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http://jdh.hamkins.org/tag/resurrection-axiom/ | # Giorgio Audrito, PhD 2016, University of Torino
Dr. Giorgio Audrito has successfully defended his dissertation, “Generic large cardinals and absoluteness,” at the University of Torino under the supervision of Matteo Viale.
The dissertation Examing Board consisted of myself (serving as Presidente), Alessandro Andretta and Sean Cox. The defense took place March 2, 2016.
The dissertation was impressive, introducing (in joint work with Matteo Viale) the iterated resurrection axioms $\text{RA}_\alpha(\Gamma)$ for a forcing class $\Gamma$, which extend the idea of the resurrection axioms from my work with Thomas Johnstone, The resurrection axioms and uplifting cardinals, by making successive extensions of the same type, forming the resurrection game, and insisting that that the resurrection player have a winning strategy with game value $\alpha$. A similar iterative game idea underlies the $(\alpha)$-uplifting cardinals, from which the consistency of the iterated resurrection axioms can be proved. A final chapter of the dissertation (joint with Silvia Steila), develops the notion of $C$-systems of filters, generalizing the more familiar concepts of extenders and towers.
# Boldface resurrection and the strongly uplifting cardinals, the superstrongly unfoldable cardinals and the almost-hugely unfoldable cardinals, BEST 2014
I will speak at the BEST conference, which is held as a symposium in the much larger 95th Annual Meeting of the American Association for the Advancement of Science, at the University of California at Riverside, June 18-20, 2014.
This talk will be for specialists in the BEST symposium.
Abstract. I shall introduce several new large cardinal concepts, namely, the strongly uplifting cardinals, the superstrongly unfoldable cardinals and the almost-hugely unfoldable cardinals, and prove their tight connection with one another — actually, they are equivalent! — as well as their equiconsistency with several natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing. This is joint work with Thomas A. Johnstone.
I am also scheduled to give a plenary General Pubic Lecture, entitled Higher infinity and the foundations of mathematics, as a part of the larger AAAS program, to which the general public is invited.
# Strongly uplifting cardinals and the boldface resurrection axioms
• J. D. Hamkins and T. Johnstone, “Strongly uplifting cardinals and the boldface resurrection axioms.” (under review, http://arxiv.org/abs/1403.2788)
@ARTICLE{HamkinsJohnstone:StronglyUpliftingCardinalsAndBoldfaceResurrection,
author = {Joel David Hamkins and Thomas Johnstone},
title = {Strongly uplifting cardinals and the boldface resurrection axioms},
journal = {},
year = {},
volume = {},
number = {},
pages = {},
month = {},
note = {under review, http://arxiv.org/abs/1403.2788},
eprint = {1403.2788},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/strongly-uplifting-cardinals-and-boldface-resurrection},
abstract = {},
keywords = {},
source = {},
}
Abstract. We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
The strongly uplifting cardinals, which we introduce in this article, are a boldface analogue of the uplifting cardinals introduced in our previous paper, Resurrection axioms and uplifting cardinals, and are equivalently characterized as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals. In consistency strength, these new large cardinals lie strictly above the weakly compact, totally indescribable and strongly unfoldable cardinals and strictly below the subtle cardinals, which in turn are weaker in consistency than the existence of $0^\sharp$. The robust diversity of equivalent characterizations of this new large cardinal concept enables constructions and techniques from much larger large cardinal contexts, such as Laver functions and forcing iterations with applications to forcing axioms. Using such methods, we prove that the existence of a strongly uplifting cardinal (or equivalently, a superstrongly unfoldable or almost hugely unfoldable cardinal) is equiconsistent over ZFC with natural instances of the boldface resurrection axioms, including the boldface resurrection axiom for proper forcing, for semi-proper forcing, for c.c.c. forcing and others. Thus, whereas in our prior article we proved that the existence of a mere uplifting cardinal is equiconsistent with natural instances of the (lightface) resurrection axioms, here we adapt both of these notions to the boldface context.
Definitions.
• An inaccessible cardinal $\kappa$ is strongly uplifting if for every ordinal $\theta$ it is strongly $\theta$-uplifting, which is to say that for every $A\subset V_\kappa$ there is an inaccessible cardinal $\gamma\geq\theta$ and a set $A^*\subset V_\gamma$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle V_\gamma,{\in},A^*\rangle$ is a proper elementary extension.
• A cardinal $\kappa$ is superstrongly unfoldable, if for every ordinal $\theta$ it is superstrongly $\theta$-unfoldable, which is to say that for each $A\in H_{\kappa^+}$ there is a $\kappa$-model $M$ with $A\in M$ and a transitive set $N$ with an elementary embedding $j:M\to N$ with critical point $\kappa$ and $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subset N$.
• A cardinal $\kappa$ is almost-hugely unfoldable, if for every ordinal $\theta$ it is almost-hugely $\theta$-unfoldable, which is to say that for each $A\in H_{\kappa^+}$ there is a $\kappa$-model $M$ with $A\in M$ and a transitive set $N$ with an elementary embedding $j:M\to N$ with critical point $\kappa$ and $j(\kappa)\geq\theta$ and $N^{<j(\kappa)}\subset N$.
Remarkably, these different-seeming large cardinal concepts turn out to be exactly equivalent to one another. A cardinal $\kappa$ is strongly uplifting if and only if it is superstrongly unfoldable, if and only if it is almost hugely unfoldable. Furthermore, we prove that the existence of such a cardinal is equiconsistent with several natural instances of the boldface resurrection axiom.
Theorem. The following theories are equiconsistent over ZFC.
• There is a strongly uplifting cardinal.
• There is a superstrongly unfoldable cardinal.
• There is an almost hugely unfoldable cardinal.
• The boldface resurrection axiom for all forcing.
• The boldface resurrection axiom for proper forcing.
• The boldface resurrection axiom for semi-proper forcing.
• The boldface resurrection axiom for c.c.c. forcing.
• The weak boldface resurrection axiom for countably-closed forcing, axiom-A forcing, proper forcing and semi-proper forcing, plus $\neg\text{CH}$.
# Resurrection axioms and uplifting cardinals
• J. D. Hamkins and T. Johnstone, “Resurrection axioms and uplifting cardinals,” Archive for Mathematical Logic, