EGMO team.. European Girls’ Mathematical Olympiad Day 2 (10 April 2019). The problems from the second paper at the eighth European Girls’ Mathematical Olympiad, sat today in Kyiv, are now available..
### geometry Ill-known/original/interesting investigations
__mathbooks_recomendedolympiad.txt TARIT GOSWAMI. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads., Geometry G1. Let ABC be an acute triangle with D,E,F the feet of the altitudes lying on BC,CA,AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that AP AQ. (United Kingdom) Solution 1. The line EF intersects the circumcircle at two points. Depending on the choice.
View Homework Help - (Maa Problem) Evan Chen-Euclidean Geometry in Mathematical Olympiads-Mathematical Association of Ame from HISTORY 101 at Princeton High. Euclidean Geometry in … 10/4/2017 · EGMO is quite useful, but has already been mentioned. I would recommend having a look at Yufei Zhao’s handouts on his website (just search up Yufei Zhao on google and you’ll find it). He has some handouts on configurations that come up in geometry...
9/19/2018 · There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of … If you're looking for a good Geometry book, you can check out Evan Chen's book, Euclidean Geometry In Mathematical Olympiads. There are no prerequisites to the book; all …
1/27/2019 · Sections 9.2–9.4 of Evan Chen’s recent book Euclidean Geometry in Mathematical Olympiads includes an ideally compact repository of useful statements. Problems, some of which veer into more challenging territory, are at the end of the section. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads.
The booklets in the series, A Taste of Mathematics, are published by the Canadian Mathematical Society (CMS).They are designed as enrichment materials for high school students with an interest in and aptitude for mathematics. Evan is also the author of a popular textbook Euclidean Geometry in Math Olympiads used by students preparing for mathematical olympiads. As a high school student, Evan was himself an IMO gold medalist and winner of the 2014 USAMO (which he took from 12:30am to 5am in Taiwan). Outside of math and teaching, Evan enjoys StarCraft and Korean pop
A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian Euclidean Geometry In Mathematical Olympiads Study group for Euclidean Geometry in Mathematical Olympiads by Evan Chen. UlTiMaTe Poll Forum! Collections of forums, tags, topics, and posts from the AoPS community. 3. V New Topic k Locked. M. Nothing matches your input.
Geometry G1. Let ABC be an acute triangle with D,E,F the feet of the altitudes lying on BC,CA,AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that AP AQ. (United Kingdom) Solution 1. The line EF intersects the circumcircle at two points. Depending on the choice Not a single complaint of any sort from anyone, anywhere in the world. I was just trying to give away - for free - PDFs of public domain texts on mathematics and mathematical astronomy - PDFs that I had either made myself, or found on other sites and "repaired" by replacing missing pages with scans from copies I had found at research libraries.
4/5/2013 · File Format: PDF/Adobe Acrobat - Quick View BMO Marking. Each question on the British Mathematical Olympiad is marked out of a total of. 10 marks. UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries. The Romanian Master of Mathematics; The European Girls' Mathematical Olympiad (EGMO). The Benelux Mathematical Olympiad. The Pan African Mathematical Olympiad. Here is specific advice about learning Euclidean Geometry. Contact Information:
While Euclidean geometry was codified almost 2,500 years ago, the field has been anything but stagnant since then. This book is an existence proof of how dynamic it is. The problems are original, challenging and have that achievable level of difficulty that makes them worthy Mathematical Olympiad UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries. The Romanian Master of Mathematics; The European Girls' Mathematical Olympiad (EGMO). The Benelux Mathematical Olympiad. The Pan African Mathematical Olympiad. Here is specific advice about learning Euclidean Geometry. Contact Information:
12/31/2016 · This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of 4/5/2013 · File Format: PDF/Adobe Acrobat - Quick View BMO Marking. Each question on the British Mathematical Olympiad is marked out of a total of. 10 marks.
If you're looking for a good Geometry book, you can check out Evan Chen's book, Euclidean Geometry In Mathematical Olympiads. There are no prerequisites to the book; all … problem collections with solutions from various International Mathematical Olympiads olympiad geometry notes, books, magazines, articles, links Here I am gonna add the full issues from free magazines of Euclidean Geometry as an online backup. geometry problems in pdf with aops links in Greek (GR) Balkan MO Geometry 1984 - 2017 GR
4/5/2013 · File Format: PDF/Adobe Acrobat - Quick View BMO Marking. Each question on the British Mathematical Olympiad is marked out of a total of. 10 marks. যখন তোমার Geometry Revisited, Plane Euclidean Geometry, Geometry Unbound, Elementary Number Theory-র মত বইগুলো শেষ হয়ে যাবে, তখন তুমি বুঝতে …
Evan Chen & Geo Book (EGMO) email: evan [at] evanchen•cc. evan chen (陳誼廷) euclidean geometry in mathematical olympiads (egmo) you can get a hard copy from amazon or the ams. you can also purchase a pdf. euclidean geometry in mathematical olympiads (often abbreviated egmo, despite an olympiad having the same name) is a comprehensive problem-solving book in we remind the reader that angle chasing is only a small part of olympiad geometry, and not to overuse it. Problem for this Section Problem 2.2. Find an example of two triangles ABC and XYZ such that AB: XY = BC: YZ, BCA= YZX,but ABC and XYZare not similar. 2.2 Power of a Point Cyclic quadrilaterals have many equal angles, so it should come as
Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads. Euclidean Geometry in Mathematical Olympiads With 248 Illustrations c 2016 by The Mathematical Association of America each with four problems to be solved in four hours EGMO The European Girls’ Mathematical Olympiad, "Analysis of Mixed Natural and Symbolic Language Input in …
2/2/2018 · See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. To compile these documents in LaTeX, you will need evan.sty. Because this style file evolves over time, your output might look a little different than the PDF's attached here. A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian
8/12/2017 · Amazon.in - Buy Euclidean Geometry in Mathematical Olympiads: 27 (MAA Problem Book Series) book online at best prices in India on Amazon.in. Read Euclidean Geometry in Mathematical Olympiads: 27 (MAA Problem Book Series) book reviews & author details and more at Amazon.in. Free delivery on qualified orders. Asia International Mathematical Olympiad Past Papers. Asia International Mathematical Olympiad Past Papers
This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of View Homework Help - (Maa Problem) Evan Chen-Euclidean Geometry in Mathematical Olympiads-Mathematical Association of Ame from HISTORY 101 at Princeton High. Euclidean Geometry in …
1. Online books by topic Number Theory by Justin Stevens Number Theory by David Santos Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer Geometry Unbound by Kiran Kedlaya Combinatorics by Pranav A. Sriram 2. Collections of notes on various topics. My Euclidean geometry class website. IMO math, from the authors of the IMO Compendium. You can also purchase a PDF. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or …
A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian 4/5/2013 · File Format: PDF/Adobe Acrobat - Quick View BMO Marking. Each question on the British Mathematical Olympiad is marked out of a total of. 10 marks.
Euclidean Geometry in Mathematical Olympiads By Evan Chen This is a challenging problem solving book in Euclidean Geometry, assuming nothing of the reader other than a good deal of courage. Topics covered include cyclic quadtrilaterals, power of a point, homothety, and triangle centers. The emphasis of the book is placed squarely on the Euclidean Geometry In Mathematical Olympiads Study group for Euclidean Geometry in Mathematical Olympiads by Evan Chen. UlTiMaTe Poll Forum! Collections of forums, tags, topics, and posts from the AoPS community. 3. V New Topic k Locked. M. Nothing matches your input.
You can also purchase a PDF. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or … Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads.
MAA Press Books American Mathematical Society Home. 9/19/2018 · There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of …, 152 8. Inversion let X be the point on closest to O (so OX⊥ ).Then X∗ is the point on γ farthest from O, so that OX∗ is a diameter of γ.Since O, X, X∗ are collinear by definition, this implies the result. In a completely analogous fashion one can derive the converse—the image of a circle passing through O is a line. Also, notice how the points on ω are fixed during the whole.
### Euclidean geometry Eventually Almost Everywhere
Evan Chen & Olympiad. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or international mathematical olympiads., 2/1/2018 · Euclidean Geometry in Mathematical Olympiads MAA Problem Book Series March 25, 2016 Textbook for students preparing for national/international olympiads such as USAMO and IMO..
### Egmo.org
Egmo.co.il site-stats.org. Geometry G1. Let ABC be an acute triangle with D,E,F the feet of the altitudes lying on BC,CA,AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that AP AQ. (United Kingdom) Solution 1. The line EF intersects the circumcircle at two points. Depending on the choice https://en.wikipedia.org/wiki/Euclid_axioms This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of.
• Egmo.org
• Is 'Euclidean Geometry in Mathematical Olympiads' (by Evan
• Geometry Problems from IMOs books (new)
• 9/19/2018 · There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of … Evan Chen & Geo Book (EGMO) Web.evanchen.cc Email: evan [at] evanchen•cc. Evan Chen (陳誼廷) Euclidean Geometry in Mathematical Olympiads (EGMO) You can get a hard copy from Amazon or the AMS. You can also purchase a PDF. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in
152 8. Inversion let X be the point on closest to O (so OX⊥ ).Then X∗ is the point on γ farthest from O, so that OX∗ is a diameter of γ.Since O, X, X∗ are collinear by definition, this implies the result. In a completely analogous fashion one can derive the converse—the image of a circle passing through O is a line. Also, notice how the points on ω are fixed during the whole we remind the reader that angle chasing is only a small part of olympiad geometry, and not to overuse it. Problem for this Section Problem 2.2. Find an example of two triangles ABC and XYZ such that AB: XY = BC: YZ, BCA= YZX,but ABC and XYZare not similar. 2.2 Power of a Point Cyclic quadrilaterals have many equal angles, so it should come as
1/27/2019 · Sections 9.2–9.4 of Evan Chen’s recent book Euclidean Geometry in Mathematical Olympiads includes an ideally compact repository of useful statements. Problems, some of which veer into more challenging territory, are at the end of the section. problem collections with solutions from various International Mathematical Olympiads olympiad geometry notes, books, magazines, articles, links Here I am gonna add the full issues from free magazines of Euclidean Geometry as an online backup. geometry problems in pdf with aops links in Greek (GR) Balkan MO Geometry 1984 - 2017 GR
we remind the reader that angle chasing is only a small part of olympiad geometry, and not to overuse it. Problem for this Section Problem 2.2. Find an example of two triangles ABC and XYZ such that AB: XY = BC: YZ, BCA= YZX,but ABC and XYZare not similar. 2.2 Power of a Point Cyclic quadrilaterals have many equal angles, so it should come as Euclidean Geometry in Mathematical Olympiads With 248 Illustrations c 2016 by The Mathematical Association of America each with four problems to be solved in four hours EGMO The European Girls’ Mathematical Olympiad, "Analysis of Mixed Natural and Symbolic Language Input in …
Euclidean Geometry in Mathematical Olympiads By Evan Chen This is a challenging problem solving book in Euclidean Geometry, assuming nothing of the reader other than a good deal of courage. Topics covered include cyclic quadtrilaterals, power of a point, homothety, and triangle centers. The emphasis of the book is placed squarely on the MAA Press books are now published and distributed by the AMS. The MAA Press has been dedicated to quality exposition since its founding in 1925. Innovative and imaginative MAA Press series encompass many areas and levels of collegiate mathematics, including biography, history, recreational mathematics, problems, textbooks, classroom resource
Euclidean Geometry In Mathematical Olympiads Study group for Euclidean Geometry in Mathematical Olympiads by Evan Chen. UlTiMaTe Poll Forum! Collections of forums, tags, topics, and posts from the AoPS community. 3. V New Topic k Locked. M. Nothing matches your input. UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries. The Romanian Master of Mathematics; The European Girls' Mathematical Olympiad (EGMO). The Benelux Mathematical Olympiad. The Pan African Mathematical Olympiad. Here is specific advice about learning Euclidean Geometry. Contact Information:
Asia International Mathematical Olympiad Past Papers. Asia International Mathematical Olympiad Past Papers 1/27/2019 · Sections 9.2–9.4 of Evan Chen’s recent book Euclidean Geometry in Mathematical Olympiads includes an ideally compact repository of useful statements. Problems, some of which veer into more challenging territory, are at the end of the section.
Asia International Mathematical Olympiad Past Papers. Asia International Mathematical Olympiad Past Papers A Mathematical Olympiad Primer, by Geoff Smith: If you have no past experience of mathematical olympiads, this is the place to begin. Plane Euclidean Geometry, by Bradley and Gardiner: This is a rigorous exploration of Euclidean geometry, including more basic techniques such as angle chasing and similar triangles in addition to vectors, Cartesian
10/4/2017 · EGMO is quite useful, but has already been mentioned. I would recommend having a look at Yufei Zhao’s handouts on his website (just search up Yufei Zhao on google and you’ll find it). He has some handouts on configurations that come up in geometry... Evan is also the author of a popular textbook Euclidean Geometry in Math Olympiads used by students preparing for mathematical olympiads. As a high school student, Evan was himself an IMO gold medalist and winner of the 2014 USAMO (which he took from 12:30am to 5am in Taiwan). Outside of math and teaching, Evan enjoys StarCraft and Korean pop
1. Online books by topic Number Theory by Justin Stevens Number Theory by David Santos Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer Geometry Unbound by Kiran Kedlaya Combinatorics by Pranav A. Sriram 2. Collections of notes on various topics. My Euclidean geometry class website. IMO math, from the authors of the IMO Compendium. You can also purchase a PDF. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or …
While Euclidean geometry was codified almost 2,500 years ago, the field has been anything but stagnant since then. This book is an existence proof of how dynamic it is. The problems are original, challenging and have that achievable level of difficulty that makes them worthy Mathematical Olympiad Geometry G1. Let ABC be an acute triangle with D,E,F the feet of the altitudes lying on BC,CA,AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that AP AQ. (United Kingdom) Solution 1. The line EF intersects the circumcircle at two points. Depending on the choice
we remind the reader that angle chasing is only a small part of olympiad geometry, and not to overuse it. Problem for this Section Problem 2.2. Find an example of two triangles ABC and XYZ such that AB: XY = BC: YZ, BCA= YZX,but ABC and XYZare not similar. 2.2 Power of a Point Cyclic quadrilaterals have many equal angles, so it should come as This challenging problem-solving book on Euclidean geometry is suitable for students preparing for national or international mathematical Olympiads. In addition to offering a guided tour through the classical results of Euler, Pascal, and others, it provides several carefully chosen worked examples and over 300 practice problems from contests
2/2/2018 · See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. To compile these documents in LaTeX, you will need evan.sty. Because this style file evolves over time, your output might look a little different than the PDF's attached here. Yuka has been added to the squad of eleven from which the team of six for the IMO in Bath will be selected at the Leeds camp. Our thanks to Man Group for their support of the EGMO team.. European Girls’ Mathematical Olympiad Day 2 (10 April 2019). The problems from the second paper at the eighth European Girls’ Mathematical Olympiad, sat today in Kyiv, are now available.
British Mathematical Olympiad, Round 1 (BMO 1) This is a 3½-hour paper with 6 problems (the first being intended to be more accessible than the rest), taken by students in their own schools. Selection is based on performance in the UK Senior Mathematical Challenge (UKSMC). Euclidean Geometry in Mathematical Olympiads With 248 Illustrations c 2016 by The Mathematical Association of America each with four problems to be solved in four hours EGMO The European Girls’ Mathematical Olympiad, "Analysis of Mixed Natural and Symbolic Language Input in …
You can also purchase a PDF. Euclidean Geometry in Mathematical Olympiads (often abbreviated EGMO, despite an olympiad having the same name) is a comprehensive problem-solving book in Euclidean geometry. It was written for competitive students training for national or … 12/31/2016 · This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of
Asia International Mathematical Olympiad Past Papers. Asia International Mathematical Olympiad Past Papers Euclidean Geometry in Mathematical Olympiads With 248 Illustrations c 2016 by The Mathematical Association of America each with four problems to be solved in four hours EGMO The European Girls’ Mathematical Olympiad, "Analysis of Mixed Natural and Symbolic Language Input in …
If you're looking for a good Geometry book, you can check out Evan Chen's book, Euclidean Geometry In Mathematical Olympiads. There are no prerequisites to the book; all … See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. If you notice any errors, please let me know! LaTeX notes: I provided the LaTeX source for most of these files as an example here.
152 8. Inversion let X be the point on closest to O (so OX⊥ ).Then X∗ is the point on γ farthest from O, so that OX∗ is a diameter of γ.Since O, X, X∗ are collinear by definition, this implies the result. In a completely analogous fashion one can derive the converse—the image of a circle passing through O is a line. Also, notice how the points on ω are fixed during the whole 2/2/2018 · See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. To compile these documents in LaTeX, you will need evan.sty. Because this style file evolves over time, your output might look a little different than the PDF's attached here.
If you're looking for a good Geometry book, you can check out Evan Chen's book, Euclidean Geometry In Mathematical Olympiads. There are no prerequisites to the book; all … Euclidean Geometry in Mathematical Olympiads By Evan Chen This is a challenging problem solving book in Euclidean Geometry, assuming nothing of the reader other than a good deal of courage. Topics covered include cyclic quadtrilaterals, power of a point, homothety, and triangle centers. The emphasis of the book is placed squarely on the
2/2/2018 · See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers. To compile these documents in LaTeX, you will need evan.sty. Because this style file evolves over time, your output might look a little different than the PDF's attached here. 8/12/2017 · Amazon.in - Buy Euclidean Geometry in Mathematical Olympiads: 27 (MAA Problem Book Series) book online at best prices in India on Amazon.in. Read Euclidean Geometry in Mathematical Olympiads: 27 (MAA Problem Book Series) book reviews & author details and more at Amazon.in. Free delivery on qualified orders.
All Issues of the Pentagon (Kappa Mu Epsilon) [1941-2015]-- (pentagon.kappamuepsilon.org)\ Annales de l'Olympiade Math matique Belge\ Base d'Exercices de Niveau Maths-Sup, Maths-Sp \ British Columbia Secondary Schools Math Contests [1999-2016]\ Cours de Math matiques (association-tremplin.org)\ Dr. Barbeau (math.toronto.edu)\ Euclidean Geometry in Mathematical Olympiads by Evan Chen (chapter 2 8/12/2017 · Amazon.in - Buy Euclidean Geometry in Mathematical Olympiads: 27 (MAA Problem Book Series) book online at best prices in India on Amazon.in. Read Euclidean Geometry in Mathematical Olympiads: 27 (MAA Problem Book Series) book reviews & author details and more at Amazon.in. Free delivery on qualified orders. | 2021-10-22 10:31:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3189185559749603, "perplexity": 1627.4216398071587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585504.90/warc/CC-MAIN-20211022084005-20211022114005-00606.warc.gz"} |
https://en.lexipedia.org/wiki/Lobachevsky_integral_formula | # Lobachevsky integral formula
In mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions. One of those is the improper integral of the sinc function over the positive real line, ∫ 0 ∞ sin x x d x = ∫ 0 ∞ sin 2 x x 2 d x = π 2 . {\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}\,dx=\int _{0}^{\infty }{\frac {\sin ^{2}x}{x^{2}}}\,dx={\frac {\pi }{2}}.}
## Words
This table shows the example usage of word lists for keywords extraction from the text above.
WordWord FrequencyNumber of ArticlesRelevance
x10298920.768
sin435650.432
frac327010.337
integral354170.306
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https://www.freemathhelp.com/forum/threads/fundamental-matrices.117412/ | # Fundamental Matrices
#### Metronome
##### New member
Here and here the Wronskian is said to be the determinant of the fundamental matrix. However, later in the same video, an example of a fundamental matrix is given where the lower elements aren't derivatives of the higher elements, which is the point of the Wronskian. Are these two different meanings of the fundamental matrix, or is there some other reason for this discrepancy? | 2019-08-18 23:32:42 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8415975570678711, "perplexity": 289.16726859622577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314353.10/warc/CC-MAIN-20190818231019-20190819013019-00169.warc.gz"} |
https://engineering.stackexchange.com/questions?tab=Votes | # All Questions
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### Why do glass windows still exist? (Why haven't they been replaced by plastics?)
Glass is fragile and impractical to transport, install and repair. Even worse, glass kills and hurts people when it breaks. Falling to the streets like guillotines during earthquakes and bomb raids. ...
35k views
9k views
### How to make smoke for a small wind tunnel?
I am making a small (desktop) wind tunnel for educational purposes, I want to have 10 fairly thick smoke-streams about 3cm apart. I have experimented with incense but the stream is not thick enough ...
4k views
### Does a roadway bridge experience more load when vehicles are parked or when they are moving?
Bridges are designed for the loads that come from the vehicles that are expected to cross them. This includes the weight the vehicle and any dynamic loads that may be introduced from movement of the ...
8k views
### Why does it take so long to restart a nuclear power plant?
I have heard a couple of times that an operating nuclear power plant which was shut down (non-emergency; e.g. for a regular check) needs over 24 hours (up to 72 hours?) to get up running again. Why ...
13k views
### How does a traffic light sense the proximity of vehicles?
Some traffic lights don't operate periodically but instead detect when a car is close by and then turns green. I have heard that they use a magnetic sensor embedded in the road to sense cars as they ...
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### How does width and thickness affect the stiffness of steel plate?
I have a 2 mm thick steel plate which is 300 mm long and 30 mm wide, supported at either end. It supports a weight-bearing wheel that can roll along the plate. It currently supports the maximum weight ...
117k views
### What's wrong with transporting a refrigerator on its side?
I tried to deliver a refrigerator and the customer saw that the refrigerator was lying down in the truck instead of standing up. He refused to accept it, claiming that the compressor would be damaged ... | 2020-02-26 16:03:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4535016417503357, "perplexity": 2122.846412099684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146414.42/warc/CC-MAIN-20200226150200-20200226180200-00483.warc.gz"} |
https://christoph-jahn.com/?tag=latex | # LaTeX: Hyphenation of \texttt
I have been using since I was 17 and still love it. But one thing has been bugging me for a long time. For a while I had been manually tweaking documents to get proper linebreaks when using the \texttt{} macro. Now I had finally come to the point where this did not work any longer. A search gave me this snippet:
\renewcommand{\texttt}[1]{%
\begingroup
\ttfamily
\begingroup\lccode~=/\lowercase{\endgroup\def~}{/\discretionary{}{}{}}%
\begingroup\lccode~=[\lowercase{\endgroup\def~}{[\discretionary{}{}{}}%
\begingroup\lccode~=.\lowercase{\endgroup\def~}{.\discretionary{}{}{}}%
\catcode/=\active\catcode[=\active\catcode.=\active
\scantokens{#1\noexpand}%
\endgroup
}`
It works great!
# Using LaTeX with Windows 7
For those who are lazy and look for a relatively quick and easy setup guide for LaTeX with Windows 7, you should have a look at http://schlosser.info/latexsystem-en . Another interesting PDF document is here (German only).
Where I have deviated is the installation of Emacs. For several years now I have been using the distribution from http://ourcomments.org/Emacs/EmacsW32.html (patched version) which auto-installs all you need, incl. gnuserv and gnuclientw. There is one caveat, however. At the end of the installation you can have Emacs started automatically (on by default). This will most likely cause problems on Windows 7 (and probably also on Vista), because the installer is usually running with admin rights. In such a case the directory ~/.emacs.d and its subdirectories will not be created with the proper account as owner. Instead the admin user or group will own things. This will show itself in two problems:
• The window “Emacs Client” will display the message “Waiting for Emacs server to start” until a timeout occurs
• You will see the message “The directory ~/.emacs.d/server is unsafe” in the status line of your main Emacs window
To correct things, go to %USERPROFILE%\AppData\Roaming and change the owner of .emacs.d/ to your own user (make sure to apply this change to all sub-directories!).
Update October 2015 | 2019-08-18 22:24:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7304236888885498, "perplexity": 2278.0182284696875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314130.7/warc/CC-MAIN-20190818205919-20190818231919-00348.warc.gz"} |
http://tex.stackexchange.com/questions/39435/how-can-i-center-a-too-wide-table/39436 | How can I center a too wide table?
I have a document containing a table which is slightly too wide for the page. But instead of growing to the right side only, I would like to have it centered on the page.
I have tried to use the center environment but this doesn't seem to help.
-
Have a look at Center flow chart horizontally, the answer works also for too wide tables, perhaps also have a look at Centering wide tables or figures. Similar solution here: How to center the minipage. – Stefan Kottwitz Dec 27 '11 at 16:30
Please compose a compilable MWE that illustrates the problem including the \documentclass and the appropriate packages so that those trying to help don't have to recreate it. – Peter Grill Dec 27 '11 at 16:30
@PeterGrill: I don't think a MWE is really necessary here. It's kind of a common, easily understandable issue. – Martin Scharrer Dec 27 '11 at 16:45
@martin Sure, but knowing, say, the class, help know which tools are readily available. – daleif Dec 27 '11 at 17:34
– Dejan Dec 29 '11 at 14:26
If a table (or any other horizontal box) is wider than the text (\textwidth) an overfull hbox warning is given and the content is placed anyway, which makes it run into the right margin. To avoid this and to suppress the error the content must be placed in a box with is equal or smaller than \textwidth. The \makebox macro with its two optional argument for the width and horizontal alignment can be used for this: \makebox[\textwidth][c]{<table>} will center the content. See Center figure that is wider than \textwidth and Place figures side by side, spill into outer margin were this is used for figures and further explained.
For more complicated tables, especially if they should contain verbatim material, you should use a different approach. \makebox reads the whole content as macro argument which does not allow verbatim content and is not very efficient (ok, nowadays the latter isn't really important any longer). The \Makebox macro or the Makebox environment from the realboxes can be used as an replacement. It reads the content as a box. Better would be the adjustbox macro or environment from the adjustbox package together with the center key.
\begin{adjustbox}{center}
Which centers the content to \linewidth (mostly identical to \textwidth) by default but also takes any other length as an optional value, e.g. center=10cm.
@Dejan: \centerline reads the content as a macro argument as well and is basically the same as \makebox[\hsize][c], using more lower-level TeX commands than LaTeX ones. It will work, but \makebox or \adjustbox is more flexible. – Martin Scharrer Dec 28 '11 at 8:52
Put your table into \centerline{}. The table will extend evenly into both margins if it's wider than \textwidth. | 2014-11-26 16:46:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9124687314033508, "perplexity": 1809.1975687191323}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007150.95/warc/CC-MAIN-20141125155647-00179-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://laustep.github.io/stlahblog/posts/rstudioAddins.html | Posted on February 3, 2022 by Stéphane Laurent
Tags: R, misc
In this blog post I introduce three small RStudio addins I did.
## ‘bracketify’
I prefer subsetting with the double brackets than with the dollar in R, because this is more readable in RStudio thanks to the syntax highlighting. That’s why I did bracketify. This addin replaces all occurrences of foo\$bar with foo[["bar"]], either in a whole file or only in the current selection.
To use carefully: if you have some dollar symbols in your code which are not used for subsetting (e.g. in a regular expression), they can be transformed by bracketify.
## ‘pasteAsComment’
Originally, I made pasteAsComment to paste the content of the clipboard as a comment:
I updated this package today. Now it also allows to paste the content of the clipboard as roxygen lines. This is particularly useful to write some code in the @examples field:
## ‘JSconsole’
JSconsole is available on CRAN. This addin allows to send some selected JavaScript code to the V8 console. This is useful when you want to test a function. | 2022-07-05 19:41:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23933292925357819, "perplexity": 2606.144502619662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104597905.85/warc/CC-MAIN-20220705174927-20220705204927-00235.warc.gz"} |
https://questions.examside.com/past-years/year-wise/gate/gate-cse/gate-cse-2002/ | ## GATE CSE 2002
Exam Held on Thu Jan 01 1970 00:00:00 GMT+0000 (Coordinated Universal Time)
Click View All Questions to see questions one by one or you can choose a single question from below.
## Algorithms
In the worst case, the number of comparisons needed to search a singly linked li...
Consider the following algorithm for searching for a given number x in an unsort...
The running time of the following algorithm Procedure A(n)<br/> If n<=2 return (...
The solution to the recurrence equation <br/> T(2<sup>k</sup>) = 3 T(2<sup>k-1</...
## Computer Organization
In $$2’s$$ complement addition, the overflow
Sign extension is the step in
The performance of a pipelined processor suffers if
Which of the following is not a form of memory?
In absolute addressing mode
Horizontal micro programming
In serial data transmission, every byte of data is padded with a $$‘0’$$ in the ...
## Data Structures
In the worst case, the number of comparisons needed to search a singly linked li...
The number of leaf nodes in a rooted tree of n nodes, with each node having 0 or...
## Database Management System
Relation $$R$$ with an associated set of functional dependencies, $$F,$$ is deco...
From the following instance of a relation schema $$R(A, B, C),$$ we can conclude...
Relation $$R$$ is decomposed using a set of functional dependencies, $$F,$$ and ...
With regard to the expressive power of the formal relational query languages, wh...
A B<sup>+</sup> - tree index is to be built on the Name attribute of the relatio...
## Digital Logic
The decimal value $$0.25$$
Sign extension is the step in
The $$2's$$ compliment representation of the decimal value $$-15$$ is
Consider the following logic circuit whose inputs are functions $${f_1},$$ $${f_... Transform the following logic circuit (without expressing its switching function...$$f\left( {A,B} \right) = A' + B$$Simplified expression for function$$f((x+y,y...
Express the function $$f( x, y, z)= xy'+ yz'$$ with only one complement operatio...
Minimum $$SOP$$ for $$f(w, x, y, z)$$ shown in karnaugh $$-$$ map below is <img ...
Consider the following multiplexer where $$10, 11, 12, 13$$ are four data input ...
## Discrete Mathematics
"If X then Y unless Z" is represented by which of the following formulas in prop...
Four fair coins are tossed simultaneously. The probability that at least one hea...
The binary relation $$S = \phi$$ (emply set) on set A = {1, 2, 3} is
Determine whether each of the following is a tautology, a contradiction, or neit...
The minimum number of colors required to color the vertices of a cycle with $$n... The rank of the matrix$$\left[ {\matrix{ 1 & 1 \cr 0 & 0 \cr } } \ri...
Maximum number of edges in a n - node undirected graph without self loops is
(a) $$S = \left\{ { < 1,2 > ,\, < 2,1 > } \right\}$$ is binary relation on set $... Let $$A$$ be a set of $$n\left( { > 0} \right)$$ elements. Let $${N_r}$$ be the ... Obtain the eigen values of the matrix $$A = \left[ {\matrix{ 1 & 2 & {34} & ... The function$$f\left( {x,y} \right) = 2{x^2} + 2xy - {y^3}$$has ## Operating Systems Which of the following scheduling algorithms is non-preemptive? Which combination of the following features will suffice to characterize an$$OS... Draw the process state transition diagram of an $$OS$$ in which <br>(i) each pr... Which of the following scheduling algorithms is non-preemptive? Which of the following is not a form of memory? The optimal page replacement algorithm will select the page that. Dynamic linking can cause security concerns because A computer system uses $$32$$-bit virtual address, and $$32$$-bit physical addre... In the index allocation scheme of blocks to a file, the maximum possible size of... ## Programming Languages The results returned by function under value-result and reference parameter pass... In the C language Consider the following declaration of a two-dimensional array in C: <br/><br/>ch... ## Theory of Computation The smallest finite automaton which accepts the language <br>$$L = \left. {\le... The Finite state machine described by the following state diagram with$$A$$as ... The language accepted by a pushdown Automation in which the stack is limited to ... The$$C$\$ language is:
Which of the following is true?
### EXAM MAP
#### Joint Entrance Examination
JEE Advanced JEE Main
#### Graduate Aptitude Test in Engineering
GATE CSE GATE EE GATE ECE GATE ME GATE CE GATE PI GATE IN | 2021-09-21 02:26:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7047455310821533, "perplexity": 1653.0044557163271}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057131.88/warc/CC-MAIN-20210921011047-20210921041047-00167.warc.gz"} |
http://mathcountsnotes.blogspot.com/2008/11/triangle-numbers-word-problems-chapter.html | ## Monday, October 26, 2015
### Triangular Numbers & Word Problems: Chapter Level
Triangular Numbers : From Math is Fun. 1, 3, 6, 10, 15, 21, 28, 36, 45...
Interesting Triangular Number Patterns: From Nrich
Another pattern: The sum of two consecutive triangular numbers is a square number.
What are triangular numbers? Let's exam the first 4 triangular numbers:
The 1st number is "1".
The 2nd number is "3" (1 + 2)
The 3rd number is "6" (1 + 2 + 3)
The 4th number is "10" (1 + 2 + 3 + 4)
.
.
The nth number is $$\frac{n(n+1)}{2}$$
It's the same as finding out the sum of the first "n" natural numbers.
Let's look at this question based on the song "On the Twelve Day of Christmas"
(You can listen to this on Youtube,)
On the Twelve Day of Christmas
On the first day of Christmas
my true love gave to me
a Partridge in a Pear Tree
On the second day of Christmas,
My true love gave to me,
Two Turtle Doves,
And a Partridge in a Pear Tree.
On the third day of Christmas,
My true love gave to me,
Three French Hens,
Two Turtle Doves,
And a Partridge in a Pear Tree
On the fourth day of Christmas,
My true love gave to me,
Four Calling Birds,
Three French Hens,
Two Turtle Doves,
And a Partridge in a Pear Tree.
The question is "How many gifts were given out on the Day of Christmas?"
Solution I"
1st Day: 1
2nd Day: 1 + 2
3rd Day 1 + 2 + 3
4th Day 1 + 2 + 3 + 4
.
.
12th Day 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
Altogether, you'll have 12 * 1 + 11 * 2 + 10 * 3 + 9 * 4 + 8 * 5 + 7 * 6 + 6 * 7 + 5 * 8 + 4 * 9 + 3 * 10 + 2 * 11 + 1 * 12 = 12 + 22 + 30 + 36 + 40 + 42 + 42 + 40 + 36 + 30 + 22 + 12 = 364
Solution II: The sum of the "n" triangular number is a tetrahedral number.
To get the sum, you use $$\frac{n(n+1)(n+2)}{6}$$
n = 12 and $$\frac{12 (13)(14)}{6}= 364$$
Here is a proof without words.
Applicable questions: (Answers and solutions below)
#1 Some numbers are both triangular as well as square numbers. What is the sum of the first three positive numbers that are both triangular numbers and square numbers?
#2 What is the 10th triangular number? What is the sum of the first 10 triangular numbers?
#3 What is the 20th triangular number?
#4 One chord can divide a circle into at most 2 regions, Two chords can divide a circle at most into 4 regions. Three chords can divide a circle into at most seven regions. What is the maximum number of regions that a circle can be divided into by 50 chords?
#5: Following the pattern, how many triangles are there in the 15th image?
#1 1262 The first 3 positive square triangular numbers are: 1, 36 (n = 8) and 1225 (n = 49).
#2 55; 220 a. $$\frac{10*11}{2}=55$$ b. $$\frac{10*11*12}{6}=220$$
#3 210 $$\frac{20*21}{2}=210$$
#4 1276
1 chord : 2 regions or $$\boxed{1}$$ + 1
2 chords: 4 regions or $$\boxed{1}$$ + 1 + 2
3 chords: 7 regions or $$\boxed{1}$$ + 1 + 2 + 3
.
.
50 chords:$$\boxed{1}$$ + 1 + 2 + 3 + ...+ 50 = 1 + $$\frac{50*51}{2}$$ =1276
#5: 120
The first image has just one triangle, The second three triangles. The third 6 triangles total.
It follows the triangular number pattern. The 15th triangular is $$\frac{15*16}{2}$$ =120 | 2018-08-17 03:02:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5104841589927673, "perplexity": 829.9924708219645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221211664.49/warc/CC-MAIN-20180817025907-20180817045907-00198.warc.gz"} |
http://nrich.maths.org/public/leg.php?code=-68&cl=3&cldcmpid=4927 | Search by Topic
Resources tagged with Visualising similar to More Magic Potting Sheds:
Filter by: Content type:
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There are 187 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Visualising
Picturing Triangle Numbers
Stage: 3 Challenge Level:
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Christmas Chocolates
Stage: 3 Challenge Level:
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Eight Hidden Squares
Stage: 2 and 3 Challenge Level:
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
Squares in Rectangles
Stage: 3 Challenge Level:
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
Convex Polygons
Stage: 3 Challenge Level:
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
Triangle Inequality
Stage: 3 Challenge Level:
ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.
Cubes Within Cubes Revisited
Stage: 3 Challenge Level:
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
Chess
Stage: 3 Challenge Level:
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
On the Edge
Stage: 3 Challenge Level:
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
Mystic Rose
Stage: 3 Challenge Level:
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Picturing Square Numbers
Stage: 3 Challenge Level:
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Painted Cube
Stage: 3 Challenge Level:
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Concrete Wheel
Stage: 3 Challenge Level:
A huge wheel is rolling past your window. What do you see?
Konigsberg Plus
Stage: 3 Challenge Level:
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Tourism
Stage: 3 Challenge Level:
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Framed
Stage: 3 Challenge Level:
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .
Tetra Square
Stage: 3 Challenge Level:
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Trice
Stage: 3 Challenge Level:
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
Stage: 3 Challenge Level:
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
One and Three
Stage: 4 Challenge Level:
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
An Unusual Shape
Stage: 3 Challenge Level:
Can you maximise the area available to a grazing goat?
Pattern Power
Stage: 1, 2 and 3
Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.
Sea Defences
Stage: 2 and 3 Challenge Level:
These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?
Zooming in on the Squares
Stage: 2 and 3
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
Isosceles Triangles
Stage: 3 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
All in the Mind
Stage: 3 Challenge Level:
Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .
A Tilted Square
Stage: 4 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
You Owe Me Five Farthings, Say the Bells of St Martin's
Stage: 3 Challenge Level:
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
Coloured Edges
Stage: 3 Challenge Level:
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
Stage: 3 Challenge Level:
How many different symmetrical shapes can you make by shading triangles or squares?
Threesomes
Stage: 3 Challenge Level:
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Reflecting Squarely
Stage: 3 Challenge Level:
In how many ways can you fit all three pieces together to make shapes with line symmetry?
Tetrahedra Tester
Stage: 3 Challenge Level:
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
Königsberg
Stage: 3 Challenge Level:
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Jam
Stage: 4 Challenge Level:
A game for 2 players
There and Back Again
Stage: 3 Challenge Level:
Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?
Masterclass Ideas: Visualising
Stage: 2 and 3 Challenge Level:
A package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on “visualising” and is designed to meet the needs. . . .
Stage: 3 Challenge Level:
Can you mark 4 points on a flat surface so that there are only two different distances between them?
Square Coordinates
Stage: 3 Challenge Level:
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
Triangles Within Pentagons
Stage: 4 Challenge Level:
Show that all pentagonal numbers are one third of a triangular number.
Cuboid Challenge
Stage: 3 Challenge Level:
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
More Pebbles
Stage: 2 and 3 Challenge Level:
Have a go at this 3D extension to the Pebbles problem.
Steel Cables
Stage: 4 Challenge Level:
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
3D Stacks
Stage: 2 and 3 Challenge Level:
Can you find a way of representing these arrangements of balls?
Auditorium Steps
Stage: 2 and 3 Challenge Level:
What is the shape of wrapping paper that you would need to completely wrap this model?
Troublesome Dice
Stage: 3 Challenge Level:
When dice land edge-up, we usually roll again. But what if we didn't...?
Triangles Within Triangles
Stage: 4 Challenge Level:
Can you find a rule which connects consecutive triangular numbers?
Conway's Chequerboard Army
Stage: 3 Challenge Level:
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
Khun Phaen Escapes to Freedom
Stage: 3 Challenge Level:
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Dice, Routes and Pathways
Stage: 1, 2 and 3
This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . . | 2014-12-22 05:37:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3065565526485443, "perplexity": 1580.7572351897973}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802774464.128/warc/CC-MAIN-20141217075254-00103-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://codereview.stackexchange.com/questions/192783/picking-the-desired-one-between-two-links | Picking the desired one between two links
I've written a script in python to scrape the link within contact us or about us from few webpages. The challenge here is to prioritize contact us over about us. For example, if any site contains both of them then my scraper should pick the link within contact us. However, if contact us is not present then only the scraper go for parsing the link within about us. My first attempt used the logic if "contact" in item.text.lower() or "about" in item.text.lower() but I could notice that in every cases while dealing with the below links the scraper picks the link within about us whereas my first priority is to get the link within contact us. I next rewrote it with the following approach (using two for loops to get the job done) and found it working.
This is what I've tried to get the links complying with the above criteria:
import requests
from urllib.parse import urljoin
from bs4 import BeautifulSoup
"http://www.innovaprint.com.sg/",
"https://www.richardsonproperties.com/",
"http://www.innovaprint.com.sg/",
"http://www.cityscape.com.sg/"
)
res = requests.get(site)
soup = BeautifulSoup(res.text,"lxml")
for item in soup.select("a[href]"):
if "contact" in item.text.lower():
return
for item in soup.select("a[href]"):
return
if __name__ == '__main__':
The two for loops defined within the above function look awkward so I suppose there is any better idea to do the same. Thanks in advance for any betterment of this existing code.
You're right that the two for-loops are possibly overkill... though, it's not all that bad.. how big are the documents really?. The alternative is to track a default about link, and if a contact link appears, to override it. I'll explain that later, but first I should mention that the function should really return a value, not just print it inside the loops.
Having said that, consider the loop:
link = None
for item in soup.select("a[href]"):
if "contact" in item.text.lower(): | 2020-02-29 10:32:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3351765275001526, "perplexity": 1930.0215362352517}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875148850.96/warc/CC-MAIN-20200229083813-20200229113813-00006.warc.gz"} |
https://www.vedantu.com/question-answer/find-the-value-of-x-that-satisfying-150x-equiv-class-10-maths-cbse-5ee0ba9fc9e6ad0795eff084 | Question
# Find the value of $x$ that satisfying $150x \equiv 35(Mod31)$ is$\ A)14 \\ B)22 \\ C)24 \\ D)12 \\ \$
Hint: To proceed with this solution we have to compare the given expression with the standard form.
Given expression is $150x \equiv 35(Mod31)$
Let us compare the expression with standard form i.e. $a \equiv b(\bmod n)$
Before comparing it lets us know what does the standard form means
Here,
$a \equiv b(\bmod n)$ Means $a - b$ is divisible by $n$
So, here if we compare we can say $a = 150x,b = 35$ and $n = 31$
So,
$150x \equiv 35(Mod31)$
$\Rightarrow 150x - 35$ is divisible by $31$
$\Rightarrow 5(30x - 7)$ is divisible by $31$
Now here let us say that $30x - 7 = 31k$ where $k$is any natural number.
From this we can equate x value as
$\Rightarrow$ $x = \dfrac{{31k + 7}}{{30}}$
$\Rightarrow \dfrac{{30k + (k + 7)}}{{30}}$
$\Rightarrow \dfrac{{30k}}{{30}} + \dfrac{{k + 7}}{{30}}$
$\Rightarrow k + \dfrac{{k + 7}}{{30}}$
Here $k + 7$ is divisible by $30$ where $k = 23$ which is smallest possible value
So, if $k = 23$ then
$\Rightarrow k + \dfrac{{k + 7}}{{30}}$=$k + \dfrac{{23 + 7}}{{30}} = k + \dfrac{{30}}{{30}} = k + 1$
And we know that $k = 23$
So, here
$\ \Rightarrow x = K + 1 \\ \Rightarrow x = 23 + 1 \\ \Rightarrow x = 24 \\ \$
From this we can say that for $x = 24$ is the value that satisfying the expression
Option C is the correct
NOTE: Problems like the above model have to be compared with standard forms and in the above problem we have to substitute k value in two places in the same equation. Instead of substituting K value in the two places at a time it is better to substitute in one place then make the equation simple so get the answer directly. | 2021-04-17 03:26:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7808569073677063, "perplexity": 281.33799439440713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038098638.52/warc/CC-MAIN-20210417011815-20210417041815-00355.warc.gz"} |
https://www.gamedev.net/forums/topic/333691-working-with-text/ | # Working with text.
This topic is 4867 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
lol, I'm really kicking myself over this one. I'm writing a channel display and I cannot get my text to align properly. Here is what it looks like... When the server speaks to the client, it sends message (Max 200 char) to the client, which the client has to break into smaller messages that fit in each display row with set amount of characters. eg... A channel is composed of about 100 rows, where each row is drawn individually from all the others. Each row holds flag_FCharacters number of characters.
/* This procedure appends the message on the channel. */
void sDealWithMessage (char lcChannel, struct gtRGB loColor, char *lacMessage)
{
struct gtChannelRow *laoChannel;
int liX, liLength, liOffset, liShifter;
/* Get which channel we are appending to. */
if(lcChannel == flag_ChnDefault)
laoChannel = gaoChannelDefault;
else if(lcChannel == flag_ChnGamechat)
laoChannel = gaoChannelGamechat;
/* Append the message onto the given channel. */
for(liX=0;liX<100;liX++)
{
if(laoChannel[liX].lucUsed == 0)
{
lblFill:
/* We need to continously post until all of the message is gone. */
liLength = strlen(lacMessage);
liOffset = 0;
while(liLength > 0)
{
/* If its the first line, no padding. */
if(liOffset == 0)
{
laoChannel[liX].lucUsed = 1;
laoChannel[liX].loColor = loColor;
for(liShifter = 0; liShifter < 10; liShifter++)
if(lacMessage[flag_FCharacters - liShifter-1] == ' ' ||
lacMessage[flag_FCharacters - liShifter-1] == 0)
break;
strncpy(laoChannel[liX].lacMessage, &lacMessage[0], flag_FCharacters - liShifter);
liLength -= (flag_FCharacters - liShifter);
liOffset += (flag_FCharacters - liShifter);
liX++;
}
/* If this text took up multiple lines, than pad! */
else
{
strcpy(laoChannel[liX].lacMessage, " ");
laoChannel[liX].lucUsed = 1;
laoChannel[liX].loColor = loColor;
for(liShifter = 0; liShifter < 10; liShifter++)
if(lacMessage[liOffset + flag_FCharacters - liShifter-1] == ' ' ||
lacMessage[liOffset + flag_FCharacters - liShifter-1] == 0)
break;
strncpy(&laoChannel[liX].lacMessage[5], &lacMessage[liOffset], flag_FCharacters - 5 - liShifter);
liLength -= flag_FCharacters - 5 - liShifter;
liOffset += flag_FCharacters - 5 - liShifter;
liX++;
}
}
break;
}
if(liX >= 89)
{
/* If we are approaching the end, memmove the bottom up. */
memset(&laoChannel[0], 0, sizeof(struct gtChannelRow) * 10);
memmove(&laoChannel[0],&laoChannel[10], 89*sizeof(struct gtChannelRow));
goto lblFill;
}
}
}
The problem is that, every line after the second isn't separating the edges properly, as shown in the screenshot. What am I doing wrong? [bawling]
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Thevein,
After a quick look at your code I suspect the problem is when you're trying to find the 'shifter' for your second and subsequent lines. I believe you scan backwards from (index) flag_FCharacters-1 to flag_FCharacters-10 looking to break the line, however in this case you've already used the first 5 characters (as your padding). This means you only have flag_FCharacters-5 characters for use in the line, therefore start and end your scan in the wrong place.
Cheers,
Tom
// Originalif(lacMessage[liOffset + flag_FCharacters - liShifter-1] == ' ' || lacMessage[liOffset + flag_FCharacters - liShifter-1] == 0)// Adjusted for paddingif(lacMessage[liOffset + (flag_FCharacters-5) - liShifter-1] == ' ' || lacMessage[liOffset + (flag_FCharacters-5) - liShifter-1] == 0)
##### Share on other sites
I cannot give any advice, but I like the layout of your game! Nice'n'tidy...
##### Share on other sites
Quote:
Original post by TomHAfter a quick look at your code I suspect the problem is when you're trying to find the 'shifter' for your second and subsequent lines. I believe you scan backwards from (index) flag_FCharacters-1 to flag_FCharacters-10 looking to break the line, however in this case you've already used the first 5 characters (as your padding). This means you only have flag_FCharacters-5 characters for use in the line, therefore start and end your scan in the wrong place.
You solved it! [wow]
Many many thanks: I wish I could give you more ratings than just a mere 8, since you clearly deserve more. [smile]
See you all in a few weeks when I make post asking why my Firefields arn't burning the objects as they are suppose to.. (Just kidding of course)[rolleyes]
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× | 2018-11-18 00:34:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32685378193855286, "perplexity": 7016.895609999005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743913.6/warc/CC-MAIN-20181117230600-20181118012014-00048.warc.gz"} |
https://control.com/textbook/control-valves/sliding-stem-valves/ | # Sliding-stem Valves
## Chapter 30 - Basic Principles of Control Valves and Actuators
A sliding-stem valve body is one where the moving parts slide with a linear motion. Some examples of sliding-stem valve body designs are shown here:
Most sliding-stem control valves are direct acting, which means the valve opens up wider as the stem is drawn out of the body. Conversely, a direct-acting valve shuts off (closes) when the stem is pushed into the body. Of course, a reverse-acting valve body would behave just the opposite: opening up as the stem is pushed in and closing off as the stem is drawn out.
### Globe valves
Globe valves restrict the flow of fluid by altering the distance between a movable plug and a stationary seat (in some cases, a pair of plugs and matching seats). Fluid flows through a hole in the center of the seat, and is more or less restricted by the plug’s proximity to that hole. The globe valve design is one of the most popular sliding-stem valve designs used in throttling service. A photograph of a small (2 inch) globe valve body appears here:
A set of three photographs showing a cut-away Masoneilan model 21000 globe valve body illustrates just how the moving plug and stationary seat work together to throttle flow in a direct-acting globe valve. The left-hand photo shows the valve body in the fully closed position, while the middle photo shows the valve half-open, and the right-hand photo shows the valve fully open:
As you can see from these photographs, the valve plug is guided by the stem to maintain alignment with the centerline of the seat. For this reason, this particular style of globe valve is called a stem-guided globe valve.
A variation on the stem-guided globe valve design is the needle valve, where the plug is extremely small in diameter and usually fits well into the seat hole rather than merely sitting on top of it. Needle valves are very common as manually-actuated valves used to control low flow rates of air or oil. A set of three photographs shows a needle valve in the fully-closed, mid-open, and fully-open positions (left-to-right):
Yet another variation on the globe valve design is the port-guided valve, where the plug has an unusual shape, projecting into the seat. Thus, the seat ring acts as a guide for the plug to keep the centerlines of the plug and seat always aligned, minimizing guiding stresses that would otherwise be placed on the stem. This means that the stem may be made smaller in diameter than if the valve trim were stem-guided, minimizing sliding friction and improving control behavior.
A photograph showing a small port-guided globe valve plug appears in the following photograph:
Some globe valves use a pair of plugs (on the same stem) and a matching pair of seats to throttle fluid flow. These are called double-ported globe valves. The purpose of a double-ported globe valve is to minimize the force applied to the stem by process fluid pressure across the plugs:
Differential pressure of the process fluid ($$P_1 - P_2$$) across a valve plug will generate a force parallel to the stem as described by the formula $$F = PA$$, with $$A$$ being the plug’s effective area presented for the pressure to act upon. In a single-ported globe valve, there will only be one force generated by the process pressure. In a double-ported globe valve, there will be two opposed force vectors, one generated at the upper plug and another generated at the lower plug. If the plug areas are approximately equal, then the forces will likewise be approximately equal and therefore nearly cancel. This makes for a control valve that is easier to actuate (i.e. the stem position is less affected by pressure drop across the valve).
The following photograph shows a disassembled Fisher “A-body” double-ported globe valve, with the double plug plainly visible on the right:
This particular double-ported globe valve happens to be stem-guided, with bushings guiding the upper stem and also a lower stem (on the bottom side of the valve body). Double-ported, port-guided control valves also exist, with two sets of port-guided plugs and seats throttling fluid flow.
While double-ported globe valves certainly enjoy the advantage of easier actuation compared to their single-ported cousins, they also suffer from a distinct disadvantage: the near impossibility of tight shut-off. With two plugs needing to come to simultaneous rest on two seats to achieve a fluid-tight seal, there is precious little room for error or dimensional instability. Even if a double-ported valve is prepared in a shop for the best shut-off possible835, it may not completely shut off when installed due to dimensional changes caused by process fluid heating or cooling the valve stem and body. This is especially problematic when the stem is made of a different material than the body. Globe valve stems are commonly manufactured from stainless steel bar stock, while globe valve bodies are commonly made of cast steel. Cold-formed stainless steel has a different coefficient of thermal expansion than hot-cast steel, which means the plugs will no longer simultaneously seat once the valve warms or cools much from the temperature it was at when it seated tightly.
A more modern version of the globe valve design uses a piston-shaped plug inside a surrounding cage with ports cast or machined into it. These cage-guided globe valves throttle flow by uncovering more or less of the port area in the surrounding cage as the plug moves up and down. The cage also serves to guide the plug so the stem need not be subjected to lateral forces as in a stem-guided valve design. A photograph of a cut-away control valve shows the appearance of the cage (in this case, with the plug in the fully closed position). Note the “T”-shaped ports in the cage, through which fluid flows as the plug moves up and out of the way:
An advantage of the cage-guided design is that the valve’s flowing characteristics may be easily altered just by replacing the cage with another having different size or shape of holes. By contrast, stem-guided and port-guided globe valves are characterized by the shape of the plug, which requires further disassembly to replace than the cage in a cage-guided globe valve. With most cage-guided valves all that is needed to replace the cage is to separate the bonnet from the rest of the valve body, at which point the cage may be lifted out of the body and swapped with another cage. In order to change a globe valve’s plug, you must first separate the bonnet from the rest of the body and then de-couple the plug and plug stem from the actuator stem, being careful not to disturb the packing inside of the bonnet as you do so. After replacing a plug, the “bench-set” of the valve must be re-adjusted to ensure proper seating pressure and stroke calibration.
Cage-guided globe valves are available with both balanced and unbalanced plugs. A balanced plug has one or more ports drilled from top to bottom, allowing fluid pressure to equalize on both sides of the plug. This helps minimize the forces acting on the plug which must be overcome by the actuator:
Unbalanced plugs generate a force equal to the product of the differential pressure across the plug and the plug’s area ($$F = PA$$), which may be quite substantial in some applications. Balanced plugs do not generate this same force because they equalize the pressure on both sides of the plug, however, they exhibit the disadvantage of one more leak path when the valve is in the fully closed position (through the balancing ports, past the piston ring, and out the cage ports):
Thus, balanced and unbalanced cage-guided globe valves exhibit similar characteristics to double-ported and single-ported stem- or port-guided globe valves, and for similar reasons. Balanced cage-guided valves are easy to position, just like double-ported stem-guided and port-guided globe valves. However, balanced cage-guided valves tend to leak more when in the shut position due to a greater number of leak paths, much the same as with double-ported stem-guided and port-guided globe valves.
Another style of globe valve body is the three-way body, sometimes called a mixing or a diverting valve. This valve design has three ports on it, with the plug (in this particular case, a cage-guided plug) controlling the degree to which two of the ports connect with the third port:
This dual illustration shows a three-way valve in its two extreme stem positions. If the stem is positioned between these two extremes, all three ports will be “connected” to varying degrees. Three-way valves are useful in services where a flow stream must be diverted (split) between two different directions, or where two flow streams must converge (mix) within the valve to form a single flow stream.
A photograph of a three-way globe valve mixing hot and cold water to control temperature is shown here:
### Gate valves
Gate valves work by inserting a dam (“gate”) into the path of the flow to restrict it, in a manner similar to the action of a sliding door. Gate valves are more often used for on/off control than for throttling.
The following set of photographs shows a hand-operated gate valve (cut away and painted for use as an instructional tool) in three different positions, from full closed to full open (left to right):
### Diaphragm valves
Diaphragm valves use a flexible sheet pressed close to the edge of a solid dam to narrow the flow path for fluid. Their operation is not unlike controlling the flow of water through a flexible hose by pinching the hose. These valves are well suited for flows containing solid particulate matter such as slurries, although precise throttling may be difficult to achieve due to the elasticity of the diaphragm. The next photograph shows a diaphragm valve actuated by an electric motor, used to control the flow of treated sewage:
The following photograph shows a hand-actuated diaphragm valve, the external shape of the valve body revealing the “dam” structure against which the flexible diaphragm is pressed to create a leak-tight seal when shut:
Some diaphragm valves are pneumatically actuated, using the force of compressed air on one side of the diaphragm to press it against the dam (on the other side) to shut off flow. This next example is of a small air-actuated diaphragm valve, controlling the flow of water through a 1-inch pipe:
The actuating air for this particular diaphragm valve comes through an electric solenoid valve. The solenoid valve in this photograph has a brass body and a green-painted solenoid coil.
• Share
Published under the terms and conditions of the Creative Commons Attribution 4.0 International Public License | 2019-12-16 10:15:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.502816379070282, "perplexity": 2296.8586057030234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541319511.97/warc/CC-MAIN-20191216093448-20191216121448-00185.warc.gz"} |
http://physics.stackexchange.com/tags/astronomy/new | # Tag Info
2
Spectroscopic parallax is the technique whereby you estimate the absolute magnitude (i.e. the brightness it would have if it were placed at 10 pc) by estimating what "type" of star it is using information fro a spectrum. It can be applied to any kind of star where (a) you have a reasonable chance of determining the type of star from its spectrum and (b) ...
3
The positions of stars on the sky are defined against a co-ordinate reference frame that is ultimately defined by the positions of very distant radio sources (quasars at high redshift) that are assumed to be "stationary" in terms of their celestial grid co-ordinates (see International Celestial Reference Frame) All stars can have their positions precisely ...
2
$\theta$ is half the difference in direction of 2 measurements of the position of the star taken 6 months apart.
5
One piece of physics that you've missed is the most pulsars spin down due to the emission of magnetic dipole radiation. For instance, the crab pulsar has a period of 33.5028 (plus a few more sig figs) milliseconds, but slows down by 38 nanoseconds per day. Furthermore, the size of several more increasing order derivatives is known accurately. So in ...
1
Sort of. This is demonstrated clearest in barred spiral galaxies, which make up about 1/2 to 2/3 of all spiral galaxies. A dramatic example is NGC 1365: Image courtesy of Wikipedia. Others, such as M95, have spiral arms that wrap even further around while still retaining the central bar: Original image courtesy of Wikipedia; color added by me in ...
2
You are correct. If you worked through the same steps on the distance ladder in a universe with a different speed of light, you would find the answer in light years would be different. Your question on whether light travels one light year in one year, though trivial sounding, opens the door to difficulties that are inherent in the nature of space-time. Your ...
1
See, first of all our eyes are not a good device to determine the main color of a group of photons. the main color that we see is actually the intensity of a specific range of frequency in the light wave. it means there are all kind of photons coming out of the sun, but the amount of "Yellow" photons are much much more. that's because we see the sun ...
0
I would like to expand here on the mechanisms of scattering. Light is scattered by particles in the air which act like dipoles, and oscillate because of the electromagnetic frequency of light. Oscillating dipoles may emit a different frequency. There are two mechanisms of scattering. Rayleigh scattering: This occurs when particle sizes are smaller than ...
-1
No, there's not a map of all the black holes in the galaxy, remember they're quite hard to detect, but there's one of stars in our solar interstellar neighborhood.
1
You have to distinguish between stellar mass and gravitating mass. The quoted Milky Way mass includes dark matter. Despite searching in the literature, I have yet to find reliable apparent magnitudes and good distance estimates for NGC 1097. The total Blue luminosity of this galaxy is near to Minus B-band absolute magnitude -21, but only ballpark ...
1
The primary reason for the asymmetry of the Ly$\alpha$ line is bulk motion of neutral hydrogen, i.e. accreting gas (causing a blueshifted line) or galactic outflows (causing a redshifted line). Mechanism of the Ly$\alpha$ double peak In general, for Ly$\alpha$ photons produced in the center of a blob of neutral hydrogen (i.e. a galaxy), the photons must ...
0
This answer is a summary, based on what I consider to be the most salient points made in the comments above. It is closely linked to the answer of my later question, Deflection of Earth directed NEOs using nuclear powered laser beams Below is the answer I received in relation to my second question: Project Excalibur The idea of a nuclear pumped X-ray ...
Top 50 recent answers are included | 2015-04-28 02:21:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6859369277954102, "perplexity": 626.7418128980709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246660493.3/warc/CC-MAIN-20150417045740-00229-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.lmfdb.org/SatoTateGroup/ | The database currently contains all rational Sato-Tate groups of weight 1 and degree up to 4, as well as all Sato-Tate groups of weight 0 and degree 1 with component group of order at most $10^{20}$. Here are some further statistics.
By weight: 0 1 By degree: 1 2 3 4 By identity component: $\mathrm{SO}(1)$ $\mathrm{U}(1)$ $\mathrm{SU}(2)$ $\mathrm{U}(1)_2$ $\mathrm{SU}(2)_2$ $\mathrm{U}(1)\times\mathrm{U}(1)$ $\mathrm{U}(1)\times\mathrm{SU}(2)$ $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $\mathrm{USp}(4)$ Some interesting Sage-Tate groups or a random Sato-Tate group | 2020-10-30 08:26:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8399507403373718, "perplexity": 2301.6402663118765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107909746.93/warc/CC-MAIN-20201030063319-20201030093319-00548.warc.gz"} |
https://physics.stackexchange.com/questions/254436/luminosity-of-an-accretion-disk | # Luminosity of an accretion disk?
With reference to Black holes in particular, how can you approximate the luminosity of an accretion disk? It is possible to quantify the temperature at a given point, but as the disk is not a black body, and this temperature is at a specific point, I am unsure how to equate this to luminosity - surely you could not do so using the Stefan-Boltzmann constant?
$$L_\lambda = 2 \int_{r_{\rm in}}^{r_{\rm out}} 2 \pi r [\pi B_\lambda(r)] dr$$
where the overall factor of 2 is for the two sides of the disk, $2 \pi r dr$ is the area of each annulus, $B_{\lambda}$ is the Planck function, which depends on the temperature at the given radius, and $\pi B_\lambda$ is the flux that arises from integrating the thermal emission over solid angle. According to the model by Shakura and Sunyaev 1973, where this is all explained in much greater detail, the temperature roughly goes as $r^{-3/4}$. To get the bolometric luminosity, integrate $L_\lambda$ over wavelength.
• Hi @NoahP, the most important point is that a single temperature doesn't characterize the entire disk, as you noted in your question. So the total emission will be a sum over different temperatures along the radius of the disk. So instead of temperature, it often helps to think of another physical quantity that sets the luminosity of the disk. Traditionally this is the rate at which mass is being fed in, or $\dot{M}$. For thin, radiatively efficient disks, the luminosity will be around $10\%$ of $\dot{M} c^2$. Depending on the exact accretion flow, that efficiency can vary. – kleingordon May 9 '16 at 22:59 | 2019-07-22 09:56:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9660608768463135, "perplexity": 246.8541863366979}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527907.70/warc/CC-MAIN-20190722092824-20190722114824-00521.warc.gz"} |
http://newsgroups.derkeiler.com/Archive/Comp/comp.text.tex/2005-12/msg01546.html | # Re: breaking of lines ? hyphenation problem or what is it?
"Peter Flynn" <peter.nosp@xxxxxxxxxxxxx> wrote in message
news:415l21F1cup5tU1@xxxxxxxxxxxxxxxxx
-snip-
> You are presumably concerned that it is not breaking COURVETTE.H
And also wondering why it doesn't break the line between include and
COURVETTE.H but I guess it's because LaTeX thinks there would otherwise be
too much empty space in that particular line (too big space between
individual words).
> I'm surprised you expect it to: it's not an [English] word, so
> there is no way it is going to break correctly, or at all.
I was sure that it would have made an estimated guess of how to hyphenate
that word, like it has done for me (in a wrong way) so many other times. I
remember that LaTeX doesn't know which words are english and which isn't
because it doesn't have such a big dictionary with english words, so I've
come to the explanation that the hyphenation pattern mechanism must have
been too confused to even make a qualified guess.... Is that correct?
Otherwise my guess would be that it has something to do with the
\textit{ }-statement, perhaps LaTeX handles hyphenation more strictly inside
the "{ }".
I'm also asking, since I can't remember ever before have had this problem
for the last couple of reports I've written....
> Adding \hyphenation{COUR-VET-TE} to the preamble seems effective.
That solves the problem, thanks. For a stupid reader perhaps though they
would think that the filename is wrong? Or perhaps not...
However, how can I make LaTeX break the line between include and
COURVETTE.H - *even though* there would be much white space. Just to see how
it looks.... And such a relaxed line breaking should only be for this
particular line - other lines should be handled with default line splitting
rules... Is that possible?
Med venlig hilsen / Best regards
Martin Jørgensen
--
---------------------------------------------------------------------------
Home of Martin Jørgensen - http://www.martinjoergensen.dk
.
## Relevant Pages
• Re: breaking of lines ? hyphenation problem or what is it?
... >> You are presumably concerned that it is not breaking COURVETTE.H ... I remember that LaTeX doesn't know which words are ... > english and which isn't because it doesn't have such a big dictionary ... I'm not an expert in TeX hyphenation, ...
(comp.text.tex)
• Re: Avoid hyphenation across pages?
... do need hyphenation for other lines. ... Is there any way to stop LaTeX ... hyphenation) before it starts page breaking. ... editions with multiple footnotes, bigfoot will be able to avoid ...
(comp.text.tex)
• Re: Where does fmtutil get its inputs?
... The reason I want to rebuild formats is to avoid using Babel, ... I want UK English hyphenation by default to avoid having to specify UK ... What then happens to these changed configuration files? ... Let's look at a concrete example: latex. ...
(comp.text.tex)
• Re: "It doesnt matter if youre a good programmer, its the syntax that matters"
... I'm not sure it's reasonable to expect LaTeX to infer logical ... to me for the writer/programmer to be explicit about logical structure. ... structure from layout-using-whitespace in the source code. ... Is it TeX making decisions about hyphenation? ...
(comp.programming)
• texlive auf xubuntu 7.04: pdflatex und fonts
... Babel and hyphenation patterns for english, usenglishmax, ... Document Class: article 2005/09/16 v1.4f Standard LaTeX document class ... Output written on test.pdf ...
(de.comp.text.tex) | 2013-05-24 11:37:02 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9269424676895142, "perplexity": 4531.99841488783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704645477/warc/CC-MAIN-20130516114405-00030-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://plainmath.net/differential-calculus/47468-find-the-length-of-the-curve-vec-r-t-equal-less-then-8t-t-2 | Danelle Albright
2021-12-21
Find the length of the curve.
$\stackrel{\to }{r}\left(t\right)=<8t,{t}^{2},\frac{1}{12}{t}^{3}>,0\le t\le 1$
esfloravaou
${\stackrel{\to }{r}}^{\prime }\left(t\right)=<8,2t,\frac{3}{12}{t}^{2}>$
${\stackrel{\to }{r}}^{\prime }\left(t\right)=<8,2t,\frac{1}{4}{t}^{2}>$
$||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\sqrt{{\left(8\right)}^{2}+{\left(2t\right)}^{2}+{\left(\frac{1}{4}{t}^{2}\right)}^{2}}$
$⇒||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\sqrt{64+4{t}^{2}+\frac{1}{16}{t}^{4}}$
$⇒||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\sqrt{\frac{1}{16}\left(\left(64\cdot 16\right)+64{t}^{2}+{t}^{4}\right)}$
$⇒||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\frac{1}{4}\sqrt{1024+64{t}^{2}+{t}^{4}}$
$⇒||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\frac{1}{4}\sqrt{{32}^{2}+\left(2\cdot 32\right){t}^{2}+{t}^{4}}$
$⇒||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\frac{1}{4}\sqrt{{\left(32+{t}^{2}\right)}^{2}}$
$⇒||{\stackrel{\to }{r}}^{\prime }\left(t\right)||=\frac{1}{4}|32+{t}^{2}|$
Now, ${t}^{2}$ is always positive. So, $\left(32+{t}^{2}\right)$ is always positive.
Cheryl King
Where is the second part of solution?
RizerMix
we know that, the length of the curve between t = 0 and t = 1 can be computed as
$\begin{array}{}L={\int }_{t=0}^{t=1}||\stackrel{\to }{r}\left(t\right)||dt\\ ⇒L={\int }_{t=0}^{t=1}\frac{1}{4}\left(32+{t}^{2}\right)dt\\ ⇒L=\frac{1}{4}{\int }_{t=0}^{t=1}\left(32+{t}^{2}\right)dt\\ ⇒L=\frac{1}{4}\left[32+\frac{{t}^{3}}{3}{\right]}_{t=0}^{t=1}\\ ⇒L=\frac{1}{4}\left[\left(32+\frac{1}{3}\right)-\left(0+0\right)\right]\\ ⇒L=\frac{1}{4}\left(32+\frac{1}{3}\right)\\ ⇒L=\frac{1}{4}\ast \frac{97}{3}\\ ⇒L=\frac{97}{12}\end{array}$
Do you have a similar question? | 2023-03-26 05:12:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 13, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8453673124313354, "perplexity": 772.6144534045304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00696.warc.gz"} |
http://www.darwinproject.ac.uk/letter/DCP-LETT-7634.xml | DCP-LETT-7634
# From F. C. Donders 28 March 1871
## 28 March 71
My dear Mister Darwin,
I am very much honoured by your last letter, which makes some other enquiries from me.1 Upon these I can give you only a provisional answer.
The idea that in meditation the eyeball might sink a little into the orbit, is not at all fanciful: I hope to be able to tell you how it is, within a few weeks.
On the influence of different passions on the pupil I cannot say you much for the moment.2 I will try to know some thing about and if I have some success, will give you an account of it. The chief point why I write provisonnally to you, is this: that it might be possible, that you are not aware of the influence of accommodation and of the associating motions of the eyeball (convergence) on the diameter of the pupil.3 I think, this is of some importance. I often saw that the pupil of parrots contract and dilate, independently of the amount of light. But I suppose, that these movements were combined with movements of the eyeball, the contraction coinciding with convergence. Perhaps in parrots exists some accommodation even without movement of the eye. I believe, H. Mueller demonstrated, that they have two points of exact (direct) vision on every eye, one for uniocular, another for binocular vision; and if they use that for uniocular vision, there is no reason, why birds could not accommodate for short distances, without moving the eyes.4 In man, that is not commonly the case; but after some exercise, I succeeded to contract my pupils, by the effort of accommodating for a near point, without any motion of the eye (of whatever of the eyes). Also the convergence, without change of accommodation, is combined with contraction of the pupil: this I could demonstrate by putting prismatical glasses before the eyes, the angles turned inwards, which require more convergence for binocular vision of the same points, and meanwhile an increase of accommodation can be avoided.— Perhaps the whole doctrine of the movements of the iris is in some respect important for your studies on physiognomy. You may find that in every good physiological book. I hope however that you will allow me to send to you a copy of my book on the anomalies of accommodation, etc., which contains a short exposition of the subject p. 572–575, followed by that of the ciliary system and its function.—5 In the same book you find (p. 197) a description of the entoptic method. It may be of some interest for you, not for the irregularities in the eye, but for the opportunity which it gives to see the changes of the diameter of the pupil in your own eye. As it is mentioned p. 573: A small opening in an opaque plate, held at about $\frac{1}{2}$ inch from the eye and turned towards the light, gives, in the vitreous humour a bundle of nearly parallel rays, of the size of the pupil, and is therefore seen as a round, illuminated disc, whose diameter increases and diminishes with that of the pupil. This method allows us to judge very accurately about motions of our pupils, which might occur, whilst we produce in (or suggest to ourselves different animi pathemata.6 The opening in the plate should not be too small for this purpose,—perhaps $\frac{1}{4}$ Millimeter;— an opening made with a needle in a piece of thick black paper will do very well.
If all this has no value at all for the sake of your researches, I implore your pardon for the trouble I give you by adressing this letter.
I ordered a copy of my book to be sent to you in England, where it is published.
I am obliged to go to Leipsick7 and Berlin for a couple of weeks. Therefore, I must pray you to give me some time more for entering, if possible, in further details into the questions, which you mention in your letter. That there is nothing, which I wish more than to remove some obstacles in your researches, as far as my special knowledge of the one or the other point may afford, I hope you are persuaded. Will you kindly excuse my bad English?—
Believe me respectfully and yours very sincerely | Donders.
## CD annotations
3.4 and of the … pupil. 3.5] scored pencil and red crayon
3.6 I often … light. 3.7] scored red crayon
Top of letter: ‘Used’ pencil, del pencil; ‘Parrots’ pencil
## Footnotes
1
See letter to F. C. Donders, 18 March 1871.
2
See letter to F. C. Donders, 18 March 1871 and nn. 5 and 6.
3
For more on Donders’s research on accommodative convergence of the eye, see Donders 1876, pp. 392–407.
4
Donders refers to Heinrich Müller and to Müller’s 1862 report of a second fovea (now called the temporal fovea) in the eyes of some birds, such as hawks and parrots (see Heinrich Müller 1872, pp. 142–3).
5
Donders refers to On the anomalies of accommodation and refraction of the eye (Donders 1864). CD’s lightly annotated copy is in the Darwin Library–CUL (see Marginalia 1: 204).
6
Animi pathemata: affections of the mind (Latin and Greek). The phrase was used by early medical practitioners to refer to emotional states.
7
Leipzig.
## Letter details
Letter no.
DCP-LETT-7634
From
Donders, F. C.
To
Darwin, C. R.
Sent from
Utrecht
Source of text
DAR 162: 228
Physical description
6pp †
## Summary
Answers to CD’s queries will take time. CD may not be aware of the influence of accommodation on the diameter of the pupil of the eye. Parrots, for example, contract or dilate the pupil independently of amount of light [see Expression, p. 304]. Sends his book on the subject [On the anomalies of accommodation and refraction of the eye (1864)].
## Subjects
### Scientific Terms
Darwin Correspondence Project, “Letter no. 7634,” accessed on 29 May 2016, http://www.darwinproject.ac.uk/DCP-LETT-7634
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http://math.stackexchange.com/questions/338998/why-does-m%C3%B6bius-transformation-fz-map-the-unit-disk-z1-to-itself | # Why does Möbius transformation $f(z)$ map the unit disk $|z|<1$ to itself?
$f(z)= a\frac{z-z_0}{\overline{z_0}\cdot z-1}$, where $\left|a\right| = 1$ and $\left|z_0\right| < 1$ .
If $\left|z\right|= 1$, it is obvious that $\left|f(z)\right| = 1$, thus $f(z)$ maps unit circle to unit circle. If $z= z_0$, $f(z)= 0$.
So why does this transformation send $\left|z\right|<1$ to itself?
-
Please see if the edit was what you intended for. – AlanH Mar 23 '13 at 18:53
Yes thanks !!!! – sarah Mar 23 '13 at 19:02
Maybe this help math.stackexchange.com/questions/318060/… – Cortizol Mar 23 '13 at 19:10
Hints:
$$|z|<1\;,\;|w|<1\;,\;\;|a|=1\Longrightarrow \left|\;a\frac{z-w}{\bar w z-1}\;\right|=\frac{|z-w|}{|\bar w z-1|}<1\iff$$ $$\iff |z-w|^2<|\bar wz-1|^2\;\;(**)$$
To make things now clearer, perhaps, put $\,z:=x+iy\;,\;\;w=a+bi\,$ , so
$$(**)\iff (x-a)^2+(y-b)^2<(ax+by-1)^2+(bx-ay)^2\iff$$
$$\iff x^2+y^2+a^2+b^2-\color{red}{2(ax+by)}<a^2x^2+b^2y^2+\color{green}{2abxy}-\color{red}{2(ax+by)}+1+b^2x^2+a^2y^2-\color{green}{2abxy}\iff$$
$$\iff (a^2+b^2-1)(x^2+y^2-1)>0$$
And since the last inequality is clearly true (why?) , then...
-
I will denote by $U(1)=\{z | |z|=1 \}$.
Note first that
$$f(z)=f(y) \Leftrightarrow \frac{z-z_0}{\overline{z_0}\cdot z-1}=\frac{y-z_0}{\overline{z_0}\cdot y-1} \Leftrightarrow (z-z_0)(\overline{z_0}\cdot y-1)=(\overline{z_0}\cdot z-1)(y-z_0)$$
$$\Leftrightarrow y-z=z-y\Leftrightarrow y=z$$
This proves that $f$ is one to one (which is something you probably already know). You can also get from here that $f(U(1))=U(1)$ [before we only knew $f(U(1)) \subset U(1)$].
Now let $z$ be so that $|z| <1$. Then the segment $z_0z$ doesn't intersect $U(1)$ and hence $f(z_0z)$ cannot intersect $U(1)$.
But $f(z_0z)$ is a closed curve which starts at $0$ and ends at $f(z)$. Since this curve doesn't meet $U(1)$, it lies entirely inside $D=\{ z | |z| <1 \}$, thus $|f(z)| <1 \,.$
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https://www.groundai.com/project/magnetoelectric-properties-of-the-multiferroic-cucro_2-studied-by-means-of-ab-initio-calculations-and-monte-carlo-simulations/ | Magnetoelectric properties of the multiferroic CuCrO{}_{2} studied by means of ab initio calculations and Monte Carlo simulations
# Magnetoelectric properties of the multiferroic CuCrO2 studied by means of ab initio calculations and Monte Carlo simulations
## Abstract
Motivated by the discovery of multiferroicity in the geometrically frustrated triangular antiferromagnet CuCrO below its Néel temperature , we investigate its magnetic and ferroelectric properties using ab initio calculations and Monte Carlo simulations. Exchange interactions up to the third nearest neighbors in the plane, inter-layer interaction and single ion anisotropy constants in CuCrO are estimated by series of density functional theory calculations. In particular, our results evidence a hard axis along the [110] direction due to the lattice distortion that takes place along this direction below . Our Monte Carlo simulations indicate that the system possesses a Néel temperature K very close to the ones reported experimentally ( K). Also we show that the ground state is a proper-screw magnetic configuration with an incommensurate propagation vector pointing along the [110] direction. Moreover, our work reports the emergence of spin helicity below which leads to ferroelectricity in the extended inverse Dzyaloshinskii-Moriya model. We confirm the electric control of spin helicity by simulating - hysteresis loops at various temperatures.
## I Introduction
Through the discovery of the mineral CuFeO in 1873, Friedel opened the door to the delafossites ABO (1); (2). Such a family crystallizes in the layered space group, see Fig. 1. The diversity of properties they exhibit raises up an ever increasing interest in this class of compounds. In particular, the discovery of simultaneous transparency and -type conductivity in CuAlO by Kawazoe et al. (3), laid ground for the development of transparent optoelectronic devices. Furthermore, depending on the chemical composition, a plethora of behaviors can be evidenced. For instance, for A in a configuration, e.g., A = Pd or Pt, highly metallic compounds with anomalous temperature dependence of the resistivity have been reported (4); (5); (6); (7). The transport in these compounds has been found to be strongly anisotropic, with a degree of anisotropy that may reach 1000 (4); (5); (8). For A in a configuration, the semi-conducting materials CuBO, with B = Cr, Fe, Rh, may be turned into promising thermoelectric ones through hole doping (9); (10); (11) — in particular, an especially high power factor has been found in the case of CuRhMgO (12), which transport coefficients served as a basis for the Apparent Fermi Liquid scenario (13). Regarding the magnetic compounds CuFeO and CuCrO, many studies point towards a strong coupling of the magnetic and structural degrees of freedom (14); (15); (16); (17); (18); (19); (20); (21); (22), that paves the way to multiferroelectricity.
With its frustrated triangular lattice CuCrO received a lot of attention since it is ferroelectric without applying magnetic fields or doping upon Cr sites, unlike CuFeO (17); (24). The emergence of ferroelectricity in CuCrO is induced by the proper-screw magnetic ordering below the Néel temperature , and the control of this ferroelectricity by an applied magnetic field is very important for new spin-based device applications. CuCrO forms a rhombohedral lattice where the edge-shared CrO layers are alternatively stacked between Cu layers along the c-axis as shown in Fig. 1. Due to the weak inter-layer interaction (Fig. 2), the material behaves as a quasi-2D magnet, which makes it even more interesting.
The magnetic properties of CuCrO have been investigated by neutron diffraction experiments (25); (20); (26); (27); (28). It was shown that the magnetic configuration of CuCrO below is proper screw with an incommensurate propagation vector q = (0.329, 0.329, 0) (28) pointing along the [110] direction. Such deviation from the commensurate magnetic configuration of q = (1/3, 1/3, 0) is due to the lattice distortion that takes place along the [110] direction below upon the spiral-spin ordering which leads to anisotropic in-plane exchange interactions and (Fig. 2) (29). Polarized neutron-diffraction measurements on single crystals of CuCrO (20) showed that the spins are oriented in a spiral plane parallel to the (110) plane suggesting that the [110] direction is a hard axis.
The electric polarization emerges upon the spiral-spin ordering (30); (20); (31), which reflects the strong coupling between non-collinear magnetic ordering and ferroelectricity in CuCrO. Within the spin-current model or the inverse Dzyaloshinskii-Moriya (DM) mechanism (32); (33); (34), the electric polarization produced between the canted spins and , located at sites and , respectively, is given by
Pij∝eij×(Si×Sj)≡p1 (1)
where is a unit vector joining the sites and . However, Eq.(1) fails to explain the emergence of ferroelectricity in CuCrO because in the proper-screw configurations, is parallel to ( is along the [110] direction due to symmetry considerations (30)) unlike the cycloid spin structures.
Based on symmetry considerations, Kaplan and Mahanti (35) introduced an additional contribution to the macroscopic polarization which contributes in both cycloid and proper-screw configurations. Therefore, within this model, now referred to as extended DM model, the total polarization is given by
P=p1+p2 (2)
In this study, we investigate the magnetoelectric properties of CuCrO by means of a combination of Density Functional Theory (DFT) calculations and Monte Carlo (MC) simulations. More precisely, we estimate a set of exchange interactions and anisotropy constants and confront it to the experimental magnetic properties and we verify the appearance of spiral spin ordering at low temperatures which can be related to the ferroelectric polarization.
In Sec. II we detail briefly the DFT method that we used to extract the coupling and anisotropy constants in CuCrO, while the model and MC method are presented in Sec. III. Sec. IV is devoted to the results where we discuss the magnetic and ferroelectric properties of CuCrO. A conclusion is given in Sec. V.
## Ii DFT computational method
We performed a series of DFT calculations using full-potential linear muffin-tin orbital (FP-LMTO) method as implemented in RSPt (36) code. An experimental crystal structure (37) was considered, taking into account a small in-plane lattic distortion, suggested in Ref. (29). Our results are in-line with earlier calculations (21). The DFT+ (38) approach was used in order to take into account the effect of strong correlations between Cr electrons. The adopted values of Hubbard and Hundâs exchange were 2.3 and 0.96 eV, which were extracted from first-principles calculations for a similar system LiCrO (23). The same computational scheme was used in a prior study on the magnetic properties of CuCrO (22). The Fully Localized Limit (FLL) (39) formed of the double-counting correction was applied. We calculated the exchange parameters between Cr ions by means of the magnetic force theorem (40); (41). The so-called muffin-tin head projection scheme was applied to construct the set of localized Cr- orbitals (for more details see Ref. (42)). The ’s were extracted from both ferromagnetic and antiferromagnetic configurations. The obtained values turned out to be insensitive to the assumed magnetic order, which implies that they can be used as fixed parameters in a Hamiltonian describing the interacting spins. The spin-orbit coupling was taken into account only for the calculation of the magnetocrystalline anisotropy, which was calculated directly from the total energies.
## Iii Model and Monte Carlo simulation
To model the magnetic properties of CuCrO, we note that Cr ions with = 3/2 spins are large enough to be treated classically, so we used the following classical three dimensional (3D) Heisenberg Hamiltonian
H= −∑⟨i,j⟩JijSi⋅Sj−Dx∑iS2ix−Dz∑iS2iz +gμBB⋅∑iSi (3)
where refers to the exchange interactions up to the 4 neighbors (Fig. 2). The -axis corresponds to the [110] direction and the -axis corresponds to the [001] direction. and correspond to the hard and easy axes anisotropy constants respectively. The fourth term corresponds to the Zeeman energy where is the applied magnetic field ( is the Bohr magneton and is the Landé factor).
To model the ferroelectric properties of CuCrO and the coupling between the spins and the electric field , we added the following term to the previous Hamiltonian
He=−A0E⋅∑⟨i,j⟩Si×Sj (4)
where the sum runs over the magnetic bonds along the [110] direction, and is a coupling constant related to the spin-orbit and spin exchange interactions. Adding this contribution leads to the model for multiferroics proposed by Kaplan and Mahanti (35).
Our MC simulations (43) were performed on 3D triangular lattices (Fig. 1 with only Cr ions) with periodic boundary conditions (PBC) using the standard Metropolis algorithm (44) and the time-step-quantified method (45) when needed.
Typically, the first MC steps were discarded for thermal equilibration before averaging over the next MC steps. Note that our results are averaged over 24 simulations with different random number sequences so that statistical fluctuations are negligible.
## Iv Results and discussions
It was reported in Ref. (29) that the lattice undergoes a tiny in-plane distortion below with and being the lattice constants along the [110] and the [100] directions, respectively. As a first step, we considered (29) to calculate the exchange interactions and anisotropy constants in CuCrO. The extracted values given in Table 1 (line 1) are very close to the ones reported in Ref. (46) concerning and as well as the single ion anisotropy constants. Note that here is very close to 1 (). Knowing that PBC favors the commensurate configuration when is close to 1, large enough sizes are required to obtain an incommensurate magnetic ground state (GS).
However, a MC simulation with 90902 unit cells was not able to reproduce an incommensurate GS with this set of interactions (). Thus larger sizes of the simulation box were required which are not accessible within reasonable computer time (47). Therefore we enhanced the lattice distortion by a factor of 30 (i.e. ). We found that the new set of ’s () and anisotropy constants (Table 1) is a good candidate to reproduce an incommensurate GS for a system of reasonable size 45452 unit cells. It is worth noting that the considered distortion mainly affect the first nearest neighbors interactions while the remaining interactions are not affected. Also it is very interesting to note that the magnitude of the in-plane anisotropy constant () increases when enhancing the lattice distortion reflecting that this anisotropy results from the lattice distortion.
### iv.1 Magnetic properties
In order to characterize the GS configuration and to estimate the Néel temperature we performed a first set of simulations without applying an external magnetic field. The following procedure has been retained: we started the simulations from random spin configurations at a high enough temperature () and we then cooled down to = 0.01 K with a constant temperature step = 0.5 K.
In order to estimate the Néel temperature, we calculated the specific heat per spin defined as
C=1N∂U∂T=⟨E2⟩T−⟨E⟩2TNkBT2 (5)
where with being the energy of each magnetic configuration, means thermal average, is the number of spins and is the Boltzmann constant. For the parameter set given in Table 1 () the phase transition as signaled by the peak of the specific heat (Fig. 3) takes place at K. This value is in a good agreement with the reported experimental values ( K) (9); (30); (28). This may be taken as a first validation of the extracted exchange interactions of Table 1.
To characterize the nearly GS configuration we considered the spin chirality defined as
Extra open brace or missing close brace (6)
where 1, 2 and 3 refer to the spins at the corners of each elementary triangular plaquette in an plane. Then we defined the order parameter per plane to be where is the number of plaquettes per plane, and finally the order parameter of the whole system was defined as where is the average of over the planes. We found that the direction of the vector chirality () of each plane is pointing along the [110] direction confirming the fact that the spins are oriented in the (110) plane as reported in Ref. (20). Fig. 4 shows the variation of the order parameter as function of temperature. At K, indicates a small deviation from the commensurate (120) configuration of . Moreover, the simulated value of q (0.322, 0.322, 0) confirms that the GS is an incommensurate configuration very close to the reported experimental configuration of q (0.329, 0.329, 0) (28). This good agreement may be taken as another validation of the parameters of Table 1.
On the other hand, the magnetic field dependence of the magnetization calculated along the easy axis (-axis) shows a linear behavior () confirming the antiferromagnetic nature of the GS (not shown here).
Magnetic properties under 0.5 T were simulated between 300 K and 2 K to estimate the CurieWeiss temperature (). Fig. 5 shows the variation of the magnetization and inverse susceptibility measured along the applied magnetic field. It can be seen that obeys well the CurieWeiss law for antiferromagnets (, with is the Curie constant) at high temperatures with K close to the measured experimental values ( K) (9); (48). The curve starts to deviate from the linear behavior at about 100 K. In order to understand the origin of this deviation we calculated the temperature dependence of the spin-spin correlation function defined as along the [100] direction. As shown in Fig. 6, short-range antiferromagnetic correlations start to develop below 100 K, which leads to the deviation from the CurieWeiss law seen in Fig. 5. Furthermore, these correlation functions exhibit inflection points close to estimated from the specific heat curve (Fig. 3). Besides, an anomaly in the magnetization curve (Fig. 5) appears at K consistent with the estimate of from the specific heat curve. We note that the ratio () reflects the frustrated nature of the GS (49); (50).
### iv.2 Ferroelectric properties
In this section, we considered the Hamiltonian . In these simulations, we applied a poling electric field during the cooling process to obtain a single ferroelectric domain. We then turned it off just before statistical averaging to calculate which is associated to the spontaneous ferroelectric polarization (Eq. (2)) according to Ref. (35). Fig. 7 shows the temperature dependence of , the projection of along the [110] direction, which starts to develop at . It is clearly seen that by switching the poling electric field, can be reversed.
Further insight into the degree of electrical polarization may be gained through the knowledge of the - hysteresis loops, which are shown in Fig. 8 at different temperatures. shows a linear dependence without hysteresis above because the system is in the paraelectric phase, while clear hysteresis loops are seen for temperatures below . This strongly suggests that ferroelectricity is induced by the out-of-plane incommensurate magnetic configuration, in agreement with Ref. (31).
Also, it can be seen that below the saturation field MV/m is independent of the temperature. The hysteresis loop simulated at K shows an electric coercive field for reversal MV/m very close to that measured experimentally ( MV/m (51)). Note that the reversal of results from the reversal of the helicity of each atomic plane. Thus our simulations confirm the electric control of spin helicity in CuCrO as reported in Ref. (20).
## V Conclusion
In this paper, we proposed estimates of the exchange interactions and single ion anisotropy constants in the multiferroic CuCrO using DFT calculations. They were checked against the experimental Néel and CurieWeiss temperatures as well as the electric coercive field, thereby proving them to be good candidates to model the magnetoelectric properties of CuCrO. We showed that the lattice distortion that takes place below is responsible for the appearance of a weak in-plane hard-axis anisotropy. Regarding the magnetic properties, we obtained a peak in the specific heat curve at K very close to the experimental observations. Furthermore the ground-state has been shown to be an antiferromagnetic incommensurate proper-screw configuration. The estimated K is in a good agreement with experimental data too. Also, our simulated - hysteresis loops confirm the electric control of spin helicity which is related to the ferroelectric polarization below .
## Acknowledgments
We gratefully thank M. Alouani and S. Hébert for stimulating discussions. We are grateful to the Centre Régional Informatique et d’Applications Numeriques de Normandie (CRIANN) where our simulations were performed as project number 2015004. We also acknowledge the computational resources provided by the Swedish National Infrastructure for Computing (SNIC) and Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX). The authors acknowledge the financial support of the French Agence Nationale de la Recherche (ANR), through the program Investissements d’Avenir (ANR-10-LABX-09-01) and LabEx EMC3.
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The feedback must be of minumum 40 characters | 2019-03-24 19:02:57 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8251132965087891, "perplexity": 2725.7709112479506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203491.1/warc/CC-MAIN-20190324190033-20190324212033-00200.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=130&t=18158&p=48241 | ## Chapter 8 HMWK Problem 8.21 Clarification
$\Delta U=q+w$
Sunny Chera 1N
Posts: 20
Joined: Wed Sep 21, 2016 2:56 pm
### Chapter 8 HMWK Problem 8.21 Clarification
The question is stated as follows:
Question: A piece of copper of mass 20.0 g at 100.0 degrees Celsius is placed in a vessel of negligible heat capacity but containing 50.7 g of water at 22.0 degrees Celsius. Calculate the final temperature of the water. Assume that no energy is lost to the surroundings.
Problem: I know that the heat lost by the metal is equal to the heat gained by the water. Therefore, I said that -q=qH20. Although I got the same answer as in the solutions manual, it stated that the heat lost by the metal is equal to the negative heat gained by water (qmetal=-qH20). Is either interpretation considered valid, or is there a flaw with my reasoning? I've had the same issue with several other problems as well.
Aishwarya_Natarajan_2F
Posts: 11
Joined: Mon Jul 11, 2016 3:00 am
### Re: Chapter 8 HMWK Problem 8.21 Clarification
In a case of heat transfer between a system and it's surroundings, we assume qsystem +qsurrounding= 0, or in other words q sys = -q sur
In this case the problem is asking us to examine heat transfer between a piece of copper at a temp of 100 deg C (the system) and the water at 22.0 deg C (the surroundings).
You are correct in your setup of -qmetal = qH2O because when you plug values into this you get:
-(20.0 g)(0.38 J/g C)(Tfinal - 100 C) = (50.7 g)(4.184 J/g C)(Tfinal - 22)
Logically, we know that the final temperature will lie somewhere in the middle of 100 and 22 when this transfer occurs, which means, ignoring the negative sign, that the q sur, or water side of the equation, will be positive because it's temperature will rise, and the q sys, or copper side, will be negative because it's temperature falls. If you put the negative sign on the water side, then both sides will become negative and if you put it on the copper side, both sides will become positive. Mathematically, having the negative sign there is like multiplying one of the sides by (-1), and so in this case it actually doesn't matter what side the negative sign is on. This is why in this problem, either interpretation would be valid. | 2020-10-30 22:50:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6084362268447876, "perplexity": 606.495690295916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107911792.65/warc/CC-MAIN-20201030212708-20201031002708-00308.warc.gz"} |
https://www.cut-the-knot.org/m/Algebra/SquareSpiral.shtml | # Exercise with Square Spiral
### Source
Hubert Shutrick has kindly communicated to me his investigation of a hand-drawn spiral, as bellow:
This is a simple construction that leads to a self-similar shape, concerning which several questions may be meaningfully asked as to the lengths and areas involved. Answers to these questions reveal that, the self-similarity notwithstanding, neither the boundary curves, nor the enclosed form, are fractals - just plain self-similar figures. The curves have finite lengths, whereas the enclosed shape has finite area. There are certainly weirder cases.
### Questions
There are several questions that may be asked concerning the resulting fractal. The shape consists of quarter circles stuck together. It is bounded by two spiral curves: the smooth outer spiral and the rectilinear inner spiral.
1. What is the length of the smooth outer spiral?
2. What is the length of the rectilinear inner spiral?
3. What is the area enclosed between the two curves?
4. What is the accumulation point of the two spirals?
For all questions the answer comes from a formula for a sum of the geometric series:
$\displaystyle a+aq+aq^2+aq^3+\ldots =a\frac{1}{1-q}=\frac{a}{1-q}.$
1. What is the length of the smooth outer spiral?
$\displaystyle a=\frac{2\pi}{4}=\frac{\pi}{2},\,$ $\displaystyle q=\frac{1}{2},\,$ $\displaystyle\frac{a}{1-q}=\pi.$
2. What is the length of the rectilinear inner spiral?
$a=1,\,$ $\displaystyle q=\frac{1}{2},\,$ $\displaystyle\frac{a}{1-q}=2.$
3. What is the area enclosed between the two curves?
$\displaystyle a=\frac{\pi}{4},\,$ $\displaystyle q=\frac{1}{4},\,$ $\displaystyle\frac{a}{1-q}=\frac{\pi}{3}.$
4. What is the accumulation point of the two spirals?
If the accumulation point - the tip of the spiral shape - is described as $(x_0,y_0),\,$ we'll compute the two separately:
$x_0=\displaystyle\frac{1}{4}-\frac{1}{16}+\frac{1}{64}-\ldots,\;$ so that $a=\displaystyle\frac{1}{4},\,$ $q=\displaystyle -\frac{1}{4}.\;$ Thus,
$\displaystyle\frac{a}{1-q}=\frac{1}{4}\cdot\frac{4}{5}=\frac{1}{5}.$
For $y_0,\,$ the calculations are the same, except $\displaystyle a=\frac{1}{2},\,$ so that $\displaystyle\frac{a}{1-q}=\frac{1}{2}\cdot\frac{4}{5}=\frac{2}{5}.$
We come up with the accumulation point $\displaystyle (x_0,y_0)=\left(\frac{1}{5},\frac{2}{5}\right).$
Hubert Shutrick's original solutions dependent on the shape at hend being self-similar. The first piece aside, the rest of the shape is constructed exactly like the whole but strating with half as large quarter circle (and this rotated $90^{\circ}\,$ degrees clockwise). This remark leads to a few simple equations:
1. What is the length of the smooth outer spiral?
If the sought length is $L\,$ then $\displaystyle \frac{\pi}{2}+\frac{L}{2}=L,\,$ implying $L=\pi.$
2. What is the length of the rectilinear inner spiral?
If the sought length is $M\,$ then $\displaystyle 1+\frac{M}{2}=M,\,$ implying $M=2.$
3. What is the area enclosed between the two curves?
If $S\,$ is the said area, $\displaystyle S=\frac{\pi}{4}+\frac{1}{4}S,\,$ implying $S=\displaystyle\frac{\pi}{3}.$
4. What is the accumulation point of the two spirals?
If $z\,$ is the accumulation we have $\displaystyle z=\frac{i}{2}-\frac{i}{2}z,\,$ implying
\displaystyle\begin{align} 2z&=i-iz,\\ (2+i)z&=i,\\ z&=\frac{i}{2+i}=\frac{i(2-i)}{5}=\frac{1}{5}+i\frac{2}{5}. \end{align}
It's possible to avoid complex variables by observing that $\displaystyle\left(\frac{3}{16},\frac{3}{8}\right),$ is the new origin of a copy size $\displaystyle\frac{1}{16}-\text{th}\,$ so the vector $v\,$ from the origin to the accumulation point satisfies
$\displaystyle v = \left(\frac{3}{16},\frac{3}{8}\right) + \frac{v}{16}.$
Finally, Hubert also observed that the homothety from the whole elephant to the one $\displaystyle\frac{1}{16}-\text{th}\,$ its size is a projection from the accumulation point which can therefore be also found as the intersection of the line of centres $y=2x\,$ and the line of starting points of the spirals $3y-x=1.$ | 2020-08-10 06:34:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8774324059486389, "perplexity": 642.0489589916684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738609.73/warc/CC-MAIN-20200810042140-20200810072140-00258.warc.gz"} |
https://expskill.com/question/if-veca-hati-hatj-hatk-and-%CE%BB5-what-is-value-of-%CE%BBveca/ | # If $$\vec{a}$$ =$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$ and λ=5, what is value of λ$$\vec{a}$$?
Category: QuestionsIf $$\vec{a}$$ =$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$ and λ=5, what is value of λ$$\vec{a}$$?
Editor">Editor Staff asked 11 months ago
If $$\vec{a}$$ =$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$ and λ=5, what is value of λ$$\vec{a}$$?
(a) $$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$
(b) 5$$\hat{i}$$ + 5$$\hat{j}$$ + 5$$\hat{k}$$
(c) $$\hat{i}$$ + 5$$\hat{j}$$ + 5$$\hat{k}$$
(d) 10$$\hat{i}$$ + 10$$\hat{j}$$ + 10$$\hat{k}$$
This question was addressed to me at a job interview.
My doubt stems from Multiplication of a Vector by a Scalar in division Vector Algebra of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience
Right option is (b) 5$$\hat{i}$$ + 5$$\hat{j}$$ + 5$$\hat{k}$$
To explain I would say: Multiplication of vector $$\vec{a}$$ =$$\hat{i}$$ + $$\hat{j}$$ + $$\hat{k}$$ by scalar value 5 results in 5$$\hat{i}$$ + 5$$\hat{j}$$ + 5$$\hat{k}$$, as in these type of questions we multiply$$\hat{i}$$, $$\hat{j,}$$
$$\hat{k}$$ with the constant given and the answer comes out to be 5$$\hat{i}$$ + 5$$\hat{j}$$ + 5$$\hat{k}$$. | 2022-11-30 18:05:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38827621936798096, "perplexity": 986.9483374632779}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710765.76/warc/CC-MAIN-20221130160457-20221130190457-00109.warc.gz"} |
https://www.physicsforums.com/threads/a-stupid-question-i-think.381720/ | # A stupid question (I think)
1. Feb 25, 2010
### LucasGB
Suppose I have a perfectly constant, uniform magnetic field extending over space. Now I imagine a circle in this space. Now I imagine the circle is growing larger. The magnetic flux through this imaginary circle is, therefore, increasing. Therefore, there must be an induced electric field. But how can I have created an electric field by imagining things?
2. Feb 25, 2010
### Born2bwire
I guess it could work. What you would do in reality is have a long wire. You would loop this wire into your loop in the plane normal to your magnetic fields and then run the wire through a slip knot so that the leftover wire is hanging down along the magnetic field vectors. This way, you could pull on the wire to open and close the loop, the excess wire being pulled down along the field vectors.
In this way you would be providing a mechanical input into the system. By moving the wire, you are accelerating the charges in the wire since they are confined by the wire. It seems to be conceptually the same as the classic case of when we have a loop of wire entering and exiting a static magnetic field.
3. Feb 25, 2010
### netheril96
There is NO electric field here.The EMF is due to the motion of the wire,or Lorentz force in micrsoscopic scale,not a newly induced e-field.
4. Feb 25, 2010
### Bob S
What you have described is very close to a particle accelerator called a betatron. Instead of an imaginary circle growing larger in a constant magnetic field, the betatron has a constant diameter toroidal vacuum chamber (your imaginary circle), and a varying magnetic field inside the toroid. The volts per turn (at 60 Hz) is given by Faraday's Law:
V(t) = -(dB/dt)∫B(t)·dA = 377·A·Bmax·sin(ωt)
The largest of these accelerators, built at the University of Chicago in 1949, had a ~1.5-m diameter toroidal vacuum chamber, and roughly 1 square meters of transformer iron that ran off the ac line frequency, such that the magnetic field cycled from ~- 1.4 Tesla to + 1.4 Tesla at 60 Hz. The peak azimuthal electric field inside the vacuum chamber was roughly 200 volts per meter. Electrons from a hot filament were accelerated in this azimuthal field inside the vacuum chamber to over 300 million volts during the half-cycle where dB/dt was the right polarity. See photo of betatron
http://storage.lib.uchicago.edu/apf/apf2/images/derivatives/apf2-00056r.jpg [Broken]
See theory
http://teachers.web.cern.ch/teachers/archiv/HST2001/accelerators/teachers notes/betatron.htm
Bob S
Last edited by a moderator: May 4, 2017
5. Feb 25, 2010
### LucasGB
Thank you all for your replies.
Yes, that seems very reasonable. But Faraday's Law of Induction clearly states that the circulation of the electric field around a curve equals minus the rate of change of the flux through a surface defined by that curve with time. The flux through my imaginary circle is changing. Where's the electric field?
6. Feb 25, 2010
### netheril96
You got it wrong.What Faraday's Law of Induction states is the electromotive force equals minus the rate of change of the flux through a surface
7. Feb 25, 2010
### LucasGB
So there's two Faraday's Laws? One for when the conductor moves and one for when the conductor stands still? Because I'm pretty sure the third Maxwell Equation is what I described.
8. Feb 25, 2010
### netheril96
9. Feb 25, 2010
### LucasGB
Check the attachment.
#### Attached Files:
• ###### Taken from Wikipedia.bmp
File size:
83.7 KB
Views:
93
10. Feb 25, 2010
### netheril96
Please type LaTex next time
The formula you give is Maxwell equations,not Faraday's law of induction
11. Feb 25, 2010
### LucasGB
I apologize, but I don't know how to use LaTeX very well. The formula I gave you is both Faraday's Law of Induction and one of Maxwell's Equations.
All 4 Maxwell's Equations have alternative names. Respectively: Gauss's Law for Electric Fields, Gauss's Law for Magnetic Fields, Faraday's law of Induction and Ampère's Circuital Law.
12. Feb 25, 2010
### bjacoby
It's obvious the IMAGINARY electric field has a circulation about your imaginary surface!
You are of course being cute, but the question in a large sense can be quite serious. We could word it this way: Are thoughts things? Since mathematics is abstract, we could ask "Is mathematics more real than reality?" The questions are real though physics at present is EXTREMELY reluctant to tackle them.
As a general introduction lets just note a couple of things about modern physics and these issues. First, one might generate a theory that the physical universe has more orthogonal dimensions than three or four. One could imagine that thoughts exist in parallel dimensions in other similar but orthogonal three dimensional spaces. Thought, for example, might exist in such a space that we might term, Oh, I don't know, maybe we could call it the "astral space" or "etheric space". Many physicists vehemently deny this possibility. And yet they then turn around and come up with string theory that proposes maybe 11 such dimensions. And then they turn around and deny it again strongly asserting that reality only has THREE dimensions but then come up with a cosmology that requires four dimensions so that the universe can be everywhere expanding!
Obviously there is a lot of confusion and conflicting thoughts here. And all that is made worse by a dogmatic attitude that rejects a variety of subjects such as extra dimensions or the reality of thought beyond mere brain functioning. Quite obviously physics is quite primitive at present and many people intend to keep it that way.
13. Feb 25, 2010
### netheril96
I use MathType to spare the pains of learning LaTex's grammar.
If I don't get it wrong,$$$\oint_{\partial S} {\vec E \cdot d\vec l = - \frac{{\partial \Psi }}{{\partial t}}}$$$is the Faraday's induction law
BUT you should notice that the E here is NOT electrostatic field strength.Instead,it is non-electrostatic strength like Lorentz force or vortex electric field strengh.And here,it is Lorentz force rather than vortex e-field
14. Feb 25, 2010
### LucasGB
Whoa. Look, the question is not metaphysical or philosophical. I'm really making a mistake in understanding the physics, and I'm asking people to help me see where the mistake is. But I liked what you wrote anyway.
Last edited: Feb 25, 2010
15. Feb 25, 2010
### LucasGB
Thank you for the MathType tip, I'll look into that.
Look, in that formula, E really isn't "electrostatic field strength". It represents the electric field vector, an induced electric field, as opposed to electrostatic. Nevertheless, it is an electric field. Therefore, if I imagine that the magnetic flux through the disk of my imaginary circle is changing because the circle is increasing, then the equation you just used says there must be an induced electric field circulating around the circle. That's what the E.dl integral means. Now, this can't be right, for I can't create an electric field by imagining things, and define its intensity by imagining how fast the imaginary circle changes size. So there's a mistake in my analysis. I don't know where it is.
16. Feb 25, 2010
### netheril96
You problem is your misunderstanding of what is an electromotive force
17. Feb 25, 2010
### LucasGB
Let's wait untill tomorrow and see what other people have to say about this. Thank you for your help, though.
18. Feb 26, 2010
### LucasGB
So, any ideas on this?
19. Feb 26, 2010
### Bob S
Faraday's Law for inducing an electric field E (NOT a current) in a loop of length L surrounding the area A of induction:
1) ∫E·dl = E·L = -(d/dt)∫B·n dA = -A·dB/dt
and
2) ∫E·dl = E·L = -(d/dt)∫B·n·dA = -B·dA/dt
The expanding virtual loop in the OP is case 2) with a constant magnetic field and increasing area A. This form of Faraday's Law applies to generators with rotating armatures in a constant stator field.
The constant area virtual loop with a changing enclosed magnetic field (case 1) represents the betatron particle accelerator in my earlier post #4. The induced azimuthal electric field is a closed (virtual) loop in vacuum, without wires, with or without free electrons.
Bob S
20. Feb 26, 2010
### LucasGB
I see. So if there's a constant magnetic field and no electric field, and I merely imagine a size-changing circle, then according to 2) there must be an induced electric field? Isn't this problematic? | 2017-08-16 17:58:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7276206016540527, "perplexity": 1111.9481081930016}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102309.55/warc/CC-MAIN-20170816170516-20170816190516-00121.warc.gz"} |
http://mathhelpforum.com/differential-geometry/186196-example-best-approximation-theorem.html | # Math Help - Example for best approximation theorem
1. ## Example for best approximation theorem
The best approximation theorem states that if (H,<,>) is an inner product space and M is a non-empty, complete, convex subset of H, then for every x in H there is a unique y in M such that d(x,M) = ||y-x||.
I don't know any good examples of inner product spaces (especially not complete IPS) with a complete convex subset. Any ideas?
Thanks!
2. ## Re: Example for best approximation theorem
An example of non complete inner product space is given by $H:=\left\{f\in\mathcal{C}^1{\left[0,1\right]}, f(0=f(1)=0\right\}$ with the inner product $\langle f,g\rangle :=\int_0^1 f(t)g(t)dt+\int_0^1 f'(t)g'(t)dt$.
An example of complete non empty convex subset is given by a finite dimensional subspace (it's works in each inner product space).
3. ## Re: Example for best approximation theorem
So in this case, any polynomial space, Pn, for n finite. | 2016-05-06 22:05:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5271926522254944, "perplexity": 714.5740066986973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461862134822.89/warc/CC-MAIN-20160428164854-00031-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=3789840 | # A problem from Artin's algebra textbook
by AbelAkil
Tags: abstract algebra
P: 9 1. The problem statement, all variables and given/known data (a)Let H and K be subgroups of a group G. Prove that the intersection of xH and yK which are cosets of H and K is either empty or else is a coset of the subgroup H intersect K (b) Prove that if H and K have finite index in G then the intersection of H and K also has finite index. 2. Relevant equations 3. The attempt at a solution The intersection of xH and yK is a subgroup of both H and K, then how to continue?
HW Helper
PF Gold
P: 2,655
Quote by AbelAkil 3. The attempt at a solution The intersection of xH and yK is a subgroup of both H and K, then how to continue?
This is not true in general. If xH and yK are not subgroups, then neither contains the identity, so their intersection also doesn't contain the identiy. So it can't be a subgroup.
Moreover, in general $xH \cap yK$ isn't even a subSET of H or K. xH and H are disjoint unless $x \in H$. Similarly for yK and K.
P: 9
Quote by jbunniii This is not true in general. If xH and yK are not subgroups, then neither contains the identity, so their intersection also doesn't contain the identiy. So it can't be a subgroup. Moreover, in general $xH \cap yK$ isn't even a subSET of H or K. xH and H are disjoint unless $x \in H$. Similarly for yK and K.
Sorry, I made some mistakes when I wrote the post. In fact, I mean the intersection of H and K is a subgroup of both H and K...Could U give me some tips to prove it?
HW Helper
PF Gold
P: 2,655
## A problem from Artin's algebra textbook
If xH and yK have nonempty intersection, then there is an element g contained in both: $g \in xH$ and $g \in yK$.
The cosets of $H \cap K$ form a partition of G, so g is contained in exactly one such coset, call it $a(H \cap K)$.
If you can show that $a(H \cap K)$ is contained in both $xH$ and $yK$ then you're done.
Hint: both $xH$ and $yK$ are partitioned by cosets of $H \cap K$.
P: 9
Quote by jbunniii If xH and yK have nonempty intersection, then there is an element g contained in both: $g \in xH$ and $g \in yK$. The cosets of $H \cap K$ form a partition of G, so g is contained in exactly one such coset, call it $a(H \cap K)$. If you can show that $a(H \cap K)$ is contained in both $xH$ and $yK$ then you're done. Hint: both $xH$ and $yK$ are partitioned by cosets of $H \cap K$.
Yeah...I get it. Thanks very much. In addition, how to prove part (b), that is how can I show that both $H$ and $K$ are partitioned by finite cosets of $H \cap K$... I appreciate your insightful answer!
Sci Advisor P: 906 the index of H in G is the number of cosets of H. if this number is finite, then if it just so happened that H∩K was of finite index in H, we get: [G:H][H:H∩K] cosets of H∩K in G in all, which would be finite. can you think of a way to show that [H:H∩K] ≤ [G:K]? perhaps you can think of an injection from left cosets of H∩K in H to left cosets of K in G?
Related Discussions Academic Guidance 3 Science & Math Textbooks 14 Linear & Abstract Algebra 15 Linear & Abstract Algebra 5 Calculus & Beyond Homework 2 | 2014-03-10 21:12:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8035949468612671, "perplexity": 370.6114011919336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011020120/warc/CC-MAIN-20140305091700-00067-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/27010/sum-of-digits-iterated | # Sum of digits iterated
Original version. I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of digits $n_2=S(n)=S_{10}(n)$ in the decimal system. If the newer number $n_2$ is greater than $10$, then compute the sum $n_3=S(n_2)$ of its digits, and continue this iteration $n_k=S(n_{k-1})$ unless you get a number $n^* =n_\infty$ in the range $1\le n^* \le 9$. Is $n^*$ uniformly distributed in the set $\lbrace 1,2,\dots,9\rbrace$? If this is not true in the decimal systems, what can be said in the other systems?
I just learned yesterday about the Feng shui system of determining what kind of problems/advantages one can get according to the house number, say $n$, of his/her home. This depends on the above $n^*$. I do not seriously count on the conclusions but I am curious whether $n^*$ is sufficiently democratic.
Edit. The question was immediately realized as obvious, because $n^*$ is the residue modulo $9$ (with 0 replaced by 9), and this works in any base as well. So the Feng shui function is really trivial, but one can deal with less trivial ones.
Let me fix $m$ and define $Q_m(n)$ as the sum of $m$th powers of decimal digits of a positive integer $n$. What can be said about the sequence of iterations $n_k=Q_m(n_{k-1})$ for a given integer $n_0$? How long can the (minimal) period be for a fixed $m$? And what can be said about the distribution of the purely periodic tails?
I hope that the question is still elementary.
• I've always wanted to respond to a question by saying this: I don't know. And it's true. – Will Jagy Jun 4 '10 at 4:56
The case $m=2$ appears in Hugo Steinhaus's "One Hundred Problems In Elementary Mathematics", problem 2(at least in Russian edition of 1986). Either sequence will come to 1 and stay here, or will enter to the cycle (145,42,20,4,16,37,58,89)
• Thanks, Nurdin! The last time I read the book (in Russian, of course) was at least 25 years ago, but I am back to it after your hint. The things are really surprising to me... – Wadim Zudilin Jun 4 '10 at 6:47
• If they end in 1, they are called "Happy Numbers". Check Wikipedia. Their distribution up to 10**7000 can be found here: <shaunspiller.com/happynumbers> – Stefan Gruenwald Jun 28 '14 at 7:00
A starting place might be http://www.oeis.org/A005188 which lists $n$-digit numbers $r$ with $Q_n(r)=r$, and has references to related oddities.
• Oh-oh, I did not expect that there was research in this direction... These are narcissistic numbers (en.wikipedia.org/wiki/Armstrong_number): In "A Mathematician's Apology", G. H. Hardy wrote: "There are just four numbes, after unity, which are the sums of the cubes of their digits: $$153 = 1^3 + 5^3 + 3^3, \quad 370 = 3^3 + 7^3 + 0^3, \quad 371 = 3^3 + 7^3 + 1^3, \quad 407 = 4^3 + 0^3 + 7^3.$$ These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician." I probably have to count myself an amateur. – Wadim Zudilin Jun 4 '10 at 6:45 | 2019-08-25 19:25:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7160472273826599, "perplexity": 380.96533440389334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330786.8/warc/CC-MAIN-20190825173827-20190825195827-00538.warc.gz"} |
https://mathsgee.com/35548/state-the-remainder-theorem-of-polynomials | 0 like 0 dislike
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State the remainder theorem of polynomials
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The remainder theorem gives us an easy way of finding the remainder when we divide a polynomial by a linear term.
Remainder Theorem: When a $(x)$ is divided by $x-k$ the remainder is $a(k)$.
by Platinum (119,114 points)
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0 like 0 dislike | 2022-05-24 09:05:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5949565768241882, "perplexity": 3997.5614371430975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662570051.62/warc/CC-MAIN-20220524075341-20220524105341-00002.warc.gz"} |
https://www.physicsforums.com/threads/laplace-transforms-involving-unit-step-and-ramp-functions.833380/ | # Laplace Transforms Involving: Unit-Step, and Ramp Functions
Tags:
1. Sep 19, 2015
### ConnorM
1. The problem statement, all variables and given/known data
Here is an imgur link to my assignment: http://imgur.com/N0l2Buk
I also uploaded it as a picture and attached it to this post.
2. Relevant equations
$u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases}$
3. The attempt at a solution
Question 1.1 -
$L[tu(t)] = \int_0^∞ tu(t)e^{-st} \,dt$
Using the definition of the step function, $t \geq 0, u(t) = 1$
*Is it right to assume that $c = 0$?*
$L[tu(t)] = \int_0^∞ t(1)e^{-st} \,dt$
$L[tu(t)] = \int_0^∞ te^{-st} \,dt$
$L[tu(t)] = 1/s^2$
I'm not sure if this is correct. Should it be solved using the rule, $L[tf(t)] = -F'(s)$
Question 1.2 -
Let $r_1 (t), r_2 (t)$ be the two ramp functions
Let $u_1 (t), u_2 (t)$ be the two unit-step functions
$r_1 (t) = \begin{cases} t & \text{if } 0 \leq t < 1 \end{cases}$
$r_2 (t) = \begin{cases} t+1 & \text{if } 1 \leq t < 2 \end{cases}$
$u_2 (t) = \begin{cases} 3 & \text{if } 2 \leq t < 4 \end{cases}$
I'm not quite sure what to do for the unit-step functions. Could someone help me figure out what they should be?
#### Attached Files:
• ###### Assignment1.JPG
File size:
37.8 KB
Views:
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Last edited: Sep 19, 2015
2. Sep 20, 2015
### Orodruin
Staff Emeritus
Your transformation is correct. For the second part, I do not think you have done what was intended. You are supposed to write the function as a linear combination, not as different expressions in different regions.
3. Sep 20, 2015
### ConnorM
So would it be something like,
Let $y(t)$ be the function on the graph that I am trying to recreate.
** Is this true, $r(t) = tu(t)$? **
So,
$y(t) = tu(t) - (t+1)u(t-1) - 3u(t-2)$
Although I still don't know what the second unit-step function should be?
Last edited: Sep 20, 2015
4. Sep 20, 2015
### Orodruin
Staff Emeritus
Have you tried plotting the function you gave? How does it differ from the one you should find?
5. Sep 20, 2015
### vela
Staff Emeritus
Try plotting the function g(t) = u(t-1)-u(t-2). You should see it's a pulse. Now consider what you'll get if you multiply g(t) by some function. Try plotting the function f(t) alone and the product f(t)g(t). For instance, try it with f(t) = t2. Do you understand the effect of multiplying by g(t) on the graph?
6. Sep 20, 2015
### ConnorM
I tried plotting it and from what I see it's not similar at all.
7. Sep 20, 2015
### ConnorM
So multiplying f(t) by g(t) gives me a point on that function?
8. Sep 20, 2015
### vela
Staff Emeritus
Which function? Which point? Can you elaborate?
9. Sep 20, 2015
### ConnorM
So for g(t) that you suggested it gave a pulse at (1,1), then when I plotted $f(t) = t^2$ I got a parabola. When I multiplied f(t) by g(t), I got a pulse again at (1,1).
10. Sep 20, 2015
### Orodruin
Staff Emeritus
I suggest that you think along the following lines:
• What happens when you pass the step of a step function? What changes? Where do these changes occur for the sought function?
• What happens when you pass the base of a function of the form $(t-a) u(t-a)$? Where do these changes occur for the sought function?
11. Sep 20, 2015
### vela
Staff Emeritus
I don't think you plotted g(t) correctly then. I'm not sure what you mean by a pulse at (1,1). I've attached a plot of what you should've gotten.
#### Attached Files:
• ###### plot.png
File size:
1.9 KB
Views:
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12. Sep 20, 2015
### ConnorM
So I did some work with graphing the equations and I think this is the right one,
$y(t) = tu(t) + u(t-1) - (t-2)u(t-2) - 3u(t-4)$
Let me know if this latex is working or not, I'm on my phone and can't see if it's showing up correctly!
13. Sep 20, 2015
### Orodruin
Staff Emeritus
Looks reasonable. | 2017-08-23 07:44:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6101802587509155, "perplexity": 1673.4948064566306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886117874.26/warc/CC-MAIN-20170823055231-20170823075231-00602.warc.gz"} |
https://meangreenmath.com/2017/02/11/my-favorite-one-liners-part-11/ | # My Favorite One-Liners: Part 11
In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.
Every once in a while, I’ll cover a theorem in class that looks utterly surprising to students at first glance. For example, in trigonometry, I might state that
$\sin^{-1} \left( \sin \pi \right) \ne \pi$,
so that the inverse function doesn’t quite behave like it’s supposed to (because of the restricted domain used to define inverse sine.)
Before explaining why $\sin^{-1} \left( \sin \pi \right)$ isn’t equal to $\pi$, I’ll get the discussion started by saying, “Don’t believe me? Just watch.”… a tip of the cap to this recent hit song (at the time of this writing, the third-most watched video on YouTube).
While on this topic, I have to tip my cap to Kelli Hauser, a sixth-grade teacher in my city who made the following motivational video for students about to take their end-of-year high-stakes test (called, here in Texas, the STAAR exam).
One more parody concerning a recent spacecraft that visited Pluto:
For further reading, here’s my series on inverse functions. | 2018-12-10 15:11:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7232275605201721, "perplexity": 2121.354546680369}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00047.warc.gz"} |
http://mathoverflow.net/questions/111528/is-every-symmetric-operator-on-the-schwartz-space-essentially-self-adjoint | # Is Every Symmetric Operator on the Schwartz Space Essentially Self-Adjoint?
More generally, suppose $S$ is a subspace of a Hilbert space $H$ that contains an orthonormal basis of $H$ (For example- the Schwartz space inside $L^2(\mathbb{R}^n)$). If $A:S \rightarrow S$ is symmetric, is $A$ necessarily essentially self-adjoint? That is, does $A$ have a unique self-adjoint extension?
This seems like it would be a standard theorem if it were true, and my inability to find such a statement on, say, Wikipedia, suggests that it is probably false.
This seems elementary- but my first attempts at a proof have not been successful. Maybe somebody knows of a good counterexample?
-
Note that by the Gram-Schmidt, every dense subspace $S$ contains an orthonormal basis. Thus here the operator $A$ is just symmetric, densely defined, with $A(S)\subset A$, and the standard counterexamples work (e.g. $i d/dx$ on the half-line). Or am I missing something? – Pietro Majer Nov 5 '12 at 7:50
One needs additional assumptions to obtain essential self-adjointness. In particular, it is sufficient that S contains a dense set of analytic vectors. The whole subject is explained very well in Vol.2 of "Methods of Modern Mathematical Physics" by Reed and Simon.
-
Thanks! I'll check it out. – Alex Zorn Nov 5 '12 at 15:44 | 2016-05-26 16:57:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9208368062973022, "perplexity": 334.5278989007285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276131.38/warc/CC-MAIN-20160524002116-00144-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://dmoj.ca/problem/ceoi19pp2 | ## CEOI '19 Practice P2 - Separator
View as PDF
Points: 10 (partial)
Time limit: 0.65s
Memory limit: 512M
Problem type
Allowed languages
Ada, Assembly, Awk, Brain****, C, C#, C++, COBOL, CommonLisp, D, Dart, F#, Forth, Fortran, Go, Groovy, Haskell, Intercal, Java, JS, Kotlin, Lisp, Lua, Nim, ObjC, OCaml, Octave, Pascal, Perl, PHP, Pike, Prolog, Python, Racket, Ruby, Rust, Scala, Scheme, Sed, Swift, TCL, Text, Turing, VB, Zig
Let be a sequence of distinct integers. An index is called a separator if the following two conditions hold:
• for all ,
• for all .
In other words, the array consists of three parts: all elements smaller then , then itself, and finally all elements greater than .
For instance, let . The separators are the indices and , corresponding to the values and .
The sequence is initially empty. You are given a sequence of elements to append to , one after another. After appending each , output the current number of separators in the sequence you have.
The input format is selected so that you have to compute the answers online. Instead of the elements you should append to , you are given a sequence .
Process the input as follows:
The empty sequence contains separators.
For each from to , inclusive:
1. Calculate the value .
2. Append to the sequence .
3. Calculate : the number of separators in the current sequence .
4. Output a line containing the value .
#### Input
The first line contains a single integer : the number of queries to process.
Then, lines follow. The -th of these lines contains the integer . The values are chosen in such a way that the values you'll compute will all be distinct.
#### Output
As described above, output lines with the values through .
#### Sample Input 1
7
30
9
20
50
79
58
89
#### Sample Output 1
1
0
0
1
2
1
2
#### Sample Input 2
10
0
0
0
0
0
0
0
0
0
0
#### Sample Output 2
1
2
3
4
5
6
7
8
9
10
#### Note
The first example equals is described in the problem statement.
The second example is decoded as . | 2020-11-01 00:29:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8929535746574402, "perplexity": 4768.683882457015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107922746.99/warc/CC-MAIN-20201101001251-20201101031251-00279.warc.gz"} |
https://dsp.stackexchange.com/tags/pitch/hot | Tag Info
Hot answers tagged pitch
12
At the core of MIDI is a representation of music as discrete note events, each of those having a static pitch. This is perfect for representing music as played on keyboard instruments. You can convert any frequency corresponding to a note on the tempered scale into a MIDI note number, using: $69 + 12 \times \log_2 \frac{frequency}{440}$ Under the ...
10
I've never seen the word "Formula" with "AMDF". My understanding of the definition of AMDF is $$Q_x[k,n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \Big| x[n+n_0] - x[n+n_0+k] \Big|$$ $n_0$ is the neighborhood of interest in $x[n]$. Note that you are summing up only non-negative terms. So $Q_x[k,n_0] \ge 0$. We call "$k$" the "lag". clearly if ...
9
I've tried to get the bin with greatest magnitude but that only give me right results for higher pitch signals, it doesn't matter which oversampling factor I use I still get bad data for low freq signals. That's because the harmonics are larger than the fundamental. Plot your spectrum and you'll see. A better method to find the true fundamental is ...
9
If you really insist on using FFT (rather than parametric methods, which wouldn't suffer from time/frequency trade-offs), you can fake a much better resolution by using the phase information to recover the instantaneous frequency for each FFT bin. Partials can then be detected by looking for plateaus in the function giving instantaneous frequency as a ...
9
You are right that the repetition is around 650 by how exactly do I compute that automatically? Seems like a peak-picking problem to me? Or is there some other methods that can be used? Yes, it's just peak-picking. Your period is the x value of the first strong peak: Your peaks are all similar in height, probably because you're doing the autocorrelation ...
9
"Is there a way to measure frequency (detect pitch) better than FFT, that is, with better resolution in less acquisition time?" yes there is. or are. there are multiple better ways to do musical pitch detection in real time that are far, far better than running an FFT. consider : Average Magnitude Difference Function (AMDF) $$Q_x[k] = \sum_n |x[n] - ... 9 This is what we call in the pitch-detection biz, the "octave problem". First of all, I would change the AMDF to ASDF. And I would not reduce the window size as the lag increases. (Also, I am changing notation to what I consider to be more conventional. "x[n]" is a discrete-time signal.) The Average Squared Difference Function (ASDF) of x[n] in the ... 8 From the (limited) description the uHz rotator algorithm sounds like one of the phase-weighted averages from this site, but it's not an algorithm I am familiar with. The Cramér–Rao lower bound^1 for estimating the frequency of sinusoid with amplitude A in white noise with variance \sigma^2 is given by:$$ \mathrm{var}(\hat{f}) \ge \frac{12}{(2\pi)^2\...
7
I don't think pitch information is relevant for what you want to do. The variation of pitch during speech is known as intonation, and can convey emotions, indicate if a sentence is a question etc. However, there is no universal rule as to how pitch variation patterns are mapped to meaning - this is quite language dependent ; and some languages sound "...
7
I'll take an orthogonal tack to answering this question from what Peter K has (validly) already proposed. I assert that the 8-significant-figure claim is little more than marketing-speak; while the software may be able to provide you an estimate with that many digits on it, that doesn't mean that they carry any real information! It appears that the software ...
7
From the ones I've been using I can recommend: YAAFE - very pleasant to work with in Python ESSENTIA - another one I like particularly due to Python integration aubio FEAPI Aquila - friend of mine used it extensively and he likes it a lot Recently I came across this paper and I believe that this should perfectly answer your question. Moffat D. et al - ...
7
Cons: Not as accurate This is just compared to the other methods. I was measuring frequency very accurately to look for clock drift, etc: 1000.000004 Hz for 1000 Hz, for instance. For guitar pitch detection it will be fine. doesn't work for inharmonic things like musical instruments I should have said "it can't find an accurate fundamental if there is ...
6
To answer the "how to shift the frequency of an audio signal up" bit: You could multiply the signal by a sine wave at a high frequency. This would shift and mirror the whole spectrum of the original signal into the high frequencies (multiplication by a sine in the time domain = convolution by a pair of symmetric Dirac in the frequency domain) - the mirror ...
6
Similar to this thread: Is there an algorithm for finding a frequency without DFT or FFT? FFT isn't a particular efficient way of building a tuner. Better (and cheaper) methods include auto-correlation, phased locked loops and delay locked loops, etc.. One example is to use tracking of local maxima and minima to roughly hone in on the fundamental ...
6
One would expect such a sequence to have a spectrum consisting of lines, as it is almost periodic (if it was periodic, it would have a Fourier series representation, even though it is not sinusoidal). As a quick example: load raw1.mat % calculate "unbiased" normalized cross-correlation; adjusted for % regions where there isn't full overlap corr = xcorr(...
6
Yes, using a peak frequency estimator for pitch is wrong. Pitch is a psychoacoustic phenomena, so pitch detection or estimation is different from frequency estimation. There have been plenty of pitch estimation methods given in previous answers to similar questions here. There's more than 1 to choose from. Here's one: https://stackoverflow.com/questions/...
6
MIDI is a protocol that allows (primarily) synthesizers to control or be controlled by other synthesizers or computers. It's a serial protocol that allows to exchange messages such as "key C1 up" "key D4 down" "key velocity, "sound change", etc. Many controllers have a "pitch wheel" that's a joystick or am modulation wheel. These allow the player to ...
6
What you are describing is very similar the the Harmonic Product Spectrum method of pitch estimation, as listed in this Stanford CCRMA paper. An FFT does not give you an "infinite sum of amplitudes", but a finite number of result bins depending on the length of the FFT. 5 mS is only 1 period of a 200 Hz note, and only a fraction of a period below 200 Hz. ...
6
This question (about "time-scaling" audio) is closely related to pitch shifting, which is time-scaling combined with resampling. But changing the speed without changing pitch is only time-scaling, so there is no resampling involved (contrary to what thomas has suggested). There are frequency-domain methods (phase-vocoder and sinusoidal modeling) that can ...
5
I have tried the following: Launch Audacity. Generate a 15000 Hz tone in the track created by default. Add a new track. Generate a 15400 Hz tone in the new track. A lower frequency tone appears during playback. The reason is that both tracks have high levels, so their sum exceeds 1.0; and Audacity applies clipping or limiting. This non-linear operation is ...
5
The two concepts are related to two different dimensions or aspects of music which might or might not be correlated. Onset detection is concerned with finding the points in time at which sounds start. Doing this does not require prior knowledge of the particular pitch (or fundamental frequency) of the sound. It may indeed rely on the property that at the ...
5
When I play A3 (220Hz) in my guitar, the fifth string which is A2 (110Hz) also vibrates a bit: it is what is called Sympathetic resonance. Besides other non-linear effects, this could be the case.
4
A frequency histogram is often used as part of the explanation of the harmonic product spectrum method of pitch estimation. A histogram that is a composite of several STFT frames over time may contain more of the harmonics of a note whose spectrum evolves over time (with various overtones appearing and/or disappearing, including even the fundamental), thus ...
4
It could have been Goertzel's algorithm, though it looks at a single frequency rather than a specific band. Another approach is to apply modulation techniques to shift the central frequency of your range of interest into the baseband, aka "zoom FFT". Your intuition about the max to average ratio is good. Another "peakedness" metric is the ratio of ...
4
Consider trying an upsampled or interpolated ASDF, AMDF, autocorrelation or other similar periodicity estimation algorithm. There in an information theoretic time versus frequency resolution versus noise trade-off. At a sample rate of 44100, estimating 440 Hz +-2 Hz might require somewhere in the range of 2 to 6 times 44100/440 samples (to determine the ...
4
Audiophiles don't hear anything unless they are told what to expect. Joke aside, this is equivalent to transposition by a bit less than a semitone. The effect won't be noticed on voices - this is way too low for any kind "chipmunk" effect to be observed. As for music, some people very familiar with the original music might detect the change of tonality and/...
4
Check out chapter 1.3 of this IRCAM paper on multi-F0 estimation. It discusses the difficulties in extracting multiple F0s from a recording, including the handling of overlapping partials, transients, and reverberation, as well as the modeling of domain-specific sources with varied spectral properties.
4
OK I did the two some time ago, TDHS in principle just apply time scale modification and to change the pitch do you need apply interpolation (resample) and it will shift the spectral envelope. For TDHS is hard to find some paper that teach how its really works, I learned the math and how it works in the Burazerovic Dzevdet paper: $N_p$ is defined as the ...
4
They may be referring to patent US3800088 by Harald Bode. I have a bunch of images from 15 years ago to explain a way to do frequency shifting so that it has the best chance of sounding good. I would call it single-sideband frequency shifting. Here the range of possible frequencies in the signal are drawn as two arrows each spanning half the circumference ...
4
Because guitar strings have non-zero stiffness, non-zero diameter, and non-zero displacement, the physical vibration frequency of the string's overtones can be slightly sharp. Thus inharmonic. But this slight inharmonicity might be part of what makes certain real stringed instruments sound more "interesting" than some simplistic additive and waveform ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2020-02-25 03:14:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5753233432769775, "perplexity": 1301.7739467340302}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146004.9/warc/CC-MAIN-20200225014941-20200225044941-00198.warc.gz"} |
https://developer.aliyun.com/article/398220 | # LSM Tree 学习笔记——本质是将随机的写放在内存里形成有序的小memtable，然后定期合并成大的table flush到磁盘
The Sorted String Table (SSTable) is one of the most popular outputs for storing, processing, and exchanging datasets.
An SSTable is a simple abstraction to efficiently store large numbers of key-value pairs while optimizing for high throughput, sequential read/write workloads.
Unfortunately, the SSTable name itself has also been overloaded by the industry to refer to services that go well beyond just the sorted table, which has only added unnecessary confusion to what is a very simple and a useful data structure on its own. Let's take a closer look under the hood of an SSTable and how LevelDB makes use of it.
### SSTable: Sorted String Table
A "Sorted String Table" then is exactly what it sounds like, it is a file which contains a set of arbitrary, sorted key-value pairs inside
Duplicate keys are fine, there is no need for "padding" for keys or values, and keys and values are arbitrary blobs. Read in the entire file sequentially and you have a sorted index. Optionally, if the file is very large, we can also prepend, or create a standalone key:offset index for fast access.
That's all an SSTable is: very simple, but also a very useful way to exchange large, sorted data segments.
+ 订阅 | 2021-10-24 22:59:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19336502254009247, "perplexity": 2047.3923539286843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587606.8/warc/CC-MAIN-20211024204628-20211024234628-00139.warc.gz"} |
https://labs.tib.eu/arxiv/?author=Cyril%20Imbert | • ### The weak Harnack inequality for the Boltzmann equation without cut-off(1608.07571)
Feb. 21, 2019 math.AP
In this paper, we obtain the weak Harnack inequality and H\"older estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cut-off can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local H\"older estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cut-off.
• ### Decay estimates for large velocities in the Boltzmann equation without cut-off(1804.06135)
April 17, 2018 math.AP
We establish pointwise large velocity decay rates for the solution of the inhomogeneous Boltzmann equation without cutoff, under the assumption that three hydrodynamic quantities (mass, energy, entropy densities) stay under control.
• ### A toy nonlinear model in kinetic theory(1801.07891)
Jan. 24, 2018 math.AP
This note is concerned with the study of a toy nonlinear model in kinetic theory. It consists in a non-linear kinetic Fokker-Planck equation whose diffusion in the velocity variable is proportional to the mass of the solution and steady states are Maxwellian. Solutions are constructed by combining energy estimates, well-designed hypoelliptic Schauder estimates, and the hypoelliptic extension of the De Giorgi-Nash H{\"o}lder estimates obtained recently by Golse, Vasseur and the two authors (2017).
• ### Effective junction conditions for degenerate parabolic equations(1601.01862)
Sept. 19, 2017 math.AP
We are interested in the study of parabolic equations on a multi-dimensional junction, i.e. the union of a finite number of copies of a half-hyperplane of dimension d + 1 whose boundaries are identified. The common boundary is referred to as the junction hyperplane. The parabolic equations on the half-hyperplanes are in non-divergence form, fully non-linear and possibly degenerate, and they do degenerate and are quasi-convex along the junction hyperplane. More precisely, along the junction hyperplane the non-linearities do not depend on second order derivatives and their sublevel sets with respect to the gradient variable are convex. The parabolic equations are supplemented with a non-linear boundary condition of Neumann type, referred to as a generalized junction condition, which is compatible with the maximum principle. Our main result asserts that imposing a generalized junction condition in a weak sense reduces to imposing an effective one in a strong sense. This result extends the one obtained by Imbert and Monneau for Hamilton-Jacobi equations on networks and multi-dimensional junctions. We give two applications of this result. On the one hand, we give the first complete answer to an open question about these equations: we prove in the two-domain case that the vanishing viscosity limit associated with quasi-convex Hamilton-Jacobi equations coincides with the maximal Ishii solution identified by Barles, Briani and Chasseigne (2012). On the other hand, we give a short and simple PDE proof of a large deviation result of Bou{\'e}, Dupuis and Ellis (2000).
• ### Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case(1410.3056)
Aug. 25, 2017 math.AP
A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic pe-nalization function |x -- y| 2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate. Mathematical Subject Classification: 35F21, 49L25, 35B51.
• ### Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks(1306.2428)
July 31, 2017 math.AP
We study Hamilton-Jacobi equations on networks in the case where Hamiltonians are quasi-convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general vertex condition is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free parameter, the flux limiter. Second, a general method for proving comparison principles is introduced. This method consists in constructing a vertex test function to be used in the doubling variable approach. With such a theory and such a method in hand, we present various applications, among which a very general existence and uniqueness result for quasi-convex Hamilton-Jacobi equations on networks.
• ### H\"older gradient estimates for a class of singular or degenerate parabolic equations(1609.01123)
Sept. 5, 2016 math.AP
We prove interior H\"older estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$u_t=|\nabla u|^{\kappa}\mbox{div} (|\nabla u|^{p-2}\nabla u),$$ where $p\in (1,\infty)$ and $\kappa\in (1-p,\infty).$ This includes the from $L^\infty$ to $C^{1,\alpha}$ regularity for parabolic $p$-Laplacian equations in both divergence form with $\kappa=0$, and non-divergence form with $\kappa=2-p$. This work is a continuation of a paper by the last two authors \cite{JS}.
• ### Estimates on elliptic equations that hold only where the gradient is large(1306.2429)
May 9, 2016 math.AP
We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the H{\"o}lder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
• ### Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation(1604.07377)
April 25, 2016 math.AP
We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in \cite{ISV}.
• ### Self-similar solutions for a fractional thin film equation governing hydraulic fractures(1403.7491)
Jan. 3, 2016 math.AP
In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient $K\geq 0$. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness ($K=0$) and the finite toughness $K\in(0,\infty)$ cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all $K\textgreater{}0$ there exists an injection rate for the fluid such that self-similar solutions exist.
• ### A junction condition by specified homogenization and application to traffic lights(1406.5283)
Dec. 22, 2015 math.AP
Given a coercive Hamiltonian which is quasi-convex with respect to the gradient variable and periodic with respect to time and space at least "far away from the origin", we consider the solution of the Cauchy problem of the corresponding Hamilton-Jacobi equation posed on the real line. Compact perturbations of coercive periodic quasi-convex Hamiltonians enter into this framework for example. We prove that the rescaled solution converges towards the solution of the expected effective Hamilton-Jacobi equation, but whose "flux" at the origin is "limited" in a sense made precise by the authors in \cite{im}. In other words, the homogenization of such a Hamilton-Jacobi equation yields to supplement the expected homogenized Hamilton-Jacobi equation with a junction condition at the single discontinuous point of the effective Hamiltonian. We also illustrate possible applications of such a result by deriving, for a traffic flow problem, the effective flux limiter generated by the presence of a finite number of traffic lights on an ideal road. We also provide meaningful qualitative properties of the effective limiter.
• ### H\"older continuity of solutions to hypoelliptic equations with bounded measurable coefficients(1505.04608)
June 19, 2015 math.AP
We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are H\"older continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and regularity transfers developed in kinetic theory. The latter tool is used repeatedly: first in the proof of the local gain of integrability of sub-solutions; second in proving that the gradient with respect to the velocity variable is $L^{2+\epsilon}_{\mathrm{loc}}$; third, in the proof of an "hypoelliptic isoperimetric De Giorgi lemma." To get such a lemma, we develop a new method which combines the classical isoperimetric inequality on the diffusive variable with the structure of the integral curves of the first-order part of the operator. It also uses that the gradient of solutions w.r.t. $v$ is $L^{2+\epsilon}_{\mathrm{loc}}$.
• ### Finite speed of propagation for a non-local porous medium equation(1411.4752)
June 12, 2015 math.AP
This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and V\'azquez (2010).
• ### Global Well-Posedness Of A Non-Local Burgers Equation: The Periodic Case(1506.02240)
This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$u_t - u |\nabla| u + |\nabla|(u^2) = 0.$$ We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.
• ### Nonlocal porous medium equation: Barenblatt profiles and other weak solutions(1302.7219)
Feb. 28, 2013 math.AP
A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles --- the well-known solutions of the classical porous medium equation.
• ### Large Time Behavior of Periodic Viscosity Solutions for Uniformly Elliptic Integro-Differential Equations(1210.5691)
Oct. 21, 2012 math.AP
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of "mixed operators" for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.
• ### Electrified thin films: Global existence of non-negative solutions(1102.0949)
Feb. 6, 2012 math.AP
We consider an equation modeling the evolution of a viscous liquid thin film wetting a horizontal solid substrate destabilized by an electric field normal to the substrate. The effects of the electric field are modeled by a lower order non-local term. We introduce the good functional analysis framework to study this equation on a bounded domain and prove the existence of weak solutions defined globally in time for general initial data (with finite energy).
• ### $C^{1,\alpha}$ regularity of solutions of degenerate fully non-linear elliptic equations(1201.3739)
Jan. 18, 2012 math.AP
In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\"older estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii-Lions method in order to get $C^{1,\alpha}$ estimates for solutions of these equations.
• ### Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations(1107.3228)
Jan. 6, 2012 math.AP
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.
• ### A Hamilton-Jacobi approach to junction problems and application to traffic flows(1107.3250)
Nov. 28, 2011 math.AP
This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.
• ### Barenblatt profiles for a nonlocal porous media equation(1001.0910)
June 6, 2011 math.AP
We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the $L^p$ decay of weak solutions of the corresponding Cauchy problem.
• ### Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations(0903.1699)
July 30, 2010 math.AP
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.
• ### Existence of solutions for a higher order non-local equation appearing in crack dynamics(1001.5105)
May 6, 2010 math.AP
In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulty associated to higher order equations (e.g. lack of maximum principle). However, there are important differences with, for instance, the thin film equation: First, our equation is nonlocal; Also the natural energy estimate is not as good as in the case of the thin film equation, and does not yields, for instance, boundedness and continuity of the solutions (our case is critical in dimension $1$ in that respect).
• ### Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations(0911.0240)
Nov. 2, 2009 math.AP
The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space.
• ### Phasefield theory for fractional diffusion-reaction equations and applications(0907.5524)
July 31, 2009 math.AP
This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both the novelty and the challenging difficulty of this work. | 2019-12-08 12:43:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.80064857006073, "perplexity": 418.17318735954325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540510336.29/warc/CC-MAIN-20191208122818-20191208150818-00097.warc.gz"} |
https://demo7.dspace.org/items/eefeb3fc-4bf5-4b8d-a2f1-c51b1c3fdc03 | ## Chaoticity Parameter Lambda in Hanbury-Brown Twiss Interferometry
Wong, Cheuk-Yin
Zhang, Wei-Ning
##### Description
In Hanbury-Brown-Twiss interferometry measurements using identical bosons, the chaoticity parameter lambda has been introduced phenomenologically to represent the momentum correlation function at zero relative momentum. It is useful to study an exactly solvable problem in which the lambda parameter and its dependence on the coherence properties of the boson system can be worked out in great detail. We are therefore motivated to study the state of a gas of noninteracting identical bosons at various temperatures held together in a harmonic oscillator potential that arises either externally or from bosons' own mean fields. We determine the degree of Bose-Einstein condensation and its momentum correlation function as a function of the attributes of the boson environment. The parameter lambda can then be evaluated from the momentum correlation function. We find that the lambda(p,T) parameter is a sensitive function of both the average pair momentum p and the temperature T, and the occurrence of lambda=1 is not a consistent measure of the absence of a coherent condensate fraction. In particular, for large values of p, the lambda parameter attains the value of unity even for significantly coherent systems with large condensate fractions. We find that if a pion system maintains a static equilibrium within its mean field, and if it contains a root-mean-squared radius, a pion number, and a temperature typical of those in high-energy heavy-ion collisions, then it will contain a large fraction of the Bose-Einstein pion condensate.
Comment: 23 pages in Latex, 13 figures
##### Keywords
High Energy Physics - Phenomenology, Nuclear Theory, Quantum Physics | 2022-11-26 08:29:10 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.857154905796051, "perplexity": 784.9741726044759}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446706285.92/warc/CC-MAIN-20221126080725-20221126110725-00788.warc.gz"} |
https://www.physicsforums.com/threads/all-my-calculus-troubles.79767/ | # All my Calculus Troubles
#### Lucretius
Well, seeing as I will be working on calculus for a long time, and I figure at the rate I've been going I'll get stuck a lot and need help, I'll just make one thread for all my calculus troubles.
I just read the chapter on Tangent Lines and Slopes. I did a problem, double-checked the math, then turned to the back of the book and found out the answer was completely incorrect. I retraced again, and found no problem, but the book is a bit confusing so I might have missed something.
The problem reads: Find a formula that gives the slope at any point P (x,y) on the given curve. $y=4-x^2$ $P:(1,-1)$
My work is as follows:
1)I begin by making a secant line from point P to a point Q, which I arbitrarily place. The coordinates are $Q:(-1+\Delta x, 4-(-1+\Delta x)^2$
2)To find my slope: $$\frac{\Delta y}{\Delta x} = \frac{4-(-1+\Delta x)^2}{\Delta x}$$
3)This becomes: $$\frac{4-1+2\Delta x-\Delta x^2}{\Delta x}$$
4)Finally, I get $5-\Delta x=m$ Getting rid of the $\Delta x$ my slope is $m=5$
The back of the book says the answer is -2x.
What happened?
Related Introductory Physics Homework News on Phys.org
#### amcavoy
I'm new here, but I might be able to help.
$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$?
Try using this formula on your original equation $$f(x)=4-x^2$$ and that will give you the tangent line at any point.
#### Lucretius
The section on limits and derivatives comes right after this one.
#### VietDao29
Homework Helper
Hmmm, I don't understand what you are doing here...
P(1, -1) is not on the curve.
Let $P(\varepsilon, 4 - \varepsilon ^ 2)$be a point on a curve.
Let $Q(\varepsilon + \Delta x, 4 - (\varepsilon + \Delta x)^ 2)$be another point on a curve.
You have as $\Delta x \rightarrow 0 => Q \rightarrow P$
Can you find $$\frac{Q_y - P_y}{Q_x - P_x}$$ as $$\Delta x \rightarrow 0$$?
Er, I think limits should be learn before tangent line.
Viet Dao,
Last edited:
#### Lucretius
Should I skip past this chapter for now, learn limits, and then return to it?
You mentioned P(1,-1) is not on the curve — and you're right. It's (-1,3). The math I did uses (-1,3) though, I just mistyped it here.
Can you make that itexed slope formulation tex-ed? It's too small, can't read it.
Should I substitute 0 for $$\Delta x$$?
Last edited:
Lucretius said:
Well, seeing as I will be working on calculus for a long time, and I figure at the rate I've been going I'll get stuck a lot and need help, I'll just make one thread for all my calculus troubles.
I just read the chapter on Tangent Lines and Slopes. I did a problem, double-checked the math, then turned to the back of the book and found out the answer was completely incorrect. I retraced again, and found no problem, but the book is a bit confusing so I might have missed something.
The problem reads: Find a formula that gives the slope at any point P (x,y) on the given curve. $y=4-x^2$ $P:(1,-1)$
My work is as follows:
1)I begin by making a secant line from point P to a point Q, which I arbitrarily place. The coordinates are $Q:(-1+\Delta x, 4-(-1+\Delta x)^2$
2)To find my slope: $$\frac{\Delta y}{\Delta x} = \frac{4-(-1+\Delta x)^2}{\Delta x}$$
3)This becomes: $$\frac{4-1+2\Delta x-\Delta x^2}{\Delta x}$$
4)Finally, I get $5-\Delta x=m$ Getting rid of the $\Delta x$ my slope is $m=5$
The back of the book says the answer is -2x.
What happened?
i dont know how far along you are, but all you need to do is find the derivative of the given function:
$$y=4-x^2$$ the derivative of this is -2x.
the derivative of a function is the slope of the tangent line at a certain point.
for the point (1,1) the equation of the tangent line is :
$$y-1=-2(x-1)$$ :surprised
#### Curious3141
Homework Helper
I think the OP is just beginning with calculus and is not allowed to "just differentiate".
So, going back to first principles, the slope of a tangent to a curve at a point is defined by $$\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$$
We have $$y = 4 - x^2$$. ---eqn(1)
Then
$$y + \Delta y = 4 - (x + \Delta x)^2 = 4 - x^2 - 2x\Delta x - (\Delta x)^2$$ ---eqn(2)
using the binomial expansion.
Take eqn(2) - eqn(1),
we have $$\Delta y = -2x\Delta x - (\Delta x)^2$$
Divide that by $$\Delta x$$,
$$\frac{\Delta y}{\Delta x} = -2x - \Delta x$$
And taking the limit as $$\Delta x$$ goes to zero, we have :
$$\frac{dy}{dx} = -2x$$
as we would expect. This is the equation that gives the slope of the tangent to the curve at any point (x,y)
If you want the equation of a line that has that slope and that passes through the point (1, -1), the equation of that line is simply,
$$y - (-1) = -2X(x - 1)$$
or $$y = -(2X)x + 2X - 1$$
Note that that line will NOT actually be a tangent to any point on the curve, it will only be parallel to the tangent to the curve at the chosen point $(X,Y)$.
If you want the equation of the actual tangent line to the curve at a point $(X,Y)$, then it would be given by
$$y - Y = -2X(x - X)$$
$$y - (4 - X^2) = -2X(x - X)$$
which rearranges to
$$y = -(2X)x + X^2 + 4$$
Last edited:
#### Lucretius
It seems as if you are all suggesting that I use derivatives and/or limits. I believe I will skip this chapter, learn how to do limits and derivatives, and then return to it at a later time. Wish me luck with derivatives and limits. I will need it.
#### geosonel
choose an arbitrary x
then the point P = (x, 4 - x^2) is on the curve
now move to another x a short distance Δx away: x + Δx
then the point Q = (x + Δx, 4 - (x + Δx)^2 ) is also on the curve
the slope of the line connecting P and Q is:
$$\mbox{slope \overline{PQ} } \ = \ \frac {\mbox{ (y value of Q) } \ - \ \mbox{ (y value of P) } } {\mbox{ (x value of Q) } \ - \ \mbox{ (x value of P) } } \ = \ \displaystyle \frac { ( 4 - (x + \Delta x)^2) \ - \ ( 4 - x^2 ) } { (x + \Delta x) \ - \ (x)} \ = \ \displaystyle \frac { - 2x\Delta x \ - \ (\Delta x)^2 } { \Delta x} \ = \ -2x \, - \, \Delta x$$
now what happens to Δx when point P approaches point Q?
if you answered that Δx → 0, then what is the slope of the tangent to the curve?
Last edited:
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• Solo and co-op problem solving | 2019-07-22 14:05:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8262148499488831, "perplexity": 534.6863342963553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195528037.92/warc/CC-MAIN-20190722133851-20190722155851-00093.warc.gz"} |
https://www.rdocumentation.org/packages/spatstat/versions/1.55-1/topics/volume | # volume
0th
Percentile
##### Volume of an Object
Computes the volume of a spatial object such as a three-dimensional box.
Keywords
spatial, math
##### Usage
volume(x)
##### Arguments
x
An object whose volume will be computed.
##### Details
This function computes the volume of an object such as a three-dimensional box.
The function volume is generic, with methods for the classes "box3" (three-dimensional boxes) and "boxx" (multi-dimensional boxes).
There is also a method for the class "owin" (two-dimensional windows), which is identical to area.owin, and a method for the class "linnet" of linear networks, which returns the length of the network.
##### Value
The numerical value of the volume of the object.
area.owin, volume.box3, volume.boxx, volume.linnet | 2019-09-23 07:33:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23383639752864838, "perplexity": 4234.544511060037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514576122.89/warc/CC-MAIN-20190923064347-20190923090347-00072.warc.gz"} |
https://mathematica.stackexchange.com/questions/249790/fokker-planck-equation-in-mathematica-giving-weird-solution | # Fokker Planck Equation in Mathematica giving weird solution
I have the following equation $$\frac{\partial\,P(\theta,t)}{\partial t} = \alpha\cos\theta\,P(\theta,t) + \beta \frac{\partial^2\,P(\theta,t)}{\partial \theta^2}$$ subject to following conditions$$P(\theta, 0)=\delta(\theta-\pi/4), P(\theta,t)=P(\theta+2\pi,t)~.$$
I tried solving this numerically in Mathematica:
alpha = 0.02
beta = 0.02
fokkerPlanck = {D[p[x, t], t]==alpha*Cos[x]*p[x,t] + beta*D[p[x,t],{x,2}], p[x,0] == DiracDelta[Pi/4], p[2Pi, t] == p[0, t]};
sol = Flatten@NDSolve[fokkerPlanck, p, {x, 0, 2Pi},{t, 0, 100}]
If I run this, I get null solution, so something is weird. Could anyone let me know what is potentially wrong here?
• From the first glance, you need to fix also the interval in x and the corresponding boundary conditions. Am I wrong? Jun 17, 2021 at 18:48
• @AlexeiBoulbitch You are right that I need to add interval in x. I was doing that in my code, but while typing it here, I missed it. This is now fixed. The boundary conditions are already included in fokkerPlanck though. Jun 17, 2021 at 18:54
• There is an example Fokker-Panck equation in the reference. Maybe that's useful. See also this Jun 18, 2021 at 5:29
• @titanium Initial condition is not periodic, while boundary condition is periodic. What kind of solution do you expect? Jun 18, 2021 at 11:09
It seems the problem is the the DiracDelta boundary condition.
If I replace that with a highly peaked Gaussian, p[x, 0] == Exp[-(x - \[Pi]/4)^2/(2*0.001)^2], I get a finite solution.
• Even if I do so, plot of p[x, 100] looks really weird; its amplitude is way too small. Jun 17, 2021 at 22:18
• Well it's a diffusion equation, the width is supposed to get larger. Also your equation does not have a $P$ on the $\alpha$ term, but your Mathematica code does? Jun 17, 2021 at 22:26
• Thanks for catching that. I corrected the problem. My point was after running the code for long enough, the distribution of p[x,100] should be a Gaussian distribution which integrates to 1 in the range of 0, 2Pi. This is because p is the probability density. At the moment I cannot recover this. Jun 17, 2021 at 22:29
• @titanium Gaussian distribution defined on $-\infty <x<\infty$. It is not periodic function on $0\le x\le 2 \pi$ Jun 18, 2021 at 11:59
First step is come out from singularity at $$t=0$$. For this we can use analytical solution in the form
sol1 =
DSolve[{D[p[x, t], t] ==
1/50*Cos[x]*p[x, t] + 1/50*D[p[x, t], {x, 2}],
p[x, 0] == DiracDelta[(x - Pi/4)]}, p[x, t], {x, t}]
Out[]= {{p[x, t] ->
ConditionalExpression[(
5 E^(-((25 (\[Pi] - 4 x)^2)/(32 t)) + 1/50 t Cos[x]))/(
Sqrt[2 \[Pi]] Sqrt[t]), Re[t] > 0]}}
This solution is not periodic on $$0\le x \le 2 \pi$$, but we can make it periodic for $$t>t_0$$ using FEM as follows
Needs["NDSolveFEM"]
alpha = 0.02;
beta = 0.02; reg = Rectangle[{0, 1/2}, {2 Pi, 100}]; mesh =
ToElementMesh[reg,
MeshRefinementFunction ->
Function[{vertices, area},
area > 0.001 (0.05 + 3 Norm[Mean[vertices]])]]
fokkerPlanck = {D[p[x, t], t] ==
alpha*Cos[x]*p[x, t] + beta*D[p[x, t], {x, 2}],
DirichletCondition[
p[x, t] == (
5 E^(-((25 (\[Pi] - 4 x)^2)/(32 t)) + 1/50 t Cos[x]))/(
Sqrt[2 \[Pi]] Sqrt[t]), t == 1/2 && 10^-2 <= x <= 2 Pi - 10^-2],
PeriodicBoundaryCondition[p[x, t], x == 2 \[Pi],
Function[x, x - 2 \[Pi]]]};
sol = NDSolve[fokkerPlanck, p, Element[{x, t}, mesh]]
There is a message about stability of solution
NDSolve::femcscd: The PDE is convection dominated and the result may not be stable. Adding artificial diffusion may help.
Visualization
• If I execute the DSolve part for obtaining analytical solution, it returns me the code I entered, not the analytical result. Any idea on why that might be the case? I cleared all the cache as well, but that did not help. Jun 18, 2021 at 16:50
• @titanium What version do you run? Jun 18, 2021 at 21:09
• It is version 12.0.0.0. Jun 18, 2021 at 22:13
• @titanium Ok! My \$Version is 12.3.0 for Microsoft Windows (64-bit) (May 10, 2021). But DSolve has been updated in 12. Could you show code you execute? Jun 19, 2021 at 11:23 | 2022-08-15 12:56:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.493513286113739, "perplexity": 2844.509089073497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572174.8/warc/CC-MAIN-20220815115129-20220815145129-00408.warc.gz"} |
https://mathspace.co/textbooks/syllabuses/Syllabus-1073/topics/Topic-20737/subtopics/Subtopic-269714/?activeTab=theory | # 8.03 Convert between fractions, decimals and percentages
Lesson
## Ideas
Let's review how to write fractions as percentages .
### Examples
#### Example 1
Express the fraction \dfrac{5}{25} as a percentage.
Worked Solution
Create a strategy
Find the equivalent fraction that has a denominator of 100.
Apply the idea
To get a denominator of 100, we need to multiply 25 by 4 since 25\times 4=100. But if we multiply the denominator by 4 we must also multiply the numerator by 4.
Idea summary
We can convert any fraction into a percentage by finding its equivalent fraction that has a denominator of 100. After this, we can write the value in the numerator followed by the \% symbol to represent the percentage.
## Convert decimals, fractions and percentages
This video looks at strategies to complete tables of equivalent values for fractions, decimals and percentages.
### Examples
#### Example 2
Convert between percentages, fractions and decimals to complete the table:
Worked Solution
Create a strategy
To convert fractions and decimals to percentages multiply by 100\%. To convert between fractions and decimals use a place value table.
Apply the idea
First we can convert the fraction to a percentage:
The fraction \dfrac{60}{100} means 60 hundredths. So we can write it in a place value table to convert to a decimal:
We can simplify this decimal by writing it as 0.6.
Idea summary
We can convert fractions and decimals to percentages by multiplying by 100\%.
## Percentages and complements
This video looks at how 1 whole is 100\%, and how we can use that knowledge to find complements, that add together to make the whole.
### Examples
#### Example 3
What is the complement of 40\%?
Worked Solution
Create a strategy
Subtract 40\% from 100\%.
Apply the idea
To find the number that makes 100\% when added to 40\%, we can just subtract 40\% from 100\%.
Idea summary
1 whole is 100\%.
If you have a percentage such as 25\%, the complement to that is 75\% because it makes up the whole.
### Outcomes
#### MA3-7NA
compares, orders and calculates with fractions, decimals and percentages | 2023-03-21 16:34:17 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8728500604629517, "perplexity": 3008.897416491024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943704.21/warc/CC-MAIN-20230321162614-20230321192614-00051.warc.gz"} |
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Root Symbol Alt Codes, HTML Code (Copy and Paste) Here is the list of alt codes for root symbols. These include the YouTube logo which can be entered by typing :yt: into any live chat.. Emotes on YouTube do not work in YouTube captions or comments, and these will convert to the text string when copied outside of live stream chats. You don't need to copy one by one. fourth root ∜ U+221C copy and paste. Yes all the math symbols are universal and there is no variation in them in any part of the world. First select the symbol then you can drag&drop or just copy&paste it anywhere you like. How to Use Symbols. Just Copy-Paste. The nth Root Symbol . You can also learn how to insert these root symbols in word and how to insert them in your phone. Best website for symbols copy and paste. For example, if you enter: \sqrt. After generating your fancy text symbols, you can copy and paste the "fonts" to most websites and text processors. Click on Math Symbols to copy it to the clipboard and paste to use on Instagram, TikTok, Facebook, Twitter, your emails, blog, etc. Copy and Paste Arabic Symbol Arabic symbol is a copy and paste text symbol that can be used in any desktop, web, or mobile applications. if you convert text into some cool fancy style using symbols to put it into Instagram bio, stories, Facebook profile, or just post on some forum. Please also also check out our font keyboard to help users easily get fonts right at the phone keyboard at … Copy & Paste the character from below: √ Enjoy! Click on Number Symbols to copy it to the clipboard and paste to use on Instagram, TikTok, Facebook, Twitter, your … If you see a square block in place of a character, then that means that that letter isn't supported on your device or perhaps your web browser. A copy and paste line symbols collection for easy access. Using it. y = nthroot(x,n) Description. example. I noticed people were trying to copy text characters on click. Kreuz symbol (aus dem Lateinischen "Knackpunkt", ein römisches Foltereinrichtung zur Kreuzigung verwendet) ist eine geometrische Figur, die aus zwei Linien oder Balken senkrecht zueinander und teilt ein oder zwei der Linien in der Mitte. microsoft word only goes up to 4th root, and when i use the nth root function on it and copy and paste it, it displays text instead of symbol. The Number Symbols is a pictogram Unicode character or emojis. ツ We have made a collection from the ones we found on the internet. so like a little 5 on the square root sign. Toggle navigation Unicode® Symbol. Every positive real number x has a single positive nth root, called the principal nth root, which is written .For n equal to 2 this is called the principal square root and the n is omitted. I am using both \mathscr and \mathcal to different meanings and expecting different characters to appear.. command-line language to enter special characters and symbols. Your text : Keyboard Layout : Arithmetic & Algebra Superscript & Subscript Fractions Statistics Measurements Calculus Greek symbols Letters symbols Logic & Theory Geometry Equivalence & Proportion Operators Other symbols Uncheck all - Check all. Instead of going through menus or trying to remember shortcuts, you can also just copy and paste the symbol below onto your spreadsheet, and then format and use it as needed. You can use it on Facebook or in Youtube comments, for example! This code point first appeared in version 1.1 of the Unicode® Standard and belongs to the "Mathematical Operators" block which goes from 0x2200 to 0x22FF. Best Popular Hashtag to use with #symbol are #brochure #logolearn #logosai #logotype #behance #iconset #visualdesign #logogrid #glyph #identity. Once you are done, you can save to file or copy [Ctrl]+[c] & paste [Ctrl]+[v] it to other documents or to your email. This is the special symbol that means "nth root", it is the "radical" symbol (used for square roots) with a little n to mean nth root. In this example I set it up to be base/n, but this was just a guess. Copy and paste text symbol letters to use with any browser or desktop and mobile application. Click icon to copy to clipboard Recently Used. Just click on a line symbol to copy it to the clipboard and paste it anywhere. Open Live Script. Are Math Symbols Universal. Letting Φgønes be Φgønes. Copy all common root symbols for use in emails, texts, letters, web pages. We could use the nth root in a question like this: Question: What is "n" in this equation? Copy and paste. 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Directly copy the HTML code ( copy and paste ) Here is the creation of images from,. | 2021-06-18 06:40:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3308608829975128, "perplexity": 4410.910283980899}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00277.warc.gz"} |
https://hal.archives-ouvertes.fr/hal-00697593 | # Entropy formulation of degenerate parabolic equation with zero-flux boundary condition
Abstract : We consider the general degenerate hyperbolic-parabolic equation: $$\label{E}\tag{E} u_t+\div f(u)-\Delta\phi(u)=0 \mbox{ in } Q = (0,T)\times\Omega,\;\;\;\; T>0,\;\;\;\Omega\subset\mathbb R^N ;$$ with initial condition and the zero flux boundary condition. Here $\phi$ is a continuous non decreasing function. Following [B\"{u}rger, Frid and Karlsen, J. Math. Anal. Appl, 2007], we assume that $f$ is compactly supported (this is the case in several applications) and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension $N\geq 1$ under a partial genuine nonlinearity assumption on $f$. Uniqueness is shown for the case $N=1$, using the idea of [Andreianov and Bouhsiss, J. Evol. Equ., 2004], nonlinear semigroup theory and a specific regularity result for one dimension.
Keywords :
Cited literature [23 references]
https://hal.archives-ouvertes.fr/hal-00697593
Contributor : Mohamed Karimou Gazibo <>
Submitted on : Thursday, October 4, 2012 - 9:29:34 PM
Last modification on : Thursday, December 26, 2019 - 12:00:08 PM
Document(s) archivé(s) le : Saturday, January 5, 2013 - 4:00:23 AM
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### Citation
Boris Andreianov, Mohamed Karimou Gazibo. Entropy formulation of degenerate parabolic equation with zero-flux boundary condition. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Wiley-VCH Verlag, 2013, 164 (5), pp. 1471-1491. ⟨10.1007/s00033-012-0297-6⟩. ⟨hal-00697593v2⟩
Record views | 2020-06-05 13:22:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.6585897207260132, "perplexity": 2465.3154592476817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348500712.83/warc/CC-MAIN-20200605111910-20200605141910-00504.warc.gz"} |
https://quant.stackexchange.com/tags/risk-neutral-measure/new | # Tag Info
3
There is a deeper relationship between the two risk-neutral measures. Take any event in the binomial model with a finite number of steps and calculate the risk-neutral probability of it. Take the same event in the Black Scholes model and calculate the risk-neutral probability of it. For most events, the two probabilities are different. Now let the number of ...
1
$q$ is not delta hedge. $q$ is determined from the fact that $S_i$ is a martingale i.e. for $S_0$ $S_0=E(S_1)=quS_0+(1-q)dS_0$ (if no rates) This equation gives the same $q$ , dependent only on $u$ and $d$ , if calculated for $S_0$ , $S_1$ etc , thus $q$ is the same for all steps.
2
Risk-neutral pricing is to help with relative value type questions: If I know the value of this what should the value of that be if it depends in some way on this. It doesn't help with absolute value type questions: Should I buy this or that, is the implied volatility too low or high etc. Those are generally "real world measure" type questions.
3
When you look at actual data from the stock market, the probability distribution that you have in mind and that would describe the likelihood of different scenarios occuring going forward is what we call the "physical" probability distribution. Intuitively the risk-neutral probability distribution is a "distorted" version of the physical probability ...
4
Let's stick to a discrete market for simplicity. So, you have a finite number of states in this type of model. The first fundamental theorem of asset pricing says that the absence of arbitrage in such markets imply the existence of (not necessarily unique) risk-neutral measure and vice-versa. The reason it works in the second direction (the existence of a ...
Top 50 recent answers are included | 2020-05-28 22:41:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8675721287727356, "perplexity": 285.22312083072745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347400101.39/warc/CC-MAIN-20200528201823-20200528231823-00021.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-6th-edition/chapter-11-cumulative-review-page-670/27d | Intermediate Algebra (6th Edition)
$x=6$
Let $2^{\log_2 6}=x$. Taking the logarithm base $2$ of both sides, then, \begin{array}{l} \log_2 2^{\log_2 6}=\log_2x \\ (\log_2 6)(\log_2 2)=\log_2x \\ (\log_2 6)(1)=\log_2x \\ \log_2 6=\log_2x .\end{array} Since the bases on both sides of the equal sign are the same, then the logarithm base $2$ can be dropped, resulting to $x=6 .$ | 2018-10-21 13:16:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 1845.1138466371508}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514005.65/warc/CC-MAIN-20181021115035-20181021140535-00369.warc.gz"} |
https://en.academic.ru/dic.nsf/enwiki/26324 | Phase angle
Phase angle
In the context of vectors and phasors, the term phase angle refers to the angular component of the polar coordinate representation. The notation $Aang ! heta,$ for a vector with magnitude (or "amplitude") "A" and phase angle θ, is called angle notation.
In the context of periodic phenomena, such as a wave, phase angle is synonymous with phase.
ee also
*Phasor (sine waves)
*Phase (waves)
* [http://www.sengpielaudio.com/calculator-timedelayphase.htm Calculation of phase angle (phase difference) from time delay (time of arrival ITD) and frequency - Connection between phase, phase angle, frequency, and time of arrival (delay)]
* [http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ACCircuit/PhaseAngle.html Phase angle]
Wikimedia Foundation. 2010.
Look at other dictionaries:
• Phase angle — (Elec.) The angle expressing phase relation. [Webster 1913 Suppl.] … The Collaborative International Dictionary of English
• phase angle — fazinis kampas statusas T sritis automatika atitikmenys: angl. phase angle vok. Phasenwinkel, m rus. фазовый угол, m pranc. angle de phase, m … Automatikos terminų žodynas
• phase angle — fazinis kampas statusas T sritis fizika atitikmenys: angl. phase angle vok. Phasenwinkel, m rus. фазовый угол, m pranc. angle de phase, m … Fizikos terminų žodynas
• phase angle — noun a particular point in the time of a cycle; measured from some arbitrary zero and expressed as an angle • Syn: ↑phase • Derivationally related forms: ↑phase (for: ↑phase) • Hypernyms: ↑point, ↑ … Useful english dictionary
• phase angle — noun 1》 Physics a phase difference expressed as an angle, 360 degrees corresponding to one complete cycle. 2》 Astronomy the angle between the lines joining a given planet to the sun and to the earth … English new terms dictionary
• phase angle — i. An angle between the sight lines to the sun and the earth measured at a remote location (e.g., other celestial bodies). ii. The number of electrical degrees between the time an AC (alternating current) voltage passes through zero moving in the … Aviation dictionary
• phase angle — the angle in which waves come to a body … Mechanics glossary
• phase angle — Physics. See under phase (def. 8). [1885 90] * * * … Universalium
• Phase angle (astronomy) — Phase angle in astronomical observations is the angle between the light incident onto an observed object and the light reflected from the object. In the context of astronomical observations, this is usually the angle Sun object observer. For… … Wikipedia
• Phase angle (disambiguation) — Phase angle may refer to one of the following.*Phase angle vectors, phasors, and periodic phenomena *Phase angle (astronomy) the angle between the incident light and reflected light … Wikipedia | 2020-07-16 05:02:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7405246496200562, "perplexity": 8783.867587944467}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657181335.85/warc/CC-MAIN-20200716021527-20200716051527-00467.warc.gz"} |
https://stats.stackexchange.com/questions/56481/how-to-prove-statistical-significance-between-the-4-seasons?noredirect=1 | # How to prove statistical significance between the 4 seasons
I have data regarding the number of deaths in a city over 35 years. I've collated the data, and now wish to prove the difference in total number of deaths between seasons is significant. Spring has 631 deaths, summer 540 deaths, autumn 502 deaths and winter has 605 deaths. Clearly there is a big difference between spring and autumn, but how do I prove that? Is it ANOVA? Or maybe t-test? I'm totally lost... Cheers
• Are the seasons the same length? i.e. same number of days. Anyway, I would create an expectation based on length of the seasons then use a chi-squared test. – waferthin Apr 18 '13 at 15:20
• Yes the seasons are all three months long (with a little variation for months eg Feb has only 28 days, making winter slightly shorter). Ok cheers! – Danny Cairns Apr 18 '13 at 15:36
You can use a Chi-Squared test to test for significant variation between seasons, under the assumption that we expect equal counts in each season. e.g. in R:
d = c(631,540,502,605)
chisq.test(matrix(c(d, rep(mean(d),4)), ncol=2))
Result:
Pearson's Chi-squared test
data: matrix(c(d, rep(mean(d), 4)), ncol = 2)
X-squared = 9.2602, df = 3, p-value = 0.02602
i.e. there is significant variation between seasons.
An analysis of variance will show that there is a significant difference somewhere. As an assist, it might be useful to plot the average number of deaths by season for the 35 years. Comparative boxplots with whiskers and notches would place the median deaths (with 95% confidence intervals) side-by-side for visual comparison. This is easily done in R. Such a plot also shows the distribution by season (answers the question about normality), and provides much more information than just significance.
You may want to do like:
1. Generate a dummy variable for each season: lets call these as s1, s2, s3, and s4.
2. Regress dependent variable y on dummy variables s2, s3, s4 plus other variables if any [Note we omit one (you can choose any) dummy variable to avoid dummy variable trap].
3. Test the significance (using t-test)on the coefficients of s2, s3 , and s4. If at least one of these is significant, it indicates the seasonal effect.
4. Since your data is time series you may want to use Newey-West standard error which is robust to autocorrelation and heteroscedasticity.
Two indicator variables contrasting spring and autumn and summer and winter respectively might be used to capture the cyclical character of seasons. I've not checked the equations, but I think you'd find that equivalent to fitting sine and cosine terms for time of year. Then it's a regression with whatever other predictors make sense. That is, don't lump the seasons; allow the data to show variability between years as well as whatever makes sense.
Periodic, trigonometric, Fourier regression are some of the names used for this. Typically statistical people forget all the trigonometry they learned when young, which comes in handy for thinking about seasonality.
CORRECTION. One variable should run winter 1, spring 0, summer -1, autumn 0 and the other winter 0, spring 1, summer 0, autumn -1. These sine and cosine terms are thus not 0, 1 indicators. | 2020-01-28 19:38:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4737619459629059, "perplexity": 1404.6768816137587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251783000.84/warc/CC-MAIN-20200128184745-20200128214745-00097.warc.gz"} |
https://proofwiki.org/wiki/Extended_Transitivity | # Extended Transitivity
## Theorem
Let $S$ be a set.
Let $\mathcal R$ be a transitive relation on $S$.
Let $\mathcal R^=$ be the reflexive closure of $\mathcal R$.
Let $a, b, c \in S$.
Then:
$(1):\quad$ $\displaystyle \paren{ a \mathrel {\RR} b } \land \paren { b \mathrel {\RR} c }$ $\implies$ $\displaystyle a \mathrel{\RR} c$ $(2):\quad$ $\displaystyle \paren { a \mathrel{\RR} b } \land \paren { b \mathrel{\RR^=} c }$ $\implies$ $\displaystyle a \mathrel{\RR} c$ $(3):\quad$ $\displaystyle \paren { a \mathrel{\RR^=} b } \land \paren { b \mathrel{\RR} c }$ $\implies$ $\displaystyle a \mathrel{\RR} c$ $(4):\quad$ $\displaystyle \paren { a \mathrel{\RR^=} b } \land \paren { b \mathrel{\RR^=} c }$ $\implies$ $\displaystyle a \mathrel{\RR^=} c$
## Proof
$(1)$ follows from the definition of a transitive relation.
$(4)$ follows from Reflexive Closure of Transitive Relation is Transitive.
Suppose that:
$\paren { a \mathrel{\RR} b } \land \paren { b \mathrel{\RR^=} c }$
By the definition of reflexive closure:
$b \mathrel{\RR} c$ or $b = c$
If $b = c$, then since $a \mathrel{\mathcal R} b$:
$a \mathrel{\RR} c$
If $b \mathrel{\RR} c$ then by transitivity of $\RR$:
$a \mathrel{\RR} c$
Thus $(2)$ holds.
A similar argument proves that $(3)$ holds as well:
Suppose that:
$\paren { a \mathrel{\RR^=} b } \land \paren { b \mathrel{\RR} c }$
By the definition of reflexive closure:
$a \mathrel{\RR} b$ or $a = b$
If $a = b$, then since $b \mathrel{\RR} c$:
$a \mathrel{\RR} c$
If $a \mathrel{\RR} b$ then by transitivity of $\RR$:
$a \mathrel{\RR} c$
Thus $(3)$ holds.
$\blacksquare$ | 2020-06-05 19:05:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.994422972202301, "perplexity": 225.13838389054274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348502204.93/warc/CC-MAIN-20200605174158-20200605204158-00282.warc.gz"} |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=DHJGII_2015_v64n11_1592 | Fiber-optic Temperature Sensor Using a Silicone Oil and an OTDR
Title & Authors
Fiber-optic Temperature Sensor Using a Silicone Oil and an OTDR
Jang, Jae Seok; Yoo, Wook Jae; Shin, Sang Hun; Lee, Dong Eun; Kim, Mingeon; Kim, Hye Jin; Song, Young Beom; Jang, Kyoung Won; Cho, Seunghyun; Lee, Bongsoo;
Abstract
In this study, we developed a fiber-optic temperature sensor (FOTS) based on a silicone oil and an optical time domain reflectometer (OTDR) to apply the measurement of a coolant leakage in the nuclear power plant. The sensing probe of the FOTS consists of a silicone oil, a stainless steel cap, a FC terminator, and a single mode optical fiber. Fresnel reflection arising at the interface between the silicone oil and the single mode optical fiber in the sensing probe is changed by varying the refractive index of the silicone oil according to the temperature. Therefore, we measured the optical power of the light signals reflected from the sensing probe. The measurable temperature range of the FOTS using a Cu-coated silica fiber is from $\small{70^{\circ}C}$ to $\small{340^{\circ}C}$ and the maximum operation temperature of the FOTS is sufficient for usage at the secondary system in the nuclear power plant.
Keywords
Fiber-optic temperature sensor;Silicone oil;OTDR;Fresnel reflection;Coolant leakage;
Language
Korean
Cited by
1.
Silicon Oil-Based 2-Channel Fiber-Optic Temperature Sensor Using a Subtraction Method, Journal of Sensor Science and Technology, 2016, 25, 5, 344
References
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J. S. Kim, J. H. Kim, H. Y. Bae, C. Y. Oh, Y. Kim, K. S. Lee, and T. K. Song, "Welding Residual Stress Distributions for Dissimilar Metal Nozzle Butt Welds in Pressurized Water Reactors", The Transactions of the Korean Society for Energy Engineering, Vol. 36, No. 2, pp. 137-148, 2012.
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S. H. Shim, J. S. Song, K. B. Yoon, K. M. Hwang, T. E. Jin, and S. H. Lee, "A Study on Managing of Metal Loss by Flow-Accelerated Corrosion in the Secondary Piping of CANDU Nuclear Plants", The Transactions of the Korean Society of Mechanical Engineers, Vol. 11, No. 1, pp. 18-25, 2002.
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J. S. Song, H. Kim, and S. Lee, "A Study on Radioactive Source-term Assessment Method for Decommissioning PWR Primary System", The Transactions of the Korean Radioactive Waste Society, Vol. 12, No. 2, pp. 153-164, 2014.
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S. B. Shimanskii, B. P. Strelkov A. N. Anan'ev, A. M. Lyubishkin, T. Iijima, H. Mochizuki, Y. Kasai, K. Yokota, and J. Kanazawa, "Acoustic method of leak detection using high-temperature microphones", Atom. Energy+, Vol. 98, No. 2, pp. 89-96, 2005.
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D. S. Kupperman, T. N. Claytor, and R. Groenwald, "Acoustic leak detection for reactor coolant systems", Nuclear Engineering and Design, Vol. 86, No. 1, pp. 13-20, 1985.
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J. R. Zhao, X. G. Huang, W. X. He, and J. H. Chen, "High-resolution and temperature-insensitive fiber optic refractive index sensor based on Fresnel reflection modulated by Fabry-Perot interference", J. Lightwave Technol., Vol. 28, No. 19, pp. 2799-2803, 2010.
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C. L. Zhao, J. H. Li, S. Q. Zhang, Z. X. Zhang, and S. Z. Jin. "A simple Fresnel reflection-based optical fiber sensor for multipoint refractive index measurement using an AWG", IEEE Photonic. Tech. L., Vol. 25, No. 6, pp. 606-608, 2013.
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J. H. Chen and X. G. Huang, "Fresnel-reflection-based fiber sensor for on-line measurement of ambient temperature", Opt. Commun., Vol. 283, No. 9, 2010.
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H. I. Sim, W. J. Yoo, S. H. Shin, J. S. Jang, J. S. Kim, K. W. Jang, S. Cho, J. H. Moon, and B. Lee, "Real-time measurements of water level and temperature using fiber-optic sensors based on an OTDR", The Transactions of the Korean Institute of Electrical Engineers, Vol. 63, No. 9, pp. 1239-1244, 2014.
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W. J. Yoo, H. I. Sim, S. H. Shin, K. W. Jang, S. Cho, J. H. Moon, and B. Lee, "A fiber-optic sensor using an aqueous solution of sodium chloride to measure temperature and water level simultaneously", Sensors, Vol. 14, No. 10, pp. 18823-18836, 2014.
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J. Chen, and X. Huang, "Fresnel-reflection-based fiber sensor for on-line measurement of ambient temperature", Opt. Commun., Vol. 283, No. 9, pp. 1647-1677, 2010.
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K. R. Sohn, "Liquid sensor using refractive intensity at the end-face of a glass fiber connected to fiber-Bragg grating", Sensor. Actuat. A-Phys., Vol. 158, No. 2, pp. 193-197, 2010.
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H. Su, and X. G. Huang, "Fresnel-reflection-based fiber sensor for on-line measurement of solute concentration in solutions", Sens. Actuators B Chem., Vol. 126, No. 2, pp. 579-582, 2007.
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L. Yuan, L. Zhou, and W. Jin, "Long-gauge length embedded fiber optic ultrasonic sensor for large-scale concrete structures", Opt. Laser Technol., Vol. 36, No. 1, pp. 11-17, 2004.
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J. Yuan, C. Zhao, M. Ye, J. Kang, Z. Zhang, and S. Jin, "A Fresnel reflection-based optical fiber sensor system for remote refractive index measurement using an OTDR", Photonic sensors, Vol. 36, No. 10, pp. 1869-1874, 1995. | 2018-05-21 11:15:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 2, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.271392285823822, "perplexity": 8718.037866505818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864063.9/warc/CC-MAIN-20180521102758-20180521122758-00195.warc.gz"} |
https://angularfixing.com/using-jest-with-angular-13-clarity/ | # Using Jest with Angular 13 + Clarity
## Issue
i am currently attempting to upgrade an existing Angular app which uses VMware Clarity.
I already managed to upgrade from 8.x to 10.x following the Angular update guidelines.
However beyond that the jest configuration breaks, as the newer Clarity versions and Angular 13 use esm.
So i tried to build a minimal working example to investigate the needed configurations.
Starting from the jest-preset-angular example app at https://github.com/thymikee/jest-preset-angular/tree/main/examples/example-app-v13 , i added Clarity as described in https://github.com/vmware-clarity/ng-clarity/blob/main/docs/INSTALLATION.md#installing-clarity-angular-
The example app’s test and test-esm run configurations in package.json work without problems.
But as soon as i add ClarityModule to the app.module.ts imports and run the test-esmconfiguration, the test suites for app.component.spec.ts and app.component.router.spec.ts fail with the same error:
ReferenceError: You are trying to import a file after the Jest environment has been torn down.
From src/app/app.component.spec.ts.
at async Promise.all (index 0)
FAIL src/app/app.component.spec.ts
● Test suite failed to run
ENOENT: no such file or directory,
open 'C:\Users\NAME\IdeaProjects\example-app-13_test\node_modules\@cds\core\index.jsicon\shapes\times.js'
at async Promise.all (index 3)
The error happens immediately after adding ClarityModule to the app.module.ts without adding any Clarity elements to the example app’s html.
To me the ENOENT part seems rather weird, as it looks like two paths for legitimate Clarity .js-files were intersected.
I tried various different combinations of adding jest.useFakeTimers, transformIgnorePatterns and other advice i found for similar problems, but these either did nothing or led to more errors. Since i am also quite inexperienced with configuring jest, I might also have used them wrong.
Could you please give me advice what might fix the above error?
The environment i am running this in is:
Angular CLI: 13.3.6
Node: 16.14.0
Package Manager: npm 8.7.0
OS: win32 x64
The versions of the Clarity packages are:
"@cds/core": "^6.0.0",
"@clr/angular": "^13.3.1",
"@clr/ui": "^13.3.1",
## Solution
After some more debugging I found out, that Jest seems to have trouble resolving ClarityIcons while processing node_modules\@cds\core\internal-components\close-button\register.js.
So i tried adding various mappings to parts of @cds/core to the moduleNameMapper in jest-esm.config.mjs and finally succeeded in running the tests of the jest-preset-angular example app without errors. After eliminating the paths one by one it seems that adding the following mapping is sufficient for the tests to work:
"@cds/core/icon": '<rootDir>/node_modules/@cds/core/icon/index.js' | 2022-09-25 17:24:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3160015344619751, "perplexity": 7347.176529187328}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334591.19/warc/CC-MAIN-20220925162915-20220925192915-00703.warc.gz"} |
https://crypto.stackexchange.com/questions/75786/total-anonymity-against-malicious-agents | # Total anonymity against malicious agents
Suppose we have $$N$$ parties, two of whom are Alice and Bob, while some fraction of the remainder are malicious agents working together. We form a network where each party can pass an encrypted message to another indirectly by routing it through other parties. Alice and Bob know a key secretly, so they can communicate privately because the other parties who look at the message just see noise and pass it on.
But now suppose the malicious agents are really malicious; they can sneak into Bob's home and learn the key, and they also can spy on the entire network to track every movement. Now it becomes easy to unmask Alice. The malicious agents check every message they get with the key, and they'll eventually notice that they always see messages meant for Bob originating with Alice.
So my question is, assuming the agents can acquire as much info through Bob and the network at large as desired, is there a way to modify this protocol such that Alice can't be unmasked? I.e., that the agents will have no way (or a computationally-infeasible way) of guessing Alice is a more likely candidate than any of the other parties? The protocol can be somewhat unreliable if necessary. Obviously it will require some second layer of encryption, but I don't know what to look for.
Dining cryptographer networks are information theoretically secure to the anonymity set size. This is ultimately a traffic analysis question; however, it's very closely related to, and somewhat hard to delineate from, cryptography. In practice a DC-net is, in numerous ways, a lot like a one time pad. Primarily useful in theory and some niche cases. Anonymity is very much dependent on having a large anonymity set size, and DC-nets aren't very scalable.
Constant rate cover traffic is something else for you to read about.
Not just the semantics to be concerned about in this area. Malicious party will delay packets and create a watermark in interpacket arrival timings, then do end to end correlation analysis looking for that actively created pattern. This rapidly explodes into the massive complexity of mix theory. Pynchon gate is the state of the art in mix theory; it uses PIR for message retrieval from nymservers.
Traffic analysis still is largely successful even if all traffic is encrypted and impossible to decrypt. The goal is maintaining -- across all of myriad dimensions -- invariance from the anonymity set. Anonymity is at least typically severely degraded by any variation from the set, other than if there is variation in the form of randomization per hop (i.e. layers of encryption being removed at each hop still has variance of ciphertexts across the anonymity set, but the variance is made statistically independent from itself at each hop by the adding or removing of a layer of encryption).
Interpacket arrival timings are far from the only thing. Need the packets all padded to the same size, or else the packet sizing disparity across streams of packets creates a variance from the anonymity set that can be followed throughout the network.
Traffic analysis is extremely meta. It isn't about the communications, it's about the signals carrying them.
Check out freehaven.net bibliography.
Almost all experts say that Tor is the best bet for anonymity, and that this will not stop being the case any time soon. There is a big disconnection between theory and practice in this area. Plethora mix network designs, and things like DC-nets, are technically much stronger and -- arguably, perhaps anyway -- more sophisticated than Tor is. It's just they have such significant constraints on them, and oftentimes such poor scalability, that in the real world Tor blows them away even though it has major variance in terms of things like stream byte counts for example.
• Thanks. I figured I could beat typical traffic analysis by having every party transmit ~3 junk messages periodically. If a party wanted to transmit a real message, he just replaces one of the junk messages, which would still look like noise. The trade-off is a slower arrival of messages. Dining cryptography seems to be what I was looking for. I found this: dedis.cs.yale.edu/dissent which seems to answer my question – HiddenBabel Nov 16 '19 at 17:27
• "Dummy messages" are what they are called in mix theory, if you want to search for more about that technique. Long term intersection attacks are among the most damning. Say that a global passive adversary knows that messages sent will arrive at their destination within 24 hours. So, every time Bob receives a message, they take a snapshot of everyone who sent a message within the past 24 hours, and assume his interlocutor among them. Over time they look for the commonality between such snapshots, which are individually called crowds. This is done simply by intersecting them together. – Gratis Nov 16 '19 at 19:33
• If you are only concerned with receiver anonymity you can use various Private Information Retrieval algorithms. These are somewhat analogous to radio broadcasts, which can be anonymously received. The most naive example, which is also the only information theoretically secure one to use a single server, is to send all messages sent to the server to all of the clients and then have clients filter out what they don't want. Obviously scales very poorly, but it is quite analogous to a radio broadcast otherwise. It provides no anonymity to senders; anonymity to the anonymity set size for receivers. – Gratis Nov 16 '19 at 19:41 | 2020-02-16 21:28:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5049275755882263, "perplexity": 1118.947354507704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875141430.58/warc/CC-MAIN-20200216211424-20200217001424-00449.warc.gz"} |
http://psychology.wikia.com/wiki/Electrical_resistance?direction=prev&oldid=52160 | # Electrical resistance
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Electrical resistance is a measure of the degree to which an object opposes the passage of an electric current. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.
The quantity of resistance in an electric circuit determines the amount of current flowing in the circuit for any given voltage applied to the circuit.
$R = \frac{\Delta V}{I}$
where
R is the resistance of the object, usually measured in ohms, equivalent to J·s/C2
ΔV is the potential difference across the object, usually measured in volts
I is the current passing through the object, usually measured in amperes
For a wide variety of materials and conditions, the electrical resistance does not depend on the amount of current flowing or the amount of applied voltage. V can either be measured directly across the object or calculated from a subtraction of voltages relative to a reference point. The former method is simpler for a single object and is likely to be more accurate. There may also be problems with the latter method if the voltage supply is AC and the two measurements from the reference point are not in phase with each other.
## Resistive loss
When a current, I, flows through an object with resistance, R, electrical energy is converted to heat at a rate (power) equal to
$P = {I^{2} \cdot R} \,$
where
P is the power measured in watts
I is the current measured in amperes
R is the resistance measured in ohms
This effect is useful in some applications such as incandescent lighting and electric heating, but is undesirable in power transmission. Common ways to combat resistive loss include using thicker wire and higher voltages. Superconducting wire is used in special applications.
## Resistance of a conductor
### DC resistance
As long as the current density is totally uniform in the conductor, the DC resistance R of a conductor of regular cross section can be computed as
$R = {l \cdot \rho \over A} \,$
where
l is the length of the conductor, measured in meters
A is the cross-sectional area, measured in square meters
ρ (Greek: rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm · meter. Resistivity is a measure of the material's ability to oppose the flow of electric current.
For practical reasons, almost any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductors such as wires.
### AC resistance
If a wire conducts high-frequency alternating current then the effective cross sectional area of the wire is reduced. This is because of the skin effect.
This formula applies to isolated conductors. In a conductor close to others, the actual resistance is higher because of the proximity effect.
## Causes of resistance
### In metals
A metal consists of a lattice of atoms, each with a shell of electrons. This can also be known as a positive ionic lattice. The outer electrons are free to dissociate from their parent atoms and travel through the lattice, creating a 'sea' of electrons, making the metal a conductor. When an electrical potential difference (a voltage) is applied across the metal, the electrons drift from one end of the conductor to the other under the influence of the electric field.
In a metal the thermal motion of ions is the primary source of scattering of electrons (due to destructive interference of free electron wave on non-correlating potentials of ions) - thus the prime cause of metal resistance. Imperfections of lattice also contribute into resistance, although their contribution in pure metals is negligible.
The larger the cross-sectional area of the conductor, the more electrons are available to carry the current, so the lower the resistance. The longer the conductor, the more scattering events occur in each electron's path through the material, so the higher the resistance. [1]
### In semiconductors and insulators
In metals the fermi level lies in the conduction band giving rise to free conduction electrons. However in semiconductors the position of the fermi level is within the band gap, exactly half way between the conduction band minimum and valence band maximum for intrinsic(undoped) semiconductors. This means that at 0 Kelvin, there are no free conduction electrons and the resistance is infinite. However, as the resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors, dopant atoms increase the majority charge carrier by donating electrons to the conduction band or accepting holes in the valence band. For both types of donor or acceptor atoms, increasing the dopant density leads to a reduction in the resistance. Highly dopped semiconductors hence behave metallic. At very high temperatures, the contribution of thermally generated carriers will dominate over the contribution from dopant atoms and the resistance will decrease exponentially with temperature.
### In ionic liquids/electrolytes
In electrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the salt concentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In biological membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, are selective to specific ions and determine the membrane resistance.
### Resistance of various materials
Material Resistivity, $\rho$ohm-meter Metals $10^{-8}$ Semiconductors variable Electrolytes variable Insulators $10^{16}$
### Band theory
Quantum mechanics states that the energy of an electron in an atom cannot be any arbitrary value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible. The energy levels are grouped into two bands: the valence band and the conduction band (the latter is generally above the former). Electrons in the conduction band may move freely throughout the substance in the presence of an electrical field.
In insulators and semiconductors, the atoms in the substance influence each other so that between the valence band and the conduction band there exists a forbidden band of energy levels, which the electrons cannot occupy. In order for a current to flow, a relatively large amount of energy must be furnished to an electron for it to leap across this forbidden gap and into the conduction band. Thus, large voltages yield relatively small currents.
## Differential resistance
When resistance may depend on voltage and current, differential resistance, incremental resistance or slope resistance is defined as the slope of the V-I graph at a particular point, thus:
$R = \frac {\mathrm{d}V} {\mathrm{d}I} \,$
This quantity is sometimes called simply resistance, although the two definitions are equivalent only for an ohmic component such as an ideal resistor. If the V-I graph is not monotonic (i.e. it has a peak or a trough), the differential resistance will be negative for some values of voltage and current. This property is often known as negative resistance, although it is more correctly called negative differential resistance, since the absolute resistance V/I is still positive.
## Temperature-dependence
Near room temperature, the electric resistance of a typical metal conductor increases linearly with the temperature:
$R = R_0(1 + aT) \,$,
where a is the thermal resistance coefficient.
The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with the temperature:
$R= R_0 e^{a/T}\,$
Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increased starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.
The electric resistance of electrolytes and insulators is highly nonlinear, and case by case dependent, therefore no generalized equations are given. | 2014-11-22 00:33:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7333354949951172, "perplexity": 716.565022565893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400374040.48/warc/CC-MAIN-20141119123254-00133-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/273307/what-is-the-probability-that-the-resulting-four-line-segments-are-the-sides-of-a | # What is the probability that the resulting four line segments are the sides of a quadrilateral?
Question: Divide a given line segment into two other line segments. Then, cut each of these new line segments into two more line segments. What is the probability that the resulting line segments are the sides of a quadrilateral?
I am stuck on this problem. I think I am close, but I am not sure if it is correct. Any help or conformation on this would be helpful.
My thoughts on the problem:
Let us say that the line segment is of length 1. The only restriction for these for line segments to form a quadrilateral is that no one side > .5 (correct me if I am wrong).
With our first cut we have two smaller line segments, one larger than the other. We only need to look at the longer on of these two line segments. Let us call $y$ the smaller line segment and $x$ the larger one. $x$ will be between 0.5 and 1.
When we cut each of these new line segments we need to find where it does not work for it to be a quadrilateral. We only have to look at $x$. Let us call $a$ the length that we cut.
If we use the example $x=0.6$ we can see that $a$ cannot be less than 0.1 or greater than 0.5.
We can generalize this for any $x$. $a$ cannot be less than $(x-1/2)$ or greater than $1/2$.
This is where I get stuck.
I believe that the probability for any $x$ value that these four line segments will not be a quadrilateral is $$\frac{2(x-1/2)}{x}$$
If this is correct is the total probability that it cannot be a quadrilateral $$\int\limits_{1/2}^1\frac{2(x-1/2)}{x}\mathrm{d}x?$$
Any help is much appreciated. Thank you
• +1 for showing your work. In the very last integral, $\mathrm dx$ should be $2\mathrm dx$. Apart from that, your reasoning looks fine. To compute the last integral, use $\int(4-2/x)\mathrm dx=4x-2\log x$, which yields the numerical value $2-2\log2$. – Did Jan 9 '13 at 6:06
• Why should it be $2dx$? – yousuf soliman Jan 9 '13 at 6:09
• Because the total mass of $\mathrm dx$ on $[1/2,1]$ is $1/2$ and you want a total mass of $1$. – Did Jan 9 '13 at 6:15
• I am sorry, I'm not sure I entirely understand this. I am fairly new to Calculus in general and have never come across a situation like that. Could you please explain a little more in depth? Thank you! – yousuf soliman Jan 9 '13 at 6:19
• @MuadDib42 Another reason why there is a 2, is because you did a symmetric argument and only considered the case where the break occurred in $[0.5, 1]$, instead of when the break occurred in the entire domain $[0,1]$. – Calvin Lin Jan 9 '13 at 7:07
## 1 Answer
In your very last integral, $\mathrm dx$ should be $2\mathrm dx$ $(*)$. Apart from that, your reasoning looks fine. To compute the last integral, use $\displaystyle\int(2−1/x)\mathrm dx=2x−\log x$, which yields the numerical value $2−2\log2\approx61\%$ for the probability of no quadrilateral.
$(*)$ To understand why, note that the total mass of $\mathrm dx$ on $[1/2,1]$ is $1/2$ and that one wants a total mass of $1$. This indicates that the probability density function of $X$ the longest of the two first lengthes is $2$ on the interval $[1/2,1]$. Once $X=x$ is known, the probability of no quadrilateral is $p(x)=2(x-1/2)/x$, as you aptly showed, hence the total probability of no quadrilateral is $$\int_{1/2}^1p(x)\,(2\mathrm dx)=2\int_{1/2}^1(2-1/x)\mathrm dx.$$ | 2019-05-21 17:36:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8154955506324768, "perplexity": 136.85406173789588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00506.warc.gz"} |
https://dmol.pub/applied/e3nn_traj.html | # 19. Equivariant Neural Network for Predicting Trajectories¶
Authors:
Sam Cox
Audience & Objectives
This chapter builds on Equivariant Neural Networks and Input Data & Equivariances. After completing this chapter, you should be able to
• Understand the importance of equivariance and how to check for equivariances in your model
• Understand how an equivariant model can be implemented in E3NN
• Be able to recognize and use the irrep notation used in E3NN
In this example, we will train an equivariant neural network to predict the next frame in the trajectory alignment example in Input Data & Equivariances. As stated in Input Data & Equivariances, for time-dependent trajectories, we do not need to concern ourselves with permutation equivariance because it is implied that the order of the points does not change. Thus, we can treat this example as point cloud, meaning that any deep learning model that we train on this data should have rotation and translation equivariance. In other words, our model should be E(3) equivariant. E3NN is a library built to create equivariant neural networks for the this group, so it’s a great choice for this problem [GS22]. We will look at E3NN in more detail later in the chapter.
## 19.1. Running This Notebook¶
Click the above to launch this page as an interactive Google Colab. See details below on installing packages.
# additional imports
import torch
import torch_geometric
import e3nn
import matplotlib.pyplot as plt
import urllib.request
import numpy as np
import jax
import dmol
## 19.2. Retrieving Data from Trajectory Alignment Example¶
First, let’s borrow a cell from Input Data & Equivariances to download our data and view the first frame.
urllib.request.urlretrieve(
)
# plot the first point
plt.title("First Frame")
plt.plot(paths[0, :, 0], paths[0, :, 1], "o-")
plt.xticks([])
plt.yticks([])
plt.show()
## 19.3. Baseline Model¶
Before we build our E3NN network, it’s always a good idea to build a baseline model for comparision.
First, let’s discuss what the input and output should be for this model. The input should be the coordinates of the 12 points: one frame. What should the output be? We want to train a neural network to predict the next trajectory for each point, the next frame, so our output should actually be the same type and size as our input.
Thus,
Inputs: 12 sets of coordinates
Outputs: 12 sets of coordinates
Note: since we are trying to build an E(3)-equivariant neural network, which should be equivariant to transformations in 3D space, we need to make these coordinates 3D. This is easy, we will just put zero for the z-coordiantes. We’ll do this now.
traj_3d = np.array([])
for i in range(2048):
for j in range(12):
TBA = paths[i][j]
TBA = np.append(TBA, np.array([0.00]))
traj_3d = np.append(traj_3d, TBA)
traj_3d = traj_3d.reshape(2048, 12, 3)
Interestingly, for this example, we want our prediction from one frame to match the following frame. So our features and labels will be nearly identical, offset by one. For the features, we want to include everything except for the final frame, which has no “next frame” in our data. We can extrapolate with our model to predict this “next frame” as a final step if we want. For our labels, we want to include everything except for the first step, which is not the “next frame” of anything in our data. We can also go ahead and split our data into training and testing sets. Let’s do an 80:20 split here. We want to make sure not to shuffle our data, as we are predicting order-sensitive data.
features = traj_3d[:-1]
labels = traj_3d[1:]
# split data 80:20
training_set = features[:1637]
training_labels = labels[:1637]
valid_set = features[1637:]
valid_labels = labels[1637:]
Let’s check to make sure our data matches up. Frame 2 in the features set should be the same as Frame 1 in the labels set.
def mse(y, yhat):
return np.mean((yhat - y) ** 2)
if mse(features[1], labels[0]) == 0:
print("success! they match!")
else:
print(mse(features[1], labels[0]))
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
axs[0].set_title("Trajectory 1 end")
axs[1].set_title("Trajectory 2 beginning")
for i in range(0, 1, 16):
axs[0].plot(features[i, :, 0], features[i, :, 1], ".-", alpha=0.2)
axs[1].plot(labels[i, :, 0], labels[i, :, 1], ".-", alpha=0.2)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
success! they match!
Great, they match! Now we are ready to build our baseline model!
@jax.jit
def baseline_model(inputs, w, b):
yhat = inputs @ w + b
return yhat
def baseline_loss(inputs, y, w, b):
return mse(y, baseline_model(inputs, w, b))
w = np.zeros((3, 3))
b = 0.0
epochs = 12
eta = 1e-6
baseline_val_loss = [0.0 for _ in range(epochs)]
ys = []
yhats = []
yst = []
yhatst = []
e = 0
for epoch in range(epochs):
e += 1
for d in range(1637):
inputs = training_set[d]
y = training_labels[d]
if e == epochs:
yhatst.append(baseline_model(inputs, w, b))
yst.append(y)
# update w & b
w -= eta * grad_bl[0]
b -= eta * grad_bl[1]
for i in range(410):
inputs_v = valid_set[i]
y_v = valid_labels[i]
if e == epochs:
yhats.append(baseline_model(inputs_v, w, b))
ys.append(y_v)
baseline_val_loss[epoch] += baseline_loss(inputs_v, y_v, w, b)
baseline_val_loss[epoch] = np.sqrt(baseline_val_loss[epoch] / 410)
plt.plot(baseline_val_loss)
plt.xlabel("Epoch")
plt.ylabel("Val Loss")
plt.show()
WARNING:jax._src.lib.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)
print("Final loss value: ", baseline_val_loss[-1])
Final loss value: 0.4302848
Now let’s view a parity plot to see if we’re learning the right trend here. Since the coordinates are so different in magnitude, we’ll look at a parity plot for the center of mass for each point.
def centerofmass(arr):
com = []
for i in arr:
for j in i:
avg1 = (j[0] + j[1] + j[2]) / 3
com.append(avg1)
return com
from sklearn.metrics import r2_score
y_s = np.stack(ys, axis=0)
yhat_s = np.stack(yhats, axis=0)
ys_centered = centerofmass(y_s)
yhats_centered = centerofmass(yhat_s)
plt.title("centered coordinates")
plt.plot(ys_centered, ys_centered, "-")
plt.plot(ys_centered, yhats_centered, ".")
plt.xlabel("Trajectory")
plt.ylabel("Predicted Trajectory")
plt.annotate(
"r-squared = {:.3f}".format(r2_score(ys_centered, yhats_centered)), (11.5, 12.4)
)
plt.show()
It looks like we are starting to get the right trend for some of the coordinates, but more training is definitely needed. Let’s look at the trajectory labels versus the predicted trajectories. To get a full picture, let’s look add the training and validation data together here.
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
y_st = np.stack(yst, axis=0)
ys_total = np.concatenate([y_st, y_s])
yhat_st = np.stack(yhatst, axis=0)
yhats_total = np.concatenate([yhat_st, yhat_s])
axs[0].set_title("Trajectory")
axs[1].set_title("Predicted Trajectory")
cmap = plt.get_cmap("cool")
for i in range(0, 2047, 40):
axs[0].plot(
ys_total[i, :, 0], ys_total[i, :, 1], ".-", alpha=0.2, color=cmap(i / 2047)
)
axs[1].plot(
yhats_total[i, :, 0],
yhats_total[i, :, 1],
".-",
alpha=0.2,
color=cmap(i / 2047),
)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
Yikes. Our model does not predict trajectories quite right. We expect this, since our baseline is simple machine learning. Importantly, as stated, we want any model that uses this data to be equivariant in 3D space. Let’s check the equivariances now.
# checking for rotation equivariance
import scipy.spatial.transform as trans
# rotate around x coordinate by 80 degrees
rot = trans.Rotation.from_euler("x", 80, degrees=True)
key = jax.random.PRNGKey(52)
input_point = jax.random.normal(key, (12, 3))
w_test1 = jax.random.normal(key, (3, 3))
input_rot = rot.apply(input_point)
output_1 = baseline_model(input_rot, w_test1, b)
output_prerot = baseline_model(input_point, w_test1, b)
output_rot = []
for xyz in output_prerot:
coord = rot.apply(xyz)
output_rot.append(coord)
output_rot = np.asarray(output_rot)
print("\033[1m" + "difference: " + "\033[0m", mse(output_1, output_rot))
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
axs[0].set_title("Rotated First")
axs[1].set_title("Rotated Last")
for i in range(0, 1, 16):
axs[0].plot(output_1[:, 0], output_1[:, 1], ".-", alpha=0.2)
axs[1].plot(output_rot[:, 0], output_rot[:, 1], ".-", alpha=0.2)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
difference: 4.4521255
So it doesn’t look like our baseline model is rotation-equivariant. This is important, because if we give our model coordinates that are rotated, we expect the output should be rotated by the same degree. Likewise, we need translation equivariance. Let’s check that now.
# checking for translation equivariance
key2 = jax.random.PRNGKey(9)
random_trans = jax.random.normal(key2, (12, 3))
input_trans = input_point + random_trans
output_2 = baseline_model(input_trans, w_test1, b)
output_trans = random_trans + baseline_model(input_point, w_test1, b)
print("\033[1m" + "difference: " + "\033[0m", mse(output_2, output_trans))
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
axs[0].set_title("Translated First")
axs[1].set_title("Translated Last")
for i in range(0, 1, 16):
axs[0].plot(output_2[:, 0], output_2[:, 1], ".-", alpha=0.2)
axs[1].plot(output_trans[:, 0], output_trans[:, 1], ".-", alpha=0.2)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
difference: 3.6074016
As expected, our model isn’t translation equviariant either. We can solve this problem a few ways. One way is to augment our data in order to teach our model equivariance. This requires more training and data storage, so let’s look at a more compact approach.
## 19.4. E3NN Basics¶
E3NN is a library for creating equivariant neural networks, specifically in E(3). E3NN is built for spatial equvariance in 3-D space, giving us equivariance with respect to the E(3) group of rotations, inversions, and translations. As discussed before, the time-dependent trajectory points do not change order, so we do not need to worry about permutation equivariance/invariance in this case; we only need E(3)-equivariance. E3NN is a great tool for this problem because we have 3-dimensional points in space, and if we transform them in space, we want the output to transform the same way.
E3NN works through the use of irreducible representations (irreps). In general, representations tell you how to interact with the data with repect to the group, and irreducible representations are the smallest and complete representations. When creating a model, we give the model the irreps so that it knows how to handle the data we will give it during trianing. It’s not necessary to understand what the irreps are; instead, just know that they are the smallest representations, which are similar to, and transform the same way as, the spherical harmonics. If you do want more information, you can read about irreps here. Any (reducible) representation can be decomposed into irreducible representations. If you want to know more, you can check out more on the E3NN documentation website [@e3nn]. Let’s take a look at how the irreps are used in this context.
For this group (O(3), which includes parity), we need to find the L and d for each piece of data, where $$d = 2L + 1$$ (d = dimension). Look at the table below.
parity
L
d
name
even
0
1
scalar
odd
0
1
pseudo scalar
even
1
3
pseudo vector
odd
1
3
vector
even
2
5
-
odd
2
5
-
The general notation is MxLp, where M is the number of 3-D coordinate per input, L is the L (spherical harmonic) from the table above, and p corresponds to the parity (e: even, o: odd).
For example, if you wanted to portray “12 scalars, 4 vectors” in this format, you would write 12x0e + 4x1o. Take a minute to make sure you understand how to use this notation, as it’s essential for E3NN. E3NN deals with equivariance by receiving the irreps as a model parameter. This allows the E3NN framework to know how each input feature/output transforms under symmetry, so that it can treat each piece appropriately. As a side note, the output of an E3NN model must always be of equal or higher symmetry than your input.
Because E3NN is built to handle 3D spatial data, we do not need to tell the model that we are going to give it 3D coordinates; it’s implicit and required. The irreps_in, instead, correspond to the input node features. In this example, we don’t have input features, but as an example, you can imagine we could want our model to predict the next set of coordinates, given the intitial coordinates and the corresponding atom types. In that case, our irreps_in would be the atom types (one scalar per input if we have one-hot vectors).
Since we don’t have input features, we’ll put “None” for that parameter, and we want our output to be the same shape as the input: 12 vectors. However, since we are trying to predict 12 vectors out for 12 vectors in, we only need to tell the model to predict 1 vector per input 1x1o. Take a minute to make sure you understand why this is the case. You can think of the model recognizing 12 input vectors and predicting a vector for each. Again, E3NN expects coordinate inputs, so we don’t specify this for the input.
## 19.5. E3NN Model¶
E3NN has several models within their library, which can be found on the E3NN github page here. For this example, we will use one of these models.
To use the E3NN model, we need to turn our data into a torch_geometric dataset. We’ll do that now. Then we can split our data into training and testing sets.
Also, instead of directly computing the next frame, we’ll change it here to predict the distance to the next frame. This is a small change, but having data centered nearer zero can be better for training. We’ll need to undo this when we look at the frames later.
feat = torch.from_numpy(features) # convert to pytorch tensors
ys = torch.from_numpy(labels) # convert to pytorch tensors
traj_data = []
distances = ys - feat # compute distances to next frame
# make torch_geometric dataset
# we want this to be an iterable list
# x = None because we have no input features
for frame, label in zip(feat, distances):
traj_data += [
torch_geometric.data.Data(
x=None, pos=frame.to(torch.float32), y=label.to(torch.float32)
)
]
train_split = 1637
traj_data[:train_split], batch_size=1, shuffle=False
)
traj_data[train_split:], batch_size=1, shuffle=False
)
Great! Now we’re ready to define our model. Since this is a pre-built model in E3NN, so we just need to import it and define the model parameters. Note that the state of this model will save automatically, so you will need to reinitialize the model every time you want to start training. To see how these models work you can look at this preprint or this video series.
The cell below sets the model parameters for the model we are using. First, we tell the model our irreps_in, which in this case is None. Then, we specify the irreps_hidden and layers, which define the width and shape of our model. These are hyperparameters. irreps_out corresponds to our output shape, 1x1o. We specify that our nodes have no attributes, and that we want to use spherical harmonics as our edge attributes. You don’t need to be too concerned with the number_of_basis, radial_layers, or radial_neurons, as they don’t change much between applications. The max_radius and num_neighbors are intuitive, just specify the average numbers in your max radius (a hyperparameter). If you do not know this, you can write a function that calculates an average number of neighbors. Lastly, the num_nodes is not important in this case since we set reduce_output to False. If we set this to True, that means we want to reduce our output over all num_nodes in our input to get a single scalar as an output.
Then we just initialize our model with our defined parameters.
from e3nn.nn.models.gate_points_2101 import Network
model_kwargs = {
"irreps_in": None, # no input features
"irreps_hidden": e3nn.o3.Irreps("5x0e + 5x0o + 5x1e + 5x1o"), # hyperparameter
"irreps_out": "1x1o", # 12 vectors out, but only 1 vector out per input
"irreps_node_attr": None,
"irreps_edge_attr": e3nn.o3.Irreps.spherical_harmonics(3),
"layers": 3, # hyperparameter
"number_of_basis": 10,
"num_neighbors": 11, # average number of neighbors w/in max_radius
"num_nodes": 12, # not important unless reduce_output is True
"reduce_output": False, # setting this to true would give us one scalar as an output.
}
model = e3nn.nn.models.gate_points_2101.Network(
**model_kwargs
) # initializing model with parameters above
Next, we set our learning rate (hyperparameter), and our optimizer. In this case, we are using the Adam optimizer, and we initialize our gradients as zero. Since we chose Adam, we have to pass in our paramters and learning rate. Adam computes adaptive learning rates for the parameters [KB14].
eta = 1e-4
optimizer = torch.optim.Adam(model.parameters(), lr=eta)
epochs = 16
val_loss = [0.0 for _ in range(epochs)]
y_values = []
yhat_values = []
y_valuest = []
yhat_valuest = []
e = 0
for epoch in range(epochs):
e += 1
for step, data in enumerate(train_loader):
yhat = model(data)
if e == epochs:
y_valuest.append(data.y)
yhat_valuest.append(yhat)
loss_1 = torch.mean((yhat - data.y) ** 2)
loss_1.backward()
optimizer.step()
for step, data in enumerate(test_loader):
yhat = model(data)
if e == epochs:
y_values.append(data.y)
yhat_values.append(yhat)
loss2 = torch.mean((yhat - data.y) ** 2)
val_loss[epoch] += (loss2).detach()
val_loss[epoch] = val_loss[epoch] / 410
v_loss = torch.tensor(val_loss)
plt.plot(v_loss, label="Validation Loss")
plt.legend()
plt.xlabel("Epoch")
plt.ylabel("Loss")
plt.show()
print("final loss value: ", val_loss[-1])
final loss value: tensor(3.4724e-05)
yhat_valuest = [y.detach().numpy() for y in yhat_valuest]
yhat_values = [y.numpy() for y in yhat_values]
yhat_total = np.concatenate([yhat_valuest, yhat_values])
original = feat.numpy()
y_arr = ys.numpy()
yhat_arr = np.stack(yhat_total + original, axis=0)
yhat_vs = np.stack(yhat_values + original[train_split:], axis=0)
y_vs = np.stack(ys[train_split:], axis=0)
yhat_centered = centerofmass(yhat_vs)
y_center = centerofmass(y_vs)
plt.title("centered coordinates")
plt.plot(y_center, y_center, "-")
plt.plot(y_center, yhat_centered, ".")
plt.xlabel("Trajectory")
plt.ylabel("Predicted Trajectory")
plt.annotate(
"r-squared = {:.3f}".format(r2_score(y_center, yhat_centered)), (11.5, 12.4)
)
plt.show()
Wow! Clearly these parity plots look better than for the baseline model. Let’s again look at the trajectory versus the trajectory predictions, to see visually how they compare. We will look at the training and the validation data together, just note that because the colors represent time, we can look at the purple section to see just the validation data.
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
ys_total = np.concatenate([y_st, y_arr])
yhat_st = np.stack(yhatst, axis=0)
yhats_total = np.concatenate([yhat_st, yhat_arr])
axs[0].set_title("Trajectory")
axs[1].set_title("Predicted Trajectory")
cmap = plt.get_cmap("cool")
for i in range(0, 2047, 40):
axs[0].plot(y_arr[i, :, 0], y_arr[i, :, 1], ".-", alpha=0.2, color=cmap(i / 2047))
axs[1].plot(
yhat_arr[i, :, 0], yhat_arr[i, :, 1], ".-", alpha=0.2, color=cmap(i / 2047)
)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
This looks pretty good! Let’s run each model on the last frame to get an extrapolated frame. Remember that for the E3NN model, we predicting displacements, so we’ll just need to add our final displacement back to our final coordinates.
last_frame_bl = y_v
extrp_bl = baseline_model(last_frame_bl, w, b)
last_frame_e3nn = yhat + data.pos
lf_e3nn = []
# format as torch geometric dataset (dummy y values)
lf_e3nn += [
torch_geometric.data.Data(
x=None,
pos=last_frame_e3nn.to(torch.float32),
y=last_frame_e3nn.to(torch.float32),
)
]
# run through model
for i in lf_loader:
extrp_e3nn = model(i)
# add extrapolated displacements back into last frame
extrp_e3nn += last_frame_e3nn
extrp_bl = np.array(extrp_bl)
extrp_e3nn = extrp_e3nn.detach().numpy()
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
axs[0].set_title("Baseline Model Extrapolated Frame")
axs[1].set_title("e3nn Model Extrapolated Frame")
cmap = plt.get_cmap("cool")
axs[0].plot(extrp_bl[:, 0], extrp_bl[:, 1], ".-", alpha=0.2)
axs[1].plot(extrp_e3nn[:, 0], extrp_e3nn[:, 1], ".-", alpha=0.2)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
We can clearly see that the e3nn model predicts trajectories much closer to our data. Again, the baseline model has predicted poorly.
Now we can check for equivariance in the same way that we did before with the baseline model. Let’s just take the final frame and rotate, then compare to just the output rotated.
# checking for rotation equivariance
import scipy.spatial.transform as trans
# rotate around x coordinate by 80 degrees
rot = trans.Rotation.from_euler("x", 80, degrees=True)
key3 = jax.random.PRNGKey(58)
input_point = np.asarray(jax.random.normal(key3, (12, 3)))
input_rot = rot.apply(input_point)
input_point = torch.from_numpy(input_point)
input_rot = torch.from_numpy(input_rot)
# format as torch geometric dataset (dummy y values)
rot_first = []
rot_first += [
torch_geometric.data.Data(
x=None, pos=input_rot.to(torch.float32), y=input_rot.to(torch.float32)
)
]
# run through model
for i in rf_loader:
output_1 = model(i)
# format as torch geometric dataset (dummy y values)
rot_last = []
rot_last += [
torch_geometric.data.Data(
x=None, pos=input_point.to(torch.float32), y=input_point.to(torch.float32)
)
]
# run through model
for i in rl_loader:
output_2 = model(i)
output_2 = output_2.detach().numpy()
output_1 = output_1.detach().numpy()
output_rot = []
for xyz in output_2:
coord = rot.apply(xyz)
output_rot.append(coord)
output_rot = np.array(output_rot)
np.set_printoptions(precision=20, suppress=True)
print("\033[1m" + "difference: " + "\033[0m", np.array([mse(output_1, output_rot)]))
fig, axs = plt.subplots(ncols=2, squeeze=True, figsize=(16, 4))
axs[0].set_title("Translated First")
axs[1].set_title("Translated Last")
for i in range(0, 1, 16):
axs[0].plot(output_1[:, 0], output_1[:, 1], ".-", alpha=0.2)
axs[1].plot(output_rot[:, 0], output_rot[:, 1], ".-", alpha=0.2)
for i in range(2):
axs[i].set_xticks([])
axs[i].set_yticks([])
difference: [0.00000000000000000646]
Great! Our random array, when rotated first, gives the same results as when we rotated last! Now we know we have rotational equivariances. I won’t go further to test translational equivariances; I will leave that as an exercise. The E3NN model outperforms the baseline significantly, and it is E(3)-equivariant, unlike our baseline model!
## 19.6. Cited References¶
GS22
Mario Geiger and Tess Smidt. E3nn: euclidean neural networks. 2022. URL: https://arxiv.org/abs/2207.09453, doi:10.48550/ARXIV.2207.09453.
KB14
Diederik P Kingma and Jimmy Ba. Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. | 2022-11-26 12:45:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.533072829246521, "perplexity": 4187.996487378175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446706291.88/warc/CC-MAIN-20221126112341-20221126142341-00109.warc.gz"} |
https://codeyarns.com/tech/2010-01-19-how-to-add-clickable-hyperlinks-and-email-addresses-in-latex.html | 📅 2010-Jan-19 ⬩ ✍️ Ashwin Nanjappa ⬩ 📚 Archive
Hyperlinks can be added to a LaTeX document by using the commands from the hyperref package:
\documentclass[a4paper]{article}
\begin{document}
\href{http://web.media.mit.edu/~minsky/}{Marvin Minsky}\\ % Text to URL
\url{http://web.media.mit.edu/~minsky/}\\ % Text is URL
\href{mailto:linus@kernel.org}{Email Linus} % Email
\end{document}
Note that the URL is displayed in a fixed-width font. This is the default font used for the \url command. If you want the URL to be displayed in the same font as that used for the rest of the document, use the \urlstyle command as follows:
\documentclass[a4paper]{article}
\end{document} | 2020-10-19 21:55:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.995903491973877, "perplexity": 6196.040642376957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107866404.1/warc/CC-MAIN-20201019203523-20201019233523-00303.warc.gz"} |
https://studyalgorithms.com/link_list/write-a-program-to-insert-a-node-in-a-sorted-linked-list/ | Home Linked List Write a program to insert a node in a sorted Linked List
# Write a program to insert a node in a sorted Linked List
0 comment
Question:
Write a program to insert a node in a given sorted Linked List.
Given List: 23 -> 32 -> 99 -> 101 -> 2222
Node to add:- 50
Output:- 23 -> 32 -> 50 -> 99 -> 101 -> 2222
To insert a node in a sorted Linked List, we need to perform a basic Linked List operation discussed in this post.
In the given post, we knew the position at which we had to insert the node. But in this case we need to find the position at which we have to insert the node.
To find the position, start traversing the list from the beginning and get to the point, where we need to insert the node. Then we can call the insert a node function.
We can implement this problem this way.
#include
struct node
{
int data;
struct node * next;
};
//defining a function to insert in sorted list
struct node * insertInSorted(struct node * head, int number)
{
//let us suppose the initial position to be zero
int pos = 0;
//always create a temporary variable as we should never loose
struct node * temp = head;
//now we need to traverse the list until we reach the node whose value
//is greater than the number input.
//If the 3rd number is greater, that means we need to insert a new number
//at the third position.
while(temp != NULL)
{
if(temp->data > number)
break; // if we get the node, means we have the position, break
temp = temp->next; // the next node
pos++; // increment the position
}
}
//helper functions
struct node * addAtPos(struct node *head, int number, int pos)
{
int initial_pos = 0;
struct node * mover = head;
while(initial_pos != pos)
{
mover = mover -> next;
initial_pos++;
}
struct node * temp = (struct node*)malloc(sizeof(struct node));
temp -> data = number;
temp -> next = mover -> next;
mover -> next = temp;
}
struct node * addElement(struct node *head, int number)
{
struct node * temp = (struct node*)malloc(sizeof(struct node));
temp -> data = number;
temp -> next = NULL;
struct node * temp2 = head;
while(temp2 -> next != NULL)
{
temp2 = temp2 -> next;
}
temp2 -> next = temp;
}
void printList(struct node * head)
{
{
}
}
//main function to test the code
int main(void)
{
//creating a list with initial sorted elements
struct node * listHead = (struct node*)malloc(sizeof(struct node));
printf("Enter a number to insert in sorted list:- ");
int num;
scanf("%d",&num);
return 0;
}
Enter a number to insert in sorted list:- 50
23 32 50 99 101 2222
0 comment
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This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept Read More | 2021-05-12 17:16:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18157587945461273, "perplexity": 5236.213081982439}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989766.27/warc/CC-MAIN-20210512162538-20210512192538-00008.warc.gz"} |
http://math.stackexchange.com/questions/157255/the-boundary-is-a-closed-set | # The boundary is a closed set
A point $p$ in a metric space $X$ is a boundary point of the set $A$, if any neighbourhood of $p$ has points of both $A$ and $X-A$.Prove that the set of all boundary points of $A$ is closed.
My attempt: By definition of an open set this means that for every $x$ in the boundary there is an open ball centred at $x$ contained in the boundary. An open ball is a neighbourhood of $x$, which implies it contains points of $A$ and $X - A$, which in turn implies there are points in both $A$ and $X - A$ that are in the boundary of $A$.
If $A$ is open, then pick any such point in $A$ that is also in the boundary. This point cannot be in $X - A$ by definition of set subtraction. Further, because $A$ is open there exists an open ball centred around this point contained in $A$. Again, an open ball is a neighbourhood, which means a neighborhood of this point does not contain points of $X - A$, implying it cannot be in the boundary, a contradiction. If A is closed then $X - A$ is open and a symmetric argument holds. Hence the boundary is closed.
Is my work correct?
-
The first paragraph of the attempt makes it seem like you're trying to prove that the boundary $\partial A$ is open. Note too that $A$ may be neither open nor closed. ["A subspace is not a door."] – Dylan Moreland Jun 12 '12 at 6:03
What if $A$ is neither open or closed? – William Jun 12 '12 at 6:03
Also, note that this result and many proofs of it will not require that $X$ be a metric space, just a topological space. – Dylan Moreland Jun 12 '12 at 6:10
The boundary of a set $A$ is defined as $\overline{A} \cap \overline{X - A}$. It is the intersection of two closed sets and hence is closed.
By the way your proof is not correct because you assumed that $A$ is either open or closed. There are sets like $(0,1] \subset \Bbb{R}$ in the usual topology that are neither open nor closed.
-
He gives a definition in the first paragraph. Of course, $p$ being in the closure of $B$ is the same as "all neighborhoods of $p$ intersect $B$", so your definition is the same. I don't know if that's clear to the OP. Anyway, +1 for a succinct, correct answer. – Dylan Moreland Jun 12 '12 at 6:06
Your very first statement simply isn’t true: there need not be any non-empty open set contained in the boundary of $A$. Suppose that $A=[0,1]$ in the space $\Bbb R$: the boundary of $A$ is the set $\{0,1\}$, which does not contain any non-empty open subset of $\Bbb R$.
I suggest that you try to show that $X\setminus\operatorname{bdry}A$ is open, from which it will follow at once that $\operatorname{bdry}A$ is closed. To do this, pick a point $x\in X\setminus\operatorname{bdry}A$, and show that some open neighborhood of $x$ is disjoint from $\operatorname{bdry}A$. You’ll need to consider two cases: if $x\in X\setminus\operatorname{bdry}A$, either $x$ has an open neighborhood disjoint from $A$, or $x$ has an open neighborhood disjoint from $X\setminus A$.
-
This is exactly what I was going to write (your second paragraph). Albeit, slightly modified (perhaps it isn't right). What do you think of using that the complement of the boundary is the union of the interior and the exterior where the exterior is the complement of the closure? Then the complement of the boundary is the union of two open sets and hence open. – Rudy the Reindeer Jun 16 '12 at 6:14
@Matt: That works fine, provided that one already knows that $\operatorname{bdry}A=\operatorname{int}A\cup(X\setminus\operatorname{cl}A)$. – Brian M. Scott Jun 16 '12 at 6:19
I guess one would have to show that and one doesn't have the beauty of a one sentence short answer anymore : ) – Rudy the Reindeer Jun 16 '12 at 6:20
It appears that your proof is not correct since you only consider the case when $A$ is either open or closed.
A set is closed if and only if it contains all its limit points.
Suppose $(x_n)$ is sequence of boundary points of $A$ which converges to some point $x$. One seeks to show that $x$ is also a boundary point. Let $U$ be an open set containing $x$. By definition of being the limit of $(x_n)$, there exists a $N$ such that $x_N \in U$. Since $X_N$ is a boundary point and $U$ is a neighborhood of $x_N$, $U$ contains a point of $A$ and $X - A$. Since $U$ is arbitrary, $x$ is a boundary point. The set of boundary points of $A$ is a closed set.
- | 2016-06-27 17:01:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9263237118721008, "perplexity": 75.70366797573939}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396100.16/warc/CC-MAIN-20160624154956-00085-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://manasataramgini.wordpress.com/ | ## The Vyomavyāpin in the Pāśupata-tantra and a discursion on nine-fold Rudra-mantra-s
The Pāśupata-tantra is a poorly understood śaiva text that is believed to be affiliated with the Pāśupata tradition of Lakulīśa. While the colophons of some manuscripts present it as “Lakulīśa-pravartita-Pāśupata-tantram”, internally, it presents itself as a teaching of Nandin to the Bhārgava sage Dadhīci upon direction by Rudra himself. While we have seen a text going by this name in certain manuscript catalogs and seen fragmentary manuscripts of it, only recently was a nearly complete version of the text partially edited. This is not the place to go into a detailed discussion of the affinities and the provenance of the text, but we will make the below observations:
1. While a text going by this name has been mentioned by South Indian Vaiṣṇava polemicists, like Yāmuna, there is no evidence that they meant the text under discussion in this note.
2. The text as we have it can be confidently said to have been composed in South India, in the greater Drāviḍa country or its surroundings because: (i) It mentions the worship of Skanda with his two śakti-s named Devasenā and Devayānī. The latter is a unique feature of certain strands of the Southern Kaumāra cult. (ii) It mentions the worship of the god Śāstṛ, a southern ectype of the god Revanta, presented as the son of Rudra and Mohinī. (iii) Several of the manuscripts display typical Drāviḍa misspellings like “taha” for “daha”.
3. There may have been a transmission to Northeastern India, perhaps Vaṅga or its surroundings, due to some versions showing spelling errors typical of the Vāṅga-s, like the “v-b” confusion.
4. It is a late text (i.e., post-mantramārga) because it shows iconographic conventions typical of the period when the mantra-mārga was dominant: e.g., the mode of worship and depiction of Vināyaka, the Saptamātṛkā-s, the Rudra-parivāra and the pentacephalic Rudra (as opposed to the tricephalic and tetracephalic Rudra-s of the earlier Pāśupata-s). This point is important to the main topic of this note.
5. It is divided into four kāṇḍa-s: jñāna, caryā, kriyā, and yoga. Such a division is typical of various mantra-mārga texts in both the śaiva and vaiṣṇava traditions.
6. The main mantra-s it treats at length are the Pañcākṣarī, Pañcabrahma, Vyomavyāpin, Śivakavaca, Aghorāstra, Pāśupatāstra and multiple Rudra-gāyatrī-s. Additionally, it extensively uses Vaidika-mantra-s indicated by pratīka-s, suggesting that its practitioners were Veda-knowing brāhmaṇa-s.
7. It has an extensive account of the Bhuvanādhvan-s and the Rudra-s of various forms in each of them.
In conclusion, a brief examination of its contents suggests that it is a text that has been influenced by the mantramārga, in particular, the siddhānta-srotas. The main reasons for this conclusion are: (i) The repeated mention of the supreme Rudra as Sadāśiva enthroned on the Yogapīṭha. (ii) The mention of several tantra-s of Paśupati following a model reminiscent of the Saiddhānitka self-image. (iii) Primacy of the Īśāna face of the pentacephalic Rudra. However, we do think there is something to its affiliation with the Pāśupata tradition. In support of this, one may point to the extensive use of Vaidika mantra-s where the Siddhānta might use tāntrika alternatives and visualizations of the supreme Rudra rather distinct from the Siddhānta versions but overlapping with the fierce Bhairava-s of the other srotas-es (also see below). One possibility is that it is a Lakulāgama associated with the South Indian Kālāmukha-s
The Pāśupata-tantra is notable for providing a full uddhāra of the famed Vyomavyāpin mantra. This is thought to be a unique mantra of the saiddhāntika-s. For instance, the Mataṅgapārameśvara-tantra of that stream states its importance multiple times. In its kriyāpāda 1.60, it states that the Vyomavyāpin is the garbha from which all mantra-s arise — like the pañcabrahma, Caṇḍeśa, the Sāvitrī, Indrādi-mantra-s etc. In its vidyāpāda 7.31 onward, it sees the mantra as the devī who constitutes the body of Sadāśiva (c.f. similar metaphor used in the Bhairavasrotas for the goddess, e.g., by Abhinavagupta). A similar view is expressed in the Pāśupata-tantra; indeed, Nandin introduces it thus to Dadhīci:
sarvamantra-samāyuktam vyoma-vyāpinam avyayam ।
mantrāṇāṃ saptakoṭīnāṃ sāraṃ tat te vadāmy aham ॥
Comprised of all the mantra-s is the imperishable Vyomavyāpin.
I shall teach you that which is the essence of the seven crore mantra-s.
Given the above, one could argue that the Pāśupata-tantra borrowed this mantra from the saiddhāntika -s. However, we believe it emerged among the later Pāśupata-s (i.e., subsequent to their Vedic representatives) but prior to the branching off of the streams of the mantra-mārga, like the saiddhāntika-s. Our reasons for holding this view are: (i) Within the saiddhāntika tradition, the Vyomavyāpin is remarkable in showing a diversity of readings despite being a central mantra, as noted above. This suggests that it emerged in the pre-saiddhāntika mantraśāstra matrix. Hence, it had already diversified within the oral prayoga traditions from which the siddhāntāgama-s inherited alternative versions of it. (ii) In terms of its structure, it is more removed from the later bīja-rich mantra-s and closer to the mantra-s of the transitional mantraśāstra, viz., at the junction between the Vaidika- and the full-blown Tāntrika-mantramārga (e.g., some of the mantra-s to Rudra in the Atharvavedīya-pariśiṣṭa-s, Viṣṇumāyā and the bauddha Mahāmāyūrī-vidyā-rājñī). 3. Its dhyAna-s describe a 14- and 10- handed Rudra distinct from Sadāśiva, the devatā of the saiddhāntika version.
The core without the kavaca and Aghorāstra- sampuṭikaraṇa-s is said to follow the 14-handed dhyāna, which is the same as that for Pañcākṣarī:
vasiṣṭha ṛṣiḥ । gāyatrī chandaḥ । parameśvaro devatā ॥
śūlāhi-ṭaṅka-ghaṇṭāsi raṇaḍ ḍamarukaṃ kramāt ।
vajra-pāśāgny abhītiṃ ca dadhānaṃ kara-pallavaiḥ ॥
kapālam akṣamālāṃ ca śaktiṃ khaṭvāṅgam eva ca ।
evaṃ dhyātvā prabhuṃ divyaṃ tato yajanam ārabhet ॥
Vasiṣṭha is the seer, gāyatrī the meter, and Parameśvara the deity.
Having visualized the lord, in order, equipped with a trident, hatchet, bell, sword, a resounding two-headed drum, the vajra, a lasso, fire, the gesture of fearlessness, a skull, a rosary, a spear and a skull-topped brand in his blossom-like hands, the [votary] may begin his worship.
With the kavaca and astra, the dhyāna is the fierce five-headed 10-handed rudra:
kalpāntārkaṃ sahasrābhaṃ raktāktaṃ raktavāsasaṃ ।
daṃṣṭrā-karāla-saṃbhinnam pañcavaktram bhayaṅkaram ॥
keśaiś ca kapilair dīptaṃ jvālamālā-samākulam ।
ṭaṅkaṃ carma kapālaṃ ca cāpaṃ nāgaṃ ca vāmataḥ ॥
śūlaṃ khaḍgaṃ yugāntāgniṃ bāṇaṃ varadam eva hi ।
dakṣiṇaiḥ svabhujair dīptaṃ rudraṃ dhyātvā yajet prabhum ॥
Having visualized the blazing Rudra with the luminosity of a thousand suns at the end of the kalpa, smeared with gore, with red clothes, displaying terrifying fangs, five frightening faces, and tawny hair like a blazing garland of flames, holding in his left hands a hatchet, a shield, a skull, a bow, and a snake, and his right hands a trident, a sword, the eon-ending fire, an arrow and the gesture of boon-giving, he may worship the lord.
The core mantra (i.e., with the saṃdhi-s in the duplications and without the 5 initial praṇava-s, the terminal ṣaḍakṣarī, the hṛllekha-s, the haṃ-kāra (prāsada), kavaca and the astra typical of the Pāśupata version) is 365 syllables. The versions from most surviving saiddhāntika texts are typically in the range of 361-374. The pristine form in the Mataṅgapārameśvara-tantra has 361 by the same reckoning as above, suggesting that it might have come to 365 with the addition of a namo namaḥ after the terminal praṇava. We believe the Pāśupata-tantra version is close to the original as the old saiddhāntika text, the Niśvāsa-guhya, associates Rudra embodied by this mantra with the phrase “saṃvatsara-śarīriṇaḥ”, i.e., of the year as the body. This form would also be consistent with 9-fold maṇḍala taught by the Kashmirian mantravādin bhaṭṭa Rāmakaṇṭha-II and his southern successors. In his Vyomavyāpi-stava, referring to the 81 segments of the mantra (see below) and the nine-fold maṇḍala Rāmakaṇṭha says: ekāśītipadaṃ devaṃ nava-parvoktidarśanāt ॥ 8b. In this regard, it is also worth noting that the Mataṅgapārameśvara-tantra defines the devī of the form the Vyomavyāpin as having a body of $9 \times 9 = 81$ segments. The same is also mentioned by Śrīkaṇṭha-sūri in his Ratnatrayaparīkṣa thus: ekāśītipadā devī vyomavyāpi-lakṣaṇā śaktiḥ । The count of 81 relates to a certain mapping that is specified in the saiddhāntika tradition to 15 classes of mantra-s. The number 15 is again likely to have temporal significance as the tithi-s of the lunar cycle. In the Pāsupata version, this division of the mantra into 81 segments mapping onto 15 sets of mantra-s goes thus:
(1) Aṅga-mantra-s (The body of Rudra): 1. oṃ 2. vyomavyāpine 3. vyomarūpāya 4. sarvavyāpine 5. śivāya (total: 5)
(2) Vidyeśvara-s: 6. anantāya 7. anāthāya 8. anāśritāya 9. dhruvāya 10. śāśvatāya 11. yogapīṭhādisaṃsthitāya 12. nityayogine 13. dhyānāhārāya (total: 8)
(3) Pañcākṣarī-vidyā (equated with the Rudra-gāyatrī by the saiddhāntika-s): 14. oṃ namaḥ śivāya (total: 1)
(4) Sāvitrī-vidyā: 15. sarvaprabhave (total: 1)
(5) Vidyeśvaropacāra: 16. śivāya (total: 1)
(6) Pañcabrahma-mantra-s: 17. īśāna-mūrdhnāya 18. tatpuruṣa-vaktrāya 19. aghora-hṛdayāya 20. vāmadeva-guhyāya 21. sadyojāta-mūrtaye (total: 5)
(7) Caṇḍeśvara: 22. oṃ namaḥ (total: 1)
(8) Caṇḍeśāṅgani (the body of Cāṇḍeśvara): 23. guhyādi-guhyāya 24. goptre 25. anidhanāya 26. sarvavidyādhipāya 27. jyotīrūpāya 28. parameśvaraparāya (total: 6)
(9) Caṇḍeśāsana: 29. acetanācetana (total: 1)
(10) Anantāsana: 30. vyomin $\times$ 2 31. vyāpin $\times$ 2 32. arūpin $\times$ 2 33. prathama $\times$ 2 34. tejas tejaḥ 35. jyotir jyotiḥ (total: 6)
(11) kesara-s (mantra-s of the 32-petaled lotus, likely corresponding to the syllables of the bahurūpī ṛk): 36. arūpa 37. anagne 38. adhūma 39. abhasma 40. anāde 41. nānā nānā 42. dhū dhū dhū dhū 43. oṃ bhūr 44. bhuvaḥ 45. svaḥ 46. anidhana 47. nidhanodbhava 48. 49. śiva 50. śarva 51. sarvapara 52. maheśvara 53. mahādeva 54. sadbhāveśvara 55. mahātejaḥ 56. yogādhipate 57. muñca muñca 58. pramatha pramatha 59. śiva śarva 60. bhavodbhava vidhya vidhya 61. vāmadeva 62. sarva-bhūta-sukhaprada 63. sarva-sāṃnidhyakara 64. brahmā-viṣṇu-rudra-para 65. anarcita $\times$ 2 66. asaṃsthita $\times$ 2 67. pūrvasthita $\times$ 2 (total: 32)
(12) Kamala (center of the lotus throne): 68. sākṣin $\times$ 2 (total: 1)
(13) Indrādi-devatā-s (the gods of the directional ogdoad): 69. turu $\times$ 2 70. piṅga $\times$ 2 71. pataṅga $\times$ 2 72. jñāna $\times$ 2 73. śabda $\times$ 2 74. sūkṣma $\times$ 2 75. śiva 76. śarva (total: 8)
(14) Vidyāṅga-s (the body of the goddess; corresponds to the 10-syllabled mantra known as the Vidyā in the early saiddhāntika text, the Niśvāsa-guhya-sūtra): 77. sarvada 78. oṃ namaḥ 79. śivāya oṃ 80. [hrīṃ] śivāya (total: 4)
(15) Vajra (the thunderbolt): [oṃ haṃ hrīṃ śivāya] oṃ [namo] namaḥ (total: 1)
This nine-fold nature implied in the original form of the Vyomavyāpin has ties with a similar nine-fold expression seen elsewhere in the śaiva world. Both the early Saiddhāntika (NGS) and Bhairava streams emphasize the importance of knowing the nine-fold form of Śiva known as Navātman and his mantra. The former states that japa of the Navātman-mantra over $10^5$ times yields magical powers. Both the early saiddhāntika and Brahmayāmala traditions speak of the 9 observances (e.g., japa of specific mantra-s wearing clothes and turbans of various colors) that seem to map to the nine-fold structure of the Navātman-mantra. On the Bhairava side, in the root Dakṣiṇa-śaiva tradition, the Svacchanda-tantra teaches the Vidyārāja, which is called ekāśitipadāḥ (81 segmented, just like the Vyomavyāpin):
ekāśitipadā ye tu vidyārāje vyavasthitāḥ ।
padā varṇātmikās te ‘pi varnāḥ prāṇātmikāḥ smṛtāḥ ॥ ST 4.252
Abhinavagupta’s cousin, Kṣemarāja informs us that this Vidyārāja is none other than the Navātman mantra. However, in the maṇḍala taught in the Svacchanda-tantra, Navātman is not the central deity, but the eighth Bhairava in the parivāra around the central Svacchanda-bhairava. The said tantra informs us that manifestation of the bhuvanādhvan-s is encapsulated in the 81 segments of the Navātman-mantra, and its prayoga-s yield siddhi-s comparable to the saiddhāntika prayoga-s. In the Paścimāṃnāya, Navātman-bhairava is the primal deity and consort of the supreme goddess Kubjikā (also seen in the combined Dakṣiṇāṃnāya-Paścimāṃnāya tradition of the Saundaryalaharī) and his $9 \times 9$-segmented mantra is taught. In the Pūrvāṃnāya (Trika), we see different formulations with Navātman-bhairava: (i) in the Siddhayogeśvarī-mata, he is the central deity of the maṇḍala of the kha-vyoman, known as the Kha-cakra-vyūha, where he is surrounded by a retinue of yoginī-s and vīra-s. (ii) in one formulation of the Tantrasadbhāva his 81 segmented Vidyārāja is presented similarly to that in the Svacchandatantra. (iii) In the classic formulation of the Tantrasadbhāva (followed by Abhinavagupta), there is an ascending series of Bhairavī-s and Bhairava-s starting with Aparā with Navātman-bhairava, Parāparā with Ratiśekhara-bhairava and Parā with Bhairava-sadbhāva. Notably, the visualization of the Rudra deity of the Vyomavyāpin conjoined with the kavaca and the astra in the Pāśupata-tantra is quite similar to that of Navātman in the Paścimāṃnāya.
Across these Bhairava traditions and certain saiddhāntika references (e.g., that of Aghoraśiva-deśika), a nine-fold composite bīja of Navātman is specified. It is given in multiple variant forms even within the same tradition, e.g., the Paścimāṃnāya. However, we see some geographical proclivities in terms of the preferred form in prayoga texts: r-h-k-ṣ-m-l-v-y-ūṃ = rhkṣmlvyūṃ (Kashmirian) or Śambhu form h-s-kṣ-m-l-v-r-y-ūṃ = hskṣmlvryūṃ / Śakti form: s-h-kṣ-m-l-v-r-y-īṃ = śkṣmlvryīṃ (Nepal, Vaṅga, South India). These 9 elements of the bīja are said to map onto 9 pada-s each yielding the 81 segments of the Vidyārāja alluded to in the Svacchandra-tantra and specified in the Dūtī-cakra (interestingly also associated with the god Viṣṇu manifesting as Ananta/the Saṃkarṣaṇa) of the Kubjikā-mata-tantra (14.62 onward). Therein, we get the below emanational series for the ekāśitipadāḥ of the Navātman mantra as: Viṣṇu $\to$ (1) Ananta $\to$ (2) Kapāla, (3) Caṇḍalokeśa/Caṇḍeśa, (4) Yogeśa, (5) Manonmana, (6) Hāṭakeśvara, (7) Kravyāda, (8) Mudreśa and (9) Diṅmaheśvara. Each of these 9 then emanates a set of 9 dūtī-s who comprise the body of Navātman:
Ananta $\to$ (1..9) Bindukā, Bindugarbhā, Nādinī, Nādagarbhajā, Śaktī, Garbhinī, Parā, Garbhā and Arthacāriṇī.
Kapāla $\to$ (10..18) Suprabuddhā, Prabuddhā, Caṇḍī, Muṇḍī, Kapālinī, Mṛtyuhantā, Virūpākṣī, Kapardinī, Kalanātmikā
Caṇḍeśa $\to$ (19..27) Caṇḍamukhī, Caṇḍavegā, Manojavā, Caṇḍākṣī, Caṇḍanirghoṣā, Bhṛkuṭī, Caṇḍanāyikā, Caṇḍīśanāyikā.
Yogeśa $\to$ (28..36) Vāgvatī, Vāk, Vāṇī, Bhimā, Citrarathā, Sudhī, Devamātā, Hiraṇyakā, Yogeśī.
Manonmana $\to$ (37..45) Manovegā, Manodhykṣā, Mānasī, Mananāyikā, Manoharī, Manohlādī, Manaḥprīti, Maneśvarī, Manonmanī.
Hāṭakeśvara $\to$ (46..54) Hiraṇyā, Suvarṇā, Kāñcanī, Hāṭakā, Rukmiṇī, Manasvī, Subhadrā, Jambukāyī, Bhaṭṭanī.
Kravayāda $\to$ (55..63) Lambinī, Lambastanī, Śuśkā, Pūtanā, Mahānanā, Gajavaktrā, Mahānāsā, Vidyut, Kravyādanāyikā.
Mudreśa $\to$ Vajriṇī, Śktikā, Daṇḍī, Khaḍginī, Pāśinī, Dhvajī, Gādī, Śūlinī, Padmī.
Diñmaheśvara $\to$ Indrāṇī, Hutāśanī, Yāmyā, Nirṛtī, Vāruṇī, Vāyavī, Kauberī, Īśānī, Laukikeśvarī.
The 9 pada-s corresponding to Ananta are 9 repetitions of the composite Navātaman-bīja with the consonantal elements resolved with an `a’-vowel. The remaining 8 sets of 9 pada-s are derived by taking the 9, 8, 7…2 of the resolved consonantal elements from the above Navātaman as the first pada followed by 8 others in the form of their respective first consonantal element conjoined with the 8 bīja-s: āṃ, īṃ, ūṃ, ṝṃ, ḹṃ, aiṃ, auṃ, aḥ. Interestingly, given the association with Viṣṇu, the Kubjikāmata also teaches that the deity might be worshiped as Navātma-Viṣṇu, suggesting potential interaction with the Pāñcarātrika tradition (c.f. Navābja-Viṣṇu-maṇḍala). This reinforces the ancient connection between the worship of Ananta/the Saṃkarṣaṇa and the śaiva traditions that we have discussed before. More generally, it also parallels the “Rudraization” of ancient deities in the śaiva-mantramarga: one striking example is the worship of the ancient I-Ir deity Mitra as a Bhairava in the Mātṛcakra of the Paścimāṃnāya, which we hope to discuss in greater detail in a separate note.
Thus, from the above discussion, it might be concluded that a nine-fold form of Rudra was likely known to the pre-mantramārga śaiva-s that expressed itself in the form of two distinct mantra-s the Vyomavyāpin and the Navātman, which were added to the more ancient set of pañcabrahma-mantra-s. It is possible that such a nonadic conception of Rudra had ancient roots in the nine-fold manifestation of Rudra mentioned in the Śatapatha-brāhmaṇa 6.1.3.18 of the Vājasaneyin-s: tāny etāny aṣṭāv agni [=rudra] -rūpāṇi । kumāro navamaḥ saivāgnis trivṛttā ॥. Both these 81-pada mantra-s continued to be important right from the beginning of the mantramārga. Interestingly, in the Paścimāṃnāya, the retinue of siddha-s worshiped in the cakra of Navātman first features Bhṛgu, the founder of the Atharvan tradition, followed by Lakulīśa. This illustrates a memory in this later śaiva stream of its roots in the Pāśupata tradition. Thus, it is not impossible that the Pāśupata-tantra, despite being influenced by the mantramārga retained a memory of the old presence of the Vyomavyāpin in the Pāśupata tradition.
There are other potentially archaic connections suggested by both these nine-fold mantra-s of Rudra: the Vyomavyāpin literally means that which pervades space. This immediately brings to mind the ancient Indo-European deity Vāyu who has one foot in the Rudra class. Indeed, in the Eastern-Iranic world the Rudra-class deity, while iconographically identical to the Indic expression, is centered on the cognate of Vāyu, Vayush Uparikairya. Even in the Indo-Aryan sphere, the Rudra-s in the atmosphere are placed with Vāyu. In the Vrātya texts of the Atharvaveda and the Sāmaveda, Rudra is described as the god who animates like Vāyu-Vāta: the verb used is sam-īr- which is also used for Vāyu-Mātariśvan. Navātman as the deity of the Kha-cakra-maṇḍala also implies his pervading of space. Remarkably, the Vyomavyāpin has a segment featuring a tetrad of the verbal root dhū of ancient IE provenance. It means to blow or to cause things to be agitated by being blown at. Elsewhere in the IE world, its cognates mean storm, breath, soul, and wafting of odors — all activities associated with Vāyu-Vāta. Thus, we posit that the Vyomavyāpin retains memories of the intimate link between the Rudra class and Vāyu seen in the Indo-Iranian borderlands.
The Vyomavyāpin with the kavaca and astra mantra-s as specified in the Pāśupata-tantra:
oṃ $\times$ 5 haṃ oṃ namo vyomavyāpine vyomarūpāya sarvavyāpine śivāya anantāya anāthāya anāśritāya dhruvāya śāśvatāya yogapīṭhādisaṃsthitāya nityayogine dhyānāhārāya । oṃ namaḥ śivāya sarvaprabhave śivāya īśāna-mūrdhnāya tatpuruṣa-vaktrāya aghora-hṛdayāya vāmadeva-guhyāya sadyojāta-mūrtaye । oṃ namaḥ guhyādi-guhyāya goptre anidhanāya sarvavidyādhipāya jyotīrūpāya parameśvaraparāya । acetanācetana । vyomin $\times$ 2 । vyāpin $\times$ 2 । arūpin $\times$ 2 । prathama $\times$ 2 । tejas tejaḥ । jyotir jyotiḥ । arūpa । anagne । adhūma । abhasma । anāde । nānā nānā । dhū dhū dhū dhū । oṃ bhūr bhuvaḥ svaḥ । anidhana । nidhanodbhava । śiva । śarva । sarvapara । maheśvara । mahādeva । sadbhāveśvara । mahātejaḥ । yogādhipate muñca muñca pramatha pramatha । śiva । śarva । bhavodbhava vidhya vidhya । vāmadeva । sarva-bhūta-sukhaprada । sarva-sāṃnidhyakara । brahmā-viṣṇu-rudra-para । anarcita $\times$ 2 । asaṃsthita $\times$ 2 । pūrvasthita $\times$ 2 । sākṣin $\times$ 2 । turu $\times$ 2 । piṅga $\times$ 2 । pataṅga $\times$ 2 । jñāna $\times$ 2 । śabda $\times$ 2 । sūkṣma $\times$ 2 । śiva । śarva । sarvada । oṃ namaḥ । śivāya oṃ hrīṃ śivāya oṃ haṃ hrīṃ śivāya oṃ namo namaḥ ॥
oṃ namaḥ sarvātmane parāya parameśvarāya parāya yogāya । yogasambhavakara sadyobhavodbhava vāmadeva sarva-karma-praśamana sadāśiva namo ‘stu te svāhā । suśiva śiva namo brahmaśirase । śiva-hṛdaya-jvālini jvālinyai svāhā । oṃ śivātmakam mahātejaḥ sarvajñam prabhum avyayam । āvartayen mahāghoraṃ kavacaṃ piṅgalaṃ śubham । āyāhi piṅgalam mahākavacaṃ śivājñayā hṛdayam bandha । jvala ghūrṇa saṃsphura kiri śakti-vajradhara vajrapāśa vajraśarīra mama śarīram anupraviśya sarvaduṣṭān stambhaya huṃ phaṭ । oṃ jūṃ saḥ jyotīrūpāya namaḥ । oṃ prasphura ghora-ghoratara-tanu-rūpa caṭa daha vama bandha ghātaya huṃ phaṭ ॥
The core Vyomavyāpin as per the saiddhāntika text, the Mataṅgapārameśvara-tantra:
oṃ namo vyomavyāpine vyomarūpāya sarvavyāpine śivāya anantāya anāthāya anāśritāya dhruvāya śāśvatāya yogapīṭhasaṃsthitāya nityaṃ yogine dhyānāhārāya । oṃ namaḥ śivāya sarvaprabhave śivāya īśāna-mūrdhnāya tatpuruṣa-vaktrāya aghora-hṛdayāya vāmadeva-guhyāya sadyojāta-mūrtaye । oṃ namaḥ guhyāti-guhyāya goptre nidhanāya sarvavidyādhipāya jyotīrūpāya parameśvaraparāya । acetanācetana । vyomin $\times$ 2 । vyāpin $\times$ 2 । arūpin $\times$ 2 । prathama $\times$ 2 । tejas tejaḥ । jyotir jyotiḥ । arūpa । anagne । adhūma । abhasma । anāde । nā nā nā । dhū dhū dhū । oṃ bhūḥ । oṃ bhuvaḥ । oṃ svaḥ । anidhana । nidhana। nidhanodbhava । śiva । sarva । paramātman । maheśvara । mahādeva । sadbhāveśvara । mahātejaḥ । yogādhipate muñca muñca prathama prathama । śarva śarva । bhava bhava । bhavodbhava । sarva-bhūta-sukhaprada । sarva-sāṃnidhyakara । brahmā-viṣṇu-rudra-para । anarcita $\times$ 2 । asaṃstuta $\times$ 2 । pūrvasthita $\times$ 2 । sākṣin $\times$ 2 । turu $\times$ 2 । piṅga $\times$ 2 । pataṅga $\times$ 2 । jñāna $\times$ 2 । śabda $\times$ 2 । sūkṣma $\times$ 2 । śiva । śarva । sarvada । oṃ namo namaḥ । oṃ śivāya namo namaḥ । oṃ [namo namaḥ] ॥
A personal note
We first heard the Vyomavyāpin as a kid being recited by a mantrin in a temple of Rudra founded by the Kālāmukha-s in the Karṇāṭa country. He had recited the pañcabrahma-s and other incantations from the Yajurveda that we knew, but this one was entirely unfamiliar to us. As soon as we heard the words vyomin $\times$ 2 । vyāpin $\times$ 2 ।, we experienced a special gnosis of the pervasion of space by Rudra in two ways. We wondered what this mantra was — we could not find it in any of the manuals our grandfather, or we had at that time. By some coincidence a fortnight or so later we happened to lay our hands on N.R. Bhat’s edition of the Mataṅgapārameśvara in the library, and we saw it right there. Unfortunately, there was no access to a copying device there, and we did not have writing material at hand. Hence, it just remained that until we met R1’s father who gave us more information about its rahasya-s.
## Bhāskara-II’s polygons and an algebraic approximation for sines of pi by x
Unlike the Greeks, the Hindus were not particularly obsessed with constructions involving just a compass and a straightedge. Nevertheless, their pre-modern architecture and yantra-s from the tāntrika tradition indicate that they routinely constructed various regular polygons inscribed in circles. Of course, the common ones, namely the equilateral triangle, square, hexagon, and octagon are trivial, and the earliest preserved geometry of the Hindus is sufficient to construct these. The pentagon and its double the decagon are a bit more involved but are still constructible by the Greek compass and straightedge method; however, few have looked into how the Hindus might have constructed it. These aside, we do have multiple examples of yantra-s with heptagons and nonagons. A particularly striking example is the wide use of the nonagon in yantra-s (likely related to the early Śrīkula tradition of 9 yoginī-s and the division of the Vyomavyāpin mantra of the saiddhāntika-s) found at the Marundīśvara temple near Chennai. The kaula tradition of the Kubjikā-mata-tantra has a sūtra that mentions a yantra with multiple regular polygons relating to pacifying the seizure by Ṣaṣṭhī and others (trikoṇaṃ navakoṇaṃ ca ṣaṭkoṇaṃ maṇḍalākṛtiḥ |). The heptagon and the nonagon cannot be constructed using just a compass and a straightedge. To construct them precisely, one would require a means of accurately drawing conics (other than circles and straight lines) of particular specifications. While Archimedes invented a machine to draw ellipses, and examples of ellipses are occasionally encountered in early Hindu architecture, the technology for the easy generation of desired conics was unlikely to have been widely available to premodern architects. Hence, the Hindus should have constructed their regular polygons, including heptagons and nonagons through other means.
Figure 1.
It is easy to see (Figure 1) that for a circle of diameter $d$ the side $s$ of an inscribed regular polygon of $n$ sides is $s=d\sin\left(\tfrac{\pi}{n}\right)$. Thus, if one does not insist on a compass and straightedge construction, one can easily draw any polygon as long as one has a sine table. The Hindus have had a long history of generating sine tables as well as algebraic functions that approximate the sine function to varying degrees of accuracy. Thus, one would expect that this was the most likely route they took. This still leaves us with the question of how exactly they did it in practice. A likely answer for this comes from Bhāskara-II’s Līlāvatī though this knowledge appears to have been lost in some parts of India in the late medieval period.
In Līlāvatī 206-209, Bhāskara gives a table for the sides of the inscribed polygons in three anuṣtubh-s (see below) followed by a numerical example (L 209). We have resolved the saṃdhi with a + for ease of reading the numbers:
tri-dvyaṅkāgni-nabhaś-chandrais tribāṇāṣṭayugāṣṭabhiḥ |
vedāgni-bāṇa-khāśvaiś ca kha-khābhrābhra-rasaiḥ kramāt || L 206
tri+dvi +aṅka +agni +nabhaś +candrais (103923)
tri +bāṇa +aṣṭa +yuga +aṣṭabhiḥ (84853) |
veda +agni +bāṇa +kha +aśvaiś (70534)
ca kha +kha +abhra +abhra +rasaiḥ (60000) kramāt ||
bāṇeṣu-nakha-bāṇaiś ca dvi-dvi-nandeṣu sāgaraiḥ |
ku-rāma-daśa-vedaiś ca vṛtta-vyāse samāhate || L 207
bāṇa +iṣu +nakha +bāṇaiś (52055) ca
dvi +dvi +nanda +iṣu +sāgaraiḥ (45922)
ku +rāma +daśa +vedaiś ca (41031)
kha-kha-khābhrārka saṃbhakte labhyante kramaśo bhujā |
vṛttāntar tryasra-pūrvāṇāṃ navāsrāntam pṛthak pṛthak || L 208
vṛtta-vyāse samāhate kha +kha +kha +abhra +arka (120000) saṃbhakte labhyante kramaśo bhujā
vṛtta-antar tri +asra-pūrvāṇām nava +asra-antam pṛthak pṛthak
Essentially, the above means that one should multiply the diameter of the circle with the numbers specified in the above table in verse form and divide them by 120000. This gives, in order, the sides of the inscribed regular polygons from a triangle to a nonagon. Thus, the ratios of these numbers provide rational approximations for $\sin\left(\tfrac{\pi}{n}\right)$ for $n=3..9$. We compare these to the actual values in the below table:
These rational approximations provided by Bhāskara are best for a triangle, square, pentagon, hexagon and octagon. These are the angles for which he derives closed forms in his Jyotpatti (On the generation of sines) and correspond to the constructible polygons of the Greek tradition. The sines of the heptagonal and nonagonal angles which have no closed forms were obtained using serial interpolations or a sine-approximating function.
In terms of the latter, Bhāskara specifies a formula for the length of a cord corresponding to an arc in a Vasantalikā verse that can be used to obtain an algebraic function approximating $\sin\left(\tfrac{\pi}{x}\right)$:
cāpona-nighna-paridhiḥ pratham-āhvayaḥ
syāt pañcāhataḥ paridhi-varga-caturtha-bhāgaḥ |
ādyonitena khalu tena bhajec catur-ghna
vyāsāhatam prathamam āptam iha jyakā syāt || L 210
cāpa +ūna-nighna-paridhiḥ prathama +āhvayaḥ
syāt pañca +āhatas paridhi-varga-caturtha-bhāgaḥ |
ādya +ūnitena khalu tena bhajet +catur +ghna-
vyāsa +āhatam prathamam āptam iha jyakā syāt ||
The circumference is reduced by the arc and multiplied by the arc: this is called the prathama.
One-fourth of the circumference squared is multiplied by 5
This is then reduced by the prathamā. The prathamā multiplied by 4
and the diameter should be divided by the above result. The fraction thus obtained is the chord.
Let the diameter of the circle be $d$, its circumference $c$, the length of the given arc $a$ and $y$ its chord. Then the above can be written in modern notation as:
$y= \dfrac{4da(c-a)}{\frac{5c^2}{4}-a(c-a)} = \dfrac{16da(c-a)}{5c^2-4a(c-a)}$
Now, the arc can be written as the $x^{\mathrm{th}}$ fraction of the circumference, $\therefore a=\tfrac{c}{x}$. By plugging this into the above equation, we get:
$y= \dfrac{16d\tfrac{c}{x}(c-\tfrac{c}{x})}{5c^2-4\tfrac{c}{x}(c-\tfrac{c}{x})}$
This allows us to eliminate $c$ and write
$y=\dfrac{16d\left(x-1\right)}{5x^{2}-4\left(x-1\right)}$
Thus, we get an algebraic function approximating $y=\sin\left(\tfrac{\pi}{x}\right)$:
$y=\dfrac{16\left(x-1\right)}{5x^{2}-4\left(x-1\right)}$
Figure 2.
In the below table we show the values of the polygon sines for $n=3..9$ generated by this formula and compare them with the earlier table provided by Bhāskara and the actual value:
We can see that the values from this function are more approximate than those provided by the table. Thus, it is clear that Bhāskara did not use this algebraic function to generate his table. However, the fact that he provides this formula after the table indicates that he meant this as an alternative method to get rational approximations for the polygonal sines. Such a method too could have been readily used by artists/artisans in their polygonal constructions in architecture and yantra preparation.
## Origins of the serpent cult and Bhāguri’s snake installation from the Sāmaveda tradition
Mathuran Nāga installations
From the few centuries preceding it down to the first few centuries of the Common Era we see numerous installations of snake deities, i.e., Nāga-s, at various archaeological sites throughout northern India (most famously at the holy city of Mathurā). Comparable, but usually smaller Nāga installations continue to this date in South India, usually in association with śaiva and kaumāra shrines. A related icon is that of the great Sātvatta Vaiṣṇava deity Balabhadra, who is depicted with a hooded snake canopy. Tradition holds that he was the incarnation or homomorph of the snake of Viṣṇu, often named Ananta. Given that Viṣṇu was the “time-god” par excellence, we hold that the snake imagery (the coils) associated with his bed is a metaphor for periodicities in time — diurnal, lunar, solar and precessional cycles. In this note, we explore the connections of these later manifestations of the serpentine cult with the Vedic roots of snake worship (Ahi Budhnya of the earliest Vedic tradition), with probable deeper Indo-European antecedents and broad Eurasiatic ramifications.
Saṃkarṣaṇa installations at Tumbavana (L) and Mathurā (R)
On one hand, we have the śrauta sarpa-sattra outlined in the Sāmaveda brāhmaṇa-s, which is modeled after a ritual supposed to have been performed by the Nāga-s to gain their venom. The sarpa-sattra in an inverted form, viz., the ritual of Janamejaya to destroy the Nāga-s who were responsible for his father’s death, is the frame story of the national epic the Mahābhārata. The core story of the Mahābhārata itself is permeated with simultaneous inter-generational conflicts and marriages between the Nāga-s and the Pāṇḍu-s. On the other hand, we have the gṛhya sarpabali that is enjoined in various gṛhyasūtra-s and certain vidhāna-s. The sarpabali or the offering to the snake deities is performed when the full moon occurs in Śravaṇā (the ecliptic division associated with the longitude of Altair). This bali usually coincides with the Indian Southwest Monsoon. Along with this bali the ritualist and his family sleep on a raised bed until the Āgrayaṇa ritual. This, along with the contents of the ritual, indicate that its primary function was protection from snakes that might enter houses during the monsoons due to the flooding of their lairs. While the rite is found in most gṛhyasūtra-s, that of the Hiraṇyakeśin school associated with the Taittirīyaka tradition gives a rather detailed account of the rite. At first, the ritualist makes oblations of unbroken grains, unbroken fried grains, coarsely ground grains, leaves and flowers of the Kimsuka tree to Agni Pārthiva, Vāyu Vibhumant, Sūrya Rohita and Viṣṇu Gaura:
namo .agnaye pārthivāya pārthivānām adhipataye svāhā । namo vāyave vibhumata āntarikṣāṇām adhipataye svāhā । namaḥ sūryāya rohitāya divyānām adhipataye svāhā । namo viṣṇave gaurāya diśyānām adhipataye svāhā ॥
After these oblations, the snakes of the earth (the real ones), the snakes of the atmosphere (lightning), the snakes of the heavens (Āśleṣā $\approx$ the constellation of Hydra), and those of the directions (the serpent ogdoad) are worshiped with the famous Yajuṣ-es beginning with namo astu sarpebhyaḥ … (TS 4.2.8.3) followed by the bali incantations ye pārthivāḥ sarpāstebhya imaṃ baliṃ harāmi । ya āntarikṣāḥ । ye divyaḥ । ye diśyāḥ ॥
Following the bali, the ritualist goes thrice around his dwelling in a circle corresponding to the radius that he wishes to keep the snakes away from sprinkling water from a pot while uttering the below incantation (the Pāraskara-gṛhya-sūtra instead prescribes drawing a line with a white pigment):
apa śveta padā jahi pūrveṇa cāpareṇa ca । sapta ca mānuṣīr imās tisraś ca rājabandhavaiḥ । na vai śvetasyābhyācareṇāhir jaghāna kaṃ cana । śvetāya vaidarvāya namo namaḥ śvetāya vaidarvāya ॥
Smite away, O white one, with your foot, fore and hind, these seven women with the three of the king’s clan. No one indeed, in the roaming ground of the white one the snakes have ever killed.
In the above incantation, the king and his clan evidently refer to the Nāga king and his folks and the women to the Nāgakanya-s. The white one is described as having fore and hind feet. This implies that he is none other than the white snake-killing horse (Paidva) given to Pedu by the Aśvin-s:
yuvaṃ śvetam pedava indrajūtam ahihanam aśvinādattam aśvam । RV 1.118.9a
paidvo na hi tvam ahināmnāṃ hantā viśvasyāsi soma dasyoḥ ॥ RV 9.88.4c
However, it is notable that the Viṣṇu deity specific to this ritual is Viṣṇu Gaura, or the white one, paralleling the color of the horse. In this regard, we may also point to the role of Viṣṇu in the Aśvamedha rite. After the horse has successfully wandered for an year, the emperor undergoes consecration. In preparation for the sacrifice, the oblations known as Vaiśvadeva culminating in the pūrṇāhuti are offered over a period of seven days. On days one and two he offers to Ka Prajāpati; on day three to Aditi; on day four to Sarasvatī; on day five to Puṣan; day six to Tvaṣṭṛ Viśvakarman; on day 7 to Viṣṇu the with the purṇāhuti. As per the Taittirīya-śruti, two Viṣṇu deities are invoked in the rite in addition to the standalone Viṣṇu:
viṣṇave svāhā । viṣṇave nikhuryapāya svāhā । viṣṇave nibhūyapāya svāhā ॥
These peculiar names of the two Viṣṇu deities, Nikhuryapa and Nibhūyapa are rather enigmatic. Since they are unique to the Aśvamedha, we posit that Nikhuryapa could be Viṣṇu as the protector of the hoofs (khura: hoof), whereas Nibhūyapa could be Viṣṇu as the protector of the stallion which makes the herds increase. These equine associations of Viṣṇu in the Aśvamedha raise the possibility that the white snake-smiting horse was also associated with the White Viṣṇu of the ritual. Interestingly, the color the Saṃkarṣaṇa is also said to be white. Moreover, the later tradition starting from the Mahābharata preserves strong equine connections for Viṣṇu as Hayaśiras. Thus, in the least, one could say that the sarpabali ritual established an early connection between Viṣṇu and offerings to the snakes, which could have presaged its augmentation in the later tradition.
Other traditions associated with the Vedic sarpabali were also expanded in the later serpent cult. Evidence for this comes from an adaptation of the ritual found in the Yajurvidhāna-sūtra-s of the Vājasaneyin-s (YVS 15.8-11):
namo .astu sarpebhya iti ghṛta-pāyasaṃ nāgasthāne juhuyāt । suvarṇam udpadyate ॥ vṛṣṭyarthe śikhaṇḍyādīñ juhuyāt vṛṣṭir bhavati । atasī-puṣpair mahāvṛṣṭir bhavati ॥
The sarpa-yajuṣ is deployed with oblations of ghee and milk pudding in the locus of the Nāga-s in order to obtain gold. For rains he offers oblations of peacock feathers; for torrential rains, he offers flax flowers. Thus, in this vidhāna deployment of the sarpabali mantra, we see a reworking for obtaining gold (a connection already mentioned in the Mahābhārata 5.114.4 “vulgate”: he guards the wealth/gold generated by Agni for Kubera) and rain (a connection possibly going back to Ahi Budhnya in the Ṛgveda: RV 4.55.6; RV 7.34.16; Taittirīya-Saṃhitā in 1.8.14). The Yajurvidhāna-sūtra-s also describe a rite with a trident and a liṅga made of cow dung in the fire-shed using this mantra for the rain-making and fearlessness (namo .astu sarpebhya iti tisṛbhir arghyaṃ dadyād agnyāgāre gomaya-liṅgaṃ pratiṣṭhāpya pañcagavyena saṃsnāpya dakṣiṇataḥ śūlaṃ nikhanet । punaḥ sahasraṃ japet । suvarṇa-śataṃl labhet siddhaṃ । karmety ācakṣate vṛṣṭau śikhaṇḍān atasīpuṣpāṇi vā yuñjantīti । mahābhaye japed abhayaṃ bhavati ॥). Similarly, a rite using an iron trident is offered for the subjugation of nāga-s with a mantra to Agni (ajījana iti rahasyo mantra (RV 3.29.13) etena nāgā vaśam upayānti । lauhaṃ triśūlagṃ sahasrābhimantritaṃ kṛtvā dakṣiṇa-pādenākramya payo-dadhi-madhu-ghṛtair ayutaṃ hutvā vikṛta-rūpā striya uttiṣṭhanti । kim asmābhiḥ kartavyam iti bruvantyo abhirucikāmena tām ājñāpayet ॥). This later rite is developed further within the śaiva context in the Jayadrathayāmala-tantra. These objectives outlined in the Vidhāna were greatly expanded in early śaiva and bauddha traditions (also seen in the Indic-influenced Cīna dragon traditions). These themes are brought together rather dramatically in the story of the Drāviḍa mantravādin, the Nāga Mahāpadma residing in a Kashmirian lake, and the king Jayāpīḍa narrated by Kalhaṇa in the Rājataraṃgiṇī (4.593 onward).
However, the question remains as to whether the sarpabali of the old Gṛhya tradition had any connection with the installation of the images of Nāga and Saṃkarṣana seen at the archaeological sites. A potential transitional rite describing a Vaidika snake installation comes from a now apparently extinct Sāmaveda tradition, namely the Gautama school, which seems to have been practiced in some form in the Karṇāṭa country till around 1600-1700 CE. The Gautama-gṛhya-pariśiṣṭa furnishes a detailed Nāga-pratiṣṭha ritual attributed to Bhāguri (GGP 2.12):
-The ritual is to be performed on the 12th tithi of a śuklapakṣa when the moon is in a devanakṣatra (i.e., Northern half of the ecliptic) or during the northern course of the sun or on an auspicious nakṣatra.
-On the day before the installation rite, the ritualist brushes his teeth, takes a bath with water from a tīrtha (holy ford) and having performed the saṃkalpa for the installation, immerses the image in water.
-He chooses an ācārya who delights in right conduct and of peaceful temperament and performs the rite via his instruction.
-Having cleansed the spot for installation, the ācārya washes his feet, performs ācamana, and having seated himself, performs prāṇāyāma and saṃkalpa.
-He recites the puṇyāha incantations (hiraṇyavarṇāḥ…) and sprinkles the image with water. He recites the triple vyāhṛti-s and lustrates the image with the five bovine products.
-He washes the images with clean water utter āpo hi ṣṭha… (SV-Kauthuma 1837) and tarat sa mandī dhāvati… (SK-K 500, 1057)
-He utters oṃ and lustrates the image with water in which gold flakes, the shoots of dūrva grass and palaśa leaves have been placed. He offers flowers and dūrva grass at the feet of the image.
-He utters the sāvitrī or oṃ and cloaks the image with newly woven unwashed clothing.
-He offers special naivedya and recites svasti na indro… incantation. Thereafter, he immerses the image in a river while singing the Varuṇa-sāman.
-He rises the next day and performs his nityakarmāṇi, he proceeds with the ācārya and assistant ritualists (like in the śrauta ritual) to the place where he has immersed the image. There, they bring out the image while reciting praitu brahmaṇas patiḥ pra devy etu sūnṛtā ।…(SV-K 56). Then they install it at the designated spot and perform prāṇāyāma and saṃkalpa.
-They again lustrate the image with the five bovine products while reciting oṃ nāgāya namaḥ. Then they wash it with clean water and cloak it with a new dress. They decorate it with scented unguents and flowers.
-Then they perform nyāsa both of the self and the image thus: oṃ nāgāya namaḥ । hṛdayāya namaḥ । oṃ nāgāya namaḥ । śirase namaḥ । oṃ nāgāya namaḥ । śikhāyai namaḥ । oṃ nāgāya namaḥ । kavacāya namaḥ । oṃ nāgāya namaḥ । netratrayāya namaḥ । oṃ nāgāya namaḥ । astrāya namaḥ ।
Then he does a dhyāna of the serpentine deity:
sarpo raktas trinetraś ca dvibhujaḥ pītavastragaḥ ।
phaṇkoṭidharaḥ sūkṣmaḥ sarvābharaṇa-bhūṣitaḥ ॥
-He then measures out a droṇa of paddy, clean rice and sesame seeds and spreads them out one over the other. On them, he draws out an eight-petaled lotus and installs a pitcher on top of it.
-Inside the pitcher, he places five each of barks, shoots, soils, gemstones, bovine products, ambrosial sweets, scents, kinds of rice, medicinal herbs, and unguent powders.
-He drapes the pitcher with a new piece of cloth and invokes Nāgeśa in it:
oṃ bhūḥ । puruṣam āvāhayāmi । oṃ bhuvaḥ । śeṣam āvāhayāmi । ogṃ suvaḥ । anantam āvāhayāmi ॥
-He then provides the deity with the 16-fold sacraments uttering oṃ anantāya namaḥ for each.
-He then worships the deity with the following mantra:
āyātu bhagavān śeṣaḥ sarva-karma-sanātanaḥ ।
ananto mat priyārthāya mad anugraha-kāmyayā ॥
-The four brāhmaṇa ritual assistants and the ācārya touch the pitcher and recite āpo hi ṣṭha…
-Then they recite the Puruṣa hymn.
-Then they sing the following Sāman-s: Sarpa, Vāmadevya, Rathantara, Bṛhat, Jyeṣṭha and Bhāruṇḍa.
-Then they recite oṃ namo brahmaṇe…bṛhate karomi (Taittirīya āraṇyaka 2.13.1).
-To the west of the pitcher, the ācārya sets up a sthaṇḍila (fire altar). To the north of the altar, he collects twelve materials for the pradhānāhuti-s (main oblations) and offers them with the following incantations into the fire followed by a svāhā and the tyāga formula: idaṃ anantāya na mama ।
aghorebhyo ‘tha…: cooked rice
tat puruṣāya vidmahe…: fried rice
īśānaḥ sarvavidyānām…: saktu flour
oṃ nāgāya namaḥ: milk
hṛdayāya svāhā: barley
śirase svāhā: sesame
śikhāyai svāhā: sugarcane
kavacāya svāhā: banana
netra-trayāya svāhā: jackfruit
astrāya svāhā: mustard
-25 oblations are made of each item. Thereafter, he offers sesame $8 \times, 28 \times, 108 \times$ with oṃ bhūr-bhuvaḥ svaḥ svāhā ।.
-He then worships the serpentine deity with the below incantation calling on him to accept all the oblations:
tvām eva cādyam puruṣam purāṇam ।
gṛhṇīṣva māṃ rakṣa jagannivāsa ॥
-Then he gives the brāhmaṇa-s their fees and sings the Vāmadevya sāman.
-Then to the singing of the Sarpasāman he lustrates the image of Nāgeśa with the contents of the pitcher, followed by the five bovine products, the five ambrosial sweets, curds, milk, coconut juice, whey, sugarcane juice and finally scented water.
-Then he recites the Mantra Brāhmaṇa 2.8.6, utters oṃ nāgāya namaḥ thrice, and offers pādya to the image.
-He recites annasya rāṣṭrir asi… (MB 2.8.9) and offers arghya.
-With yaśo .asi… (MB 2.29.16) he offers ācamana.
-With yaśaso yaśo .asi… (MB 2.8.11) he offers madhuparkam.
-With oṃ nāgāya namaḥ he successively offers, lower garments, an upavīta, upper garments, and ornaments.
-With gandhadvārāṃ durādarṣām… he offers scents.
-With īḍiṣvā… (SV-K 103) he offers incense.
-With pavamānaḥ… dyad (SV-K 484) he offers a lamp.
-The ācārya drapes the image with a new robe and also himself.
-With śukram asi jyotir asi tejo .asi (in TS 1.1.10) he takes up a golden needle. With viśvataścakṣur uta viśvatomukho viśvatobāhur uta viśvataspāt । (RV 10.81.3a) and uttering oṃ he activates the eyes of the image with the golden needle.
-He touches the heart of the image and recites the prāṇapratiṣṭha incantation invoking the goddess Anumati $28 \times$ to infuse the image with consciousness:
asunīte punar asmāsu cakṣuḥ
punaḥ prāṇam iha no dhehi bhogam ।
jyok paśyema sūryam uccarantam
anumate mṛḻayā naḥ svasti ॥ (RV 10.59.6)
-Maidens of good disposition display lamps to the image and a cow is led before it.
-The image is placed over a deposit of a gemstone, pearl, coral, gold and silver atop which a white cloth has been spread.
-Having decorated the image, the yajamāna worships the deity with the incantations: oṃ śeṣāya namaḥ । oṃ bhūdharāya namaḥ । om anantāya namaḥ ।
-He then offers naivedya of milk pudding, cooked rice, sesame rice, turmeric rice, apūpa cake, pūrikā bread, and the śarkarāḍhya sugar pastry. Thereafter, he offers betel leaves.
-Having given gifts to the ācārya and his assistant brāhmaṇa-s, he takes the image and has it permanently installed at a temple of Rudra or Viṣṇu, or under a pipal tree while reciting the mantras:
udgāteva śakune sāma gāyasi
brahmaputra iva savaneṣu śaṃsasi ।
vṛṣeva vājī śiśumatīr apītyā
viśvato naḥ śakune puṇyam ā vada ॥ (RV 2.43.2)
-He then worships the serpentine deity performing 12 namaskāra-s with the following incantations:
anantāya namaḥ । nāgāya namaḥ । puruṣāya namaḥ । sarpebhyo namaḥ । viśvadharāya namaḥ । śeṣāya namaḥ । viśvambharāya namaḥ । saṃkarṣaṇāya namaḥ । balabhadrāya namaḥ । takṣakāya namaḥ । vāsukaye namaḥ । śivapriyāya namaḥ ।
-He concludes by feeding 12 brāhmaṇa-s of good learning and character and educating children.
-He who does such a snake installation obtains 8 children, whatever he prays for, and the higher realms.
There are several notable points regarding this ritual:
-Its essential details, including the new mantra-s specifically spelt out in the text, closely relate to other iconic sthāpana rites specified in the late Vedic texts. These include: 1. The installation and worship of Skanda (AV Skandayāga and Dhūrtakalpa of the Bodhāyana-pariśiṣṭa); 2. The black goddess of the Night, Rātrī-devī (AV-pariśiṣṭa 6); 3. The Bhārgava Brahma-yāga (AV-par 19b), where an image of the god Brahman is installed; 4. Gośānti (AV-par 66), where an image of Rudra fashioned out of cow dung is installed in the midst of a maṇḍala for the protection of cattle. Similarly, a metal/stone image of Rudra is installed in the Bodhāyana-pariśiṣṭa and also deployed by the Vādhūla-s in their Vādhūlagṛhyāgama, a versified version of their Gṛhyasūtra-s and pariśiṣṭa-s. 5. Installations of the images of Viṣṇu and Durgā according to the Bodhāyana-pariśiṣṭa-s. The former is also specified in the Vādhūla collection. Thus, it may be inferred that the Nāga-pratiṣṭha of the Gautama-gṛhya-pariśiṣṭa is of the same genre and likely the same temporal period marking the tail end of the Vedic age and the transition to the Tantro-Paurāṇic age.
-Here the character of the Nāga has evolved from that seen in the earlier gṛhya sarpabali. While the sarpa-s are venerated in the sarpa-yajuṣ they are also expelled by means of the white horse of Pedu and the perimeter of safety is established. However, in the Nāga-pratiṣṭha the snake deity is not just clearly positive but is also identified with the Puruṣa himself.
-The text presents an early example of the ṣoḍaśopacāra-pujā that was to become dominant in the Tantro-Paurāṇic iconic worship. It may also mark the earliest account of the eye-opening rite that became prominent in the later āgamika strand of the religion.
-Several mantra-s which are provided only by pratīka-s are missing in the Kauthuma-Rāṇāyanīya and Jaiminīya texts and their auxiliary mantra collections. This suggests that the Gautama Sāmavedin-s had their own auxiliary mantra collection that was distinct from the extant texts. It is conceivable that they had the pañcabrahma-mantra-s, which today are only found as a complete group in the Taittirīya and AV Mahānārāyaṇa texts.
-The text rather remarkably combines both śaiva and vaiṣṇava elements. The former is seen in the form of the pañcabrahma-mantra-s and the latter is seen in the form of the explicit identification of the serpentine deity with the Puruṣa and also Saṃkarṣaṇa/Balabhadra. Both these aspects persisted in the subsequent layers of the religion. The serpentine form remained a key aspect of the iconography of the Saṃkarṣaṇa and Ananta figure as the bed of Viṣṇu. The Nāgapratiṣṭha continued as a ritual with new śaiva accretions in the Saiddhāntika stream in Rudrālaya-s (e.g., the Raurava tantra). Notably, it was also continued with modifications in the Bauddha practice of the Mūlamantra-sūtra (where it is combined with the old rain-making ritual) that was preserved in a rather pristine form among the Chinese ritualists.
We see a convergence of philology and archaeology with respect to Nāga-pratiṣṭha-s, offset by 2-3 centuries, perhaps due to preservation bias. In the bauddha lore, we hear of the famous conflict between the Tathāgata and the Vaidika brāhmaṇa Urubilva Jaṭila Kāśyapa (Vinaya 1.25). The latter had evidently installed a Nāga in his fire-shed which the Tathāgatha is claimed to have subjugated. This would be consistent with some version of the rites as recorded in the Nāga-pratiṣṭha from the Sāmaveda tradition being in place by around the time of the Shākya. Alternatively, it could be an allusion to the snake deity Ahi Budhnya being stationed at the Gārhapatya fire altar upon the conclusion of rituals in it (upasthāna). Subsequently, as noted above, by the Mauryan-Śuṅga age we see evidence for such installations and also images of Balabhadra in archaeology continuing down to the age of the Kuṣāṇa-s. Notably, both the early bauddha and jaina texts mention the worship of Balabhadra providing approximately coeval philological evidence for the same. Further, some of the early Pāśupata śaiva shrines like that of Bhogyavardhana (modern Bhokardhan in Maharashtra state) and Viṣṇukuṇḍin temple (at Devunigutta, Kothur, modern Andhra Pradesh) depict the Saṃkarṣaṇa suggesting further development of the potential links indicated by the use of the pañcabrahma-mantra-s in the installation of the snake.
## Two simple stotra-s, sectarian competition, and the Varāha episode from the archaic Skandapurāṇa
The Ur-Skandapurāṇa (SkP) or the “archaic” Skandapurāṇa ( $\approx$ the Bhaṭṭārāi edition known as the Ambikā-khaṇḍa) is a Śaiva text with affinities to the Pāśupata branch of that tradition. Though it is aware of the mantra-mārga traditions like the Mātṛ-tantra-s and the Yāmala-tantra-s as being part of the Śaiva scriptural corpus, its emphasis on the Pāśupata-vrata makes it clear that the core affiliation of the text was with the Pāśupata-mata that later Śaiva tradition identified as Atimārga. Nevertheless, it shows imprints of a three-way struggle for dominance between the major Hindu sects — Śaiva, Kaumāra and Vaiṣṇava. As the Skandapurāṇa, the existence of a Kaumāra layer is unsurprising. However, in the text, as it has come down to us, the Kaumāra elements are largely subordinated to the “Vīra” (as in strongly sectarian)-Śaiva elements. The subordination of the Vaiṣṇava-mata is primarily directed against the great deeds of Viṣṇu in his Nṛsiṃha and Varāha forms. In this regard, the Ur-SkP has a rather unprecedented ordering of the Daitya dynasty and the corresponding incarnations of Viṣṇu:
Hiraṇyakaśipu $\to$ Hiraṇyākṣa $\to$ Andhaka $\to$ Prahlāda $\to$ Virocana $\to$ Bali.
While Vipracitti is mentioned as assisting Hiraṇyākṣa in his battle with Viṣṇu, and being overthrown by the latter, it is not clear if he ever occupies the Daitya throne. The thus ordered Daitya-s are respectively slain by Viṣṇu as Nṛsiṃha; Viṣṇu as Varāha; Rudra; Viṣṇu (and Indra); Indra; Viṣṇu as Vāmana-Trivikrama. The main battle with Prahlāda as the Daitya emperor is situated in the episode of the churning of the World-ocean, during which Viṣṇu manifests as the gigantic turtle Kūrmā and also the bewitching female form Mohinī. As per the Ur-SkP, while Viṣṇu suppresses Prahlāda in an epic battle during this episode, he continues leading the Daitya-s in several further fights till Viṣṇu assisted by Indra destroys him. However, the Padmapurāṇa (13.186) places his slaying by Indra (an incident alluded to in the śruti itself) right in the episode of the churning of the World-ocean followed by the slaying of his son Virochana by Indra during the Tārakāmaya devāsura-yuddha. Correspondingly, in contrast to most other Purāṇa-s, in the Ur-SkP, the vibhava-s of Viṣṇu come in the order: Nṛsiṃha, Varāha, Kūrma/Mohinī, Vāmana/Trivikrama.
As we have seen before, Nṛsiṃha is shown as being subdued by Rudra in his dinosaurian Śarabha form after he has slain Hiraṇyakaśipu. The Ur-SkP has several parallels to the Vāmana-purāṇa, but in the latter, the Śarabha-Nṛsiṃha is given a Smārta resolution rather than a demonstration of Rudra-paratva. Upon being subdued by Śarabha, in the Ur-SkP, Nṛsiṃha is said to have recited the below stotra to Rudra. A votary who recites the stotra is said to attain the state of a gaṇa of Rudra.
The stotra to Śarabha by Nṛsiṃha:
namaḥ śarvāya rudrāya senānye sarvadāya ca ।
namaḥ parama-devāya brahmaṇe paramāya ca ॥54॥
kālāya yamarūpāya kāladaṇḍāya vai namaḥ ।
namaḥ kālānta-kartre ca kālākāla-harāya ca ॥55॥
namaḥ pinākahastāya raudra-bāṇa-dharāya ca ।
namo vidyādhipataye brahmaṇaḥ pataye namaḥ ।
namo ‘suravaraghnāya kālacakrāya vai namaḥ ॥57॥
saṃvartakāgni-cakrāya pralayāntakarāya ca ।
naranārāyaṇeśāya naranārāyaṇātmane ॥58॥
śarabhāya surūpāya vyāghra-carma-suvāsase ॥59॥
nandīśvara-gaṇeśāya gaṇānāṃ pataye namaḥ ।
indriyāṇāmatheśāya manasāṃ pataye namaḥ ॥60॥
namaḥ pradhānapataye surāṇāṃ pataye namaḥ ।
namo ‘stu bhāvapataye tattvānāṃ pataye namaḥ ॥61॥
carācarasya pataye bhūtānāṃ pataye namaḥ ।
trailokyapataye caiva lokānāṃ pataye namaḥ ॥62॥
bhagavaṃs tvatpratiṣṭho .asmi tvan niṣṭhas tvat parāyaṇaḥ ।
śaraṇaṃ tvāṃ prapanno .ahaṃ prasīda mama sarvadā ॥64॥
We shall discuss below some notable epithets used in this stotra:
1. The first three epithets: Śarva, Rudra and Senāni, betray the influence of the Śatarudrīya; this influence is seen in several later Śaiva stotra-s.
2. Parama-deva and brahman indicate the identification of Rudra with the supreme deity, keeping with the Pāśupata affiliation of the text.
3. kāla, yamarūpa, kālānta-kartṛ: These epithets associated with Yama and the end of time bring to mind the epithets in the opening mantra-s for liṅgasthāpanā: nidhanapati and nidhanapatāntika.
4. raudra-bāṇa-dhara: evidently a reference to the Pāśupatāstra.
5. Kālacakra: While Viṣṇu was the original time deity, within the Śaiva tradition, Rudra gradually began expanding into that domain. This is one of the early references to Rudra as the Kālacakra – a term that was to be used by the Vajrayāna bauddha-s for their eponymous Bhairava-like deity. On the Hindu side, the original Kālacakra-tantra was a saura text. We have philological and iconographic evidence for a prolonged interaction between the Saura-s and Śaiva-s. Interestingly, the Paśupata shrines at Kāmyakeśvara and Harṣanātha combine Śaiva and Saura elements. Most striking are two shrines near Kāmyakeśvara: Lakulīśa is shown on the lintel of the Saura temple, and Sūrya is shown on the lintel of the Rudrālaya. Thus, we posit that the syncretic or interacting Śaiva-Saura tradition influenced the emergence of the Bauddha deity Kālacakra.
6. Saṃvartakāgni-cakra: The fire of the dissolution of the universe — this is the epithet used for Navātman-bhairava in the Kaula Paścimāṃnāya tradition emerging from the Bhairavasrotas in the mantramārga. Indeed, the foundational sūtra-s of the Paścimāṃnāya are known as the Saṃvartāmaṇḍala-sūtra-s.
7. Nara-nārāyaṇeśa, Nara-nārāyaṇātman: The Nara-nārāyaṇa tradition is very prominent in the Mahābhārata and appears to be a quasi-humanized ectype of the Indra-Viṣṇū dyad of the Veda. This dyad, while important in the early Nārāyaṇīya Pāñcarātra of the Mahābhārata, faded away in the later Vaiṣṇava tradition. However, its presence here shows that this dyad was still important in the contemporaneous stream of the Vaiṣṇava tradition with which the Ur-SkP interacted (A tradition with connections to the Harivaṃśa; see below).
8. Vyāghra-carma-suvāsas: The wearer of the tiger-skin robe — an epithet related to Kṛttivāsas found in the Śatarudrīya.
9. Śarabha: While the whole stotra is to Śarabha there is little description of him in it beyond a single mention of his name.
10. Nandīśvara-gaṇeśa: The lord of the gaṇa Nandīśvara. This gaṇa’s association with Rudra goes back to the single mention in the Pratyaṅgirā-sūkta of the RV Khila (also seen in the AV saṃhitā-s). He subsequently rises to great prominence in the Saiddhāntika tradition. His presence here indicates that this was already presaged in the Pāśupata tradition.
After the Nṛsiṃha cycle, the Ur-SkP moves to the Varāha cycle. At the beginning of that cycle, the gods praise Viṣṇu with the below stotra to urge him to take on the Varāha Nandivardhana form which they constitute with their own bodies. The votary who recites it is said to become free of sins and sorrow.
The Viṣṇu Janārdana installed at the śaiva temple of Viśveśvara at Raghapura, Odisha.
namaḥ sarva-ripughnāya dānavāntakarāya ca ।
namo ‘jitāya devāya vaikuṇṭhāya mahātmane ॥15॥
namo nirdhūta-rajase namaḥ satyāya caiva ha ।
namaḥ sādhyāya devāya namo dhāmne suvedhase ॥16॥
namo yamāya devāya jayāya ca namo namaḥ ।
namaś cāditi-putrāya nara-nārāyaṇāya ca ॥17॥
namaḥ sumataye caiva namaś caivāstu viṣṇave ।
namo vāmanarūpāya kṛṣṇa-dvaipāyanāya ca ॥18॥
namo rāmāya rāmāya dattātreyāya vai namaḥ ।
namaste narasiṃhāya dhātre caiva namo namaḥ ॥19॥
namaḥ śakuni-hantre ca namo dāmodarāya ca ।
salile tapyamānāya nāgaśayyā-priyāya ca ॥20॥
namaḥ kapilarūpāya mahate puruṣāya ca ।
namo jīmūtarūpāya mahādeva-priyāya ca ॥21॥
namo rudrārdharūpāya tathomārupiṇe namaḥ ।
cakra-mudgara-hastāya maheśvara-gaṇāya ca ॥22॥
caturbhujāya kṛṣṇāya ratna-kaustubha-dhāriṇe ।
trivikrama-viyat-sthāya pīta-vastra-suvāsase ॥24॥
yogine yajamānāya bhṛgupatnī-pramāthine ॥25॥
vṛṣarūpāya satataṃ ādityānāṃ-varāya ca ।
cekitānāya dāntāya śauriṇe vṛṣṇibandhave ॥26॥
purāśvagrīva-nāśāya tathaivāsura-sūdine ।
namo jayāya śarvāya rudra-datta-varāya ca ॥28॥
namaḥ sarveśvarāyaiva naṣṭa-dharma-pravartine ।
puruṣāya vareṇyāya namaste śatabāhave ।
tava prasādāt kṛcchrān vai tarāmaḥ puruṣottama ॥29॥
We discuss below some of the notable epithets found in this stotra:
1. Vaikuṇṭha: This distinctive epithet first appears in the Mahābhārata and is repeatedly used in the early Pāñcarātrika section of that text (parvan 12). There it appears as a name of the god both in Viṣṇusahasranāma and the 171-epithet early Pāñcarātrika mantra of Viṣṇu composed by Nārada. It also appears in a similar mantra in a stava composed by Kaśyapa in the Harivaṃśa. In later iconography, the epithet is usually taken to mean Viṣṇu caturātman with anthropomorphic, leonine, porcine and Kapilan heads. Viṣṇu is specifically addressed by this name in the Ur-SkP as he prepares to slay Hiraṇyākṣa with the cakra (see below).
2. Nirdhūta-rajas: One who has freed himself from the dust. The dust here might be seen as the particulate bonds — or the ātman bound to the evolutes of Prakṛti.
3. Sādhya deva: In the Puruṣa-sūkta we are enigmatically informed of an ancient class of deities known of the Sādhya-s alongside the deva-s — nothing more is said of the former. They appear episodically in various brāhmaṇa texts and are generally seen as a class of celestial deities. By making Viṣṇu a sādhya, the stotra expands his domain to include these obscure deities.
4. Yama deva: Interestingly, like Rudra, Viṣṇu too is identified with Yama.
5. Aditi-putra, Ādityānāṃ-vara: Viṣṇu membership in the Āditya class of deities is not just cemented, but he has risen to be the chief of them.
6. Vāmanarūpa, Trivikrama-viyat-stha, Nara-Nārāyaṇa, Kṛṣṇa-dvaipāyana, the two Rāma-s (Rāmacandra Aikṣvākava and Rama Bhārgava or the Saṃkarṣaṇa is unclear), Dattātreya, Narasiṃha, Kapila, Śaurin, Vṛṣṇibandhu, Kṛṣṇa, Saubha-Sālva-vighātin: The late daśāvatara has not yet crystallized, but the tendency in that direction is clear in the list. We have Narasiṃha, Vāmana, two Rāma-s, and Kṛṣṇa who figure in the classic lists. Varāha is specifically avoided because that incarnation is about to occur in the current narrative. Yet, anachronistically, there is a clear acknowledgment of the Sāttvata religion with the identification of Viṣṇu with Kṛṣṇa and various Kārṣṇi/Sāttvata epithets in the above list. These include the famous act of Kṛṣṇa Devakīputra, i.e., the killing of Sālva and the destruction of his airplane the Saubha. Some other incarnations that are widely accepted, but not in the classic list of 10, are also mentioned such as Kṛṣṇa-dvaipāyana and Dattātreya. This shows that the incarnational model pioneered by the Sāttvata religion had already been expanded to include a wider range of figures.
7. Śakuni-hantṛ: This epithet is peculiar because, at first sight, people take Śakuni to mean the eponymous prince of Gandhāra. However, this is not the case because that Śakuni was not killed by Viṣṇu or Kṛṣṇa. The Harivaṃśa tells us that:
pūtanā śakunī bālye śiśunā stanapāyinā ।
stanapānepsunā pītā prāṇaiḥ saha durāsadā ॥ HV 65.26
also:
pūtanā nāma ghorā sā mahākāyā mahābalā ।
viṣa-digdhaṃ stanaṃ kṣudrā prayacchantī mahātmane ॥ HV 96.31
Here, the mighty and terrible Pūtanā, whom Kṛṣṇa slew when he drank her milk as she tried to breast-feed him in his infancy with her poisoned breast, is described as Śakunī and a rākṣasī. Hence, the epithet Śakuni-hantṛ records this episode. We know from the Kaumāra tradition that Śakunī, Pūtanā and Revatī are the names of pediatric avicephalous Kaumāra goddesses who are invoked for freeing a child from various diseases. Indeed, this identification was known even in the HV in the ancient hymn to the transfunctional goddess, the Āryā-stuti:
śakunī pūtanā ca tvaṃ revatī ca sudāruṇā । HV (“appendix”) 1.8.39
Thus, the Pūtanā-Śakunī episode represents an example of ancient sectarian competition between the Kaumāra-s and the Sāttvata stream of the Vaiṣṇava-s who portray their hero as slaying the demonized Kaumāra avicephalous goddess and thus expanding into the domain of pediatric apotropaism that belongs to the god Skanda.
Avicephalous and therocephalous Kaumāra goddesses from Kuśana age Mathurā
9. Madhu-kaiṭabhaghātin, Dhundhumāra, Aśvagrīva-nāśa, Bhṛgupatnī-pramāthin: These epithets concern the ancient Asura/Asurī-s slain by Viṣṇu. Of these Dhundhu, the son of Madhu, is said to have caused landslides or earthquakes and was killed by the Ikśvāku hero Kuvalāśva, the son of Bṛhadaśva, into whom the tejas of Viṣṇu had entered (HV, chapter 9). It is possible that this epithet implies that the said king was seen as an incarnation of Viṣṇu (a parallel to the later Ikṣvāku incarnation as Rāma). In contrast to this more widespread legend, a parallel myth alluded to in the Liṅgapurāna suggests that Viṣṇu himself slew the Asura by acquiring the cakra from Rudra. The killing of Aśvagrīva is alluded to in both the Itihāsa-s and the later Purāṇa-s either connect it with the Pravargya-like tale of the beheading of Viṣṇu by the rebound of his bow or the Matsya incarnation. The ancestress of our clan, Paulomī, the wife of Bhṛgu, is said to have been an Asurī or a partisan of the Dānava-s. She was killed by Viṣṇu for aiding them — this is already known in the Rāmāyaṇa.
10. Rudrārdharūpa: An acknowledgment of the Harihara form. The first surviving icons of this form are known from the Kuṣāṇa age.
11. Jīmūtarūpa: Of the form of a cloud — this is an unusual name. It likely indicates the expansion of Viṣṇu into the domain of Parjanya via a specific myth found earlier in the Ur-SkP (chapter 31). There the personified Vedic ritual, Yajña, was designated by Brahman to do good to the world. He soon found himself possessing insufficient power to do that. Hence, he performed tapas and pleased Rudra. Rudra granted him the boon of becoming a cloud (Jīmūta) and delivering life-giving waters to the world. Before Rudra acquired his bull, Yajña as the cloud also became his vehicle – it is stated by becoming the abode of lightning (which as per the Veda is a manifestation of Rudra – 11 lightnings of the Yajurveda; also the name Aśani) he carried Rudra on his back. Given the Vedic incantation: yajño vai viṣṇuḥ ।, the cloud is identified with Viṣṇu.
12. Vṛṣarūpa — In the Harivaṃśa, Vāsudeva slays a son of Bali named Kakudmin Vṛṣarūpa. However, here given that it is the name of Viṣṇu, it might imply an identification with Rudra’s bull, who was his next vehicle after Yajña as the cloud.
13. Umārūpin: This is an unusual identification that was to have a long life in the later tradition all the way to the late Śrikula system of Gopālasundarī and parallels the coupling of Mohinī and Rudra or the Harihara iconography.
14. Mahādeva-priya, Maheśvara-gaṇa, Rudra-datta-vara: In the Ur-SkP, Viṣṇu is not outright antagonistically demoted vis-a-vis Rudra. He is instead cast as a mighty god who is, however, second to Rudra. This is made clear by calling him dear to Mahādeva (or even equating him with Umā: the above epithet), while at the same time subordinating him as a gaṇa of the god and one receiving boons from Rudra.
15. Salile Tapyamāna: This is again a rather peculiar epithet because it applied to Rudra in the Ur-SkP and goes back to the Mahābhārata where it occurs in the stotra uttered by Kṛṣṇa and Arjuna to Rudra in order to obtain the Pāsupata missile. Thus, its application to Viṣṇu here might indicate his identification with Rudra. This idea of a deity heating the waters as part of the evolutionary process is an idea going back to the Veda. In the context of Rudra, it related to his liṅga form – i.e., Sthāṇu. That said, there are other clear links between Viṣṇu and the primordial waters – he is termed Nārāyaṇa – typically interpreted as the abode of the waters. Moreover, the same stotra also refers to him as Nāgaśayyā-priya – i.e., fond of this serpentine bed. Tradition unequivocally places this bed in the midst of the ocean. His Hayagrīva form is also mentioned in the Mahābhārata as dwelling in the waters consuming oblations (related to the ancient motif of the submarine equine fire). Thus, this epithet could specifically apply to that form.
In the Varāha cycle of this purāṇa, this sectarian tension plays out in the battle between the Daitya and Viṣṇu and the events that follow it. A brief synopsis of it is provided below:
-The gods formed the boar body for Viṣṇu with their own bodies. Thus, he advanced against the Daitya-s by diving into the ocean — an account is given of his encounter with various marine life, like different kinds of whales, sharks, fishes and molluscs. Visiting the various nether realms, he advanced to Rasātala where the Asura-s lived.
-There, a submarine Daitya guard Nala sighted Varāha and in fear rushed to inform his lord. But Varāha followed him and thus discovered the Asura stronghold.
-Prahlāda informed Hiraṇyākṣa that he had a bad dream that someone in a man-boar form might kill him.
-Hiraṇyākṣa says that he too had a dream in which Rudra asked him to surrender to Indra, give up his dominion, and come to dwell near him. The Daitya-s suggested that Hiraṇyākṣa not go to battle. Instead, they suggested that they would head out to the battle with Andhaka as their head. Vipracitti suggested that he would go himself to destroy the gods.
-Nala came in and informed the Asura-s that he had sighted a terrifying boar coming to attack them.
-Prahlāda urged Hiraṇyākṣa to take some action as he realized that the boar was none other than Viṣṇu who has come to destroy them with the aid of his māyā.
-Hiraṇyākṣa responded that he wished to avenge his brother’s death by killing Viṣṇu and offering his boar-head as a bali for Rudra.
-He sent forth his Asura troops to attack Varāha. At first, Varāha ignored them saying that he was just searching for his wife and the one who had kidnapped her. The Asura-s launched a massive attack on him, and he retaliated by demolishing them.
-Hearing of their defeat Hiraṇyākṣa asked the great Asura generals Prahlāda, Andhaka, Vipracitti, Dhundhu, Vyaṃsa and others to get ready to confront Varāha.
-Varāha made an anti-clockwise circuit of the Asura stronghold and stormed it via the southern gate. He destroyed the śataghni missiles fired from the gate and also the Kālacakra missiles that were hurled at him. The Asura-s made a great sally at him. He feigned a retreat and drew them out of their fortifications. However, the Asura-s realized his plan and attacked him from the rear. The deva-s in his body were able to detect this attack and oriented him towards the Asura-s attacking him from behind.
-Varāha challenged them to one-on-one duels. Andhaka agreed that it was the right thing to do. However, Prahlāda informed them that the vile Varāha was none other than the wicked Viṣṇu using his māyā because he was afraid to fight them with his own form. Then Prahlāda showered astra-s on him and asked the other Asura-s to join him in a combined attack.
-Varāha then smashed Prahlāda’s chariot and hammered him with his own standard on his head. The daitya retaliated with his mace, but it had no effect on Varāha.
-He then attacked fought Andhaka and Vipracitti in a great battle. In the end, he carried both like Garutmat carrying the elephant and the tortoise and hurled them down like bolts of lightning.
-He then destroyed and slew the divisions of the remaining Asura-s.
-Vipracitti returned to the battle having rearmed himself, but after a strong fight Varāha whirled him around and sent him crashing into the fortress of Hiraṇyākṣa.
-Hiraṇyākṣa alarmed by the noise went to check things out and found his general unconscious. After reviving him, the Daitya emperor asked him who could possibly defeat him. Vipracitti then told him that it was the invincible Varāha and perhaps it was similar to Nṛsiṃha who had earlier crushed them. He suggested that the Asura-s should abandon their stronghold and flee.
-Disregarding Vipracitti, Hiraṇyākṣa set out for battle himself. He is said to have been of the complexion of a heap of collyrium but with a blond beard and four fangs.
-His advance is described using two astronomical allegories: He is said to be like a great comet and Vipracitti who accompanies him is like a reflection of that comet. He is also described as being like the Sun, with Prahlāda, Andhaka and others surrounding him like the planets — an interesting heliocentric simile.
-Varāha scattered the other Daitya-s and Dānava-s and rushed at Hiraṇyākṣa, who, however, paralyzed him by piercing his joints with his arrows. The deva-s removed those arrows with magical incantations, and Varāha resumed the attack. This time he came close to striking the Asura’s car, but the Daitya’s charioteer steered it away, and Hiraṇyākṣa bound Varāha with the Nāgāstra.
-Then the Asura-s massed around him and tried to chop him up with their weapons. However, Garuḍa came to his aid and released him from the Nāgāstra. Thus revived, he smashed the Daitya-s and resumed his attack on Hiraṇyākṣa.
-The Daitya then pierced him in the heart with an astra causing him to faint. On regaining consciousness, he called on the deva-s to reinforce him, and they filled him with their tapas. Thus, he shone like seven suns, resembling Rudra preparing to destroy the worlds.
-Varāha then displayed several māyāvin tactics and overcoming the nāgāstra-s of the Asura king destroyed his chariot. He continued fighting on foot and struck Varāha with the Mohanāstra, which stunned the boar. The deva-s in his body countered it, and Varāha returned to the battle.
-Varāha uprooted a tree (axial mytheme) and struck the Asura lord on his head. The latter fell unconscious, and his bow with five arrows slipped from his hand. The other daitya-s and dānava-s wailed thinking he was dead and fell upon the boar with their weapons. Varāha simply swallowed all those weapons.
-As Varāha was engaged with the daitya-s, Hiraṇyākṣa recovered, and uttering the mantra rudrāya vai namaḥ ।, he hurled a mighty spear at his enemy. Varāha was struck in the heart by that and fell down as if dead.
-The sun then lost its luster, and the planets were on collision course. Brahman at that point invoked Rudra. Varāha rose up again, and the tejas of Rudra entered him. Pulling out the spear stuck in him, blazing like a thousand fires, he pierced Hiraṇyākṣa like Skanda striking Krauñca. However, the Asura was unfazed by that blow.
-The Daitya returned the blow with his sword, but Varāha felt no pain and struck the sword away with the back of his palm.
-Then the two engaged in a prolonged wrestling bout at the end of which an incorporeal voice told Varāha that he can kill the Daitya only with Rudra’s cakra.
-Invoking Rudra and calling the cakra that was born of the “waters” (an oblique reference to the Jalaṃdhara episode, where Rudra killed the Asura using a cakra that he drew from water: the whirlpool mytheme), Varāha assumed a gigantic form pervading the triple-world.
-Hiraṇyākṣa fought him with various astra-s and displays of māyā, but Varāha destroyed all of them with the cakra and finally cut off the Daitya’s giant head.
-Varāha then searched for Pṛthivī by destroying the parks and tanks of the Asura-s and uprooting mountains. Going south he uprooted Śaṅkha mountain and found her bound there, and guarded by dānava-s. He hurled the Śaṅkha mountain and slew the dānava-s and drove away the Nāga-s. He then seized the jewels of the Asura-s.
Pṛthivī clinging to Varāha’s tusk from Gupta age Udayagiri.
-He carried Pṛthivī clinging to his tusk even as Brahman had carried the former earth Vasudhā when he had assumed a boar form (It is notable that the Śaiva-s revived the memory of this old Vedic narrative of Prajāpati’s boar form probably to obliquely indicate that Viṣṇu’s Varāha form was only second to that of Brahman).
-He then handed over the triple-world to Indra and reaffirmed their eternal friendship.
-Varāha indicated that he wished to enjoy the pleasures of his boar form in fullness. Thousands of Apsaras-es become sows to consort with him even as the brāhmaṇa-s lauded him with their hymns.
-Mating with his wife in the form of the sow Citralekhā he birthed a lupine son known as Vṛka (Temples of Varāha as the father of the lupine Vṛka – Kokamukhasvāmin – seem to have been there in Nepal and from there transmitted to Bengal).
-Vṛka roved around the world with his pack eating various animals. Finally, he arrived at the forest of Skanda at Gaurīkūṭa with medicinal plants, minerals and gems. At that time Skanda was away visiting the Mandara mountain and had deputed his avicephalous or therocephalous gaṇa Kokavaktra (himself with a lupine head or with a cuckoo or waterfowl head) to guard his forest.
-Vṛka ravaged Kumāra’s forest. At first Kokavaktra tried to be good to him and told him that he was happy with his power. Kokavaktra asked Vṛka to stop and told him that he would repair the damage and put in a word with Skanda to make him a gaṇa.
-Vṛka refused and attacked the Skandapārṣada. After a fight, Kokavaktra knocked down Vṛka and bound him with pāśa-s (A rare reflex of the Germanic Fenris wolf motif in the Hindu world).
-When Skanda returned he sentenced Vṛka to be subject to Narakatrāsa-s by his gaṇa-s.
-Nārada informed Varāha about what had happened and told him that due to his childish arrogance, Skanda does not bow before the great god and has bound his lupine son.
-Infuriated, Varāha proceeded to fight Skanda. Skanda and his gaṇa-s neutralize the cakra and other weapons of Varāha. Finally, Guha pierced Varāha’s heart with his saṃvartikā spear. Viṣṇu at that point abandoned his Varāha body and resumed his usual form.
Skanda wearing the tusks of Varāha on his necklace, Gupta age.
-Viṣṇu then praised Rudra who conferred a boon to him. Viṣṇu asked him to teach him the Pāśupata-vrata. Rudra mounted his bull and went to Sumeru to teach Viṣṇu the said vrata.
Here we see a three-way competition between Śaiva-s, Vaiṣṇava-s and Kaumāra-s. The normally accommodating relationships between the Kaumāra-s and Vaiṣṇava-s (barring some conflicts as the Pūtanā case alluded to above), seem to have broken down probably under Śaiva influence. The incident of the defeat of Varāha by Kumāra is seen in both the South Indian Skandapurāṇa and the Ur-SkP, suggesting it was there in an ancestral SkP. It has some Gupta-Puṣyabhūti age iconographic representation in the form of Skanda wearing the tusks of the boar in his necklace. However, in this text, the Śaiva-s trump both of them with the final flourish of Viṣṇu ultimately asking Rudra to teach him the Pāśupata-vrata. We believe that the Varāha episode in the Ur-SkP is a genuine early version of this famous mytheme, but it was strategically tweaked at certain points by the Śaiva-s to downgrade Viṣṇu and exalt Rudra.
## The zombie obeys: a note on host manipulation by parasites and its ecological consequences
In 1858-59, as AR Wallace, one of the founders of the modern evolutionary theory, was exploring the Sulawesi Islands, he collected an ant, Polyrachis merops, that he sent over to England. Years later, the naturalist W Fawcett studying these ants collected by Wallace and others from South America, realized they were attacked by a fungus that is today known as Ophiocordyceps. In 1869 when Wallace learnt of mycologists discussing these insect-killing fungi, he was much surprised and even expressed doubt if it was a genuine fungus. However, those doubts of the great man aside, the fungus was to have a bright future as a beacon for studies on the manipulation of host behavior by parasites. It is today widely known that Ophiocordyceps fungi infect ants, such as the carpenter ants (of genus Camponotus) and spiny ants (Polyrachis), and alters their behavior making them leave their colonies and wander onto leaves. Here it makes them clamp down on the leaves in the canopy above the ant trails with their mandibles. They remain stuck there until the fungus kills them from within; then, the fungus grows out of them, often bursting out from their head, and sporulates. The spores rain down on the unsuspecting ants scurrying below on their trail; thus, the fungus infects a new set of victims. This peculiar adaptation has evidently evolved as part of the arms race to keep up with the emergence of hygiene in the ants – they regularly groom themselves, and when they find a corpse, they quickly break it down and take it out of their nest. Thus, by showering spores on them when they are on the trail, the hygienic practice of the ants is breached by the fungus.
Today we know that this behavioral manipulation of ants is not unique to Ophiocordyceps, an ascomycete, but is also evinced by the fungus Pandora that belongs to a distant lineage of insect-specialist fungi (Entomophthoromycota) in a distinct genus of ants, Formica. Even more remarkably, the same type of behavior is also induced in Formica ants by the trematode brain fluke Dicrocoelium dendriticum which has a remarkably complex life cycle. It begins with sexual reproduction in the bile duct of a cow and excretion via its dung of fertilized eggs bearing developing embryos. The eggs are eaten by snails (e.g., genus Cochlicopa), where the worm emerges as miracidia larvae. The miracidia drill through the gut wall and enter the respiratory system of the snail, where they protect themselves from the host immunity by forming sporocysts. The snail extrudes them in slime balls through the respiratory pore, and they emerge as cercaria larvae from the sporocysts. These cercariae infect an ant when it feeds on the snail slime balls. In the ant, they develop into the next stage, the metacercariae, which start controlling their new host’s behavior and causes it to desert its colony and climb up leaves and clamp down on them with their mandibles. Once on the leaves, they might get eaten by cows to resume the cycle.
A parallel strategy has evolved in the parasitic insect, the myrmecolacid strepsipteran – here, the male takes formicid ants as hosts while the female takes locusts as their hosts. The male strepsipteran alters the ant’s behavior to again desert its colony and climb up leaves and hold on to them with its legs. The strepsipteran then emerges from the ant and flies off in search of the female. By leading the ant onto the leaves, he can better sense the female’s pheromones wafting in and also have a launchpad for his final flight to find his mate. On finding the infected locust from which the morphologically degenerate female protrudes out, he mates with her by piercing the brood-canal in her cephalothorax with his spiny tube-like sperm delivery organ, the aedeagus. Interestingly, we can also find this behavioral manipulation in a more general sense in baculoviruses, which cause the caterpillars they infect to “summit”, i.e., climb the outer branches of the trees and stay there. The virus then kills them and liquefies their corpses so that the virions are spread on the leaves allowing new caterpillars to consume them with their meal. The virus achieves this by a UDP-sugar glycosyltransferase enzyme that it encodes, which modifies the insect molting hormone ecdysteroid to inactivate it, and thus prevents it from molting on the trunk of the tree. Thus, a virus, two fungi, a fluke, and a strepsiteran insect, each with a distinct life cycle, have all evolved broadly convergent behavioral manipulations of their hosts to enhance their spread.
Rather remarkably, this broad strategic category of altering host behavior to favor transmission to a new host furnishes several other examples of channeling of convergent manipulations by evolutionarily distant parasites. One of the best known of these is the induction of erratic behavior leading to suicide by drowning in various insects and crustaceans by the nematomorph and mermithid nematode parasites that need to access water for the next stage of their development. In the case of the nematomorphs, like Paragordius varius, they induce their cricket/grasshopper host to jump into water and drown, allowing them to come out and mature in the aquatic environment and lay eggs. The larvae that hatch from the hosts then burrow into the guts of aquatic insect larvae, like mosquitoes, and form a cyst. This cyst survives into the adult of the mosquito that returns to land. On land, when the mosquitoes die, they might be eaten by crickets leading to the transmission to the new host. Similar suicide by drowning is driven by mermithid nematodes, such as Mermis nigrescens in the earwig Forficula auricularia and the ant Colobopsis, and by Thaumamermis zealandica in the crustacean sandhopper Bellorchestia quoyana. Here again, the drowning seems to allow the nematodes to ultimately access their secondary hosts in the form of aquatic larvae. In molecular terms, this suicidal behavior appears to be induced by the upregulation of Wnt proteins in the head of the infected orthopterans.
A conopid fly
Apart from manipulating host behavior to allow the parasite to reach a new host, there are several instances of convergent evolution of manipulations that alter the host behavior to make the parasite more secure. This was observed early on in the braconid parasitoid wasp Aphidius ervi, which may undergo two alternative larval programs, namely one of uninterrupted development to pupation and adulthood and the second involving a dormant phase known as the diapause. One of their hosts is the aphid Acyrthosiphon pisum, into which they inject an egg. The emergent larva eats the aphid from within and leaves its bronzed exoskeleton as a puparium for the final stage of its development. If the wasp larva opts for a diapause program, they manipulate their host aphid to abandon the aphid colony and go either into a curled leaf or entirely leave the plant and go to an obscure site where they are “mummified”. In contrast, the larvae opting for uninterrupted development cause their host to leave the aphid colony and climb onto the upper surfaces of leaves prior to mummification. A comparable adaptation is seen in the case of the parasitoid conopid flies, such as Physocephala rufipes, which are morphological wasp mimics that target bumble bees. When the conopid fly comes upon a bumble bee foraging among the flowers, it attacks it and inserts its ovipositor between the abdominal cuticular sternites to deliver eggs into the bee. The fly larvae grow within the bee, feeding on it from within and altering its behavior. They cause it to desert the hive and limit their nectar collection activity. Finally, when the larva is close to pupation, it causes the bee to bury deep into the soil – evidently, here, it induces in workers a behavioral program that executes in the queen when it hibernates over winter. There the fly larva kills the bee and uses its exoskeleton as a puparium to overwinter and emerge in spring as an adult. Those flies which develop in such underground bee carcasses, on an average, develop better than those which end up killing their host above the ground, clearly indicating a fitness gain accrued from the manipulation of host behavior.
Reclinervellus nielseni larva manipulates Cyclosa argenteoalba
A related form of parasitic manipulation was discovered by the naturalists Takasuka et al. among spiders that spin webs in Japanese shrines. Here, the host spider Cyclosa argenteoalba weaves two kinds of webs — a normal orb web to catch prey and a resting web where it molts. The larva of the ichneumonid ectoparasitoid wasp Reclinervellus nielseni manipulates the spider host by injecting it with a toxic mixture. This causes the spider to make a version of the resting web with more threads so that it is better reinforced and also add decorations that reflect UV light allowing it to be avoided by birds or large insects in their flight. Thus, the wasp larva induces its host to create a resilient cocoon for it, where it pupates after killing the host. Since removing the ectoparasitoid larva causes the spider to return to its normal web-weaving, it is clear that the altered behavior is induced by molecules in the wasp’s venom. Another component of this venom also prevents the molting of the host spider. Notably, this behavioral manipulation has also convergently evolved in another ichneumonid ectoparasitoid Hymenoepimecis argyraphaga, which, on the evening it will kill its host, the spider Plesiometa argyra, alters its host’s behavior to spin a comparable cocoon web. However, in this case, rather than making the spider weave a resting web, the wasp toxin appears to induce it to repeat a subset of the early steps of normal orb construction while suppressing the remaining steps resulting in a cocoon for the larva.
The above classes of behavioral manipulations broadly fall under the rubric of host behavioral manipulation for reaching new hosts or for providing suitable “housing” for pupation or dormancy. A further class has been recognized in the form of manipulation to make the host provide policing services. A good example of this was described several years ago for the braconid parasitoid wasp Glyptapanteles sp., which lay their eggs in caterpillars of the geometrid moth Thyrinteina leucocerae. After developing within their host, they exit it by piercing its lateral body wall but do not kill it; instead, it heals from the trauma. One or two wasp larvae remain behind inside the caterpillar and apparently manipulate the latter to act as a bodyguard for the egressed larvae that start pupating. Under the remaining larvae’s influence, the caterpillar stops feeding, hangs around with the pupae, and shows behaviors not seen in uninfected caterpillars — it knocks off predators such as the bug Supputius and other hyperparasitoid wasps by violently swinging its head. However, it never matures into a moth and dies once it has done its policing job for the parasitoid. It appears that the 1-2 larvae that remain behind to manipulate the host sacrifice their own fitness for the sake of their egressed kin. Field studies in Brazil showed that this protection significantly increased the survival of the wasps supporting the adaptive nature of the behavior manipulation and its potential evolution under kin selection. In a dramatic lepidopteran on hymenopteran reversal, a convergent evolution of the bodyguard strategy is seen in the case of the caterpillars of the lycaenid butterfly Narathura japonica that intoxicates the workers of the ant Pristomyrmex punctatus with secretions from its dorsal nectary organ found in the abdomen. These reduce the locomotory activities of the ants by acting on their dopaminergic circuit, turning them into defensive bodyguards for the caterpillar. However, at least in the case of certain related lycaenid butterfly caterpillars and the ant Formica japonica, the former might also provide some benefit to the bodyguards in the form of a sucrose+amino acid shot from the dorsal nectary organ.
The Toxoplasma gondii-wolf-puma system as illustrated by Meyer et al.
This finally brings us to a relative of Sarcocystis, another apicomplexan parasite Toxoplasma gondii, which illustrates the macroscopic ecological consequences of the multi-directional fitness consequences of interlocking host-parasite and prey-predator interactions. The best-studied aspect of this is the cat (Felis catus)-rodent cycle of Toxoplasma, where the rodents are the intermediate hosts and the cat the definitive host (where the parasite completes its sexual cycle). Here the parasite changes the neurotransmitter concentrations in the mice and rat brains to make them attracted to the odorants in feline urine — it is believed that the male rodents are induced by the parasite to experience sexual arousal to cat odorants. Needless to say, this draws the rodents towards the cats and makes them easier prey, thereby allowing the parasite to complete its cycle. More recent studies have found similar results with other cats. For example, in our close cousin, the chimpanzee, toxoplasmosis causes a morbid attraction towards leopard urine, thus, increasing their chances of being killed and eaten by one. Another study found that hyena cubs infected by Toxoplasma tend to lose their fear of lions and approach them more closely than uninfected ones. Thus, they tend to be killed more often by cats. These studies were capped up by the recent publication of a multi-year study on Toxoplasma’s role in the wolf-puma (cougar; Puma pumoides) interactions in North America. The authors found evidence that toxoplasmosis in wolves makes them greater risk-takers, thereby increasing their tendency to break off and found their own packs or become leaders of packs. They propose that this behavior brings them in contact with pumas that the wolves normally avoid. On one hand, this results in an increased propensity for them being infected by the parasite from puma feces, and on the other, it increases the propensity of Toxoplasma transmission to the cat, where the parasite completes its sexual cycle. Sarcocystis neurona, which resides in the neurons of its intermediate host, is proximally positioned to alter its behavior in ways similar to Toxoplasma but its ecological consequences remain poorly explored.
In each of the above cycles, the behavioral alterations of the intermediate host clearly advantage the parasite by increasing its probability of reaching the definitive host. Like with the Sarcocystis example, it is apparent that toxoplasmosis in the definitive host does not cause it to die — it seems to be a mild infection with no serious sequelae. Studies on domestic cats indicate that most infected with T. gondii show no signs of disease. In fact, it only seems to flare up as a serious condition if the cat is also infected by a retrovirus, like FIV or FeLV, which compromises its immune system. Thus, in balance, it is conceivable that Toxoplasma actually confers a fitness advantage to the cats by “bringing” prey to them. In rodents, chimpanzees and hyenas, the manipulation seems to obviously depress the fitness of the intermediate hosts. However, a closer look suggests that the picture might be more complex. The above study on the wolf-puma system suggests that, at least in some intermediate hosts, the manipulation by the parasite might not be entirely fitness-reducing. Studies on male rats suggest that Toxoplasma might make male rats more sexually active by increasing testosterone production. In domestic dogs, sheep, goats, rabbits, rats, and probably humans, there is evidence for Toxoplasma being sexually transmitted between mating partners and also to their progeny (congenital toxoplasmosis). Hence, it might also be similarly transmitted within a wolf pack via sex. This, taken together with the manipulation resulting in testosterone elevation, suggests that the parasite also attempts to increase its range within intermediate hosts via a sexual and congenital cycle. The testosterone effect with the behavioral changes suggests that it might not be all bad for the intermediate host — potentially contributing to their fitness via increased sexual activity. In the wolf example, behavioral changes, like pack founding and new territory acquisition, seem to have a positive effect on fitness too. Thus, the net balance of the fitness consequences of toxoplasmosis might be harder to evaluate, even for the intermediate host.
In parallel with the evidence from the extant chimpanzee, we have fossil evidence that the human lineage was prey for large felids: e.g., the Sterkfontein Paranthropus with leopard canine marks on its skull; the Olduvai OH 7 Homo habilis leg with leopard tooth marks (other hominins in the same site were eaten by crocodiles); the Dmanisi Homo georgicus skeletons were likely accumulated by a big cat such as Megantereon megantereon, Homotherium crenatidens or Panthera gombaszoegensi; the Asian Homo erectus eaten by a large cat at Zhoukoudian; the Cova Negra Homo neanderthalensis whose skull was punctured by a leopard; at least one of the Sima de los Huesos hominins, who were related to Neanderthals and maternally to Denisovans, was consumed by a large cat; tigers, lions, and leopards have been recorded as eating numerous humans in India and Africa until 100 years ago — this was likely a far more common occurrence in earlier times though we do not have good records for it. Thus, it can be said that for much of its history, the hominin clade was an intermediate host for Toxoplasma and transmitted it to cats that preyed on them. However, things changed as, with their growing brains, H. sapiens managed to turn the tables on the big cats and nearly exterminate them. Thus, today humans are practically dead-end hosts for Toxoplasma. This does not mean that the behavioral manipulations have ceased. There is some evidence that it might alter sexual behavior and aggression in both human males and females. There are correlational studies suggesting that it might foster entrepreneurial tendencies and road rage in human males and generally aggressive behavior and neuroticism in women. There is also evidence for association with personality disorders on the schizophrenia spectrum. In the past, some of these behaviors might have reduced the fear in humans and made them venture closer to big cats in the environment that then preyed on them. However, today a subset of these altered behaviors, like enhance entrepreneurship, might provide some fitness benefit.
It should be noted that today millions of humans are infected by Toxoplasma primarily due to their contact with domestic cats. Nevertheless, not all of them become more neurotic or entrepreneurs. This suggests that perhaps the strain that infects domestic cats does not affect its human host strongly. Moreover, it is likely that the humans who are more affected by the behavioral modifications induced by Toxoplasma have some genetic predisposition for the same. Nevertheless, even if a dead end for the parasite, we wonder if it might have played a role in human ecology with respect to cats. Cats were domesticated somewhere in West Asia during the Neolithic. It is generally believed that this was a symbiotic relationship because human settlements allowed for increased rodent populations, and the domestic cat could control them. Nevertheless, it needs to be considered if the infection of humans by Toxoplasma as a result of increased proximity with the proto-domestic cats resulted in some kind of behavioral alteration that made humans attracted to cats and increased their bonding. It is possible this goes back even deeper in the Paleolithic, where the attraction towards large cats provided the germs for the “man-cat” hybrid imagery that is widely seen across human cultures. This idea is worth considering because, unlike the domestic dog, which usually exhibits much greater emotional overlap with humans, the cat is a mostly aloof animal.
Other apicomplexan parasites also manipulate their hosts with potentially differential fitness consequences for their intermediate and definitive hosts. For instance, while the malarial parasite Plasmodium primarily resides in the gut (ookinete stage) and the salivary gland (sporozoites) of Anopheles mosquitoes, it manages to alter the host odorant response, which is localized to the antennae, such that the mosquito is more attracted towards vertebrate odors. It is not clear if the odorant manipulation is done by a few sporozoites that enter the brain or remain behind elsewhere in the mosquito to act on behalf of their kin. It is conceivable that this action might confer some fitness benefit for the mosquito in terms of getting it to a vertebrate host for a blood meal. A convergent evolution of this manipulation is suggested in the case of the kinetoplastid Trypanosoma cruzi, which appears to make its bug host Triatoma both more active and responsive to human odors. A complementary manipulation is mediated by the related apicomplexan, Hepatozoon, which has a complex life cycle alternating between Culex mosquitoes and a single vertebrate host like a frog or two vertebrate hosts like a frog followed by a snake which eats the former. Here, Hepatozoon manipulates its vertebrate hosts to make their smell more attractive to the mosquito. This adaptation has convergently evolved in the kinetoplastid parasite Leishmania, which makes their mammalian hosts’ smell more enticing to the sandfly. While we still poorly understand how these manipulations are achieved at the molecular level, the genomes of some of these apicomplexans show that they encode remarkable arrays of effectors that bear the signs of a long evolutionary history of meddling with host systems. This is providing glimmers of how these parasites might comprehensively hijack various host systems. However, the mechanisms of deployment and targets of the effectors of even well-known apicomplexan parasites still remain poorly understood.
The manipulation of host odors and behaviors brings us to the more general macro-ecological consequences of parasites that are also not clearly understood. Several researchers like Zahavi, Hamilton, Thornhill and Fincher have proposed hypotheses that are dependent on parasite load in a species. Both Zahavi’s handicap principle and Hamilton’s proposal regarding the strength of expression of secondary sexual characters derive from the idea that these are honest signals for a strong intrinsic immunity against parasites in the possessors (typically males) to their potential mates. Indeed, in support of Zahavi’s hypothesis, the high-ranked male mice with increased testosterone were more susceptible to the apicomplexan parasite Babesia microtii suggesting that maintenance of top-tier male behavior in the face of parasites needs a stronger intrinsic immunity. In contrast, Thornhill’s hypothesis suggests that societies with a higher parasite load tend to display behaviors that are more aligned with conservative/xenophobic tendencies, and those with lower parasite loads tend to develop more liberal/xenophilic tendencies — this generally matches the caricature of the left-liberal as a shabby and unkempt individual (e.g., their father Karl Marx himself). Given that genome-wide association studies in humans have uncovered linkages between political orientations and certain odorant receptors, one must also bring into the picture the possibility that odor manipulations by parasites might be at the heart of such connections — for example, an odorant receptor variant with the capacity to “smell” infection might trigger a xenophobic response. Similarly, behavior manipulations, such as increased xenophilia, might allow the parasite to spread. Thus, beyond the Thornhill hypothesis, one needs to consider the possibility of direct manipulation by parasites leading to certain political orientations. Indeed, one cannot avoid seeing parallels to the behavioral manipulations induced by memetic parasites such as West Asian monotheisms and their secular mutations. Therein, a multiplicity of behavioral consequences can be seen, ranging from a totivirus-fungus-type association to suicidal behavior induced by several parasites.
## Cārucitrābhisambodhi
Chakkalal and Mundalal saw that Gannaram Dakiya, the owner of the little eatery, had taken a bit too much of an ethanolic beverage and had forgotten to lock the safe with his phone, cards and some cash. They broke into his shop and made away with those. As they were sneaking out, they were seen by Gannaram’s cook, who was washing vessels in the vicinity. However, he did not make much of their presence, as they were familiar idlers who lazed around near his stall, getting a free meal from the leftovers of Gannaram’s customers. When Gannaram returned to his mindfulness, he was shocked to discover his loss and cast about trying to find out who had robbed him. His cook was quick to point out that he had seen Chakkalal and Mundalal and suggested that it might be them. Gannaram soon assembled a band of fellow jāti-folks, some of whom were particularly rough, and set out in search of the two wastrels. They found the two wasted from the aftermath of a heavy celebratory libation and dope sourced with their ill-gotten gains and thrashed the hell out of them. Chakkalal died and Mundalal went comatose.
The founder of the Nīladrāpeya-dala, while born a Hindu himself, sought the eradication of the Hindu Dharma, for which, as stated by the old Monier-Williams, he wished to start with the liquidation of the V1s. However, the most immediate enemies of the NDs were the service jāti-s that stood just above them in the hierarchy. Their recent electoral success had brought the powerful Nīladrāpeya politician Mhaisasur from Ashmanvati into the ruling circle. Chakkalal and Mundalal’s relatives now applied to their coethnic Mhaisasur, who had just been elected to power, to avenge their fate. He had earlier assembled a band of rowdies to slay some sādhu-s, an act that had gone unpunished by the Pratapa Simha government as part of their effort to woo the NDs. Buoyant in his power, Mhaisasur thought there was little anyone could do to stop him from taking the law into his own hands. Thus, he unleashed his goons to go postal on the jāti of Gannaram.
With the junior college exams behind her, Charuchitra and her mother traveled to several Viṣṇu shrines among the ancient hills. Having completed those pious visits, they took a bus to proceed to Somakhya’s city. There she was to join her cousin in taking a critical entrance exam for the best schools across the country. Due to the reservation policy for various jāti-s claiming a depressed status, there was no guarantee that a V1 girl like her could make it to the course of her choice in her own city. However, by taking this common entrance exam, where merit was still valued to a slightly greater degree, she could increase her chances for the same. Her parents fervently hoped that between the college and the common entrance tests, she would make it to a reasonable institution in their own city. As a backup, they were also hedging on Somakhya’s or Babhru’s city so that she could stay in the safety of one of her close clansfolk’s homes to complete her education.
As they were so caught up with her exams, they had hardly paid any attention to the news. Thus, largely ignorant of the unexpected electoral results, they were on the bus, which was to stop briefly at Ashmanvati and pick up a few passengers before proceeding to Somakhya’s city. Charuchitra’s mind was filled with expectant thoughts – Somakhya was the cousin she was closest to and had not seen him in a couple of years. She was also hoping that he might provide some key solutions for questions in the impending exam that had perplexed her – after all, Somakhya’s city and college were perhaps among the most competitive in the nation. Even as they were nearing Ashmanvati, the bus suddenly came to a halt. At first, they thought there might be an accident downstream, but ere long, the halt had already stretched to half an hour. The passengers began irately asking the driver and the conductor what was happening. Finally, the conductor announced that a major riot was underway in Ashmanvati, and they were waiting for their sources to give them the green light to enter. On hearing this, the anger turned into a wave of disquiet among the passengers. Charuchitra wanted to text her father and siblings, but her mother did not want to get her husband worried and pressed her not to do so.
To their fortune, that did not come to pass as the driver finally found his way out of the riot-torn Ashmanvati. Charuchitra could not take her mind off the bloody sights she had just seen, and they kept reappearing before her eyes each time she would doze off, only to reawaken her. As she looked out, she now saw the more calming sights of the irrigated fields spreading out in front of the window, punctuated by the occasional derelict rustic house or granary – a far cry from the comforts of urban modernity she was familiar with. Unable to shake off the visions of Ashmanvati, she looked at her mother: “Mom, that was truly gruesome.’’ C.M: “Yes dear, those are indeed the marks of our age … or perhaps any age for that matter … for that is the nature of men.’’
C: “I find it very unfair that a temple sevaka, a V1, full of piety, should have been killed in that manner. At least a soldier signs up for that as part of his job, and we entrust our safety to his sacrifice. But why have the gods deserted the sevaka thus when he was proceeding to his duty?’’
C.M: “Our itihāsa-s have taught us lessons in that regard. In the first one, we learn how the princes of Ayodhyā, who were like an earthly Indra and Viṣṇu, had to spend 14 long years in the forest, full of suffering, for no fault of their own. In the second one, we hear of a similar trial for the Pāṇḍu-s – so tragedies happen to people though it seems they apparently do not deserve them.’’
C: “Why do you say “apparently’’? Do you suspect there is some hidden cause we do not know?’’
C.M: “I don’t know for sure if there is a meta-causality for a person’s fate or if it is just the probabilistic nature of things. As you know, our people believe that one is reborn again and again. Hence, if they cannot find a cause for a prasaṅga in this janman they project it into the previous one.”
C: “While that definitely satisfies our urge for completion of the causal chain of a prasaṅga, I have no way of knowing if it is true. Moreover, I don’t know what is the conversion table for the dharma of one species to another…’’
C.M: “That latter is something I definitely do not know. Given that a soothsayer said that in your last birth you were born as a cat and Somakhya as a rhinoceros, I really cannot say what karman-s in your non-human past janman-s yielded this human birth.’’
The rest of their journey, though somber, was eventless, and their disquiet eased a bit as they reached Somakhya’s house late that evening. Despite the traumatic sights, the fear of the impending exam turned Charuchitra’s mind entirely to it. The next day when Somakhya awoke, he found his cousin already all clean, prim and busy with her books. Somakhya on the contrary, was considering spending part of the day studying the wildlife in a dry well he and Lootika had located. Seeing his cousin so lost in her books, he slipped away on this venture. That evening he returned with Lootika and introduced her to his cousin, who seemed pretty happy that she had clocked a nearly uninterrupted study of eight hours that day.
Lootika: “Charu, if only we had your focus, we would probably rank among the greatest scientists of our age. Unfortunately, the great god Indra separates the guṇa-s among the folks.’’
C: “I’m shocked you guys did not even look at your books the whole of today, and now you are inviting me for a session of microscopic examination of your specimens! Thankfully, I’m pretty safe for the biology papers, I believe, thanks to our friend Indrasena. While younger than us, he has given such extensive notes that I could possibly write graduate-level exams with them.’’ Lootika smirked at Somakhya: “You should tell my sister Vrishchika that.’’
C: “My mom has stiffened me in statistics, but chemistry is the weakest link. I hope you shore me up a bit there.’’
S: “Actually, the chemistry is not really too stiff for this exam.’’
C: “That may be so for you. I want to ask you a bit about the basic electronic wave functions.’’
S: “We don’t have to solve any version of Schrödinger’s equation in 3D for this test. We just need to know how to deal with the radial distance wave functions and know the shapes of the orbitals by rote.’’
C: “Ah, there you are! That brings me straight to what I wanted. I was looking at questions collected from previous exams by our seniors and there was this one: Draw the radial wave probability distributions for the orbitals 1s to 3p and use it to explain why $sp^3, sp^2$, etc. hybridization happens. I know that these distributions have some weird humps but how do we get those shapes exactly?’’
S: “You get those shapes by solving Schrödinger’s equation for the wave functions of electrons at different excitation levels in a hypothetical atom. For this exam, all you need to know is the form of these wave functions. If $x$ is the radial distance variable, we can write the shape of the wave functions thus. Of course, these are to be normalized by factors of $\tfrac{Z}{\sqrt{\pi}}$, where $Z$ is the atomic number, and we set the Bohr radius $a_0=1$ But for our purposes just the shape matters:
$f_{1s}\left(x\right)=e^{-x}\left\{0\le x\right\}$
$f_{2s}\left(x\right)=\left(2-x\right)e^{-\frac{x}{2}}\left\{0\le x\right\}$
$f_{2p}\left(x\right)=xe^{-\frac{x}{2}}\left\{0\le x\right\}$
$f_{3s}\left(x\right)=\frac{1}{6}\left(27-18x+2x^{2}\right)e^{-\frac{x}{3}}\left\{0\le x\right\}$
$f_{3p}\left(x\right)=\left(6x-x^{2}\right)e^{-\frac{x}{3}}\left\{0\le x\right\}$
From these, you get the shape of the probability distributions as $x^2f_{o}(x)^2$, where $o$ is your desired orbital. Lootika could you draw these out on your tablet for her? So all you need for this exam are the above 5 equations.’’
L: “As you can see from their plots, there is considerable overlap in the 2s and 2p which allows their hybridizations.’’
C: “No wonder you guys seem so relaxed! But I have a bunch of other questions and some puzzlers from the previous years’ math papers.’’
S & L: “Sure, let’s work them out!’’
ↈↈↈↈↈ
The exams were over, and Charuchitra returned home with Somakhya and his friends Lootika and Sharvamanyu after dropping off the bike her uncle had rented for her. Somakhya and Charuchitra’s mothers accosted them and asked them about their prospects. They said they need to do a “post-match’’ analysis and they would let them know after that what their chances might be. Somakhya’s mother: “Lootika, call your mom right away and tell her you are here.’’ Lootika asked them to wait for her sister Vrishchika to come over: “My sis is way more systematic than I’m. She asked me to call her so that she can collect all the questions we remember and note them down!’’
C: “I fully commiserate, as I did the same with my seniors.’’
L: “Sharva and Vidrum got them for us from a bunch of seniors last week and we did a quick survey – definitely that helped as some of the notorious questions were the same as the previous years.’’
C: “I was impressed by your collection. That session you guys gave me has really boosted my hopes.’’
Shortly thereafter, Vrishchika sauntered in: “How did it go? Hope you’ll survived!’’.
L: “Forget about us; it will be your turn soon!’’; thus, they started giving Vrishchika whatever questions they remembered.
Vrishchika: “How many bonds are there between carbon and oxygen in Carbon Monoxide? What’s the answer here?’’
L: “Vrishchika, either pay attention in class this year or ask our sister Varoli; she’ll definitely give you the answer’’. However, Somakhya passed her a chit of paper with a drawing: “That should give you the answer.’’
They then started trying to recall the math questions. Sharvamanyu remembered the below problem:
What is the geometric figure defined by the convergence of the sum:
$\displaystyle \sum_{j,k=0}^{\infty} x^j y^k$
Charuchitra: “What answer did you guys get?”
Somakhya: “An area bounded by a square of side with length 2 defined by the diagonal points $(\pm 1, \pm 1)$.’’
C: “Oh no! I put it down as a circle with radius 1; How foolish I have been.’’
S: “Charu, your mom will give you a shelling if she hears this!’’
C: “Without using a calculator, approximate $\sqrt[3]{2}$ to 5 places after the decimal point.’’
Vr: “Ah, I think I can do that by setting up some expression for binomial expansion! But how do I break up $\sqrt[3]{2}$?’’
C: “You can use $\tfrac{5}{4}\left(1+\tfrac{3}{125}\right)$’’
Vr: “Yes! Should have thought of that!’’
Looking at the math questions they had given her, Vrishchika pointed to one: “What about this one: Show $1-1+1-1+1-1+1 \cdots=\tfrac{1}{2}$ — this is a ridiculous question – are they out of their senses? ’’
L: “Dear, it has an easy answer; go home and work it out. Ask our little sister Jhilleeka, she might solve this. You still need to fortify several lacunae.’’
Sharvamanyu: “How did you’ll answer this strange one? What is the first metallic acid? I wrote Permanganic acid.”
C: “I believe the correct answer is $\textrm{Al(OH)}_3$’’
Sh: “Hey, but that is Aluminium hydroxide, a base!’’
C: “yes but is amphoteric; Al still retains some of its homolog Boron’s tendencies. So it forms aluminates similar to borates in addition to behaving as a base. I too was puzzled by this question which had appeared in a previous year, but Somakhya had filled me in on this the other day.’’
Sh: “Hell! I will be losing that one.’’
Vrishchika suddenly felt that she was not really up to speed with the seniors. Looking at her sister, she felt like Bhīmasena before Karṇa. Her teachers and some others, like Somakhya’s mother, thought she was smarter than her sister Lootika based on her curricular performance, but now she could see what she sensed all along – it would take her much more effort to measure up to her sister. It was just that Lootika, like her friend Somakhya, did not invest much in curricular achievements. With these thoughts crowding her mind, she got up to return home with the questions she had gathered: “I think I really need to be spending some time gaming these exams. I’m not yet ready – you guys seem to be on top of it.’’
C: “Don’t worry, Vrishchika; when I was in your place, I was much worse off than you. It took me a whole two years of effort and the last-minute boost from bro Somakhya and your sis to feel relatively safe. I’m sure you will get there.’’
Vrishchika took a silent mental oath to strive with her studies to outdo her elder sister when her turn came.
Sh: “Don’t forget to pass on the math and physics questions to Abhirosha; she couldn’t make it as she is attending a preparatory course.’’
Vr: “Sure, I would.’’
They then tallied up their answers and made estimates of their total marks. Despite some slips here and there, at the end of the exercise, they felt confident that they would probably get enough to be admitted to the courses of their choice. Somakhya and Sharvamanyu then called Vidrum to check on him – he had to hurry to catch a train to his native village and was speeding away towards his destination: “I wish I could have joined you’ll for the postmortem, but I’m just glad it is all over. I could have gotten a more accurate measure of where I stand had I been with you all. In any case, I estimated my performance several times and feel I’d probably make the cut. But for now, I just need a break from all this – I hope to be sipping coconut and palm juices in my grandfather’s backyard soon. If I fail, I may as well continue as an agriculturist in my ancestral land. I just hope the mayhem from Ashmanvati does not spread to my village. See you later.’’ As the boys were talking to their friend, Lootika and Charuchitra were trying out decorative plaits on each other’s tresses.
S: “Girls, it seems you are rather gainfully employed, so we’ll leave you to that, and I will ride up with Sharva to his place, see him off and come back.’’
C: “No, there was something I have been wanting to talk to you’ll about. I just overheard you talking to your friend about Ashmanvati. I have been struggling to keep it out of my mind till the exams were over.’’
S: “You know, Charu and my aunt were passing through Ashmanvati en route here even as the violence broke out?’’
C: “It was a very close brush. What I witnessed has been gnawing away at the back of my mind, but I have been pushing it away for I did not want it to come in the way of the exams.’’ Charuchitra then proceeded to tell them what she had seen.
Sh: “That sounds bad. While you were in the thick of the action, it seems you are not aware of what actually transpired in Ashmanvati.’’
C: “Apart from hearing that there was inter-caste violence, I did not have the time to follow the news over the past few days. But I can swear to you’ll that I saw a dreadful band of marūnmatta-s marching down the street!’’
Sh: “Yes, you are yet another witness to part of what really unfolded there. The news media has only been reporting a fight between a scheduled tribe and the “upper castes,” making it appear as if the V1s and V2s have been oppressing the former because their leader Mhaisasur got elected in the recent ill-fated elections. However, via social media, we know the reality – the original fight was between the former oil-presser service jāti and the scheduled tribe. Mhaisasur, from the latter, belongs to the Nīladrāpeya Dala, and was inspired by the ideology of the founder of his movement, aided and abetted by the foundation of the Mahāmleccha unmatta, Gregory Kun. Thus, he used it as an opportunity to attack the savarṇa-s, in addition to settling scores relating to the original fight. However, in the process, Mhaisasur either accidentally or wittingly attacked the men of his election ally, the IML leader, Shaikh Badi ad-Durubi bin Darboos. Ad-Durubi retaliated with a massive show of strength, and Mhaisasur’s gullet was bisected in the clash. Now the media has been blaming it on none other than you guys – the reactionary Brahminical forces as they would have it!”
C: “Wow, you seem to be politically really well informed despite the exam!’’
Sh: “You better be; as you just experienced, it could be a matter of life and death.’’
L: “Was ad-Durubi not in jail for attempting a bombing during the Ārdrā fair at the Kāśiliṅga temple?’’
S: “Indeed, but he was let off by the legal activism of the woman who became the candidate of the SJWP party with the aid of the judge Udup Sandha, who has now become the Chief Justice!’’
C: “The common man has to wait for ages to get a hearing in court. How did they pull it off for him? Something sounds fishy?’’
Sh: “Well, they have an endless credit line extended by Gregory Kun, who puts mahāmleccha presidents on the gaddi.’’
C: “Hmm… so, there is more to these recent developments than it meets the layperson’s eye. Our friend Indrasena had told me that we might be headed towards a major clash of men!’’
S: “Absolutely. As everywhere else, the parties like the SJWP have become wildly popular among the screen-addicted urban elite, seized by a disease of the mind the pañcanetraka-mleccha-s have exported to the H. By subscribing to their ways, the upper savarṇa elite, which has internalized the false guilt imputed to them by the mleccha-s, feels a certain sense of holiness. Using the said credit line from Kun, Schwarzstein, the Gulliame Glympton foundation, and the like, they have been extensively converting the deracinated H, who cannot distinguish Skanda from Vināyaka, to this secular self-loathing ideology. One can say that many a neuron in the head of the puruṣa is badly misfiring. This has also meant that Pratapa Simha’s government has had little chance to uphold the laws they enacted in face of the protests from the ND and the Paṭṭa-dala as they had no real public support from their base. This has only allowed those parties to pursue the agendas set for them by their puppeteers in Bahukṣālapura, Navyarkapura, Bhallūkapura and Gajalanḍapura. As I have told you before, the farther a group diverges from the Hindu dharma, the more its propensity to act towards destroying the Indian state. The end result of all this is paving the path for the marūnmatta, who is quite resistant to the memetic diseases spread by the mleccha! We are seeing the first steps in the enactment of that cycle whose natural conclusion will be a clash of men where H will have to pay an enormous human price either way – whether we survive or become extinct.’’
Sh: “And I tell you of the two options, I would rather choose to fight for survival, whatever it might take.’’
L: “If we don’t fight for the glorious tradition our ancestors founded on the steppe and extended all the way from there across Jambudvīpa to the eastern lands and the archipelago pointing towards the Pacific, then who will? The mleccha-s would rather see us as museum pieces, while the navyonmatta-s and marūnmatta-s would send us back to the soil!’’
C: “I wonder how we should place ourselves with respect to our predecessors in such a clash. We can look at former H attempts in what I see in its essence as the same battle. When we were nearly extinct, Vijayanagara allowed us to come back from the ashes. After a good run, they fell in their attempt, but they had laid the foundation for a new attempt in the form of the Marāṭhā-s. Maybe that attempt nearly made it – we can say they almost had a golden age, even if it might have lasted just a decade. Despite all the criticism launched at this attempt by its critics, there is little doubt a clear vision was there — the objective was to reach Gandhāra and sweep the marūnmatta-s and mleccha-s out of Jambudvīpa and demolish their disputed structures, restoring our prāsāda-s. Of course, mistakes were made, and some of those proved too costly, resulting in their ultimate fall to the Christian nation with superior cunning. But I would say that attempt of the Marāṭhā-s was not an entire failure – the country which we have today can be largely attributed to their effort. What we lost can be seen as the last triumphs of the monstrous Durr-e-Durran and the evil Mogol. But from what you say, it seems we are headed to play that cycle once more. But are we in a weaker state than our predecessors?’’
Sh: “I agree that there were touches of sheer brilliance in the Marahaṭṭa assault that seem rather out of the reach of our current stock. The great offensive against the Mogols in Feb-Jun of 1670 CE by the Mahārāja was among the greatest military efforts in recorded history, only to be rivaled by the great Khan of the Mongols or the Qara Khitai knocking down the Seljuks and Ghurids. In that great war, the Marahaṭṭa-s almost took one fort every six days from the Moslems, culminating in the bloodbath in June of 1670 when 4 strong forts were taken in the space of 9 days! The rājan followed this up by reverting the economic warfare to the Mogol territory through the sack of Surat and the rout of the army of Islam near Nashik in the autumn of that year. Would the H forces be able to pull off something like that today when the clash comes upon us?’’
C: “Sadly, such a clash will need much more than a little punch knife or a tactical rod.’’
Sh: “Of course, no one is calling on you to fight the mahāyuddha with a gravity knife. Moreover, don’t forget, Charuchitra, you’re a V1 girl and are to be playing a different role unless you are pushed against the wall. If things come to that dire pass, something is indeed better than nothing, and that punch knife might be the difference between life and death as long as you have learnt to wield it correctly in a real situation. I’m totally with you when you said that even if you fall, you should at least have the satisfaction of having taken one of your enemies with you. But given the grip of the mental disease H are under, you V1s have a lot of work to do in other domains – you need to be like the dog that awakened the legendary sleeping goat. Of course, that doesn’t mean you should not train in arms and keep your body functional in the event you have to join us V2s in the hard fighting, as it happened when Pratāpasiṃha of Citrakūṭa had to face the tyrant Ghāzi Akbar.’’
S: “Our true situation and how we got here needs a more careful assessment. Remember, it was not just the cunning of the Christian nation but, as Lootika and I would often remark, the fact that the less Christian among them were studying snails in the Western Ghats when the Marahaṭṭa did not know that they even existed until he was asked to collect them by his English overseer. It was that which culminated in a Maxwell and a Darwin around the time our people were desperately fighting them in the first war of independence. It took some time for the brawny Jute, the Saxon of flaxen mane and the belligerent Angle to get there; to rise and then fall before passing the spoils accrued by his collective race to his cousins in the New World in a confluence with the uparimarakata project. There is definitely something like “the character of a nation” that manifests even if the individuals who constitute it vanish into the sands of time. We see that character repeatedly express itself in various peoples – the fates of the Cīna, the Atiprāchya, the khaghanate of the Rūs have all played out as per their character. In the case of the H nation, one may ask why, despite their brilliance, did Vijayanagara and the Marahaṭṭa ultimately stumble? Hence, on one side, the character of our nation might imply that, as in the past, we would stumble when the crisis comes upon us. But we could also look at the positive side of it. I’d be the first to agree with you, Charu, that the large modern Indian state would not exist but for the Marahaṭṭa effort, even if the path to it was hardly direct. We share our Indo-European ancestry with many glorious peoples, almost all of whom were conquered by West Asian diseases of the mind, but we still perform the same rites as those of our ancestors on the steppe with the old, accented language. We could find some affirmation in the fact that we are still upholding the way of the gods. This is the only glimmer we have of the hope that we might eventually find a way out of the crisis as in the past. But this time around, there is a palpable sensation that we might have run out of our luck unless the crisis brings out something that we have not shown so far.’’
L: “All I’d say is that my biggest fear is the lack of an unrelenting attitude toward the enemy for that is exactly what their doctrine has for us.’’
Just then, Somakhya’s mother called Lootika: “Your mom wants to take you and your sisters for a garment offering ritual at the little shrine of Mahiṣamardinī that she has commissioned this evening. So she wants you to get home right away.’’
L: “I hope you’ll are coming too.’’
S.M: “No, dear, I had already accepted the invitation of a neighbor to take my sister-in-law to their place. In any case I’ll see your mom at the temple tomorrow for our purāṇa reading.’’
L: “But let Charu come along with me.’’
S.M: “How will she get to your place? You have come by bike, and she has given away the bike my husband had rented for her.’’
L: “She can use Somakhya’s. He said he’ll be doing some research this evening.’’
C.M: “It will be late when you are done. She doesn’t know the city well enough, and it would be risky for her to come back by herself in the dark.’’
L: “She could stay at my place and we’ll come back together tomorrow morning.’’
L: “You’re being very formal but it is no big deal for my father to drop her off by ratha when we return along with Somakhya’s aśva.’’
Thus, after some haggling with the elders and assuring Somakhya that she would make sure that good care was taken of his bike, Lootika got to take Charuchitra along with her. After the garment rite at the little shrine, the four sisters persuaded their parents with some effort to get dinner from the main temple’s annakūṭa with vaṭaka-s, pāyasa-s and other delightful bhakṣaṇa-s. A little distance from the temple, they saw the statue of a warrior with a bow and a quiver by his waist. Charuchitra: “Who is this?’’ Lootika took her close to it and asked her to read the inscription below it. Saṃrāṭ Pṛthivīrāja Chāhamāna, the last Hindu emperor of Dilli; śaka 1244-1270; died at age 26 defending the dharma against Islam. Lootika: “Technically, that is not right as the Gujarati Khusroo, and the brave Hemacandra Vikramāditya sat on the throne of Dilli in brief H interregnums.’’ As Charuchitra was taking in the inscription, Lootika’s sisters decided to take some pictures of themselves sporting their new hair plaits and clips beside the statue and the brightly lit fountain next to it. Suddenly, Charuchitra felt the noise of the merry evening revelers, the patter of the fountain, the hum of the river beside the temple, and the racket of the birds roosting on the banks all die down. At that instant, she felt the spirit of the Saṃrāṭ of the long-gone past leap out from the statue, even as he is supposed to have done when he claimed his bride Saṃyogitā. She suddenly felt that her life was to soon take a different turn establishing a deeper connection to “the last Hindu emperor of Dilli’’ in more than one way. Then that mysterious affectation passed away even as it had come upon her. As she snapped out of it, she felt alarmed as Lootika, who was leaning on Charuchitra with her arm on her shoulder, remarked: “the spirit of the Saṃrāṭ lives on.’’ C: “What! Why do you say so?’’ L: “You know why and you will learn more soon.’’ Before she could press her any further, Lootika’s parents hurried them along to return home.
The next day Charuchitra went over to wake her cousin up with the intention of telling him about the strange incident by the statue. However, before she could get to it, Somakhya took her down the path of talking about the wars of the Chahamāna-s and Calukya-s with the Ghaznavids and Ghurids, and the battles of the latter with Seljuqs, Khwarazm Shahs and the Khitans. Lootika was to spend the whole day with them and joined them for breakfast. However, at the back of her mind, Charuchitra was still thinking about the apparition and other issues like her discussion with her mother on the bus. Thus, when they were done with breakfast, she returned to the topic with her cousin and his friend: “I ain’t pulling a fast one here. But I had a strange experience while standing before the statue of the Chahamāna.’’ Before she could go on, Lootika jumped in: “I believe you felt an apparition of the king manifest before your eyes accompanied by a suppression of other aural stimuli. The apparition seemed to connect somewhere within you, indicating a turn or a new path in your life’s course.’’
C: “I knew you were cognizant that I had experienced something out of the ordinary from your remarks immediately after it. But heavens! You seem to have exactly captured what I went through in the first person.’’ Charuchitra intently looked at her cousin to see if he was surprised – he seemed interested but not really surprised. C: “I must reiterate, we are not trying to set you up for some prank.’’
S: “I know Lootika likes pranks, but they are quite earthly.’’
C: “Lootika? Perchance, did you also experience the same?’’
L: “Not exactly. If you recall, I was leaning on your shoulder then. That allowed me to capture your experience.’’
C: “How is that even possible?’’
L: “Not something we can easily elaborate. Some people have that experience naturally on rare occasions. Others might be able to achieve something like that through hypnosis and yet others through mastery of certain prayoga-s. It is usually easier with physical contact or proximity. Given the environs, I probably would not have achieved that yesterday if I was not in contact with you. I was also fresh from a successful puraścaraṇa I had just performed beside the Mahiṣamardini shrine as the garment offering rite was being conducted.’’
C: “I can rationalize my own experience of the apparition as a purely internal process probably triggered by the rather grave discussion we had earlier that day and the experience I had on my journey. But is thought transference even possible – I see this as also potentially intersecting with the whole question of whether there could be reincarnation and whether thought, memory or karman transference could happen in that case?’’
L: “While there could be a connection running through all of them, we have to be careful and consider them case by case. First, both Somakhya and I can empirically attest to experience transference of two kinds: one is perception in distance of another person’s experience; viz., we do not experience it as the distant person has, but we get a sense of what that person has gone through. The second is a more direct type where we more or less see in the first person what the other person is experiencing. Because of the widespread auto- and objective ethnography supporting this across very different cultures, we tend to accept it as a genuine phenomenon. What we do not know is if this relates to the perception of phantoms of the living or the deceased, the possibility of reincarnation, and any transference which might occur during that process. Our hunch is that it is related to the former, as for the latter, we personally do not have enough empirical evidence to say anything definitive.’’
C: “But does this not go against our very understanding of the world?’’
S: “Yes, the world as we understand it today. But that does not mean we should deny and ignore what we can empirically arrive at. There are several phenomena that are just beyond the reach of a controlled study – they could have commonplace explanations that we do not know, or they could have other explanations relating to facts about the nature of existence that we do not know. Whatever the case, we do not shut ourselves off from the observations and the utility they might have in our lives.’’
C: “OK, but let us break this down. What are the limits of transference that we can currently infer from biology? Thought? memories?’’
S: “To the extent we understand these things, we can say that thought relates to the more dynamic processes within and between neurons. The “connectome,” i.e., the graph of neuronal linkages by itself, is not going to give you thought. Instead, that probably lies in the dynamics of that network, namely the neurotransmitter release at the synapses and the electrical conductance across the neurons – this is likely what constitutes the bulk of it. It is indeed very difficult to see how one could possibly transfer the signals corresponding to the neurotransmitter releases from one neural network to another. However, it is not impossible that there might be some way to sense and reproduce the patterns of electrical conductance, even if it seems out of the reach of our current understanding. Memory, while linked to the above processes, is a different thing in its essence. One class of theories seeks them in patterns of the connectome; however, we see this as only a preliminary step in the actual formation of memories. Based on the correlation between various neural phenotypes of genes encoding proteins involved in the epigenetic modification of chromatin proteins, like histones, and DNA, we believe that memory is hard coded in the form of such modifications. It is also possibly encoded via other epigenetic information purveyors like modifications of cytoplasmic proteins or RNA modifications.’’
L: “Indeed, there are some interesting experiments that suggest the potential transfer of memory in at least some organisms like snails and planaria. The latter are capable of rather remarkable feats of regeneration – if their head is cut off, they can make a new one. Interestingly, it was observed that their learning was transferred to the remaining body even after the original brain was cut off and regenerated. There are some studies that indicated that this transfer happened via RNA. In the snails, a similar transfer was observed via RNA, but the effect of the RNA appears to have been evinced via an epigenetic modification – i.e., methylation of DNA. We don’t know how airtight these experiments are, but, at least in invertebrates, memory transference seems likely and indicative of an epigenetic hard coding. Early studies claimed the same in rats, and it was attributed to a small peptide termed scotophobin, which was believed to transfer the memory of the fear of the dark. However, these experiments were not really reproducible. Thus, we can ultimately say that memory is very tangibly biochemically encoded – something we’ll understand better in the near future. Thought is more dynamic, and we are, to a degree, able to externally control it by electrical means but its experimental transference remains dubious.’’
S: “That said, I believe what we are able to empirically apprehend has a leg outside the domain of objective science in the more poorly understood realm of first-person experience.’’
C: “Well, given all I have seen in the past week and your prognosis for the future, I wonder if this experience forebodes something I need to fear – death or danger? I really don’t have a feel for where it will take me.’’
L: “Being a coparticipant, I can tell you that it is definitely going to mark a change in your path that might happen as early as the end of today. Perhaps it will even be rewarding and you might find your true calling.’’
## RV 10.78
RV 10.77 and 10.78 are similarly themed sūkta-s to the Marut-s by our ancient clansman Syūmaraśmi Bhārgava. He is mentioned twice by authors within the RV – in RV 1.112.16 by Kutsa Āṅgirasa and in RV 8.52.2 by Āyu Kāṇva. In the first instance, he is mentioned as being aided by the Aśvin-s, and in the second he is mentioned as performing a soma sacrifice where he made offerings to Indra. Of his two sūkta, we shall only consider 10.78 below. While the anukraṃani lists it as being composed of triṣṭubh-s and jagati-s, several ṛk-s do not conform to those meters (the syllable count is given in brackets). Instead, in several of them, one hemistich is triṣṭubh-like and the other is jagati-like. Some, like the first ṛk, conform to neither. It was perhaps an unusual meter that was lost in later Indo-Aryan tradition. It has been suggested that it might have been a mātra meter like those from the later register of the language.
viprāso na manmabhiḥ svādhyo
devāvyo na yajñaiḥ svapnasaḥ । (18)
rājāno na citrāḥ susaṃdṛśaḥ
kṣitīnāṃ na maryā arepasaḥ ॥ 1 (21)
Well-minded like vipra-s with mantra-thoughts,
wealthy like those seeking the gods with rituals,
beautiful in appearance like splendid kings,
spotless like the young warriors of the nations…
agnir na ye bhrājasā rukma-vakṣaso
vātāso na svayujaḥ sadya-ūtayaḥ । (24)
prajñātāro na jyeṣṭhāḥ sunītayaḥ
suśarmāṇo na somā ṛtaṃ yate ॥ 2 (23) (hypometrical jagati)
Who with golden ornaments on their chests blaze like Agni,
like winds with their own yokemates bring instant aid,
guides who like elders provide good council,
who provide good protection like soma offerings to seekers of the law…
vātāso na ye dhunayo jigatnavo
.agnīnāṃ na jihvā virokiṇaḥ । (21)
varmaṇvanto na yodhāḥ śimīvantaḥ
pitṝṇāṃ na śaṃsāḥ surātayaḥ ॥ 3 (21) (doubly hypometrical triṣṭubh)
Who like roaring winds move quickly,
like the tongues of fires shine forth brightly,
striving like armored warriors,
liberal like the ancestors at the ritual lauds…
rathānāṃ na ye .arāḥ sanābhayo
jigīvāṃso na śūrā abhidyavaḥ । (21)
vareyavo na maryā ghṛtapruṣo
.abhisvartāro arkaṃ na suṣṭubhaḥ ॥ 4 (21) (doubly hypometrical triṣṭubh)
Who, like the spokes of wheels, have the same nave (navel=source),
like conquering brave warriors facing heaven,
showering ghee like the young warriors wooing [their bride= Rodasī],
like chanters reciting the arka incantation…
aśvāso na ye jyeṣṭhāsa āśavo
didhiṣavo na rathyaḥ sudānavaḥ । (22) (triṣṭubh-like)
āpo na nimnair udabhir jigatnavo
viśvarūpā aṅgiraso na sāmabhiḥ ॥ 5 (24) (jagati-like)
Who are swift like the best horses,
good givers like the charioteers seeking a common bride [=Rodasī]
like waters constantly moving with dense moisture,
multiform like the Aṅgiras-es with their Saman-s…
grāvāṇo na sūrayaḥ sindhumātara
ādardirāso adrayo na viśvahā । (24) (jagati-like)
śiśūlā na krīḻayaḥ sumātaro
mahāgrāmo na yāmann uta tviṣā ॥ 6 (22) (triṣṭubh-like)
Liberal ones like soma-pressing stones, with the river as their mother,
repeatedly smashing everything like rocks,
playful like little children, they with a good mother,
move like a great troop imbued with impetuosity…
śubhaṃyavo nāñjibhir vy aśvitan । (22) (triṣṭubh-like)
parāvato na yojanāni mamire ॥ 7 (24) (jagati-like)
Imparting auspiciousness to the ritual like the rays of the dawns,
Shining forth with brilliance as if seeking auspiciousness,
rushing like rivers, with blazing spears,
as if they have measured out the yojana-s of the yonder realm…
subhāgān no devāḥ kṛṇutā suratnān
asmān stotṝn maruto vāvṛdhānāḥ । (23) (hypometrical jagati-like)
sanād dhi vo ratnadheyāni santi ॥ 8 (21) (hypometrical triṣṭubh-like)
O gods, make us the possessors of good shares and good gems,
us reciters of chants to you O Marut-s, who have been eulogized,
May you attend to our chant and friendship,
for indeed since ancient times the gifting of gems has been yours.
The sūkta has the structure of a riddle hymn, or a brahmodya, where the first 7 ṛk-s are a series of similes. There are a total of 28 similes using na as the comparator, one per foot, each presenting an attribute of the deities of the sūkta. This 4 x 7 pattern is perhaps an implicit acknowledgement of the 7-fold troops of the Marut-s. The sūkta finally culminates in the answer to the riddle in ṛk-8, where the name of the deities is revealed as the Marut-s. To cap it off, the pronoun naḥ (us) is used in the last ṛk. to pair with the comparator na found in the rest. Another striking feature of the sūkta is the repeated (12 times) use of words with the prefix su-, i.e., good or auspicious. Its count in each of the ṛk-s is provided below:
1 3
2 2
3 1
4 2
5 1
6 1
7 0
8 2
While the 7th does not feature such a word, it has two successive words, adhvaraśriyaḥ and śubhaṃyavaḥ, which respectively feature śrī and śubham, both of which imply auspiciousness. We suspect this is intentional, with the build-up of 6 ṛk-s with the su- prefix leading to ṛk-7, where the author reveals his purpose by stating that they confer auspiciousness to the ritual. He then concludes by returning to the su- prefix in ṛk-8 now that he has made apparent his intention in the previous one.
There are a few other notable features in this sūkta:
1. In ṛk-5 the Marut-s are compared to the Aṅgiras-es singing sāman-s. This brings to mind the riddle sūkta of father Manu, where the same metaphor is used for the Marut-s: arcanta eke mahi sāma manvata tena sūryam arocayan ।
2. There are several direct and suggestive “linkages” between the ṛk-s: 1 and 4 are linked by the word marya describing the Marut-s are young warriors. Ṛk-s 2 and 3 are linked by double similes comparing them to both Agni and the Vāta-s. The coupling of the Marut-s with Agni is an important feature of their membership in the Raudra-class, reflecting the duality of Agni and their father Rudra. This is presented in ritual in the form of the offerings accompanying the Agnimāruta-śastra (see RV 1.19). Their connection to the Vāta-s, is emphasized in the post-Vedic traditions starting with the Rāmāyaṇa – Māruti as the son of Vāyu-Vāta and the paurāṇika identification of the Marut-s with the winds. This potentially reflects a parallel early IE tradition (c.f., the Greek reflex of the sparkling or swift-moving wind-deity Aeolus/Aiolos with this 12 stormy children). On the other hand, the connection to Agni (and also Vāyu in the Southern Kaumāra tradition) is retained in the Kaumāra tradition of Skanda, the para-Marut. Further, the accouterments of the warrior (marya) also ṛk-s 2 and 3 – the first has śarman – implying a helmet and the second has varman – armor. Ṛk-4 refers to the arka and ṛk-5 to the sāman – this probably reflects the combination of the śastra and stotra recitation occurring in the soma offering to the Marut-s.
3. Ṛk-s 6 and 7 are linked by riverine similes. Ṛk-6 speaks of the matriline of the Marut-s – they are said to have good maternity, implying Pṛṣṇī. However, remarkably, they are also said to have the river as their mother. This is a rare phrase and in a non-metaphorical sense is only applied elsewhere in the RV to the other sons of Rudra, i.e., the Aśvin-s (RV 1.46.2: yā dasrā sindhu-mātarā manotarā rayīṇām ।). This strikingly parallels the birth of Skanda, often in a hexadic form, from the river in the later tradition. Notably, this motif also occurs in one of the narrations of the birth of Gaṇeśa, where he is born from the bathwater of Pārvatī cast into the Gaṅgā and drunk by the riverine elephant-headed goddess Mālinī (e.g., the Kashmirian mantravādin Jayaratha’s Haracaritacintāmaṇi). Further, like the hexad of Kumāra-s and the other Kumāraka-s born of Rudra, in this ṛk, the Marut-s are referred to as śiśūla-s (c.f., Śiśu, the red-eyed, fierce companion of Skanda in the Skandopākhyāna of the Mahābhārata). Hence, we postulate that even in the core Vedic tradition there was an association between Rudra’s progeny and the river mother. This could merely be a metaphor for Pṛṣṇī, given her atmospheric nebular connections or represent her terrestrial ectype in the form of a river. This riverine connection also extends to the aquatic goddess Saravatī, who is explicitly called the friend of the Marut-s (Marutsakhā in RV 7.96.2) and epithet otherwise only applied to one other goddess, i.e., Indrāṇī (RV 10.86.9).
4. Ṛk-s 4 and 5 are linked by the similes of the Marut-s wooing a bride – vareyavaḥ and didhiṣavaḥ. This is an allusion to their wooing of their common bride, Rodasī, who elsewhere in the RV is mentioned as riding in the chariot along with the Marut-s, gleaming like a beautiful lightning (RV 1.64.9) or the spears they bear (RV 1.167.3). This common wife of the Marut-s is reflected in the para-Marut Kaumāra tradition by the name of Skanda, Bhrātṛstrīkāma (AV pariśiṣṭha Skandayāga), i.e., an allusion to Ṣaṣṭhī as the shared wife of Skanda and Viśākha.
## The turning of the yugacakra
As the wheel turns, what goes up comes down and what is down comes up, again and again. There is a symmetry to the process in the downward and the upward movements, albeit in opposite directions. The old Hindus, right from the days of the śruti (e.g., the Asyavāmīya and the Vivāhasūkta), saw the passage of historical time as such a wheel; indeed, the Bhārata states:
kālacakraṃ jagaccakraṃ yugacakraṃ ca keśavaḥ ।
ātmayogena bhagavān parivartayate ‘niśam ॥
The lord Keśava, by the means of his own yoga, causes the wheel of time, the wheel of the world, and the wheel of the eon to turn constantly.
This triple mention of the wheel likely signifies the three cycles that enamored the old Hindus – the quotidian one, the annual one, and the great cycle of axial precession – the scale on which history occurs – the yugacakra. This wheel of time is worshiped as the supreme god Vāsudeva in early Vaiṣṇava thought and as a Bhairava-like figure, Kālacakra, in late vajrayāṇa bauddha thought. It is described thus:
āvartamānam ajaraṃ vivartanaṃ
ṣaṇ-ṇemikaṃ dvādaśāraṃ suparva ।
yasyedam āsye pariyāti viśvaṃ
tat kālacakraṃ nihitaṃ guhāyām ॥
Eternally turning forth and turning back,
with a six-sectored felly, twelve spokes, and a good linchpin
into whose mouth all existence rushes forth,
that wheel of time is stationed in the [secret] cave of existence.
In the days of yore, the upward turn was seen as the creative expansion or sarga, and the downward one as the decadent pratisarga. The followers of the nagna called the same utsarpiṇi and avasarpiṇi. However, the same “level”, i.e., distance from the lowest or highest point is attained both in the downward and the upward turn. This symmetry in the turn of history is perhaps what some closer to our times have termed the “rhyming of history”. It also relates to the Spenglerian conception of the unfolding of history. A unifying monarch or dynasty, who brings glory to his people and places them in history might be seen in the ascending turn. Likewise, in the descending turn as conditions are worsening people might fumble around for a great leader. A figure or a dynasty might arise to fill in that emptiness, but it is more like the helium flash of a dying star. Indeed, such a figure/dynasty might oversee the end of a civilization. The humble onlooker finds it difficult to tell the difference between the two figures respectively from the utsarpiṇi and avasarpiṇi turns because the timescale of history exceeds that of the mere mortal.
Many religions, both organic and pathological, and the ones in between have some form of millenarianism. This is implicitly related to the turning of the yugacakra, even in counter-religious traditions that have a rather linear view of history. In its simplest form, it may be seen as the expectation among the beholders that the cycle will imminently reach the low point and turn up again. In several versions of the millenarian narrative, it is superimposed onto a savior figure who turns the wheel past its lowest point. In early Indo-Aryan thought, it was expressed as the incarnation of the Vāsudeva to reestablish dharma when it has decayed: “ dharma-saṃsthāpanārthāya saṃbhavāmi yuge yuge ।’’. In the Iranian world, we have the Saoshyant who will come holding the weapon of Verethraghna to restore the Zoroastrian vision of the world. While at some point, both the Āryan branches might have believed that such a coming was at hand, they soon realized the “long arc of history’’ and placed these figures in the remote future. However, when the Iranian counter-religion infected the West Semitic world, the imminency of the coming of the savior figure or the upward turning of the wheel became the dominant theme in many counter-religions coming out of that substratum. Indeed, this lies at the heart of their secular mutations such as rudhironmāda and its subsequent mutations like navyonmāda.
Perhaps, it is a widespread human tendency to think that we live at the cusp of the upward turn of the giant wheel of history. Thus, in every age, there are reports of such a claim, even as some figures are hailed as or try to play the Saoshyant. The lay onlooker possibly hears this voice more prominently in some epochs than others. The current age is one where the rise in its loudness is perceptible. However, there are several distinct directions the expressions of this voice might take:
1. A diverse group of voices can be broadly classified as utopianists of the “techno-optimist’’ type. An extreme and well-known voice of this type is the American Kurzweil who believes that a technological singularity following on the lines of what John von Neumann originally envisaged is almost at hand. But there are several others who place their bets on more limited but directionally similar bets for the near future – the emergence of artificial general intelligence, augmented reality, the realization of quantum computational supremacy, and hyper-longevity/biological freedom/trans-humanism. While most of these see the current state of human biology, behavior, and economics as an impediment, or as defective or inferior in some way, their visions are (at least to us) quite unclear about how their techno-optimism would result in a superior world. Another version of this is a vision of techno-freedom wherein distributed network architectures spanning everything from property and currency to healthcare defeat the ability of traditional regimes to impose their power on the lesser mortals thus ushering in a state closer to utopia.
2. An alternative vision, related to the above, sees the future in interplanetary exploration – literally leaving the earth for potentially greener pastures. Most realistic proponents of this view still see this as more remote than the more immediate technological singularity postulated by those from the above camp. In their reckoning, once technological hyperintelligence is achieved, a superior physics might be discovered, allowing them to break free from the planetary constraints. Of course, they do not bother about the Fermi paradox or the possibility that the superior future physics tells us even more emphatically that interstellar travel is a no-go. It appears that most proponents of this view are nevertheless not extreme utopianists unlike many in the above group, rather, they see planetary escape more as a means for surviving a disaster or resource crunch on earth.
3. If the above visions are on the optimistic side, we also have those who prefer something more like a doomsday track. The most common movement of this type is centered on the possibility of anthropogenic climate change bringing an “end to the world’’. Its proponents seek to bring an end to climate change by acting as the savior figures and reversing the arrow of human industry, agriculture and animal husbandry. While the reality of climate change and its consequences are valid topics to debate, the activists pushing this cause are plainly millenarian.
4. Navyonmāda: This is the successor of the socialist millenarianism, a secular ekarākṣasonmāda, that started with the duṣṭadāḍhika and his śūlapuruṣa sidekick. We have extensively covered navyonmāda on these pages before and alluded to its classical utopian belief system. It has embedded within it a characteristic feature of millenarian ekarākṣasonmāda, in the form of trying to will “critical consciousness” into being by rejecting or inverting pratyakṣa truth, which then will result in the upward turn of the wheel leading to an utopia. In this regard, it also shares features with strands of techno-optimistic millenarianism in seeing biology as essentially bad or limiting. The jātivāda lineage within navyonmāda sees biology as fundamentally bad because it clashes with the samavāda central to its theology. However, ironically, in the process, it ends up reinforcing jātivāda through overpitching and creating “sacred” jāti-s (mostly kṛṣṇa-s and sometimes marūnmatta-s, who are not a jāti per se) that are different from those of its primary proponents (mostly yuropaka-s and mūlavātūla-s). The ṣaṇḍavāda lineage within navyonmāda, like the techno-optimists, sees biology as limiting and seeks to transcend it through interventions that bypass natural selection. This in part explains the enthusiastic alliance we see among the Mahāmleccha-s between tech and navyonmāda – the alliance that helped overthrow the Nāriñgapuruṣa and put Piṇḍaka on the āsandi. At least in the case of ṣaṇḍavādin-s, unlike their co-lineage, the samaguhyānveṣṭṛ-s, natural selection will mostly nullify their fitness in a single generation. Thus, biology would get better of them but not before they have ravaged society with their religion.
The inter-utopianist alliance between the navyonmatta-s and Big-Tech has resulted in this unmāda being deeply embedded among the Mahāmleccha-s. Further, by capturing the seat of power in the government, they have also come in control of the most powerful enforcement organizations in the world, the Mahāmlecchasenā and the spaśālaya-s. Thus, they are poised to bring misery to the world much like their predecessors, the marūnmatta-s and pretonmatta-s. Of the original unmāda-s, marūnmāda is rather resistant to penetration by navyonmāda or any of the many strands of millenarianism; some strands of the mūlarug and pretonmāda might also survive it. However, the Hindus are rather susceptible to some of these utopian movements, especially the most pernicious of them, navyonmāda. While we have been talking of this for ages, only now the general populace seems to be waking up to the fact that key centers of education in India have been captured by navyonmatta-s. Indeed, several families are reporting that their kids have succumbed to this disease from their exposure at educational institutions. Thus, as we have remarked several times on these pages, instead of the expected Satyayuga, the adoption of navyonmāda will bring immense harm to the H, who unlike the mūlasthāna of navyonmāda (the Mahāmleccha), lack the resources in terms of human capital, energy and mineral wealth, to weather the pandemic. Thus, like all utopian movements to date, we see the signs that the marriage of tech utopianism to navyonmāda will also bring misery to many.
There are two key lessons from biology that we have emphasized before on these pages. First, most innovation arises from conflict in biology. Likewise, most true innovation in human technology is downstream of conflict, and it will set off an arms race. Second, there is frequent regime change in the network hubs of a biological system over evolution. This is particularly well-illustrated where we first discovered it – the transcription factor-target gene networks. A similar dynamic plays out with technological hubs. As a result, there will necessarily be inequality – some players will amass immense resources and others will lose the resources they had. A combination of these two parallels to biology means that conflict and inequality of resource distribution will remain the way of the future. Indeed, some of the dramatic new technologies which excite the techno-optimists will create a profound gulf between the haves and the have-nots – a point that arouses the navyonmatta-s. For now, the two have cozied up into an alliance so that this aspect is ignored. In a purely tech-ascendant scenario, the programmer will try to be king. However, his ascendancy will directly clash with the reality-denying navyonmatta who insists on wrong answers for even the most basic operations like summation. Thus, their current coziness would eventually hit the point of a paradox where clashes between state power and a more democratic and/or meritocratic tech-derived power might start.
All this will play out against a backdrop that most techno-optimists apparently ignore – energy. The cognizant are well aware that we are living off a one-time bonanza of fossil fuels that have stored solar energy over a period of several millions of years. Once they are used up, there is no way to replenish them for that process would take millions of years. Thus, even as past civilizations have collapsed or downgraded from resource limitations, the current one too will go down the same path. The techno-optimists hope that the dawning of hyperintelligence with the technological singularity will solve this issue as the real end of fossil fuels is still some time into the future. Others hope that nuclear energy will keep the yugacakra turning. However, simple numerical considerations will show that even nuclear energy cannot sustain future growth on the same exponential track, which a biological species tends to follow whenever it comes upon a new resource. Importantly, the other material resources needed for tapping nuclear energy might place even more drastic limits on the density at which it becomes available. Thus, singularity or otherwise the energetic limitations necessarily imply that the downward turn of the yugacakra awaits us in the future (probably after the time of the people alive today). Some, like Hagens, have called this “the great simplification’’ – the idea that problem-solving mechanisms (tech) will falter from a paucity of cheap and readily available energy (vide Tainter) triggering a possible economic collapse. We suspect it will not be pretty by any stretch of imagination. We all know how wars were and are being fought over fossil fuels and the one who controls them holds the key to winning a long war. The current vassalage of old Europe to the American empire is a direct consequence of this. In the future, with other technologies, like nuclear energy, the same trend will continue for only a few nations have the wherewithal to harness this form of energy safely and efficiently.
Nevertheless, we shall end this note by going back to the idea that the same height is attained repeatedly on both the upward and downward turns of the wheel. Our conception of the yugacakra is a more fractal one – like a Fourier series, wheels turning within wheels. Thus, there are more local arcs of history that we can see and larger ones to which we tend to be blind. With respect to the local arc, we see some remarkable parallels in the turning of the wheel that happens on the order of a century. The most recognizable of these are: 1. the Wuhan corruption of 2019 and the Spanish flu of 1918 (yes, people were masking even then) 2. The great economic downturn we are entering and the Great Depression that started in 1929. 3. The rise of navyonmāda revolutions starting in late spring of 2020 in the USA and the European Marxian revolutions of 1917-1923. These Marxian revolutions laid the ground for major future conflicts even as navyonmāda is doing the same now. 4. The Occidental potentates baiting Russia into a major conflict in 2022 and the same with Japan in the 1930s. There are potentially more events one could align if one went back to doing a more careful analysis.
Given this alignment of events, are we on the cusp of a great war? Briefly, from a geopolitical viewpoint, it is easy to see that there is a fairly high probability of this happening: we continue to stick with our estimate of 15% for the near future while some others with no connection to our thinking or ideology have placed it as high as 20-25%. One thing is clear – the Rūs are by themselves not looking great. Their reliance on Iran for things like gas turbines and drones, the loss of most of their Jewish intellectuals, not quickly producing much tactical machinery to arm their mobilized troops, and bad demographics suggest that they might not have the substance for large-scale military operations. Even some Rūs nationalists are hoping for help from the Han (!) – a rather optimistic view in our opinion given their demographics and that the latter have made themselves even more hated than the Rus in Asia. Yet the Rūs have made some good strategic moves that have rattled the Euro-vassals of the Mahāmleccha. Thus, how far the Mahāmleccha can pursue their aim of destroying the Rus remains in balance as of now. Finally, when a nation is faced with an existential threat, then all stops will be pulled, and we still estimate that the Rūs might have a fight left in them in such a situation that can ultimately prove rather dangerous for the Mahāmleccha. A key to this is when greater disunity will emerge among the Mahāmleccha, who are currently fairly united against the Rūs. However, this will not be forever. We are already noticing irreconcilable differences emerging among the two mleccha-pakṣa-s on the ground that they might be unable to live with each other in the future. Our own model is that, like with some chaotic systems which we have discussed on these pages, the current conflict is not yet the maximal cycle. That might follow in the coming 3-7 years – then the possibility of the replay of the great wars that sandwiched the influenza epidemic of 1918 CE will be higher. Time will tell if there is any truth to this.
## A sampler of Ramanujan’s elementary results and their manifold ramifications
As we have remarked before, Ramanujan seemed as if channeling the world-conquering strides of Viṣṇu, when he single-handedly bridged the lacuna in Hindu mathematics from the days of the brāhmaṇa-s of the Cerapada to the modern era. Starting around the age of 16, he started recording his results in his now famous notebooks. Till that point, Ramanujan had access to only two primary educational sources: Plane trigonometry by Sydney Luxton Loney of Surrey and Synopsis of Pure Mathematics by George Shoobridge Carr of Middlesex which were used in the English educational system. The influence of Loney’s opening pages filled with essential formulae and Carr’s laconic presentations are writ large over the notebooks. It can be inferred that he had obtained many of the results while he was still at school, before the time he started writing them down in his notebooks. For instance, we hear that he had derived for himself the series expansions for trigonometric functions, which he only later learned to be common knowledge – the is said to have then thrown away that early discovery in embarrassment. His first three notebooks were largely completed over a period of six years during his youth in India, with only some entries recorded in the time he was in England. When in England, he mentions that he was only intending to publish his current research, rather than those in the notebooks, until the World War I came to an end. Much of the so called “Lost Notebook” appears to have been written down in the final year of his short but momentous life. Unfortunately, it is believed that several other unpublished works of Ramanujan were entirely lost. Given the time range covered in the notebooks, we do find several elementary results that gives non-mathematicians like us a glimpse into the great man’s mind. We provide below some discussion on a sampler of elementary results from his notebooks. The original entries of Ramanujan discussed in this note can be found in “Ramanujan’s Notebooks, Part I-IV” by Bruce Berndt; here we express them or their corollaries in our own way.
$\pi$ and squaring of a circle
Ramanujan is well-known for his numerous approximations of $\pi$ both in a paper he published on the subject and the various entries in his notebooks. One of those leads to an approximate squaring of the circle that is eminently suitable for a modern śrauta ritualist to construct an āhavanīya that is equal in area to the gārhapatya (Figure 1).
Figure 1. Approximate quadrature of the circle by Ramanujan’s formula
The said construction goes thus:
1. Divide the radius of the circle into 5 equal parts.
2. Extend the radius by 4 of these parts. This gives a segment of length $\tfrac{9}{5}$. Use that segment to construct a circle with diameter $1+\tfrac{9}{5}$.
3. Apply the geometric mean theorem on that circle (Figure 1) to obtain a segment of length $\sqrt{\tfrac{9}{5}}$. Use that to construct a segment of length $\tfrac{9}{5}+\sqrt{\tfrac{9}{5}}$. With that segment construct a circle of diameter $1+ \tfrac{9}{5}+\sqrt{\tfrac{9}{5}}$ (Figure 1).
4. Apply the geometric mean theorem on that segment to obtain the side of the desired square.
One can see that this construction corresponds to Ramanujan’s approximation: $\pi \approx \tfrac{9}{5}+\sqrt{\tfrac{9}{5}}$. This is very close to Āryabhaṭa’s approximation: $\pi \approx \tfrac{62832}{20000} = \tfrac{3927}{1250}$. People have claimed that Āryabhaṭa arrived at his value by using polygons to approximate a circle. There is absolutely no evidence for this claim making one wonder if he had somehow arrived at a formula like that of Ramanujan. It remains unknown to me if Ramanujan had found some special connections relating to this value. The same value is also recommended as a correction to the very approximate Bronze Age values by Dvārakānātha Yajvan, a medieval śrauta ritualist and commentator on the Śulbasūtra of Baudhāyana. This value is somewhat less accurate than another ancient value $3\tfrac{16}{113}$ recorded by Vīrasena and in some Bhāskara-II manuscripts (Ramanujan also provides a construction for the quadrature using that approximation):
vyāsaṃ ṣoḍaśa-guṇitaṃ tri-rūpa-rūpair-bhaktam ।
vyāsaṃ triguṇitaṃ sūkṣmād api tad bhavet sūkṣmam ॥
The relationship between the reciprocals of odd numbers and $\pi$
$4n-3$ defines the alternate odd numbers: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37…
$4n-1$ defines the remaining odd numbers absent in the above sequence: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39…
There is an interesting relationship between the reciprocals of these two sets of odd numbers and $\pi$:
$\displaystyle \pi = 4\sum_{n=1}^{\infty} \left( \dfrac{1}{4n-3} - \dfrac{1}{4n-1}\right)$
It is an interesting though not very efficient formula for $\pi$ reaching 3.1 after 13 terms. This relationship can be obtained from the general cotangent relationship, which is valid for any number $z$ (including the complex plane) that Ramanujan discovered for himself:
$\displaystyle \pi\cot(\pi z) = \sum_{n=1}^{\infty} \left(\dfrac{1}{n-1+z} - \dfrac{1}{n-z}\right)$
Remarkably, Ramanujan records this after a result related to the zeta function, which in turn implies the famous series for the digamma function $\psi(x)$, i.e., the ratio of the derivative of the gamma function to the gamma function:
$\displaystyle \psi(x+1)=\dfrac{\Gamma'(x+1)}{\Gamma(x+1)}=\sum_{n=1}^{\infty} \left(\dfrac{1}{n}-\dfrac{1}{n+x}\right) -\gamma$
Here $\gamma$ is Euler’s constant.
$\sqrt{10}$, cubes of triangular numbers and $\pi$
Figure 2. $10-\pi^2$
In old India (e.g., Brahmagupta and the Jaina Prajñāpti texts) we find $\sqrt{10}$ as an approximation for $\pi$. This is interestingly close to the Egyptian approximation $\left(\tfrac{16}{9}\right)^2$. We can ask the converse question of how close is $\pi^2$ to 10 (Figure 2). Ramanujan discovered an interesting answer for this.
The triangular numbers, $T_n$, are the sums of successive integers up to $n$, i.e, $T_n = \tfrac{n^2+n}{2}$: 1, 3, 6, 10… Then,
$\displaystyle 10-\pi^2 = \dfrac{1}{8}\sum_{n=1}^{\infty} \dfrac{1}{T_n^3}$
Reciprocals of the cubes of odd numbers and $\zeta(3)$
The zeta function elicits an almost mystical experience in us — when you realize how it connects what were seen as disparate branches of mathematics you get a sense of the deep order in the Platonic realm. The function was first discovered by Euler in course of solving what was called the Basel problem. It was known (probably since antiquity) that the sum of the reciprocal of integers slowly diverges to $\infty$:
$\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{n} =\infty$
This can be proved easily with a basic school level mathematics using the comparison test to the reciprocals of the powers of 2 bounding each interval of integer reciprocals (i.e., $\tfrac{1}{3}>\tfrac{1}{4}; \tfrac{1}{5}, \tfrac{1}{6}, \tfrac{1}{7} > \tfrac{1}{1/8}$ so on). However, the question of the sum of the reciprocals of the squares of integers defied attempts of brilliant mathematicians, such as the Bernoulli clan (hence, the Basel problem), until it fell to Leonhard Euler in 1734 CE. Thus, the zeta function can be defined as a generalization of such sums for any number $z$ on the complex plane:
$\displaystyle \zeta(z) = \sum_{n=1}^{\infty} \dfrac{1}{n^z}$
While Euler originally defined it for positive integers, it was generalized as above by Chebyshev on the real line and then by Bernhard Riemann on the complex plane. Thus, the Basel problem is essentially the value of the zeta function at 2, which Euler proved to be $\zeta(2) = \tfrac{\pi^2}{6}$. Euler subsequently established that the reciprocal of $\zeta(2)$ gives the probability of two integers drawn at random from the interval between $n_1$ and $n_2$ being mutually prime (i.e., having GCD=1). This suggested the link between the zeta function and primality, and finally, three years after solving the Basel problem Euler showed the explicit link between prime numbers and the zeta function by his product formula:
$\displaystyle \zeta(z) = \sum_{n=1}^{\infty} \dfrac{1}{n^z} = \prod_{p} \dfrac{1}{1-\dfrac{1}{p^z}}$,
where the product is over all primes.
From the subsequent work culminating in Riemann’s famous hypothesis, the relationship between the zeros of the zeta function and the prime number distribution became clear. Remarkably, Ramanujan discovered many aspects of the zeta function all by himself unaware of the developments in the west, such as those of Chebyshev and Riemann. Among other things, was his much-ridiculed result, where he provided the sum of integers (1+2+3+4…) as a finite negative number $-\tfrac{1}{12}$ (found in his first notebook and communication with Godfrey H. Hardy) — this was essentially his auto-discovery of the value of $\zeta(-1)$. He also discovered for himself the connection between the zeta function and prime numbers. He discovered that the zeta function displays an oscillatory behavior on the negative real line, taking the value 0 at all even negative numbers (-2, -4, -6…). He used these zeros to derive the distribution of primes, paralleling the work of Riemann. However, he overestimated the accuracy of his results for he was unaware of further zeros discovered by Riemann on the complex plane that he only learnt of from Hardy when he went to England. From his notebooks, we learn that during the Indian phase of his career, like Chebyshev, he also explored the values of the zeta function on the real line beyond 2. Thus, Ramanujan discovered a general formula, one of whose special cases is a series specifying $\zeta(3)$ that is today sometimes called Apéry’s constant. After the days of Ramanujan, this constant has appeared in several areas of physics.
$\displaystyle \zeta(3)=\dfrac{8}{7} \sum_{n=0}^{\infty} \dfrac{1}{(2k+1)^3}$
Now, $\zeta(3)$ is also the area under the below curve for positive $x$ (Figure 3), thereby giving an alternative integral formula for the sum based on Ramanujan’s formula.
$y= \dfrac{x^{2}}{2\left(e^{x}-1\right)}$
Figure 3. $\zeta(3)$
However, Ramanujan’s formula more generally goes on to provide several other series that link the cubes of the reciprocals of numbers separated by 3 (1, 4, 7, 10…), 4 (1, 5, 9, 13…) so on which have the general form $k_1 \pi^3+k_2 \zeta(3)$, where $k_1, k_2$ are constants specific to each sum:
$\displaystyle \sum_{n=0}^{\infty} \dfrac{1}{(3k+1)^3} = \dfrac{1}{27}\left(\dfrac{2}{3\sqrt{3}}\pi^3+13\zeta(3)\right)$
$\displaystyle \sum_{n=0}^{\infty} \dfrac{1}{(4k+1)^3} = \dfrac{1}{16}\left(\dfrac{1}{4}\pi^3+7\zeta(3)\right)$
We were puzzled by what might be the sum of the reciprocals of cubes of numbers separated by 5: 1, 6, 11, 16… Taking the limit as $n \to \infty$ of the digamma derivative-based series formula we established that this can be expressed in a rather compact form with the tetragamma function, i.e., the second derivative of $\psi(x) \Rightarrow \psi^{(2)}(x)$:
$\displaystyle \sum_{n=0}^{\infty} \dfrac{1}{(5k+1)^3} = -\dfrac{1}{250} \psi^{(2)}\left(\dfrac{1}{5}\right) \approx 1.0059121444577$
The Ramanujan primorial plus one fourth sequence
The prime numbers 2, 3, 5, 7, 11… are denoted by $p_1, p_2, p_3, p_4, p_5 \dots$. By analogy to the factorial product, one can define the primorial as the product of successive primes:
$\displaystyle p_n\# = \prod_{k=1}^{n} p_k$
Ramanujan defines a sequence such that $2 \sqrt{p_n\# + \tfrac{1}{4}}$ is an odd integer. This holds for the below values of $n$:
It remains unknown to us if there is any further term in this sequence and if there is one, how many more exist?
Elementary results relating to powers of numbers
Ramanujan provides numerous elementary results relating to the sums of the powers of numbers that he likely derived when he was still in school. One of the simplest is the following, which, however, was apparently unknown until his discovery:
if $ad=bc$ then for $n=2, 4$:
$(a+b+c)^n+(b+c+d)^n+(a-d)^n = (c+d+a)^n +(d+a+b)^n+(b-c)^n$
Thus, it is a parametrization that allows one to find six numbers such that the sum of the squares and the fourth powers respectively of the first 3 is equal to those of the last 3.
1 9 10 5 6 11 2 11 13 7 7 14 3 13 16 9 8 17 4 15 19 11 9 20 5 17 22 13 10 23 6 19 25 15 11 26 7 21 28 17 12 29 8 23 31 19 13 32 9 25 34 21 14 35 10 27 37 23 15 38 11 29 40 25 16 41 12 31 43 27 17 44
With this parametrization, we can obtain the above hexad where the first term is every positive integer; the second term is every odd integer starting with 9; the fourth term is every odd integer starting with 5; the fifth term is every positive integer starting with 6. The third and sixth terms are the sums of the previous two terms. The sum of the squares of the two triads constituting these hexads will be defined by: $14n^2 + 70n + 98; \; n=1, 2, 3 \dots$ The sum of the 4th powers is given by $98 (n^4+ 10 n^3 + 39 n^2+ 70 n +49 )$.
The next problem in this genre is to find rational solutions to the indeterminate equations:
$2w^2=x^4+y^4+z^4 \; ; \; 2w^4=x^4+y^4+z^4 \; ; \; 2w^6=x^4+y^4+z^4$
Ramanujan gives parametrizations to solve such equations: if $a+b+c=0$ then,
$2(ab+bc+ac)^2 = a^4 +b^4+c^4 \dots \S 1$
$2(ab+bc+ac)^4 = (a(b-c))^4+(b(c-a))^4+(c(a-b))^4 \dots \S 2$
$2(ab+bc+ac)^6 = (a^2b+b^2c+c^2a)^4+(ab^2+bc^2+ca^2)^4+(3abc)^4 \dots \S 3$
The equation $\S 1$ is quite trivial. For the equation $\S 2$, using Ramanujan’s parametrization one can obtain several sets of tetrads. Below is an example where we take $a$ to be successive integers starting from 0 and $b$ to be 1 more than $a$:
$a=0,1,2,3\dots; b=a+1$
w x y z
1 0 1 1
7 5 3 8
19 16 5 21
37 33 7 40
61 56 9 65
91 85 11 96
127 120 13 133
169 161 15 176
217 208 17 225
271 261 19 280
For this tetrad, one sees that $y$ is the sequence of odd numbers. $w$ (first column) are the hex numbers, i.e., the centered hexagonal numbers given by the quadratic expression $3n^2+3n+1$. The sequence defined by $w$ also defines the maximum number of bounded areas you can obtain by drawing triangles on a plane: With 1 triangle you can obtain at most 1 bounded area; with 2 you can obtain at most 7 (hexastar); with 3 you obtain 19 and so on. $x$ (second column) is the square star numbers (Figure 4), i.e., square numbers with triangular numbers on each side, given by the quadratic expression $3n^2+2n$, while $z$ (fourth column) are the square grid numbers given by the expression $3n^2-2n$ (Figure 5).
Figure 4. Square star numbers.
Figure 5. Square grid numbers.
Notably, these three columns are also linearities on the hexadic spiral (Figure 6).
Figure 6. The hexadic spiral. The 3 linearities in blue boxes correspond to 3 of the columns in the above parametrization
We can again use Ramanujan’s parametrization for $\S 2$ with $a=1$ and $b$ as successive integers starting with 0. This yields the below sequence of tetrads:
w x y z
1 1 1 0
3 3 0 3
7 5 3 8
13 7 8 15
21 9 15 24
31 11 24 35
43 13 35 48
57 15 48 63
73 17 63 80
91 19 80 99
The sequence corresponding to $w$ in this tetrad is specified by $n^2 - n + 1$. Remarkably, this sequence appears in multiple geometric contexts. One is the Euler (Venn) diagram problem. It is an analog of the problem of the maximum number of bounded areas obtained with triangles. What is the maximum number of bounded compartments you can represent using circles (Figure 7)? The answer is this sequence.
Figure 7. The Euler diagram problem.
A further geometric context relates to the sequence of triangles where one side is $1$, the second side is $1, 2, 3 \dots$ and the angle between these two sides is $\tfrac{\pi}{3}$. Then the squares on the third sides of this sequence of triangles will have an area equal to the sequence $w$ (Figure 8). In this tetrad $x$ is the sequence of odd numbers, $y$ takes the form $n^2 - 2n$ and $z$ is the same sequence as $y$ offset by 1 in the backward direction. This sequence appears in the so-called Monty Hall problem discussed by Martin Gardner many years ago illustrating the difficulty of understanding probability even in simple problems.
Figure 8. The length of the third side of the 60-degree triangle problem.
## A catalog of attractors, repellors, cycles, and other oscillations of some common functional iterates
One of the reasons we became interested in functional iterates was from seeking an analogy for the effect of selective pressure on the mean values of a measurable biological trait in a population. Let us consider a biological trait under selection to have a mean value of $x_n$ at a given point in time in a population. Under the selective pressure acting on it, in the next generation, it will become $x_{n+1}$. Thus, the selective pressure can be conceived as a function that brings about the transformation $x_{n+1} = f(x_n)$. Thus, iterating this function with its prior value with give us the trajectory of the measure of the said trait in the population. While it might be difficult to establish the exact function $f$ for a real-life biological trait under selection, we can imagine it as being any common function for a simplistic analogical model. This led us to the geometric representation of the process — the cobweb diagram.
Figure 1. Cobweb diagram for the functional iterates of $f(x)=\tfrac{1+x}{2+x^2}$
For example, let us take the function acting on $x_n$ to be $f(x)=\tfrac{1+x}{2+x^2}$ (Figure 1). We can see that the iterative application of this function on any starting $x_0$ (point B in Figure 1) eventually leads to convergence to a fixed point that can be determined by obtaining the intersection between $f(x)$ and the line $y=x$. In this case, one can prove that it will be 0.6823278…, the only real root of the equation $x^3+x-1=0$. Thus, 0.6823278… can be described as the attracting fixed point or attractor of this functional iteration.
Figure 2. Cobweb diagram for the functional iterates of $f(x)=\tfrac{1}{x}-x$
Instead, consider the same process under the function $f(x)=\tfrac{1}{x}-x$ (Figure 2). Here, we can show that there would be two points that emerge as a result of the intersection between $f(x)$ and $y=x$: the two roots of the equation $2x^2-1=0$, $\pm \tfrac{1}{\sqrt{2}}$. These two points draw the iterates towards themselves but the competition between them results in the outcome being chaos unless $x_0$ is exactly at one of them. Thus, these two fixed points can be described as repelling fixed points or repellors. Thus, exploring different simple functions, we realized that there can be three possible broad outcomes for functional iterates: (1) Convergence to an attractor; (2) Convergence to a cyclic attractor, where the endpoint is to cycle between 2 or more fixed points; (3) Chaotic oscillations driven by repellors. Hence, we conjectured that even in the evolutionary process under selection we will see these three outcomes. Convergence to an attractor is commonly observed when populations starting with different mean values of the trait are driven by selection to a similar endpoint. The cycle is less common but might be seen in situations like the coexistence of different morphs of males and females, each with a distinct mating strategy, e.g., in beetles, damselflies and lizards. Finally, the absence of convergence but chaotic wandering of the trait is less-appreciated but we believe is also manifested in nature. We shall see below that there are different forms of chaos and each of them might have rather different consequences.
One can find some of the fixed points or other consequences of functional iteration in certain mathematical volumes or online resources. However, we did not find any of those to be comprehensive enough for easy reference. Hence, we thought it would be useful to provide such a catalog covering a subset of the common functions we have explored. We provide these by stating the function and the consequence of the iteration (attractors, cycles or chaos with associated repellors), followed by comments in some cases. We omit trivial cases like $\sin(x)$, which shows a gradual convergence to 0. The gradual convergence in cases like this is related to their limit as $x\to 0$; e.g., $\lim_{x \to 0} \tfrac{\sin(x)}{x} =1$. In the below catalog, $\phi$ denotes the Golden Ratio and $\phi'$ its reciprocal.
(1) Simple algebraic functions. Here the attractors or repellors can be easily determined by solving the polynomial equations defined by the difference equation specifying the map.
$\sqrt{1+|x|}$: $\phi$
$1+\dfrac{1}{x}$: $\phi$; This attractor also extends to the complex plane. For more discussion of this system see our earlier note.
$2+\dfrac{1}{x}$: $1+\sqrt{2}$; This attractor also extends to the complex plane.
$1+\dfrac{1}{2x}$: $\dfrac{1+\sqrt{3}}{2}$
$2+\dfrac{1}{2x}$: $1+\sqrt{\frac{3}{2}}$
$\dfrac{1+x}{2+x}$: $\phi'$
Figure 3. Chaotic functional iterates of some simple algebraic functions
$\dfrac{1}{x}-x$: symmetric sawtooth chaos: $\phi, \phi'$ are repellors.
$x-\dfrac{1}{x}$: sawtooth chaos: $\phi, \phi'$ are repellors.
The two above systems (Figure 3, first two panels) show chaotic behavior with a peculiar pattern. In the first one, there are rapid oscillations giving an overall symmetric appearance. In the second one, there is a sharp rise to the local peak or valley followed by a slower, convex return towards 0. The profiles of these maps have a tooth-like appearance, though the first is constituted by oscillations fitting into a similar profile as the second.
$2x^2-1$ (Chebyshev 2): chaotic (-1,1)
$4x^3 -3x$ (Chebyshev 3): chaotic (-1,1)
These next two functions are the Chebyshev polynomials 2 and 3, which show chaotic behavior if $x_0$ lies in the interval (-1,1). At -1,1 they remain stationary and beyond those they diverge. Despite the chaos, the values of the iterates show a characteristic U-shaped distribution, with the highest density close to the boundaries, -1, 1, and low densities throughout the middle of the interval (Figure 4). This type of distribution is typical of many chaotic iterates of polynomial functions, e.g., the famous logistic map.
Figure 4. Distribution of the functional iterates of $4x^3 -3x$
(2) Circular trigonometric functions
Figure 5. Number of iterations to convergence or divergence to $\infty$ of iterates of $\cos(x)$
$\cos(x)$: 0.73908513321516 (the solution of the equation $x=\cos(x)$) is the attractor for all real values. On the complex plane, other than those values in the white region (Figure 5), all values within a fractal boundary converge at different rates (indicated by coloring) to the same attractor.
Figure 6. Iterates of $\tan(x)$ from different starting points.
$\tan(x)$: chaotic (Figure 6). The oscillations are generally of low amplitude but are punctuated by rare “explosions” of huge amplitude (hence, shown in $\mathrm{arcsinh}$ scale in the figure). See our earlier note on functions with comparable behavior. Such behavior is analogous to what have been termed Levy flights.
$\sin(2x)$: 0.94774713351699
Figure 7. Distribution of the functional iterates of $\cos(2x)$
$\cos(2x)$: chaotic. The iterates are contained in the interval $(-1,1)$ with certain exclusion zones. The most prominent exclusion zone contains the primary repellor 0.514933264661… (solution of the equation $x=\cos(2x)$; red point in Figure 7).In the negative part of real line, the exclusion begins at $\cos(2)$ (purple point in Figure 7). The points of the other exclusions zones (black points) are more mysterious.
$\tan(2x)$: chaotic
$\sin(x)-\cos(x)$: -1.25872817749268
$\sin(x)+\cos(x)$: 1.2587281774927
$\cos(x)-\sin(x)$: bicycle: -0.83019851706782, 1.41279458572762; These attractors are also valid in the complex plane.
Figure 8. Functional iterates of 160801 starting points of $\sec(x)$ in the complex plane
$\sec(x)$: chaotic for both real and complex values. Interestingly, in the complex plane, the iterates show certain preferred regions of density that are symmetric about the real axis (Figure 8). The centers of these regions of density appear to be close to the multiple of $\pi$ Figure 8; red points).
$\cot(x)$: While it is chaotic on the real line, on the complex plane it converges to either $\pm 1.1996786402577i$ depending on the initial point.
$\csc(x)$: 1.1141571408719
Figure 9. Regions of convergence or divergence to $\infty$ of iterates of $\cos^2(x)$. The light-yellow regions converge to the attractor indicated as a blue point
$\cos^2(x)$: 0.6417143708 is the attractor for real starting points. In the complex plane all initial points withing the fractal boundary converge to the same attractor while the rest diverge (Figure 9).
Figure 10. Regions of convergence of iterates of $\csc^2(x)$ or divergence to $\infty$. The light yellow regions converge to the attractor indicated as a blue point
$\csc^2(x)$: 1.17479617129 is the attractor for real starting points. In the complex plane all initial points withing the fractal boundary converge to the same attractor while the rest diverge (Figure 10).
$\sec^2(x)$: chaotic
Figure 11. Number of iterations of function $\tfrac{x}{\tan(x)}$ for convergence or divergence to $\infty$
$\dfrac{x}{\tan(x)}$: The attractor on the real line is $\dfrac{\pi}{4}$. On the complex plane, the points within a fractal boundary (Figure 11) converge to the same point at different rates (the contours in Figure 11).
$\sin(\cos(x))$: 0.69481969073079
$\tan(\sin(x))$: 1.5570858155247
$\sin(\tan(x))$: -0.99990601241267
$\sin(\sec(x))$: 0.97678326638014
$\cos(\sec(x))$: 0.44604767999913 (root of the equation $\cos(x)= \tfrac{1}{\arccos(x)}$) it the attractor both on the real line and the complex plane.
$\sin(\csc(x))$: $\pm 0.94403906661161$ is the attractor both of the real line and the complex plane depending on the starting point defined by (root of the equation $\sin(x)= \tfrac{1}{\arcsin(x)}$
$\tan(\cos(x))$: bicycle: 0.013710961966803, 1.55708579436399; These values are remarkably close to but not identical to the solution of the equation $\arccos(x)= \tan(\cos(x))$, i.e., $r=0.01371006057$ and $\arccos(r)=\tan(\cos(r))=1.55708583668$. Thus, the sum of these two values is close to $\tfrac{pi}{2}$.
$\cos(\tan(x))$: bicycle: 0.013710102886935, 0.999906006233481; These values are remarkably close to but not identical to the solution of the equation $\arccos(x)= \cos(\tan(x))$, i.e., $r=0.999906018592$ and $\arccos(r)=\cos(\tan(r))=0.01371006057$.
$\cos(\csc(x))$: octocycle: 0.366798375086067, -0.938273127439933, 0.324922488718667, -0.999958528842272, 0.373119965761099, -0.921730305866654, 0.310327826505175, -0.991153343837468; this cycle appears to be associated with oscillations close to $r=\arcsin(\tfrac{1}{\pi})=\mathrm{arccsc}(\pi)=0.323946106932$ and $\cos(\csc(r))=-1$
(3) Hyperbolic trigonometric functions
$\coth(x)$: converges to either $\pm 1.19967864026$ (solutions of the equation $x= coth(x)$) depending on the starting point.
$\mathrm{sech}(x)$: 0.7650100
$\dfrac{1}{\mathrm{arcsinh}(x)}$: $\pm 1.07293831517215$ depending on the starting point.
(4) Exponential functions
$e^{-x}$: 0.5671433; remarkably this is $\mathrm{W}(1)$, where $\mathrm{W}(x)$ is the function discovered by the polymath Johann Heinrich Lambert, in the 1700s. This value can be computed using the below definite integral:
$k= \displaystyle\int_{-\pi}^{\pi}\log\left(1+\dfrac{\sin(x)}{x}e^{\tfrac{x}{\tan(x)}}\right) dx$
Then the fixed point of the exponential function, $\textrm{F}(e^{-x})=\dfrac{k}{2\pi} \approx 0.5671433 \cdots$latex
Figure 12. Number of iterations for convergence of functional iterates of $e^{-x}$ or divergence to $\infty$
In the complex plane, all points within the fractal boundary (Figure 12) converge to the same attractor at different rates or diverge to $\infty$ (white regions).
$2^{-x}$: 0.64118574450499; comparable behavior as above in the complex plane. The closed form for this fixed point can be derived from the Lambert function: $\textrm{W}(x)$:
$\displaystyle \textrm{W}(x)=\dfrac{1}{2\pi}\int_{-\pi}^{\pi}\log\left(1+\frac{x\sin\left(t\right)}{t}e^{\frac{t}{\tan\left(t\right)}}\right)dt$
Then the $\textrm{FP}(2^{-x})= e^{-\textrm{W}(\log(2))}$
$e^{-\tan(x)}$: 0.54522571736464
Figure 13. Number of iterations for convergence of functional iterates of $e^{-x^2}$ or divergence to $\infty$
$e^{-x^2}$: the attractor 0.652918640419 is the solution to the equation $x^2+\log(x)=0$. We can again find a closed form for this fixed point using $\textrm{W}(x)$:
$\textrm{FP}(e^{-x^2})= e^{-\frac{\textrm{W}(2)}{2}}$
In the complex plane, all points within the fractal boundary (Figure 13) converge to the same attractor at different rates or diverge to $\infty$ (white regions). It is interesting to see that one of the convergence contours recapitulates the curve $y=e^{-x^2}$ reflected about the real axis (Figure 13).
$\dfrac{1}{e^x-x}$: 0.7384324007018
$\dfrac{1}{(e^x-x)^2}$: 0.63654121332649
Figure 14. Number of iterations for convergence of functional iterates of $\tfrac{1}{\log(x^2)}$
$\dfrac{1}{\log(x^2)}$: This function is interesting in that it is chaotic on the real line with a repellor at 1.4215299358831… As the iterates approach 1 from below they are prone to negative explosions; if they do so from above, they undergo a positive explosion. The distribution of the iterates shows a preponderance of small values but when extreme values occur they are very large (explosions). Interestingly, in the complex plane, it converges to -0.32447650840966+0.31470495550992i (Figure 14). The number of iterations to convergence reveals a fractal pattern of interlocking circles.
While the fixed points can be determined by numerical solving the equations specifying them, the closed forms, if any, remain unknown for many of them. Finding if they exist would be a good exercise for the mathematically minded.
## The wink of the Gorgon and the twang of the Lyre
The discovery of the archetypal eclipsing binary Algol
The likes of Geminiano Montanari are hardly seen today. This remarkable Italian polymath aristocrat from the 1600s penetrated many realms of knowledge spanning law, medicine, astronomy, physics, biology and military technology. Having fled to Austria after a fight over a woman, he took doctoral degrees in law and medicine. As a result, he obtained a number of aristocratic patronages in return for services as a legal adviser, econometrician and military engineer. In course of these duties, he invented a megaphone to amplify sounds, worked on desilting of lagoons for the state of Venice, prepared a manual for artillery deployment, and composed a tract on fortifications. Like his junior contemporary Newton, he spent a while working as the officer of the mint. These duties also brought him in contact with astronomy and mathematics while interacting with aristocrats at Modena and as a result, he became absorbed in their study, eventually turning into a Galilean. However, he kept quiet about his thoughts on this matter in the initial period owing to the muzzle placed by the church on “things that were obvious” and the “claws of the padres.” This period also led him to go against the church doctrines by becoming an “eclectic corpuscularian”, i.e., atomist and he used the “atomistic” principles to explain physical phenomena, such as his observations on capillarity and the paradoxical strength and explosiveness of the peculiar glass structures known as Prince Rupert’s tears.
By the time Montanari was thirty, he was already an accomplished astronomer and eventually, went on to succeed the famous astronomer and mathematician Cassini of oval fame as the professor of astronomy at Bologna. He was remarkably productive in his thirties and started off by observing two comets in 1664 and 1665. It was through these observations that he presented clear empirical evidence for the first time in the west that these comets were farther from the earth than the moon and were part of the Galilean solar system (contra Aristotelian physics which saw them as atmospheric phenomena). His accurate observations of meteors led him to calculate their speed for the first time also. He also used that to estimate the thickness of the Earth’s atmosphere. As a skilled optician, he also invented a telescope eyepiece with a micrometer grid to construct the first accurate map of the Moon. Montanari was also a friend of the noted biologist Marcello Malpighi and conducted pioneering work on blood transfusion in dogs, noting that in some animals it had a positive impact on their health, whereas it was not so in others. Like a lot of his work, this was largely forgotten and the proper understanding of this phenomenon lay in the distant future. In another foray into biology, he studied the role of temperature in the artificial incubation of chicken eggs.
In our opinion, one of Montanari’s most remarkable discoveries came in 1667 CE when he observed that the star $\beta$-Persei (Algol) had changed its brightness. In his own words:
“And if you look at the scary head of Medusa, you will see (and now without the danger of being petrified, unless the wonder makes you immobile) that the brightest star that shines there, surprised by frequent mutations, possesses the greatest luminosity only sometimes. I had already observed it for many years as of third magnitude. At the end of 1667, it declined to the fourth magnitude, in 1669 it recovered the original rays of the second magnitude, and in 1670 it passed a little over the fourth.”
We could say that this was the first clearly defined report on the variability of Algol. A couple of years earlier his fellow Italian, Pietro Cavina had noted that:
“The Head of Medusa was second [magnitude], agreeing with the ancient catalogs [evidently that of Ptolemaios] and globes and Aratus of Colonia, although Tycho, and other Moderns have placed it at the third [magnitude].”
It is not clear if this was somehow known to Montanari, but in any case, as far as we can tell, there was no evidence that Cavina recognized the variability as Montanari clearly did. He communicated his observations on stellar variability, which included a list of stars for which he had observed differences in magnitude with respect to Galileo’s observations and older catalogs, to the Royal Society in England. In this, he speculated that the different reports of the numbers of the bright Pleiades (6 or 7) might stem from their variability. While most of the differences he reported for the other stars were probably due to inaccurate magnitude determinations in the older catalogs, his observation of Algol was definitely a clear demonstration of stellar variability adding to the earlier discovery of Mira (o) Ceti by Fabricius in Germany. While Montanari got much praise for his observations on stellar variability at the Royal Society and his prolific observations of comets eventually led to a citation in The Principia of Newton, he seems to have been largely forgotten and the renewed study of the variability of Algol had to wait for more than a 100 years.
The rediscovery of Algol’s variability was due to another remarkable man, the farmer Johann Palitzsch, from Dresden (today’s Germany). Early on, he acquired a deep interest in botany, agricultural economics, astronomy and mathematics. As an autodidact, he amassed a vast collection of literature on these topics by writing down whole books by hand. As a farmer he was the first to introduce the New World crop, the potato, to his regions, and conducted regular meteorological observations, leading him to devise a lightning rod that came to be used in Dresden. Palitzsch reported his weather observations to the local mathematical and physical center at Dresden. This allowed him to access the latest literature on astronomy and inspired his own study. As a result, he beat the veteran Messier in recovering the Halley’s comet in 1758 CE (while observing Mira Ceti’s variability) and confirmed the eponymous English astronomer’s prediction regarding its orbital period. In 1761, he studied the solar transit of Venus and discovered that the planet had an atmosphere. Starting September 12th, 1783, Palitzsch carried a remarkable series of observations on Algol and showed that it varied from the 3rd to the 4th magnitude with a periodicity of 2 days 20 hours and 51-53 minutes (today’s period: 2 days 20 hrs and 48.9 minutes). These observations were communicated to the Royal Society in London by Count Hans Moritz von Brühl and were published as: “Observations on the Obscuration of the Star Algol, by Palitch, a Farmer. Philosophical Transactions of the Royal Society of London, Vol. 74, p. 4 (1784).” It is said that Palitzsch correctly inferred that this variability was likely due to an eclipse by a dark companion that was revolving around the star. We see this as a momentous event in modern astronomy – a rather remarkable accuracy of observation for a naked eye autodidact. We may conclude this account of Palitzsch’s great discovery by citing a translation of a copper engraving made in the Latin in his honor:
“Johann Georg Palitzsch, farmer in Prolitz near Dresden, the most diligent cultivator of his paternal farms, a preeminent astronomer, naturalist, botanist, almost in no science a stranger, a man who was his own teacher, pious, sincere, a sage in his whole life. Born on 11th of June 1723.”
However, the story of the rediscovery of Algol’s variability did not end there. As if an Über-mind was in action, coevally with Palitzsch, over in England, the young astronomer Edward Pigott decided to systematically observe stars that might vary in brightness. For this, he roped in his relative, the 18-year-old deaf John Goodricke, to whom he suggested Algol as a target. Goodricke noted that Algol was variable in brightness by observing the star from his window but had initial doubts that it might be a problem with his eyes or due to poor atmospheric conditions. However, using the conveniently located stars around Algol, Goodricke confirmed that it was indeed the star that was variable. He initially thought it might have a period of 17 days but after prolonged observations arrived at a period of 2 days, 20 hours and 45 minutes — close to what Palitzsch had independently reported. Both their observations were reported in back-to-back communications in the Philosophical Transactions of the Royal Society. Goodricke, reasoned that unlike the previously favored star-spot hypothesis of Frenchman Bullialdus and his compatriot Newton for Mira Ceti, the variability of Algol was due to an eclipse by a planet:
“The opinion I suggest was, that the alteration of Algol’s brightness was maybe occasioned, by a Planet, of about half its size, revolving around him, and therefore does sometimes eclipse him partially.”
We do not exactly know what prompted Pigott to ask Goodricke to study Algol; however, it seems that after its variability was confirmed, he checked the older literature and realized that Montanari had described its variability though not its period. It is possible he was already aware of Montanari’s work in the first place and that prompted him to pay attention to the star. In any case, this story ended tragically — Goodricke was awarded the Copley medal for his momentous finding and elected a Fellow of the Royal Society, but he died shortly thereafter due to pneumonia aggravated by the cold from exposure from his observation sessions. Before his death, at the age of 21, he had discovered the variability of Algol, $\beta$ Lyrae and $\delta$ Cephei. The former two will take the center-stage in this note, while the latter was covered in an earlier note. While Baronet Goodricke’s triumph and tragedy earned him his place in history, the farmer Palitzsch, despite recognition from his coethnics Wilhelm and John Herschel faded away into obscurity. His home and observatory were destroyed by Napoleon’s assault.
In 1787, an year after Goodricke’s death and an year before that of Palitzsch, the 19 year old Daniel Huber (in Basel) of the Bernoullian tradition generated the first light curve of Algol. Using this, he definitively demolished the star-spot theory for Algol and presented evidence that it had to vary due to an eclipsing mechanism with predictions regarding the form of the two components. However, this work of Huber, even like his work on least squares (preceding Gauss) was almost entirely forgotten. Thus, it took until 1889, when the German astronomer Hermann Vogel using the spectroscope and his discovery of spectral line shifts from the Doppler effect showed that Algol was a system of two stars that eclipsed each other. Together, with the light curve, he constructed the first physical model of this binary star system with his landmark publication “Spectroscopic observations on Algol.”
We began our observations on Algol starting in the 13th year of our life as Perseus appeared rather conveniently from our balcony and the air was still tolerably unpolluted. Its dramatic variability, like the wink of the Gorgon, has a profound impression on us. We wondered, given its repeated rediscovery, if its variability might have been known to the ancients. Indeed, some have suggested that the number of Gorgons — three — with two being immortal and one (Medusa) being mortal (slain by Perseus) might reflect the $\approx$ 3 day period of Algol with the mortal Medusa representing the dimming of the star. The myth also has a reflection in that of the sisters of the Gorgon, the Graeae, who are described as three hags, who shared a single eye which they passed from one to another before it was seized by Perseus who desired to know the secret of the Hesperides from them. The seizure of that single eye has again been suggested to be an allusion to the three-day period and dimming of Algol in the language of myth. Some others have proposed that this knowledge might have been known to the Egyptians and that the Greeks probably inherited the myth from them. However, the Egyptian case seems even less direct and we remain entirely unconvinced.
After the Vedic age, the Hindus showed a singular character defect in the form of their negligence of the sky beyond the ecliptic (other than an occasional nod to Ursa Major). However, from the Vedic age, we have the sūkta of Skambha (world axis) from Atharvaveda (AV-vulgate 10.8), which pays some attention to the Northern sky. The ṛk 10.8.7 describes the rotation of the sky around the polar axis. In ṛk 10.8.8 we see the following:
pañcavāhī vahaty agram eṣāṃ praṣṭayo yuktā anusaṃvahanti ।
ayātam asya dadṛśe na yātaṃ paraṃ nedīyo .avaraṃ davīyaḥ ॥ AV-vul 10.8.8
This cryptic ṛk talks of the 5-horsed car, which is said to move in the front of the celestial wheel, with two flanking horses yoked to the remaining ones. The second hemistich might be interpreted as its circumpolar nature, as no path is seen untraveled. Hence, we interpret it as the constellation of Cassiopeia with its 5 main stars. In support of such an interpretation, it is juxtaposed in ṛk-9 with a clear mention of Ursa Major (also mentioned in ṛk 5 where the 7 stars of Ursa Major are juxtaposed with the 6 of the Pleiades; derived from Dirghatamas’ giant riddle sūkta in the Ṛgveda) described as an upward facing ladle:
tiryagbilaś camasa ūrdhvabudhnas tasmin yaśo nihitaṃ viśvarūpam ।
tad āsata ṛṣayaḥ sapta sākaṃ ye asya gopā mahato babhūvuḥ ॥ AV-vul 10.8.9
We believe that ṛk 11 again talks about another near polar constellation, which it curiously describes as shakes, flies and stands (3 verbs), breathing or non-breathing, and importantly which while manifesting, shuts its eye:
tad dādhāra pṛthivīṃ viśvarūpaṃ tat saṃbhūya bhavaty ekam eva ॥ AV-vul 10.8.11
Given the remaining near-polar constellations and other stellar allusions in the sūkta, this could be interpreted as the sole ancient Hindu allusion to Algol. However, we should state that we find this or the Greek allusion in the language of myth to be relatively weak evidence for the variability of Algol being known prior to the discovery of Montanari. While we have some direct ancient Greco-Roman allusions to new stars, e.g., the one supposedly seen by Hipparchus (remembered by Pliny the Elder) and one seen in the 130s during Hadrian’s reign, which was taken to be the ascent of his homoerotic companion Antinuous to the heavens, we do not have the same kind of direct testimony for Algol. Hence, while it is conceivable that there was some ancient knowledge of its variability with a roughly three-day period preserved in the language of myth, we believe that there was no direct testimony for that in any tradition.
A look at eclipsing binaries using modern data
Interestingly, two of the variables reported/discovered by Goodricke, Algol and $\beta$ Lyrae, became the founding members of two major classes (respectively EA and EB) of eclipsing binaries in the traditional classification system. The third class EW, typified by W Ursae Majoris, was discovered much later. These traditionally defined classes were primarily based on the shape of the light curve and the period of variability. The most recognizable of these are the EA type binaries. We provide below (Figure 1) the mean light curve of Algol, the founder member of the EA class from the photometric data collected by NASA’s TESS mission as a phase diagram.
Figure 1. Light curve of Algol as a phase diagram from TESS photometric data
The characteristic of EAs is the relatively sharp transitions from the eclipses. In the case of Algol, the secondary eclipse is relatively shallow. This indicates that one of the two stars in the binary system is bright while the other one is dim relative to it. Thus, when the dim star eclipses the bright star, there is the deep primary eclipse, whereas when the bright star eclipses its dim companion, there is the shallow secondary eclipse. In the case of Algol, the brighter star is of spectral type B8V of 3.7 $M_\odot$ (solar masses) and 2.90 $R_\odot$ (solar radii); the dimmer star is of spectral type K2IV of 0.81 $M_\odot$ and 3.5 $R_\odot$. An approximate depiction of an Algol-like system is shown in Figure 2.
Figure 2. An Algol-like binary system
Figure 3 shows the TESS light curve of $\beta$ Lyrae the founder member of the EB type. As this data has a bit of a break, we also present the TESS light curve for another well-known EB binary $\delta$ Pictoris a $\approx 4.72$ magnitude star near Canopus.
Figure 3. Light curves of $\beta$ Lyrae and $\delta$ Pictoris as phase diagrams from TESS photometric data. The magnitudes automatically inferred from the fluxes are inaccurate in this case.
It is immediately apparent that the transitions between the eclipses are much smoother in the EB class. A closer look shows that $\delta$ Pictoris (with a bit of sharpness) is in between the EAs and a full-fledged EB like $\beta$ Lyrae with a smooth light curve. These curves provide a view into the geometry of this system, i.e., the distortion of the two components of the EBs by the massive tidal force they exert on each other. The sides of the stars which face each other are pulled towards the center of mass of the system by the gravitational force. However, the gravitational force declines as the inverse square law. Hence the opposite sides experience a correspondingly lower force and due to inertia move less towards the center of mass — the principle of tides. As a result, the binary stars get elongated into ellipsoids (Figure 4) and that geometry influences the luminous surface area presented by the system, resulting in smoother light curves.
Figure 4. An $\beta$ Lyrae-like binary system
Finally, we have the EW systems, the TESS photometric light curve of whose founder member W Ursae Majoris is provided below in Figure 5.
Figure 5. Light curve of W Ursae Majoris as a phase diagram from TESS photometric data.
Like the EB systems, the EW systems have smooth light curves with one eclipse almost immediately leading to the next. This indicates that the stars in this system too are likely geometrically distorted. However, they differ in having very short periods — e.g., W UMa has a period of just 0.3336 days (nearly exactly 8 hrs) and low amplitudes for the eclipses. This implies that the stars are really close together — so close that they are fused together (Figure 6).
Figure 6. A W Ursae Majoris-like binary system
With these traditional types in place, we can take a brief look at some light curves of eclipsing binaries discovered by the high-quality photometry of the Kepler Telescope (Figure 7), whose original mission was to discover exoplanet transits (see below). We had participated in the crowd-sourced phase of the project and kept the light curves of stars we found interesting. However, the curves here are plotted from the official post-publication data release by Kirk et al.
Figure 7. The blue and red are the deconvolved and reconvolved fitted normalized fluxes.
The first 5 can be classified as being of Algoloid or EA type. Algol itself would be comparable to KIC 09366988 or KIC 12071006 (4 and 5 in the above plot), whereas the shape of KIC 09833618 (6 in above) is in between another EA star $\lambda$ Tauri and the EB $\delta$ Pictoris. In KIC 04365461, KIC 03542573 and KIC 05288543 (1, 2 and 3 in the above) the two eclipses are nearly the same or the secondary eclipse is in the least rather deep. This implies that both stars are comparable in luminosity. Stars 7..12 in Figure 7 show more EB- and EW-like smooth curves and/or short periods. Thus, the traditional classification is something of a spectrum. However, that there is some valid signal in this classification suggested by the period-amplitude diagram, where the amplitude is defined with respect to the deepest eclipse. We first drew this diagram for the 532,990 eclipsing binaries from the VSX catalog of variable stars in which the traditional classification is available for a large fraction (Figure 8). The EWs are clearly distinguished from the rest by the narrow band to the left that they occupy — mostly low in amplitude and short in period. The EAs are pretty much seen across amplitude and period range but are under-represented in the left band where the EWs dominate. They are also less frequent in the right zone with less than 1 mag amplitude but a long period (10-100 days). The EBs overlap with the central zone of the EAs but have a tighter amplitude distribution. They are also more common in the mid-amplitude-long period right zone where the EAs are somewhat under-represented. In fact, the EBs appear to form 3-4 overlapping populations.
Figure 8. The period amplitude diagrams for the traditional types of eclipsing binaries in the VSX catalog.
We next plotted the same diagram for the 425,193 eclipsing binaries from the galactic bulge at the center of the Milky Way photometrically recorded by the Polish OGLE project (Figure 9). We see that the general shape of the period-amplitude plot is the same for both datasets indicating that this pattern is an intrinsic feature of eclipsing binaries that can be used for their classification. The OGLE stars were classified by Bodi and Hajdu on the basis of the shape of their light curves using locally linear embedding, an unsupervised dimensionality reducing classification method (first developed in the Kepler Project), which projects all the stars in the data as a one-dimensional curve. This allowed their classification by a single number the morphology parameter. As can be seen in Figure 7 (M is the morphology parameter for each of the depicted Kepler stars), when this parameter is less than $\approx 0.62$ then the stars are typically EAs. A morphology parameter greater than $\approx 0.62$ includes EBs and EWs, with those close to 1 being mostly EWs. The stars in the period-amplitude diagram in Figure 9 are colored according to their morphology parameter (Figure 9). One can see that it approximately recapitulates a separation between the EAs and the EWs+EBs. However, the EBs and EWs can only be separated to a degree based on the period axis.
Figure 9. The period amplitude diagram for the Milky Way galactic bulge colored by the morphology parameter (categories: $0 \le x \le 0.25$ etc). The contours being 2D distribution densities
One of the major correlates of the morphology parameter is the period of the binary. When we plot a period-morphology diagram for the 2877 eclipsing binaries detected by the Kepler mission (Figure 10) we find that the period declines with the increasing morphology parameter and the majority of stars fall in a fairly narrow band. Only for morphology $\ge 0.75$, we start seeing the emergence of two populations belonging to distinct period bands.
Figure 10. Period-morphology plot for the Kepler eclipsing binaries (colored as above).
However, the selection of the Kepler stars was biased towards shorter periods. Hence, a similar plot for the much larger OGLE Milky Way bulge set shows a truer version of the period-morphology diagram (Figure 11). It largely recapitulates the Kepler plot for morphology $\le 0.66$. However, for values $\ge 0.66$ it shows an interesting trifurcation with 3 distinct bands corresponding to those with a period of 1 day or lesser; with a period of 10s of days; with a period in the 100 days range. Given that the morphology parameter captures the shape of the light curve, this trifurcation evidently reflects the separation between the EWs and the different populations of EBs in the traditional classification.
Figure 11. Period-morphology plot for the OGLE galactic bulge eclipsing binaries (colored as above)
The histogram of the eclipsing binary systems from the OGLE data by the morphology parameter also presents some interesting features. First, the number of stars appears to non-linearly increase with morphology. This is potentially not entirely surprising, given that from the earthly viewpoint, the probability of eclipses occurring increases in very close or contact binary systems that are characterized by morphologies closer to 1. Second, remarkably, the histogram shows 6 distinct peaks, which indicate that there are apparently certain preferred types of geometry among these systems (Figure 12).
Figure 12. Histogram of stars by morphology for the OGLE galactic bulge eclipsing binaries
The 6 peaks approximately occur at morphology values of 0.047, 0.43, 0.52, 0.74, 0.76, and 0.86. The first three of these would be squarely in Algoloid territory. The first and lowest peak would correspond to EAs with sharp, narrow and similarly deep minima. This would imply that one relatively rare but preferred type of geometry is of well-separated, similarly luminous small stars. The next two peaks would correspond to more conventional EAs with broader minima and a clearer distinction between the primary and secondary minimum. These would correspond to stars with clear distinct luminosities belong to different spectral classes as seen in the Algol system. The final sharp peak at around 0.86 is likely dominated by EWs with the two stars in contact. The closely spaced peaks at 0.74 and 0.76 are likely dominated by EBs with the lower peak potentially closer to $\delta$ Pictoris like EBs and the higher one closer to $\beta$ Lyrae itself.
These peaks in the distribution of morphologies suggest that there are some preferred evolutionary pathways among eclipsing binaries (or binaries more generally). To probe this more we looked at the spectral class/temperature data for eclipsing binaries. Unfortunately, this is not readily available for both the stars in the binary for bigger datasets. The only dataset that we found to be amenable for such an analysis was the Russian eclipsing binary catalog, which has 409 systems with spectral types for both components (Figure 13). This is a relatively measly set and skewed towards EAs: 56.6\% EAs; 13.1\% EBs; 15.7\% EWs (In the large VSX database roughly 75\% of the eclipsing binaries are EW).
Figure 13. Distribution of eclipsing binary systems by the spectral types of the two stars. The Wx category is a composite bin holding both Wolf-Rayet stars and hot white dwarfs.
In this dataset, the spectral type B-B pairs are the most common. Whereas only 10.5\% of the EAs in this set are B-B pairs, 28.2\% of the EBs are B-B pairs, suggesting that there is a greater propensity for $\beta$ Lyrae type systems to be hot B-B pairs (Figure 4). That this is a genuine difference specific to the B spectral type is suggested by the observation that the spectral type A-A pairs are in similar proportions among both the EAs and EBs, respectively 8.3\% and 7.1\%. In contrast, the spectral type A-G/A-K pairs, which are another over-represented group are almost entirely EAs and constitute about 22\% of the EAs in the above plot. While the EWs are underrepresented in this set, we still find that 36\% of the EWs are spectral type G-G pairs and constitute a little over 58\% of such pairs in this set. Thus, it establishes that just as B-B pairs are a specialty of the $\beta$ Lyrae, the G-G pairs are typical of W Ursae Majoris stars, whereas the Algols tend to be enriched in hot-cool pairs.
While the spectral classification of the individual stars is not available for the OGLE galactic bulge data, an intrinsic color (V-I) is available. Here, it seems that the V-I color was determined using filters equivalent to the Johnson 11-color system. Thus, one could plot period versus color to see if there might be any features of note (Figure 14).
Figure 14. Period versus color diagram for the galactic bulge eclipsing binaries. The stars in the ranges corresponding to the 6 peaks in the morphology distribution are colored distinctly.
One can see that the systems from the first morphology peak (i.e., those with sharp, narrow and similar eclipses) tend to have long periods and are concentrated in a V-I range that would approximately correspond to the G-K spectral types. We also see that the mid-morphology peaks (2, 3 in Figure 12), which are enriched in more typical EAs, tend to have a broader spread with much greater representation in the higher V-I range corresponding to the M spectral type. In the case of the subsequent two peaks (3, 4 in Figure 12), we see that they show an extension in the lower V-I range $(\le 0.5)$, which indicates the inclusion of hotter stars. This seems consistent with this morphology range being enriched in EBs. The last morphology peak as a color profile similar to the first but at a lower period range. This would be consistent with it being primarily composed of EW stars, which in the Russian eclipsing binary dataset was enriched in G-G pairs.
Though Kepler used its own distinct broad bandpass filter, the effective temperature was calculated for the catalog of Kepler stars. We can use this temperature to study how the Kepler stars are distributed in a period versus temperature diagram — effectively a variant of the period-color diagram (Figure 15).
Figure 15. Period versus effective temperature diagram for the Kepler eclipsing binaries. Stars in 3 distinct morphology bands which are over-represented in the Kepler data are colored distinctly.
Here, we notice that the low morphology parameter stars are again in the longer period range and occur in a relatively narrow temperature band (1st-3rd quartile range: 5937K-5219K) corresponding to G to early K spectral types. The stars over-represented in the middle of the morphology band, i.e., mainly conventional EAs, have a broader 1st-3rd quartile range of 6422K-5197K — from F to early K. Finally, those with a high morphology parameter have a 1st-3rd quartile range of 6590K-5426K, which is the F-G spectral range. This last group, which is enriched in the EW eclipsing binaries (periods less than a day), is notable in showing a fairly tight period-temperature relationship (Figure 15) that is most clearly visible in the temperatures corresponding to the F-K range. Evidently, this corresponds to the period-luminosity-color relationship that was uncovered for the EW stars in the 1990s by Rucinski. Thus EWs, which are rather numerous, can be used as a tool for statistical distance estimation.
Finally, we take a brief look at what the eclipsing binaries offer for our understanding of stellar evolution. For example, some obvious questions that emerge from the above observations are: 1) When we look at systems like Algol we have more massive and hotter stars which are in an earlier evolutionary state than their dimmer, cooler companions which are in a later stage of evolution. Why is this paradoxical situation observed, given that one would expect the more massive star to have evolved faster according to the usual stellar evolutionary trajectory? 2) Why do EW systems show a period-color/temperature relationship similar to pulsating variables like Cepheids?
To address the above, we need to take a closer look at the gravitational geometry of binary systems, i.e., the basics of the Euler-Lagrange gravitational potential curves (Figure 16). Let us consider a binary system with stellar masses $m_1, m_2; \; m_1 \ge m_2$ in the $x-y$ plane with the origin in rectangular coordinates, $(0,0)$, at the center of the more massive of the two stars. We then take the distance of the center of the less massive star from the more massive one $a$ to be a unit distance. This yields its dimensionless coordinates as $(1,0)$. Then the magnitude of the position vectors to a point on this $x-y$ plane from the two stellar centers will be:
$s_{1}\left(x,y\right)=\sqrt{x^{2}+y^{2}}$
$s_{2}\left(x,y\right)=\sqrt{\left(x-1\right)^{2}+y^{2}}$
We define the stellar mass ratio: $q=\dfrac{m_2}{m_1}$
Then, the distance of the center of mass $C$ of the two stars from the origin will be:
$\dfrac{m_2}{m_1+m_2} =\dfrac{q}{1+q}$
Thus, the coordinates of $C$ would be $(\dfrac{q}{1+q}, 0)$
The gravitation potential $\phi$ at a point on the $x-y$ plane is specified thus:
$\phi= -G\left (\dfrac{m_1}{s_1(x,y)} + \dfrac{m_2}{s_2(x,y)} + \dfrac{(m_1+m_2)r(x,y)^2}{2a^3} \right)$
Here, $G$ is the gravitational constant and the first two terms are the gravitational potentials from the two stars respectively. The third term is the centrifugal force, which needs to be accounted for as the two stars are revolving around their common center of mass $C$: here $r(x,y)$ is the magnitude of the position vector from $C$ and $a$ is the distance between the centers of the two stars. Since we have already set $a=1$, i.e., taken it as the distance unit, and computed the coordinates of $C$, we write the equation of $\phi$ after factoring out $\dfrac{m_1+m_2}{2}$ in a dimensionless form in $-G\dfrac{m_1+m_2}{2}$ units on the $x-y$ plane as:
$\phi\left(x,y\right)=\dfrac{2}{\left(1+q\right)s_{1}\left(x,y\right)}+\dfrac{2q}{\left(1+q\right)s_{2}\left(x,y\right)}+\left(x-\dfrac{q}{1+q}\right)^{2}+y^{2}$
With this equation, we can plot the Lagrangian equipotential curves for $k$ a given potential value (Figure 16):
$\dfrac{2}{\left(1+q\right)s_{1}\left(x,y\right)}+\dfrac{2q}{\left(1+q\right)s_{2}\left(x,y\right)}+\left(x-\dfrac{q}{1+q}\right)^{2}+y^{2}=k$
Figure 16. The Lagrangian equipotential curves for an Algol-like system with the five Lagrangian points.
The $(x,y)$ for which the equipotential curve first takes on a real value, i.e., it appears as just two points, define the two Lagrangian points $L_4, L_5$. These can also be found using the equilateral triangle with the two stellar centers. From these two points, the equipotential curves expand as two disjoint lobes lying on either side of the X-axis. Finally, the two lobes intersect at a point on the X-axis to the left of the star with the larger mass. This point of intersection defines the point $L_3$ (Figure 16). The equipotential curves then become closed curves with two inflection points that advance towards each other. They finally meet on the X-axis to the right of the lower mass star. This point of intersection is the point $L_2$. After this, the curve becomes two loops, with an inner loop with two inflections and an outer loop that tends towards a circle (Figure 15). The inflections in the inner loop then intersect at a point on the X-axis between the two stars. This point is $L_1$. After this intersection, the curve becomes 3-looped, with two oval loops around the two stars and the outer loop surrounding both of them. At these points, $L_1-.L_5$, the gravitational forces exerted by the two stars cancel each other. Based on the potential equation one can derive an equation whose solution gives the $x$ values for which the gravitational forces cancel each other yielding $L_1, L_2, L_3$ (Figure 16):
$f\left(x\right)=x-\dfrac{q}{1+q}-\dfrac{x}{\left(1+q\right)\left|x\right|^{3}}-\dfrac{q\left(x-1\right)}{\left(1+q\right)\left|x-1\right|^{3}}$
The inner loop of the equipotential curve defining $L_1$ has two lobes, one around each star, which are known as the Roche lobes. If the stars are far enough, such that each is within the Roche lobe then we have a detached binary. However, if they get close enough such that one of the stars occupies its Roche lobe then it becomes a semi-detached binary. In this case, gas from that star flows out via $L_1$ and falls on the more massive star. The residual escaped gas forms a disk around the more massive star of the system. This kind of mass transfer is seen in the case of Algol from the dimmer, distended K star, which fills its Roche lobe, to the B star. The differential evolution of the stars in such systems, contrary to what is expected from their mass, is believed to occur due to this mass transfer.
As the stars get closer together both stars might occupy their respective Roche lobes. This happens in the case of the EW systems which are believed to have evolved from detached/semi-detached eclipsing binaries with periods less than 2.24 days winding closer and closer together. Thus, these systems are known as contact systems, with the outflow from both stars forming a common envelope whose shape is defined by the infected inner loop of the equipotential curves (Figure 16). This contact will result in the formation of a single body with temperature equilibration. Thus, the radiating surface area (hence luminosity) of the EW stars will scale with their period given Kepler’s third law. As EWs are mostly in the main sequence on the Hertzsprung-Russell diagram their period will also be related to their temperature/color. From the Kepler data (Figure 15) it appears possible that a loose version of such a relationship emerges first in the semi-detached systems with periods in the 2.25 days to just under a day range, which becomes tight in the contact systems represented by the EWs. Thus, remarkably, a subset of the eclipsing binaries has joined the pulsating stars as potential candles for measuring cosmological distances.
## Some poems
Below are some poems in English by our brother. He sends us his compositions in a much more transient medium making them hard to preserve or share. Hence, we decided to anthologize those we could recover and present them here as a record on the internet. Sometimes, they are accompanied by a bit of a “bhāṣya”, which we provide in the cases we were able to salvage it. We also provide some comments of our own.
The Beetle and the Milky Way
From thy curls flows the heavenly stream,
beacon to all creatures big and small;
A scarab scurries under that milky gleam,
homeward bound, rolling her ball.
Danger lurks in the inky dark shadows,
So, the straight path o’er the veldt is best,
But all cardinal points the night swallows;
Who now will guide Titibhā to her nest?
Mounting her ball, as little Titibhā dances,
Her dorsal eye catches the cosmic light —
From a million miles what are the chances
that she could glimpse so distant a sight?
Yet, before long emerge her larvae,
Under the haze of the Milky Way.
The poets “bhāṣya”: Gaṅgā emerges from Hara’s matted locks. In the first quatrain, I have imagined Akaśa-gaṅgā, the Milky Way, emerging from the cosmic body of Rudra. Now, scientists have found that some beetles called scarabs to navigate using the light of the Milky Way. In the dark, they roll their balls of dung away from the source. Second quatrain: This beetle lives in the veldt of southern Africa. After the beetle has collected its forage it must quickly travel in a straight line. If it does not, it risks going in circles and being eaten or its pile stolen by other beetles, or simply going back to the original pile where the competition from other beetles is intense. So, it is imperative that it must take the straight path. But at night, the darkness swallows all the cardinal points; there is no way for it to know where it is going. Third quatrain: Now the beetle does something very interesting. It mounts its ball of forage and does a little dance. As it does that, its eyes catch the Milky Way. Using that as a cue and the small differences in light, it holds a straight-line course. She then buries her eggs in the dung pile. This poem tries to express the awe of how even small creatures are capable of navigating using cosmic cues.
The goddess Ambikā
Mother, these ogres ne’er seem to learn;
Flushed with pride,
every new enterprise seems
to raise their hopes
Only to end in humiliation.
Poet’s Vision: “I see Ambika now seated upon her lion on the brow of a hillock, boisterously laughing, her lips reddened with wine, her roving eyes mocking them.”
When their chief tried to capture thee,
They hurl their best missiles at thee
And not one came within a yard of thee!
what this really means,
I have truly known!
O Ambikā I see you now
on the brow of a hillock, boisterously laughing,
and your reckless eyes mocking them.
Mater familias of three-eyed One,
swarthy as the nimbus on June’s first day,
Mother of the storm troop!
Comment: The last two quarters indicate her manifestation as Pṛṣṇi, the wife of Rudra, and the mother of the Marut-s.
The gods Saṃkarṣaṇa and the Vāsudeva manifest as the Nandakumāra-s
I saw two boys playing in the mead,
frolicking yearlings followed them everywhere,
drawn by their laughter,
with happy lowing to rapturous notes filling the bright glade.
One lad was fair as marble and wore bright blue,
marking the ground for boisterous play,
with his tiny plow;
The other boy, dark as marble, decked in yellow;
The whole world seemed
to be splashed with joy
They were themselves joy all pure —
like word and meaning tied forever.
The best books were books with pictures:
lilac castles ‘n golden mornings,
pretty princesses with dainty glass shoes,
pining princes or ones in frogs;
brave seamen ‘n stormy seas,
for many a rainy evening.
Who’d need Andersen’s flying trunk
or Uderzo’s magic carpet
to travel to the farthest lands
fed by the undying well springs
of childhood’s imagination?
The best books were books with words:
fluttered pennons proud ‘n royal hearts;
while dashing seamen braving wind-kissed surfs
‘n brazen buccaneers
leapt out of the pages,
ruffled by untamed gales,
beating upon windows frail.
Who’d need a flying trunk
or a magic carpet
when words could weave
Tabrizian tapestries with the silken threads
of youthful imagination?
O unputdownable novella,
had drowned the cock’s crow at dawn
but I can scarce recall your title now,
let alone the pretty pictures of castles
like the dreams of my youth, long faded now.
The best books were the books that whispered ‘n spoke:
Faintly at first:
like the tentative chirping of starlings
on spring’s first morn;
And then like the cuckoo’s full-throated ‘n raucous
at midsummer’s high noon.
As I closed my eyes to listen,
the years seemed to fall away!
Proud banners flew o’er the citadel again,
And to the beating of kettle drums marched my tin soldiers,
five and twenty in all,
and astride a dappled mare
tossing her rufous mane,
rode the spirit of story herself,
and even the swaggering buccaneers
with cutlasses drawn,
all came rushing into the mind’s glade
to watch their queen as she cantered.
I smiled.
Through childhood, boyhood, youth
and even in the somber twilight
Ever watching all go by and pass beyond the bend,
reliving the ages now with my own little reader,
who poked at the words
with her chubby dainty finger —
a little wand that turned them into pictures.
A quatrain to the god Kāma
The slender maids of the Kuntala country sweet n fair,
betwixt shy kanakāmbara blossoms trellised o’er their hair,
seem to sing thy triumph from upright turrets.
The visions of the god Viṣṇu
He has a slender waist,
And he’s blue all over;
All riches dwell in his chest —
Our world-strider ‘n soul-saver!
Who could imagine thee —
in the wee fry scooped up
in Satyavrata’s arghya;
Or, bearing mighty Mandara
or, in womanhood’s highest excellence,
ever keeping the greatest secrets
out of demonic reach;
Or, hiding within that pillar,
but the Mantrarāja’s knowers
have seen thee waiting to spring;
Or, crossing the wide ocean,
armed with mighty bow
hastening to the Aśoka grove —
“Aśoka” — coz there’s hope.
I know you were there in all those times.
How can I repay?
O Muses will ye carry these words of praise to Him.
Comment: the verses reflect the poet’s meditative visions of the god.
Blank verse benediction invoking Kumāra
Victory to the reed-born son of Gauri,
whose lance point cleft a hole in the looming darkness of Krauñca,
where birds of light and insight
now chirp and dart in joy;
Impelled by his grace,
may the spear of your intellect too
give us a window to peer
into the secrets of the cell and its denizens.
Who is the thief of life?
Night after night I lay awake,
beset by worry and fear
that your retinue should be near.
In every ache, malaise, and niggle
I heard your herald’s menacing bugle.
Small mercy – you didn’t come!
Yet, I felt my life was stolen
ere the fun had even begun.
So, I’ve come myself to your great hall
to settle the matter once and for all.
I took my courage from the little boy,
who’d waited three days at your gates
in the quest for the fount of eternal joy
He now shines bright like the flame
you named after his own name [1].
All resplendent you seem
like the thunder cloud.
No offense do I mean,
but are you a thief?
On my way here I saw many a sight
that turned the blood cold in my veins —
Ten thousand pyres all alight
after unending pointless pains.
Heap upon heap of broken dreams,
Families left with no means,
Mangled bodies and minds,
hollowed out long before the end
Ghastly tragedies of all kinds
And wounds that none can mend.
Then I grew numb to it all
‘Tis all absurd as Sisyphus’ curse —
No matter what that downhill fall
in a meaningless universe
Tell me, what are you hoarding here sir?
I ask you squarely “are you a thief?”
Then spoke the resplendent Death,
resting his mace upon his shoulder
“I am no thief.”
It’s true I come when it’s my time.
Yet I did not commit this crime.
Long before were you robbed
by anxious thoughts all your own,
The present moment quietly slipped
like a rug beneath your feet tugged,
I was nowhere in the scene.
Yet you hardly lived these years passed
Why blame me sir?
Granted, sir, you’re not a thief.
Still, I have been robbed every night.
Who will return my precious days,
lost to worry and despair?
I do not have another life to spare.
Resplendent Death thought a bit
And then said: “I think there is One”
But He’s a thief too. [2]
“What? You’ll send me to another thief?”
Then he pointed to his chest mighty
You see this three-pronged scar of old?
I was once young and haughty
And paid dearly when hurled my stranglehold [3].
Perhaps only He can recover what you’ve lost
Hasten, sir. There’s no time.
He lives in the mountains.
Take the winding path.
up the snowy slopes.
The road goes beyond the great river’s womb.
Ignore the goblins and ghouls –
He keeps strange company.
On that path you must trudge,
You will then see his two boys playing [4].
And their mother knitting a shawl [5].
She is the great queen of all,
Yet she won him by austerity —
No greater love story for posterity.
“How will I know him?”
“You cannot mistake Him”
who wears the moon in his tiara.
1. An allusion to the journey of Naciketas the Gautama to the realm of Mṛtyu that is prominently mentioned in the literature of the Kaṭha-s. The final line in this verse alludes to the iṣṭi that is named after him.
2. Rudra is said to manifest as various criminals (e.g., taskara= thief) in the Śatarudrīya from Yajurveda-saṃhitā-s.
3. The conquest of Mṛtyu/Yama by Rudra — the liṅgasthāpanā-mantra “OM nidhanapatāntikāya namaḥ |” alludes to this.
4. Skanda and Vināyaka.
5. The motif of the goddess weaving time.
## The Kaumāra cycle in the Skandapurāṇa’s Śaṃkara-saṃhitā
Many khaṇḍa-s, māhātmya-s and saṃhitā-s attach themselves to the sprawling “Mega-Skandapurāṇa”. We use this term to distinguish it from the “Ur-Skandapurāṇa”, which was first published by Bhaṭṭārāi in the late 1980s and is now known to survive as three related recensions, one of which is represented by rather early manuscripts from Nepal. Of the texts associated with the “Mega-Skandapurāṇa”, the Śaṃkara-saṃhitā, remains relatively poorly known. It is unclear if there was a pan-Indian understanding of its constituent texts and if a complete version was ever extant in any part of the Indosphere. As far as we can tell, one of its khaṇda-s known as the Śivarahasya is preserved only in South India and is likely of South Indian origin. It was most likely composed in the Drāviḍa country; though one cannot entirely rule out the Southern Andhra country or parts of Southern Karṇāṭa as its original source. It was edited by a maternal śrauta-ritualist- and paurāṇika-clansman of ours in the 1950s-1960s. Upon completing its editing, he offered it to the shrine of Skanda housing the kuladevatā of our clan. The text as available still has some corruptions, several of which might have been introduced while typesetting. The Śivarahasya presents its relationship to the Mega-Skandapurāṇa thus:
teṣv api+idam muni-śreṣṭhāḥ skāndaṃ sukhadam uttamam ।
sarva-vedānta-sārasvaṃ pañcāśat khaṇḍamaṇḍitam ॥
ādyā sanatkumārīyā dvitīyā sūta-saṃhitā ।
brāhmī tu saṃhitā paścāt turīyā vaiṣṇavī matā ॥
pañcamī śāṃkarī-jñeyā saurī ṣaṣṭhī tu saṃhitā ।
ādyā tu pañca-pañcāśat sahasraiḥ ślokakair yutā ॥
dvitīyā saṃhitā viprāḥ ṣaṭsahasrair alaṃkṛtā ।
trisāhasrair yutā brāhmī pañcabhir vaiṣṇavī-yutā ॥
triṃśatbhiḥ śāṃkarīyuktā khaṇḍair dvādaśabhis tathā ।
ṣaṣṭhī tu saurī saṃyuktā sahasreṇaika kenasā ॥
grantha-lakṣair yutaṃ skāndaṃ pañcāśat khaṇḍa-maṇḍitam ।
tat trayā saṃhitā proktā śāṃkarī veda-sammatā ।
triṃśat sahasrair granthānāṃ vistareṇa suvistṛtā ॥
tat trayodaśa-sāhasraiḥ saptakāṇḍair alaṃkṛtam ॥
The Mega-Skandapurāṇa is divided into 6 saṃhitā-s that have a total of 50 khaṇḍa-s among them. These are listed as follows with their corresponding verse counts: 1. Sanatkumāra: 55,000; 2. Sūta: 6000; 3. Brāhmī: 3000; 4. Vaiṣṇavī: 5000; 5. Śāṃkarī: 30,000; Saurī: 1000. Thus, the entire text is said to be of 100,000 verses. Within it, the Śaṃkara-saṃhitā (Śāṃkarī) is said to have 12 khaṇda-s of which the Śivarahasya of 13,000 verses is one. The Śivarahasya itself is divided into 7 kāṇḍa-s, which are: 1. Sambhava; 2. Āsura; 3. Māhendra; 4. Yuddha; 5. Deva; 6. Dakṣa; 7. Upadeśa.
The published Mega-Skandapurāṇa does not align precisely with this tradition and has 7 khaṇḍa-s: 1. Māheśvara; 2. Vaiṣṇava; 3. Brahma; 4. Kāśī; 5. Avanti; 6. Nāgara; 7. Prabhāsa. The Māheśvara-khaṇḍa in this compendium is not the same as the Śāṃkarī Samhitā under consideration in this discussion. However, they share many common themes that include the central thread gathered around the destruction of Dakṣa’s sacrifice, the marriage of Pārvatī and Rudra, the birth of Kumāra and the killing of Tāraka by him, the birth of Gaṇeśa, the Śivarātri ritual and the worship of Rudra at Aruṇācala. The tale of Skanda and the Tāraka war is repeated twice in the Māheśvara-khaṇḍa of the Mega-Skandapurāṇa.
The first 5 kāṇḍa-s and parts of 6 and 7 of the Śivarahasya in the Śāṃkarī Samhitā comprise a narration of the Kaumāra cycle partly modeled after the Rāmāyaṇa of Vālmīki. Much of the kāṇḍa-s 6 and 7 are primarily śaiva material relating to the observation of vrata-s and Śiva-dharma — these thematically overlap with the material in the Māheśvara-khaṇḍa of the Mega-Skandapurāṇa. The Kaumāra portions of the Śivarahasya were rendered in Tamil by the saiddhāntika guru Kāśyapaśiva in the medieval period as the Tamil Skandapurāṇa. His version has some differences from the extant Sanskrit text of the Śivarahasya — it is unclear if these differences are due to his reformulation of the narrative or because he was using a distinct recension of the text. A Telugu rendering of the text also exists but we do not have much familiarity with it. While the ancient versions of the Kaumāra cycle have the killing of the dānava/daitya Mahiṣa or Tāraka by the god Skanda as their centerpiece (Rāmāyaṇa and Mahābhārata), this text presents an unusual version of it: after the initial section culminating in Tāraka’s killing, there are two extended sections dealing with the elder brothers of Tāraka. These culminate in the great battle in which Skanda slays these demons, Siṃhamukha and Śūrapadma, along with their vast horde of Asura-s. So far, we have not seen any record of these demons outside of South India. Long before Kāśyapaśiva’s Tamil rendering, Śūrapadma appears in the South Indian tradition as represented by the earliest surviving Tamil texts, such as the Puṟanānūṟu (Puṟanānūṟu 23, a poem probably roughly contemporaneous with the Kuṣāṇa age in the North given that it describes the early Pāṇḍya king Neṭuñceḻiyaṉ), and a subsequent Tamil poetic anthology, the Paripāṭal. This suggests that the South Indian tradition had a deep history of certain unique elements of Kaumāra mythology.
As far as archaeology goes, we know that there was an active Kaumāra tradition in the Andhra country starting from the Andhra empire down to their smaller successor states, such as the Ikṣvāku-s and Viṣṇukuṇḍin-s among others, which had Nāgārjunakoṇḍa, as one of its foci. In the Tamil country, clear-cut archaeological evidence for strong Kaumāra traditions can be seen from the Pallava period onward. We believe this temporal period stretching from the Andhra empire down to the rise of the Pallava-s overlaps with the period during which the Puṟanānūṟu and the later Paripāṭal were composed in the Tamil country. The Paripāṭal displays a distinctive combination of the worship of Viṣṇu with his Pāñcarātrika vyūha-s and Kumāra — this pattern is seen in the Northwest, i.e., Panjab/Gandhara, and in Mathura during the Śaka-Kuṣāṇa age. This was mirrored in the South Indian Maturai (approximately the same longitude as its Northern namesake Mathura), the cultic locus of the Paripāṭal. Thus, one could argue that the core Kaumāra tradition in the Tamil country was a transmission of this Mathuran tradition.
Apart from the references to Śūrapadma, the themes in the Paripāṭal, while clearly linked to the ancient Kaumāra narratives, such as those seen in the Mahābhārata, show certain unique archaisms which have not survived in the Sanskrit tradition. For example, in Paripāṭal-5 by Kaḍuvan Iḷaveyinanār we encounter an incorporation of the Paurāṇika Marut mytheme into the tale of the birth of Kumāra. Here, after a prolonged dalliance with Rudra, mirroring the Sanskrit sources, Pārvatī becomes pregnant with Kumāra. Then Indra, who had acquired a boon from Rudra, cut the developing embryo into pieces with his Vajra (the number seven is implied by the repeated mention of seven in this verse) — the Paurāṇika Marut-motif. Then the pieces were placed in the three ritual fires by the seven ṛṣi-s (allegorically identified in the text with the seven brightest stars of Ursa Major), who realized that they would form the future commander of the deva-s. The pieces were purified by Agni and placed in the wombs of six of the wives (Kṛttikā-s=Pleiades), barring Arundhatī, of the seven ṛṣi-s (c.f. archaic Mahābhārata version). Thus, this South Indian tradition preserves a memory of the connection between the Vedic Marut-s, who are the sons of Rudra, and Skanda that was largely forgotten elsewhere (except for the reference to Kumāra as leader of the seven Marut troops in the oldest version of the cycle in the Mahābhārata).
Some of those mythic elements strongly persisted in the Tamil country and found their way into the Śivarahasya narrative, which the evidence presented below indicates is a later text:
1) In the Śivarahasya, the gaṇeśvara Nandin is prominent. Our textual analysis (to be presented later) has revealed that this is a strong marker of a text influenced by the Saiddhāntika Śaiva tradition. There are several other allusions throughout the text that point to its affiliation with the Saiddhāntika rather than any other Śaiva school of the mantramārga or the atimārga. This would also explain why the saiddhāntika Kāśyapaśiva chose to render it Tamil. Whereas in North India (outside of Nepal) and Vañga, the rise of the Siddhānta resulted in considerable erosion of the Kaumāra tradition from the 700s of CE, in the Drāviḍa country, the strong Kaumāra tradition was co-opted and incorporated within a Saiddhāntika framework. For example, this is seen in the works of the great polymath Aghoraśiva-deśika, who in addition to his numerous Saiddhāntika treatises also composed a work on the sthāpanā of Kaumāra shrines. This places the Śivarahasya in a distinct stratum from the Paripāṭal era (and even perhaps the Tirumurukārruppaṭai period) when Siddhānta was dominant in the Tamil country.
2) Its narration of the birth of Kumāra omits the coitus of Rudra and Pārvatī, which indicates a “sanitization” of the sexual elements of that narrative, which, for example, are an important aspect of its presentation in the Rāmāyaṇa, Mahābhārata, Śivapurāṇa and Kālidāsa’s Kumārasaṃbhava. This change in attitude again points to a relatively late date for Śivarahasya.
3) None of the early narrations of the Kaumāra cycle in the Iitihāsa-s or the Purāṇa-s attempt to model themselves after the Rāmāyaṇa. In fact, the Kumārākhyāna was seen as one of those old, independent mythic motifs of Hindu tradition that formed the basis of numerous retellings by different narrators, even as it was with the Rāmāyaṇa. Thus, the modeling of parts of the Śivarahasya, namely those concerning the war with Śurapadma and his clan (and possibly the arrangement in seven kāṇḍa-s), after the Rāmāyaṇa betrays a late “reconstruction” following the loss of continuity with the old Kaumāra Paurāṇika tradition.
4) The text acknowledges an already large Skandapurāṇa of the size of 100,000 verses. This implies that it comes from a period when the accretion of texts to form a mega-Skandapurāṇa was common knowledge.
While these elements point to a relatively late date for the Śivarahasya, we should point out that like all Paurāṇika corpora it does preserve several notable elements that have ancient roots going back to the Indo-European past. While the kāṇḍa-s 6 and 7 are dominated by the Śaiva material, its core is primarily a Kaumāra text intent on the aggrandizement of Skanda. Beyond the distinctive form of the Kaumāra cycle, there are multiple elements that indicate a southern locus for its immediate origin:
1) It presents a prominent role for the god Śāstṛ or Ārya as Hariharaputra. This transmogrified southern ectype of Revanta (commonly seen as Hariharaputra) was prominently worshiped at least since the time of the composition of the famous Tamil epic Śilpādhikāra.
2) It presents Vināyaka as elder to Skanda. While this is the position adopted by the text, its core Kaumāra narrative of the conquest of the demons still clearly indicates a tradition where Gaṇeśa was not yet born/in place.
3) The text describes two marriages of Skanda — one to Devasenā, seen across the Indosphere, and the other to Valli (related to the Dravidian term for tubers such as the tapioca and the sweet potato), that emerged in the Southern folk traditions and spread through the Southern zone of influence in the Indosphere.
4) The presence of the Kāverī-Agastya myth, which specifically points to the Drāviḍa country.
5) The staging ground of Kumāra in course of his campaign is called Śentīpura, which in the Tamil version of Kāśyapaśiva is identified as Tiruceñdūru, a major Kaumāra center, in the Drāviḍa country. It is already mentioned as a shrine of Skanda by the sea with a beautiful beach in Puṟanānūṟu 55.
6) The shrine of Aruṇācala in the Drāviḍa country is praised as an important Śaiva-kṣetra. Several other shrines in the Drāviḍa country as mentioned throughout the text, e.g., the Tyāgarāja and the Madhyārjuna shrines.
With this background, we shall briefly examine the contents of the Śivarahasya and a few of its notable points:
1) The Sambhava kāṇḍa
This section opens with a maṅgalācaraṇa seeking succor from Rudra, Umā, and their sons:
maṅgalaṃ diśatu me vināyako maṅgalaṃ diśatu me ṣaḍānanaḥ ।
maṅgalaṃ diśatu me maheśvarī maṅgalaṃ diśatu me maheśvaraḥ ॥
This is followed by short stotra-s with invocations of Gaṇeśa and Skanda by a set of 16 names each.
Gaṇeśa:
omkāra-nilayaṃ devaṃ gajavaktraṃ caturbhujam ।
picaṇḍilam ahaṃ vande sarvavighnopaśāntaye ॥
sumukhaś caikadantaś ca kapilo gajakaraṇakaḥ ।
lambodaraś ca vikaṭo vighnarājo vināyakaḥ ॥
dhūmaketur gaṇādhyakṣaḥ phālacandro gajānanaḥ ।
vakratuṇḍaḥ śūrpakarṇo herambaḥ skandapūrvajaḥ ॥
Skanda:
subrahmaṇyam praṇamyāhaṃ sarvajñaṃ sarvagaṃ sadā ॥
abhīpsitārtha siddhy arthaṃ pravakṣye nāma ṣoḍaśa ।
prathamo jñānaśaktyātmā dvitīyaḥ skanda eva ca ॥
agnibhūś ca tṛtīyaḥ syāt bāhuleyaś caturthakaḥ ।
gāṅgeyaḥ pañcamo vidyāt ṣaṣṭhaḥ śaravanodbhavaḥ ॥
saptamaḥ kārttikeyaḥ syāt kumāraḥ syād athāṣṭakaḥ ।
navamaḥ ṣaṇmukhaś caiva daśamaḥ kukkuṭa-dhvajaḥ ॥
ekādaśaḥ śaktidharo guho dvādaśa eva ca ।
trayodaśo brahmacārī ṣāṇmāturś caturdaśaḥ ॥
etat ṣoḍaśa nāmāni japet saṃyak sadādaram ॥
These stotra-s are popular in South India in Gaṇeśa- and Skanda-pūjā-s. However, it is notable that the names of Skanda do not mention Śūrapadma or Siṃhamukha; instead, they only utilize the pan-Indospheric Kaumāra material.
This is followed by the following topics:
-An account of the origin of the Purāṇa as narrated by the sūta, the student of Vyāsa, to the brāhmaṇa-s at Naimiśāraṇya and the nature of the Skandapurāṇa.
-An account of Kailāsa the abode of Rudra. This is followed the by usual Śaiva cycle of Pārvatī and her marriage that includes the below events.
-Kāma approaches Rudra who is in meditation.
-The incineration of Kāma by the fire from Rudra’s third eye.
-The lament of Rati.
-Rudra tests Pārvatī by appearing to her as an old man.
-Rudra reveals his true form to Pārvatī.
-Rudra sends the seven ṛṣi-s/stars of Ursa Major as his emissaries to seek the hand of Pārvatī in marriage.
-The construction of the marriage hall.
-The makeup and jewelry of Pārvatī.
-The gaṇeśvara Nandin leads the gods to the marriage of Rudra and Pārvatī.
-The names of the Rudra-s and an account of their vast hordes in the marriage procession. This is followed by an account of the retinue of Rudra. Below is a notable section of this text:
sahasrāṇāṃ sahasrāṇi ye rudrāḥ pṛthivīṣadaḥ ।
sahasra-yojane lakṣya-bhedinaḥ saśarāsanāḥ ॥
te rudrās tridaśa-śreṣṭhās trinetraṃ saṃsiṣevire ।
asmin mahati sindhau ye ye ‘ntarikṣe divi-sthitāḥ ॥
nīlagrīvās trinetrās te ‘saṃkhyātāś cāpurīśvaram ।
aghaḥ kṣamācarāś cānye sarve te nīlakandharāḥ ॥
girīśayo ‘stu kalyāṇaṃ siṣeviṣava āpire ।
vṛkṣeṣu piñjarā rudrāḥ nīlakaṇṭhā vilohitāḥ ॥
bhūtānāṃ cādhipatayo vikeśāś ca jaṭadharāḥ ।
sahasrair apy asaṃkhyātāḥ sāyudhāḥ prāpurīśvaram ॥
anneṣu ye vividhyanti janān pātreṣu bhuñjataḥ ।
ye pathāṃ pathi rakṣanti tīrthāni pracaranti ca ॥
ye rudrā dikṣu bhūyāṃsas tiṣṭhanti satataṃ ca te ।
gaurī-kalyāna-sevāyai giriśaṃ samupāśrayan ॥
Here the account of the hordes of Rudra is adapted from that of the great multitude of Rudra-s provided in the final anuvāka (11) of the Yajurvedic Śatarudrīya. Apart from these, a great retinue of goddesses and natural phenomena is said to accompany Rudra on his marriage procession. The bluish violet Viṣṇu is said to have joined them with his four forms, i.e., Pāñcarātrika vyūha-s, and was introduced by Nandin.
-Rudra enters the marriage hall and the marriage is concluded.
-Brahman and the other gods send Vāyu as their messenger to urge Rudra to produce a son with Pārvatī. However, Nandin turned him back asking him not the break the marital privacy of the deities.
-All the gods went to Kailāsa themselves and beseeched Rudra, whose half was occupied by Ambikā, to produce the promised son who would relieve them from the Asura-s.
-Rudra assumed a six-headed form blazing like a crore suns and enveloped the realms of the universe terrifying all beings. Then, from the third eye of each of his six heads, the upward seminal flow (ūrdhvaretas) exploded as six flashes of intense light that vaporized the directions (dudravaḥ sarvato diśaḥ). Terrified by this manifestation, all sought refuge in Rudra, praising him with hymns.
-He gathered back those six blazes and they came together as six pacified minute sparks. He then instructed Vāyu and Agni to take them to the arrow-reed forest on the banks of the Gaṅgā and vanished along with Ambikā. Thereafter, Vāyu and Agni, each getting tired after a while, with much effort bore the sparks to the Gaṅgā and deposited them there. The other gods eager to see what would happen also arrived there.
-The Marut-s with their joyful selves filled the quarters with a pleasant breeze (diśaḥ prasedur maruto vavuś ca sukhamātmanāṃ). Then, in the midst of a lotus in the arrow-reed forest, a six-headed, twelve-armed, two-footed divine boy took shape (note recurrence of the ancient motif of the birth of Agni in the lotus: Ṛgveda 6.16.13).
-Viṣṇu called upon the six Kṛttikā-s to nurse him. Instantaneously, becoming six separate kids he drank from their breasts.
-Even as the six flashes from Rudra were vaporizing the directions, Pārvatī too was startled and jumped away. As a consequence, the anklet fell from her feet and broke spilling the gems within it. Ambikā was reflected on those nine gems and appeared tenfold — herself and the 9 reflections. These became the Kālikā goddesses, who were fertilized by the rays emanating from Rudra and became pregnant.
-The droplets of the sweat of the startled goddess were also fertilized by Rudra. From them were born a 100,000 fierce gaṇa-s (who became the retinue of Skanda).
-Ambikā was displeased by seeing these goddesses pregnant and cursed them that they would have an unending and painful pregnancy. They went to Rudra seeking his aid and upon his counseling Ambikā released them from her curse and each gave birth to a mighty son of the complexion of their respective mothers.
-Goddesses and the corresponding sons were: Raktā (ruby) — Vīrabāhu; Taralā (pearl) — Vīrakesarin; Pauṣī (topaz) — Vīramahendra; Gomedā (garnet) — Vīramaheśvara; Vaiḍūryā (beryl) — Vīrapuraṃdara; Vajramaṇi (diamond) — Vīrarākṣasa; Marakatā (emerald) — Vīramārtāṇḍa; Pravālā (coral) — Vīrāntaka; Indranīlā (sapphire) — Vīradhīra. These nine Vīra-s became the companions of Skanda and were known as his brothers.
-Then Rudra told Ambikā that they have actually generated a mighty son and asked her to come along on his bull vehicle to see him.
-They set out with thousands upon thousands of Rudrakanyā-s, Mātṛ-s, gaṇa-s and the Marut-s.
-Then Umā hugged the six separate kids who became a single Ṣaṇmukha and fed him with her milk.
The narrative of the birth of Kumāra up to this point presents several interesting points:
1. There is a prominent role for Vāyu along with the usual Agni in the birth of Skanda. We believe that this is the survival of an ancient motif that is already seen in the Veda, where on rare occasions, apart from the usual Rudra, Vāyu is presented as the father of the Marut-s. This is not a mere slip, because in the Indo-Iranian world we see an overlap in the Rudra- and Vāyu class deities. On the Indian side that goes back to the worship of Rudra in the context of the rites of the Proto-Śaiva-s, the vrātya-s, and in the Eastern Iranian world in the character of the deity Vāyu Uparikairya, who is iconographically identical to the Hindu Rudra.
2. We see the subliminal presence of the Marut-s, even in this late reflex of the Kaumāra origin myth suggesting a long survival of this memory in the circles conversant with the Veda.
3. A variant of the “fertilizing sweat motif” attested in this myth presents the origin of the 100,000 Skanda-gaṇa-s from the sweat of Gaurī.
4. The Nava-vīra-s are a unique feature of the South India Kaumāra cycle. However, the number nine is also mentioned as the total of the Kaumāra-vīra-s even in one of the most ancient surviving variants of the Kaumāra cycle, which is seen in the Mahābhārata:
kākī ca halimā caiva rudrātha bṛhalī tathā ।
āryā palālā vai mitrā saptaitāḥ śiśumātaraḥ ॥
etāsāṃ vīrya-saṃpannaḥ śiśur nāmātidāruṇaḥ ।
skandaprasādajaḥ putro lohitākṣo bhayaṃkaraḥ ॥
eṣa vīrāṣṭakaḥ proktaḥ skandamātṛgaṇodbhavaḥ ।
chāga-vaktreṇa sahito navakaḥ parikīrtyate ॥
This account in the Mbh states that by the grace of Skanda, the 7 goddesses (Skandammātṛ-s), i.e., Kākī, etc., gave birth to the terrifying red-eyed deity Śiśu, who was called the eighth vīra. However, when Nejameṣa = Bhadraśākha with the head of a ram, generated by Agni is taken into account, Śiśu is said to be the ninth vīra. Then the question arises as to who were the remaining seven? From the preceding account in the Mbh we can infer that these were Viśākha and other Kumāraka-s who were emanated by Skanda when struck by Indra’s vajra. We believe that these vīra-s were ectypes of the Marut-s filtering down through later mythic overlays. It also appears likely that in the Śivarahasya, the most prominent of the nine vīra-s, Vīrabāhu, is essentially an ectype of Viśākha as the younger brother of Skanda. This connection to one of the oldest surviving versions of the Kaumāra cycle suggests that this aspect of the Southern tradition was a memory coming down from its ancient layer originally brought from the North.
-Thereafter Skanda displayed his childhood līlā-s, some of which bring out his roguish (dhūrta) aspects that are known from the oldest layers of the Kaumāra tradition. One notable set of such cosmic sports is expressed in beautiful Mandākrāntā verses:
chitvā bālaḥ prathita-mahimā svānugānāṃ karāgraiḥ ।
dikṣv aṣṭāsu svayam api dadhan dhārayan vyoma-gaṅgā
The boy putting forth his greatness, having taken in his hands the self-moving ones (planets) split the reins of propellant force (wind ropes) which bind them to the polar rays around which the celestial wheel revolves. Giving to himself the eight quarters, he then took on the Celestial Gaṅgā (Milky Way); binding the crocodiles (the constellation of Scorpius) he released them again into the Sun.
paścād ūrdhvam mahar api janas tat tapaḥ satya-lokaṃ
līlā-lolo nava ca kalayan vedhaso bhīmabhūtān
lokālokaṃ girim api mudā prāpa cikrīḍa bālaḥ ॥
Thereafter, he ascended upwards to the Mahar, Jana, Tapa and Satya realms [of the universe], and kept going on to the excellent dwellings of the Viśvedeva-s, Sādhya-s, the immortals and Indra. Making anew celestial fierce beings, joyfully attaining the boundaries of the universe (lokāloka mountain), the playfully sporting boy sported.
This displacement of the celestial bodies by Skanda already has a germ in the Mbh account: pracyutāḥ sahasā bhānti citrās tārāgaṇā iva ।. The celestial army of Indra attacked by Skanda is said to have been like the clusters of stars thrown off their orbit. Another notable point is a subtle astronomical allegory in the first verse. The constellation of Scorpius is called the nakra in Hindu tradition. Hence, we take the account of Skanda seizing the nakra-s and releasing them into the solar blaze as an allusion to the Kārttika month (when Skanda was born) when the Sun is opposite to the Kṛttikā-s in Scorpius.
-Alarmed by Skanda’s sports the gods fought him.
-Skanda defeated the gods and shows them his viśvarūpa (macranthropic) form.
The viśvarūpa of Skanda, while comparable to other viśvarūpa-s in the Itihāsa-purāṇa tradition, has some interesting cosmic verses:
tasmin tejasi te devā vaiśvarūpe jagat-trayam ।
koṭi-brahmāṇḍa-piṇḍānām mahā-vapuṣi romasu ॥
yūkāṇḍānīva koṭīni caikaikasmin sahasraśaḥ ।
tat-tad āvaraṇaiḥ sārdhaṃ tatratyair bhuvanair janaiḥ ॥
bhūtair bhavyair bhaviṣyadbhir brahma-viṣṇavādibhiḥ suraiḥ ।
jānu-pradeśa-mātre ‘sya dṛṣṭvā vismayam āgatāḥ ॥
In his radiance, were the gods of all forms and the triple world. In the hairs of his great body were the crores of spheres of galactic realms (brahmāṇḍa-piṇḍā-s). They were like the eggs of lice [of the hairs], in each one of the crores there were thousands of world-systems, each with its own set of orbits and local inhabitants of those worlds. Seeing the past, the present and the future, the Brahman, Viṣṇu and like of gods all coming only to the height of his knees, they reached bewilderment.
-Realizing who he was, the gods crowned Skanda as the commander of their army.
-The incident of the runaway sacrificial ram of Nārada. Skanda dispatched Vīrabāhu to capture it and bring it back. Skanda then took it as one of his vehicles. A homologous episode is found in the Ajopākhyāna of the Śivapurāṇa; there, instead of a ram, it is a goat. It may be noted that the Tamil allusion to this myth in the poem by the Saṅgam poet Maturai Nakkīranār (Tirumurukārruppaṭai 200-210) also records a caprine animal that might be interpreted as either a goat or a ram.
-Skanda chastised and imprisoned Brahman for his lack of gnosis of the praṇava.
-He then stationed himself at Kumāraparvata. At Rudra’s behest, Nandin tried to get him released, but Skanda warned Nandin that he might have him join Brahman and asked him to leave right away. Then Rudra and Umā finally made their son release Brahman and he taught Brahman the secrets of the praṇava, associating it with the Yajurvedic/Sāmavedic incantation subrahmaṇyom । (the mantra used in the Soma ritual to invite Indra for the libation).
-Kumāra initiated the campaign against the Asura-s by marching against the fortress of Tāraka.
-Skanda sent Vīrabāhu to launch an attack on Tāraka and Krauñca (an asura who had assumed a mountainous form due to a curse of Agastya).
-Being informed by his spies of the assault, Tāraka sallied forth to meet the gaṇa-s led by the nine vīra-s. Fierce encounters took place between Vīrakesarin, Vīrabāhu and Tāraka. Vīrabāhu repelled Tāraka’s māyā with the Vīrabhadrāstra. Tāraka drew Vīrabāhu into a feigned retreat and traps him in the mountainous cavern of Krauñca, putting him to sleep. He then routed the gaṇa-s showering missiles on them. Tāraka is described as having an elephantine head.
-Skanda entered the field to rally his gaṇa-s with Vāyu as his charioteer. Skanda routed the Asura-s. Taraka said that while he is a foe of Indra and Viṣṇu, he had no enmity with Rudra. But Skanda pointed to his sins and crimes against the deva-s and attacked him. After a fierce fight, Skanda cut off his trunk and tusks and pierced his head. He fell unconscious but on getting up he hurled the Pāśupata missile at Skanda. Skanda caught it with his hand and took it for himself. Tāraka then asked Krauñca to aid him with his māyā. After repulsing their magic, Skanda finally killed both Tāraka and Krauñca with his śakti.
-Skanda took his station at Devagiri and gifted the Pāśupata missile he had caught to Vīrabāhu.
-Rājanīti section where the court suggested to Śūrapadma, who was enraged by the death of Tāraka, that they should avoid a confrontation with Rudra’s party.
-Skanda goes on a mostly Śaiva (apart from Kañci and Veṅkaṭācala, where Skanda is said to have run away when Umā did not give him the mango) pilgrimage.
-Skanda releases the Pārāśara-s from a curse they had gotten from their father due to their cruelty towards fishes in their youth.
-Skanda goes to Śentīpura.
The Sambhava-kāṇḍa ends here. We believe that the core of this kāṇḍa derives from an older Kaumāra tradition that was of pan-Indospheric distribution. The structure of the narrative is such that Śūrapadma and Siṃhamukha, who are unique to the Southern tradition, only have a minor role in it. The break in the narrative between the killing of Tāraka and Krauñca on one side of the remaining Asura-s on the other side supports that part as being an accretion to this archaic core.
2) The Asura-kāṇda. From here on the narrative starts paralleling the Rāmāyaṇa in several ways.
-Asurendra, the lord of the Asura-s and his wife Maṅgalakeśī birthed a daughter named Surasā. She became a student of Uśanas Kāvya and acquired the moniker Māyā due to her proficiency in Māyā. When she reached adulthood, Kāvya lamented the condition of the Asura-s due to their crushing defeats at the hands of Indra and Viṣṇu. He asked her to have sons of great might through Kaśyapa and have them learn the praxis of ritual from him.
-Seduced by the beauty of Surasā, Kaśyapa abandoned his austerities and cohabitated with her. From their coitus when they assumed celestial forms Śūrapadma was born. When they engaged in coitus as lions, Siṃhamukha with a leonine head was born. From their coitus as elephants Tāraka was born with an elephantine head. When they mated in the form of goats they birthed the demoness Ajāmukhī. Taking many other animal forms they birthed several other fierce Asura-s. From their sweat, during each intercourse, numerous other demons arose.
-Kaśyapa then taught his sons the Śaiva lore.
-Then abandoning Kaśyapa, Surasā took her sons away and instructed them to perform a great sacrifice to Rudra to gain boons from him.
-Having pleased Rudra with his mighty ritual, where Śūrapadma offered himself as the oblation, he obtained the boons of the overlordship of a 1008 galactic realms (aṣṭottara-sahasrāṇām aṇḍānāṃ sarvabhaumatām ।), overlordship over the gods with enormous equipment and wealth, an adamantine body, invulnerability and the Pāśupata missile. Rudra granted him and his brothers such boons with the condition that no force except that originating from Rudra himself could destroy them.
-Armed with these boons and blessed by Kāvya, the Asura-s attacked Kubera and conquered his realm, taking him prisoner.
-They then conquered the realms of the gods and subjugated them.
-Viṣṇu fought Tāraka for long but realizing his invulnerability from Rudra’s boons retreated after congratulating him.
-Śūrapadma then had Tvaṣṭṛ build great forts for himself and his brothers.
At this point the narrative takes detour into Agastya cycle.
-Ajāmukhī forced Durvāsas to engage in congress with her. As a result, she birthed two sons Ilvala and Vātāpi. They went to Durvāsas and asked him to transfer his tapas power to them. He refused but offered them an alternative boon. They remained adamant and got into an altercation with him. He escaped from the place with his magic after cursing them that someday Agastya will slay them.
-They obtained their peculiar boon of resurrection from Brahman.
-They slew many brāhmaṇa-s through their well-known goat trick.
-They were finally slain by Agastya, who digested Vātāpi and hurled the Pāśupata that he had obtained from Rudra at Ilvala slaying him.
-Śūrapadma tried to abduct Indrāṇī. The Asura-s caused a drought; as a consequence they dried up Indra’s gardens.
-With the aid of Vināyaka, Indra caused the Kāverī river to flow out of Agastya’s pot and water the gardens.
-After the churning of the ocean, Rudra wanted to engage in coitus with Mohinī, the alluring female form of Viṣṇu. Viṣṇu indicated that it was impossible. Rudra pointed out that Viṣṇu was actually one of his śakti-s and thus a valid female. They copulated beneath a Sāla tree in Northern Jambudvīpa. The sweat from their passionate intercourse gave rise to the Gaṇḍakī river. In it, the mollusks known as vajradanta-s gave rise to the Sālagrāman-s used in the worship of Viṣṇu.
-From their conjugation, the god Śāstṛ was born.
-Rudra placed Śāstṛ in charge of protecting Indrāṇī from abduction.
-Śāstṛ in turn brought in Mahākāla as the bodyguard for Indrāṇī.
-Ajāmukhī and her friend Durmukhī tried to abduct Indrāṇī for Śūrapadma. However, Mahākāla swung into action and chopped off their hands.
-Ajāmukhī complained to Śūrapadma of her dismemberment.
-Śūrapadma forced Brahman to heal her and Durmukhī.
-Śūrapadma’s son Bhānukopa seeking revenge attacked the realm of Indra.
-He fought a fierce battle with Jayanta, the son of Indra, and eventually took him prisoner.
-Unable to find Indra or Indrāṇī, Bhānukopa destroyed the realm of Indra.
3) The Vīramāhendra-kāṇḍa.
-Before attacking the fortress of Vīramāhendra, Skanda sent Vīrabāhu as a messenger to Śūrapadma to ask him to peacefully surrender, release the deva-s he had incarcerated and return their realms that he had occupied.
-Vīrabāhu went to the Gandhamādana mountain and prepared to fly to Vīramāhendra. He mounted the massif assuming a gigantic and fierce form and laughed terrifyingly (aṭṭahāsa). He felt like extending his arms so that he could crush the Asura-s fortifications and cities with his hands like an oil-press crushing sesame seeds.
-As he pressed on the mountain the surviving warriors of Tāraka who were hiding in the caves came forth and were crushed by Vīrabāhu.
-Thinking of his guru, Skanda, Vīrabāhu flew into the sky causing the world to quake as he sped through the welkin.
-As he arrived at Lankā, which was the capital of Śūrapadma’s general Vyālimukha (who was visiting his lord), he was challenged by his deputy Vīrasiṃha and his troops. After a quick fight, Vīrabāhu chopped off Vīrasiṃha’s arms and head with his sword.
-Vīrabāhu leapt on Lankā and pushed it under the ocean.
-Vīrabāhu was then attacked by Vyālimukha’s son Ativīra and his troops who emerged out of the water. Vīrabāhu cut down the daitya troops and took on Ativīra who fought with a cleaver obtained from Brahman. However, Skanda’s Vīra cut his head off.
-Flying a thousand yojana-s he reached Vīramāhendra. As he was wondering which might be the best gate to make an entrance he arrived at the southern gate. There, he was challenged by Gajāsya, a monstrous demon with a thousand trunks and two thousand arms. After a closen fight, Vīrabāhu chopped off his trunks and hands and slew him with a kick.
-Realizing that this fight would bring more Asuras into the fray, he used his magic to become minute and entered via the eastern gate.
-There he saw the enslaved gods and the dwellings of mighty demons.
-Kumāra appeared in the dreams of Jayanta and the gods who were being subject to indignities by the Asura-s. Skanda told them that he had already killed Tāraka and Krauñca and that he had sent Vīrabāhu as his emissary who would wreak havoc among the Asuras. He assured them that thereafter he himself would attack the Dānava stronghold and slaughter them.
-Vīrabāhu met Jayanta and the other imprisoned gods and told them that their sufferings were due to their siding with Dakṣa during his ritual. He assuaged them by stating that the spear-wielding god would destroy the Asura-s and relieve them shortly.
-Vīrabāhu audaciously appeared before Śūrapadma and intimated him of the conditions for his surrender placed to him by Skanda.
-Śūrapadma rejected the terms and sent his general Śatamukha to capture Vīrabāhu. A fierce battle broke out between them during which Vīrabāhu demolished 20,000 Asura fortifications. In the end, he slew Śatamukha by thrashing him.
-Taking a giant form he crushed many other Asura warriors.
-Uprooting a mountain he smashed the Asura city.
-After another fierce fight, Vīrabāhu slew Śūrapadma’s ten-headed son Vajrabāhu by chopping off his heads with his sword.
-As Vīrabāhu was flying away, the polycephalous Vyālimukha challenged him. Another fierce fight ensued and Vīrabāhu finally killed him by chopping off his heads.
-Returning to Śentīpura, Vīrabāhu bowed thrice to Skanda.
-The Asura-s rebuilt their capital demolished by Vīrabāhu and prepared to fight Skanda.
-Skanda built the fort of Hemakūṭa as the base for this attack on Vīramāhendra.
-Śūrapadma’s spies informed him that Kumāra was gearing for an attack on Vīramāhendrapura.
-He called his son Bhānukopa and asked him to attack Skanda and his troops right away as they had approached the Asura city and fortified themselves at Hemakūṭa.
4) The Yuddha-kāṇḍa.
ṣaṇmukhaṃ dvādaśa-bhujaṃ triṣaṇṇayana-paṅkajam । kumāraṃ sukumārāṅgaṃ kekī-vāhanam āśraye ॥
-Bhānukopa donned his armor and mounting his car sallied forth with numerous Asura heroes.
-Skanda sent heavily armed, ruby-colored Vīrabāhu at the head of the bhūtagaṇa-s to intercept him. Wielding a bow like the pināka of Rudra he sallied forth. The two armies met amidst a shower of arrows from warriors on both sides.
-The fight was evenly poised but eventually, the gaṇa-s of Skanda started gaining an upper hand as the bhūta commanders slew their Asura counterparts.
-Seeing his forces retreating from the assault of the bhūtagaṇa-s of Skanda, Bhānukopa rallied them back. Bending his bow to a circle he released an unending stream of arrows on them striking down many and causing the gaṇa-s to fall back. Seeing this, the gaṇa Ugra got close to Bhānukopa and attacked him with a rod. Bhānukopa destroyed that rod with his arrows and struck down the gaṇa with the Brahma-spear. Then the gaṇa Daṇḍin attacked Bhānukopa with a mountain and struck his chariot and driver. Bhānukopa furious felled Daṇḍin with a shower of arrows. Next, he fought the gaṇa Pinākin who had rushed at him and struck him down with a shower of thousand arrows. Thereafter, several other gaṇa-s mounted a massed attack on Bhānukopa who routed them with his unending shower of arrows.
-He then fought the navavīra-s of Skanda. Vīrasiṃha was struck down by the Nārāyaṇa weapon of Bhānukopa.
-Then Vīramārtāṇḍa attacked the Asura. He got close to Bhānukopa and broke his bow; however, Bhānukopa struck him down with his cleaver.
-Vīrarākṣasa next attacked him from close quarters and both fell to the ground. Bhānukopa recovered consciousness and mounting a fresh car resumed the battle. Five of the remaining Vīra-s surrounded him and fought a great bow-battle for a while. Striking some of them down and brushing aside the rest he rallied the Asura troops and marched straight at Vīrabāhu.
-They fought a terrific battle in which both lost their bows but taking up new bows resumed the assault. Vīrabāhu smashed Bhānukopa’s helmet but he donned a new one and continued. Finally, tiring from the fight, Bhānukopa swooned.
-Immediately, the Asura forces surrounded their hero to protect him even as the gaṇa-s recovering from his assault surged forward. By then Bhānukopa stood up and continued the battle with Vīrabāhu.
-Despite all his attempts he could not dispel Vīrabāhu’s showers of arrows. Hence, he deployed the Mohāstra, a missile that caused the gaṇa-s to be paralyzed and fall to the ground. Even Vīrabāhu was paralyzed in his car. Taking advantage of this Bhānukopa shot numerous arrows at Skanda’s forces drenching them in blood.
-Skanda seeing his gaṇa-s in dire danger fired the Amoha missle from Hemakūṭa. This destroyed the Mohāstra and revived all the gaṇa-s.
-Seeing the nullification of all his tactics, Bhānukopa retreated for the day deciding to resume the battle later.
-Realizing the seriousness of the situation, Śūrapadma himself decided to join the battle. He sallied forth along with Atiśūra, the son of Siṃhamukha and Asurendra, the son of Tāraka.
-Indra saw them and informed Skanda of the impending attack. Having saluted Indra, Skanda decided to lead his forces personally in the battle. Indra asked Vāyu to be Skanda’s charioteer.
-A great battle broke out between the gaṇa Ugra and Atiśūra. Atiśūra discharged numerous divine missiles at his adversary but they failed because Skanda provided immunity against those missiles to his follower. Finally, Atiśūra discharged the Pāśupata, but it did not harm its own party and returned to Mahādeva. Frustrated thus, he leapt out of his car and struck Ugra with a rod. However, Ugra survived that blow and snatched the weapon from the demon and pummeled him to death with it.
-Tāraka’s son Asurendra rushed to shore up the ranks after his cousin’s death. He faced the gaṇa-s Kanaka, Unmatta, Siṃha, Daṇḍaka, and Vijaya in fierce encounters, defeating each of them.
-Then Vīrabāhu flew in on his car and started showering arrows on the demon. After a prolonged encounter, Vīrabāhu smashed his enemy’s chariot. The two engaged in a great battle flying in the skies, with Asurendra wielding a rod and Vīrabāhu, a sword. The latter finally beheaded Asurendra.
-Seeing his nephews slain, Śūrapadma launched a fierce attack on the bhūtagaṇa-s. He was attached by the nine Skanda-vīra-s. Fierce fights took place between him and Vīrarākṣasa, Vīramahendra, Vīradhīra, and Vīramāheśvara in that order and he overcame all of them. Vīrabāhu then entered the fray to shore up his half-brothers. He deployed the Aindra and Vāruṇa, Brahma, Vaiṣṇava missiles on the demon and Śūrapadma countered those with his Māyāsurāstra. Then both deployed the Pāśupata missile but it returned back to the respective users. Then they both deployed śakti-s that neutralized each other. Finally, Śūrapadma struck Vīrabāhu on his chest with a daṇḍa.
-Seeing his champion’s discomfiture, Kārttikeya attacked Śūrapadma and a great fight ensued. Bending his bow Skanda fired a profusion of arrows at the Asura lord. They cut each other’s shafts in mid-flight. Śūrapadma hurled his śakti at Skanda, who cut it off with 14 arrows. Then Śūrapadma uttering a loud roar hurled a trident at Skanda who cut it off with four and the seven arrows thereafter. Then with another missile, Skanda smashed the helmet of Śūrapadma and destroyed his armor with further shafts.
-Thereafter, the six-headed son of Ambikā hurled a cakra and cut down the Asura hordes that accompanied Śūrapadma and the piśāca-s feasted on their corpses.
-Śūrapadma replenished his gear and returned to the fight. He tried deploying many divine missiles but they were all nullified by Kumāra with his śakti missile. Thus, on the brink of defeat, Śūrapadma vanished and flew back to his fortified city thinking he would return later to fight Kumāra.
-Skanda ordered his bhūtagaṇa-s to storm the fortifications of the Asura stronghold.
-The gaṇa-s launched a massive assault on the fortifications of Vīramāhendrapura. In course of this assault, they killed the demon Atighora.
-Bhānukopa sallied forth again to fight Vīrabāhu, this time armed with his grandmother Māyā’s Sammohana missile.
-After prolonged use of various missiles both tried to get better of the other with the Pāśupata. Both Pāśupata missiles neutralized each other.
-At this point, Bhānukopa deployed his grandmother’s Sammohana missile, which not only made Vīrabāhu but also the rest of the Skandapārṣada-s unconscious and hurled them into the ocean.
-Bhānukopa returned to this father to tell him of his great victory and promised him that he would head out the next day to slay Skanda.
-On receiving intelligence that his army had been drowned by Bhānukopa, the six-headed son of Rudra launched his śakti, which sped to the ocean, and, destroying the Sammohana weapon, led his forces out back to the Asura capital.
-The Skandapārṣada-s now launched a ferocious assault on the defenses of the city. The Asura Vyāghrāsya advanced to defend the fortifications. He was slain after a trident fight by the Skandagaṇa named Siṃha.
-As the Skandapārṣada-s demolished the defenses of Vīramāhendrapura, Vīrabāhu hurled the Āgneya and Vāyava missiles and set the city on fire.
-Śūrapadma’s men informed him of the reversal and the impending destruction of his city by the fires. He brought in the mahāpralaya clouds to douse the fires. As they were putting off the fires, Vīrabāhu struck back with the Vāḍava missile to vaporize the clouds. Seeing the havoc in his city, Śūrapadma wanted to head out himself to fight Skanda and his forces.
-Just then Śūrapadma’s son Hiraṇya came to him and told him that it might be prudent to surrender to the six-headed son of Rudra, reminding him that there was no one who could counter him and his dreadful gaṇa-s.
-Śūrapadma warned his son that he would kill him if continued speaking thus. Hiraṇya calmed him down and decided to enter the battlefield himself.
-Hirāṇya baffled the Skandapārṣada-s for a while with his display of māyā. Finally, his māyā was overcome by the Cetana missile shot by Vīrabāhu. Hirāṇya fought Vīrabāhu for a while but the latter cut his bow and smashed his car with his astra-s. Hirāṇya knew that he would be killed shortly. He also realized that nobody would be left to perform the funeral rites for his family once Skanda’s troops stormed the city. Hence, he took the form of a fish and vanished into the ocean.
-Śūrapadma’s son Agnimukha next took the command of the Asura forces and led them against the Skandagaṇa-s with a vast force of Asura-s. In the battle that followed, Vīrapuraṃdara slew the Asura-s Somakaṇṭaka and Megha with his arrows. After a pitched battle, Agnimukha slew seven of the Vīra-s, barring Vīrapuraṃdara and Vīrabāhu with the Pāśupata. Agnimukha the advanced on Vīrabāhu. After an even astra-fight, Agnimukha got Bhadrakālī to fight on his behalf. Vīrabāhu defeated her and she left saying no one could stop the Skandapārṣada-s. Then Agnimukha returned to the fight showering thousands of arrows. Vīrabāhu hurled the Vīrabhadrāstra which burst Agnimukha’s head.
-As Vīrapuraṃdara and Vīrabāhu were lamenting the fall of their brothers, the latter flared up and shot an arrow into Yamaloka engraved with the message that, he the younger brother of Kārttikeya, wanted Yama to release his seven half-brothers. Duly Yama released their ātman-s and they reanimated their bodies.
-A great battle next took place between three thousand Asura-s born of Śūrapadma and a thousand of the chief Skandapārṣada-s. The battle was evenly poised for a while, but the Asura-s started gaining the upper hand as their severed limbs were restored by a special gift they had obtained from Brahma. Hence, the gaṇa-s turned to Skanda, who appeared in their midst, and gave one of their leaders, Vijaya, the Bhairava missile. As he deployed it, one of three thousand Asura-s named Matta deployed the Māyāstra. However, the Bhairavāstra destroyed it and cut off the heads of all three thousand Asura-s.
-Next, Dharmakopa, another son of Śūrapadma advanced with his troops against the Vīra-s. His troops were destroyed by the Vīras-s even as Dharmakopa closed in on Vīrabāhu. After an exchange of missiles, Dharmakopa struck down Vīrabāhu’s charioteer. Then they engaged in a closen fight where Dharmakopa tried to strike Vīrabāhu with a rod and then a thunderbolt. Evading both, Vīrabāhu killed him with a kick.
-Bhānukopa received the news that while he thought he had won, the forces of Skanda had returned and wreaked enormous destruction on the Asura-s. He told his father perhaps it was a futile attempt, and they should surrender to Skanda. His father refused; hence, Bhānukopa set forth again to fight the Skandapārṣada-s. After a great fight Bhānukopa overthrew the bhūtagaṇa-s attacking him and rushed forward with a shower of arrows. Vīrabāhu bent his bow and destroyed the missiles hurled by Bhānukopa. Thereupon, Vīrabāhu hurled a śakti and struck Bhānukopa. However, recovering from the blow he resumed the fight. For a while, neither could get better of the other in their exchange of astra-s. Bhānukopa then baffled Vīrabāhu with his māyā powers. Vīrabāhu destroyed the māyā display with his own magic. As their fight raged on, Bhānukopa smashed Vīrabāhu’s chariot. But Vīrabāhu retaliated by breaking the Asura’s chariot and cutting his bow with his shafts. Finally, they closed in for a sword fight. Vīrabāhu chopped off the right hand of Bhānukopa but he took the sword in his left and continued. Vīrabāhu cutoff that hand too. Handless, he tried to deploy the Māyā missile but Vīrabāhu swept his head away with a blow from his sword.
-Śūrapadma on hearing of the death of his son fainted. He lamented much when he received his son’s mutilated body parts. He called his brother Siṃhamukha to come from his city of Āsurapura to join the fight. Shocked by the news of the defeat of the Asura-s, and the death of his sons and nephews, Siṃhamukha flew over from his city to aid his brother.
-Donning his armor, he set forth for battle with his great troop of Asura-s. Skanda called Vāyu to get his chariot ready. Vīrabāhu with his half-brothers wanted to lead the 100,000 Skandapārṣada-s to the encounter first. Skanda let him do so and a fierce fight ensued. Showering balls, arrows, axes, and plows on the gaṇa, the great lord of the Asura-s advanced. His demons fought several gaṇa-s in melee as they exploded each other’s weapon discharges to smithereens. Vīramārtāṇḍa used the Jñānāstra to counter the māyā being deployed by the demons. Siṃhamukha cut through the gaṇa ranks like a great mountain on the move. He put to flight the 100,000 Skandapārṣada-s with showers of missiles. He then proceeded to crush them as though one would crush mosquitoes. Seeing this, Vīrabāhu counter-attached showering thousands of arrows with his bow. The hundred sons of Siṃhamukha surrounded him and returned the showers of arrows. With his missiles, Vīrabāhu smashed their chariots and broke their bows. Then they rushed at him with their swords but Vīrabāhu slew all hundred with an arrow and his sword.
-Infuriated and saddened by this, Siṃhamukha rushed at Vīrabāhu. They ground each other’s missiles to dust and had a prolonged astra fight. Vīrabāhu slew his charioteer. Siṃhamukha hurled a gadā at him, but it burst on striking his adamantine form born of Rudra. Vīrabāhu then demolished the Asura’s car. He resumed the battle taking new cars over and over again and deploying thousands of bows. Vīrabāhu kept breaking them repeatedly. Seeing himself unable to get better of Skanda’s warrior, Siṃhamukha deployed the Māyāpāśa. The great lasso weapon immediately bound Vīrabāhu and the gaṇa-s who were on the field and hurled them atop the Udaya mountain in a state of paralysis. Sensing victory, Siṃhamukha roared loudly and thought that Skanda too had fled. However, his spies informed him that Skanda was still very much there with the remaining gaṇa-s at Hemakūṭa. Mātariśvan informed Skanda of the events and Siṃhamukha’s advance towards their fort.
-Mounting his car driven by Vāyu, Skanda led his forces into battle. In the battle that ensued the Asura-s began to retreat. At that point, Siṃhamukha assumed a gigantic form with two thousand arms. Grabbing all the gaṇa-s of Skanda he swallowed them. Śūrapadma’s spies informed him of his brother’s deeds and he ascended an observation turret to see the great form of his brother.
-Skanda then strung his bow and twanged the string causing the whole universe to resound and the Asura vehicles fell to the ground from the sound emanating from the bow twangs of the son of Umā. Siṃhamukha rushed at him to do battle. Kumāra fired a mighty missile that split open the firm belly of the demon and from the fissures through which blood was pouring out, some of the gaṇa-s who had been swallowed emerged forth. Stanching the slits in his belly with his many arms, the demon hurled a dreadful rod at Skanda. He split it up with four missiles, which then proceed the strike the demon on his forehead. Losing his strength, he lifted his hands off his belly and the remaining gaṇa-s too escaped.
-Skanda then fired a missile that proceeded to the Udaya mountain and destroyed the Māyāpāśa. Then turning into an airplane the missile brought Vīra-s and gaṇa-s back to the field and returned to the six-headed god’s quiver.
-To shore up their leader, the Asura-s who had retreated returned to attack Skanda upon hearing his terrifying roar. As they surrounded the god, he hurled thousands of projectiles and also attacked the demons with rods, swords, spears and axes. Vāyu maneuvered the car with great speed even as Skanda dispatched his missiles that lit up the entire universe like the Vaḍava fire. The great god pierced the many galactic realms with his weapons and rent asunder the limbs of the demons and shattered their vehicles. The cluster missiles shot by Skanda branched repeatedly giving rise to crores upon crores of arrows and penetrated all the galactic realms slaying numerous Dānava-s wherever they were. Seeing these weapons being fired by their commander, all the deva-s sang the praises of the son of Rudra.
-The whole field was filled with corpses of the demons. Wielding a thousand bows Siṃhamukha again attacked the commander of the deva-s.
-He then struck Vāyu on the chest with numerous arrows and the god fainted. But Skanda controlling his own car destroyed the Asura’s chariot with a hundred arrows. Then with a thousand arrows, the son of Mahādeva, cut down all the bows of the demon at once. Siṃhamukha hurled a trident at the god, who cut it down with fourteen arrows. Then he rushed at the god with a rod who powdered it with seven shafts. The demon then sent the death-dealing pāśa but Skanda cut it up with a thousand projectiles. Then he cut up the two thousand arms of the Asura. The great demon (termed māhāmada here) uttered a “mahākilikilārāva” and rushed at Skanda. The god sliced off his thousand heads with an equal number of arrows.
-Siṃhamukha regenerated his severed limbs and reentered the fray. Skanda let this happen eight times as part of his battle sport. Then he cut all his hands except for a pair and heads except for one. The demon roared that he would slay the god with just those and uprooted a mountain and rushed at his adversary. Kārttikeya rent asunder that mountain with a single arrow. The Asura now attacked him with a terrifying daṇḍa. Thereupon, Guha hurled his vajra which blazed up like several crores of suns. It destroyed that daṇḍa and striking the Asura on his chest annihilated him.
-Having bathed in the Celestial Gaṅgā Kārttikeya returned to the Hemakūṭa fort with his Vīra-s and gaṇa-s.
-Hearing of the death of his brother from his agents, Śūrapadma fell down from the observation turret he had mounted to witness the battle. Regaining his composure, he decided to himself lead the Asura-s in the war against Guha. He ordered all the surviving Asura-s from the numerous galactic realms that he controlled to come over and join him for the battle.
-Mounting his special battle car armed with all the divine missiles, with a great force of Dānava-s he headed out of his fortified city with their roars filling the whole universe.
-The gods called on their commander Skanda to enter the field against the evil demon. Worshiped by all the gods and his 100,009 Vīra-s, Skanda took up all his weapons and set forth for battle on the car driven by the god Vāyu:
ādāya paraśuṃ vajraṃ śūlaṃ śaktiṃ vibhīṣaṇāṃ ।
khaḍgaṃ kheṭaṃ bṛhac cakraṃ daṇḍaṃ musalam eva ca ॥
dhanuś-śarān mahāghorāṃs tomarāṇi varaṇy api ।
vinirgatya rathaṃ ramyaṃ vāyunā nītam agnibhūḥ ॥
āruroha surais sarvaiḥ pūjitaḥ puṣpa-vṛṣṭibhiḥ ।
nava-saṃkhyādhikair lakṣair vīrair api mahābalaiḥ ॥
-The other gods asked Viṣṇu if Kārttikeya might meet with success in the impending encounter. Viṣṇu assuaged their doubts saying that their troubles would end soon as Skanda would definitely triumph.
-As the battle was joined, Śūrapadma’s demons charged with a great shower of weapons. Skanda twanging his dreadful bow, which resounded like the flood at the end of the Kalpa, entered the field. He launched into an orgy of slaughter with his missiles reaching the limits of the universe. Wherever the demons went, his bolts cruised after them. Breaking through the walls of the galactic realms they entered whichever region the Asura-s flew to and slew them. Floods of blood and mountains of corpses of the Asura warriors started to pile all around.
-Furious, Śūrapadma joined the fray and laid low most of the Skandapārṣada-s with his terrifying missiles. Vīrabāhu rushed forth to engage him and cut his bow with his cleaver. But Śūrapadma punched with his fist and he fainted. Deciding not to kill him for he was just a messenger of Skanda, Śūrapadma seized by his feet and hurled him into the sky.
-He then charged at Skanda and engaged him in a fierce bow battle with the exchange of innumerable arrows. Finally, Śūrapadma struck down the flag of Skanda with a shower of arrows and blew his conch as a mark of victory. Skanda however quickly retaliated cutting and hurling Śūrapadma’s banner into the sea with seven shafts. Then, the thousand-headed gaṇa Bhānukampa blew on a thousand conchs and Viṣṇu too blew on his. Agni turned into the cock banner and went to adorn the car of Guha. The cock crowed loudly. All this created a terrifying din.
-Angered, Śūrapadma now turned to the gods were and attacked them with his weapons. Skanda followed him with Vāyu driving his car at top speed. Skanda protected the gods and attacked the demon with a shower of weapons. As the chariots of the two of them wheeled around in battle the whole universe to vibrated violently and whole mountains were reduced to atoms.
-Śūrapadma attacked his enemy with halāyudha-s, bhindipāla-s, kuliśa-s, tomara-s and paraśu-s. Then Skanda destroyed his vehicle completely with fourteen missiles.
-The demon then mounted the Indralokaratha (the space-station he had captured from the gods) and started tunneling into the various galactic realms he had conquered. However, he found the tunnels into them blocked by the arrows of Mahāsena and found that many of his demons were trapped in each one of them. He broke down the obstructions with his weapons and let out his demons. Those Asura-s came out and mounted a furious attack on Skanda. With his cakra, paraśu, daṇda and musala the son of Umā slaughtered them, and chasing them to each of the realms, he burnt them down with his weapons to a fine ash.
-The Asura then deployed the deadly Sarvasaṃhāraka-cakra, but Mahāsena sportingly captured it for himself. Next, he tried māyā tactics but Skanda easily overcame those with the Jñānāstra.
-Thus, defeated he finally went to his mother and asked if she might have a means of victory. She told him that she did not see a way out against the son of Rudra, but the only thing he could do is to get the Sudhāmandara mountain to revive the dead demons.
-He mounted a lion-vehicle and sent the Indralokaratha to bring the said mountain. The craft brought the Sudhāmandara to the field and the wind blowing from it started reanimating the dead demons. Seeing this, Skanda deployed the Pāśupata missile that started branching and emitting numerous Rudra-s, the Marut-s, Agni-s and vajra-s. These destroyed all the reanimated demons and also the vivifying mountain. Thereupon, the Asura sent the Indralokaratha to scoop up Vīrabahu and the remaining 100,008 troops of Skanda and stupefy them. Guha retaliated with a series of missiles that grounded the craft and brought it back to him. The gaṇa-s got out of it safely and the space-craft became the property of Skanda.
-Then the Asura injured Vāyu with his shafts and briefly Skanda’s chariot was out of control. But regaining control he cut the bow of the demon. Śūrapadma thereafter attacked him with the terrifying missile known as the Sarvasaṃhāraka-śūla. Skanda shot numerous arrows to destroy his lion-vehicle and then hurled the Ghora-kuliśa that neutralized the said śūla and brought it back to Skanda.
-The Asura assumed the form of a gigantic fierce bird and started pecking at Skanda’s car and blowing his gaṇa-s away with the blasts from his wings.
-Skanda then looked at Indra and the latter took the form of a peacock. Mounting the peacock Skanda and Indra fought the demon. Indra pecked him and clawed him in the form of the peacock, even as Skanda pierced him with many arrows. Śūrapadma dived in his bird form and broke Skanda’s bow, but Guha drew out his sword and hacked the bird-formed demon to pieces.
-The demon then took the form of the earth. Skanda taking a new bow drowned that earth-formed demon with the oceanic missile. The demon then took the form of the sea. With a hundred fiery missiles Skanda dried up that form. He successively took various forms, including the gods, but Skanda destroyed all of those with his missiles.
-Finally, to reveal that those forms of the demon were merely fictitious forms, Skanda assumed his macranthropic form with all existence and all the gods comprising his body — the planets and stars his feet; Varuṇa and Nirṛti his ankle joints; Indra and Jayanta his thighs; Yama and time his hips; the nāga-s and the ambrosia his genitals; the gods his ribs; Viṣṇu and Brahman his arms; the goddesses his fingers; Vāyu his nose; Rudra his head; the numerous galactic realms his hair follicles; the omkāra his forehead; the Veda his mouth; the Śaivāgama-s his tongue; the seven crores of mantra-s his lips; all knowledge his yajñopavīta.
-Seeing this macranthropic form of Kumāra, Śūrapadma had the doubt for the first time in his existence if after all this god might be undefeatable — many great Asura-s had fought him and many great missiles, which were previously infallible had been, used yet he triumphed over all of those. However, the demon brushed aside these feelings and assumed a gigantic form with numerous arms, heads and feet, and casting great darkness that enveloped the whole universe he rushed forward to eat Skanda and the other deva-s.
-Skanda immediately hurled his mighty śakti. Blazing like crores of suns it destroyed the overpowering darkness of the Asura and cruised after him. He dived into the sea of the fundament even as the śakti chased him there. The Dānava immediately became the “world-tree” — a giant mango tree stretching across the limits of the universe. Blazing like the trident of Rudra, the śakti split that tree in half. Śūrapadma assumed his own form and drawing his sword rushed at Skanda. The śakti struck him immediately and slew him.
-His remains were transformed into a peacock and a cock that respectively replaced Indra and Agni as the mount and the banner of Skanda, even as the two gods returned to their prior state. Thereafter, the gods praised Mahāsena for his glorious acts.
-Śūrapadma’s primary wife expired on hearing the news of his death and his son Hiraṇya who had hidden as a fish in the ocean performed the last rites of the dead demons with help of Uśanas Kāvya.
The final battle between Skanda and Śūrapadma has some mythic motifs of interest. First, the many transmogrifications of the demon in course of his battle with the god are reminiscent of the shamanic transformations. This motif is encountered in several distant traditions — most vividly in the folk Mongolian account of Chingiz Khan’s final fight with the Tangut emperor composed by his descendant Sagang Sechen (Also note the motif of the nine heroes of the Khan in that account). There, Khasar, the brother of the Khan (one of the nine heroes) plays a role similar to Vīrabāhu in destroying a witch who was guarding the Tangut capital and preventing the entry of Subetei. Khasar killed her with his arrows allowing the Mongols to storm it. Then the Khan and the Tangut lord fought a magical battle with both of them taking on many forms like a snake, Garuḍa, tiger, lion, and the like. Finally, the Khan took the form of Khormusta Khan Tengri (the great Mongol god) himself and put an end to the shape-shifting of the Tangut. Though he struck the Tangut with many arrows and swords he still could not kill him. ṭhe Tangut let slip the secret of his death in the form of a magical wootz steel sword hidden in his boot that the Khan seized and slew him. Thus, we suspect that the shape-shifting of Śūrapadma in the final battle is from an ancient shamanic layer of the Kaumāra tradition that is attested in the Saṅgam Tamil tradition (the muruka-veri, e.g., Tirumurukāṟṟuppaṭai 200-210).
Second, the final dive of Śūrapadma into the ocean and/or his transformation into a mango tree is a motif that has deep roots in Tamil Kaumāra tradition. It is mentioned in multiple Saṅgam texts such as: 1) Pattiṟṟupattu: here the Cēra king Neṭuñcēralātaṉ, who built a fleet to fight a naval battle with the Romans, is said to have destroyed his enemies even as Skanda cut down the tree of the Asura-s. 2) The Paripāṭal 18.1-4 mentions how Skanda pursued the demon into the deep ocean. Paripāṭal 5.1-4 mentions both his pursuit into the ocean and his destruction by the Skandaśakti in the form of a mango tree. 3) Tirumurukāṟṟupaṭai 45-46 mentions his pursuit into the ocean. Tirumurukāṟṟupaṭai 58-61 mentions his destruction in the form of the mango tree and also his centaur transmogrification, which is absent in the Purāṇa. We take these mythemes to be reflexes of the famous precessional myth — i.e., the shattering of the old world axis that is widespread across Hindu tradition and beyond. A variant of this myth is associated with the submerging of the old equinoctial constellation beneath the equator (the world ocean, e.g., the Varāha myth): thus, both of these are combined here in the final fight of Śūrapadma. Notably, both in the Pattiṟṟupattu and the Paripāṭal, Skanda is described as riding an elephant rather than a peacock in the final battle against Śūrapadma. This is probably an archaism as this elephant vehicle is mentioned in the ancient Kaumāra material found in the medical Kāśyapa-saṃhitā (129) as having been generated from Airāvata by Indra for Skanda.
5. The Deva Kāṇḍa.
-The rule of the gods is restored.
-Skanda is engaged to Indra’s daughter Devasenā.
-The marriage of Skanda and Devasenā.
-Skanda seeks the daughter of Viṣṇu born as Vallī in a sweet potato excavation among the pulinda hunter-gatherers.
-He appears as an old man to her. Gaṇeśa frightens her as an elephant, and she comes into the hands of Skanda in the form of the old man seeking help from the elephant.
-Vallī recognizes Skanda and starts a clandestine affair with him.
-When she elopes with Guha, the hunters, including Vallī’s brothers and father chase and attack Skanda, who strikes them down with his arrows.
-Kumāra revives the dead huntsmen and marries Vallī and returns to his abode with his two wives.
-The praise of emperor Mucukunda.
-Mucukunda installs the image of Rudra known as Tyāgarāja.
6. The Dakṣa-kāṇḍa
This section is mostly Śaiva material relating to the cycle of Dakṣa. One notable point is that here Rudra generates Vīrabhadra and Umā generates Bhadrakālī to destroy Dakṣa’s ritual. Notably, this parallels the Ur-Skandapurāṇa wherein Umā generates Bhadrakālī by rubbing her nose. There Rudra generates Haribhadra. This might indicate a connection to that ancient Skandapurāṇa version of the Dakṣa cycle. However, we may note that a similar situation is also seen in the Brahmapuraṇa ((39.51), where Rudra created the lion-formed Vīrabhadra, whereas Umā creates Bhadrakālī to accompany him. Further, interestingly, here as Vīrabhadra destroyed the male partisans of Dakṣa, Bhadrakālī destroyed the females of Dakṣa’s clan. This symmetry appears to be an ancient motif — in the Greek world we have Apollo kill the male Niobids while Artemis killed the female ones. Apart from the Dakṣa cycle, this kāṇḍa contains:
-The ṛṣi-s’ wives at Dārukavana run after Rudra. They attack him with various beings and he destroys them.
-The killing of Gajāsura by Rudra.
-Churning of the ocean and Rudra consumes the Hālāhala.
-The appearance of eleven crore Rudra-s at the Madhyārjuna shrine (Tiruvidaimarudur in the Drāviḍa country).
-The beheading of Brahman by Bhairava.
-Dharma becomes Rudra’s bull.
-Rudra destroys the universe and smears the residue as his ash.
-Rudra slays Jalandhara.
-The birth of Gaṇeśa.
-Jyotirliṅga-s and Aruṇācala.
-Paurāṇika geography.
The only Kaumāra-related material in this kāṇḍa are: 1. The praise of Skanda’s peacock and chicken. 2. The “backstories” of the birth of Śurapadma, his mother and his clan. 3. The Skandaṣaṣṭhī festival.
The rest of this kāṇḍa is again largely Śaiva material pertaining to Śivadharma for lay devotees. Indeed, chapters in this section seem to self-identify as a Śiva-purāṇa. It begins with an account of the Rudra-gaṇa in Kailāsa. It contains several accounts of humble animals (including men like cora-s) attaining higher births from acts of Śaiva piety. Similarly, sinners who defile/steal from Śaiva shrines or take even things like lemons or bananas from them attain hell. The observance of key festivals of Rudra and the Bhairava-Vīrabhadra festival are laid out. Further, it gives the 1008 names of Rudra, the practice of Aṣṭāṅga-yoga, the theological principles of Siddhānta and the iconography of the twenty five images of Rudra that are displayed in Śaiva shrines.
It also contains accounts of: 1. Rudra destroying the Tripura-s. 2. Rudra slaying Andhaka. 3. The rise of the most terrible demon Bhaṇḍāsura. Rudra performs a fierce ritual, offering Brahman, Viṣṇu, Indra and other gods as samidh-s in a fire altar where he himself was the fire. From it arose the youthful goddess Tripurā who slew Bhaṇḍāsura. This minimal account lacks the details seen in the Lalitopākhyāna, where this myth takes the center-stage. 4. The killing of Mahiṣa by Durgā through the grace of Rudra as Kedāreśvara. 5. The killing of Raktabīja by Kālī and her dance with Rudra.
## Some notes on the runiform “Altaic” inscriptions and the early Turk Khaghanates: Orkhon and beyond
The early Turkic inscriptions from Mongolia and their discovery
On February 27th, 731 CE (17th day of the Year of the Sheep), Kül Tegin, the great hero of the second Gök Türk (Blue Turks) empire, passed away in his 47th year (literally flew away to the realm of Tengri). He was greatly mourned by his clansman — his elder brother, Bilge Khaghan, the ruler of the Turks, had his elegy inscribed on the now-famous Kül Tegin funerary stele. On the northern face of the stele, we read:
“My younger brother Kül Tegin passed away. I mourned. My bright eyes seemed unable to see and my sharp knowledge seemed unable to know. I mourned. Tengri arranges the lifespan. Humans are born to die. I mourned thus: when tears were running out of my eyes, I restrained them; when lamentation was coming out of my heart, I held it back. I thought [of him] deeply. [I feared] the eyes and eyelashes of the two shad-s (title of generals; derived via contraction from Old Iranic kshāyathiya= warrior via Sogdian xshedh= chief), of my brothers, my sons, my officials and my people were to be ruined [because of tears]. I mourned.” (translation based on Talat Tekin, Hao Chen and Denison Ross]
On November 1 of 731 CE, Bilge Khaghan held a grand funerary ritual for his brother. It is said to have been attended by several dignitaries from East and West. Again, the inscription on the northern face of the monument states:
“General Udar, representing the people of Khitan and Tatabi, came to attend the funeral feast and mourned. From the Chinese Khaghan came the secretary Likeŋ. He brought ten thousand pieces of silk, gold, silver and various things. From the Tibetan Khaghan came Bölün. From Sogdiana, Bercheker (i.e., Persia) and Bukhara in the sunset west came General Enik and Oghul Tarkan. From the On Ok, from my son [-in-law] the Türgish Khaghan, came Makarach and Oghuz Bilge, who were officials holding seals. From the Kirghiz Khaghan came Tardush Inanchu Chor. The shrine-builders, fresco-painters, memorial-builders and the nephew of the Chinese emperor, General Zhang, came.”
The artisans who arrived with General Zhang helped furnish the Kül Tegin marble stone under the direction of Toyghut Elteber and funerary inscriptions were composed by Yollugh Tegin, son of Kül Tegin’s sister. This funerary stele bearing these inscriptions was erected the following year on August 1, 732 CE and the posthumous title Inanchu Apa Yarghan Tarkhan was conferred on him (c.f. the conferment of a comparable posthumous title on prince Tolui, the younger brother of the second Mongol Khan, Ogodei). On the stele, the great acts of Kül Tegin in raising the floundering Turk empire are narrated in the words of his brother Bilge Khaghan in almost epic terms. At this juncture, it is worth mentioning that the Ashina clan of Turks from which the Blue Turks hailed were married into the Tang royal family. Despite denials in some modern Chinese quarters, Taizong himself likely had immediate Turkic and/or Mongolic ancestry. They had played a central role in raising Taizong to the apex — a time when Turkic fashion was the rage in China — Taizong himself took on the title of the Khaghan graced by Tengri and had a funerary monument with imitations of the Turko-Mongolic balbal stones. However, on the other hand, he pursued an aggressive policy to annex the Turkic khaganate to the Tang empire. The events described on the stele follow the destruction of the original Turkic Khaghanate by the Tang. Thus, Bilge Khaghan, while narrating the biography of his brother, first talks of how the divinities aided their father to salvage the Turks when the Tang emperor decided to exterminate them:
“Without thinking of how Turks had fulfilled their allegiance, [the Tang emperor] said: “I shall kill the Türk people! I shall leave them no descendants!” The Turks were perishing. [However], Tengri of the Turks above, along with the goddess Yer (Earth) of the Turks and the water deity held my father Elterish Khaghan and my mother Elbilge Khatun at the top of the sky and raised them up so that the Turks would not perish and would become a nation.”
After describing the numerous campaigns of his father on the steppes, Bilge Khaghan moves on to talk of the achievements of himself and Kül Tegin:
“I did not ascend the throne over a prosperous people. The people over whom I ascended the throne were without food inside and without clothes outside, bad and evil. I discussed this with my younger brother Kül Tegin. So that the name and fame of the people, for whom my father and uncle had striven, should not disappear. I neither slept at night nor sat down in the daytime by reason of the Turk nation. I, together with my younger brother Kül Tegin and the two shad-s, toiled to exhaustion. Having thus toiled, I prevented the united people from being like water and fire. The people who had gone elsewhere when I was ascending the throne came back again, exhausted, on foot and naked. In order to feed these people, I campaigned twelve times with a sizable army northwards against the Oghuz people, eastwards against the Khitan and Tatabi people, and southwards against the Chinese. I battled against them there. Then, as Tengri blessed [me]; because of my good fortune and fate, I revived and fed the dying people. I clothed the naked people, and made the poor people rich. I enlarged the small population and made my people superior to those who had a strong realm and a powerful khaghan.”
Giving the biography of Kül Tegin he says:
“When my father, the khaghan [Elterish], died, my younger brother Kül Tegin was only seven years old. Thanks to the kindness of my mother Khatun, like the goddess Umai, my younger brother Kül Tegin became a grown-up man. When [Kül Tegin] was sixteen years old, my uncle, the khaghan, was working hard for his realm and laws. [When Kül Tegin was seventeen years old,] we went on a campaign against the Sogdians in the Six Prefectures and destroyed them. The Chinese prince commander came with fifty thousand. We fought. Kül Tegin fiercely charged on foot. He caught the prince commander’s brother-in-law with his own armored hands. Still armored, he presented [the captive] to the khaghan. We wiped out that army there.”
After giving a long account of his many battles he concludes with the great act of Kül Tegin during the siege by the Oghuz Turks:
“The enemy Oghuz laid siege to [our] royal camp. On a white Ögsüz (horse caught on the steppe?) [horse], Kül Tegin speared nine men. He did not lose the royal camp. [Otherwise] my mother the Khatun, together with my [step-] mothers, aunts, sisters-in-law, princesses, and the [other] surviving [women] would have become slave-maids, or their corpses would have remained lying in the abandoned campsites and on the road. If it had not been for Kül Tegin, you would all have died!”
An Altai petroglyph showing the Turks on a campaign clad in lamellar armor. Above the deer one can faintly see the ibex, the animal on the tamgha of the Ashina clan.
The Kül Tegin funerary stele is one of the five famous early Turkic stelae that record the history and great deeds of key figures from the second Turkic Khaghanate. Kül Tegin stele and that of his brother Bilge Khaghan were erected in Khoshoo-Tsaidam, Central Mongolia, by the Orkhon River in the 730s of the CE. A third monument describing the deeds of Tonyukuk, the prime minister and commander of the Turkic army, also the father-in-law of Bilge Khaghan, was erected at Bain Tsokto near modern Ulaanbataar, in the Tuul River basin. Given that it mentions the deeds of Tonyukuk as though he were speaking, it is disputed if this was a funerary stele or merely an autobiographical record of the deeds of the prime minister. Tonyukuk was prime minister and commander through the reigns of the Khaghan-s Elterish, Qapaghan, Inal and Bilge, dying at an advanced age in 725-726 CE. A record in Chinese prepared for a remote descendant of Tonyukuk during the Chingizid period mentions that he lived for 120 years (see below). This was almost 6 centuries after his time; hence, it can only be taken to mean that there was a clear memory of this long life rather than an exact record of his lifespan. The deeds recorded on the stele are more or less till 716 CE. This might mean that the stele was erected then or in the year of his death. It is possible he was a semi-retired in his last years during the reign of his son-in-law and thus had no additional deeds to record after 716 CE. The fourth monument, the Ongi monument (now badly damaged), was located at the confluence of the Tarimal River and Ongi Rivers. This was erected by Īshbara Tamghan Tarkhan, a cousin of Bilge Khaghan, for his father Eletmish Yabghu who may have died in an intra-family battle in 716 CE between the supporters of Inal Khaghan and Kül Tegin, who was trying to seize the throne for his brother Bilge Khaghan. This stele was likely erected between 725-732 CE. The fifth is the funerary inscription of Īshbara Bilge Küli Chur, who also appears to have had the title of Chikhan Tonyukuk. He is said to have lived to a full 80 years and “grown old” during the reign of Elterish Khaghan. His stele mentions him killing “nine ferocious men”, probably while still in his teens. He is also recorded as “fighting the Chinese so many times that he gained much fame by virtue of his courage and manly qualities…” As a minister and general of the Turk Khaghan who grew old in the reign of Elterish Khaghan, he might have preceded the famous Tonyukuk as the highest minister. The Küli Chur inscription does not mention Bilge Khaghan or Kül Tegin, suggesting that stele bearing it was likely erected during the late 600s or early 700s of CE.
The Bain Tsokto stele with the Tonyukuk inscription
The discovery of these monuments was among the greatest moments in archaeology as they are the earliest substantial written records of the history and the deeds of the Turkic people in their own words. The German, Philip von Strahlenberg, fighting on behalf of the Swedes, was captured by the Russians in 1709 CE during the battle of Poltava. As a prisoner in Siberia, he carried out an extensive ethnographic and geographic survey of the eastern possessions of the Rus. In course of this exploration, he observed runiform inscriptions on stones in the upper course of the Yenisei River in Southern Siberia (see below) — his account was the first notice of the old Turkic inscriptions in the modern era. However, their script and contents remained mysterious to him and they were mostly ignored thereafter. More than a century later, the Finnish explorers in Siberia rediscovered them in 1887-1888 CE. An year later, the Russian-born Siberian separatist, Yadrintsev, heard from Mongol pastoralists of the presence of inscribed stelae by the Orkhon river — these were what later came to be known as the Kül Tegin and Bilge Khaghan monuments. In the 1890s, the Finnish explorers and the Germanized Russian, Vasily Radloff, along with Yadrintsev conducted further separate explorations of the Mongolian sites. In 1891 CE, some Mongols led Yadrintsev to the Ongi monument and he made a realistic drawing of the same, recording the inscribed stele and the several balbal stones erected beside it — an important record, given the subsequent damage it suffered. These explorations made it clear that the Yenisei, Orkhon and Ongi inscriptions were all in a similar runiform script recording an ancient language.
The key to breaking the mystery of these inscriptions was offered by the Kül Tegin monument, which had a subsidiary inscription in old Chinese. It contained the condolence letter written by the Tang emperor on being informed of Kül Tegin’s death — evidently, he saw him both as a worthy rival and some kind of “colleague” given the links the Tang had with the Turks. The German Sinologist Georg von der Gabelentz was able to immediately recognize based on the work of the Finnish exploration that the inscription on the stele honored a Turkic prince. He published a translation of the Chinese inscription albeit replete with errors. This led to the Danish scholar Vilhelm Thomsen deciphering the runiform inscription based on his knowledge of the Turkish language in late 1893 CE. On the Russian side, in the same year, their ambassador in Peking showed the inscription to the Ching scholar Shen Zengzhi who provided similar suggestions regarding its identity. Subsequently, Radloff made a better translation of the Chinese inscription with the help of the Ching ambassador to Moscow and published the Turkic runiform Orkhon inscriptions. These were followed by editions and translations by Thomson, Radloff, and others. In the following years, Aurel Stein discovered a Turkic book on omenology-based dice prognostication (Irk Bitig) written in the same runiform script along with two Chinese bauddha hymns, evidently based on Sanskrit originals, in the hall of the thousand Buddha-s at Dun Huang. This text, either from the 700s or 800s of the common era, offered a further body of old Turkic material in the same script. Since then, thousands of papers have been written on these old Turkic texts leading to much improved readings of them.
The preservation of this book hints that the script was not just used for inscriptions but also in books. The Irk Bitig is unique in preserving purely Turkic content even if its author was a bauddha Turk — he says he wrote it for his elder brother, the general Itachuk, in a vihāra after having listened to a bauddha guru. The dice omenology of the text relies on using three Indo-Iranian style dice with four faces each. Thus, one gets $4 \times 4 \times 4 =64$ combinations and one combination with two corresponding omens giving a total of 65 readings. Such dice have been recovered in the pre-Turk Kuṣāṇa site at Khayrabad Tepe, Uzbekistan. It seems these omens are dreams — that leads to the question as to what function the roll of the dice played? We suspect there was a correspondence between the prognosis of the two — you either got a prognosis by the dice roll or if you had the corresponding dream. Alternatively, there was something coded in the omen that is lost to us. For example, the 6th omen reads thus with the corresponding dice roll:
$\circ\circ\;\;\circ\circ\;\;\circ$ A bear and a boar met on a mountain pass. (In the fight) the bear’s belly was slit open (and) the boar’s tusks were broken, it says. Know thus: (The omen) is bad. (translation by Talat Tekin)
This omen reminds one of the statement regarding the dog and the hog in the Mahābharata; however, there it is good for the śvapāca. Another interesting point is the word üpgük = hoopoe in omen 21. While onomatopoeic, it seems like a cognate of the IE word for the bird suggesting an ancient “Nostratic” origin for it.
Imprints of the Ashina clan and the Blue Turks beyond the Khaghanate
The history of the steppes teaches us that great clans have deep impacts over time and space both at the genetic and the memetic level. This is well-known for the founding fathers of the Ārya-s, Chingiz Khan and the founder of the Tungusid Manchu empire. Was there any comparable impact of the Türk Khaghanate? A comprehensive genetic study by Yunusbayev of the impact of the Turkic expansion indicates that it is hard to assess the early signals of the Turkic expansion relative to the later ones where it was coupled with the expansion of Mongolic populations. Moreover, even though the Altaic monophyly looks increasingly unlikely, the Mongolic and Turkic peoples emerged from the same region and their languages show signs of prolonged interactions. This is also apparent in their genetics. In any case, the above study found the first signals for Turkic admixture outside the core Mongolian domain starting around 600-800 CE — this appears to correspond to the rise of the Blue Turk and Uighur Khaghanates.
A neighbor-joining tree based the single nucleotide polymorphism from ancient Central Asian samples indicting the relationship between Altaic groups speaking Turkic and Mongolic languages
On the philological side, there is strong evidence for the long-term persistence of the clan of Tonyukuk. To understand that, below we briefly recap the history of the fall of the second Blue Turk Khaghanate. On the Mongolian steppe, in 742-743 CE, three Turkic tribes, the Uighurs (originally from the region of the Selenge river), the Qarluks and the Basmyls, sensing the weakness of their Gök Türks overlords began asserting their independence. The Basmyls moved first to capture the Gök Türk capital and slay their Khaghan. The next year, the Uighurs and the Qarluks followed them to overthrow and destroy the Basmyls. The Uighurs then asserted themselves by driving the Qarluks towards Kazakhstan. Thereafter, the Uighurs moved on the remnants of the Blue Turk Khaghanate in a tacit alliance with the Chinese and beheaded their last Khaghan in 745 CE thereby erasing their empire off the eastern steppes. The Uighur lord declared himself the Khaghan under the title Qutlugh Bilge Köl Khaghan. The other branch of the Blue Turks descending from the first Khaghanate, Türgish, the “in-laws” of the second Khaghanate, had valiantly fought the Islamic Jihad and Chinese expansionism in Central Asia under their brilliant Khan Su-lu. When they encountered the Qarluks fleeing from the Uighur onslaught, they were in a weakened state from those struggles. After a prolonged fight lasting around 22 years, the Qarluks overthrew the Türgish, thus ending the line descending from the western branch of the first Turkic Khaghanate. With the old empire now gone, the famous clan of Tonyukuk, shifted their allegiance to the Uighur overlords of the Turkic world. It is notable that in this period the Kashmirian emperor Lalitāditya of the Kārkoṭa-s appointed a Turk (Cankuna) as his minister and general. We speculate that he too could have been a member of the Tonyukuk clan looking for new opportunities (though, one cannot rule out a high-ranked Türgish).
The story of the survival of the Tonyukuk clan goes back to the discovery of the earliest Turkic writings and its more recent re-investigation. In 1909 CE, a fragment of an old runiform manuscript from the period of Uighur ascendancy was procured in Khocho (Idiqutshahri). Radloff published the same the next year but he felt its contents were largely uninteresting. However, more recently, Erdal and Hao noted its parallels to another manuscript fragment from the same place in the Manichaean script that explicitly talks about the same events recorded by Tonyukuk on his stele — i.e., the revival of the Turk Khaghanate after its fall to the Chinese assault by Elterish Khaghan with his wise advice. Based on these parallels Erdal was able to interpret the contents of the first runiform manuscript as talking of the role played by Tonyukuk in the nomination and enthronement of Elterish Khaghan during the revival of the empire. Hao brought to light a work composed during the reign of the Chingizid rulers Temür Khan (son of Qubilai Khan) or his son Külük Khan that records the history of a remote descendant of Tonyukuk, Xie Wenzhi (name as recorded in Chinese), an Uighur official under the Mongols. The text states that:
1) Tonyukuk married his daughter to Bilge Khaghan.
2) After the death of Bilge Khaghan, his wife (i.e., daughter of Tonyukuk) led the Turks for some time.
3) After the conquest of Mongolia by the Uighurs, who were from the Selenge river (i.e., where three rivers join to form it), Tonyukuk’s descendants switched allegiance to them as their ministers.
4) The Uighurs saw the Tonyukukids as being “swift as falcons”.
5) The Uighurs of Khocho had a long tradition of worshiping the 20 deva-rāja-s and used Sanskrit in their liturgy.
6) Tonyukuk and Kül Tegin aided the Tang during the An Lushan convulsion in China. This is clearly an anachronistic and an ahistorical record. However, it suggests that possibly a descendant of Tonyukuk along with the Uighur Khaghan had aided the Tang during the rebellion of An Lushan and this was superimposed onto the founder Tonyukuk and Kül Tegin.
7) A certain Kezhipuer is mentioned as being a prominent minister from the Tonyukuk clan several generations after the An Lushan rebellion.
During the Chingizid Mongol rule of China, Xie Wenzhi, Xie Zhijian and other descendants of Tonyukuk were part of the elite and were prominent as scholars, artists and administrators. At the fall of the Mongol empire in China, some of these fled to Korea where they founded a prominent clan. Other members of the clan persisted under the Ming as ministers and officials in Liyang and Southeast China despite the nationalist backlash against the Mongols and their officials. Thus, the clan of Tonyukuk is a remarkable example of the human capital of a great founder lasting for over 700 years across Central Asia, China and Korea.
Looking backward in time, a major question is the provenance of the influential Ashina clan from which the Blue Turk Khaghanate, the Basmyls and the Qarluks arose. They were characterized by the famous ibex tamgha, which is seen on their inscriptions in runic script, both in Mongolia, along the Yenisei River and the Altai mountains. The clan also gave rise to Turkic elite that had intermarried with the Tang elite and conquered the western territories for the Tang emperor Taizong. It is also likely that the Ashina clan gave rise to other influential Turkic lineages of later Khaghanates like those of the Bulgars and the Khazar Khaghans. Their elite status seems to be repeatedly emphasized in their textual sources as they are distinguished from bodun — the Turkic word for the plebeians. Based on the Chinese sources one may infer that the Ashina clan might have been already present in the early Hun period of the Xiongnu Khaghanate. They were definitely present as vassals of the Rouran Hun Khaghanate and are mentioned in multiple Chinese sources as being their iron smiths. These sources also hint that the conflict between the Uighur branch of the Turks and Ashina clan might have begun in this period itself. In 546 CE, the Oghuz Turkic confederation, at whose head were the Uighurs, rebelled against their Mongolic Rouran Hun overlords. The Ashina clan is said to have aided the Huns in suppressing this revolt. However, it appears to have weakened the Rouran state and six years later, as the land thawed in the spring of 552 CE, the Ashina clan, which had risen in power from their recent exploits, overthrew their Hun overlords and drove them westwards from Mongolia. The leader of the Ashina clan declared himself the new Khaghan. Thus, there was a history of the Turkic peoples under early Mongolic rule that remains poorly understood. However, it may be reasonably inferred that there was already some diversification among them. We already see the Oghuz alliance with which the Uighurs were associated and the On Oq (10 arrows) alliance led by members of the Ashina clan. Indeed, the ethnogonic myths of the clan repeatedly mention the 10 sons of the founder, which is consistent with the On Oq having 10 sub-clans within it.
There has been a string of discordant theories regarding the origin of the Ashina clan. However, the majority of the plausible theories posit that the etymology of Ashina was not originally Turkic but Indo-European. Among the Indo-European etymologies, we have:
1) Beckwith proposed a Tocharian origin from Arśilas = noble kings. It is also related to one of the self-designations of the Tocharians for themselves (Ārśi). In further support of such a proposal, Golden noted the Turkic word for ox as öküz (note Kentum state) is likely derived from Tocharian B: oxso or Tocharian A: okās. While their probable homeland in the southern slopes of the Altai mountains would not be inconsistent with some late-surviving Tocharian imprint, there is no other evidence for a connection between the Turks and the Tocharian elite in the region.
2) Atwood proposed a similar root form, but with an Indo-Aryan etymology: ṛṣi > ārṣa > ārṣila. He notes the parallel rendering of ṛṣi as Arsilas in Greek. While an interesting proposal, it is odd that a ruling warrior clan would have such a typically brahminical etymology, unless, like certain Hindu dynasties, they sought to present ancestry from a ṛṣi.
3) Another Indo-Aryan etymology proposed by Klyashtornyj (along with proposals of Golden, Beckwith and Mair) is: Aśvin (one with a horse)>Ashina. A key point in this proposal is the status of the Wusun, who were an Indo-Iranian steppe people recorded in Chinese sources. Therein, the ethnogonic myth of the Wusun mentions that they believed that their ancestor was orphaned in an attack by the Huns (the first Khaghanate of the Huns, i.e., Chinese Xiongnu). This ancestor was then raised on the steppe by a female wolf and ravens. Multiple versions of the ethnogenesis of the Ashina clan of the Blue Turks also mention that their ancestress was a she-wolf and that they were feudatories of the Xiongnu first and the Rouran Huns thereafter (a version of this wolf motif was remembered long after the fall of the Turks to Mohammedanism by Gardīzī, the minister of the monstrous sultan Mahmud of Ghazna. He states that the Turks have sparse facial hair and a dog-like nature due to their ancestor, as per Abrahamistic tradition, Japeth, being fed wolf’s milk and ant eggs as a medicine. His teacher al Bīrūnī also records that ancestor of the Turks of Afghanistan was a long-haired dog-prince. Victor Mair proposes that wolf’s milk might have meant the slime mold Lycogala). The wolf motif is also found in the origin myth of the Uighur lineage of Turks (the Chinese sources mention their origins from the coitus of the ancestor wolf with the daughter of the Hun [Xiongnu] Khan). As per Golden, the Uighur Oghuznāma mentions “Blue Wolf” as being their war cry. The same pattern is again seen in the case of the Chingizid Mongols, where the male ancestor is the wolf. Thus, even though the wolf motif is widespread in the Turko-Mongol and Indo-European world (e.g., the founding of Rome), the Wusun and the Ashina clan share the female nurse/ancestress. Thus, the etymology of Wusun and Ashina is seen as deriving from a common root Aśvin. Beckwith correctly reasons that this group was likely a steppe Indo-Aryan remnant rather than Iranian, given the root form Aśva as opposed to Aspa (e.g., in steppe Iranian Arimaspa).
4) Finally, we have Bailey’s suggestion that it derives from steppe Iranian Śaka word āṣṣeiṇa for blue. This would match the Blue Turk appellation of the clan. However, we suspect that, while there might be something to this etymology, it is more likely an instance of retro-fitted etymologizing based on the Śaka word in one branch of the Ashina clan. There is no evidence that all branches of the clan called themselves Blue Turks.
Thus, we cautiously posit that the most likely origin of the Ashina clan was via the Turkification of an originally Indo-European (likely Indo-Aryan) steppe people that retained its elite status through multiple admixtures with East Asian groups that spoke a Turkic language. We suspect that this Turkification of the Ashina-s probably occurred over a prolonged period ranging from the Xiongnu Khaghanate all the way to the early Rouran period. Yet some imprints of the IE affinities can be gleaned even as they become more prominent on the historical landscape. It is likely that cremation was the primary funerary practice among the Ashina elite as opposed to the traditions of the Hunnic elite, linking them to an old IE tradition. In further support of a specific Indo-Aryan connection, we may point out that the names of the founding brothers of the Blue Turk Khaghanate Bümin and Ishtemi do not have an explicit Turkic etymology. However, Bümin can be transparently derived from Indo-Aryan Bhūmin (note Indic bodhisattva > early Mongolic bodi-satva on Khüis Tolgoi Brāhmī inscription) or Iranic Būmin = “the possessor of the land”. Similarly, the name Īśbara kept by multiple early Turkic Khans can be derived from Indo-Aryan Īśvara. However, the apparent decipherment of the Khüis Tolgoi, Bugut and the short Keregentas (Kazakhstan) Brāhmī inscriptions by Vovin, Maue and team suggest that there was Indic influence on the steppe which might have gone along with the missionary activity of the Bauddha-s (as opposed to remnants of steppe Indo-Aryans like the Wusun). One cannot rule out the role it might have had in transmitting Indic names and terms to the early Turkic and Mongolic groups. In the Khüis Tolgoi Brāhmī inscription we already encounter a Blue Turk Khaghan if Vovin’s reading is correct: Niri Khaghan türüg khaghan: Niri Khaghan, the Khaghan of the Turks. This would point to contact with Indic cultural elements early in their history.
In this regard, we would like to point out one further, more tenuous connection. The Chinese sources, like the Zhoushu mention that the Khaghan of the Turks performs a ritual at the ancestral cave in Ötükän mountain where Ashina was born from/suckled by the female wolf. Suishu further adds that on the eighth day of the 5th month the Turks perform a great sacrifice and send a ritualist into the cave to make offerings to their ancestors. Ethnological investigations have indicated that the Siberian Turks make offerings to the gods and ancestors with the incantation cök usually coupled with a formula. For example, Inan notes the following (in translation):
O my ancestor Kayra Khan, the Protector! cök! Here it is, offering to you Kayra Khan!
Cök! Here it is, offering to you! My mother (like fire) with thirty heads.
My old mother with forty heads; when I recite cök! Have mercy!
The latter two appear to be offerings to polycephalous female deities one of whom is associated with the fire (c.f. the Mongol fire goddess). Similarly, Anohin also recorded several formulae with offerings made with the cök incantation, including to the ancestor Kayra Khan.
Ak-it purul piske polush, cök! = Grey and white dog! help us! Cök!
Interestingly, Aydin found that this incantation found in modern Turkic formulae is already seen in several runiform Yenisei inscriptions from around the time of the Blue Turk Khaghanate and Erdal interpreted it (in our opinion correctly) as something that implies “I offer my sacrifice”. For example, we have: “Tengrim cök! bizke” = To Tengri cök!; [may he favor] us (in the Yenisei inscription cataloged as Tuba II [E 36], 2). In another Yenisei inscription, we encounter a similar formula invoking Tengri in the context of a holy rock and a cliff — perhaps a parallel to the cave offerings of the Blue Turks.
A closer examination of the known exemplars of the Turkic cök incantations reveals a parallel to the mantra incantations that end in svāhā, sometimes with an additional phrase reminiscent of idaṃ [devāya etc.] na mama. Zhang He noted (following the Song dynasty scholar Shen Kuo) that the “sai” incantation, which was usually present at the ends of the formulae deployed by the mysterious Chu kingdom (from 300 BCE or before) was likely originally svāhā or a derivative thereof (Chinese sa-po-he). The Chu kingdom is believed to have originally had a non-Cīna soma- and fire-sacrificing elite, likely of steppe Indo-Iranian origin who might have been absorbed by the Huns. Thus, it is not far-fetched to propose that the Turkic cök formula was also inspired by or derived from svāhā — something that would be compatible with the proposed Indo-Iranian roots of the elite Ashina clan.
The runiform scripts origins and spread beyond the “Orkhon” horizon
Shortly after his decipherment of the Turkic runiform inscriptions, Thomsen proposed that the runiform script was probably derived from Aramaic via Sogdian or additional Iranic intermediaries. This hypothesis came to be widely accepted in a manner parallel to the Aramaic hypothesis for the origin of the Indian Brāhmī script. However, it should be noted that some of the same problems confront the Aramaic hypothesis for both Brāhmī and runiform. Talat Tekin notes that the Orkhon inscriptions contain 38 characters and there are two additional characters that he takes to represent syllables in the Bain Tsokto stele of Tonyukuk. Of these, there are 4 vowel signs — something that Aramaic does not use. Brāhmī has an even more elaborate vowel system based on the Indo-Aryan grammatical tradition that is necessary to encode Indian languages — something which is unparalleled in the Aramaic family. The runiform script does not distinguish some long from short vowels and totally devotes 4 signs for these (/a, ä/; /i, ï/; /o, u/; /ö, ü/). 20 signs represent either a plain consonant or a/ä+consonant; e.g., at/ät. The remaining 16 signs represent various other consonants that are neutral with respect to the vowels, syllables like “ash” and sounds like ich, uk etc. Thus, the organization is quite unlike the Iranic scripts derived from Aramaic or old Aramaic itself. Now, we know that Aramaic was used in southern Central Asia within the Indosphere — e.g., Aśoka Maurya’s inscription in the northwest. There was also Kharoṣṭhī which appears to have represented a genuine Indian adaptation of Iranic administrative Aramaic with vowel diacritics for better encoding of Indo-Iranian languages. Thus, while Aramaic and Aramaic-inspired scripts were in vogue in Central Asia and India, directly deriving Brāhmī and runiform from Aramaic is not well-supported. Instead, both seem to be scripts that were probably inspired by the “presence of writing” rather than being direct adaptations of other scripts. In the Indian situation, the possibility of some memory of the Harappan signs (believed by most to be a script) is another factor, whereas in the Central Asian situation there were multiple local scripts, including possibly Aramaic, that could have provided some indirect influence.
The Issyk silver bowl inscription
If we set aside the Issyk inscription, we are still left with the question as to when the runiform script started being used? An interesting clue in this regard comes from the recent discovery of bone plates used as a grip for a composite bow that were found in an Avar grave at Szeged-Kiskundorozsma, Hungary. These plates are inscribed with a runiform script that is related to but not identical to that used in the Eastern Turkic texts. The bone plates gave calibrated radiocarbon dates ranging from 660-770 CE, whereas thermoluminescence dating of a pot from the grave site gave a central date of 695 CE. These are in the general age range of the second Turkic Khaghanate’s inscriptions. These join a relatively small set of comparable short runiform inscriptions that have been found in Eastern Europe from the Avar horizon but without the secure dating of the above. All of these remain undeciphered. However, the recent discovery of several runiform inscriptions in the Altai has uncovered signs that are similar to those in these Eastern European exemplars. Further, we know that the Khazar Khaghanate also used a runiform script. At least one clear example of this found at the end of a letter written by a Jew has a short Khazar phrase (interpreted as “I have read”) in runiform (probably by a Turk in response) that can be read largely on the basis of the Eastern Turkic Khaghanate’s runiform script. Beyond this, there are several other short Khazar inscriptions that remain undeciphered — in part because the exact Khazar dialect of Turkic remains poorly understood, the inscriptions are short, and some signs are distinct from those of the Eastern runiform corpus. However, some of these signs overlap in form with those seen in the Eastern European inscriptions attributed to the Avars.
Mammoth bone runiform inscription from Yakutia — the northern reaches of the Turk domain
Recent genetic evidence has strongly established that the Avars are the remnants of the Rouran Hun Khaghanate that was overthrown by their Turk feudatories to establish the first Turk Khaghanate. Thus, it would be reasonable to propose that the script was invented in some form before the destruction of the Rouran Khaganate, most likely among the Turkic tribes. Given that the Rouran Huns themselves appear to have preferred Brāhmī, it is possible that this was a “national” script that Turks devised to specifically distinguish themselves and their language. However, it is likely that the script was also known to at least some of the Rouran Huns or Turkic groups allied to them that carried it west as they fled. Thus, in part, the inability to decipher the Eastern European Avar exemplars might come from the fact that they encode an early branch of the Mongolic language rather than a Turkic language (c.f. the Chingizid use of the Uighur script for Mongolian). In this regard, the case of the mysterious jug inscriptions can also be considered. Before the Bratsk Reservoir in Russia was flooded, six silver jugs were found on the Murujskij island by a fisherman before it went under. The form of these jugs is similar to the silver/gold jugs found at the funerary monuments of Bilge Khaghan and other members of his family. Only two of the Murujskij jugs survive and the bottom of one of them has an interesting inscription in the runiform script that can be completely transcribed on the basis of the Eastern Turkic runiform script like that seen on the Orkhon stelae. However, the transcription cannot be deciphered as Turkic (and so far as anything else). Nevertheless, the inscription is associated with the Ashina clan’s ibex tamgha (also seen on the second jug). This indicates that even though the jugs with the inscription are from the Turkic Khaghanate, the runiform script on them was used to encode a language other than Turkic. Recently, another inscription was found on a mammoth bone amulet far north in Yakutia, indicating the spread of the script and possibly the extent of the Turkic Khaghanate. Such a scenario of expansion and subsequent splintering of the Khaghanate would be consistent with: 1) the runiform script being adopted by languages unrelated to but geographically proximal to Turkic; 2) The divergence in form between different Turkic groups (e.g., those in the West which eventually gave rise to the Khazar version and those in the east which gave rise to the Uighur version); 3) Loss — once the Turks lost their self-identity as a nation and became satellites or vehicles of the Abrahamistic religions.
The mysterious Murujskij silver jug and runifrom inscription
Finally, we can say that as more inscriptions are found in Mongolia, the Yenisei and Altai we might still learn some poorly understood facets of early Turkic history. Recently (in 2017), a further monument with 14 pillars was discovered at Dongoin Shiree in Eastern Mongolia with several inscriptions and tamghas that are yet to be published in detail. The preliminary report by Japanese researchers indicates that it was the monument of the one of the shad-s of Bilge Khaghan, the yabhgu or viceroy of the eastern territories. With respect to the Yenisei inscriptions, Klyashtornyj has recently read evidence for some facets of the history of the Western Ashina Khaghans. For example, he believes that the branch of the Türgish who left the inscriptions found in Minusinsk basin near the lake of Altyn-köl were the predecessors of the Kirghiz Khaghans who eventually conquered the Uighur Khaghanate. One of the inscriptions mentions a certain general Chabysh Ton-tarqan from this clan — the name Chabysh seems to be an early attestation of the root of Chebyshev, the famous Russian mathematician who is supposed to have descended from a Chingizid Mongol chief. Another runiform funerary inscription from the Yenisei (Uibat VI) commemorates a certain hero named Tirig-beg who is said to have fought like a wild boar when the mighty Uighurs were overthrown. These complement the Suja inscription found in the early 1900s by the Finnish expedition in Northern Mongolia, which was commissioned by Boila Qutlugh-yargan who also participated in the great Kirghiz-Uighur clash of 840 CE. It is likely that this event and the subsequent Kirghiz invasion of the Chinese territories was what formed the core of their oral epic of the hero Manas.
## Vikīrṇā viṣayāḥ: India and the Rus
$\S \star$ Our sleep was disturbed by a dream with a circulating motif whose exact story line, if any, was lost upon awakening. It started with a tall elderly man of West African ancestry playing cricket (batting) with the swagger of a young star at the peak of his career. Striking the ball along the ground or smashing it high into the stands he scored on with ease reminding one of the great black emperors of yore from the Caribbean. Then the dream entered the motif phase with the same man, rather paradoxically, in a classroom repeatedly explaining a single-peaked statistical distribution he claimed to have discovered — we tried hard to capture the equation of that distribution but failed at every repetition of the motif. This kind of REM sleep can be rather troubling, and we tossed and turned around before settling into another scene that seemed to have no connection to the above (unless of course, we forgot that snatch upon awakening). In this scene was an elderly Russian Jewish woman — we would estimate her age as being around 85-90 years — who sat on a chair with a table in front of it. Soon another bearded man appeared beside her — he seemed to be in his 60s but in great health. He exuded a profound ambivalence that strongly impressed upon us — while a part of him presented features consistent with a good character, the rest of him was filled with rapacity, cunning and a taskara spirit. He told the old woman in an unusual accent that seemed either German or Russian that we spoke German. The woman responded in a feeble voice: “Die beiden Grenadiere.” We then saw ourselves in the dream reading out the famous poem of Heinrich Heine:
Die waren in Russland gefangen.
Und als sie kamen ins deutsche Quartier,
Sie liessen die Köpfe hangen.
Two grenadiers were marching back to France
They had been held captive in Russia,
And when they reached German lands
They hung their heads in shame.
Da hörten sie beide die traurige Mär:
Dass Frankreich verloren gegangen,
Besiegt und geschlagen das tapfere Heer—
Und der Kaiser, der Kaiser gefangen.
For here they learnt the sorry tale
That France had been conquered in war,
Her valiant army beaten and shattered,
And the Emperor, the Emperor captured.
“Dann reitet mein Kaiser wohl über mein Grab,
Viel Schwerter klirren und blitzen;
Dann steig ich gewaffnet hervor aus dem Grab—
Den Kaiser, den Kaiser zu schützen!”
“That will be my Emperor riding by my grave;
Swords will be clashing and flashing;
And armed, I’ll rise up from the grave
To defend the Emperor, my Emperor!”
The old woman said: “Sollen wir mit Russland oder Frankreich sein, das war die Frage…” We either did not catch or forgot the rest of her words except for the very last: “Russen und die heidnischen Indianer”. We awoke soon thereafter and the memories of the rest of the dream were lost. It was the 100th day of the Russian invasion of Ukraine.
$\S \star$ There are two things have are remained fairly stable in the politics of our times: the mleccha-marūnmattābhisaṃdhi and Galtonism. Much else of what concerns the Hindu nation develops around these two. The concluding words that we remembered from our dream brought to mind a clear parallel that has long existed but recently raised its head again in the H world. In our previous note, from shortly after the start of the current European war, we traced the path of the rise of the new mleccha religion, navyonmāda, and its role in the overthrow of the Nārīṅgapuruṣa and beyond. Indeed, any sane, politically aware heathen living in mahāmleccha-land will get of sense of how it might have been for the last heathens of the classical world as the frenzied followers of the śūlaprotapreta were starting to gain the upper hand in enforcing their cult. We observed that when it comes to the Rus, the navyonmatta backers of Vṛddhapiṇḍaka and their internal opponents among the mleccha-s, i.e., the more protapreta-aligned folks or some of the more secularized uparimarakata-s are quite aligned in their rhetoric. As we have noted several times on these pages, the mleccha-marūnmatta-yāśu overrides even the yauna-sambandha between the paśchima-mleccha alliance and the Rus. Briefly, the inclusion of the Turuṣka in the rotten soybean soup has its deep roots in the Crimean war against the Rus. This continued to the recent times in the form of the first Afghan war, the “soft underbelly strategy”, the Chechnyan war, and the support for the marūnmattātaṅka in Rus cities. Similarly, when it comes to the H both sides of the mleccha political spectrum are quite aligned. In the case of navyonmāda, given its natural attraction for marūnmāda, it usually proceeds via active fostering of the mleccha-marūnmattābhisaṃdhi. In the case of the mleccha “right” and uparimarakata “classical liberals” or “neoconservatives” it proceeds either via support for the kīlitapreta-bhānaka-s or “enlightenment values” (e.g., the anti-H garbage spouted by the scientist Stuart Kauffman in a talk is just one example among the many from that group).
Both among the Rus and H there exist many who are truly in love with the mahāmleccha sphere (i.e., the pañcanetra-s) and identify closely with them. For the Rus the path of assimilation in the mahāmleccha mass is trivial but for the H it is formidable. Yet, the H have tried hard to do so. Among the Rus who managed to immigrate this process is mostly complete, but among those who could not, for one reason or another, different levels of yearning still exist. We could argue that even the pro-Rus elite, including Putin, wanted to be accepted as respectable members of the Occidental sphere; however, their being spurned resulted in a return to antipathy. Among the H who have immigrated and those who hold the hope to do the same, the yearning is more like that of a guy pining for a beautiful girl who does not cast a glance his way. Ironically, both the anti-Rus or anti-H policies of the Occident end up hurting those who are mostly friendly to them. However, the Indian situation is more complex. There is a sizable pro-mleccha class in India that finds work that is metaphorically not very different from that of the sepoys under the English tyrants. The Indian system and its deep penetration by the Occident, has meant that this class will actually aid the Occident in implementing any anti-H moves. It seems this class was largely eliminated or defanged among the Rus by Putin. Those of that class who were mūlavātūla-s have mostly left for their own or to the mahāmleccha lands. In contrast, the Indian equivalents of that class are going nowhere and the government neither has the awareness nor the courage to defang them as of now.
As we have again noted on these pages, the pañcanetra-s are master shadow warriors — their conquest of India and humiliation of the Cīna-s was a masterly exposition of the same. In the case of the Rus, when their marūnmatta allies failed to play the proxy role successfully they had to personally intervene in the form of the Crimean war. But even there, the core of the pañcanetra of the age (the English) lost fewer men while letting France to take the heaviest losses. Since the conclusion of the Crimean war, the (proto)pañcanetra has vigorously sought to obtain useful rentier states in the region that can do their bidding against the Rus. The current Ukrainian state was the fructification of this dream. In the subcontinent, when WW2 forced the English to retreat, following the usual doctrine of the mleccha-marūnmattābhisaṃdhi, they created TSP as the rentier state to the keep the H in check. Though the English retreated, the mantle of pañcanetra power was now in the hands of the mahāmleccha-s, who continued that policy with respect to TSP. The people of the rentier state itself might suffer, but it will be kept afloat as long as it serves the purpose of the pañcanetra-s in the realist goals against the target state. As result, these rentier states are among the most corrupt systems in the world. So far, the mleccha-s seem to have a high tolerance for the blowback that comes from these volatile pets of theirs. The mahāmleccha-s even accepted an assault of the magnitude of 9/11 to keep TSP afloat. The blowback from Ukraine has been much less, but the role it has played in frauds and cybercrimes in mleccha-land going all the way to Vyādha-piṇḍaka is immense. It is possible in the future is brings in more terrorism with all the arms the mleccha-s have pumped into the state.
$\S \star$ Given that the history of the current conflict goes back to at least the Crimean war, we cannot but help get to its essence: the pañcanetra-s and their vassals essentially wanted to fulfill the aims of that war, viz. degrade the Rus to the point that they are no longer a great power. This is essentially the basis for the expansion of the rotten soybeans confederation right into the land of the “Mother of all Russian cities”. A parallel to the H world cannot be missed — aiding TSP, that festering rump of the Mogol empire, to take over much of the pāñcanada, which was the mother of all the Indian cities was not very different. Thus, the Rus were confronted with an existential predicament: were they going to resist this encroachment of the Occident into their natural domain by means of a rentier state or were they going to back down with a whimper. Like the Hādi-śūlapuruṣa in the old days, Putin too desired to be accepted by the pañcanetra-backed West as a part of their world. However, as it became increasingly clear that the Occident had no such intention but the contrary, he decided to take back Crimea first, and try the strategy of a low-key war in the Donbas. That later strategy seemed to have worked poorly. Second, he probably sensed that the duṣṭa-Sora-bandhu in Danu-Apara-deśa might have more aggressive plans (likely backed by Sora himself, especially given that his anuyāyin-s have successfully taken control of the mahāmleccha government). Of course, the more cynical mahāmleccha-s think it might have been triggered by the realization of his impending mortality from a cancer — in a sense, he personally had nothing to lose in some big stakes gamble. However, we believe it is a very rational fork the Rus were confronted with and had to take one of the two paths mentioned above.
The expectations and the commentary on the conflict have been wild. The mleccha-s have been claiming exaggerated victories for duṣṭa-Sora-bandhu and his paradoxical allies from the Hādi-puruṣa-pakṣa On the other hand, many expected the Rus to overrun the Dānu-Apara-deśa within a month. However, that has not happened, and the fourth month of the conflict will soon dawn. This was the limit placed by the Rus nationalist Karlin as the boundary beyond which discontent might arise in Rus against their lord. Hence, many have shifted to the mahāmleccha view of things. However, as we had remarked earlier, neither of these paths should have been expected. Historically, the Rus have not shown overwhelming military dominance from the get-go and have tended to have spotty performance in battle (Crimean war, loss to Japan, Afghanistan). However, over time they have repeatedly shown the capacity to doggedly stick to and achieve their military goals. To reiterate, they initially floundered against both the Napoleonic French (Heine’s poem) and the Germans but they came back strongly on each occasion. Thus, their performance in the current war is consistent with this past. In our assessment, while they initially lost impetus, they have subsequently made steady progress. While you may not hear it in the Occidental media, there are clear indications of this: First, the Hādi-puruṣa-pakṣin-s, whose existence the Occident grudgingly accepted, appear to have faced heavy losses and many have been taken alive. These were some of the most committed fighters in the East of that deśa. Second, if one heard the latest interviews of the sora-bandhu with his backers in the Western press, one could hear between the lines that he is hard-pressed. Third, and importantly, the mahāmleccha-s are growing increasingly silent in their news coverage of the glorious wins of their Hādi-puruṣa-pakṣin allies. The mahāduṣṭa Cumbaka, even paradoxically noted that the Dānuka-s may have to cede territory to the Rus. We still do not know how far the Rus would advance. However, it is clear that the Rus-majority regions have now been or will be soon lost by the Dānuka-s despite the spectacular victories claimed on their behalf by the mahāmleccha-s. Will the Rus be able to hold on once their strongman lord attains Vaivasvata or will the mleccha-backed rump of the Kievans make a new advance to recover their losses? That remains to be seen.
Finally, it should be noted that for whatever inconvenience the sanctions of the mleccha-s have caused to the Rus, the mahā-mleccha economy itself is floundering under its navyonmatta leadership. In the end, any sane person would realize that as of today there is no way to maintain the comforts of a modern society without consuming liquid fossil fuels. Beyond being an energy source, they are also the industrial raw material for a wide range of products that are the quintessence of modern life. Indeed, the rout of Germany in WW2 was due to their limitations in accessing liquid fossil fuels. While they captured the French reserves and managed to obtain some from Romania after their eastward thrust, they simply could not match the Soviet supplies. Nor could they capture the Soviet oil fields. The Japanese initially secured their fuel supply after the conquest of the archipelago. However, the American fightback and defeat of the Japanese in the naval battle of Midway limited their safe transportation of fuel in face of the American assaults. After their rout in WW2 at the hands of the Rus in Manchuria, the Japanese decided to surrender to the mahāmleccha-s to save their sacerdotal monarch. Thus, they learnt the hard way that the key to maintaining a modern economy was to have a reliable and proximal fossil fuel supply. Hence, they decided to restore better relationships with the Rus to access oil via Sakhalin. The mahāmleccha-s are now pressurizing them to get off Rus fuel. However, the Japanese industrial leaders have correctly realized the serious negative impact this would have and called on their government to continue dealing with the Rus. The śūlapuruṣa-s too depend heavily on Rus fuel and could lose their preeminent status as the industrial powerhouse of continental Europe if they decide to go along with the mahāmleccha directives. We even suspect that the aṅglamleccha-uparimarakata alliance might be seeing this as a means to kill two birds with one stone — sink both their old enemies the śūlapuruṣa-s and the Rus. Hungary too, which knows well of the evil of duṣṭa-Sora, seems unwilling to sacrifice its comforts by going all out against the Rus. Thus, we remain skeptical as to whether the maṇḍala-dhvaja-s and śulapuruṣa-s would really decouple from the Rus. Moreover, so far the Rus scheme for ruble payments in return for fuel, grain and fertilizers continues despite the sanctions. Hence, we hold that the Occident has failed to achieve the victory it desired in its proxy war with the Rus. That said, we accept this conflict is far from over.
$\S \star$ In late Hindu antiquity, H thinkers realized that the restoration of the dharma-raṣtra cannot occur without a decisive and complete victory over the ekarākṣasonmāda-s. This was presented metaphorically as the kali being brought to a close only upon the uccāṭana of the unmāda-s by Kalkin. The tāthāgata-s recognized the same even as their centers were being reduced to cinders by the bearded ruffians. The catastrophic first war of independence in 1857 CE was fought on fundamentally unsound foundations on the part of the H. After that they have not really fought for the reestablishment of the dharma state as they continued with the same or worse premises on which the 1857 effort was founded. Moreover, freedom came only because the English had already sucked India dry and for practical purposes, they lost the bigger war elsewhere as they had to cede their preeminence in the pañcanetra system to their mahāmleccha cousins. To add to the H woes, while they had freedom from the English tyrants, they had lost key tracts of their land to their old ekarākṣasa enemies, who had not yet been completely overthrown when the English struck. Thus, the H had merely kicked the can down the road in a world where few could act independently without being policed by the pañcanetra confederation and its vassals. The one power that gained the capacity to act with some independence via a combination of the old Galtonian bond and the mleccha rapacity for cheap manufactures was the Cīna-s, who too had become an enemy of the H. Thus, just like the Rus, the H too were presented with a fork on the road: either die with a whimper like a śvan strangled for a Yulin feast or attempt to regain the dharma state by the overthrow of the ekarākṣasa yoke on their necks. The latter path would mean fighting the combined power of the mleccha-marūnmattābhisaṃdhi with the Hans potentially fishing in the troubled waters. The H leadership decided to simply postpone any confrontation of the question as it was too painful to even contemplate. Neither road was pleasant, and the human cost was going to be huge.
But nations without power do not have the choice of their battlefields. Even as we woke from the strange dream the news reached us that the Indian state had abjectly capitulated to the marūnmatta-s, with the mleccha-s and first responders cheering them on. The details of this need no elaboration as they are rather well known to all. Nevertheless, just for the historical record, we would simply say that, as is usual of them, the marūnmatta-s are baying for the head of a $V_1$ government official for speaking the truth about the rākṣasa-mata. There is nothing new in that, but the following are notable: 1) The Lāṭeśvara was brought into power with the hope that he would deal firmly with the marūnmatta-s, even as he did so when they burnt the H alive in his province. However, he meekly caved to the pressure from the West Asian marūnmatta hellholes even as his predecessor the nāmamātra-vājapeyin had done when the marūnmatta-s hijacked the Indian plane to occupied Gandhāra. Then the mistake was done of keeping those three ghāzi-s above the ground after their capture when they should have been promptly dispatched to one of Citragupta’s chambers (it seems the security forces have mostly learnt their lesson since). 2) Moreover, the capitulation of the Lāṭeśvara took place against the backdrop of the renewed ghāzi activity in Kaśyapa-deśa. Residual Vaṅga and Cerapada are tottering under regular marūnmatta assaults too. 3) Most galling thing was that the Lāṭeśvara’s government sent a message to the H that they were more concerned about their enemies who seek to annihilate them rather than the H themselves. 4) It is rather telling that the government even abdicated its mandate for law enforcement under the secular constitution to which they cleave – simple cut and dry cases of freedom of expression and incitement of violence – that could put the ruffianly marūnmatta-s in place (thankfully a couple of state leaders are following that in the least). One could go into any number of explanations (and few of them are entirely valid) of why the Lāṭeśvara capitulated but the bottom line is that the Indian government under electoral politics is too weak to confront the foes of the H. While one could raise parallels to the Mūlasthāna Sūrya temple hostage situation with the Pratihāra-s, a modern state aspiring great power status should have the means of countering such blackmail – they are quite obvious though they cannot be mentioned in public. Hence, the Lāṭeśvara and his court should have at least made that honest confession to the H people that they and probably their army are too weak to confront the mleccha-marūnmattābhisaṃdhi; hence, they would need to capitulate.
We believe that, as with the Rus, the H have been taken to the fork in the road. The Lāṭeśvara, the only patriotic leader with a mass appeal, has shown the weakness of his position. This has cast serious doubt on his ability to take the H through the confrontation — rather he has stuck with the old practice of kicking the can down the road. The government’s hope is everything will be hunky-dory after some cycles of Freitag Eruptionen, but, make no mistake, the marūnmatta-s have sensed that the aging Lāṭeśa is no longer the man he was when he held sway in Lāṭa. If the $V_1$ woman is killed, then it will embolden them even further. They have won this round and will come back for more. Duṣṭa-sora and the navyonmatta-s also want to overthrow the Ānartapa — hence, their natural alliance will swing into action. They have already planted the deśī equivalent of the Dānu-apara’s sora-bandhu along with his band of uśnīśātatāyin-s. Sora and his agents have also succeeded in corrupting the judiciary along the lines of what they have done in mahāmleccha land. Hence, we believe that whether H like it or not they will find themselves on one or the other fork sooner than later and they may not even have a choice. The default endpoint would be that of a camel garroted by a marūnmatta.
## Alkaios’ hymn to the Dioskouroi: Hindu parallels
In this note we shall see how even a short “sūkta” of the yavana Alkaios to the Dioskouroi (individually named Kastor and Polydeukes), the Greek cognates of the Aśvin-s, offers several parallels to the Hindu tradition in the Veda. In the Veda, the Aśvin-s are the sons of Rudra hinting at his overlap with Dyaus (tvam agne rudra asuro mahodivaḥ | or bhuvanasya pitṛ). In the Greek and Roman traditions, they are the son of Zeus or Jupiter maintaining that old connection going back to the Proto-Indo-European tradition and probably beyond to prehistoric times. In Greece, the memory of their Rudrian character is recorded in a 600-500 BCE stele from Sellasia in the Spartan realm where Plestiadas, a pious votary of the deities, inscribed a verse stating that he erected it “out of fear of the fury of the Tyndarid twins (the Dioskouroi)”.
Figure 1. A denarius of the Roman emperor Antoninus Pius showing the twin gods Castor and Pollux with the eagle of Jupiter between them. This iconography closely parallels that of their para-Vedic relatives Skanda and Viśākha. The stars above them signify their association with the constellation of Gemini — an ancient association also reflected in the Taittirīya Brāhmaṇa.
deũté moi nãson Pélopos lípontes,
paĩdes íphthimoi Díos ēdè Lḗdas,
eùnóōi thúmōi prophánēte, Kástor
kaì Polúdeukes,
oì kát eùrēan khthóna kaì thálassan
paĩsan èrkhesth’ ṑkupódōn ep’ ìppōn,
rē̃a d’ anthrṓpois thanátō rúesthe
zakruóentos,
eúsdúgōn thrṓiskontes ep’ ákra náōn
pḗlothen lámproi próton’ ontrékhontes
argaléai d’ en núkti pháos phérontes
nãï melaínai;
Come to me all the way from Pelops’ island,
powerful sons of Zeus and Leda,
make your appearance with a kindly soul, Kastor
and Polydeukes!
You ride over the wide earth and the entire sea
you rescue men with ease from death
due to freezing,
leaping from afar to the tops of their well-benched ships,
shining brightly as you run up the forestays;
to that in trouble in the night you bring light,
to the ship in darkness.
We shall now consider both linguistic and philological equivalences with Sanskrit usages:
• paĩdes = putra $\to$ son; This occurs in the phrase “paĩdes íphthimoi Díos” $\to$ the powerful sons of Zeus (and Leda). That parallels the phrase: divo napātā vṛṣaṇā: the manly offspring of Dyaus.
• We render thúmōi as soul. The thumos is a cognate of dhūma is Skt (= smoke/steam going back to PIE with same meaning). In Greek, one of its meanings, breath, is related to the original meaning, from which we get soul. The equation of soul and breath is also seen in H tradition: For instance, prāṇa is called the “soul”. The other Skt word ātman is related to an old IE word for breath (e.g., German Atem= breath).
• eùrēan khthóna $\approx$ uruvyachasam pṛthivīm. The first word is an exact cognate of uru = wide. The khthóna= kṣmā (kṣa) $\to$ earth;
• ṑkupódōn ep’ ìppōn parallels the phrase used for the Aśvin-s in RV 1.117.9 and RV 7.71.5: āśum aśvam: swift horse; podon = padam = foot; āśu = ōku $\to$ swift; ippos = aśvaḥ $\to$ horse. A comparable phrase is used by Gṛtsamada Śaunahotra in his spell for the chariot: āśavaḥ padyābhiḥ in RV 2.31.2: with swift steps/feet.
• rúesthe $\approx$ rakṣathaḥ; c.f. rakṣethe dyubhir aktubhir hitam in RV 1.34.8: you protect through day and night. The protection at night is also mentioned in the Greek hymn(below).
• núkti = nakta (0-graded to aktā) = night; pháos = bhAs $\to$ shine/light. phero > phérontes = bhara = to bear; náōn= nāvam $\to$ boat. This protection offered to sailors by the Dioskouroi is mirrored in the marine rescue of Bhujyu stranded at sea that is mentioned in the śruti: yad aśvinā ūhathur bhujyum astaṃ śatāritrāṃ nāvam ātasthivāṃsam | RV1.116.5: when, Aśvin-s, you ferried Bhujyu to the shore after he mounted your ship of a hundred oars.
Figure 2. Castor and Pollux on a coin of the Roman republic with the ship on reverse.
## Some notes on the Indo-European aspects of the Anatolian tradition
Prolegomenon
This section is primarily for students of the old religion who approach it from the Indo-Aryan direction and tend to be less aware of the West Asian material. The Anatolian branch of “Indo-European” (in quotes because this appellation becomes inaccurate once Anatolian is brought into the picture; see below) has no living representatives. Modern linguists usually recognize 5 major branches: Hittite, Luwic, Palaic and Lydian, of which the Luwic dialect of Pisidian is the last attested from around 1-200 CE. In 1812 CE, Burckhardt discovered Anatolian monuments with strange hieroglyphic inscriptions in an unknown language. Eventually, this language was deciphered as Luwian. The archaeological excavations conducted by Winckler and Makridi at Boghazköy in 1906 CE led to the discovery of the erstwhile capital of the great Anatolian power, the Hittites. This site yielded a massive library of thousands of clay tablets in the cuneiform script originally developed for the Sumerian language. However, the language of these texts was clearly neither Sumerian nor Akkadian (a Semitic language that adopted the cuneiform script). In 1915 CE, Hrozny made a major breakthrough in the grammatical structure of the language by recognizing that it had vibhakti-s similar to old Indo-European (preserved in an archaic form in Sanskrit). These corresponded to sambodhana (vocative), prathamā (nominative), dvitīyā (accusative), tṛtīyā (instrumental), caturthī+saptamī (a common dative-locative), pañcamī (ablative), and ṣaṣṭhī (genitive). This finding led to the realization that the Hittite language might be an Indo-European one. In the following years, relatively easy texts were deciphered, and over time an increasing diversity of texts, spanning religion, politics and administration, were at least partially understood.
These developments have led to the unequivocal realization that Anatolian is a branch of the Indo-European family. However, its grammatical structures and linguistic features suggest that it was the earliest known branch of the “Indo-European family”; hence, the more correct term for the hypothesis describing the family would be Indo-Hittite. Linguistic phylogenetic analysis strongly suggests that the next branch to diverge from the stem was Tocharian. Archaeogenetic evidence is consistent with the progenitor of this branch corresponding to the Afanasievo Culture, which branched off from the early Indo-European Yamnaya Culture in the Caspian-Pontic region and rapidly moved eastwards by around 3300 BCE. On the western steppe, the remaining early Indo-Europeans interacted and admixed with the European farmers, represented by the Globular Amphora culture, to give rise to the clade that might be termed “core Indo-European”. These had begun rapidly splitting into the stems of the other major Indo-European lineages (e.g., Italo-Celtic, Greco-Armenian, Germanic, Balto-Slavic, and Indo-Iranian) latest by 3000 BCE. Over the next 1000 years, they launched several invasions radiating out of their homeland to cover much of mainland Eurasia. Together, these observations would mean that Hittites were not part of these Indo-European expansions but represent an early movement that happened prior to 3300 BCE.
So far, neither archaeogenetics nor archaeology has given any definitive clues regarding how the Anatolians reached their destination from the steppes. While both routes, via the Balkans and the Caucasus, have been proposed, there is currently sparse evidence in support of either scenario. Given that the above-mentioned branches of Anatolian are restricted to Anatolia and its immediate environs, the divergence likely happened in situ. Given the degree of their divergence, one may conservatively infer that they had arrived in the region sometime between 2800-2300 BCE, if not earlier. However, the actual records of Anatolian are later than that — Hittite words are first seen as loans in the records of Akkadian (an extinct Semitic language) businessmen operating in Anatolia from around 1900-1800 BCE. The Hittite kingdom emerged even later — only around 1700 BCE, with the names of their first great kings, Labarna I and Hattusili I, being recorded a little after that. This first kingdom of the Hittites lasted till around 1500 BCE. Between 1500-1380 BCE, the Hittite lands were dominated by Hurrian rulers, who were probably aided by Indo-Aryan warriors from the Sintashta-Andronovo expansion. In 1380 BCE, the Hittites made a comeback and waged war against the Hurrian state of the Mitanni led by an Indo-Aryan elite (e.g., their king *Sātavāja>Sattivaza), who were likely in an alliance with the Egyptians and had arisen to considerable power between 1600-1500 BCE. The treaty between Suppiluliuma I and Sattivaza is famous for listing the Indo-Aryan gods, Mitra, Varuṇa, Indra and the Nāsatya-s.
The aggressive military action of these new Hittite kings eventually led to the collapse of the Mitanni kingdom to their east; however, their growing power brought them new rivals, such as the Egyptians. The transport of Egyptian prisoners to their capital is suspected of having transmitted a disease. As the Hittites were weakened by the epidemic, which lasted 20 years, an alliance formed against them in western Anatolia led by the Arzawa, who spoke a distinct branch of Anatolian (Luwic or Lydian), several Hittite vassals and the Mycenaean Greeks. While the Hittites were wilting from the disease and the attack, they are believed to have used biological warfare by sending infected rams to the Arzawan alliance. In the aftermath of this event, the Hittites finally turned the tables on these rivals in the final phase of the reign of Mursili II. With this victory and the epidemic drawing to a close, the Hittites reached the climax of their power around 1300 BCE. However, this intensified their conflict with the imperial Egyptians — they fought a great chariot battle at Kadesh, but neither side could gain a decisive victory in the war. Thus, they settled for a marriage treaty in 1270 BCE. New enemies arose in the East in the form of the aggressive Assyrians, who had occupied the former Mitanni lands and waged destructive wars on the Hittites. In 1237 BCE, the Assyrians led by Shalmaneser I and Tukulti-Ninurta I defeated the Hittites in a major showdown at Nihriya, which was perhaps in the vicinity of the upper reaches of the Balih River. The Assyrian emperor Tukulti-Ninurta I then forced the Hittites to stop aiding the Kassites and conquered Babylon. While the Hittites continued to retain control over the Anatolian heartland, their power declined after this rout, and they were destroyed around 1170 BCE by unknown invaders. It is conceivable these invaders had some connection to an Iranic group (perhaps related to the Hakkari stelae) that came down from the steppes to the North.
The Anatolian languages were proximal to several distinct languages. When they arrived in Anatolia, they appear to have conquered a pre-Indo-European people, the Hattians, who spoke the Hattic language. This language might have descended from the ancestral language of the Anatolian farmers. Hattic influenced Hittite and was used alongside it. There are bilingual texts; for example, in one called “When the Storm-God thunders frightfully” following the ritual injunctions in Hittite, the ritualist is called to recite some Hattic incantations. Then there were the Urarto-Hurrian languages of unclear affinities that were spoken by the Hurrians. Several texts were translated from Hurrian into Hittite. The use of the cuneiform script and geographic proximity brought them in contact with the Sumerian language; Sumerian logograms were often used for Hittite words. To the East, the successors of the Sumerians, the Akkadians, who spoke an East Semitic language also influenced the Hittites and they deployed Akkadian logograms in their written language. To their south, their contacts with Egypt brought them into the sphere of Egyptian, a distant cousin of Semitic within the Afro-Asiatic family.
In addition to these local languages that preceded the presence of the Hittites in the region, there were the two core Indo-European languages that appeared in the locale as a result of their later expansions. To the West, the Greeks appear to have closely interacted with the Lydian and Luwian branches as part of the Arzawan alliance (probably the Greek memory of this event relates to the Trojan war that many believe relates to their attack on the Hittite province of Wilusha). To the East, the Hurrian state of the Mitanni had an Indo-Aryan elite, which appeared in the region by at least 1800 BCE (probably the western branch of the same Indo-Aryan group that conquered India). In the Kizzuwatna kingdom (today’s southern Turkey), which was allied with the Mittani before their conquest by the Hittites, we again find some of the kings or elites, such as Pariyawatri (<Paryavatri) and Śūnaśūra of likely had Indo-Aryan ancestry. Similarly, other Indo-Aryan (*Devātithi, *Subandhu, *Sumitra and *Suvardāta) and Iranic chiefs (Vidarṇa) were also operating in the Levant and Syria to the East and the Armenian states of Hayasa and Isuwa through the period of 14-1200 BCE. The direct contact with the Hittites is indicated by the Indo-Aryan loans seen in the famous equestrian manual of Kikkuli from the Hittite lands. Moreover, as suggested by Mayrhofer and Petrosyan, the theonym Akni found in a Hittite source and identified with the Sumerian Nergal/East Semitic Erra (fiery god; literally the scorcher) was most probably the Indo-Aryan Agni. Given the evidence for the Indo-Aryans in the Pontic steppes (Sindoi and Maeotians), it is not clear if they arrived in West Asia in a single invasion or multiply via the Caucasus (given their Armenian presence) from a base in the North.
Thus, in addition to their early divergence (usually linked to their retention of the laryngeals), their long presence in Anatolia with several neighboring cultures resulted in the Anatolian languages acquiring some peculiarities setting them apart from the rest of the Indo-Europeans. One example of this is the ergative formation (like Hindi and other Apabhramśa-s in India) that was probably acquired from Hattic. This influence also probably resulted in the loss of the feminine gender and the development of a new saptamī-like vibhakti, which has been termed the allative (could also be Semitic influence). Other simplifications are also seen in parallel with some of the later IE languages, such as the loss of the dual number and a reduced verb gradation — for instance, Hittite has a verbal distinction comparable to that between parasmaipada and atmanepada but does not have a true passive. Likewise, Hittite has only a single preterite and lacks the complex gradation of the past tense seen in the ancestral core IE. Moreover, most verbs conjugate comparably to Sanskrit asmi. Nevertheless, the Indo-European form is quite recognizable for several words. Below, we tabulate some well-known examples (it is not clear if the Hittite s was pronounced as s or ś; hence we simply render it as s):
Hittite Sanskrit Comment
ĕsmi asmi I am
ĕssi asi you are
eszi asti s/he is
asanzi santi they are
estu astu may he be (Skt loṭ: imperative)
asantu santu may they be (loṭ)
esun āsam was (Hittite preterite; Skt laṅ)
paah-si pāhi protect (loṭ)
dah-hi dhiye take
frequentative)
hartkas ṛkṣas bear (Ursus)
yugan yugam yoke
tāru dāru wood
nĕpis nabhas cloud
hastai asthi bone
The dynamics of the IE conquests were evidently related: 1) the mass of the mobilization in each of the invasions; 2) the density of the local populations and the resources they could command; 3) Potential military alliances with local groups. The core IE conquests in Asia and Europe can be loosely compared to those of the much later Chingizid Mongols — they were rapid and vast in their scale, often overthrowing and dominating deeply entrenched and densely populated agrarian centers. This evidently implies an effective military apparatus, even though we do not fully understand all its dimensions and how it was applied. In the first phase of the conquest of Europe, it is conceivable that a mixed economy combining some farming (probably related to the interaction with the Globular Amphora Culture) and mobile pastoralism provided the backbone for their military strategy. The latter evidently involved a degree of horse- and cattle-drawn transport. The second phase of the expansion, which also provided a new impetus throughout the rest of the IE world, was probably dependent on the invention of the spoked-wheel chariot and the breeding of superior horses by the Aryan branch. Both waves of core IE expansions were associated with either large scale replacement of the pre-IE populations (in places like Scandinavia or Central Asia) or the incorporation of the pre-IE populations (accompanied by admixture) within a new IE framework (e.g., Southern and Central Europe, India and East Asia). In contrast, the Anatolian conquest was apparently more gradual. This might reflect the fact that the Anatolians diverged at a relatively early stage before the more effective versions of the IE “military package” were in place. Moreover, they were potentially a smaller invading force entering a territory with long-established sedentary populations with aggressive military capabilities. Nevertheless, even the Anatolian version of the IE package was sufficient to allow their eventual dominance in the region.
Due to the above elements the Anatolian tradition, as it has come down to us, will necessarily be somewhat less recognizably IE in its form. This is also influenced by the workers in the field who are strongly affiliated with the study of West Asian and North African languages and traditions and have a strong Afro-Asiatic bias. While Sanskrit (starting with Hrozny) played an important role in the decipherment and apprehension of the Hittite language, the Hittitologists have paid less attention to Aryan philology in understanding the Anatolian tradition. Instead, there has been a much greater emphasis on interpreting Hittite tradition from an Afro-Asiatic perspective. There has also been a long-standing tendency of connecting the Hittite and the Greek tradition — the latest in this direction are the works of Archi, Bachvarova and Rutherford, who continue on the foundation laid by the earlier scholar Singer. This has also overlapped with the tendency to find West Asian or North African roots for various Greek traditions, even when obvious IE parallels exist — a misapprehension going back to Herodotus. While Bachvarova has correctly emphasized the need to turn to Aryan philology for understanding the later West Asian religious traditions, this aspect is quite under-appreciated in Anatolian studies, despite the repeated finding of a proximal, even if subtle, Indo-Aryan presence, in West Asia during the Hittite period.
A leader in Hittitology, Harry Hoffner, Jr, stated in the introduction to his landmark tome on Hittite mythology:
“The key to understanding any society is its living context. No amount of research into the events that transpired during its history, examination of its material remains, or analysis of its language can substitute for the intuitive understanding which comes from being a part of that era and society. Obviously, it is impossible for us to have this experience for any society of the past.”
We agree that this intuitive understanding is a key — no amount of linguistic palavering can substitute for it. While we do not belong to the Bronze Age steppe, we should emphasize that we are the only surviving practitioners of a reflex of the old IE religion quite close to its ancestral state. Thus, we are indeed in possession of a share of that intuitive understanding, which is key to the understanding of these texts. Hence, even though we are no Hittitologist, we believe that looking at the Anatolian texts with a comparative lens from an Aryan perspective is of considerable value in understanding that tradition and more generally the early IE religion. Before we move on with that, we must acknowledge that our presentation owes a debt to the translations and textual work by scholars such as C Watkins, I Singer, HA Hoffner Jr, B-J Collins, M Bachvarova, I Rutherford, JD Hawkins, J Puhvel, C Karasu and D Schwemer among many other contemporary and earlier ones. When we present their translations, we use the terms adopted them by such as “Sun God” or “Storm God”; however, it should be understood that the literal meaning of these translated terms does not carry the valence of the original deities hiding under those terms. However, we cannot do much in that regard as most of these terms stem from Sumerograms or Akkadograms whose actual Hittite equivalents might be unknown unless there are further attributes in the text.
How IE is the Anatolian tradition? We address this question by taking up many aspects of the religion as it has come down to us.
Thousands of gods
The first thing that strikes one about the Anatolian religion is that the Hittites have a large number of named gods, even by the standards of complete IE pantheons, like those of the Indo-Aryans. Now, there are three theological facets to this:
1) IE tradition acknowledges that there are a large number of gods, several thousands or more, even though only tens of them are actually named and distinctly recognized in ritual. Thus, in the Ṛgveda, Viśvāmitra states that:
trīṇi śatā trī sahasrāṇy agniṃ triṃśac ca devā nava cāsaparyan । RV 3.9.9
Thus, the number of gods is given as 3339 (also given in the Vaiśvadeva-nivid) — a number related to the synchronizing of the eclipse cycle and moon phase cycles. However, elsewhere in the RV, this number is given as 33 (with the corresponding goddesses):
patnīvatas triṃśataṃ trīṃś ca devān anuṣvadham ā vaha mādayasva । RV 3.6.9
This latter number is closer to the count of actually named gods. Hence, one could state that the thousand gods of the Hittites are merely a reflection of this. Indeed, we see a reference to a 1000 gods in a similar sense in a Hittite incantation against an imprecatory deployment (CTH 429.12):
“And you, O Sun-god, O Storm-god, O Patron-god, O [all] go[ds], with bow (and) arrow sho[ot the evil tongue], drive away the ev[il] tongues made [before the gods?]! And to the mar[iyani]-field we will take th[e]m, and bur[y] them there. And [let] them disappear from the sight of the gods: away from the Sun-god, the Storm-god, the Pa[tro]n-god, [a]nd from the Thousand Gods let them disappear.” Translation by Haroutunian.
Here, the 1000 gods appear to be a reference to the large number of unnamed gods — only three gods are explicitly named.
2) From the Indo-Aryan and Greek tradition we know that the same god might manifest as a distinctly named deity (devatā) specific to a particular incantation or a specific ritual. Thus, in the different Vedic rituals belonging to the ādhvaryava tradition of the Yajurveda the one god Indra might manifest as a multiplicity of deities, each specific to the ritual like: Indra Kṣetraṃjaya (for conquest of pastures); Indra Gharmavat (Pravargya); Indra Gharmavat Sūryavat (for prosperity); Indra Dātṛ (for amicability of subjects); Indra Punardātṛ (recovery of lost goods); Indra Prababhra (overthrow of rivals); Indra Vajrin (for abhicāra); Indra Vaimṛdha (victory in battle); Indra Indriyāvat (for attaining Indrian strength/senses); Indra Amhomuc (freedom from distress); Indra Manyumat (for performing a heroic deed in battle or capture of foes); Indra Manasvat (godly intelligence); Indra Prasahvan (when the yajamāna’s ritual cow might be seized by a raiding force or victory in Aśvamedha battles); Indra Vṛtrahan (if the new moon ritual is performed after the new moon time); Indra Marutvat; Mahendra; Indra Ṣoḍhaśin (in multiple rituals); Indra Sutrāman (Rājasūya and Sautrāmaṇi); Indra Arkavat; Indra Aśvamedhavat (if one is facing destruction or loss of power); Indra Svarāj (supremacy among rulers). This does not mean that there are 22 different gods but merely that the same god manifests as 22 devatā-s specific to the respective incantations and rites. Further, incantation-specific deification might be extended to items that are not gods, such as the soma-pounding stones or the ritual grass. A comparable tendency is also recorded in the Hittite tradition. In the above list of Indra devatā-s, those with the ancient Indo-European -vant/-mant suffixes are most common. This usage is also seen for other devatā-s (e.g., for Agni devatā-s we have Agni Anīkavat and Agni Tantumant) in the ādhvaryava tradition. We also observe similar theonyms of Hittite deities that we believe stem from a comparable principle. For instance, we have Inarawant (note parallelism to Vedic Indravant; see below), Assunawant (=endowed with excellence?) and Hasauwant (we believe this is a cognate of Skt asu-vant = endowed with life force; Prajāpati is called a related name Asumant in the Taittirīya Brāhmaṇa).
3) In Indo-Aryan, Iranian and Germanic traditions we have the many names of a god — the 300 names of Rudra in the Śatarudrīya incantation and the names of Vāta-Vāyu in the Vāyavya incantation; the incantation of the 101 names of Dātar Ahura Mazdha in the Zoroastrian tradition; the 54 names of Odinn preserved in the Gylfaginning (totally the North Germanic kennings feature at least 207 names of Odinn). This was greatly expanded in the nāmāvali-s of the later Hindu tradition starting from the epics. Thus, one unacquainted with this ancient tendency and the equivalence of the names might mistake their multiplicity for an actual multiplicity of the gods.
From a historical viewpoint, the early Hittite texts contain fewer named gods than the later ones from close to their high point, where the list keeps growing in size. This can be seen as pantheonic accretion from associated cultures, with the addition of Hattian, Sumerian, Akkadian, Hurrian and even Indo-Aryan deities to the mix. However, this does not mean that the accretion proceeded without any identification or syncretism. One could say that a pathway for identification and syncretism was always latent in IE tradition. For example, far removed in space and time, we hear Odinn explain the multiplicity of his names in the Gylfaginning thus:
“It is truly a vast sum of knowledge to gather together and set forth fittingly. But it is briefest to tell you that most of his names have been given to him by reason of this chance: there being so many branches of tongues in the world, all peoples believed that it was needful for them to turn his name into their own tongue, by which they might the better invoke him and entreat him on their own behalf.”
When we take this into account, the Hittites probably had a relatively circumscribed pantheon of specifically recognized gods. The evidence for this comes from the Yazilikaya temple from 1300-1230 BCE. While the iconography and the name-markers of many of these deities are obviously Hattic, Hurrian or Semitic, their organization is unlike anything else in West Asia, indicating a Hittite organizing principle, which is likely of IE provenance. A total of over 80 reliefs are carved in two main chambers, A and B, of the rock-cut shrine. The more elaborate chamber A seems to have originally contained 64 figures (2 of which are largely lost), all or most of which can be identified as gods of the celestial Hittite pantheon. Chamber B with 12+3 figures and is identified with the nether world. Of these, the 12 gods seem to be identical to the 12 in chamber A and the remaining 3 are deities (apart from the Hittite king) which may also be represented in chamber A. Thus, conservatively, we may see the core Hittite pantheon as featuring 64 deities. In chamber A, the pantheon of gods and goddesses are shown as though in procession a towards the central deities, the Storm God and the Chief Goddess placed in the northern direction, from either side of the chamber. A pyramidal crag rises above these central gods — the site was evidently chosen to represent the mountain of the world axis. Chamber B in contrast represents the netherworld. One of the prominently displayed Chamber B deities is indicated by iconography related to the Sumerian Nergal, who was likely associated with Fire God (=Agni) on one hand the lord of the netherworld on the other (c.f., the Iranian relief from Parthian age Hatra where Nergal is syncretized with an Arabian netherworld deity (Zqyqa) and an Iranic deity and shown holding the tricephalic Kerberos in a manner similar to the Greek Herakles. Like Agni, he holds an axe — a characteristic IE feature).
Figure 1. The Yazilikaya pantheon from “Celestial Aspects of Hittite Religion: An Investigation of the Rock Sanctuary Yazılıkaya” by Zangger and Gautschy, JSA 5.1 (2019) 5–38
Keeping with the world axis symbolism, as has been proposed before (e.g., Zangger et al., most recently), we agree that the organizing principle is astronomical, with symbolism likely derived from IE tradition. On the god side, the procession opens with 12 identically depicted gods — these have been identified with the deities of the 12 months of the year — a number also reflected in other IE traditions, like the Greek Dodecad of Olympians and the 12 Āditya-s of the para-Vedic Hindu tradition. RV 10.114.5 also mentions the offering of 12 soma cups, implying that they are for a count of 12 gods. The 28th and 29th figures of this pantheon are identified as the bulls of heaven (Hurris and Seris in Hurrian), who draw the chariot of the Storm God. These hold up a large lunar symbol; thus, they likely represent the point of the full moon and the duration of the lunar month (since both 28 and 29 hold up the moon symbol, it is likely that both the synodic and sidereal months are implied). This mapping of the gods with the lunar cycle is also seen in the Indo-Iranian world; hence, the Yazilikaya frieze is likely a depiction of the Hittite reflex of the same ancestral tradition. The goddess side of the procession opens with 18 or 19 female deities. Zangger et al propose that this corresponds to the 18/19 year eclipse/lunar cycle — this might again present a mapping related to the number of gods in the RV.
The Storm God
The Storm God of the Anatolians went by the name: Tarhunna (Hittite); Tarhuwant>Tarhunz (Luwian). His name is a cognate of the Sanskrit Tūrvant (e.g., applied to Indra: sanīḻebhiḥ śravasyāni tūrvan marutvān no bhavatv indra ūtī । RV 1.100.5). Some have proposed that, while it has a clear IE etymology, it might have been adopted to mimic Taru the name of a functionally similar Hattian deity. However, we propose (also apparently favored by Schwemer) that it was transferred from IE to Hattic. We suspect that the Anatolian theonym has an etymological equivalence to the Germanic Indra-class deity. As the Indra-class deity of the Anatolian branch he was identified with a wide range of local, functionally similar deities of cities. In terms of the more widely distributed gods, we can see his identification with the Hurrian Teshub and Semitic (H)Adad. The Hittite exemplar in the Yazilikaya temple is not shown with prominent horns. However, elsewhere his Hittite images (e.g., Mursili III’s seal) and the Luwian depictions frequently show the characteristic bovine horns. While we cannot be sure where this iconographic convention originated, it is clear that it was already widespread across bronze age Eurasia, encompassing, the Bactria-Margiana complex in Central Asia, the Harappan civilization in India, and Mesopotamia and the Anatolian-Hurrian world in West Asia. The same iconography is also textually alluded to in the RV for Indra and other deities (Agni, Rudra), e.g., yas tigmaśṛṅgo vṛṣabho na bhīma ekaḥ kṛṣṭīś cyāvayati pra viśvāḥ ।. Hence, we can say that even if the specific features might have been local, the horned iconography for this deity was likely rather naturally adopted by the Anatolians as they might have had a certain “pre-adaptation” for the same from the ancestral IE tradition. He is also often shown standing on a bull, which is aligned with the frequent references to Indra as the bull. Indra was decoupled from this iconography in the later Hindu world; however, it persisted in association with Rudra who also shows that connection even in the śruti.
Figure 2 Anatolian and Mittani depictions of the Storm God
In terms of weapons, he is depicted as bearing a mace (comparable in form to the classic Indo-Iranian gadā) in the Yazilikaya temple and on the famous seal of the Hittite king Mursili III. A comparable mace is also held by the Hurrian Storm God from a seal from the early Mitanni realm. In this version, he also holds a spear and is shown trampling mountains, suggesting the possible influence of the Indo-Aryan Indra, the terror of the mountains. He also holds a spear while fighting the famous serpent demon in the Luwian site of Malitiya (Arslantepe). The Luwian versions from Malitiya and elsewhere, and the version from the Aleppo temple in Syria show him as holding a trident and sometimes also an axe in the other hand. The axe is reminiscent of one of the types seen on the Yamnaya anthropomorphic stelae suggesting potential IE influence. The trident on the other hand with its wavy prongs is a representation of the famous thunderbolt. We posit that both the mace and the trident are alternative visualizations of the same weapon — the cognate of the Aryan vajra. Some of these iconographic conventions first seen in the Anatolian exemplars persisted till much later in India (the trident-like vajra of Indra, the axe and the triśūla of Rudra) and the Roman empire (Jupiter the thunderer slaying the anguipedian = snake demon and Jupiter Dolichenus; see below). The repeated adoption of this iconographic convention by different IE branches supports an IE inspiration or, in the least a compatibility, following the ancient spread of the convention similar to the horned headgear of the deity. The version from the Aleppo temple also shows him bearing a sword on his belt in addition to the axe and trident. This is reminiscent of the later anthropomorphic stelae from IE sites on the steppes.
Both in Luwian iconography and Hittite mythological texts we have depictions of the Storm God slaying the serpent demon (Hittite: Illuyanka). This myth is found in every branch of IE; thus, it unambiguously belongs to the ancestral stratum of IE mythology. Its Hittite variants mention: 1) baiting of the serpent demon with food: A parallel is found in the Kaṭha Saṃhitā where Indra takes the form of a glob of honey to draw the serpent demon Śuṣṇa to eat it up. 2) At least one Hittite version states that the serpent demon has stolen the heart and the eyes of the Storm God, i.e., something essential for life. He has to then be tricked into giving those back. The Kaṭha Saṃhitā similarly implies Śuṣṇa had stolen the ambrosia (amṛta) of the gods. Indra takes it back by entering his maw in the form of a glob of honey. He then flies out with it in the form of an eagle (a famous IE myth). 3) A preserved Hittite myth mentions an eagle being sent to search for the vanished Storm God. However, a more direct depiction of their connection is seen on the seal of Mursili III, where an eagle is placed in front of the Storm God on his bullock cart (He is also shown holding the eagle on a silver rhyton; see below). Finally, one could also point to the reuse of the West Asian eagle wing symbol with a solar disc in IE contexts, like as the emblem above the Storm God on Luwian stelae.
Finally, the Hittites also preserved a myth of the disappearance of the sun resulting in paralyzing hahhimas (ice; cognate of Skt hima of PIE provenance). While one could imagine a winter frost in Anatolia, the concomitant “disappearance” of the sun is a motif specifically associated with more northern latitudes and is again seen across the IE world. Thus, the appearance of this myth in Anatolia is a clear sign of its IE provenance. In other IE traditions, the Indra-class deity recovers the sun, often doing battle with his vajra-like weapon against the demons, who have hidden the sun. While its details are poorly preserved, the Storm God is repeatedly mentioned in that Hittite text as confronting the freeze with other gods.
The consort and the sister of the Storm God and West Asian syncretism
The Yazilikaya temple pairs the Storm God with his consort who stands on a lion. This chief goddess of the Hittite pantheon is usually identified with the Hurrian deity Hebat and Hattic Wurusemu. We have a remarkable sūkta-like incantation (CTH 384) composed by the ritualist-princess Puduhepa (wife of king Hattusili III), the “rājarṣikā” among the Hittites:
1. O my lady, Sun Goddess of Arinna, lady of the Hatti lands,
2. Queen of the heaven and the earth!
3. Sun Goddess of Arinna, my lady, queen of all the lands!
4. In the Hatti land you take (for yourself) the name of the Sun Goddess of Arinna,
5. but besides (in the land) that you made the Cedar Land (Hurri),
6. you take (for yourself) the name of Hebat.
7. However I, Puduhepa, (am) your maid from the outset…
(translation from Karasu)
Thus, we see that Puduhepa identifies the Hittite Sun Goddess, the queen of heaven and earth (a dvandva like Dyāvāpṛthivī), with Hebat of the Hurrians. On the Hurrian side, we see no evidence for Hebat being the Sun Goddess. On the Semitic side, epithets comparable to those used for the Anatolian Sun Goddess are used in Akkadian for Shamash the solar god rather than for a goddess. However, in the IE world, we see multiple manifestations of the solar goddess (e.g., the whole Indo-Aryan marriage ritual is centered on her). Thus, we posit that the Sun Goddess was inherited from the Anatolian IE tradition, and Puduhepa identified her with Hebat, not due to solar connotations, but because she was the supreme female deity of the Hurrian tradition. Hence, it is probable that the Hittite interpretation of the consort of the Storm God corresponded to their Sun Goddess. In terms of her iconography, she rides the lion — this convention, like that of the horned headdress of the Storm God, has also spread widely across Eurasia encompassing BMAC, Sumeria and its Semitic successors, and Anatolia. A direct parallel can be seen in an Akkadian seal, where the consort of the Semitic Storm God rides in front of his cart on a lion hurling rain or lightning. In textual terms, the large felines (lion, tiger and leopard) are associated with the supreme mother goddess Aditi in the early Vedic layer of the Indo-Aryan tradition.
Figure 3. The consort of the Storm God and the mirror-wielding goddesses
The pairing of the bull-riding Storm God and the lion-riding goddess was an iconographic convention that traveled widely over space and time. In the East, it manifested in the iconography of Rudra and his consort Umā (Rudrāṇī) in India. In the West, it formed the basis of images of Jupiter Dolichenus in the Roman empire. Another Anatolian goddess, who rode a lion, was identified with the ancient goddess Kubaba of the Mesopotamian world. Here name is likely also behind the theonym Cybele, a later goddess from the region, who is iconographically comparable. Interestingly, she is associated with the Anatolian Rudra-class archer deity Santa (see below) in certain texts. Her distinctive feature in the Anatolian world is the mirror, which she shares with Rudrāṇī in India, Tapatī (Tabiti) in the steppe Iranic world, and Juno Regina Dolichena, the consort of Jupiter Dolichenus in the Roman empire. In the Far East, the mirror as an attribute of the goddess was also transferred to the Japanese solar goddess probably from a steppe Iranic source. This mirror iconography is primarily seen in the Luwian reflexes of the Anatolian religion (e.g., at Carchemish), where this goddess might have been identified or syncretized with the supreme Hittite goddess. Consistent with this, like her consort, she may be shown with the cow horns in some depictions. Indeed, such a Luwian pairing might have been the ultimate inspiration for the Dolichenian deities. Given that the mirror is not typical of Mesopotamian or North African goddesses, we posit that the mirror was probably acquired from an Aryan source relatively late in the development of the Anatolian religion. Nevertheless, its eventual wide adoption across the IE world suggests that it resonated with a deeply rooted solar aspect of the goddess.
Finally, we come to the third major Eurasian goddess called Innana in the Sumerian realm, Ishtar by their East Semitic successors (= West Semitic Ashtart) and Shaushka/Shaushga by the Hurrians. She was evidently functionally related to a comparable goddess from the BMAC in Central Asia and probably also to the horned pipal tree goddess of the Harappans. Right from her Sumerian manifestation, she is a transfunctional goddess associated with war, love and medicine. This transfunctionality made her easy to syncretize with high goddesses sharing some of these functionalities from across diverse traditions. Her transfunctionality is amply testified in the historical record: Her Hurrian iconography depicts her heavily armed, emphasizing her military nature. The Indo-Aryan Mitanni ruler Tushratta (<Tveṣaratha) sent such an image of hers with a maninnu necklace having the form of a “bed of her plant” to the Egyptian Pharaoh Amenhotep III perhaps to heal him of his illness — this exemplifies her healing aspect. Finally, the Hittite monarch Hattusili III writes that Shaushga led him to his future wife Puduhepa, the ritual expert, when he was returning from the Egyptian campaign as the commander of the Hittite army under his brother. He specifically mentions that the goddess brought them together in mutual love — exemplifying her sexual facet.
Figure 4. Shaushga and Ishtar. The drawing of Shaushga from the Aleppo is an accurate reproduction by Gestoso Singer in “Shaushka, the Traveling Goddess”, TdE 7 (2016) 43–58
Innarawant
Whether the Hittite Innara (Inara) is related to the Vedic Indra has been subject to some debate. The daughter of the Storm God named Inara is well known in the Hittite mythic tradition. A text coeval with the Yazilikaya names Inar(a) as the male god from the Hurrian land, suggesting that the knowledge of a male equivalent existed in the Anatolian world. Thus, Innarawant, which has been taken to mean strong/manly/majestic as a masculine theonym might tie the two together — Hittite often maintains homosemy between the base form and the old IE -vant augmentations. Keeping with the meaning of the name, the ritual in which the singular deity Innarawant is invoked is related to restoring the strength or manliness of the patron. This association calls to mind the Vedic term nṛmṇa (manly) used for Indra. It also reminds one of the ādhvaryava ritual invoking Indra Indrīyāvat for special strength. We list below some of the incantations and ritual actions relating to this theonym (CTH 393: “Anniwiyani’s Rituals”, transcribed and translated by B-J Collins in “Hittite Rituals from Arzawa and the Lower Land”; upper case are Sumerograms or Akkadograms):
$\S 2$ I take blue wool, red wool, barley, karsh-grain, and coriander and they roast them. One pitcher of beer, sixteen small thick breads, one goat, one puppy, fourteen pegs of poplar, two small NUNUZ-stones, fourteen small cups, and twelve small pitchers. They make all of the birds out of clay. Whichever bird the augurs observe, they do not omit any.
$\S 3$ As soon as night falls, she ties blue wool to the ritual patron—first to his feet, his hands, and his neck, his middle; to his bed (and) the four bedposts the first time. She [the auguress] ties (it) in the same way to his chariot, his bow, and his quiver.
$\S 4$ Afterwards she ties red wool in the same fashion. Then the roasted seeds, the thick breads, the implements of fired clay, the pegs, and the clay birds and the small pitchers she arranges in a pitcher. She places it under the bed on behalf of the ritual patron and it remains under the bed for him.
$\S 5$ At dawn they cut the blue and red wool off the ritual patron entirely and she places them in the basket. They bring a consecrated girl into the inner house, and they situate her in the entrance. She holds a bird of dough in her hand. The consecrated girl calls, “Go away Protective Deity Lulimi! Come in Protective Deity Innarawant!”
This is followed by dog and goat sacrifices to Innarawant.
The Hittite texts also show a plural form of the Innarawant — the Innarawantes deities — these accompany the fierce archer deity Santa/Sanda who is the bringer of epidemics. This is recorded in the ritual of Zarpiya, the physician of Kizzuwatna (CTH 757), in Hittite and Luwian that is performed when an epidemic strikes the land (which might be related to the great epidemic that swept through the Hittite empire). In that, the ritualists utter an incantation (translation based on those by Collins and Schwartz): “$\S 11$ O Santa (indicated by the Marduk Akkadogram) and the Innarawantes deities, do not approach my gate again.” The Luwian part of the text calls upon these gods to evidently eat the sacrificial sheep or cattle and not the men: “$\S 17$ Do not again approach this door in malice. Eat sheep and cows; do not eat a man, zaganin, tuwiniya.”. The Innarawantes accompanying Santa are rather notably described thus:
$\S 8$ They bring in one billy-goat and the master of the estate libates it with wine before the table for Santa. Then he holds out the bronze ax and recites: “Come Santa! Let the Innarawantes-deities come with you, (they) who are wearing blood-red (clothes), the mountain-dwellers, who are wrapped in the huprus garments;
$\S 9$ who are girt (?) with daggers, who hold strung bows and arrows. “Come and eat! We will swear (an oath to you)…”
In addition to an animal sacrifice, the ritual involves the offering of 9 libations of wine and 9 offerings of bread. Then 8 virgin boys are called in and one wears a goatskin cloak (c.f. cloak of the vrātya) and howls like a wolf. The others follow him, and they eat the sacrificial meat like wolves. This suggests that the total number of deities in this part of the ritual is 9 = 1 Santa + 8 Innarawantes.
Thus, the cast of the epidemic-associated archer deity Santa and his fierce Innarawantes companions brings to mind the Indo-Aryan Rudra and the Rudra-s or Marut-s. Some specific points include: 1) The term Innarawantes in the plural brings to mind the epithet of the Marut-s, Indravant: ā rudrāsa indravantaḥ sajoṣaso hiraṇyarathāḥ suvitāya gantana । RV 5.57.1; 2) The Innarawantes are described as being like mountain-dwellers, an epithet used for the Marut-s in the RV: pra vo mahe matayo yantu viṣṇave marutvate girijā evayāmarut । RV 5.87.1; 3) The special mention of their garments in which they are wrapped reminds one of the RV epithets for the Marut-s focusing on their armor and their ornaments: varmaṇvanto na yodhāḥ śimīvantaḥ pitṝṇāṃ na śaṃsāḥ surātayaḥ । RV10.78.3; naitāvad anye maruto yatheme bhrājante rukmair āyudhais tanūbhiḥ । RV 7.57.3; 4) Their being heavily armed again matches the descriptions of the Marut-s: vāśīmanta ṛṣṭimanto manīṣiṇaḥ sudhanvāna iṣumanto niṣaṅgiṇaḥ । RV5.57.2; 5) More tenuously, the participation of 8 lupine youths in the ritual might be a mimicry of the Innarawantes. This brings to mind the repeated emphasis on the youth of the Marut-s in the Veda and the old count of 8 for the Rudra-s.
In conclusion, while the term Innarawant refers to both singular and plural deities we believe that the usage is consistent and reflective of an ancient connection inherited from a PIE tradition. We believe that in the singular form it reflects characteristics inherited from the archetypal Indrian deity and in the plural reflects the Rudrian archetype found in the Marut-s who show an intimate connection with the Indra-class.
Other Anatolian manifestations of the Archer deity
Santa is not the only manifestation of the archer deity in the Anatolian world. Collins points to the Hittite ritual text of the female ritualist Āllī (CTH 402) from the Arzawan locus for countering abhicāra that mentions an Archer deity likely associated with the Orion region of the sky. The opening incantation of the rite goes thus (Collins’ translation):
$\S 4$ “Then the wise woman speaks as follows: “O Sun God of the Hand, here are the sorcerous people! If a man has bewitched (lit. treated) this person, herewith he is carrying it (the sorcery) with (his own) back. May he take them back! He is carrying (var. May he carry) it with (his own) back!
$\S 5$ If however, a woman has bewitched him, you O Sun God know it, so it should be a headdress for her, and she is to put it on her head. May she take them back for herself! It should be a belt for her, and she is to gird herself; it should be for her a shoe, and she is to put it on!”
These incantations are followed by the invocation of the Hunter:
$\S 8$ The Sun God of the Hand and the (divine) Huntsman (are) in front. He (the Huntsman) has his bow [and] he has his [arr]ows. For his dogs let it be bread. [For] the [h]orses let it be fodder. And for the ritual patron [let it be] figurines of clay.” The wis[e woman] puts [them] (the ritual figurines) down.
This is followed by the winding of the blue-red wool (see above) around ritual figurines and their burial. The ritual figures are shown carrying “kursa-s”, which are thematically equivalent to valaga-packages in the Indo-Aryan world.
This is followed by the below ritual actions and incantations:
$\S 21$ She steps a little away from there, and at the side of the pit breaks one flatbread for the Dark Ones. Those who turn before the Huntsman, (for them) she (the wise woman) breaks a flatbread with the miyanit tongue. She breaks one flatbread for the dark earth; she breaks one flatbread for the Sun God and recites: “You must guard this!” She breaks one flatbread for the Sun God and places it on the ground. She libates beer before the gods. And she says: “You must keep this evil witchcraft fastened (in the earth)!”
$\S 22$ She steps back a little and breaks one flatbread for Ariya and places it to the right of the road. She libates beer and says: “You, seize this evil and do not let it go!” She breaks one flatbread for the crossroad and places it to the left of the road. She libates beer and says: “You, gods of the road — the evil — guard it! Do not let it return!”
$\S 23$ She steps forward a little and breaks one flatbread to the salawana-demons of the gate. She sets it down, libates beer, and recites: “Upward [ … ] may you always say good things! GALA-priests, [you] lock up the evil (words/things)!” She breaks a pitcher, and they enter the city.
$\S 24$ She puts kars-grain, passa-breads, a bow, and three arrows in a basket and places them under the bed. It remains under the bed (overnight). She ties a strip of wool to the head and foot of the bed.
$\S 25$ On the second day, when it becomes light, she takes the basket out from under the bed, waves it back and forth over the person, and speaks: “O Huntsman, you return the sorcery to the sorcerer! Let it be your cure!” She cuts the wool from the bed and places it in the basket.
In general terms, these incantations are notable for the following points: 1) It invokes a solar deity translated as “Sun God of the Hand” — this brings to mind the major Indo-Aryan solar deity Savitṛ whose hands are a prominent feature (c.f. Yajus incantation: devo vaḥ savitā hiraṇyapāṇiḥ pratigṛhṇātu ।; hiraṇyapāṇim ūtaye savitāram upa hvaye । RV 1.22.5 ). 2) Multiple incantations in this ritual have a resemblance to the Atharvan pratyaṇgirā incantations where the kṛtyā is sent back to the sender (e.g., the yāṃ kalpayanti… ṛk). 3) The statements, “you O Sun God know it” and “You must guard this!” are reminiscent of the Atharvan anti-kṛtyā incantation invoking the sun: sūrya iva divam āruhya vi kṛtyā bādhate vaśī । AV-vulgate 8.5.7 (Like the Sun ascended the heaven, blocks sorcery with might.) 4) Here again, we see the use of the blue-red threads; this is similar to the use of the nīla-lohita wool is used in the Indo-Aryan marriage ritual to block the kṛtyā (sorcery): nīlalohitaṃ bhavati kṛtyāsaktir vy ajyate ।AV-vulgate 14.1.26 (Tr: The sorcery becomes the blue-red thread; the sorcery which clings [to the bride] is driven off).
The most notable feature of this ritual is the invocation of the archer deity who goes by the epithet the “Huntsman”. We cautiously follow Collins in accepting Ariya as the likely name of the “Huntsman”, which, in turn, is related to the Greek Orion. The etymology of the Hittite Ariya and Greek Orion remains unclear. However, it is possible that both are related to the PIE root, which is behind forms such as: Hittite arāi (rise up); Tocharian A ar- (bring forth); Avestan ar- (set into motion) $\to$ comparable to Sanskrit iyarti (liṭ form āra or bhāvakarman for arye; go forth); Greek ornūmi (set into motion); Latin orior (to proceed from source), orīgo (origin). Over a century ago, Lokamanya Tilak had boldly proposed that on the Indo-Aryan side the terms Āgrayaṇa or Agrahāyana might represent a cognate of Orion suggesting, just like Collins for Hittite, that “a” in Indo-Aryan can be seen as validly corresponding to the Greek “o”. While one could question the direct etymological homology of Āgrayaṇa and Orion, Tilak’s semantic equivalence might still be valid. The term Āgrayaṇa arose because Orion in the PIE days stood close to the equinoctial colure in the PIE days — it was the leader of the constellations even as Kṛttikā ( $\sim$ Pleiades) was in the later times. Thus, the Orion/Ariya could have derived from the root related to the “origin” or the point from which the sun goes forth on its journey starting with the vernal equinox. Hence, it is even possible that the terms Āgrayaṇa or Agrahāyana were adopted as semantically appropriate homophones of an ancient word that was a cognate of Orion/Ariya. In this regard, we should point out that the constellation of Mṛgaśiras ( $\sim$ Orion; see below) was apparently known by the name Āryikā in Sanskrit lexicographic manuscripts āryikāstu mṛgraśiraḥ śiraḥ sthāḥ pañca-tārakāḥ ।). However, this manuscript has not been published to confirm the reading (it was also recorded by German Indologist Albrecht Weber). If this reading is upheld, then it might represent the survival of a name of the constellation linking it to the Hittite and Greek versions.
The evidence from the Greek, Iranian, and Indo-Aryan sources suggest that the association of the Orion region of the sky with the Rudrian deities goes back to the ancestor of core IE. Even if the Ariya etymological link does not hold up, there are other features of the Hittite ritual which link the Huntsman to the Orion region of the sky and to the core IE archetype of the Rudra-class deity. Greek, Iranian and Indo-Aryan sources concur that this part of the sky was associated with dogs and an archer/hunter — this association is recapitulated in the Hittite incantation. Rudra is both specifically associated with dogs and is the hunter of the god Prajāpati ( $\sim$ constellation of Orion), who may take the form of a deer. In the Greek tradition, Orion’s death is brought about by the Rudrian deities Artemis and/or Apollo. In one well-known narration of the myth, Apollo directs his sister Artemis to shoot Orion with an arrow. On a painted Greek pot, Apollo is shown killing Orion as he tries to assault Artemis. In other versions, Artemis shoots him down on her own or apparently kills him with a cakra; in yet another, either she or Apollo kills him with a scorpion (constellation of Scorpio) or a snake (depicted on Greek pottery).
There are further parallels between the Greek and Indo-Aryan traditions regarding Orion. The first relates to the myth wherein the goddess Eos and Orion were to join in a liaison. The gods objected to this and directed Artemis to shoot down Orion. This again presents a remarkable parallel to the Vedic tradition: The god Prajāpati was to join in an illicit incestuous liaison with the goddess Ushas (cognate of Eos). The enraged gods sent Rudra to slay Prajāpati, whose corpse is represented in the sky by the constellation of Orion. This cognate Greek and Indo-Aryan mytheme evidently preserves an astronomical allegory relating to the sun being in the vicinity of the constellation at the vernal equinox in ancient times. The other Greek-Hindu parallel relates to the myth of the blindness of Orion. Orion is said to have been blinded by Oinopion when he tried to assault a Pleiad. He then walks eastwards hoping to catch the rays of the sun so that it would cure his blindness. The Śāntikalpa of the Atharvan tradition invokes the constellation under the name the blind one (Andhakā):
ehi me andhake devī mṛdu-karmasu śobhane ॥
I invoke the boon-granting consort of the Moon, the [goddess of the] Andhakā constellation. May the auspicious goddess of the Andhakā constellation come to me for the gentle rites.
This name for the constellation evidently comes from the “blindness” demon Andhaka who was killed by Rudra. Thus, the constellation of Orion is not identified with the Rudra-class deity himself/herself, but with the target of that deity in both Hindu and Greek traditions. Hence, we cannot automatically assume that the Huntsman of the Hittite ritual is the constellation of Orion, but rather the Rudra-class deity who is linked to that part of the sky. Both the Indian and Iranian branches of the Aryan tradition concur in identifying the Rudra-class deity with the adjacent star $\alpha$ Canis Majoris (the brightest star as seen from the earth) while also identifying the asterism containing that star with a dog.
Beyond, the astronomical connection, even this relatively meager Hittite incantation offers several key connections to the Rudra-class deities in the Anatolian world and beyond: 1) As in the Hittite rite, Rudra-class deities are frequently invoked to repel/hurl back abhicāra in the Atharvan tradition (e.g., in the yāṃ kalpayanti sūkta and the bhavā-śarvīya offerings in the Mṛgāreṣṭi). 2) The horses of the Huntsman are specifically mentioned in addition to his dogs. This is mirrored in the incantation to invite Rudra to the ritual of the Īśāna-bali or Śūlagava, where his horses are specifically mentioned: ā tvā vahantu harayaḥ sucetasaḥ śvetair aśvaiḥ saha ketumadbhiḥ । vātājirair mama havyāya śarvom ॥ 3) In addition to the Huntsman, and the Sun-god of the Hand, the ritual invokes the Dark Ones (marwayanza) and the salawana demons. These two are also associated with another notable manifestation of the Archer God in the Anatolian world going by the name Runta (Dark Ones in CTH 433.2; salawana-demons in CTH 433.3). The epidemic-causing Archer God also receives another name, Iyarri, in Dandanku’s Arzawan plague ritual, where he is again accompanied by the Dark Ones. Finally, the Dark Ones are also mentioned together with Santa in a Hieroglyphic Luwian inscription. This suggests that Santa, Runta and Iyarri are all likely manifestations of the same Rudra-class deity, with the Dark Ones either being cognates of the Innarawantes of Zarpiya’s ritual or a group of beings possibly paralleling the Marut-s or the gaṇa-s or Rudra. In this regard, it might be noted that in some Kṛṣṇa-yajurveda traditions (e.g., Maitrāyaṇīya and Kaṭha) the constellation of Mṛgaśiras is assigned to the Marut-s. The association with the demons is also mirrored in Rudra being called the Asura (tvam agne rudro asuro mahodivaḥ । RV 2.1.6). Likewise, on the Greek side, Apollo is called a Titan in the incantation from the Magical Papyrus for the ritual that was performed at sunrise when the moon is in Gemini.
4) A key connection to the Rudra-class deities is seen in the injunction to make the beer and bread offering to the deity at crossroads. This has a close parallel in the autumnal, disease-curing Vedic Tryambaka-homa:
tānt sārdham pātryāṃ samudvāsya । anvāhārya-pacanād ulmukam ādāyodaṅ paretya juhoty; eṣā hy etasya devasya dik; pathi juhoti; pathā hi sa devaś carati; catuṣpathe juhoty; etad dha vā asya jāṃdhitam prajñātam avasānaṃ yac catuṣpathaṃ tasmāc catuṣpathe juhoti ॥ Śatapatha Brāhmaṇa 2.6.2.7
Having collected all (the cakes from the potsherds) into one dish, and taken a fire-brand from the Anvāhārya-fire, he walks aside towards the north and offers — for that is the direction of the god (Rudra). He offers on a road — for on roads the god roves. He offers on a cross-road — for the cross-road, indeed, is known to be his customary haunt. This is why he offers on a cross-road.
This connection is also seen on the Greek side: The Apollo devatā, Apollo Agyieus (literally Apollo of the road), was worshiped as the manifestation of that deity associated with the road. Like the Hindu Rudra in the classical age, he tended to be worshiped aniconically in the form of liṅga-s. Further, the goddess Hecate, who likely emerged as an ectype of Artemis, is specifically associated with crossroads. 5) The use of a bow and three arrows in the ritual has a specific parallel in the ritual for the Rudra-class deity in the Indo-Aryan soma ritual. After the five-layered altar is piled in the somayāga, a major series of oblations are offered to Rudra with Yajuṣ-es and Sāman-s. In course of this, after the Śatarudrīya oblations are made, another is offered with the famous mantra “yo rudro agnau…” Then the sacrificer or another brāhmaṇa takes up a bow and three arrows and goes around the altar even as the incantation paying homage to Rudra to ransom the sacrificer from the god is recited. The Yajus texts explain it thus:
rudro vā eṣa yad agnis; tasya tisraḥ śaravyāḥ pratīcī tiraścy anūcī । in Taittirīya Saṃhitā 5.5.7
This fire is indeed him, Rudra. His missiles are three — one that comes straight on, one that strikes transversely, and one that follows up.
Indeed, this triplicity of Rudra’s arrow is explicitly connected with the slaying of Prajāpati (Orion) — he was pierced by the trikāṇḍa (tripartite or triple-headed) arrow standing for the 3 stars of Orion’s belt (Skt: Invakā-s) in Aitareya Brāhmaṇa 3.33 and:
atha yasmān nā mṛgaśīrṣa ādadhīta । prajāpater vā etac charīraṃ; yatra vā enaṃ tad āvedhyaṃs tad iṣuṇā trikāṇḍenety āhuḥ sa etac charīram ajahād; vāstu vai śarīram ayajñiyaṃ nirvīryaṃ tasmān na mṛgaśīrṣa ādadhīta ॥ Śatapatha Brāhmaṇa 2.1.2.9
Now, on the other hand (it is argued) why one should not set up his fire under Mṛgaśīrṣa (Orion). This [constellation] is indeed Prajāpati’s body. Now, when they (the gods) on that occasion pierced him with what is called a tripartite arrow he abandoned that body. As that body is a mere husk, unfit for worship and sapless, he should therefore not set up his fires under Mṛgaśīrṣa.
Figure 5. Depictions of the Anatolian deity Runta
This finally brings us to a key association of the Rudra-class deities seen both in the Greek and Hindu worlds — the deer — often their target in their role as huntsmen-archers. This animal figures in the Anatolian world in the context of the archer deity going by the name Runta/(Ku)Runtiya, sometimes identified with Inar. As noted above, the iconographic correspondence and the association with the Dark Ones establishes the equivalence between Runta on one hand and on the other Santa and the Huntsman/Ariya of the above ritual. Runta is indicated by the stag-horn hieroglyph making his connection to that animal explicit. There are several notable depictions of this deity making his connection to the deer explicit:
1) In a scene depicted on an Anatolian silver rhyton, ritualists offer libations and bread to Runta standing on a stag with an aṅkuśa and the Storm God. Both gods hold eagles. The insignia of Runta, namely his quiver, two spears and the slain stag are also shown again separately.
2) In the Aleppo temple, he is shown in a procession of gods and goddesses (including the Shaushga image depicted above) holding a bow and a spear and is labeled prominently with the deer-horn hieroglyph.
3) At Yazilikaya temple Chamber A he is shown with what might be a bow and labeled again with a prominent deer-horn hieroglyph.
4) Altinyayla stele depicts him in the mountains standing on a stag with a bow and holding a stag antler even as a worshiper pours out a libation in front of him.
5) Collins also notes several seals from Nişantepe on which the same deity is similarly depicted.
6) These depictions also suggest that the deity holding a bow and spear behind the storm god on Mursili III’s seal is likely to be the same Archer God.
In conclusion, this web of connections and iconography establishes the deer-associated Archer/Hunter God of the Anatolians as the likely reflex of the Rudra-class deity inherited from the PIE tradition.
Conclusion
While Sanskrit and IE linguistics played a central role in the decipherment of the Anatolian language texts, the prevalent tendency has been to interpret the Anatolian religion quite independently of its IE background based on local West Asian and North African models. This is rather evident in the leading Hittitologist Hoffner’s tome on Hittite myths. While there is no doubt the Hittite religion was imbrued with elements from the West Asian substrata and neighbors, we hold that, with some diligence in the comparative method, one can pick out a clear IE “signal”. However, this signal might be complicated by the interactions with other IE groups such as the Indo-Aryans and Greeks who were also operating in the vicinity during the height of Anatolian power. More recently, workers such as Archi, Bachvarova, Rutherford and Collins admit the Greek connection and explore it further. However, they (to a degree, Bachvarova is an exception) tend to ignore the rest of the IE material, especially Indo-Iranian, when approaching this issue. Here, we present a preliminary redressal of that. We believe that it helps better understand the Anatolian religion and also helps reconstruct the ancestral IE tradition. We propose that while understanding the great diversity of names among Hittite deities we have to be guided by iconographic parallels and the principle of a god presenting as a multiplicity of devatā-s — an important feature of the ādhvaryava tradition within the Vedic layer (subsequently pervasive across traditions) of the Hindu religion. Thus, by the comparative method, we propose that this ādhvaryava tendency had roots in the PIE religion.
It also helps better understand some elements of the Anatolian religion, like the Rudra-class deities. The Hittitologist Archi noted several key features of the Anatolian archer deities and suggested that they inspired the Greek Apollo. Collins hinted at a possible pre-Greek origin for the Ariya/Orion tradition in the Anatolian locus. However, we think these are misapprehensions coming from ignoring the Indo-Iranian parallels. Orion region of the sky is indeed associated with the Rudra-class deity right from the early Indo-Iranian tradition. Once those connections are considered along with their Greek parallels, the Anatolian manifestations are best seen as a PIE inheritance. We are thus led to the conclusion that the association of the Orion region of the sky with the Rudra-class deity was probably a PIE tradition with ancient calendrical associations noted over a century ago by Tilak.
## The death of Miss Lizzie Willink
Late that spring, Somakhya and Lootika were visited by their mleccha friend Irmhild. Letting her sleep off the jet lag, they left for work. Given the good weather, Lootika returned early to check on their friend and go out with her prospecting spiders in the nearby woods for their work on endosymbionts. L: “Hope you had some good sleep and are all set to lead our battle-charge? Let me get you something to eat and we’ll head out when the sun goes down a bit. Somakhya and a student will join us in the woods.” Even as Lootika brought Irmhild some snacks, she said: “Restful, but strange. At least I can confess this to you without any embarrassment — After a while, I had a visitation from Lizzie — I’m sure you have heard of her from Somakhya and your folks. In any case, it may be a good sign given what we intend to do.” L: “Ah! they mentioned her in passing, but my recollection is they were not successful in getting her to say anything.” Ir: “I would not be so negative — I learned her name from that really exciting sitting they arranged.” L: “I’d like to hear it straight from your mouth — ain’t it interesting we never got to talk about that in length?”
Lootika agreed that it would be unwise to try anything aggressive with such a phantom: “I suspect she doesn’t want to make a visual impression as she does not want to scare you with that injured manifestation of hers.” Ir: “That makes sense. However, for some reason, I’ve been feeling a pressing curiosity to discover more about this mysterious Lizzie.” L: “I could try again to get her to speak.” Accordingly, Lootika performed a bhūtākarṣaṇa and waited to see if her friend might experience an āveśa. Irmhild suddenly stopped talking and after a couple of minutes asked for writing material. She slowly wrote out a few words and drew something. Seeing her remain in that state for some time doing nothing, Lootika sprinkled some water on her from her kamaṇḍalu and brought her out of it. Ir: “It looks as though she did not say much even this time, but this is interesting. I seem to have written just a single line though it felt as though I was writing quite a bit. It says: ‘Tombstone 66, Surat European cemetery.’ Lootika, what do you making of this drawing?” L: “Hmm… well, it looks like the map of the said cemetery. I’m sure she is referring to an old cemetery in a city in India, in a state known as Gujarat. Hence, we can look up the map and locate that grave if it still survives. But did you have any other sensation of her?”
Ir: “I must confess to being a bit shocked by her visual apparition again. She seemed very cheerful, but I could not take my eyes off another dab of blood on her collar. The strange thing was she sat just beside you and pointed to her neck and tried to say something that I could not hear. Lootika, did you experience something — you just did not seem to react?” L: “That is part of performing these procedures safely. While we draw in the ghosts, we shield ourselves from them for you never know what they might spring at you. These apparitions from the days of the English tyranny often have a particular hate for my people not unlike their modern counterparts — we have had more than one encounter with such phantoms that needed us to exert all our defenses. However, I too tend to believe this girl is a good phantom.” Ir: “Now the tales of your encounters only make me more curious about this Lizzie. Let us search for this place called Surat. Ain’t it strange she points to a place in India? Could it merely be a projection of me being with you guys?” L: “I think this is genuine. As for Surat, I can take you there on the map in a moment.” Soon Lootika was able to locate the likely cemetery in the satellite image and using the map Irmhild had drawn out they seemed to locate the stated grave. L: “At least that grave seems to be still there — apparently the cemetery is in the care of the Archaeological Survey. Unfortunately, I don’t have anyone in my immediate circle with associations with that city, else I could have gotten more direct information. Let us see if Somakhya or Vrishchika might give us some leads but now it is time for us to make our foray.”
All their initial inquires came to naught in the knotty tangle of the Byzantine bureaucracy surrounding the old records from that dark phase of Indian history. Sometime later, Irmhild called Somakhya and Lootika to ask if they could help with a course she was conducting. Before concluding the conversation, she asked if they had any new leads on her phantom visitor. She mentioned that when Indrasena and Vrishchika had visited her a little while back, they had tried the planchette once again and it had issued two words — “Krishnan” and “Charuchitra” — they were taken to be nonsense words, especially given that the second was merely the name of one of Somakhya’s cousins. Nevertheless, she preserved them wondering if it was after all a genuine clue. S: “Dear Spidery, what do you make of those. I have a feeling this is not nonsense.” L: “Why? Charuchitra is a historian. She might be able to find us something about that grave via her connections, But who is this Krishnan?” S: “Indeed. I believe this chap Krishnan is the fellow who maintains the annelid and mollusc collection at the Zoological Survey. Have you forgotten that we had once gone through a torturous series of inquiries to get him to show us their museum collection? Given that we did tarpaṇa to him on that occasion, he might prove helpful in accessing these Chennai archives if they still survive. Let us activate these connections and see if can give Irmhild something when we meet her.”
In the evening after the classes, Somakhya and Lootika were hanging out with Irmhild. L: “We have big news for you. We have unraveled the mystery of your phantom clanswoman!” Ir: “What? I cannot wait to hear what you have gotten! Why do you say clanswoman? I’m not aware of any such ancestor as far as our records go.” Somakhya: “From her story, we can say that she cannot be your direct ancestor, but you may have to search your family records for a collateral line which would feature her.” Lootika handed over a copy of the document found among the papers of Blyth that had an autobiography of the phantom. L:“Irmhild, given the inferred connection to your clan, I must warn you that parts might be difficult to read. Nevertheless, it seems to bring some closure and solace too.” It was preceded by the following prefatory note from Blyth:
I must now turn to a most singular experience while in my camp near Rayghur, a fort of the chieftain of the Morettos, who had fought our men with much distinction during the mutiny. LW, who had been deceased for nearly 2 years then, suddenly appeared before me in her phantom form on the evening of March 13th, 1872. It was the first and only time in my life I have had an auditory or visual hallucination — I certainly have never experienced anything so vivid and prolonged as this. I affirm that I am stout of heart and of a most unimaginative constitution — yet, this apparition felt as real as anything from this world. She commanded me to record the story of her life and inquired if I had fitted her grave at Surat with the most abominable Hindoo grotesques she desired. I felt in no position to disobey her command. Below, I record her words as I noted them before she vanished and have not attempted to insert any parenthetical notes regarding my own appearance in the third person in the narrative. I can vouch that whatever she said with regard to the events concerning me is entirely veridical.
The words of LW’s phantom:
I was born in what was to soon be the colony of Victoria in Australia where my father JW was then the military surgeon. I was the second of four siblings; my elder brother was Robert; my younger siblings were Edward and Minnie. It was a rough place as we started taking in convicts, but I have considerable gratitude for having spent my early youth there. An important consequence was that I became a skilled equestrian early in life. The second consequence came about when I rode out to the cliffs and discovered fossil shells of cowrie snails. I compared these to the cowries we have today and realized that those from the past were notably different. I began wondering — why had they vanished? From where did the ones we have today come? I asked my mother about this. She said that the Lord the God was unhappy with some of his ante-antediluvian creations and destroyed them in their entirety. But that did not answer how the ones we have today came into being — after all, had the Lord not finished his creation within the first seven days of existence? I got some answers when the naturalist Mr. Sowerby came visiting. He became interested in my collection and in return for them gave me some coins and lent me some books by Sir Lyell and Mr. Owen. I labored through them with much interest. Later I learnt that Mr. Sowerby described the fossil cowries I had found under his name. Shortly, thereafter I found a few more new giant cowries but my family left Australia for the Bombay Presidency in our Indian possessions. My parents insisted that I should go to finishing school and sent me back to England. I abhorred the regimental order of the finishing school and was most certainly amongst the worst of their pupils. Thankfully, my father’s friend, Dr. Parkinson, was rather kindly and took interest in my shells and introduced me to the latest intricacies of natural history. He helped me publish my discovery of the Australian fossil cowries as an appendix to his own tome on fossils.
As I disembarked the smooth-sailing Fairlie at Bombay, the warm air lifted my spirits. I felt a sudden sense of purpose and eagerly scanned the quay for my parents. I finally joined my father and his koelie Joognoo Raum Pondee who took care of my luggage. As we were returning to his post to the south of Bombay, a frightening riot had broken out among the natives. The tillers known as the Ryots wished to rid themselves of their debts and turned on their native bankers known as the Mawrwarees. The Bombay Army under Sir Rose, who had formerly played a pivotal role in crushing the Mutiny, along with some natives of the Scinde Division were deployed to put down the rowdy Ryots. Unfortunately, our convoy came upon a large band of hideous Ryots who were throatily screaming cries that could blanch the stoutest heart. I froze as they threw the bleeding corpse of a decapitated Mawrwaree on the path ahead of us. Our koelie Pondee suggested that we mount the horses that were conveyed by the Scindes and make our way home swiftly via the hills. However, he worried about my safe conveyance as the Ryots closed in. Everyone in our party heaved a sigh of relief when they learnt that I was a skilled equestrian. Thus, after quite an adventure I reached home with my father. Soon, I found myself pampered by more than one dashing suitor, but my mind-numbing job as the governess to the magistrate’s children abraded any joy I might have felt from the ample attention I was receiving.
Thankfully, Pondee, who also worked as a native assistant to Mr. Blyth, put in a word to him about my abilities as a naturalist. Ere long, I had an interview with Mr. Blyth and provided him a letter of reference from Dr. Parkinson. Thus, I became his assistant, and he suggested to me the most interesting possibility of systematically discovering and recording the mantids, hemipterans and coleopterans from the Western Ghats in the Bombay Presidency. I set out twice every week on my horse with Mr. Blyth or Edward and prospected the ravines and hills where the Alexander of the warlike Morettos had once held sway and fought the armies of the Mahometans. I found considerable success in discovering hexapods new to science. Following his advice, I started classifying the insects and increasingly saw the truth in the theories of Mr. Wallace and Mr. Darwin. I had intended to describe these observations together my mentor Mr. Blyth and had never felt happier before. Unfortunately, my mission met with an unexpected interruption as little Minnie caught a cold and went into decline. I helped my mother in nursing her. One day, when she had to be confined to the bed, I heard the peculiar blare of a strange instrument followed by a strange vocal song. I looked around — neither my mother nor my brother who were in the room with Minnie heard it; nor did Minnie herself. However, our maid, Pondee’s wife, and our native cook, Tauntia, affirmed hearing the same. Pondee’s wife informed me that a great disaster was impending — it was the conch-blare and the dolorous dirge of the Yum-doots — the agents of the Indian Hades who whisk souls away. The next day poor Minnie expired.
Then, as I went to mount my horse, I saw a most dreadful apparition. I now know that it was a mātṛ from the retinue of the great god Rudra. That most frightful divine lady said to me that my allotted term of life was drawing to a close. I asked if I would be joining Minnie and Robert. She responded that due to my act of piety I would join her host and vanished. I brushed it aside as a mere hallucination from the heat and rode on towards the spot where Pondee had spotted the horned beetles going up a narrow path. In retrospect, I should have dismounted but, as the Hindoos say, who can escape what the god Bruhmah has written out for you? For some reason, my seasoned horse bolted and threw me off into the defile bristling with bamboos. I was severally skewered through my arm, neck and eye and could not extricate myself. However, Mr. Blyth heard my cry and was able to locate me after a search. He had to get Pondee along before he could finally get me down from my hellish impalement: by then, I had lost consciousness. Finally, I was taken home and my father started treating my wounds. After the initial treatment, I regained consciousness briefly and spoke once to bid my family, friends, and my suitor Captain Atkinson goodbye for the last time. Instructed them to decorate my tomb with the tridents and drums of the great god Seeva. Only my brother Edward assented but he too expired last year after being hit by a ball while playing cricket. My grieving parents left for England shortly thereafter. My life’s work will not see the light of the day. Hence, as I rejoice in the retinue of the great gods, I will aid a future member of my clan realize more of it than I did.
This was followed by a concluding note from Mr. Blyth:
I had no intention of fulfilling the delirious requests of the dying Ms. LW to place the symbols of the Hindoo Termagants and Baphomets on her tomb. I suspected that she had come under the evil influence of my assistant Pondee’s wife, who clouded her otherwise logical intellect with ghastly superstitions. However, this apparition near Rayghur filled me with such terror that I commissioned a blacksmith to make the needful auxiliaries and decided to fit them on poor Ms. LW’s tomb when I got a chance to visit Surat.
Somakhya: “Irmhild, here is a picture of her tomb. My cousin Charuchitra was able to obtain it via her connections to the Archaeological Survey. Evidently, Blyth never got to furnish it with the symbols of Rudra — he himself passed away a few months later with a fever following a cut to his thumb. The epitaph has not survived in its entirety, but it gives her name as Lizzie Willink — this matches the initials in Blyth’s account. Also note, while they did not furnish it with the tridents and the ḍamaru-s she wanted, they engraved a beautiful copy of one of the fossil cowries she discovered — it bears the unmistakable siphon and whorl peculiar to the Australian exemplar. No doubt she was able to grasp an evolutionary lesson from that. These indicate that the grave pertains to the very same person whose initials are in Blyth’s document.” Ir: “Tragic! The epitaph says that she was only 22 when she died. It now strikes me that the aunt she mentioned in her narrative must be a lineal ancestor of mine.”
[Any resemblance to real incidents or people should be taken as merely convergence in story creation under constraints]
## Indo-European expansions and iconography: revisiting the anthropomorphic stelae
Was there an early Indo-European iconography? The anthropomorphic stelae
There is no linguistic evidence for the presence of iconic or temple worship among the early Indo-Europeans. However, after their migrations, when they settled in the lands of sedentary peoples, they adopted a range of religious icons often stylistically influenced by local traditions. Nevertheless, there are some clear iconographic features of their deities and other divine entities that shine through these local styles (to be discussed in later notes). This suggests that, even if iconic worship was not the central focus of their religion, they had definitive visualizations for their deities that emerged early in IE tradition. Moreover, barring the Iranian counter-religion, most branches of IE people adopted iconic and temple worship in the later phases of their tradition. This observation, together with some of the textual features of the early iconic worship of Hindu deities (e.g., caitya-yāga and gṛhya-pariśiṣṭa-s), suggest that early IEans probably did have iconic worship on the steppes itself; it was just not a major expression of the religiosity of their elite.
The archaeology of the IEans was fraught with much confusion until archaeogenomic studies over the past decade greatly clarified the situation. Hence, we can now say with some confidence that we do have a body of archaeological records for early IEan iconography, even if we do not fully understand it. The earliest evidence for this comes from the Yamnaya horizon on Pontic–Caspian steppe that is associated with early IEans. The striking body of iconic images from this locus and time is comprised of the so-called anthropomorphic stelae (Figure 1). These stelae caught the attention of researchers right from the early work of Gimbutas and were subsequently discussed at length by Telegin and Mallory. While their iconographic content and functions continue to be debated, several authors, starting from Gimbutas, have proposed an IEan interpretation. Further, most of these authors have tended to explicitly or implicitly invoke Indo-Aryan themes (e.g., Vassilkov most recently) to provide the imagery with an IE interpretation.
Figure 1. Examples Anthropomorphic stelae from different parts of Africa and Eurasia (2, 4, 6 from Vierzig).
That said, it should be noted that the basic form of these anthropomorphs is widely distributed across Eurasia starting from the chalcolithic-Bronze Age transition, with a core temporal window of 3500-1800 BCE. A survey by Vierzig indicates that apart from commonly occurring in the Yamnaya horizon, they are also densely present in the Northern Italy-Alpine region, Iberia and Sardinia. Moderate to sparse occurrences of such anthropomorphic stelae are also seen in France, Germany, the Italian and Greek peninsulas, Sicily, Caucasus and Northern Arabia (e.g., Hai’l and Tayma’ in modern Saudi and also probably Jordan — the Israeli exemplar). A miniature terracotta version was also reported by Sarianidi in Bactria (see below for more on this). In the East, they are found in the Dzungarian Basin associated with the Chemurchek Culture (2500–1700 BCE) that succeeded the Afanasievo, the Far Eastern offshoot of the Yamnaya. Another successor of the Afanasievo, to the north of the Chemurchek culture, is the Okunevo culture in the Minusinsk Basin. This culture shows some remarkable menhirs that seem to have been influenced in some features by the classic anthropomorphic stelae. These also share features with the stelae from Shimao (roughly 2300 BCE) and later expressions of this theme such as the deer stones of Mongolia (see below) and even the totemic structures in the North American horizon. Finally, we could also mention the menhirs from Dillo in South Ethiopia that might be seen as sharing some general features with the Eurasian stelae under consideration. However, their iconography is again too distinct to be included in this discussion. The distribution of the anthropomorphic stelae suggests that, like certain other iconographic conventions (e.g., the horned deity), this convention too spread widely, even if some versions might have a convergent origin. Thus, a priori, it cannot be identified with a specific culture, though specific versions of them might show a narrower cultural affinity (see below).
The typical anthropomorphic stele under consideration is a simplistic depiction of a human form — usually only a basic outline of the body. The minority of the stelae are furnished with more elaborate embellishments. Despite their simplicity, they display a certain unity that distinguishes most of them from the more divergent menhirs with human features. Further, across the above-mentioned zone, the better preserved and more elaborate versions display some common features: 1) The male figures (which tend to be the majority) are often shown as ithyphallic. This feature is shared by the Arabian, Yamnaya (and its western successors Corded Ware) and possibly at least some of the Iberian versions. 2) The figures often wear a belt around the waist reminiscent of the Iranic avyaṅga. This feature is definitely shared by exemplars from across the above-stated distribution zone. 3) The arms and legs when shown are always presented in a static manner, even when associated objects, like weapons, are depicted. 4) Across their distribution zone, the stelae are frequently but not always associated with graves (this is also true of the Ethiopian anthropomorphic menhirs of Dillo). Even the most elaborate early versions of these anthropomorphic stelae appear simpler than the coeval religious icons of Egypt, West Asia and possibly also the Harappans. Thus, we believe that the anthropomorphic stelae did not have their primary origin in the Egypt-West Asia-Harappan corridor but in the steppes or among the Early European Farmers or in the Caucasus.
Anthropomorphic stelae IE heartland and their dispersal
The early Arabian and steppe versions show sufficient divergence to suggest memetic diffusion rather than direct transmission via invading groups; however, from the time of the Yamnaya expansion onward there are specific features to suggest the presence of an iconographic convention governing their production that was likely transmitted by expanding IE groups. We will first consider this in the context of the Yamnaya artefacts and the western expansion of the IEans. It can be best understood by comparing some famous stelae namely: 1) the so-called Kernosovskiy and Federovsky (Poltava region) idols from what is today Ukraine. 2) The Natalivka stele, again from Ukraine. 3) Cioburciu stele from what is today Moldavia. 4) The Hamangia stele from what is today Romania. 5) The Floreşti Polus stele from interior Romania. All these stelae depict male figures that are unified by the presence of a common weapon the battle axe. Importantly, in the Kernosovskiy, Federovsky, Cioburciu, Hamangia and Floreşti Polus stelae at least one axe is secured via the waist belt of the anthropomorph. These stelae (barring Floreşti Polus: fragmented? and Natalivka: not clear), as well as several others from the Yamnaya horizon (Novoselovka, Svatovo, Kasperovka, Novocherkassk and Belogrudovka), are also unified by the depiction of the outlines of the feet (Skt: pādukā-s). The Kernosovskiy, Natalivka and Svatovo stelae from Ukraine display a bow as an additional weapon. The profile of the axe common to all these stelae is boat-shaped and corresponds to the battle axe seen in the western successor of the Yamnaya, viz., the Corded Ware culture. Such axes are frequently buried in the Corded Ware graves believed to belong to elite males. One of the ārya words for the axe is paraśu, which has cognates going back to proto-Indo-European. It is quite possible this type of axe was indeed known by that ancestral IE word. On the whole, these features support the IEan provenance and westward movement of this type of anthropomorphic stele into Europe.
Figure 2. Yamnaya-associated stelae.
The Kernosovskiy idol depicts a second kind of axe with a distinct head profile. This may be compared to a recently reported massive metal axe, weighing just shy of a kilo and blade length of about 21 cm from the Abashevo culture (in the middle Volga and adjacent Ural region), which likely represented the Aryans before their southward expansion. The Ṛgveda mentions two distinct types of axes, the paraśu and the vāśī. The word vāśī does not appear to have cognates outside the Indo-Iranian branch among the IE languages. It is possible that the eastern movement of the Corded Ware-like cultures acquired a distinct type/word for axe from other local populations. But the presence of two distinct types of axes on the Kernosovskiy idol from the Yamnaya period suggests that a second type of axe might have been acquired even earlier but only used in certain descendant IE cultures.
Figure 3. Chemurchek stelae. 1 and 2 from Kovalev. 3 from Betts and Jia.
Turning to the eastern transmission, we find that a bow held in a manner similar to the Yamnaya stelae is featured in at least three anthropomorphic stelae at Chemurchek sites (2750-1900 BCE). One of these (published by Kovalev) holds another weapon, which could be either an axe or an aṅkuśa paralleling the Yamnaya stelae. A similar axe or aṅkuśa is held without a bow in the hands of three other stelae from the Chemurchek culture and can be seen on the Belogrudovka stele in the Yamnaya group. One of the Chemurchek anthropomorphs holds something like a mace comparable to what is found on the Kernosovskiy idol. Until recently the affinities of the Chemurchek people were uncertain. However, the archaeogenetic study of Zhang et al provides some clarity in this regard. First, the eastern offshoot of the Yamnaya, the Afanasievo underwent local admixtures with the Tarim early Bronze Age population (the source population of the famous Tarim mummies) and to a smaller degree with the East Asian “Baikal Early Bronze Age” giving rise to the “Dzungarian Early BA1” population. Next, this mixed, again with the Tarim BA, and a Namazga/Anau-I chalcolithic-related population (Geoksyur) to give rise to the Chemurchek people. Thus, the genetic evidence supports an ultimate link between these cultures and the Yamnaya derived populations, suggesting that iconographic similarities in the eastern anthropomorphic stelae are related to the Indo-European movement to the east. In this scenario, a Chemurchek-related population rather than the Tarim mummies population likely gave rise to Tocharian languages. However, a wrinkle remains regarding the Afanasievo situation: while Mallory claims that stelae have been recovered in that horizon, we have found no evidence for such so far in the literature. This may point to greater diversity within the early Eastern extension of Yamnaya than previously appreciated (see below section).
Figure 4. Okunevo and Shimao stelae. 1-4 Okunevo stelae (from Polyakov et al and Leontiev et al). 5 Shimao stelae (from Sun et al).
Before we leave the footprints of the Yamnaya expansion on the eastern anthropomorphic stelae, it would be remiss if we do not touch upon the Okunevo menhirs and the Shimao stelae. Like the Chemurchek culture, the Okunevo culture (2600-1700 BCE) represents a bronze age admixture between the IE and East Asian populations that arose from a comparable, but distinct, admixture of the Afanasievo with Tarim BA and Baikal BA populations, probably driven primarily by males. The Okunevo anthropomorphic stelae/menhirs share the general similarity of the round facial profiles with some of the Chemurchek stelae from the Kayinar Cemetery. However, beyond this they show much diversity and several striking and unique features, such as: 1) a halo of elements emanating from the faces, like rays, waves (often terminating in lunes) and dendritic structures. 2) A frequent motif featuring a central dyad of concentric circles surrounded by four cusps shaped like an astroid. 3) Peculiarly curved mouth on the anthropomorph. 4) Depictions of stylized animals like wolves and elk (both of which acquire mythic significance in the much later Turko-Mongol world). 5) Unlike the Yamnaya stelae they lack a belt. The earliest Okunevo specimens are close in form to the simplest versions of the Chemurchek stelae. Some of these early Okunevo versions also share a bovine motif with the Yamnaya stelae (e.g., Kernosovskiy). They acquire the greatest complexity in the middle Okunevo period with all the above-mentioned distinctive features. The first half of the middle Okunevo period is marked by the unusual tall menhirs (up to 5m tall) often depicting multiple anthropomorphic or theriomorphic faces. The Okunevo stelae/menhirs are distinguished from the Yamnaya and Chemurchek versions in almost entirely lacking weapons — we are aware of only a single exemplar from the middle Okunevo period where the anthropomorph is flanked by two tridents. To our knowledge, tridents are unknown in any of the other early steppe stelae.
We propose that the Afanasievo-like founder populations of Chemurchek and Okunevo were probably either the same or close, but distinct from other sampled Afanasievo groups that seem to have lost the ancestral stelae (contra Mallory?). This population introduced relatively simple stelae to founders of both these populations at the time of the admixture with local groups. The Chemurchek retained these in a largely conservative form, whereas Okunevo innovated upon it probably drawing on the mythemes coming from their Northeast Asian founder population with links to Siberia. This might explain the parallels between the later and the Inuit and American totems. The weaponless Okunevo stelae with exaggerated facial features and expressions are also mirrored in the iconography of the stelae from the early neolithic urban site from Shimao, Shanxi province of China (2250-1950 BCE). As these have no Chinese antecedents, it is quite possible that they were influenced by the contemporary IE-admixtured cultures to their north (a contact possibly also responsible for the dawn of the metal age in China). Currently, the archaeogenetic information from the Shimao is limited but suggests affinities to the Northern East Asian populations related to that contributing to the ethnogenesis of the Okunevo. Thus, it points, in the least, to a role for a similar population as that involved in the emergence of the Okunevo and the potential diffusion of iconographic elements.
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https://www.nature.com/articles/s43247-020-00048-9?error=cookies_not_supported | ## Introduction
Heatwaves (HWs) are among the most concerning extreme meteorological events, as they have a wide range of impacts, including human health (e.g. increased mortality and morbidity)1,2 and significant socio-economic and ecological effects, such as wildfires and poor air quality events3,4, droughts5 and peaks in energy consumption demand6,7. Within the context of global warming, an increased frequency in extremely warm events is foreseen, comprising HWs of unprecedented extension and duration8,9,10. 2019 was the second warmest year at the global scale, only surpassed by the strong El-Niño year of 201611. Unsurprisingly, summer 2019 presented exceptional HWs in Europe, exceeding notorious episodes which occurred just 1 year before in the also very hot summer of 201812,13. In terms of affected areas, the 2019 HW events resembled to a large extent the 2003 summer HW14,15 and in many places temperature extremes even shattered those of 2003. In late June an outstanding HW began in southwestern Europe12, and extended towards most of France and parts of central Europe. During this event the city of Vérargues in southeastern France reached an astonishing daily maximum temperature (TX hereon) of 46 °C on June 28th. This was the first time temperature measurements exceeded 45 °C in France. Just a few weeks later, another exceptional HW set new historical values in France and other European countries. For example, Paris registered a TX of 43 °C, surpassing the previous record standing since 1947 by ~2 °C. Furthermore, for the first time since the beginning of meteorological observations, Belgium and the Netherlands exceeded the 40 °C barrier. Fortunately, summer 2019 caused considerably less mortality excess than previous HWs, including the devastating 2003 event16. This might result from the combination of human factors, which include the lessons learned from the 2003 HW (i.e. early warning systems, better preparedness and societal awareness, deployment of sheltering and water-cooling facilities, use of air conditioning, etc.), and the shorter duration of both 2019 HWs.
Most extreme temperature events are partially driven by anomalous large-scale atmospheric circulation. However, the current rate of warming (i.e. thermodynamic changes) is sufficient to produce exceptional HWs, even without unprecedented anomalies in the large-scale circulation. Contrasting with the lack of robust projections in dynamical changes17, recent works indicate robust and significant increases in maximum HW magnitude over large regions, even for 1.5 °C global warming targets8. Moreover, anthropogenic forcing has already caused a 7‐fold increase in the likelihood of extreme heat events18. In addition to direct radiative effects of increasing greenhouse gases concentrations, the potential contribution of enhanced local land–atmosphere feedbacks has also been acknowledged19. Recent studies have further explored non-local feedbacks in recent mega-HW events20,21. Local drying and subsequent enhanced surface heat fluxes, together with horizontal warm advection and heat accumulation in the atmospheric boundary layer, have been shown to contribute to the magnitude of temperature anomalies during the August 2003 HW22. Very similar processes were observed in the 2010 Russian mega-HW22,23. Recently, the same events have been explored to introduce the concept of upwind soil dryness21 (also referred to as self-propagation20). These conceptual models illustrate how air masses warmed by sensible heat fluxes (due to pre-conditioning dry soil conditions) can be advected downwind, stimulating land–atmosphere feedbacks in nearby regions that contribute to a progressive set-up for HW occurrence.
Nevertheless, the combined effect of regional scale feedback processes and large-scale atmospheric circulation is crucial for understanding the development of extreme heat events24. Regarding the latter, several studies have described the weather systems associated with European HWs, highlighting the key role of blocking/ridge occurrence12,25,26,27. Given this major control of the dynamics, other studies have used flow-analogue approaches to quantify the contribution of thermodynamic changes to the observed magnitude of outstanding recent HW events28,29,30. Here we aim to investigate the combined roles of large-scale atmospheric circulation and soil moisture conditions in the occurrence and severity of the summer 2019 HWs in Europe. The synoptic conditions are analyzed for both events to characterize large-scale circulation features. Thermodynamic aspects are also considered, including regional feedback processes and their lagged effects. In particular, we investigate soil-atmosphere processes observed since the June 2019 HW and their potential role in amplifying the magnitude of the July 2019 HW.
## Results
### How anomalous was summer 2019 in Europe?
To place the 2019 European summer into a long-term context, we first derived estimates of the European mean summer temperature anomalies since 1500 using a multi-proxy reconstruction (1500–1900)31 and an observational dataset (1901–2019)32. Further details are provided in the Data and Methods section. Summer 2019 falls within the top five warmest recorded in Europe since the early 16th century, only surpassed by other recent devastating summers (2018, 2010 and 2003; Fig. 1a). Although extremely warm conditions for the 2019 summer were mainly confined to western and some central areas of the continent, temperature anomalies were large enough to produce an overall continental anomaly close to +2 °C (with respect to 1981–2010). Recent results estimate a return period of nearly 300 years for a similar event, taking into account recent climatic conditions18. As seen in Fig. 1b, warm summers have become more frequent since the last decades of the 20th century, and the 21st century concentrates an unusual frequency of extreme summers when compared to the long-term variability (1500–2019). This is reflected by the dominance of post-2000 events in the high-end tail of the European summer temperature distribution (histogram, Fig. 1a), and the pronounced shift in the 30-year Gaussian fitted distributions between 1960–1989 and 1990–2019 (Fig. 1a, light and dark grey shading).
Interestingly, the European mean temperature anomaly of summer 2019 falls very close to that of summer of 2003 (Fig. 1a). The spatial distribution of TX anomalies averaged for summer 2019 was also somewhat similar to that observed in 2003, according to the E-OBS dataset (see Fig. 1c). Taking into account the comparable magnitude and spatial signatures of the 200314,33 and 2019 summers, we used the former as a “benchmark” to evaluate the exceptionality of the 2019 summer temperature anomalies. Given the fast pace of current atmospheric warming, the comparison of 2003 and 2019 summers was performed by defining temperature anomalies with respect to two distinct baselines: (i) the full available period (1950–2018) and (ii) the previous 30-year period at the time these summers occurred (i.e. 1973–2002 for 2003 and 1989–2018 for 2019), which leads to a warmer climatological baseline for 2019 than for 2003. As shown in Fig. 1c, summer mean TX anomalies were in general larger for 2003 than in 2019 with respect to their corresponding climatological conditions. However, the highest daily TX anomalies registered in 2019 exceeded those observed in 2003, even if anomalies are computed with respect to their corresponding climatologies (Fig. 1d). Specifically, daily TX in northeastern France and Benelux surpassed climatological values by nearly 20 °C during summer 2019, compared with the maximum TX anomalies observed in the same regions during summer 2003 (~16 °C). Note that these values are also the warmest TX anomalies of the continent in both summers. Therefore, the dichotomy between summer averages and daily values reflects that while summer 2003 was more anomalous for the climatic conditions expected at that time (essentially due to the longer nature of the August 2003 HW), the 2019 HWs were generally more intense than the 2003 HW at daily time scales in many places of western and central Europe. Similar conclusions are obtained when using the full period (1950–2018) as baseline (Supplementary Fig. 1). To illustrate the contribution of the non-stationary climate conditions to the magnitude of recent record-breaking events, Supplementary Fig. 2a shows the difference between the summer mean TX 30-year climatologies as of 2019 and 2003 in Europe. In just a ~15-year period, TX normals have increased by more than 1 °C in summer over most areas (and even by ~2 °C at some locations of southern Europe). Additionally, the rate of record-breaking events has also been increasing over the European continent (see Supplementary Fig. 2b), consistent with the rise in European summer mean temperatures during that period (see Supplementary Fig. 2c): Actually, approximately 2/3 of historical European TX extremes have been observed in the last two decades (i.e. post-2000).
To stress the exceptionality of the 2019 HWs, Fig. 2a presents the areas where all-time records in TX were hit during that summer. The spatial pattern shows a strong resemblance with the corresponding map for summer 2003, in particular over France and Benelux (see Supplementary Fig. 3). This means that a substantial part of the records set in summer 2003 was broken during 2019, with the exception of some areas in southwestern Europe (e.g. western Iberia, where 2003 temperatures were shattered during summer 201812).
The unprecedented temperatures during summer 2019 were associated with two clearly distinct HWs in late June and late July. In Fig. 2b, c the spatial distribution and duration of these HW events are depicted, as diagnosed from a novel HW tracking algorithm (see “Data and Methods”). The panels show that the spatial distribution of areas under HW conditions during July extended much further north than those during the June HW. This is in agreement with the timing of new all-time records presented in Fig. 2a. During July, unprecedented TX was reported over larger areas and dominated higher latitudes, including more than half of the French territory, the Benelux, western Germany, southeastern England and parts of Scandinavia. In contrast, daily all-time records in June were essentially restricted to southeastern France and northeastern Iberia. In spite of this, the highest absolute values (TX > 45 °C) were observed during the June HW. The persistence of HW conditions was also more prominent over land during the June HW (cf. dots in Fig. 2b, c), with large areas of western Europe experiencing an extremely high number of HW days. These differences in the spatial signatures of the 2019 HWs are also reflected in the latitudinal location of the 500 hPa geopotential height (Z500) anomalies (contour lines in Fig. 2b, c), suggesting distinct atmospheric circulation patterns.
### Atmospheric circulation during the 2019 HWs
In this section, we describe the large-scale atmospheric circulation configurations behind the summer 2019 HWs. The temporal evolution and spatial tracking of the two HWs (summarized in Supplementary Fig. 4) show that the initial location of the HW centre was detected much further south in June than in July. In the former, HW conditions originated over northern Africa and then migrated towards northern France, before affecting eastern Europe during its later stages. In contrast, the July HW onset was detected over France and then moved to higher latitudes, reaching areas close to the Arctic towards the end of its lifecycle.
Despite these differences, a relevant common factor can be identified. Both events displayed a classical pattern of Z500 positive anomalies over the affected areas, accompanied by the presence of a low-pressure system in the eastern Atlantic (see also Supplementary Fig. 4). 1000–500 hPa geopotential height thicknesses averaged for the HW periods are presented in Fig. 3a, b, revealing pronounced ridge-like patterns in both events, extending from northern Africa towards western Europe. However, the ridge affecting southwestern Europe during the June HW was stronger and better defined (i.e. sharper zonal gradients). As a result, a stronger southerly wind component characterized this first HW, when compared to relatively more stagnant conditions during the late July episode. This is in agreement with the strong intensity of the Saharan intrusion (see “Data and Methods” for details) observed during the first event (Fig. 3a), when an air mass with desertic features reached unprecedented latitudes over France. Saharan intrusions have been shown to be associated with extreme heat events in southwestern Europe12,34,35, as they present very high potential temperatures and low moisture content, favoring intense surface warming under anticyclonic conditions. During the July HW, air masses with such thermodynamic properties were not detected further north than the western Mediterranean (Fig. 3b). In consequence, these results (supported by the HWs evolution) suggest a more pronounced influence of warm advection during the June HW.
Following the Eulerian methodology developed in previous studies12,26, in Fig. 3c, d the main physical mechanisms contributing to the temperature anomalies in the lower troposphere are examined for both HWs (see “Data and Methods”). Air motions, both horizontal (warm advection, red line) and vertical (strong adiabatic heating due to subsidence, blue line) are often important for the establishment and maintenance of HWs over Europe, although their relative contributions may differ22,36. This is also the case for the 2019 HWs. For the June HW, horizontal advection (red) was relevant before and at the onset of the HW, while vertical descent (blue) was essential for its maintenance (Fig. 3c). Differently, diabatic processes (green line) played a more important role for the setup of the July HW (Fig. 3d). This diversity in the underlying processes of the HW events is even more noticeable considering the fraction of areas (within western Europe, WEU, [43°– 53° N, 0°–10° E]) where each contributing factor accounted for the largest temperature changes during the HWs lifecycles (Fig. 3e, f). Accordingly, as discussed in detail below, regional diabatic processes played a key role during the July HW. In Supplementary Fig. 5 (top panels) these differences are reinforced by the day-to-day evolution of the vertical profiles of temperature anomalies and horizontal wind averaged over WEU. During the June HW air masses presented reduced vertical gradients (presumably associated with the presence of the vertically homogeneous Saharan warm air intrusion) as compared to the July HW. Also, wind vectors support the major role of horizontal advection in the onset of the June HW, contrasting with more stagnant conditions at the peak of both HWs. This is further evidenced by the mean WEU vertical profiles of absolute temperature averaged over the HWs duration, as well as the instantaneous profiles at the peak of the HWs (bottom panels of Supplementary Fig. 5). Moreover, during the July HW, temperature anomalies seem to propagate upwards from the surface a few days prior to the HW onset, suggesting a progressive surface-atmosphere coupling, building up in the lower troposphere towards the HW peak. A similarly gradual warming process has been reported in the boundary layer, leading to self-intensification of near-surface temperatures during the well-known mega-HWs observed in western (2003) and eastern (2010) Europe22. In the next section, we further explore whether the diabatic processes that dominated the establishment of the July HW were influenced by land–atmosphere feedbacks.
### Amplification of the late July 2019 HW due to soil desiccation
To explore the presence of land–atmosphere feedbacks during the July HW, we first analyzed the temporal evolution of a set of relevant variables averaged over a box covering the region with the highest TX anomalies (northeastern France and Belgium) as presented in Fig. 4. The preceding June HW contributed to strong losses in soil moisture content in that region, with this drying also reinforced by above-average radiative fluxes at the surface and low precipitation throughout July (see Supplementary Fig. 6). Consequently, persistent soil desiccation occurred between the two HWs. This resulted in anomalous surface heat fluxes, as shown in the lower panel of Fig. 4a. The three-week period in between the two HWs was characterized by an approximate doubling (halving) of the sensible (latent) heat fluxes when compared to the corresponding climatological values for that time of the year. This is reflected in the recurrence of days with Bowen ratio values above 1, indicating that energy partition was dominated by sensible heat fluxes from the surface, due to soil moisture limited latent fluxes (see also Supplementary Fig. 6). These results point to a contribution from regional soil moisture deficit to near-surface warming that persisted until the onset of the July HW (i.e. local land–atmosphere processes).
Figure 4b illustrates relevant fields for the land–atmosphere coupling on a larger spatial domain than the regional box over NE France/Belgium considered in Fig. 4a. The areas with significant soil dryness (dots) in the weeks preceding the July HW are in good spatial agreement with subsequent large sensible heat flux positive anomalies (shading) during the build-up of the July HW over NE France/Belgium. Collocated large anomalies of both fields extended over large areas, suggesting land–atmosphere coupling beyond NE France/Belgium, particularly to the south of the region hit by the July HW. This, along with the mean near-surface wind direction observed in the days preceding the July HW, suggests similar processes to those describing the concept of self-propagation21,22. Under the presence of southerly winds, dry air masses in central France warmed by anomalous sensible heat fluxes prior to the July HW were advected further north, likely enhancing remote land–atmosphere feedbacks and local sensible heat fluxes over the box displayed in Fig. 4b. This process arguably contributed to the amplification of the July HW. The circulation analogue exercise conducted further ahead supports these conclusions.
A similar analysis was performed for the region with the highest temperature anomalies during the June HW (southeastern France, Supplementary Fig. 7). During the intense June HW, energy transfer from the surface by sensible heat fluxes was below climatological values. In addition, in this area close to the Mediterranean, soil desiccation was not as intense as observed further north. These results suggest land–atmosphere coupling did not substantially contribute to the June HW, thus reinforcing the contrast between the two events, i.e. the more advective nature of the June HW compared to the dominance of diabatic processes associated with land–atmosphere coupling during the July HW.
To further deepen the process analysis discussed above, Figs. 5 and 6 present two distinct analogue exercises (see “Data and Methods”) with the aim of evaluating: (i) the potential contribution of the June HW to the subsequent soil desiccation observed in July; (ii) the level of amplification of surface temperature anomalies during the July HW as a result of the preceding soil moisture deficits.
The results of the flow analogues for the June HW indicate that recent circulation conditions similar to those reported during the June HW have some drying imprints in the subsequent soil moisture conditions of western Europe (Fig. 5b). Indeed, the soil moisture content over WEU (dashed box in Fig. 5c) is significantly lower for flow analogues of the June HW (Fig. 5d, dark boxes) than for random circulation conditions (light boxes; p < 0.05; t-test and Kolmogorov–Smirnov test), portraying the role of the atmospheric circulation pattern in driving subsequent soil moisture deficits. These differences are even larger when using ERA5 (1979–2019) or ERA20C (1900–2010) data as a pool of analogues (see Supplementary Figs. 8 and 10), suggesting reduced variability of soil moisture in NCEP/NCAR (i.e. weaker responses to atmospheric forcing) and/or differences related to the thickness of the uppermost soil layer (0–10 cm in NCEP/NCAR vs. 0–7 cm in ERA reanalyses). These results lend support to the hypothesis that the atmospheric circulation associated with the June HW contributed to the subsequent desiccation that preceded the July HW. Note that drying was not so obvious during the actual June HW in observations (Fig. 4a), arguably because soils were replenished throughout a deeper layer (0–2 m) prior to the event (upper panel of Supplementary Fig. 6), which might have contributed to an initial dampening of the desiccation process. On the other hand, the results of the analogue exercise also indicate that flow analogues precede drier conditions in the present than in the recent past (Fig. 5a, b). Part of this difference is associated with a generalized regional drying over the analyzed period, since a comparable soil moisture decrease is also observed between the random circulation distributions of both subperiods (Fig. 5d), which are not constrained by the atmospheric circulation. Qualitatively similar results are obtained for ERA reanalyses (see Supplementary Figs. 8d and 10a), although trends and patterns are overall weaker in ERA20C, arguably due to the lack of soil moisture-related observations in the assimilation process. The temporal differences between soil moisture distributions are consistent with the reported occurrence of more severe European droughts due to enhanced atmospheric evaporative demand by recent warming trends37,38. In summary, our results support that the June HW, together with the precipitation deficits and high radiative fluxes that followed it, contributed to the soil moisture deficits preceding the July HW. Furthermore, we also find that this drying signal has been amplified in recent decades. While this result should not be interpreted as a formal attribution to anthropogenic factors, it is in agreement with recent studies attributing dry-season water imbalance changes to human-induced climate change39.
To support the above-mentioned amplifying role of the observed soil moisture deficits in the magnitude of the July HW, we have searched for flow analogues of each day of the July HW and reconstructed the associated TX anomalies (Fig. 6). We account for the role of soil desiccation by distinguishing between analogue days preceded by dry and wet conditions over WEU, as inferred from regional mean anomalies of soil moisture averaged for the previous 15 days (see “Data and Methods”). The results indicate that similar flow patterns to those recorded during the July HW tend to cause warmer conditions when they are preceded by dry conditions (Fig. 6a, b). In other words, for similar atmospheric circulation, soil moisture deficits promote warming (Fig. 6d, dark boxes; see also Fig. 6c). By construction, this warming should be interpreted as a response to drying, and not the other way around, since trends have been removed and the use of time lags minimizes misattributions of cause and effect. Random distributions (unconstrained by the atmospheric circulation) indicate similar warming levels following short-term soil moisture deficits (Fig. 6d, light boxes). Accordingly, regional drying seems to favour above-normal temperatures, regardless of the atmospheric circulation. Interestingly, additional analyses reveal atmospheric circulation differences between the flow analogues of dry and wet years (Fig. 6e, dark whiskers), involving larger positive Z500 anomalies for flow analogues preceded by soil moisture deficits (see Fig. 6c), which translate to lower RMSE (i.e. closer patterns to the actual circulation) than during wet conditions. These differences in RMSE are also observed for the unconditional distributions (Fig. 6e, light whiskers), indicating an overall tendency for dry periods to precede higher pressure anomalies. This could reflect methodological issues (e.g. limited sampling, residual trends, autocorrelation issues), although we obtain similar results for dry and wet periods of the ERA5 (1979–2019) and ERA20C (1900–2010) reanalyses (see Supplementary Figs. 9 and 10). Alternatively, the Z500 differences between dry and wet conditions may also indicate feedbacks of soil moisture deficits on the atmospheric circulation anomalies. If the latter is the case, such effect herein involves somehow weak high-pressure patterns, whose spatial details depend on the considered dataset (cf. Fig. 6c and Supplementary Fig. 9c) and methodological choices. Previous studies have suggested atmospheric circulation responses to soil moisture deficits, including local effects through thermal expansion by enhanced sensible heat fluxes40, and remote effects caused by a thermally-induced low41 or changes in cloud cover42. In short, our results indicate that soil moisture deficits in western Europe intensified the warming already expected from the circulation observed during the July HW and might even have contributed to amplifying the circulation anomalies. Additional studies are warranted to explore and quantify the contribution of atmospheric circulation responses induced by land–atmosphere coupling to the intensity of HWs.
### Summary and discussion
Two distinct HWs affected widespread areas of western Europe in June and July 2019, contributing to placing that summer within the top five warmest since 1500 at the European scale. While the spatial distribution of the affected areas strongly resembled the historical HW of August 2003, the relatively shorter-lived 2019 HWs were more intense on daily time scales, shattering previous all-time records in many places (some of them standing since 2003). Here we have dissected these events with recently developed tools to provide an assessment of different relevant factors: (i) the role of the dynamics (synoptic setups associated with the 2019 HWs); (ii) underlying physical processes (warm advection vs. diabatic fluxes, including enhanced near-surface heating due to soil moisture deficits); (iii) recent thermodynamic changes (the steady regional warming trend).
We have applied recent novel methodologies to track the two 2019 HWs and the occurrence of subtropical warm air intrusions. The June HW displayed a clear fingerprint of a Saharan intrusion. The advection of exceptionally warm and dry air, together with enhanced subsidence under pronounced Z500 anomalies, are the main features of this event that triggered all-time temperature records in southern France and northeastern Spain. Differently, the July HW extended much further north, and unprecedented temperatures hit a comparatively larger domain of Europe. Diabatic processes, rather than temperature advection associated with a Saharan intrusion, played a dominant role for the setup of this event. Some studies have found different relative contributions of horizontal advection, subsidence and diabatic processes in shaping European HWs22,36. Our analysis attests to the distinct nature of two HWs that took place over the same region within a few weeks, supporting the coexistence of distinct dominant forcing mechanisms for their onset and maintenance.
We have shown evidence supporting a contribution of land–atmosphere coupling to the temperature anomalies during the July HW, potentially involving the self-propagation mechanism discussed for previous HWs20,21. The atmospheric conditions prevailing since the onset of the June HW, i.e. prolonged periods with no rain and persistently high solar radiative fluxes and temperatures, significantly contributed to anomalous soil moisture deficits over large areas, in particular over France during the transition period between both HWs. This resulted in strong energy transfer between the soil and atmosphere, via increased (decreased) sensible (latent) heat fluxes before the onset of the July HW. These diagnostics of land–atmosphere feedbacks were not restricted to the region most affected by the July HW (northeastern France and Belgium). They were also observed further south in areas under the influence of sustained southerlies, which suggests a northward propagation of dryness through the advection of warm air masses. By using atmospheric flow analogues, our analysis further supports the role of soil desiccation on the amplification of the July 2019 HW. In this context, lessened soil-atmosphere feedbacks have been reported in areas where shallow groundwater is available43. Accordingly, we argue that an event occurring under similar synoptic patterns to those observed in July 2019 would result in lower temperature anomalies if preceding soil conditions were wetter. Our results also suggest non-negligible effects of the June HW on the July HW through an imprint of the former on the soil moisture deficits that influenced the latter. However, further modelling studies are warranted to support this conclusion, as well as to address land–atmosphere feedbacks on the atmospheric circulation. In this regard, recent ensemble experiments provide evidence about complex local and remote effects of soil moisture deficits up to two months, including non-local responses in atmospheric circulation that can further amplify HW magnitudes42.
Our results also indicate that previous colder climatic conditions would have resulted in less soil desiccation than observed. This effect is found regardless of the atmospheric circulation, pointing to atmospheric warming effects as a consequence of anthropogenic forcing44, probably enhanced by land-use and land-cover changes43,45. We acknowledge limitations (e.g. limited sample size, transient climate conditions over the analyzed period or biases in the reanalysis datasets) and assumptions (e.g. event definition) in our analogue experiments. Moreover, their results should not be interpreted as a formal attribution to anthropogenic forcing, despite the overall consistency based on independent evidence. For example, following earlier studies on European HWs46, recent findings also indicate that hot summer temperatures in the Mediterranean area are often shortly preceded by the occurrence of dryness in spring or even early summer47, thus favoring temporally compounding events48. These facts highlight that future extreme heat episodes will probably be even further exacerbated by the increasing severity of drought events37,49, as stronger losses by evaporation are expected in a warming climate50 whenever soil moisture is still available51.
## Data and methods
### E-OBS dataset
Daily minimum and maximum 2 m temperature from the E-OBS gridded dataset (v21.0) was used to characterize anomalies and extremes during the 2003 and 2019 events, as well as trends and record-breaking values during the available period (since 1950). Anomalies are computed by removing the daily climatological mean (1981–2010). E-OBS is a European land-only high-resolution gridded observational dataset, using the European Climate Assessment and Dataset (ECA&D) blended daily station data52. It is presented on a horizontal resolution of 0.25°×0.25°. Files are replaced in monthly updates and in updated versions of the E-OBS dataset. Accordingly, small changes might occur between these releases after new data and/or stations are added.
### NCEP/NCAR dataset
Meteorological fields were retrieved from the NCEP/NCAR reanalysis daily dataset53, starting from 1948. The following variables were considered for pressure levels between 1000 and 500 hPa on a 2.5° × 2.5° horizontal resolution grid: air temperature, geopotential height, zonal/meridional wind components, vertical velocity. We also analyzed other fields represented in a Gaussian grid: surface net radiation fluxes (long-wave and short-wave), latent and sensible heat fluxes, precipitation, 2 m temperature, 10 m wind, potential evapotranspiration and soil moisture fraction (0–10 cm and 10–200 cm). These fields were used to: (i) characterize and track the HW events, (ii) derive a catalogue of Saharan intrusions, (iii) generate vertical profiles, (iv) compute the contributing terms to the temperature tendency equation, and (v) perform the analogue exercises. Specific methods for products derived from these variables are explained below. In all cases, anomalies are computed with respect to the climatological seasonal cycle (1981–2010).
### ERA5 and ERA20C datasets
Meteorological fields were extracted from two ECMWF (European Centre of Medium-range Weather Forecast) reanalyses to replicate the analogue exercises (see methodology further ahead) performed with the NCEP/NCAR dataset. The ERA554 and ERA20C55 datasets were considered, using the highest horizontal resolution available for the latter (1.25° × 1.25°), for the 1979–2019 and 1900–2019 periods, respectively. Daily time series of 2 m temperature, Z500 and soil moisture fraction (0–7 cm) were retrieved.
### European temperature reconstruction since 1500
We use a near-surface temperature reconstruction on a 0.5° × 0.5° regular grid over [35°–70° N, 25° W – 40° E] based on long instrumental series and different proxies (including Greenland ice cores, tree rings and documentary sources)31,56. This reconstruction covers the period 1500–2002, although data for 1901–2002 comes from instrumental datasets. Near-surface temperature analyses of the Goddard Institute for Space Studies (GISS, data.giss.nasa.gov/gistemp/)32 were herein used at 2° × 2° spatial resolution to update the temperature reconstruction over the period 1901–2019. This monthly observational dataset was herein used at 2° × 2° spatial resolution to update the temperature reconstruction over the period 1901–2019. To do so, reconstructions and instrumental observations were linearly interpolated into a common 2.5° × 2.5° resolution grid over land, as that employed in Barriopedro et al.57. Afterwards, seasonal mean temperature anomalies were computed with respect to their respective 1981–2010 climatologies. Finally, the European mean temperature anomaly of each summer in the 1500–2019 period was computed as the area-weighted mean of all land 2.5° grid cells.
To assess whether the decadal frequency of extreme European summers is significantly higher than that expected by random chance we performed a 1000-trial bootstrap, each containing a randomly resampled series of the European summer temperature anomalies over 1500–2019. For each trial, the maximum running decadal frequency was retained, with the 95th percentile of the resulting distribution identifying the value whose one-tailed likelihood of occurring by chance is less than 5%.
### C3S soil moisture dataset
The C3S dataset from Copernicus provides estimates of volumetric soil moisture (in m3 m−3) in a layer of 2 to 5 cm depth, retrieved from a large set of satellite sensors. Data is presented on a 0.25° × 0.25° regular grid with some gaps in space and time. Climate Data Records (CDR) and interim-CDR (ICDR) products are generated using the same software and algorithms. CDR is intended to have sufficient length, consistency, and continuity to characterize climate variability and change. ICDR provides a short-delay access to current data where consistency with the CDR baseline is expected but has not been extensively checked. The dataset contains the following products: “active”, “passive” and “combined”. The “active” and “passive” products are created by using scatterometer and radiometer soil moisture products, respectively. The “combined” product results from a blend based on the two previous products. Here we used the “combined” dataset, which is available for Europe since 1978. Climatological means for each calendar day and grid cell were computed in order to derive local and regional anomalies during the 2019 HW events.
Data is accessible online trough: https://cds.climate.copernicus.eu/cdsapp#!/dataset/satellite-soil-moisture.
### HW algorithm
To perform a spatio-temporal tracking of the 2019 summer HWs, we have adopted a semi-Lagrangian perspective. The 850 hPa temperature (T850) from the NCEP/NCAR dataset was used to analyze the spatio-temporal evolution of extreme temperature patterns, instead of considering HWs as isolated local surface extremes, thus enabling the temporal monitoring of the spatial extent of HWs affecting distinct areas during their lifecycle. The algorithm identifies HW events, defined as areas larger than 500,000 km2 with daily mean T850 above the local daily 95th percentile (with respect to 1981–2010) that persist for at least four consecutive days and fulfil some predefined conditions on spatial overlap during those days. Additional information on this methodology can be found in Sánchez-Benítez et al.58.
### Saharan intrusions
A catalogue of air masses with subtropical desertic characteristics was obtained relying on simple thermodynamic air properties, considering the following conditions:
1. (i)
1000–500 hPa geopotential height thickness higher than 5800 m 59;
2. (ii)
925–700 hPa potential temperature (θ) above 40 °C.
Grid cells satisfying both criteria correspond to low density, warm, stable and very dry air masses, with the potential to be additionally warmed by subsidence60. Using the NCEP/NCAR reanalysis dataset, we have classified the mean climatological (1948–2019) location and extension of Saharan air masses during summer, identifying temporary intrusions towards higher latitudes for each grid cell on a daily basis. Further details on the methodology can be found in Sousa et al.12.
### Temperature tendency and related processes
The contributions of horizontal advection and vertical descent to temperature tendency were determined as
$$\left( {\frac{{{\Delta}T}}{{{\Delta}t}}} \right)_h\left( {\lambda ,\phi ,t} \right) = - \vec v \cdot \nabla _pT,$$
(1)
$$\left( {\frac{{{\Delta}T}}{{{\Delta}t}}} \right)_v\left( {\lambda ,\phi ,t} \right) = - \omega \frac{T}{\theta }\frac{{\partial \theta }}{{\partial p}},$$
(2)
where (1) is the temperature advection by the horizontal wind, and (2) the temperature tendency by vertical motion. Equations (1) and (2) are computed from daily mean fields in constant pressure coordinates, according to the pressure levels available in the NCEP/NCAR dataset, with , ϕ, t) representing latitude, longitude and time, respectively, and v being the horizontal wind, T the temperature, ω the vertical velocity and θ the potential temperature. The daily mean temperature rate due to other diabatic processes (e.g. radiative and heat fluxes) is estimated as a residual from the previous two terms based on the temperature tendency equation:
$$\left( {\frac{{{\Delta}T}}{{{\Delta}t}}} \right)_d\left( {\lambda ,\phi ,t} \right) = \frac{{{\Delta}T}}{{{\Delta}t}} - \left( {\frac{{{\Delta}T}}{{{\Delta}t}}} \right)_h \, - \, \left( {\frac{{{\Delta}T}}{{{\Delta}t}}} \right)_v,$$
(3)
where the first term on the right-hand side of (3) is the daily mean temperature tendency (in °C day−1). It must be kept in mind that different factors such as sub-grid turbulent mixing, analysis increments and other numerical errors may contribute to the residual term. This bulk analysis is performed for the 1000–850 hPa layer. The relative contribution of each term to the temperature tendency is used to identify the dominant mechanism for each day and grid cell. For further details on the methodology the reader is referred to Sousa et al.26.
### Analogue method
We use the analogue method, which infers the probability distribution of a target field from the atmospheric circulation during a considered time interval28. Herein, two analogue exercises were designed, one for each HW event, but with different target fields, as explained below. In both cases, flow analogue days are defined from their root-mean-square errors (RMSE) with respect to the actual Z500 anomaly field at the time of the HW event over a given domain ([35°–65°N, 10° W–25°E] for the June HW, and [40°–70°N, 10° W–25°E] for the July HW, following the regions with the largest Z500 anomalies; Fig. 2b, c). For each day of the considered HW events, the search of flow analogues was restricted to the [−31,31] day interval (i.e. 62-day window) around the corresponding calendar day, excluding the year of occurrence of the HW. Similar results are obtained for 15- and 31-day windows. Analogue days are used to reconstruct the target field by randomly picking one of the N best flow analogues for each day of the HW event. This number was determined based on the pool size of eligible days (Y × L × D, where Y is the number of years, L the length of the window and D the duration of the event) and their associated RMSE distribution, with a minimum value of 20. This N value ranges from 20 to 40, depending on the available period of the dataset. In all cases, the mean RMSE of these N best analogues averaged over all days of the event was below the 10th percentile of the RMSE distribution. For each day, the random selection of flow analogues was repeated 5000 times to derive flow-conditioned distributions. To test whether the dynamics played a significant role in the reconstructed anomalies of the target field, unconditional distributions were also retrieved by repeating the whole process with a random selection of days (instead of restricting the search to N days with similar flow configurations). Different choices in the spatial domain or the number of circulation analogues (e.g. N values ranging between 10 and 50) were tested, yielding similar results. For both analogue exercises, Z500 and volumetric soil moisture content at 0–10 cm were obtained from daily means of the NCEP/NCAR reanalysis (1950–2019). EOBS (1950–2019) was employed for daily maximum temperature at 2 m in the second analogue exercise, although the conclusions remain unchanged if NCEP/NCAR is used instead. We also repeated the analogue exercises with equivalent fields from different periods and reanalyses: ERA5 (1979–2019) and ERA20C (1900–2010; herein taking the 2019 fields from ERA5). Note that in both ERA reanalyses the uppermost soil layer spans 0–7 cm, and that ERA20C only assimilates surface and mean sea level pressure and surface marine winds. In all cases, anomalies are defined with respect to the 1981–2010 period.
In the first analogue exercise, we reconstructed the expected mean volumetric soil moisture fraction for the 15-day period ([1,15] day interval) after each day of the June HW (24 June–1 July 2019) by using daily flow analogues from the present and past subperiods separately, defined as 1984–2018 (1999–2018 and 1951–2010) and 1950–1983 (1979–1998 and 1900–1950) in NCEP/NCAR (ERA5 and ERA20C), respectively. This 15-day period is similar to the temporal interval between the end of the June HW and the beginning of the July HW (Fig. 4a), but we obtain similar results for other choices (e.g. [5,20] or [1,30] day intervals). In addition to spatial fields, flow-conditioned and random distributions of the mean soil moisture content over WEU ([43°–53° N, 0°–10° E]) were computed for each subperiod. As the atmospheric circulation is constrained, the difference between the reconstructions of the past and present should largely be ascribed to overall climatological differences between the two subperiods, enabling the estimation of the effect of recent changes in the soil moisture distributions, howsoever caused.
A second flow analogue exercise was performed to address whether the previously accumulated soil moisture deficits over WEU could have contributed to intensifying the temperature anomalies over that region at the time of the July HW. In this case, we reconstructed the maximum 2 m temperature anomalies expected from the circulation during the July HW, distinguishing between analogue days preceded by dry and wet conditions. Wet and dry conditions are defined as summer days of the full period with 15-day mean regional anomalies for the previous [−15,−1] day interval staying above the 66.6th percentile and below the 33.3rd percentile of the climatological distribution, respectively. That way, soil moisture departures of a given analogue day represent previously accumulated values and are not the direct response to the actual atmospheric circulation conditions. We tested the robustness of the results with respect to the percentile employed for the classification of dry and wet years (e.g. the first and last deciles or quartiles), reporting larger differences for the most extreme definitions. To avoid the effects of long-term trends that may further complicate the causality of the relationships between soil moisture and temperature, these fields were detrended by removing the local trends. For Z500, we removed the regional mean linear trend over the considered domain in order to keep the spatial gradients when searching for flow analogues. Flow-conditioned and random distributions of regional mean temperature anomalies over WEU were also derived for wet and dry conditions.
The choice of reanalysis, periods and other methodological aspects can affect the results quantitatively, as well as the spatial details of the reconstructed patterns. However, for the datasets employed and sensitivity tests described above, we did not report substantial differences that affect the main conclusions of the text, therefore adding confidence on the results. | 2023-02-08 13:34:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45669683814048767, "perplexity": 3455.12153455253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500813.58/warc/CC-MAIN-20230208123621-20230208153621-00361.warc.gz"} |
https://astronomy.stackexchange.com/questions/11594/what-is-the-largest-hydrogen-burning-star?rq=1 | # What is the largest hydrogen-burning star?
I am wondering what is the largest known core hydrogen-burning star? A look at the list of largest known stars on Wikipedia seems to indicate VV Cephei B (at the bottom of the list), but I would like to know for sure if it is the largest known. In addition to knowing which star it is, I would also like to know its temperature, size, and expected lifetime.
I am also curious to know if the largest known core hydrogen-burning star is similar to what astrophysicists theorize is the largest possible core hydrogen-burning star (given current metallicity conditions in the universe; I know star-formation timescales have a metallicity dependence) and the expected temperature, size, and lifetime of such an object.
• I'm not crystal clear on what you mean by hydrogen burning star, or why you picked VV Cephei B. All young stars burn hydrogen, even very large ones. Do you mean burn primarily hydrogen or only hydrogen (ie, not hot enough to burn helium) or haven't run out of hydrogen fuel? – userLTK Aug 20 '15 at 2:23
• I mean main sequence, I think. – NeutronStar Aug 20 '15 at 2:38
• You probably do, since even when stars go off the main sequence they still tend be burning hydrogen in outer shells. – zibadawa timmy Aug 20 '15 at 7:00
• Thanks for pointing that out. I updated the question to be core hydrogen burning. – NeutronStar Aug 20 '15 at 11:15
I assume by largest, you mean largest radius.
Well it won't be VV Cep B since this is merely a B-type main sequence star.
O-type main sequence stars are known and these have both larger masses and larger radii on the main sequence (when they are burning hydrogen in their cores).
A selection of the most massive objects can be found in the R136 star forming region in the Large Magellanic Clouds. If you look at this list (though I recommend having a look at the primary literature), you will see that O3V stars are listed. Such objects are also present in our Galaxy, for instance in the supercluster NGC 3603 (Crowther & Dessart 1998).
Such stars have masses of maybe $100 M_{\odot}$, luminosities of $2\times 10^{6} L_{\odot}$ and temperatures of 50,000 K. Using Stefan's law, we can deduce radii of $\sim 20 R_{\odot}$.
There are suggestions that even more massive main sequence stars have existed in R136 and NGC 3603 (see Crowther et al. 2010), which are now seen as evolved Wolf-Rayet objects, possibly up to $300 M_{\odot}$ on the main sequence (though this is a model-dependent extrapolation), and these would have had radii $>20 R_{\odot}$.
In the very early universe, population III main sequence stars without metals could have been much more massive and larger. | 2020-02-18 12:46:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.596785306930542, "perplexity": 1048.3411829988822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143695.67/warc/CC-MAIN-20200218120100-20200218150100-00036.warc.gz"} |
https://plastid.readthedocs.io/en/latest/FAQ.html | ## Installation and runtime¶
### Installation fails in pip with no obvious error message¶
Installation can fail for multiple reasons. To figure out what is responsible, repeat installation passing the --verbose flag to pip:
$pip install --no-cache-dir --verbose plastid | tee 2>&1 plastid_install_log.txt Then find the corresponding error message below. If the error is not listed, let us know by filing a bug report at our issue tracker. Please attach plastid_install_log.txt to your report to help us figure out what is going on. ### Installer quits with an error message about numpy, pysam and Cython¶ The most common cause of installation failure is that the setup script for plastid requires numpy, pysam, and Cython to be installed before it can be run. In this case, the following message should appear: *** IMPORTANT INSTALLATION INFORMATION *** plastid setup requires numpy>=1.9.0, pysam>=0.8.4, and cython>=0.22 to be preinstalled. Please install these via pip, and retry:$ pip install --upgrade numpy pysam cython
$pip install plastid If this is the problem, simply install numpy, pysam and Cython first, then repeat the installation: $ pip install numpy pysam cython
$pip install plastid Check if install worked: $ pip list
You should see numpy, pysam, and Cython in the list.
### numpy, pysam and Cython are up to date, but install still fails¶
If install fails when numpy, pysam and Cython are already up to date, then there may be multiple versions of these libraries installed on your system, with plastid seeing an earlier version than pip. A few users have come across this problem when installing plastid in a conda/Anaconda environment.
A solution is to install inside a vanilla environment in a fresh virtualenv (note: not a conda/Anaconda environment). See instructions at Install inside a virtualenv.
If install succeeds in a virtualenv, this suggests that there are in fact multiple versions of one or more of plastid’s dependencies installed on your system. In this case, plastid can be used inside the virtualenv.
### Install fails inside a conda/Anaconda environment¶
Two users have reported difficulties installing inside conda/Anaconda environments. Despite having up-to-date versions of numpy, pysam, and Cython installed in the conda environment, during the build process pip found an incompatible version of Cython or of pysam.
A workaround is to install inside a virtualenv, instructions for which can be found at Install inside a virtualenv.
### Locale error when installing or running plastid on OSX¶
This is known to occur on OSX. In this case, you should see a stack trace ending with something like:
from docutils.utils.error_reporting import locale_encoding, ErrorString, ErrorOutput
File "/Applications/anaconda/lib/python2.7/site-packages/docutils/utils/error_reporting.py", line 47, in <module>
locale_encoding = locale.getlocale()[1] or locale.getdefaultlocale()[1]
File "/Applications/anaconda/lib/python2.7/locale.py", line 543, in getdefaultlocale
return _parse_localename(localename)
File "/Applications/anaconda/lib/python2.7/locale.py", line 475, in _parse_localename
raise ValueError, 'unknown locale: %s' % localename
ValueError: unknown locale: UTF-8
### Install fails on OSX with error code 1¶
If installing on OSX and you find an error message that resembles the following:
Command "/usr/local/opt/python/bin/python2.7 -c "import setuptools, tokenize;\
__file__='/private/var/folders/8y/xm0qbq655f1d4v20kq5vvfgm0000gq/T/pip-build-0bVdPy/pysam/setup.py';\
exec(compile(getattr(tokenize, 'open', open)(__file__).read().replace('\r\n', '\n'), __file__, 'exec'))"\
install --record /var/folders/some-folder/install-record.txt --single-version-externally-managed \
--compile --user --prefix=" failed with error code 1 in /private/var/folders/some-folder/pysam
Before installing, type:
$export CFLAGS=-Qunused-arguments$ export CPPFLAGS=-Qunused-arguments
and then retry.
### command not found: I can’t run any of the command line scripts¶
If you receive a command not found error from the shell, the folder containing the command-line scripts might not be in your environment’s PATH variable.
Command-line scripts will be installed wherever your system configuration dictates. On OSX and many varities of linux, the install path for a single-user install is ~/bin or ~/.local/bin. For system-wide installs, the path is typically /usr/local/bin. Make sure the appropriate location is in your PATH by the following line adding to your .bashrc, .bash_profile, or .profile (depending on which your system uses):
export PATH=~/bin:~/.local.bin:/usr/local/bin:$PATH ### A script won’t run, reporting error: too few arguments¶ If you see the following error: <script name>: error: too few arguments Try re-ordering the script arguments, so that all of the required arguments (the ones that don’t start with -) come first. For example, change: $ cs count --fiveprime --offset 13 --min_length 23 --max_length 35 \
--count_files ../some_file.bam some_file.positions some_sample_name
to
$cs count some_file.positions some_sample_name \ --fiveprime --offset 13 --min_length 23 --max_length 35 \ --count_files ../some_file.bam Alternatively, put a -- before the required options: $ cs count --fiveprime --offset 13 --min_length 23 --max_length 35 \
--count_files ../some_file.bam \
-- some_file.positions some_sample_name
Things should then run.
### I get an ImportError or DistributionError when using plastid¶
If you get an error like the following:
Traceback (most recent call last):
File "/home/user/Rib_prof/venv/bin/crossmap", line 5, in <module>
File "/home/user/Rib_prof/venv/lib/python2.7/site-packages/pkg_resources/__init__.py", line 2970, in <module>
working_set = WorkingSet._build_master()
File "/home/user/Rib_prof/venv/lib/python2.7/site-packages/pkg_resources/__init__.py", line 567, in _build_master
ws.require(__requires__)
File "/home/user/Rib_prof/venv/lib/python2.7/site-packages/pkg_resources/__init__.py", line 876, in require
needed = self.resolve(parse_requirements(requirements))
File "/home/user/Rib_prof/venv/lib/python2.7/site-packages/pkg_resources/__init__.py", line 761, in resolve
raise DistributionNotFound(req)
pkg_resources.DistributionNotFound: scipy>=0.12.0
One or more dependencies (in this example, SciPy is not installed). Please see Installer quits with an error message about numpy, pysam and Cython, above.
## Analysis¶
### Can I use plastid with reverse-complemented sequencing data, like dUTP sequencing?¶
Yes.
Kits like Illumina’s Truseq Stranded mRNA Library Prep Kit, yield reads that are anti-sense to the mRNA from which they were generated, so the data coming off the sequencer will be reverse-complemented compared to the original strand that was cloned.
To use this data in plastid, reverse-complement your FASTQ file using the fastx_reverse_complement tool from the Hanon lab’s fastx toolkit. Then align the reverse-complemented data using your favorite aligner.
### Can plastid be used with paired-end data?¶
Yes, but two points:
• Because there are few nucleotide-resolution assays that used paried-end sequencing, it has been unclear what sorts of mapping functions might be useful.
If you have a suggestion for one, please submit your suggestion with a use case on our issue tracker. It would be helpful!
• For simple gene expression counting, it is possible to use the --fiveprime (implemented in FivePrimeMapFactory) mapping function with zero offset. Accuracy can be improved by counting a single read from each pair in which both reads are mapped.
However, it is critical to retain the read that appears on the same strand as the gene from which it arose.
If the library was prepared using dUTP chemistry, as in many paired-end prep kits, select read2 from each pair using samtools:
$samtools view -f 129 -b -o single_from_pair.bam paired_end_file.bam Otherwise, select read1: $ samtools view -f 65 -b -o single_from_pair.bam paired_end_file.bam
Then use single_from_pair.bam with plastid as usual.
### The P-site and/or Metagene scripts show few or zero read in their output¶
This occurs in datasets with few counts, because psite and metagene plots the median density at each position. In this case, there are a few options:
• increase the minimum counts required to be included in the metagene / P-site estimate. Set --min_counts argument to a high number (e.g. for a 100 nt normalization region, choose >= 100 counts)
• the metagene profile or P-site can be estimated from aggregate counts (as opposed to median density) at each position using the --aggregate argument, as shown here.
This might add some noise to the data, but it should still be interpretable if the gene models are good
### cs, counts_in_region, or some other part of plastid reports zero counts for my gene, even though there are read alignments there¶
The default behavior for all of the scripts and tools in plastid is to exclude reads that are antisense to any given genomic feature when calculating coverage over that feature.
Paired end libraries, and single-end libraries that have been prepared with dUTP sequencing or a number of other protocols will contain read alignments antisense to the original mRNAs, causing these reads to be considered antisense to genes, and therefore excluded from gene expression totals.
See Can I use plastid with reverse-complemented sequencing data, like dUTP sequencing? for instructions on how to reverse-complement your data for single-end dUTP data; or Can plastid be used with paired-end data? for info on using plastid with paired-end data.
### Why do some scripts report fractional count numbers?¶
Fractional counts for read alignments arise when using a alignment mapping rule that maps reads fractionally over multiple positions (such as --center mapping). See the discussion of Read mapping functions, where these are discussed in depth.
### Why does IGV report higher coverage at a given nucleotide than the file exported from make_wiggle?¶
When IGV calculates coverage of a nucleotide, it counts the number of alignments covering that nucleotide. So, a 30-nucleotide read would contribute 30 counts to a dataset.
While it is possible to write any mapping rule in plastid, the mapping rules included by default count each read only once (e.g. at their 5’ end, 3’ end, et c). Even when using center or entire mapping, each position covered by a read alignment is only incremented by $$1.0/\ell$$, where $$\ell$$ is the length of the read. So, in this case, a 30-nucleotide read would only contribute 1 count to a dataset. See Read mapping functions for more information.
### What are the differences between counts_in_region and cs?¶
counts_in_region very simply counts read coverage (or any data) over regions of interest, and reports those numbers in terms of counts and RPKM. It can optionally take a mask file, if there are genomic positions in the regions of interest which should be excluded from analysis. Otherwise, it makes no corrections.
cs is more complex, and is principally designed to make rough estimates of gene expression at the gene, rather than transcript, level. In so doing, it makes several heuristic corrections to regions before tabulating their counts and RPKM. Specifically:
1. Genes that have transcripts that share exons are merged into single entities
2. Gene areas are defined for each merged geen by including all positions occupied by all transcripts from that merged gene
3. Regions occupied by two or more merged genes on the same strand are excluded from the calculation of expression values for both genes
4. Optionally, a mask file can be used to exclude any other positions from analysis.
5. Expression values (in counts and RPKM) are tabulated for the entire gene area (reported as exon_counts and exon_rpkm) as well as for sub regions, if the gene is coding. Specifically, cds_counts and cds_rpkm are calculated from counts that cover positions in the gene area that are annotated as CDS in all transcripts in the merged gene. Ditto for 5’ and 3’ UTRs
Either one can be an appropriate starting place for a pipeline, depending upon your needs. See the documentation and/or source code for cs and counts_in_region for further discussion.
### Why does SegmentChain.as_gff3() sometimes throw errors?¶
The incredible fle flexibility of the GFF3 file format introduces ambiguities for representation of discontinuous features: some sort of parent-child relationship needs to exist, and, except in the case of transcripts, plastid doesn’t know which one to use.
See this advice on how to handle this.
### How do I prepare data for differential gene expression analysis in DESeq?¶
See Gene expression analysis in the Tutorials section.
## Tests¶
### The tests won’t run¶
In order to run the tests, you need to download the test dataset and unpack it into plastid/test/.
We decided not to include the test data in the main package in order to keep the package download and the github repository small. | 2020-08-15 08:35:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31022050976753235, "perplexity": 6918.832760755218}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740733.1/warc/CC-MAIN-20200815065105-20200815095105-00105.warc.gz"} |
https://www.springerprofessional.de/grammatical-inference-theoretical-results-and-applications/3356460 | scroll identifier for mobile
main-content
## Inhaltsverzeichnis
### Grammatical Inference and Games: Extended Abstract
This paper discusses the potential synergy between research in grammatical inference and research in artificial intelligence applied to games. There are two aspects to this: the potential as a rich source of challenging and engaging test problems, and the potential for real applications.
Simon M. Lucas
### Molecules, Languages and Automata
Molecular biology is full of linguistic metaphors, from the language of DNA to the genome as “book of life.” Certainly the organization of genes and other functional modules along the DNA sequence invites a syntactic view, which can be seen in certain tools used in bioinformatics such as hidden Markov models. It has also been shown that folding of RNA structures is neatly expressed by grammars that require expressive power beyond context-free, an approach that has even been extended to the much more complex structures of proteins. Processive enzymes and other “molecular machines” can also be cast in terms of automata. This paper briefly reviews linguistic approaches to molecular biology, and provides perspectives on potential future applications of grammars and automata in this field.
David B. Searls
### Inferring Regular Trace Languages from Positive and Negative Samples
In this work, we give an algorithm that infers Regular Trace Languages. Trace languages can be seen as regular languages that are closed under a partial commutation relation called the independence relation. This algorithm is similar to the RPNI algorithm, but it is based on Asynchronous Cellular Automata. For this purpose, we define Asynchronous Cellular Moore Machines and implement the merge operation as the calculation of an equivalence relation. After presenting the algorithm we provide a proof of its convergence (which is more complicated than the proof of convergence of the RPNI because there are no Minimal Automata for Asynchronous Automata), and we discuss the complexity of the algorithm.
Antonio Cano Gómez
### Distributional Learning of Some Context-Free Languages with a Minimally Adequate Teacher
Angluin showed that the class of regular languages could be learned from a Minimally Adequate Teacher (
mat
) providing membership and equivalence queries. Clark and Eyraud (2007) showed that some context free grammars can be identified in the limit from positive data alone by identifying the congruence classes of the language. In this paper we consider learnability of context free languages using a
mat
. We show that there is a natural class of context free languages, that includes the class of regular languages, that can be polynomially learned from a
mat
, using an algorithm that is an extension of Angluin’s
lstar
algorithm.
Alexander Clark
### Learning Context Free Grammars with the Syntactic Concept Lattice
The Syntactic Concept Lattice is a residuated lattice based on the distributional structure of a language; the natural representation based on this is a context sensitive formalism. Here we examine the possibility of basing a context free grammar (
cfg
) on the structure of this lattice; in particular by choosing non-terminals to correspond to concepts in this lattice. We present a learning algorithm for context free grammars which uses positive data and membership queries, and prove its correctness under the identification in the limit paradigm. Since the lattice itself may be infinite, we consider only a polynomially bounded subset of the set of concepts, in order to get an efficient algorithm. We compare this on the one hand to learning algorithms for context free grammars, where the non-terminals correspond to congruence classes, and on the other hand to the use of context sensitive techniques such as Binary Feature Grammars and Distributional Lattice Grammars. The class of
cfg
s that can be learned in this way includes inherently ambiguous and thus non-deterministic languages; this approach therefore breaks through an important barrier in
cfg
inference.
Alexander Clark
### Learning Automata Teams
We prove in this work that, under certain conditions, an algorithm that arbitrarily merges states in the prefix tree acceptor of the sample in a consistent way, converges to the minimum
DFA
for the target language in the limit. This fact is used to learn automata teams, which use the different automata output by this algorithm to classify the test. Experimental results show that the use of automata teams improve the best known results for this type of algorithms. We also prove that the well known Blue-Fringe EDSM algorithm, which represents the state of art in merging states algorithms, suffices a polynomial characteristic set to converge.
Pedro García, Manuel Vázquez de Parga, Damián López, José Ruiz
### Exact DFA Identification Using SAT Solvers
We present an exact algorithm for identification of deterministic finite automata (DFA) which is based on satisfiability (SAT) solvers. Despite the size of the low level SAT representation, our approach is competitive with alternative techniques. Our contributions are fourfold: First, we propose a compact translation of DFA identification into SAT. Second, we reduce the SAT search space by adding lower bound information using a fast max-clique approximation algorithm. Third, we include many redundant clauses to provide the SAT solver with some additional knowledge about the problem. Fourth, we show how to use the flexibility of our translation in order to apply it to very hard problems. Experiments on a well-known suite of random DFA identification problems show that SAT solvers can efficiently tackle all instances. Moreover, our algorithm outperforms state-of-the-art techniques on several hard problems.
Marijn J. H. Heule, Sicco Verwer
### Learning Deterministic Finite Automata from Interleaved Strings
Workflows are an important knowledge representation used to understand and automate processes in diverse task domains. Past work has explored the problem of learning workflows from traces of processing. In this paper, we are concerned with learning workflows from
interleaved
traces captured during the concurrent processing of multiple task instances. We first present an abstraction of the problem of recovering workflows from interleaved example traces in terms of grammar induction. We then describe a two-stage approach to reasoning about the problem, highlighting some negative results that demonstrate the need to work with a restricted class of languages. Finally, we give an example of a restricted language class called
terminated languages
for which an accepting deterministic finite automaton (DFA) can be recovered in the limit from interleaved strings, and make remarks about the applicability of the two-stage approach to terminated languages.
Joshua Jones, Tim Oates
### Learning Regular Expressions from Representative Examples and Membership Queries
A learning algorithm is developed for a class of regular expressions equivalent to the class of all unionless unambiguous regular expressions of loop depth 2. The learner uses one
representative
example of the target language (where every occurrence of every loop in the target expression is unfolded at least twice) and a number of membership queries. The algorithm works in time polynomial in the length of the input example.
Efim Kinber
### Splitting of Learnable Classes
A class
$\mathcal{L}$
is called mitotic if it admits a splitting
$\mathcal{L}_0,\mathcal{L}_1$
such that
$\mathcal{L},\mathcal{L}_0,\mathcal{L}_1$
are all equivalent with respect to a certain reducibility. Such a splitting might be called a symmetric splitting. In this paper we investigate the possibility of constructing a class which has a splitting and where any splitting of the class is a symmetric splitting. We call such a class a symmetric class. In particular we construct an incomplete symmetric BC-learnable class with respect to strong reducibility. We also introduce the notion of very strong reducibility and construct a complete symmetric BC-learnable class with respect to very strong reducibility. However, for EX-learnability, it is shown that there does not exist a symmetric class with respect to any weak, strong or very strong reducibility.
Hongyang Li, Frank Stephan
### PAC-Learning Unambiguous k,l-NTS ≤ Languages
In this paper we present two hierarchies of context-free languages: The
k
,
l
-NTS languages and the
k
,
l
-NTS
≤
languages.
k
,
l
-NTS languages generalize the concept of Non-Terminally Separated (NTS) languages by adding a fixed size context to the constituents, in the analog way as
k
,
l
-substitutable languages generalize substitutable languages (Yoshinaka, 2008).
k
,
l
-NTS
≤
languages are
k
,
l
-NTS languages that also consider the edges of sentences as possible contexts. We then prove that Unambiguous
k
,
l
-NTS
≤
(
k
,
l
-UNTS
≤
) languages be converted to plain old UNTS languages over a richer alphabet. Using this and the result of polynomial PAC-learnability with positive data of UNTS grammars proved by Clark (2006), we prove that
k
,
l
-UNTS
≤
languages are also PAC-learnable under the same conditions.
Franco M. Luque, Gabriel Infante-Lopez
### Bounding the Maximal Parsing Performance of Non-Terminally Separated Grammars
Unambiguous Non-Terminally Separated (UNTS) grammars have good learnability properties but are too restrictive to be used for natural language parsing. We present a generalization of UNTS grammars called Unambiguous Weakly NTS (UWNTS) grammars that preserve the learnability properties. Then, we study the problem of using them to parse natural language and evaluating against a gold treebank. If the target language is not UWNTS, there will be an upper bound in the parsing performance. In this paper we develop methods to find upper bounds for the unlabeled
F
1
performance that any UWNTS grammar can achieve over a given treebank. We define a new metric, show that its optimization is NP-Hard but solvable with specialized software, and show a translation of the result to a bound for the
F
1
. We do experiments with the WSJ10 corpus, finding an
F
1
bound of 76.1% for the UWNTS grammars over the POS tags alphabet.
Franco M. Luque, Gabriel Infante-Lopez
### CGE: A Sequential Learning Algorithm for Mealy Automata
We introduce a new algorithm for sequential learning of Mealy automata by
congruence generator extension
(CGE). Our approach makes use of techniques from term rewriting theory and universal algebra for compactly representing and manipulating automata using finite congruence generator sets represented as
string rewriting systems
(SRS). We prove that the CGE algorithm correctly learns in the limit.
Karl Meinke
### Using Grammar Induction to Model Adaptive Behavior of Networks of Collaborative Agents
We introduce a formal paradigm to study global adaptive behavior of organizations of collaborative agents with local learning capabilities. Our model is based on an extension of the classical language learning setting in which a teacher provides examples to a student that must guess a correct grammar. In our model the teacher is transformed in to a workload dispatcher and the student is replaced by an organization of worker-agents. The jobs that the dispatcher creates consist of sequences of tasks that can be modeled as sentences of a language. The agents in the organization have language learning capabilities that can be used to learn local work-distribution strategies. In this context one can study the conditions under which the organization can adapt itself to structural pressure from an environment. We show that local learning capabilities contribute to global performance improvements. We have implemented our theoretical framework in a workbench that can be used to run simulations. We discuss some results of these simulations. We believe that this approach provides a viable framework to study processes of self-organization and optimization of collaborative agent networks.
### Transducer Inference by Assembling Specific Languages
Grammatical Inference has recently been applied successfully to bioinformatic tasks as protein domain prediction. In this work we present a new approach to infer regular languages. Although used in a biological task, our results may be useful not only in bioinformatics, but also in many applied tasks. To test the algorithm we consider the transmembrane domain prediction task. A preprocessing of the training sequences set allows us to use this heuristic to obtain a transducer. The transducer obtained is then used to label problem sequences. The experimentation carried out shows that this approach is suitable for the task.
Piedachu Peris, Damián López
### Sequences Classification by Least General Generalisations
In this paper, we present a general framework for supervised classification. This framework provides methods like boosting and only needs the definition of a generalisation operator called
lgg
lgg
is a learner that only uses positive examples. We show that
grammatical inference
has already defined such learners for automata classes like
reversible automata
or
k-TSS automata
. Then we propose a generalisation algorithm for the class of balls of words. Finally, we show through experiments that our method efficiently resolves sequence classification tasks.
Frédéric Tantini, Alain Terlutte, Fabien Torre
### A Likelihood-Ratio Test for Identifying Probabilistic Deterministic Real-Time Automata from Positive Data
RTI
) for identifying (learning) a
deterministic real-time automaton
(DRTA) to the setting of positive timed strings (or time-stamped event sequences). An DRTA can be seen as a deterministic finite state automaton (DFA) with time constraints. Because DRTAs model time using numbers, they can be exponentially more compact than equivalent DFA models that model time using states.
We use a new likelihood-ratio statistical test for checking consistency in the
RTI
algorithm. The result is the
RTI
+ algorithm, which stands for
real-time identification from positive data
.
RTI
+ is an efficient algorithm for identifying DRTAs from positive data. We show using artificial data that
RTI
+ is capable of identifying sufficiently large DRTAs in order to identify real-world real-time systems.
Sicco Verwer, Mathijs de Weerdt, Cees Witteveen
### A Local Search Algorithm for Grammatical Inference
In this paper, a heuristic algorithm for the inference of an arbitrary context-free grammar is presented. The input data consist of a finite set of representative words chosen from a (possibly infinite) context-free language and of a finite set of counterexamples—words which do not belong to the language. The time complexity of the algorithm is polynomially bounded. The experiments have been performed for a dozen or so languages investigated by other researchers and our results are reported.
Wojciech Wieczorek
### Polynomial-Time Identification of Multiple Context-Free Languages from Positive Data and Membership Queries
This paper presents an efficient algorithm that identifies a rich subclass of multiple context-free languages in the limit from positive data and membership queries by observing where each tuple of strings may occur in sentences of the language of the learning target. Our technique is based on Clark et al.’s work (ICGI 2008) on learning of a subclass of context-free languages. Our algorithm learns those context-free languages as well as many non-context-free languages.
Ryo Yoshinaka
### Grammatical Inference as Class Discrimination
Grammatical inference is typically defined as the task of finding a compact representation of a language given a subset of sample sequences from that language. Many different aspects, paradigms and settings can be investigated, leading to different proofs of language learnability or practical systems. The general problem can be seen as a one class classification or discrimination task. In this paper, we take a slightly different view on the task of grammatical inference. Instead of learning a full description of the language, we aim to learn a representation of the boundary of the language. Effectively, when this boundary is known, we can use it to decide whether a sequence is a member of the language or not. An extension of this approach allows us to decide on membership of sequences over a collection of (mutually exclusive) languages. We will also propose a systematic approach that learns language boundaries based on subsequences from the sample sequences and show its effectiveness on a practical problem of music classification. It turns out that this approach is indeed viable.
### MDL in the Limit
We show that within the Gold paradigm for language learning an informer for a superfinite set can cause an optimal MDL learner to make an infinite amount of mind changes. In this setting an optimal learner can make an infinite amount of wrong choices without approximating the right solution. This result helps us to understand the relation between MDL and identification in the limit in learning: MDL is an optimal model selection paradigm, identification in the limit defines recursion theoretical conditions for convergence of a learner.
### Grammatical Inference Algorithms in MATLAB
Although MATLAB has become one of the mainstream languages for the machine learning community, there is still skepticism among the Grammatical Inference (GI) community regarding the suitability of MATLAB for implementing and running GI algorithms. In this paper we will present implementation results of several GI algorithms, e.g., RPNI (Regular Positive and Negative Inference), EDSM (Evidence Driven State Merging), and k-testable machine. We show experimentally based on our MATLAB implementation that state merging algorithms can successfully be implemented and manipulated using MATLAB in the similar fashion as other machine learning tools. Moreover, we also show that MATLAB provides a range of toolboxes that can be leveraged to gain parallelism, speedup etc.
Hasan Ibne Akram, Colin de la Higuera, Huang Xiao, Claudia Eckert
### A Non-deterministic Grammar Inference Algorithm Applied to the Cleavage Site Prediction Problem in Bioinformatics
We report results on applying the OIL (Order Independent Language) grammar inference algorithm to predict cleavage sites in polyproteins from translation of Potivirus genome. This non-deterministic algorithm is used to generate a group of models which vote to predict the occurrence of the pattern. We built nine models, one for each cleavage site in this kind of virus genome and report sensibility, specificity, accuracy for each model. Our results show that this technique is useful to predict cleavage sites in the given task with accuracy rates higher than 95%.
Gloria Inés Alvarez, Jorge Hernán Victoria, Enrique Bravo, Pedro García
### Learning PDFA with Asynchronous Transitions
In this paper we extend the PAC learning algorithm due to Clark and Thollard for learning distributions generated by PDFA to automata whose transitions may take varying time lengths, governed by exponential distributions.
Borja Balle, Jorge Castro, Ricard Gavaldà
### Grammar Inference Technology Applications in Software Engineering
While Grammar Inference (GI) has been successfully applied to many diverse domains such as speech recognition and robotics, its application to software engineering has been limited, despite wide use of context-free grammars in software systems. This paper reports current developments and future directions in the applicability of GI to software engineering, where GI is seen to offer innovative solutions to the problems of inference of domain-specific language (DSL) specifications from example DSL programs and recovery of metamodels from instance models.
Barrett R. Bryant, Marjan Mernik, Dejan Hrnčič, Faizan Javed, Qichao Liu, Alan Sprague
### Hölder Norms and a Hierarchy Theorem for Parameterized Classes of CCG
We develop a framework based on Hölder norms that allows us to easily transfer learnability results. This idea is concretized by applying it to Classical Categorial Grammars (CCG).
Christophe Costa Florêncio, Henning Fernau
### Learning of Church-Rosser Tree Rewriting Systems
Tree rewriting systems are sets of tree rewriting rules used to compute by repeatedly replacing equal trees in a given formula until the simplest possible form (normal form) is obtained. The Church-Rosser property is certainly one of the most fundamental properties of tree rewriting system. In this system the simplest form of a given tree is unique since the final result does not depend on the order in which the rewritings rules are applied. The Church-Rosser system can offer both flexible computing and effecting reasoning with equations and have been intensively researched and widely applied to automated theorem proving and program verification etc. [3,5].
M. Jayasrirani, D. G. Thomas, Atulya K. Nagar, T. Robinson
### Generalizing over Several Learning Settings
We recapitulate inference from membership and equivalence queries, positive and negative samples. Regular languages cannot be learned from one of those information sources only [1,2,3]. Combinations of two sources allowing regular (polynomial) inference are MQs and EQs [4], MQs and positive data [5,6], positive and negative data [7,8]. We sketch a meta-algorithm fully presented in [9] that generalizes over as many combinations of those sources as possible. This includes a survey of pairings for which there are no well-studied algorithms.
Anna Kasprzik
### Rademacher Complexity and Grammar Induction Algorithms: What It May (Not) Tell Us
This paper revisits a problem of the evaluation of computational grammatical inference (GI) systems and discusses what role complexity measures can play for the assessment of GI. We provide a motivation for using the Rademacher complexity and give an example showing how this complexity measure can be used in practice.
Sophia Katrenko, Menno van Zaanen
### Extracting Shallow Paraphrasing Schemata from Modern Greek Text Using Statistical Significance Testing and Supervised Learning
Paraphrasing normally involves sophisticated linguistic resources for pre-processing. In the present work Modern Greek paraphrases are automatically generated using statistical significance testing in a novel manner for the extraction of applicable reordering schemata of syntactic constituents. Next, supervised filtering helps remove erroneously generated paraphrases, taking into account the context surrounding the reordering position. The proposed process is knowledge-poor, and thus portable to languages with similar syntax, robust and domain-independent. The intended use of the extracted paraphrases is hiding secret information underneath a cover text.
Katia Lida Kermanidis
### Learning Subclasses of Parallel Communicating Grammar Systems
Pattern language learning algorithms within the inductive inference model and query learning setting have been of great interest. In this paper an algorithm to learn a parallel communicating grammar system in which the master component is a regular grammar and the other components are pure pattern grammars is given.
Sindhu J. Kumaar, P. J. Abisha, D. G. Thomas
### Enhanced Suffix Arrays as Language Models: Virtual k-Testable Languages
In this article, we propose the use of suffix arrays to efficiently implement
n
-gram language models with practically unlimited size
n
. This approach, which is used with synchronous back-off, allows us to distinguish between alternative sequences using large contexts. We also show that we can build this kind of models with additional information for each symbol, such as part-of-speech tags and dependency information.
The approach can also be viewed as a collection of virtual
k
-testable automata. Once built, we can directly access the results of any
k
-testable automaton generated from the input training data. Synchronous back-off automatically identifies the
k
-testable automaton with the largest feasible
k
. We have used this approach in several classification tasks.
Herman Stehouwer, Menno van Zaanen
### Learning Fuzzy Context-Free Grammar—A Preliminary Report
This paper takes up the topic of a task of learning fuzzy context-free grammar from data. The induction process is divided into two phases: first the generic grammar is derived from the positive sentences, next the membership grades are assigned to the productions taking into account the occurrences of productions in a learning set. The problem of predicting the location of promoters in
Escherichia coli
is examined. Language of bacterial sequence can be described using formal system such as context-free grammar, and problem of promoter region recognition can be replaced by grammar induction. The induced fuzzy grammar was compared to other machine learning methods.
Olgierd Unold
### Polynomial Time Identification of Strict Prefix Deterministic Finite State Transducers
This paper is concerned with a subclass of finite state transducers, called
strict prefix deterministic finite state transducers
(
SPDFST
’s for short), and studies a problem of identifying the subclass in the limit from positive data. After providing some properties of languages accepted by SPDFST’s, we show that the class of SPDFST’s is polynomial time identifiable in the limit from positive data in the sense of Yokomori.
Mitsuo Wakatsuki, Etsuji Tomita
### Backmatter
Weitere Informationen
## BranchenIndex Online
Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.
## Whitepaper
- ANZEIGE -
### Product Lifecycle Management im Konzernumfeld – Herausforderungen, Lösungsansätze und Handlungsempfehlungen
Für produzierende Unternehmen hat sich Product Lifecycle Management in den letzten Jahrzehnten in wachsendem Maße zu einem strategisch wichtigen Ansatz entwickelt. Forciert durch steigende Effektivitäts- und Effizienzanforderungen stellen viele Unternehmen ihre Product Lifecycle Management-Prozesse und -Informationssysteme auf den Prüfstand. Der vorliegende Beitrag beschreibt entlang eines etablierten Analyseframeworks Herausforderungen und Lösungsansätze im Product Lifecycle Management im Konzernumfeld. | 2018-08-21 19:51:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45527541637420654, "perplexity": 1554.4487451786547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221218899.88/warc/CC-MAIN-20180821191026-20180821211026-00270.warc.gz"} |
http://mathhelpforum.com/trigonometry/152094-how-find-segments-length-print.html | # How to find a segment's length
• July 27th 2010, 05:12 AM
Alhazred
How to find a segment's length
http://img72.imageshack.us/img72/9281/snap1f.jpg
Given the greatest angle and the circle radius I need to calculate the length of A segment.
The circle is tangent to both lines.
How can I do?
• July 27th 2010, 05:50 AM
Unknown008
Let's call the centre of the smaller circle O, the point where the small circle touches the semi circle on the horizontal A, the other point which touches the semicircle B and the centre of the semicircle X.
Angle AXB = 116 degrees.
Angle OAX = angle OBX = 90 degrees.
Find angle BOA.
Once found, use the formula for the length of arc AB, that is $s = r\theta$
A segment is an area. (Circular segment - Wikipedia, the free encyclopedia)
• July 27th 2010, 06:12 AM
Alhazred
Probably all those lines could confuse.
I have 2 lines, S1 and S2, with a given angle beetween them (in this case is 116 degrees), then I have a circle tangent to both the lines with a given radius (in this case 20).
I need to find a way to calculate the length of the segment AB.
http://img191.imageshack.us/img191/6669/snap1do.jpg
• July 27th 2010, 06:18 AM
Unknown008
Oh, that's what you mean. Ok, no problem.
You now can find the angle AOB from your current diagram as I told you earlier. Well, let's call the point where the circle touches the other tangent X.
You know that angle XAB = 116 degrees.
That means angle XOB = 180 - 116 = 64 degrees.
(or [360 - 90 - 90 - 116] since both tangents are 90 degrees)
From there, the angle AOB becomes 64/2 = 32.
From there, you use the trigonometrical ratio tan.
tan 32 = Opp/Adj = AB/20
Solve for AB. (Happy)
• July 27th 2010, 08:20 AM
Alhazred
Thenks :) | 2014-04-17 22:01:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.676123857498169, "perplexity": 1267.7048610967909}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00192-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://lilypond.org/doc/v2.12/Documentation/user/lilypond/Curves | ### 1.3.2 Curves
This section explains how to create various expressive marks that are curved: normal slurs, phrasing slurs, breath marks, falls, and doits.
#### Slurs
Slurs are entered using parentheses:
```f4( g a) a8 b(
a4 g2 f4)
<c e>2( <b d>2)
```
Slurs may be manually placed above or below the notes, see Direction and placement.
```c2( d)
\slurDown
c2( d)
\slurNeutral
c2( d)
```
Simultaneous or overlapping slurs are not permitted, but a phrasing slur can overlap a slur. This permits two slurs to be printed at once. For details, see Phrasing slurs.
Slurs can be solid, dotted, or dashed. Solid is the default slur style:
```c4( e g2)
\slurDashed
g4( e c2)
\slurDotted
c4( e g2)
\slurSolid
g4( e c2)
```
#### Predefined commands
`\slurUp`, `\slurDown`, `\slurNeutral`, `\slurDashed`, `\slurDotted`, `\slurSolid`.
#### Selected Snippets
Using double slurs for legato chords
Some composers write two slurs when they want legato chords. This can be achieved by setting `doubleSlurs`.
```\relative c' {
\set doubleSlurs = ##t
<c e>4( <d f> <c e> <d f>)
}
```
Positioning text markups inside slurs
Text markups need to have the `outside-staff-priority` property set to false in order to be printed inside slurs.
```\relative c'' {
\override TextScript #'avoid-slur = #'inside
\override TextScript #'outside-staff-priority = ##f
c2(^\markup { \halign #-10 \natural } d4.) c8
}
```
Music Glossary: slur.
Learning Manual: On the un-nestedness of brackets and ties.
Notation Reference: Direction and placement, Phrasing slurs.
Snippets: Expressive marks.
Internals Reference: Slur.
#### Phrasing slurs
Phrasing slurs (or phrasing marks) that indicate a musical sentence are written using the commands `\(` and `\)` respectively:
```c4\( d( e) f(
e2) d\)
```
Typographically, a phrasing slur behaves almost exactly like a normal slur. However, they are treated as different objects; a `\slurUp` will have no effect on a phrasing slur. Phrasing slurs may be manually placed above or below the notes, see Direction and placement.
```c4\( g' c,( b) | c1\)
\phrasingSlurUp
c4\( g' c,( b) | c1\)
```
Simultaneous or overlapping phrasing slurs are not permitted.
Phrasing slurs can be solid, dotted, or dashed. Solid is the default style for phrasing slurs:
```c4\( e g2\)
\phrasingSlurDashed
g4\( e c2\)
\phrasingSlurDotted
c4\( e g2\)
\phrasingSlurSolid
g4\( e c2\)
```
#### Predefined commands
`\phrasingSlurUp`, `\phrasingSlurDown`, `\phrasingSlurNeutral`, `\phrasingSlurDashed`, `\phrasingSlurDotted`, `\phrasingSlurSolid`.
Learning Manual: On the un-nestedness of brackets and ties.
Notation Reference: Direction and placement.
Snippets: Expressive marks.
Internals Reference: PhrasingSlur.
#### Breath marks
Breath marks are entered using `\breathe`:
```c2. \breathe d4
```
Musical indicators for breath marks in ancient notation, divisiones, are supported. For details, see Divisiones.
#### Selected Snippets
Changing the breath mark symbol
The glyph of the breath mark can be tuned by overriding the text property of the `BreathingSign` layout object with any markup text.
```\relative c'' {
c2
\override BreathingSign #'text = \markup { \musicglyph #"scripts.rvarcomma" }
\breathe
d2
}
```
Inserting a caesura
Caesura marks can be created by overriding the `'text` property of the `BreathingSign` object. A curved caesura mark is also available.
```\relative c'' {
\override BreathingSign #'text = \markup {
\musicglyph #"scripts.caesura.straight"
}
c8 e4. \breathe g8. e16 c4
\override BreathingSign #'text = \markup {
\musicglyph #"scripts.caesura.curved"
}
g8 e'4. \breathe g8. e16 c4
}
```
Music Glossary: caesura.
Notation Reference: Divisiones.
Snippets: Expressive marks.
Internals Reference: BreathingSign.
#### Falls and doits
Falls and doits can be added to notes using the `\bendAfter` command. The direction of the fall or doit is indicated with a plus or minus (up or down). The number indicates the pitch interval that the fall or doit will extend beyond the main note.
```c2-\bendAfter #+4
c2-\bendAfter #-4
c2-\bendAfter #+8
c2-\bendAfter #-8
```
The dash `-` immediately preceding the `\bendAfter` command is required when writing falls and doits.
#### Selected Snippets
Adjusting the shape of falls and doits
The `shortest-duration-space` property may have to be tweaked to adjust the shape of falls and doits.
```\relative c'' {
\override Score.SpacingSpanner #'shortest-duration-space = #4.0
c2-\bendAfter #+5
c2-\bendAfter #-3
c2-\bendAfter #+8
c2-\bendAfter #-6
}
``` | 2013-06-19 18:46:01 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8463873267173767, "perplexity": 11046.096945635118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368709006458/warc/CC-MAIN-20130516125646-00045-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.fact-archive.com/encyclopedia/Quantum_field_theory | ## Online Encylopedia and Dictionary Research Site
Online Encyclopedia Search Online Encyclopedia Browse
# Quantum field theory
Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework widely used in particle physics and condensed matter physics. In particular, the quantum theory of the electromagnetic field, known as quantum electrodynamics, is one of the most well-tested and successful theories in physics. The fundamentals of quantum field theory were developed between the late 1920s and the 1950s, notably by Dirac, Fock, Pauli, Tomonaga, Schwinger, Feynman, and Dyson.
Contents
## Shortcomings of ordinary quantum mechanics
Quantum field theory corrects several deficiencies of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equation, in its most commonly encountered form, is
$\left[ \frac{|\mathbf{p}|^2}{2m} + V(\mathbf{r}) \right] |\psi(t)\rang = i \hbar \frac{\partial}{\partial t} |\psi(t)\rang$
where |ψ> denotes the quantum state of a particle with mass m, acted on by a potential energy V.
There are two problems with this equation. Firstly, it is not relativistic, reducing to classical mechanics rather than relativistic mechanics in the correspondence limit. To see this, we note that the first term on the left is only the classical kinetic energy p²/2m, with the rest energy mc² omitted. It is possible to modify the Schrödinger equation to include the rest energy, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy spectra which extend to -∞, so that there is no ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory.
The second problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as
$|\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j}{N!}} \sum_{p} |\phi_{p(1)}\rang \cdots |\phi_{p(N)} \rang$
where i> are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases.
## Quantum fields
### Second quantization
Both of the above problems are resolved by moving our attention from a fixed set of particles to a quantum field. The procedure by which quantum fields are constructed from individual particles was introduced by Dirac, and is (for historical reasons) known as second quantization.
We should mention two possible points of confusion. Firstly, the aforementioned "field" and "particle" descriptions do not refer to wave-particle duality. By "particle", we refer to entities which possess both wave and point-particle properties in the usual quantum mechanical sense; for example, these "particles" are generally not located at a fixed point, but have a certain probability of being found at each position in space. What we refer to as a "field" is an entity existing at every point in space, which regulates the creation and annihilation of the particles. Secondly, quantum field theory is essentially quantum mechanics, and not a replacement for quantum mechanics. Like any quantum system, a quantum field possesses a Hamiltonian H (albeit one that is more complicated than typical single-particle Hamiltonians), and obeys the usual Schrödinger equation
$H \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle$
(Quantum field theories are often formulated in terms of a Lagrangian, for reasons of convenience. However, the Lagrangian and Hamiltonian formulations are believed to be equivalent.)
In second quantization, we make use of particle indistinguishability by specifying multi-particle wavefunctions in terms of single-particle occupation numbers. For example, suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states |φ1>, |φ2>, |φ3>, and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N = 3, with one particle in state |φ1> and two in state |φ2>. The normal way of writing the wavefunction is
$\frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang |\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang |\phi_2\rang |\phi_1\rang \right]$
In second quantized form, we write this as
$|1, 2, 0, 0, 0, \cdots \rangle$
which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."
Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. For example, the annihilation operator a2 and the creation operator a2 have the following effects:
$a_2 | N_1, N_2, N_3, \cdots \rangle \equiv$ $\sqrt{N_2}$ $|N_1, (N_2 - 1), N_3, \cdots \rangle$ $a_2^\dagger | N_1, N_2, N_3, \cdots \rangle \equiv$ $\sqrt{N_2 + 1}$ $| N_1, (N_2 + 1), N_3, \cdots \rangle$
We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called "vacuum" state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them.
The creation and annihilation operators obey the commutation relation
$\left[a_i , a_j \right] = 0 \quad,\quad \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad \left[a_i , a_j^\dagger \right] = \delta_{ij}$
where δ stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.
The creation and annihilation operators for fermions obey an anticommutation relation,
$\left\{c_i , c_j \right\} = 0 \quad,\quad \left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad \left\{c_i , c_j^\dagger \right\} = \delta_{ij}$
One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the Pauli exclusion principle.
The final step toward obtaining a quantum field theory is to re-write our original N-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is
$H = \sum_k E_k \, a^\dagger_k \,a_k$
where Ek is the energy of the k-th single-particle energy eigenstate.
### Significance of creation and annihilation operators
When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator ak destroys a particle during an infinitesimal time step, the creation operator ak to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.
On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e., linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by ck and ck, we could add a "potential energy" term to our Hamiltonian such as:
$V = \sum_{k,q} V_q (a_q + a_{-q}^\dagger) c_{k+q}^\dagger c_k$
This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k+q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case.
In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the fact that its quantum state is a superposition of states with different particle numbers.
### Field operators
We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator φ(r) is
$\phi(\mathbf{r}) \equiv \sum_{i} e^{i\mathbf{k}_i\cdot \mathbf{r}} a_{i}$
The bosonic field operators obey the commutation relation
$\left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})$
where δ(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say
$H = - \frac{\hbar^2}{2m} \sum_i \nabla_i^2 + \sum_{i < j} U(|\mathbf{r}_i - \mathbf{r}_j|)$
where the indices i and j run over all particles, then the field theory Hamiltonian is
$H = - \frac{\hbar^2}{2m} \int d^3\!r \; \phi(\mathbf{r})^\dagger \nabla^2 \phi(\mathbf{r}) + \int\!d^3\!r \int\!d^3\!r' \; \phi(\mathbf{r})^\dagger \phi(\mathbf{r}')^\dagger U(|\mathbf{r} - \mathbf{r}'|) \phi(\mathbf{r'}) \phi(\mathbf{r})$
This looks remarkably like an expression for the expectation value of the energy, with φ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.
### Quantization of classical fields
So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon, so a quantum field theory must be formulated right from the start. The process is known as "first quantization of a classical field equation."
The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as
$\left[ q_i , p_j \right] = \delta_{ij}$
For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential φ(x) and the vector potential A(x) at every point x. This is an uncountable set of variables, because x is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory.
## Axiomatic Approach
There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. The most prominent of these are the Wightman axioms and the Haag-Kastler axioms.
The classic results gained from the axiomatic approach are the PCT Theorem (stating that the combination of parity, time and charge inversion is an unbroken symmetry) and the spin-statistics theorem (stating that particles of integer valued spin follow the Bose-Einstein statistics and particles of half-integer spin follow the Fermi statistics).
Loudon, Rodney. The Quantum Theory of Light
Peskin, M. and D. Schroeder. An Introduction to quantum field theory.
Weinberg, Steven. The Quantum theory of fields (3 volumes)
• Warren Siegel. Fields (a free 796-page e-book) http://arxiv.org/abs/hep-th/9912205
• Fresh http://insti.physics.sunysb.edu/%7Esiegel/Fields2.pdf
• Others http://insti.physics.sunysb.edu/%7Esiegel/plan.html
General subfields within physics
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http://relliktheseries.com/freebooks/category/system-theory/page/3/ | # A Concise System of Mathematics in Theory and Practice for
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Learn more about the arc of Capra’s work, from the Dance of Shiva to Leonardo’s Vitruvian Man. In " The Romance Resonance ", Sheldon makes a great scientific discovery of a new super-heavy element which thrills Sheldon until he realizes he made a bone-headed error in his calculations. Alpha particles are helium atom particles. Please note that all abstracts and articles in the ADS are copyrighted by the publisher, and their use is free for personal use only.
# Positive Polynomials in Control (Lecture Notes in Control
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I hope that they will fulfill the promises! 14 Field theory for disordered systems Giorgio Parisi References [1] A. Wheaton appears again in " The 21-Second Excitation " at a showing of the movie, " Raiders of the Lost Ark ," with 21 seconds of deleted footage. Areas of climate science that can particularly benefit from input by physicists are emphasised. Please feel free to write to us if you wish to discuss any of this article, or if you are able to help in getting this experiment performed.
# Digital Control
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This is a word best avoided entirely in physics except when placed in quotes, or with careful qualification. By altering the wavelength and the direction of propagation, the results could be delivered to any terrestrial point. The physical domain is thus completely closed causally, impervious to influence from modes of other attributes and to intervention of divine will, and fully deterministic (Ip29). The difficulties begin when you want to know X at a time T which is no more infinitesimaly close to t0 or in other words when you want to integrate the ODE system for any t.
# On Moment Theory and Controllability of One-Dimensional
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It appears to me, therefore, that the formation of the concept of the material object must precede our concepts of time and space. (Albert Einstein, 1954) Metaphysics, as a true description of Reality, must be based on a priori causes AND these must be united back to one common thing that causes and connects the many things (matter). For more discussion of this topic in metaphysics, see (Carroll and Markosian 2010, pp. 173-7). The magnetic force was found to depend on the inverse square of the distance but was more complex due to the subtle vector relations between the currents and distances.
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~>~{{k(\epsilon)}\over{s^{2.5}}},~\lim_{\epsilon~\rightarrow~0}k(\epsilon)~\rightarrow~0 the orbit is more stable. While the advice and information in this journal is believed to be true and accurate at the date of its publication, neither the authors, the editors, nor the publisher can accept any legal responsibility for any errors or omissions that may have been made. These are the same relativistic principles that make sense in the everyday world, that most people equate with 'common sense'.
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There is no experimental evidence that the character of physical time is affected in any way by the presence or absence of consciousness or the presence or absence of any biological phenomenon. MARIA SPIROPULU: So, it's a big discovery to find supersymmetry. In both cases, the buffer manager will note the cause of material being delayed coming into the buffer and the most frequent of these causes will be high on the list of things to correct; buffer management thus helps to drive continuous improvement.
# Stochastic Control of Partially Observable Systems
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The point(s) from which light rays diverge as they enter a lens or mirror. When sound waves strike the microphone, the coil moves and a electrical signal is induced. The course is intended for students in biomedical engineering, physics, and medical sciences. If you look at transformations in general, you notice that there is always the unit transformation that changes nothing, like E above. Over the centuries, a small but significant body of scientists and philosophers—each working independent from the other but building on the ideas of his predecessors—slowly began chipping away at the Aristotelian framework.
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Technically, quantum theory is actually the theory of any objects isolated from their surroundings but, because it is very difficult to isolate large objects from their environments, it essentially becomes a theory of the microscopic world of atoms and sub-atomic particles. Columbia, one of the leading university centers for training in plasma physics, offers a graduate program leading to the Master of Science (MS), Master of Philosophy (MPhil), Doctor of Philosophy (PhD) and Doctor of Engineering Science (Eng.
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The Origin of Mass and the Feebleness of Gravity: Try out this lecture where theoretical physicist Frank Wilczek shares his expertise on the origins of mass. Amy goes straight to his house and demands him to take it down. Best of all, you can calculate the theoretical value for RT60 using the formula RT60 = 0.161V/ S(a×s) where V is the volume of the room in m3, a is the area of each absorbing material in m2, and s is the absorption coefficient for each material.
# Psychology of System Design (Advances in Human
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Unlike Einstein's theory of relativity which reinterpret's the meaning of classical concepts such as time, position, velocity, mass and so on, quantum theory introduces brand new and radical ideas which have no pre-existing counterpart in classical physics. Now the idea is that, first, the whole CM is moving, and in the CM there is a relative velocity $\FLPw$, and the molecules collide and come off in some new direction. Shorter projects with higher on time completion rates has been the experience of many companies employing the critical chain approach. | 2018-11-14 07:38:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2753288745880127, "perplexity": 1874.4842131472783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741660.40/warc/CC-MAIN-20181114062005-20181114084005-00554.warc.gz"} |
http://peeterjoot.com/tag/vector-potential/page/2/ | ## Notes for Balantis chapter 4: linear wire antennas.
These are notes for the UofT course ECE1229, Advanced Antenna Theory, taught by Prof. Eleftheriades, covering ch. 4 [1] content.
Unlike most of the other classes I have taken, I am not attempting to take comprehensive notes for this class. The class is taught on slides that match the textbook so closely, there is little value to me taking notes that just replicate the text. Instead, I am annotating my copy of textbook with little details instead. My usual notes collection for the class will contain musings of details that were unclear, or in some cases, details that were provided in class, but are not in the text (and too long to pencil into my book.)
## Magnetic Vector Potential.
In class and in the problem set $$\BA$$ was referred to as the Magnetic Vector Potential. I only recalled this referred to as the Vector Potential. Prefixing this with magnetic seemed counter intuitive to me since it is generated by electric sources (charges and currents).
This terminology can be justified due to the fact that $$\BA$$ generates the magnetic field by its curl. Some mention of this can be found in [4], which also points out that the Electric Potential refers to the scalar $$\phi$$. Prof. Eleftheriades points out that Electric Vector Potential refers to the vector potential $$\BF$$ generated by magnetic sources (because in that case the electric field is generated by the curl of $$\BF$$.)
## Plots of infinitesimal dipole radial dependence.
In section 4.2 of [1] are some discussions of the $$kr < 1$$, $$kr = 1$$, and $$kr > 1$$ radial dependence of the fields and power of a solution to an infinitesimal dipole system. Here are some plots of those $$k r$$ dependence, along with the $$k r = 1$$ contour as a reference. All the $$\theta$$ dependence and any scaling is left out.
The CDF notebook visualizeDipoleFields.cdf is available to interactively plot these, rotate the plots and change the ranges of what is plotted.
A plot of the real and imaginary parts of $$H_\phi = \frac{j k}{r} e^{-j k r} \lr{ 1-\frac{j}{k r} }$$ can be found in fig. 1 and fig. 2.
fig 1. Radial dependence of Re H_phi
fig 2. Radial dependence of Im H_phi
A plot of the real and imaginary parts of $$E_r = \inv{r^2} \lr{1-\frac{j}{k r}} e^{-j k r}$$ can be found in fig. 3 and fig. 4.
fig 3. Radial dependence of Re E_r
fig 4. Radial dependence of Im E_r
Finally, a plot of the real and imaginary parts of $$E_\theta = \frac{ j k }{r} \lr{1 -\frac{j}{k r} -\frac{1}{k^2 r^2} } e^{-j k r}$$ can be found in fig. 5 and fig. 6.
fig. 5. Radial dependence of Re E_theta
fig. 6. Radial dependence of Im E_theta
## Electric Far field for a spherical potential.
It is interesting to look at the far electric field associated with an arbitrary spherical magnetic vector potential, assuming all of the radial dependence is in the spherical envelope. That is
\label{eqn:chapter4Notes:20}
\BA = \frac{e^{-j k r}}{r} \lr{
\rcap a_r\lr{ \theta, \phi }
+\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}.
The electric field is
\label{eqn:chapter4Notes:40}
\BE = – j \omega \BA – j \frac{1}{\omega \mu_0 \epsilon_0 } \spacegrad \lr{\spacegrad \cdot \BA }.
The divergence and gradient in spherical coordinates are
\label{eqn:chapter4Notes:80}
\begin{aligned}
&=
\inv{r^2} \PD{r}{} \lr{ r^2 A_r }
+ \inv{r \sin\theta } \PD{\theta}{} \lr{A_\theta \sin\theta}
+ \inv{r \sin\theta } \PD{\phi}{A_\phi}
\end{aligned}
\label{eqn:chapter4Notes:100}
\begin{aligned}
&=
\rcap \PD{r}{\psi}
+\frac{\thetacap}{r} \PD{\theta}{\psi}
+ \frac{\phicap}{r \sin\theta} \PD{\phi}{\psi}.
\end{aligned}
For the assumed potential, the divergence is
\label{eqn:chapter4Notes:120}
\begin{aligned}
&=
\frac{a_r}{r^2} \PD{r}{} \lr{ r^2 \frac{e^{-j k r}}{r} }
+ \inv{r \sin\theta } \frac{e^{-j k r}}{r} \PD{\theta}{} \lr{\sin\theta a_\theta}
+ \inv{r \sin\theta } \frac{e^{-j k r}}{r} \PD{\phi}{a_\phi} \\
&=
a_r
e^{-j k r}
\lr{
\inv{r^2}
-j k \inv{r}
}
+ \inv{r^2 \sin\theta } e^{-j k r} \PD{\theta}{} \lr{\sin\theta a_\theta}
+ \inv{r^2 \sin\theta } e^{-j k r} \PD{\phi}{a_\phi} \\
&\approx
-j k \frac{a_r}{r}
e^{-j k r}.
\end{aligned}
The last approximation dropped all the $$1/r^2$$ terms that will be small compared to $$1/r$$ contribution that dominates when $$r \rightarrow \infty$$, the far field.
The gradient can now be computed
\label{eqn:chapter4Notes:140}
\begin{aligned}
&\approx
-j k
\lr{
\frac{a_r}{r}
e^{-j k r}
} \\
&=
-j k \lr{
\rcap \PD{r}{}
+\frac{\thetacap}{r} \PD{\theta}{}
+ \frac{\phicap}{r \sin\theta} \PD{\phi}{}
}
\frac{a_r}{r}
e^{-j k r} \\
&=
-j k \lr{
\rcap a_r \PD{r}{} \lr{
\frac{1}{r}
e^{-j k r}
}
+\frac{\thetacap}{r^2}
e^{-j k r}
\PD{\theta}{a_r}
+
e^{-j k r}
\frac{\phicap}{r^2 \sin\theta}
\PD{\phi}{a_r}
} \\
&=
-j k \lr{
-\rcap \frac{a_r}{r^2} \lr{
1
+ j k r
}
+\frac{\thetacap}{r^2}
\PD{\theta}{a_r}
+
\frac{\phicap}{r^2 \sin\theta}
\PD{\phi}{a_r}
}
e^{-j k r} \\
&\approx
– k^2 \rcap \frac{a_r}{r}
e^{-j k r}.
\end{aligned}
Again, a far field approximation has been used to kill all the $$1/r^2$$ terms.
The far field approximation of the electric field is now possible
\label{eqn:chapter4Notes:160}
\begin{aligned}
\BE
&= – j \omega \BA – j \frac{1}{\omega \mu_0 \epsilon_0 } \spacegrad \lr{\spacegrad \cdot \BA } \\
&=
– j \omega
\frac{e^{-j k r}}{r} \lr{
\rcap a_r\lr{ \theta, \phi }
+\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}
+ j \frac{1}{\omega \mu_0 \epsilon_0 }
k^2 \rcap \frac{a_r}{r}
e^{-j k r} \\
&=
– j \omega
\frac{e^{-j k r}}{r} \lr{
\rcap a_r\lr{ \theta, \phi }
+\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}
+ j \frac{c^2}{\omega }
\lr{\frac{\omega}{c}}^2 \rcap \frac{a_r}{r}
e^{-j k r}
\\
&=
– j \omega
\frac{e^{-j k r}}{r} \lr{
\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}.
\end{aligned}
Observe the perfect, somewhat miraculous seeming, cancellation of all the radial components of the field. If $$\BA_{\textrm{T}}$$ is the non-radial projection of $$\BA$$, the electric far field is just
\label{eqn:chapter4Notes:180}
\boxed{
\BE_{\textrm{ff}} = -j \omega \BA_{\textrm{T}}.
}
## Magnetic Far field for a spherical potential.
Application of the same assumed representation for the magnetic field gives
\label{eqn:chapter4Notes:220}
\begin{aligned}
\BB
&=
&=
\frac{\rcap}{r \sin\theta} \partial_\theta \lr{A_\phi \sin\theta}
+ \frac{\thetacap}{r} \lr{ \inv{\sin\theta} \partial_\phi A_r – \partial_r \lr{r A_\phi}}
+ \frac{\phicap}{r} \lr{ \partial_r\lr{r A_\theta} – \partial_\theta A_r} \\
&=
\frac{\rcap}{r \sin\theta} \partial_\theta \lr{
\frac{e^{-j k r}}{r} a_\phi
\sin\theta}
+ \frac{\thetacap}{r} \lr{ \inv{\sin\theta} \partial_\phi \lr{
\frac{e^{-j k r}}{r} a_r
} – \partial_r \lr{r
\frac{e^{-j k r}}{r} a_\phi
}
}
+ \frac{\phicap}{r} \lr{ \partial_r\lr{r
\frac{e^{-j k r}}{r} a_\theta
} – \partial_\theta
\lr{
\frac{e^{-j k r}}{r} a_r
}
} \\
&=
\frac{\rcap}{r \sin\theta}
\frac{e^{-j k r}}{r}
\partial_\theta \lr{
a_\phi
\sin\theta}
+ \frac{\thetacap}{r} \lr{ \inv{\sin\theta}
\frac{e^{-j k r}}{r}
\partial_\phi
a_r
– \partial_r \lr{
e^{-j k r}
}
a_\phi
}
+ \frac{\phicap}{r} \lr{
\partial_r
\lr{
e^{-j k r}
}
a_\theta
\frac{e^{-j k r}}{r}
\partial_\theta
a_r
}
\approx
j k \lr{ \thetacap a_\phi
\phicap a_\theta
}
\frac{e^{-j k r}}{r} \\
&=
-j k \rcap \cross \lr{
\thetacap a_\theta
+\phicap a_\phi
}
\frac{e^{-j k r}}{r} \\
&=
\inv{c} \BE_{\textrm{ff}}.
\end{aligned}
The approximation above drops the $$1/r^2$$ terms. Since
\label{eqn:chapter4Notes:240}
\inv{\mu_0 c} = \inv{\mu_0} \sqrt{\mu_0\epsilon_0} = \sqrt{\frac{\epsilon_0}{\mu_0}} = \inv{\eta},
the magnetic far field can be expressed in terms of the electric far field as
\label{eqn:chapter4Notes:260}
\boxed{
\BH = \inv{\eta} \rcap \cross \BE.
}
## Plane wave relations between electric and magnetic fields
I recalled an identity of the form \ref{eqn:chapter4Notes:260} in [3], but didn’t think that it required a far field approximation.
The reason for this was because the Jackson identity assumed a plane wave representation of the field, something that the far field assumptions also locally require.
Assuming a plane wave representation for both fields
\label{eqn:chapter4Notes:300}
\boldsymbol{\mathcal{E}}(\Bx, t) = \BE e^{j \lr{\omega t – \Bk \cdot \Bx}}
\label{eqn:chapter4Notes:320}
\boldsymbol{\mathcal{B}}(\Bx, t) = \BB e^{j \lr{\omega t – \Bk \cdot \Bx}}
The cross product relation between the fields follows from the Maxwell-Faraday law of induction
\label{eqn:chapter4Notes:340}
0 = \spacegrad \cross \boldsymbol{\mathcal{E}} + \PD{t}{\boldsymbol{\mathcal{B}}},
or
\label{eqn:chapter4Notes:360}
\begin{aligned}
0
&=
\Be_r \cross \BE \partial_r e^{j\lr{ \omega t – \Bk \cdot \Bx}}
+
j \omega \BB e^{j \lr{\omega t – \Bk \cdot \Bx}} \\
&=
-j \Be_r k_r \cross \BE e^{j \lr{\omega t – \Bk \cdot \Bx}}
+
j \omega \BB e^{j \lr{\omega t – \Bk \cdot \Bx}} \\
&=
\lr{ – \Bk \cross \BE + \omega \BB } j
e^{j \lr{\omega t – \Bk \cdot \Bx}},
\end{aligned}
or
\label{eqn:chapter4Notes:380}
\begin{aligned}
\BH
&= \frac{ k}{k c \mu_0 } \kcap \cross \BE \\
&= \inv{ \eta } \kcap \cross \BE,
\end{aligned}
which also finds \ref{eqn:chapter4Notes:260}, but with much less work and less mess.
## Transverse only nature of the far-field fields
Also observe that its possible to tell that the far field fields have only transverse components using the same argument that they are locally plane waves at that distance. The plane waves must satisfy the zero divergence Maxwell’s equations
\label{eqn:chapter4Notes:420}
\label{eqn:chapter4Notes:440}
so by the same logic
\label{eqn:chapter4Notes:480}
\Bk \cdot \BE = 0
\label{eqn:chapter4Notes:500}
\Bk \cdot \BB = 0.
In the far field the electric field must equal its transverse projection
\label{eqn:chapter4Notes:520}
\BE = \textrm{Proj}_\T \lr{-j \omega \BA
– j \frac{1}{\omega \mu_0 \epsilon_0 } \spacegrad \lr{\spacegrad \cdot \BA } }.
Since by \ref{eqn:chapter4Notes:140} the scalar potential term has only a radial component, that leaves
\label{eqn:chapter4Notes:540}
\BE = -j \omega \textrm{Proj}_\T \BA,
which provides \ref{eqn:chapter4Notes:180} with slightly less work.
## Vertical dipole reflection coefficient
In class a ground reflection scenario was covered for a horizontal dipole. Reading the text I was surprised to see what looked like the same sort of treatment section 4.7.2, but ending up with a quite different result. It turns out the difference is because the text was treating the vertical dipole configuration, whereas Prof. Eleftheriades was treating a horizontal dipole configuration, which have different reflection coefficients. These differing reflection coefficients are due to differences in the polarization of the field.
To understand these differences in reflection coefficients, consider first the field due to a vertical dipole as sketched in fig. 7, with a wave vector directed from the transmission point downwards in the z-y plane.
fig. 7. vertical dipole configuration.
The wave vector has direction
\label{eqn:chapter4Notes:560}
\kcap = \zcap e^{\zcap \xcap \theta} = \zcap \cos\theta + \ycap \sin\theta.
Suppose that the (magnetic) vector potential is that of an infinitesimal dipole
\label{eqn:chapter4Notes:580}
\BA = \zcap \frac{\mu_0 I_0 l}{4 \pi r} e^{-j k r} %= \frac{A_r}{4 \pi r} e^{-j k r}
The electric field, in the far field, can be computed by computing the normal projection to the wave vector direction
\label{eqn:chapter4Notes:600}
\begin{aligned}
\BE
&= -j \omega \lr{\BA \wedge \kcap} \cdot \kcap \\
&= -j \omega \frac{\mu_0 I_0 l}{4 \pi r} \lr{\zcap \wedge \lr{\zcap \cos\theta
+ \ycap \sin\theta} } \lr{\zcap \cos\theta + \ycap \sin\theta} \\
&= -j \omega \frac{\mu_0 I_0 l}{4 \pi r} \lr{ \zcap \ycap \sin\theta }
\lr{\zcap \cos\theta + \ycap \sin\theta} \\
&= -j \omega \frac{\mu_0 I_0 l}{4 \pi r} \sin\theta \lr{-\ycap \cos\theta +
\zcap \sin\theta} \\
&= j \omega \frac{\mu_0 I_0 l}{4 \pi r} \sin\theta \ycap e^{\zcap \ycap \theta}.
\end{aligned}
This is directed in the z-y plane rotated an additional $$\pi/2$$ past $$\kcap$$. The magnetic field must then be directed into the page, along the x axis. This is sketched in fig. 8.
fig. 8. Electric and magnetic field directions
Referring to [2] (\eqntext 4.40) for the coefficient of reflection component
\label{eqn:chapter4Notes:620}
R
=
\frac{
n_t \cos\theta_i – n_i \cos\theta_t
}
{
n_i \cos\theta_i + n_t \cos\theta_t
}
This is the Fresnel equation for the case when
that corresponds to
$$\BE$$ lies in the plane of incidence, and the magnetic field is completely parallel to the plane of reflection). For the no transmission case, allowing $$v_t \rightarrow 0$$, the index of refraction is $$n_t = c/v_t \rightarrow \infty$$, and the reflection coefficient is $$1$$ as claimed in section 4.7.2 of [1]. Because of the symmetry of this dipole configuration, the azimuthal angle that the wave vector is directed along does not matter.
## Horizontal dipole reflection coefficient
In the class notes, a horizontal dipole coming out of the page is indicated. With the page representing the z-y plane, this is a magnetic vector potential directed along the x-axis direction
\label{eqn:chapter4Notes:640}
\BA = \xcap \frac{\mu_0 I_0 l}{4 \pi r} e^{-j k r}.
For a wave vector directed in the z-y plane as in \ref{eqn:chapter4Notes:560}, the electric far field is directed along
\label{eqn:chapter4Notes:660}
\begin{aligned}
\lr{ \xcap \wedge \kcap } \cdot \kcap
&=
\xcap – \lr{ \xcap \cdot \kcap } \kcap \\
&=
\xcap – \lr{ \xcap \cdot \lr{
\zcap \cos\theta + \ycap \sin\theta
} } \kcap \\
&= \xcap.
\end{aligned}
The electric far field lies completely in the plane of reflection. From [2] (\eqntext 4.34), the Fresnel reflection coefficients is
\label{eqn:chapter4Notes:680}
R =
\frac{
n_i \cos\theta_i – n_t \cos\theta_t
}
{
n_i \cos\theta_i + n_t \cos\theta_t
},
which approaches $$-1$$ when $$n_t \rightarrow \infty$$. This is consistent with the image theorem summation that Prof. Eleftheriades used in class.
### Azimuthal angle dependency of the reflection coefficient
Now consider a horizontal dipole directed along the y-axis. For the same wave vector direction as avove, the electric far field is now directed along
\label{eqn:chapter4Notes:700}
\begin{aligned}
\lr{ \ycap \wedge \kcap } \cdot \kcap
&=
\ycap – \lr{ \ycap \cdot \kcap } \kcap \\
&=
\ycap – \lr{ \ycap \cdot \lr{
\zcap \cos\theta + \ycap \sin\theta
} } \kcap \\
&=
\ycap – \kcap \sin\theta \\
&=
\ycap – \sin\theta \lr{
\zcap \cos\theta + \ycap \sin\theta
} \\
&=
\ycap \cos^2 \theta – \sin\theta \cos\theta \zcap \\
&= \cos\theta \lr{ \ycap \cos\theta – \sin\theta \zcap } \\
&= \cos\theta \ycap e^{ \zcap \ycap \theta }.
\end{aligned}
That is
\label{eqn:chapter4Notes:720}
\BE =
-j \omega \frac{\mu_0 I_0 l}{4 \pi r} e^{-j k r}
\cos\theta \ycap e^{ \zcap \ycap \theta }.
This far field electric field lies in the plane of incidence (a direction of $$\thetacap$$ rotated by $$\pi/2$$), not in the plane of reflection. The corresponding magnetic field should be directed along the plane of reflection, which is easily confirmed by calculation
\label{eqn:chapter4Notes:740}
\begin{aligned}
\kcap \cross
\lr{ \ycap \cos\theta – \sin\theta \zcap }
&=
\lr{ \zcap \cos\theta + \ycap \sin\theta } \cross
\lr{ \ycap \cos\theta – \sin\theta \zcap } \\
&=
-\xcap \cos^2 \theta – \xcap \sin^2\theta \\
&= -\xcap.
\end{aligned}
The far field magnetic field is seen to be
\label{eqn:chapter4Notes:721}
\BH =
j \omega \frac{I_0 l}{4 \pi r} e^{-j k r}
\cos\theta \xcap,
so a reflection coefficient of $$1$$ is required to calculate the power loss after a single ground reflection signal bounce for this relative orientation of antenna to the target.
I fail to see how the horizontal dipole treatment in section 4.7.5 can use a single reflection coefficient without taking into account the azimuthal dependency of that reflection coefficient.
# References
[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley \& Sons, 3rd edition, 2005.
[2] E. Hecht. Optics. 1998.
[3] JD Jackson. Classical Electrodynamics. John Wiley and Sons, 2nd edition, 1975.
[4] Wikipedia. Magnetic potential — Wikipedia, The Free Encyclopedia, 2015. URL https://en.wikipedia.org/w/index.php?title=Magnetic_potential&oldid=642387563. [Online; accessed 5-February-2015].
## Notes for ece1229 antenna theory
I’ve now posted a first set of notes for the antenna theory course that I am taking this term at UofT.
Unlike most of the other classes I have taken, I am not attempting to take comprehensive notes for this class. The class is taught on slides that match the textbook so closely, there is little value to me taking notes that just replicate the text. Instead, I am annotating my copy of textbook with little details instead. My usual notes collection for the class will contain musings of details that were unclear, or in some cases, details that were provided in class, but are not in the text (and too long to pencil into my book.)
• Reading notes for chapter 2 (Fundamental Parameters of Antennas) and chapter 3 (Radiation Integrals and Auxiliary Potential Functions) of the class text.
• Geometric Algebra musings. How to do formulate Maxwell’s equations when magnetic sources are also included (those modeling magnetic dipoles).
• Some problems for chapter 2 content.
## Recovering the fields
This is a small addition to Phasor form of (extended) Maxwell’s equations in Geometric Algebra.
Relative to the observer frame implicitly specified by $$\gamma_0$$, here’s an expansion of the curl of the electric four potential
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:720}
\begin{aligned}
&=
\inv{2}\lr{
} \\
&=
\inv{2}\lr{
\gamma_0 \lr{ \spacegrad + j k } \gamma_0 \lr{ A_{\textrm{e}}^0 – \BA_{\textrm{e}} }
\gamma_0 \lr{ A_{\textrm{e}}^0 – \BA_{\textrm{e}} } \gamma_0 \lr{ \spacegrad + j k }
} \\
&=
\inv{2}\lr{
\lr{ -\spacegrad + j k } \lr{ A_{\textrm{e}}^0 – \BA_{\textrm{e}} }
\lr{ A_{\textrm{e}}^0 + \BA_{\textrm{e}} } \lr{ \spacegrad + j k }
} \\
&=
\inv{2}\lr{
– 2 \spacegrad A_{\textrm{e}}^0 + j k A_{\textrm{e}}^0 – j k A_{\textrm{e}}^0
– 2 j k \BA_{\textrm{e}}
} \\
&=
– \lr{ \spacegrad A_{\textrm{e}}^0 + j k \BA_{\textrm{e}} }
\end{aligned}
In the above expansion when the gradients appeared on the right of the field components, they are acting from the right (i.e. implicitly using the Hestenes dot convention.)
The electric and magnetic fields can be picked off directly from above, and in the units implied by this choice of four-potential are
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:760}
\BE_{\textrm{e}} = – \lr{ \spacegrad A_{\textrm{e}}^0 + j k \BA_{\textrm{e}} } = -j \lr{ \inv{k}\spacegrad \spacegrad \cdot \BA_{\textrm{e}} + k \BA_{\textrm{e}} }
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:780}
c \BB_{\textrm{e}} = \spacegrad \cross \BA_{\textrm{e}}.
For the fields due to the magnetic potentials
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:800}
\lr{ \grad \wedge A_{\textrm{e}} } I
=
– \lr{ \spacegrad A_{\textrm{e}}^0 + j k \BA_{\textrm{e}} } I
so the fields are
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:840}
c \BB_{\textrm{m}} = – \lr{ \spacegrad A_{\textrm{m}}^0 + j k \BA_{\textrm{m}} } = -j \lr{ \inv{k}\spacegrad \spacegrad \cdot \BA_{\textrm{m}} + k \BA_{\textrm{m}} }
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:860}
Including both electric and magnetic sources the fields are
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:900}
\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:920}
## Dual-Maxwell’s (phasor) equations in Geometric Algebra
These notes repeat (mostly word for word) the previous notes Maxwell’s (phasor) equations in Geometric Algebra. Electric charges and currents have been replaced with magnetic charges and currents, and the appropriate relations modified accordingly.
In [1] section 3.3, treating magnetic charges and currents, and no electric charges and currents, is a demonstration of the required (curl) form for the electric field, and potential form for the electric field. Not knowing what to name this, I’ll call the associated equations the dual-Maxwell’s equations.
I was wondering how this derivation would proceed using the Geometric Algebra (GA) formalism.
## Dual-Maxwell’s equation in GA phasor form.
The dual-Maxwell’s equations, omitting electric charges and currents, are
\label{eqn:phasorDualMaxwellsGA:20}
\spacegrad \cross \boldsymbol{\mathcal{E}} = -\PD{t}{\boldsymbol{\mathcal{B}}} -\BM
\label{eqn:phasorDualMaxwellsGA:40}
\label{eqn:phasorDualMaxwellsGA:60}
\label{eqn:phasorDualMaxwellsGA:80}
Assuming linear media $$\boldsymbol{\mathcal{B}} = \mu_0 \boldsymbol{\mathcal{H}}$$, $$\boldsymbol{\mathcal{D}} = \epsilon_0 \boldsymbol{\mathcal{E}}$$, and phasor relationships of the form $$\boldsymbol{\mathcal{E}} = \textrm{Re} \lr{ \BE(\Br) e^{j \omega t}}$$ for the fields and the currents, these reduce to
\label{eqn:phasorDualMaxwellsGA:100}
\spacegrad \cross \BE = – j \omega \BB – \BM
\label{eqn:phasorDualMaxwellsGA:120}
\spacegrad \cross \BB = j \omega \epsilon_0 \mu_0 \BE
\label{eqn:phasorDualMaxwellsGA:140}
\label{eqn:phasorDualMaxwellsGA:160}
These four equations can be assembled into a single equation form using the GA identities
\label{eqn:phasorDualMaxwellsGA:200}
\Bf \Bg
= \Bf \cdot \Bg + \Bf \wedge \Bg
= \Bf \cdot \Bg + I \Bf \cross \Bg.
\label{eqn:phasorDualMaxwellsGA:220}
I = \xcap \ycap \zcap.
The electric and magnetic field equations, respectively, are
\label{eqn:phasorDualMaxwellsGA:260}
\spacegrad \BE = – \lr{ \BM + j k c \BB} I
\label{eqn:phasorDualMaxwellsGA:280}
\spacegrad c \BB = c \rho_m + j k \BE I
where $$\omega = k c$$, and $$1 = c^2 \epsilon_0 \mu_0$$ have also been used to eliminate some of the mess of constants.
Summing these (first scaling \ref{eqn:phasorDualMaxwellsGA:280} by $$I$$), gives Maxwell’s equation in its GA phasor form
\label{eqn:phasorDualMaxwellsGA:300}
\boxed{
\lr{ \spacegrad + j k } \lr{ \BE + I c \BB } = \lr{c \rho – \BM} I.
}
## Preliminaries. Dual magnetic form of Maxwell’s equations.
The arguments of the text showing that a potential representation for the electric and magnetic fields is possible easily translates into GA. To perform this translation, some duality lemmas are required
First consider the cross product of two vectors $$\Bx, \By$$ and the right handed dual $$-\By I$$ of $$\By$$, a bivector, of one of these vectors. Noting that the Euclidean pseudoscalar $$I$$ commutes with all grade multivectors in a Euclidean geometric algebra space, the cross product can be written
\label{eqn:phasorDualMaxwellsGA:320}
\begin{aligned}
\lr{ \Bx \cross \By }
&=
-I \lr{ \Bx \wedge \By } \\
&=
-I \inv{2} \lr{ \Bx \By – \By \Bx } \\
&=
\inv{2} \lr{ \Bx (-\By I) – (-\By I) \Bx } \\
&=
\Bx \cdot \lr{ -\By I }.
\end{aligned}
The last step makes use of the fact that the wedge product of a vector and vector is antisymmetric, whereas the dot product (vector grade selection) of a vector and bivector is antisymmetric. Details on grade selection operators and how to characterize symmetric and antisymmetric products of vectors with blades as either dot or wedge products can be found in [3], [2].
Similarly, the dual of the dot product can be written as
\label{eqn:phasorDualMaxwellsGA:440}
\begin{aligned}
-I \lr{ \Bx \cdot \By }
&=
-I \inv{2} \lr{ \Bx \By + \By \Bx } \\
&=
\inv{2} \lr{ \Bx (-\By I) + (-\By I) \Bx } \\
&=
\Bx \wedge \lr{ -\By I }.
\end{aligned}
These duality transformations are motivated by the observation that in the GA form of Maxwell’s equation the magnetic field shows up in its dual form, a bivector. Spelled out in terms of the dual magnetic field, those equations are
\label{eqn:phasorDualMaxwellsGA:360}
\spacegrad \cdot (-\BE I)= – j \omega \BB – \BM
\label{eqn:phasorDualMaxwellsGA:380}
\spacegrad \wedge \BH = j \omega \epsilon_0 \BE I
\label{eqn:phasorDualMaxwellsGA:400}
\spacegrad \wedge (-\BE I) = 0
\label{eqn:phasorDualMaxwellsGA:420}
## Constructing a potential representation.
The starting point of the argument in the text was the observation that the triple product $$\spacegrad \cdot \lr{ \spacegrad \cross \Bx } = 0$$ for any (sufficiently continuous) vector $$\Bx$$. This triple product is a completely antisymmetric sum, and the equivalent statement in GA is $$\spacegrad \wedge \spacegrad \wedge \Bx = 0$$ for any vector $$\Bx$$. This follows from $$\Ba \wedge \Ba = 0$$, true for any vector $$\Ba$$, including the gradient operator $$\spacegrad$$, provided those gradients are acting on a sufficiently continuous blade.
In the absence of electric charges,
\ref{eqn:phasorDualMaxwellsGA:400} shows that the divergence of the dual electric field is zero. It it therefore possible to find a potential $$\BF$$ such that
\label{eqn:phasorDualMaxwellsGA:460}
-\epsilon_0 \BE I = \spacegrad \wedge \BF.
Substituting this \ref{eqn:phasorDualMaxwellsGA:380} gives
\label{eqn:phasorDualMaxwellsGA:480}
\spacegrad \wedge \lr{ \BH + j \omega \BF } = 0.
This relation is a bivector identity with zero, so will be satisfied if
\label{eqn:phasorDualMaxwellsGA:500}
\BH + j \omega \BF = -\spacegrad \phi_m,
for some scalar $$\phi_m$$. Unlike the $$-\epsilon_0 \BE I = \spacegrad \wedge \BF$$ solution to \ref{eqn:phasorDualMaxwellsGA:400}, the grade of $$\phi_m$$ is fixed by the requirement that $$\BE + j \omega \BF$$ is unity (a vector), so
a $$\BE + j \omega \BF = \spacegrad \wedge \psi$$, for a higher grade blade $$\psi$$ would not work, despite satisfying the condition $$\spacegrad \wedge \spacegrad \wedge \psi = 0$$.
Substitution of \ref{eqn:phasorDualMaxwellsGA:500} and \ref{eqn:phasorDualMaxwellsGA:460} into \ref{eqn:phasorDualMaxwellsGA:380} gives
\label{eqn:phasorDualMaxwellsGA:520}
\begin{aligned}
\spacegrad \cdot \lr{ \spacegrad \wedge \BF } &= -\epsilon_0 \BM – j \omega \epsilon_0 \mu_0 \lr{ -\spacegrad \phi_m -j \omega \BF } \\
\end{aligned}
Rearranging gives
\label{eqn:phasorDualMaxwellsGA:540}
\spacegrad^2 \BF + k^2 \BF = -\epsilon_0 \BM + \spacegrad \lr{ \spacegrad \cdot \BF + j \frac{k}{c} \phi_m }.
The fields $$\BF$$ and $$\phi_m$$ are assumed to be phasors, say $$\boldsymbol{\mathcal{A}} = \textrm{Re} \BF e^{j k c t}$$ and $$\varphi = \textrm{Re} \phi_m e^{j k c t}$$. Grouping the scalar and vector potentials into the standard four vector form
$$F^\mu = \lr{\phi_m/c, \BF}$$, and expanding the Lorentz gauge condition
\label{eqn:phasorDualMaxwellsGA:580}
\begin{aligned}
0
&= \partial_\mu \lr{ F^\mu e^{j k c t}} \\
&= \partial_a \lr{ F^a e^{j k c t}} + \inv{c}\PD{t}{} \lr{ \frac{\phi_m}{c}
e^{j k c t}} \\
&= \spacegrad \cdot \BF e^{j k c t} + \inv{c} j k \phi_m e^{j k c t} \\
&= \lr{ \spacegrad \cdot \BF + j k \phi_m/c } e^{j k c t},
\end{aligned}
shows that in
\ref{eqn:phasorDualMaxwellsGA:540}
the quantity in braces is in fact the Lorentz gauge condition, so in the Lorentz gauge, the vector potential satisfies a non-homogeneous Helmholtz equation.
\label{eqn:phasorDualMaxwellsGA:550}
\boxed{
\spacegrad^2 \BF + k^2 \BF = -\epsilon_0 \BM.
}
## Maxwell’s equation in Four vector form
The four vector form of Maxwell’s equation follows from \ref{eqn:phasorDualMaxwellsGA:300} after pre-multiplying by $$\gamma^0$$.
With
\label{eqn:phasorDualMaxwellsGA:620}
F = F^\mu \gamma_\mu = \lr{ \phi_m/c, \BF }
\label{eqn:phasorDualMaxwellsGA:640}
G = \grad \wedge F = – \epsilon_0 \lr{ \BE + c \BB I } I
\label{eqn:phasorDualMaxwellsGA:660}
\grad = \gamma^\mu \partial_\mu = \gamma^0 \lr{ \spacegrad + j k }
\label{eqn:phasorDualMaxwellsGA:680}
M = M^\mu \gamma_\mu = \lr{ c \rho_m, \BM },
Maxwell’s equation is
\label{eqn:phasorDualMaxwellsGA:720}
\boxed{
}
Here $$\setlr{ \gamma_\mu }$$ is used as the basis of the four vector Minkowski space, with $$\gamma_0^2 = -\gamma_k^2 = 1$$ (i.e. $$\gamma^\mu \cdot \gamma_\nu = {\delta^\mu}_\nu$$), and $$\gamma_a \gamma_0 = \sigma_a$$ where $$\setlr{ \sigma_a}$$ is the Pauli basic (i.e. standard basis vectors for \R{3}).
Let’s demonstrate this, one piece at a time. Observe that the action of the spacetime gradient on a phasor, assuming that all time dependence is in the exponential, is
\label{eqn:phasorDualMaxwellsGA:740}
\begin{aligned}
\gamma^\mu \partial_\mu \lr{ \psi e^{j k c t} }
&=
\lr{ \gamma^a \partial_a + \gamma_0 \partial_{c t} } \lr{ \psi e^{j k c t} }
\\
&=
\gamma_0 \lr{ \gamma_0 \gamma^a \partial_a + j k } \lr{ \psi e^{j k c t} } \\
&=
\gamma_0 \lr{ \sigma_a \partial_a + j k } \psi e^{j k c t} \\
&=
\gamma_0 \lr{ \spacegrad + j k } \psi e^{j k c t}
\end{aligned}
This allows the operator identification of \ref{eqn:phasorDualMaxwellsGA:660}. The four current portion of the equation comes from
\label{eqn:phasorDualMaxwellsGA:760}
\begin{aligned}
c \rho_m – \BM
&=
\gamma_0 \lr{ \gamma_0 c \rho_m – \gamma_0 \gamma_a \gamma_0 M^a } \\
&=
\gamma_0 \lr{ \gamma_0 c \rho_m + \gamma_a M^a } \\
&=
\gamma_0 \lr{ \gamma_\mu M^\mu } \\
&= \gamma_0 M.
\end{aligned}
Taking the curl of the four potential gives
\label{eqn:phasorDualMaxwellsGA:780}
\begin{aligned}
&=
\lr{ \gamma^a \partial_a + \gamma_0 j k } \wedge \lr{ \gamma_0 \phi_m/c +
\gamma_b F^b } \\
&=
– \sigma_a \partial_a \phi_m/c + \gamma^a \wedge \gamma_b \partial_a F^b – j k
\sigma_b F^b \\
&=
– \sigma_a \partial_a \phi_m/c + \sigma_a \wedge \sigma_b \partial_a F^b – j k
\sigma_b F^b \\
&= \inv{c} \lr{ – \spacegrad \phi_m – j \omega \BF + c \spacegrad \wedge \BF }
\\
&= \epsilon_0 \lr{ c \BB – \BE I } \\
&= – \epsilon_0 \lr{ \BE + c \BB I } I.
\end{aligned}
Substituting all of these into Maxwell’s \ref{eqn:phasorDualMaxwellsGA:300} gives
\label{eqn:phasorDualMaxwellsGA:800}
which recovers \ref{eqn:phasorDualMaxwellsGA:700} as desired.
## Helmholtz equation directly from the GA form.
It is easier to find \ref{eqn:phasorDualMaxwellsGA:550} from the GA form of Maxwell’s \ref{eqn:phasorDualMaxwellsGA:700} than the traditional curl and divergence equations. Note that
\label{eqn:phasorDualMaxwellsGA:820}
\begin{aligned}
&=
&=
+
&=
\end{aligned}
however, the Lorentz gauge condition $$\partial_\mu F^\mu = \grad \cdot F = 0$$ kills the latter term above. This leaves
\label{eqn:phasorDualMaxwellsGA:840}
\begin{aligned}
&=
&=
\gamma_0 \lr{ \spacegrad + j k }
\gamma_0 \lr{ \spacegrad + j k } F \\
&=
\gamma_0^2 \lr{ -\spacegrad + j k }
\lr{ \spacegrad + j k } F \\
&=
-\lr{ \spacegrad^2 + k^2 } F = -\epsilon_0 M.
\end{aligned}
The timelike component of this gives
\label{eqn:phasorDualMaxwellsGA:860}
\lr{ \spacegrad^2 + k^2 } \phi_m = -\epsilon_0 c \rho_m,
and the spacelike components give
\label{eqn:phasorDualMaxwellsGA:880}
\lr{ \spacegrad^2 + k^2 } \BF = -\epsilon_0 \BM,
recovering \ref{eqn:phasorDualMaxwellsGA:550} as desired.
# References
[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley \& Sons, 3rd edition, 2005.
[2] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.
[3] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999.
## Maxwell’s (phasor) equations in Geometric Algebra
In [1] section 3.2 is a demonstration of the required (curl) form for the magnetic field, and potential form for the electric field.
I was wondering how this derivation would proceed using the Geometric Algebra (GA) formalism.
## Maxwell’s equation in GA phasor form.
Maxwell’s equations, omitting magnetic charges and currents, are
\label{eqn:phasorMaxwellsGA:20}
\label{eqn:phasorMaxwellsGA:40}
\spacegrad \cross \boldsymbol{\mathcal{H}} = \boldsymbol{\mathcal{J}} + \PD{t}{\boldsymbol{\mathcal{D}}}
\label{eqn:phasorMaxwellsGA:60}
\label{eqn:phasorMaxwellsGA:80}
Assuming linear media $$\boldsymbol{\mathcal{B}} = \mu_0 \boldsymbol{\mathcal{H}}$$, $$\boldsymbol{\mathcal{D}} = \epsilon_0 \boldsymbol{\mathcal{E}}$$, and phasor relationships of the form $$\boldsymbol{\mathcal{E}} = \textrm{Re} \lr{ \BE(\Br) e^{j \omega t}}$$ for the fields and the currents, these reduce to
\label{eqn:phasorMaxwellsGA:100}
\spacegrad \cross \BE = – j \omega \BB
\label{eqn:phasorMaxwellsGA:120}
\spacegrad \cross \BB = \mu_0 \BJ + j \omega \epsilon_0 \mu_0 \BE
\label{eqn:phasorMaxwellsGA:140}
\label{eqn:phasorMaxwellsGA:160}
These four equations can be assembled into a single equation form using the GA identities
\label{eqn:phasorMaxwellsGA:200}
\Bf \Bg
= \Bf \cdot \Bg + \Bf \wedge \Bg
= \Bf \cdot \Bg + I \Bf \cross \Bg.
\label{eqn:phasorMaxwellsGA:220}
I = \xcap \ycap \zcap.
The electric and magnetic field equations, respectively, are
\label{eqn:phasorMaxwellsGA:260}
\spacegrad \BE = \rho/\epsilon_0 -j k c \BB I
\label{eqn:phasorMaxwellsGA:280}
\spacegrad c \BB = \frac{I}{\epsilon_0 c} \BJ + j k \BE I
where $$\omega = k c$$, and $$1 = c^2 \epsilon_0 \mu_0$$ have also been used to eliminate some of the mess of constants.
Summing these (first scaling \ref{eqn:phasorMaxwellsGA:280} by $$I$$), gives Maxwell’s equation in its GA phasor form
\label{eqn:phasorMaxwellsGA:300}
\boxed{
\lr{ \spacegrad + j k } \lr{ \BE + I c \BB } = \inv{\epsilon_0 c}\lr{c \rho – \BJ}.
}
## Preliminaries. Dual magnetic form of Maxwell’s equations.
The arguments of the text showing that a potential representation for the electric and magnetic fields is possible easily translates into GA. To perform this translation, some duality lemmas are required
First consider the cross product of two vectors $$\Bx, \By$$ and the right handed dual $$-\By I$$ of $$\By$$, a bivector, of one of these vectors. Noting that the Euclidean pseudoscalar $$I$$ commutes with all grade multivectors in a Euclidean geometric algebra space, the cross product can be written
\label{eqn:phasorMaxwellsGA:320}
\begin{aligned}
\lr{ \Bx \cross \By }
&=
-I \lr{ \Bx \wedge \By } \\
&=
-I \inv{2} \lr{ \Bx \By – \By \Bx } \\
&=
\inv{2} \lr{ \Bx (-\By I) – (-\By I) \Bx } \\
&=
\Bx \cdot \lr{ -\By I }.
\end{aligned}
The last step makes use of the fact that the wedge product of a vector and vector is antisymmetric, whereas the dot product (vector grade selection) of a vector and bivector is antisymmetric. Details on grade selection operators and how to characterize symmetric and antisymmetric products of vectors with blades as either dot or wedge products can be found in [3], [2].
Similarly, the dual of the dot product can be written as
\label{eqn:phasorMaxwellsGA:440}
\begin{aligned}
-I \lr{ \Bx \cdot \By }
&=
-I \inv{2} \lr{ \Bx \By + \By \Bx } \\
&=
\inv{2} \lr{ \Bx (-\By I) + (-\By I) \Bx } \\
&=
\Bx \wedge \lr{ -\By I }.
\end{aligned}
These duality transformations are motivated by the observation that in the GA form of Maxwell’s equation the magnetic field shows up in its dual form, a bivector. Spelled out in terms of the dual magnetic field, those equations are
\label{eqn:phasorMaxwellsGA:360}
\spacegrad \wedge \BE = – j \omega \BB I
\label{eqn:phasorMaxwellsGA:380}
\spacegrad \cdot \lr{ -\BB I } = \mu_0 \BJ + j \omega \epsilon_0 \mu_0 \BE
\label{eqn:phasorMaxwellsGA:400}
\label{eqn:phasorMaxwellsGA:420}
\spacegrad \wedge (-\BB I) = 0.
## Constructing a potential representation.
The starting point of the argument in the text was the observation that the triple product $$\spacegrad \cdot \lr{ \spacegrad \cross \Bx } = 0$$ for any (sufficiently continuous) vector $$\Bx$$. This triple product is a completely antisymmetric sum, and the equivalent statement in GA is $$\spacegrad \wedge \spacegrad \wedge \Bx = 0$$ for any vector $$\Bx$$. This follows from $$\Ba \wedge \Ba = 0$$, true for any vector $$\Ba$$, including the gradient operator $$\spacegrad$$, provided those gradients are acting on a sufficiently continuous blade.
In the absence of magnetic charges, \ref{eqn:phasorMaxwellsGA:420} shows that the divergence of the dual magnetic field is zero. It it therefore possible to find a potential $$\BA$$ such that
\label{eqn:phasorMaxwellsGA:460}
\BB I = \spacegrad \wedge \BA.
Substituting this into Maxwell-Faraday \ref{eqn:phasorMaxwellsGA:360} gives
\label{eqn:phasorMaxwellsGA:480}
\spacegrad \wedge \lr{ \BE + j \omega \BA } = 0.
This relation is a bivector identity with zero, so will be satisfied if
\label{eqn:phasorMaxwellsGA:500}
\BE + j \omega \BA = -\spacegrad \phi,
for some scalar $$\phi$$. Unlike the $$\BB I = \spacegrad \wedge \BA$$ solution to \ref{eqn:phasorMaxwellsGA:420}, the grade of $$\phi$$ is fixed by the requirement that $$\BE + j \omega \BA$$ is unity (a vector), so a $$\BE + j \omega \BA = \spacegrad \wedge \psi$$, for a higher grade blade $$\psi$$ would not work, despite satisifying the condition $$\spacegrad \wedge \spacegrad \wedge \psi = 0$$.
Substitution of \ref{eqn:phasorMaxwellsGA:500} and \ref{eqn:phasorMaxwellsGA:460} into Ampere’s law \ref{eqn:phasorMaxwellsGA:380} gives
\label{eqn:phasorMaxwellsGA:520}
\begin{aligned}
-\spacegrad \cdot \lr{ \spacegrad \wedge \BA } &= \mu_0 \BJ + j \omega \epsilon_0 \mu_0 \lr{ -\spacegrad \phi -j \omega \BA } \\
\end{aligned}
Rearranging gives
\label{eqn:phasorMaxwellsGA:540}
\spacegrad^2 \BA + k^2 \BA = -\mu_0 \BJ – \spacegrad \lr{ \spacegrad \cdot \BA + j \frac{k}{c} \phi }.
The fields $$\BA$$ and $$\phi$$ are assumed to be phasors, say $$\boldsymbol{\mathcal{A}} = \textrm{Re} \BA e^{j k c t}$$ and $$\varphi = \textrm{Re} \phi e^{j k c t}$$. Grouping the scalar and vector potentials into the standard four vector form $$A^\mu = \lr{\phi/c, \BA}$$, and expanding the Lorentz gauge condition
\label{eqn:phasorMaxwellsGA:580}
\begin{aligned}
0
&= \partial_\mu \lr{ A^\mu e^{j k c t}} \\
&= \partial_a \lr{ A^a e^{j k c t}} + \inv{c}\PD{t}{} \lr{ \frac{\phi}{c} e^{j k c t}} \\
&= \spacegrad \cdot \BA e^{j k c t} + \inv{c} j k \phi e^{j k c t} \\
&= \lr{ \spacegrad \cdot \BA + j k \phi/c } e^{j k c t},
\end{aligned}
shows that in \ref{eqn:phasorMaxwellsGA:540} the quantity in braces is in fact the Lorentz gauge condition, so in the Lorentz gauge, the vector potential satisfies a non-homogeneous Helmholtz equation.
\label{eqn:phasorMaxwellsGA:550}
\boxed{
\spacegrad^2 \BA + k^2 \BA = -\mu_0 \BJ.
}
## Maxwell’s equation in Four vector form
The four vector form of Maxwell’s equation follows from \ref{eqn:phasorMaxwellsGA:300} after pre-multiplying by $$\gamma^0$$.
With
\label{eqn:phasorMaxwellsGA:620}
A = A^\mu \gamma_\mu = \lr{ \phi/c, \BA }
\label{eqn:phasorMaxwellsGA:640}
F = \grad \wedge A = \inv{c} \lr{ \BE + c \BB I }
\label{eqn:phasorMaxwellsGA:660}
\grad = \gamma^\mu \partial_\mu = \gamma^0 \lr{ \spacegrad + j k }
\label{eqn:phasorMaxwellsGA:680}
J = J^\mu \gamma_\mu = \lr{ c \rho, \BJ },
Maxwell’s equation is
\label{eqn:phasorMaxwellsGA:700}
\boxed{
}
Here $$\setlr{ \gamma_\mu }$$ is used as the basis of the four vector Minkowski space, with $$\gamma_0^2 = -\gamma_k^2 = 1$$ (i.e. $$\gamma^\mu \cdot \gamma_\nu = {\delta^\mu}_\nu$$), and $$\gamma_a \gamma_0 = \sigma_a$$ where $$\setlr{ \sigma_a}$$ is the Pauli basic (i.e. standard basis vectors for \R{3}).
Let’s demonstrate this, one piece at a time. Observe that the action of the spacetime gradient on a phasor, assuming that all time dependence is in the exponential, is
\label{eqn:phasorMaxwellsGA:740}
\begin{aligned}
\gamma^\mu \partial_\mu \lr{ \psi e^{j k c t} }
&=
\lr{ \gamma^a \partial_a + \gamma_0 \partial_{c t} } \lr{ \psi e^{j k c t} }
\\
&=
\gamma_0 \lr{ \gamma_0 \gamma^a \partial_a + j k } \lr{ \psi e^{j k c t} } \\
&=
\gamma_0 \lr{ \sigma_a \partial_a + j k } \psi e^{j k c t} \\
&=
\gamma_0 \lr{ \spacegrad + j k } \psi e^{j k c t}
\end{aligned}
This allows the operator identification of \ref{eqn:phasorMaxwellsGA:660}. The four current portion of the equation comes from
\label{eqn:phasorMaxwellsGA:760}
\begin{aligned}
c \rho – \BJ
&=
\gamma_0 \lr{ \gamma_0 c \rho – \gamma_0 \gamma_a \gamma_0 J^a } \\
&=
\gamma_0 \lr{ \gamma_0 c \rho + \gamma_a J^a } \\
&=
\gamma_0 \lr{ \gamma_\mu J^\mu } \\
&= \gamma_0 J.
\end{aligned}
Taking the curl of the four potential gives
\label{eqn:phasorMaxwellsGA:780}
\begin{aligned}
&=
\lr{ \gamma^a \partial_a + \gamma_0 j k } \wedge \lr{ \gamma_0 \phi/c + \gamma_b A^b } \\
&=
– \sigma_a \partial_a \phi/c + \gamma^a \wedge \gamma_b \partial_a A^b – j k
\sigma_b A^b \\
&=
– \sigma_a \partial_a \phi/c + \sigma_a \wedge \sigma_b \partial_a A^b – j k
\sigma_b A^b \\
&= \inv{c} \lr{ – \spacegrad \phi – j \omega \BA + c \spacegrad \wedge \BA }
\\
&= \inv{c} \lr{ \BE + c \BB I }.
\end{aligned}
Substituting all of these into Maxwell’s \ref{eqn:phasorMaxwellsGA:300} gives
\label{eqn:phasorMaxwellsGA:800}
\gamma_0 \grad c F = \inv{ \epsilon_0 c } \gamma_0 J,
which recovers \ref{eqn:phasorMaxwellsGA:700} as desired.
## Helmholtz equation directly from the GA form.
It is easier to find \ref{eqn:phasorMaxwellsGA:550} from the GA form of Maxwell’s \ref{eqn:phasorMaxwellsGA:700} than the traditional curl and divergence equations. Note that
\label{eqn:phasorMaxwellsGA:820}
=
=
+
=
however, the Lorentz gauge condition $$\partial_\mu A^\mu = \grad \cdot A = 0$$ kills the latter term above. This leaves
\label{eqn:phasorMaxwellsGA:840}
\begin{aligned}
&=
&=
\gamma_0 \lr{ \spacegrad + j k }
\gamma_0 \lr{ \spacegrad + j k } A \\
&=
\gamma_0^2 \lr{ -\spacegrad + j k }
\lr{ \spacegrad + j k } A \\
&=
-\lr{ \spacegrad^2 + k^2 } A = \mu_0 J.
\end{aligned}
The timelike component of this gives
\label{eqn:phasorMaxwellsGA:860}
\lr{ \spacegrad^2 + k^2 } \phi = -\rho/\epsilon_0,
and the spacelike components give
\label{eqn:phasorMaxwellsGA:880}
\lr{ \spacegrad^2 + k^2 } \BA = -\mu_0 \BJ,
recovering \ref{eqn:phasorMaxwellsGA:550} as desired.
# References
[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley & Sons, 3rd edition, 2005.
[2] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.
[3] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999. | 2023-01-27 18:33:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9799699783325195, "perplexity": 5573.691830753746}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764495001.99/warc/CC-MAIN-20230127164242-20230127194242-00743.warc.gz"} |
http://clay6.com/qa/40300/in-right-angled-triangle-the-difference-between-the-two-acute-angles-is-lar | Browse Questions
# In right-angled triangle,the difference between the two acute angles is $(\large\frac{\pi}{15})^c$.Find the angles in degrees
$\begin{array}{1 1}(A)\;51^{\large\circ},39^{\large\circ},90^{\large\circ}\\(B)\;50^{\large\circ},40^{\large\circ},95^{\large\circ}\\(C)\;60^{\large\circ},40^{\large\circ},70^{\large\circ}\\(D)\;80^{\large\circ},50^{\large\circ},90^{\large\circ}\end{array}$
Toolbox: | 2016-12-08 02:07:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7014577388763428, "perplexity": 6356.19299484463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542323.80/warc/CC-MAIN-20161202170902-00459-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/134135-integration-problem-print.html | # Integration Problem
• March 16th 2010, 04:35 PM
Integration Problem
Integrate:
$
6(e^{4-3x})^2
$
Is it correct if i do this: $6(e^{8-6x})$ and then solve?
and then let u = 8-6x and du=-6
then the answer comes to $-e^{8-6x} + C$
• March 16th 2010, 04:37 PM
Moo
Hello,
It's all correct (Nod)
• March 16th 2010, 04:47 PM
thank you, here is another one (I have a test on Friday so I'm trying to do as many problems as I can)
Integrate: $\frac{3}{4}-\frac{1}{\sqrt{6x}}$
I let u = 6x and du = 6dx
the answer I end up with is:
$
-\frac{1}{4}\sqrt{6x}+C
$
• March 16th 2010, 04:52 PM
Moo
Hello,
And what about integrating the constant ? The integral of 1 is x, the integral of 3/4 will be...
But according to your answer, would the integral rather be $\int \frac 34\times \frac{1}{\sqrt{6x}} ~dx$ ? In which case, it's correct - again :p
• March 16th 2010, 05:01 PM
oops no you were right I did it wrong but thanks for pointing that out :)
• March 16th 2010, 05:10 PM
$ | 2016-02-14 03:12:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9567912220954895, "perplexity": 1619.6620464482294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701168076.20/warc/CC-MAIN-20160205193928-00163-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://eips.ethereum.org/EIPS/eip-205 | 🚧 This EIP had no activity for at least 6 months.
# EIP-205: ENS support for contract ABIs Source
Author Nick Johnson Stagnant Standards Track ERC 2017-02-06 137, 181
## Simple Summary
This EIP proposes a mechanism for storing ABI definitions in ENS, for easy lookup of contract interfaces by callers.
## Abstract
ABIs are important metadata required for interacting with most contracts. At present, they are typically supplied out-of-band, which adds an additional burden to interacting with contracts, particularly on a one-off basis or where the ABI may be updated over time. The small size of ABIs permits an alternative solution, storing them in ENS, permitting name lookup and ABI discovery via the same process.
ABIs are typically quite compact; the largest in-use ABI we could find, that for the DAO, is 9450 bytes uncompressed JSON, 6920 bytes uncompressed CBOR, and 1128 bytes when the JSON form is compressed with zlib. Further gains on CBOR encoding are possible using a CBOR extension that permits eliminating repeated strings, which feature extensively in ABIs. Most ABIs, however, are far shorter than this, consisting of only a few hundred bytes of uncompressed JSON.
This EIP defines a resolver profile for retrieving contract ABIs, as well as encoding standards for storing ABIs for different applications, allowing the user to select between different representations based on their need for compactness and other considerations such as onchain access.
## Specification
### ABI encodings
In order to allow for different tradeoffs between onchain size and accessibility, several ABI encodings are defined. Each ABI encoding is defined by a unique constant with only a single bit set, allowing for the specification of 256 unique encodings in a single uint.
The currently recognised encodings are:
ID Description
1 JSON
2 zlib-compressed JSON
4 CBOR
8 URI
This table may be extended in future through the EIP process.
Encoding type 1 specifies plaintext JSON, uncompressed; this is the standard format in which ABIs are typically encoded, but also the bulkiest, and is not easily parseable onchain.
Encoding type 2 specifies zlib-compressed JSON. This is significantly smaller than uncompressed JSON, and is straightforward to decode offchain. However, it is impracticalfor onchain consumers to use.
Encoding type 4 is CBOR. CBOR is a binary encoding format that is a superset of JSON, and is both more compact and easier to parse in limited environments such as the EVM. Consumers that support CBOR are strongly encouraged to also support the stringref extension to CBOR, which provides significant additional reduction in encoded size.
Encoding type 8 indicates that the ABI can be found elsewhere, at the specified URI. This is typically the most compact of the supported forms, but also adds external dependencies for implementers. The specified URI may use any schema, but HTTP, IPFS, and Swarm are expected to be the most common.
### Resolver profile
A new resolver interface is defined, consisting of the following method:
function ABI(bytes32 node, uint256 contentType) constant returns (uint256, bytes);
The interface ID of this interface is 0x2203ab56.
contentType is a bitfield, and is the bitwise OR of all the encoding types the caller will accept. Resolvers that implement this interface must return an ABI encoded using one of the requested formats, or (0, "") if they do not have an ABI for this function, or do not support any of the requested formats.
The abi resolver profile is valid on both forward and reverse records.
### ABI lookup process
When attempting to fetch an ABI based on an ENS name, implementers should first attempt an ABI lookup on the name itself. If that lookup returns no results, they should attempt a reverse lookup on the Ethereum address the name resolves to.
Implementers should support as many of the ABI encoding formats as practical.
## Rationale
Storing ABIs onchain avoids the need to introduce additional dependencies for applications wishing to fetch them, such as swarm or HTTP access. Given the typical compactness of ABIs, we believe this is a worthwhile tradeoff in many cases.
The two-step resolution process permits different names to provide different ABIs for the same contract, such as in the case where it’s useful to provide a minimal ABI to some callers, as well as specifying ABIs for contracts that did not specify one of their own. The fallback to looking up an ABI on the reverse record permits contracts to specify their own canonical ABI, and prevents the need for duplication when multiple names reference the same contract without the need for different ABIs.
Copyright and related rights waived via CC0. | 2021-10-18 08:04:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24739770591259003, "perplexity": 5212.617442117341}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585199.76/warc/CC-MAIN-20211018062819-20211018092819-00025.warc.gz"} |
https://plainmath.net/45032/solve-5x-3-3x-2-20x-plus-12-equal-0 | # Solve 5x^{3}-3x^{2}-20x+12=0
Solve $5{x}^{3}-3{x}^{2}-20x+12=0$
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scoollato7o
We have $5{x}^{3}-3{x}^{2}-20x+12=0Z$
${x}^{2}\left(5x-3\right)-4\left(5x-3\right)=0$
$\left({x}^{2}-4\right)\left(5x-3\right)=0$
$\left(x+2\right)\left(x-2\right)\left(5x-3\right)=0$
Thus, $x=±2,\frac{3}{5}$
redhotdevil13l3
$5{x}^{3}-3{x}^{2}-20x+12=0Z$
${x}^{2}\left(5x-3\right)-4\left(5x-3\right)=0$
$\left({x}^{2}-4\right)\left(5x-3\right)=0$
$\left(x+2\right)\left(x-2\right)\left(5x-3\right)=0$
Using the Zero Factor Principle
$x=-2$, $x=2$, $x=\frac{3}{5}$ | 2022-10-03 15:46:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 57, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8641345500946045, "perplexity": 4521.837398326553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337421.33/warc/CC-MAIN-20221003133425-20221003163425-00703.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=0380790 | MathSciNet bibliographic data MR380790 55E45 Miller, Haynes R.; Ravenel, Douglas C.; Wilson, W. Stephen Novikov's ${\rm Ext}\sp{2}$${\rm Ext}\sp{2}$ and the nontriviality of the gamma family. Bull. Amer. Math. Soc. 81 (1975), no. 6, 1073–1075. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | 2017-09-21 23:20:24 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9990538954734802, "perplexity": 5327.632503479997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687938.15/warc/CC-MAIN-20170921224617-20170922004617-00520.warc.gz"} |
https://cglab.ca/seminar/2019/michielSpanner.html | Region Spanner for Disks
Michiel Smid
Carleton University
Consider $n$ pairwise disjoint disks. Inside any disk, we can travel at infinite speed, whereas outside all disks, we can travel at speed one. The shortest travel times between any pair of disks defines a metric space. I will present a $(1+\varepsilon)$-spanner for this metric space having $O(n/\varepsilon)$ edges. The spanner is a natural variant of the Yao-graph. | 2023-01-28 14:08:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7929274439811707, "perplexity": 934.608002445407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499634.11/warc/CC-MAIN-20230128121809-20230128151809-00664.warc.gz"} |
https://www.esaral.com/q/the-hcf-of-two-numbers-is-23-and-their-lcm-is-1449-71132 | # The HCF of two numbers is 23 and their LCM is 1449.
Question:
The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.
Solution:
Let the two numbers be and b.
Let the value of be 161.
Given: HCF = 23 and LCM = 1449
we know, × b = HCF × LCM
⇒ 161 × b = 23 × 1449
$\Rightarrow \quad \therefore b=\frac{23 \times 1449}{161}=\frac{33327}{161}=207$
Hence, the other number b is 207. | 2023-03-23 04:42:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8594445586204529, "perplexity": 910.018188349912}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00632.warc.gz"} |
http://mathoverflow.net/revisions/72820/list | The key point is that you're confusing uniform convergence and $L^2$ convergence ; indeed as $\mathcal{C}([0;1])$ is both a subspace of $\mathcal{B}([0;1])$ with $|.|_\infty$ and of $L^2([0;1])$ with $|.|_2$, you get two norms on the same vector space.
But as it isn't a finite-dimensional space, it can have non-equivalent norms - and indeed, those two norms definitely aren't equivalent, which in particular means that a sequence which has a good behaviour for the $L^2$ norm (the partial sums of the Fourier series) doesn't necessarily have a good $|.|_\infty$ behaviour.
EDIT: I should have said a little more ; there's an obvious inequality between the two norms (the mean inequality) so they are not that unrelated. But there is no reverse inequality, as can be shown by considering a sequence of piecewise linear functions : for $n\in\mathbb N$, consider $f_n$ as $t\mapsto n^\alpha-n^{\alpha+\beta}t$ on $[0;n^{-\beta}]$ and zero elsewhere ; if you choose $\alpha,\beta>0$ carefully, then you'll get a sequence which converges to zero for the $L^2$ norm, and won't converge uniformly.
The key point is that you're confusing uniform convergence and $L^2$ convergence ; indeed as $\mathcal{C}([0;1])$ is both a subspace of $\mathcal{B}([0;1])$ with $|.|_\infty$ and of $L^2([0;1])$ with $|.|_2$, you get two norms on the same vector space.
But as it isn't a finite-dimensional space, it can have non-equivalent norms - and indeed, those two norms definitely aren't equivalent, which in particular means that a sequence which has a good behaviour for the $L^2$ norm (the partial sums of the Fourier series) doesn't necessarily have a good $|.|_\infty$ behaviour. | 2013-05-26 05:48:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9236490726470947, "perplexity": 129.60101887426302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706631378/warc/CC-MAIN-20130516121711-00034-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/10804/keypoint-detection-with-color-images | # Keypoint detection with color images
Are there any known methods to use existing keypoint detectors (which generally work on grayscale images) on color images? of course, one could always add the results of the detector on all three channels to get $3N$ keypoints, but I was wondering if there exist other established methods.
Specifically, I am looking to use a FAST like detector.
Note, that I am not looking for a color based descriptor, but rather for a color based detector.
Short answer is no, there are none, or maybe none that made it until becoming a de facto standard like FAST or SIFT. This is still an open area of research by the way if you plan to work in this field.
Mostly, the issues are linked to the fragility of color representation in digital images. The common RGB color space is not really relevant (distances between colors in RGB do not match the perceptual intuition of color distance), so you probably have to go Lab or YCbCr first, then work on the luminance channel. Also, be aware of the white balance that can vary a lot between different cameras, or with a given camera but varying light conditions (sunlight, artificial light...).
Furthermore, in most cases the detection and/or description have good reasons to be color agnostic. For example, when you watch a black-and-white movie, your brain can still recognize objects although they have lost their color. | 2020-01-29 08:13:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29567185044288635, "perplexity": 734.7700979980334}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251789055.93/warc/CC-MAIN-20200129071944-20200129101944-00231.warc.gz"} |
https://plainmath.net/93132/consider-the-complex-function-f-s-1 | Consider the complex function f(s)=(1)/(l/c sqrt((s(s+r_0))) where r_0,l,c are positive real number and s is a complex variable.
Jensen Mclean 2022-09-06 Answered
Consider the complex function $f\left(s\right)=\frac{1}{\frac{l}{c}\sqrt{\left(s\left(s+{r}_{0}\right)}}$ where ${r}_{0},l,c$ are positive real number and s is a complex variable. How I can obtain the inverse Laplace transformation of this function?
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Jordan Owen
I have found the following formula about my question.
${L}^{-1}\left\{\frac{1}{\sqrt{s+a}\sqrt{s+b}}\right\}={e}^{-\frac{\left(a+b\right)t}{2}}{I}_{0}\left(\frac{a-b}{2}t\right)$
where ${I}_{0}\left(x\right)$ is modified Bessel function. | 2022-11-26 09:26:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 50, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7575082182884216, "perplexity": 1228.3649774993069}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446706285.92/warc/CC-MAIN-20221126080725-20221126110725-00673.warc.gz"} |
https://cs.stackexchange.com/questions/101685/finding-the-longest-path-in-an-undirected-node-weighted-tree | # Finding the longest path in an undirected node-weighted tree
I have a tree where each node is assigned a weight (a real number that can be positive or negative). I need an algorithm to find a simple path of maximum total weight (that is, a simple path where the sum of the weights of the nodes in the path is maximum). There's no restriction on what node the path starts or ends.
I have a possible algorithm, but I am not sure it works and I am looking for a proof. Here it is:
1)Select an arbitrary node u and run DFS(u) to find the maximum weight simple path that starts at u. Let (u, v) be this path.
2)Run DFS(v) to find the maximum weight simple path that starts at v. Let this path be (v, z).
Then (v, z) is a simple path of maximum weight. This algorithm is linear in the size of the graph. Can anyone tell me if it works, and if so, give a proof?
Note: The Longest Path Problem is NP-Hard for a general graph with cycles. However, I only consider trees here.
• Trees don't have cycles, a path from a node to another node in a tree is unique as long as you are not allowing some redundant moves - visit a node and come back. Dec 17, 2018 at 18:30
Here is a counterexample. Let graph $$G$$ have five weighted nodes, $$A\mapsto 1$$, $$B\mapsto -1$$, $$C\mapsto 0$$, $$D\mapsto 1$$, $$E\mapsto 1$$. There are four edges, $$AB$$, $$BC$$, $$CD$$ and $$CE$$.
1. Select node $$B$$ and run DFS($$A$$). We may get the maximum weight simple path $$B, A$$, whose weight is 0.
2. Run DFS($$A$$), we may find the maximum weight simple path that starts at $$A$$, which is $$A, B, C, D$$, whose weight is 1. That path is returned.
However, the simple path with the maximum weight is actually $$D,C,E$$ whose weight is $$2$$. | 2022-05-25 20:37:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9070634245872498, "perplexity": 155.2863875258012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00389.warc.gz"} |
https://gamedev.stackexchange.com/questions/27843/physics-motor-on-servers-how-a-billiard-server-works/27845 | # Physics motor on servers, How a billiard server works?
i'm trying to figure how a phisics game like billiard works in server side . I'm asking this cause i know how to achieve this in a client , but if i use a phisics motor i think it could be very heavy for a server when i want to have many people in the server.
When i see this type of games, it use to be really fast , thinking that server is controlling the movements of course . Cause any other way maybe from client you could have been doing cheats.
Then my questions are:
The server sends in this case, all ball movements? Are the server trusting in clients? In this case : How to be sure a phisics motor doesn't have little error. (float decimals differences)
And globaly : How this type of server works?
• I don't have a full answer, but I'll say that billiards is a really good game for physics online. That's because only one player at a time is active, and you don't need to worry about multiple players interacting at the same time with physics objects. Also, billiards has a relatively limited number of objects in motion at once, meaning you can probably perform some very efficient calculations for ball collisions that can keep your server CPU load low. – Tim Holt Apr 23 '12 at 2:39
As with most things in game development, there are a number of ways to tackle this issue. The most common way I have seen is to have the client-side physics be completely "dumb."
The simulation on the client side is merely reacting to data delivered by the server and inputs from the client side are sent to the server, interpreted and distributed to all connected clients as data. You can easily do very basic cheat detection during this "interpretation" step by ensuring that the move the client is attempting to make fits within the rules of your game. For instance, in the context of your billiards game, if the server currently has ball1's position as (x:26, y:50) in the game world and suddenly fields a request to move ball1 from (x:35, y:50), you know something strange is going on. Now "something strange" could be either network latency, dropped packets or player tampering and it's up to you to decide how to determine the cause and handle it.
This article by Gaffer on Games (really great source), can explain in much more detail than I ever could, but these are the essential steps:
• Player presses a key or invokes an action.
• Action is sent to the server.
• Server interprets the action for the player.
• Server checks if the action is valid.
• If valid, server applies action to all relevant game objects.
• Server sends updated data on all affected objects to all connected players.
With this implementation, the drawback is that because real game data is only flowing one way (server to player), and network transmission is never perfect, the client has to do some guesstimating. This is called client-side prediction and is covered in the section of the same title in the article linked above.
Even though you're not working with the UnrealEngine, their Networking Overview article does an excellent job of explaining how their server-client system works and how they tackled certain challenges that exist in managing multiple objects and states over network.
Hope this helps! | 2020-08-14 03:29:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2390289604663849, "perplexity": 1086.1204972254843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739134.49/warc/CC-MAIN-20200814011517-20200814041517-00406.warc.gz"} |
https://www.physicsforums.com/threads/finding-the-limit-and-a-differential-equation.164714/ | # Finding the limit and a differential equation
1. Apr 8, 2007
### cokezero
i cant seem to figure this out...
if the differential equation dy/dx= y-2y^2 has a solution curve y=f(x) contianing point (0, 0.25) , then the limit as x approaches infinity of f(x) is
a)no limit
b. 0
c. 0.25
d. 0.5
e. 2
i usually just separate the variables and find f(x) then take the limit, but i cant seem to find f(x) b/c it would require the integral of 1/(y-2y^2)
2. Apr 8, 2007
### Data
so, integrate $\frac{1}{y-2y^2}$! Partial fractions will do it.
3. Apr 8, 2007
### cokezero
yeah i know...
i get the equation
y= 1/(e^-x + 2) +C
without the c value it is 1/2 for the limit
but with the c value which is -1/12 i get a limit of 5/12 which is not an answer choice...
so the question becomes, does the limit depend on the c value or not?
4. Apr 8, 2007
### Data
That's not the answer I get for y(x). Try checking your work again. If you still can't figure it out, post what you've done and I'll try to tell you what's wrong! | 2017-01-19 19:26:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9006102085113525, "perplexity": 1958.6002282485165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280730.27/warc/CC-MAIN-20170116095120-00378-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://ibn.idsi.md/en/vizualizare_articol/75020/dublincore | About physical transformers to create toxicity biosensors
Articolul precedent Articolul urmator 500 0 SM ISO690:2012ORLOV, Anatolii; MAKSIMCHUK, N.; SPIVAK, Victor; MESHANINOV, S.. About physical transformers to create toxicity biosensors. In: Health Technology Management3rd International Conference. Editia 3, R, 6-7 octombrie 2016, Chișinău. Chișinău, Republica Moldova: Technical University of Moldova, 2016, p. 80. EXPORT metadate: Google Scholar Crossref CERIF DataCiteDublin Core
Health Technology Management
Editia 3, R, 2016
Conferința "Health Technology Management"
3, Chișinău, Moldova, 6-7 octombrie 2016
About physical transformers to create toxicity biosensors
Pag. 80-80
Orlov Anatolii1, Maksimchuk N.1, Spivak Victor1, Meshaninov S.2 1 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute",2 Dneprodzerzhinsk State Technical University Disponibil în IBN: 2 aprilie 2019
Rezumat
There are proposed to solve the task of improving noise immunity biosensor systems and simultaneous determination of several toxic substances proposed to solve through the use of two or more individuals converters (transducers), which is measured simultaneously change multiple parameters of information caused by various physical and chemical effects. Obtained, that To create the biosensor system that can measure the toxicity immediately after several informative parameters offered to pick up such biological signal converters that were manufactured to be modern microelectronic technology, and therefore:− would be able to miniaturization;have high sensitivity, reliability, stability, reproducibility of measurement; used a small number of samples (max 0,1 ml); characterized by low cost; could be part of an integrated circuit, which also includes analog-to-digital converter and a microprocessor for measurement and calculation of the analytical signal analysis results. It shows that the ceria is not a classical semiconductor material for biosensors. In the study of ten commercially available biosensors toxicity from enzymes and cells of vertebrates none of the individual sensor does not react to distilled water and only one sensor gave a response to the heavy water [1]. No one reacted sensor not more than six chemicals in a given range response. In addition, none of the presented sensor is not identified with the desired nicotine sensitivity. However, a combination of three selected sensors (bioluminescent, fluorescent and impedansomesurement (conductometric) gave the best results. In [2] describes attempts to solve the problem of simultaneous determination of several toxic substances by creating a multisensor system based on different enzymes, selective to certain substances. Famous cases of different enzymes. However, the creation of multisensors is a very difficult task, since all enzymes are used to operate simultaneously on the same conditions. In addition, there is the problem of stability of each individual enzyme. To create the biosensor system that can measure the toxicity immediately after several informative parameters offered to pick up such biological signal converters that were manufactured to be modern microelectronic technology, and therefore:− would be able to miniaturization; - Have high sensitivity, reliability, stability, reproducibility of measurement; - Used a small number of samples (max 0,1 ml); - Characterized by low cost; - could be part of an integrated circuit, which also includes analog-to-digital converter and a microprocessor for measurement and calculation of the analytical signal analysis results. Experience with ceria showed that while its effect on the bacteria Escherichia coli strain TG1, the intensity of bioluminescence is correlated with the enzymatic activity of these bacteria, ceria is not toxic in relation to the test cultures even at maximum concentration in the aqueous suspension (20,000 mg / l).
### Dublin Core Export
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<dc:creator>Orlov, A.T.</dc:creator>
<dc:creator>Maksimchuk, N.V.</dc:creator>
<dc:creator>Spivak, V.M.</dc:creator>
<dc:creator>Meşaninov, S.K.</dc:creator>
<dc:date>2016</dc:date>
<dc:description xml:lang='en'><p>There are proposed to solve the task of improving noise immunity biosensor systems and simultaneous determination of several toxic substances proposed to solve through the use of two or more individuals converters (transducers), which is measured simultaneously change multiple parameters of information caused by various physical and chemical effects. Obtained, that To create the biosensor system that can measure the toxicity immediately after several informative parameters offered to pick up such biological signal converters that were manufactured to be modern microelectronic technology, and therefore:− would be able to miniaturization;have high sensitivity, reliability, stability, reproducibility of measurement; used a small number of samples (max 0,1 ml); characterized by low cost; could be part of an integrated circuit, which also includes analog-to-digital converter and a microprocessor for measurement and calculation of the analytical signal analysis results. It shows that the ceria is not a classical semiconductor material for biosensors. In the study of ten commercially available biosensors toxicity from enzymes and cells of vertebrates none of the individual sensor does not react to distilled water and only one sensor gave a response to the heavy water [1]. No one reacted sensor not more than six chemicals in a given range response. In addition, none of the presented sensor is not identified with the desired nicotine sensitivity. However, a combination of three selected sensors (bioluminescent, fluorescent and impedansomesurement (conductometric) gave the best results. In [2] describes attempts to solve the problem of simultaneous determination of several toxic substances by creating a multisensor system based on different enzymes, selective to certain substances. Famous cases of different enzymes. However, the creation of multisensors is a very difficult task, since all enzymes are used to operate simultaneously on the same conditions. In addition, there is the problem of stability of each individual enzyme. To create the biosensor system that can measure the toxicity immediately after several informative parameters offered to pick up such biological signal converters that were manufactured to be modern microelectronic technology, and therefore:− would be able to miniaturization; - Have high sensitivity, reliability, stability, reproducibility of measurement; - Used a small number of samples (max 0,1 ml); - Characterized by low cost; - could be part of an integrated circuit, which also includes analog-to-digital converter and a microprocessor for measurement and calculation of the analytical signal analysis results. Experience with ceria showed that while its effect on the bacteria Escherichia coli strain TG1, the intensity of bioluminescence is correlated with the enzymatic activity of these bacteria, ceria is not toxic in relation to the test cultures even at maximum concentration in the aqueous suspension (20,000 mg / l).</p></dc:description>
<dc:source>Health Technology Management (Editia 3, R) 80-80</dc:source>
<dc:title><p>About physical transformers to create toxicity biosensors</p></dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
</oai_dc:dc> | 2023-03-24 11:06:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3822190761566162, "perplexity": 4136.682458505617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945279.63/warc/CC-MAIN-20230324082226-20230324112226-00746.warc.gz"} |
http://www.acmerblog.com/hdu-2488-tornado-3952.html | 2014
01-26
Professor Jonathan is a well-known Canadian physicist and meteorologist. People who know him well call him “Wind Chaser”. It is not only because of his outstanding tornado research which is the most influential in the academic community, but also because of his courageous act in collecting real data of tornados. Actually he has been leading his team chasing tornado by cars equipped with advanced instruments hundreds of times.
In summer, tornado often occurs in the place where Professor Jonathan lives. After several years of research, Wind Chaser found many formation rules and moving patterns of tornados. In the satellite image, a tornado is a circle with radius of several meters to several kilometers. And its center moves between two locations in a straight line, back and forth at a fixed speed. After observing a tornado’s movement, Wind Chaser will pick a highway, which is also a straight line, and chase the tornado along the highway at the maximum speed of his car.
The smallest distance between the Wind Chaser and the center of the tornado during the whole wind chasing process, is called “observation distance”. Observation distance is critical for the research activity. If it is too short, Wind Chaser may get killed; and if it is too far, Wind Chaser can’t observe the tornado well. After many times of risk on lives and upset miss, Wind Chaser turns to you, one of his most brilliant students, for help. The only thing he wants to know is the forthcoming wind chasing will be dangerous, successful or just a miss.
Input contains multiple test cases. Each test case consists of three lines which are in the following format.
xw1 yw1 xw2 yw2 vw
xt1 yt1 xt2 yt2 vt
dl du
In the first line, (xw1, yw1) means the start position of Wind Chaser; (xw2, yw2) is another position in the highway which Wind Chaser will definitely pass through; and vw is the speed of the car. Wind chaser will drive to the end of the world along that infinite long highway.
In the second line, (xt1, yt1) is the start position of tornado; (xt2, yt2) is the turn-around position and vt is the tornado’s speed. In other words, the tornado’s center moves back and forth between (xt1, yt1) and (xt2, yt2) at speed vt .
The third line shows that if the observation distance is smaller than dl , it will be very dangerous; and if the observation distance is larger than du, it will be a miss; otherwise it will lead to a perfect observation.
All numbers in the input are floating numbers.
-2000000000 <= xw1, yw1, xw2, yw2, xt1, yt1, xt2, yt2 <= 2000000000
1 <= vw, vt <= 20000
0 <= dl, du <= 2000000
Note:
1. It’s guaranteed that the observation distance won’t be very close to dl or du during the whole wind chasing process. There will be at least 10-5 of difference.
2. Wind Chaser and the tornado start to move at the same time from their start position.
Input contains multiple test cases. Each test case consists of three lines which are in the following format.
xw1 yw1 xw2 yw2 vw
xt1 yt1 xt2 yt2 vt
dl du
In the first line, (xw1, yw1) means the start position of Wind Chaser; (xw2, yw2) is another position in the highway which Wind Chaser will definitely pass through; and vw is the speed of the car. Wind chaser will drive to the end of the world along that infinite long highway.
In the second line, (xt1, yt1) is the start position of tornado; (xt2, yt2) is the turn-around position and vt is the tornado’s speed. In other words, the tornado’s center moves back and forth between (xt1, yt1) and (xt2, yt2) at speed vt .
The third line shows that if the observation distance is smaller than dl , it will be very dangerous; and if the observation distance is larger than du, it will be a miss; otherwise it will lead to a perfect observation.
All numbers in the input are floating numbers.
-2000000000 <= xw1, yw1, xw2, yw2, xt1, yt1, xt2, yt2 <= 2000000000
1 <= vw, vt <= 20000
0 <= dl, du <= 2000000
Note:
1. It’s guaranteed that the observation distance won’t be very close to dl or du during the whole wind chasing process. There will be at least 10-5 of difference.
2. Wind Chaser and the tornado start to move at the same time from their start position.
0 0 1 0 2
10 -5 12 7 4
1.3 2.7
0 0 1 0 2
10 -5 12 7 1
0.3 0.4
Dangerous
Perfect
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
int path[88][88], vis[88][88], p, q, cnt;
bool flag;
int dx[8] = {-1, 1, -2, 2, -2, 2, -1, 1};
int dy[8] = {-2, -2, -1, -1, 1, 1, 2, 2};
bool judge(int x, int y)
{
if(x >= 1 && x <= p && y >= 1 && y <= q && !vis[x][y] && !flag)
return true;
return false;
}
void DFS(int r, int c, int step)
{
path[step][0] = r;
path[step][1] = c;
if(step == p * q)
{
flag = true;
return ;
}
for(int i = 0; i < 8; i++)
{
int nx = r + dx[i];
int ny = c + dy[i];
if(judge(nx,ny))
{
vis[nx][ny] = 1;
DFS(nx,ny,step+1);
vis[nx][ny] = 0;
}
}
}
int main()
{
int i, j, n, cas = 0;
scanf("%d",&n);
while(n--)
{
flag = 0;
scanf("%d%d",&p,&q);
memset(vis,0,sizeof(vis));
vis[1][1] = 1;
DFS(1,1,1);
printf("Scenario #%d:\n",++cas);
if(flag)
{
for(i = 1; i <= p * q; i++)
printf("%c%d",path[i][1] - 1 + 'A',path[i][0]);
}
else
printf("impossible");
printf("\n");
if(n != 0)
printf("\n");
}
return 0;
}
1. #include <cstdio>
int main() {
int n, u, d;
while(scanf("%d%d%d",&n,&u,&d)==3 && n>0) {
if(n<=u) { puts("1"); continue; }
n-=u; u-=d; n+=u-1; n/=u;
n<<=1, ++n;
printf("%dn",n);
}
return 0;
}
2. 这道题这里的解法最坏情况似乎应该是指数的。回溯的时候
O(n) = O(n-1) + O(n-2) + ….
O(n-1) = O(n-2) + O(n-3)+ …
O(n) – O(n-1) = O(n-1)
O(n) = 2O(n-1)
3. Excellent Web-site! I required to ask if I might webpages and use a component of the net web website and use a number of factors for just about any faculty process. Please notify me through email regardless of whether that would be excellent. Many thanks | 2017-02-27 11:44:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3839327394962311, "perplexity": 4090.139834913457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501172783.57/warc/CC-MAIN-20170219104612-00193-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://docs.mosek.com/8.1/capi/intro_info.html | # 1 Introduction¶
The MOSEK Optimization Suite 8.1.0.33 is a powerful software package capable of solving large-scale optimization problems of the following kind:
• linear,
• conic quadratic (also known as second-order cone),
• semidefinite,
• and general convex.
Integer constrained variables are supported for all problem classes except for semidefinite and general convex problems. In order to obtain an overview of features in the MOSEK Optimization Suite consult the product introduction guide.
The most widespread class of optimization problems is linear optimization problems, where all relations are linear. The tremendous success of both applications and theory of linear optimization can be ascribed to the following factors:
• The required data are simple, i.e. just matrices and vectors.
• Convexity is guaranteed since the problem is convex by construction.
• Linear functions are trivially differentiable.
• There exist very efficient algorithms and software for solving linear problems.
• Duality properties for linear optimization are nice and simple.
Even if the linear optimization model is only an approximation to the true problem at hand, the advantages of linear optimization may outweigh the disadvantages. In some cases, however, the problem formulation is inherently nonlinear and a linear approximation is either intractable or inadequate. Conic optimization has proved to be a very expressive and powerful way to introduce nonlinearities, while preserving all the nice properties of linear optimization listed above.
The fundamental expression in linear optimization is a linear expression of the form
$A x - b \in \K$
where $$\K = \{y: y \geq 0\}$$, i.e.,
$\begin{split}\begin{array}{l} A x - b =y,\\ y \in \K. \end{array}\end{split}$
In conic optimization a wider class of convex sets $$\K$$ is allowed, for example in 3 dimensions $$\K$$ may correspond to an ice cream cone. The conic optimizer in MOSEK supports three structurally different types of cones $$\K$$, which allows a surprisingly large number of nonlinear relations to be modelled (as described in the MOSEK modeling cookbook), while preserving the nice algorithmic and theoretical properties of linear optimization.
## 1.1 Why the Optimizer API for C?¶
The Optimizer API for C provides low-level access to all functionalities of MOSEK from any C compatible language. It consists of a single header file and a set of library files which an application must link against when building. This interface has the smallest possible overhead, however other interfaces might be considered more convenient to use for the project at hand.
• Linear Optimization (LO)
• Conic Quadratic (Second-Order Cone) Optimization (CQO, SOCO)
• Semidefinite Optimization (SDO)
• General and Separable Convex Optimization (SCO)
as well as additional interfaces for:
• problem analysis,
• sensitivity analysis,
• infeasibility analysis,
• BLAS/LAPACK linear algebra routines. | 2017-11-22 02:01:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5388702750205994, "perplexity": 1173.9370581332998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806447.28/warc/CC-MAIN-20171122012409-20171122032409-00763.warc.gz"} |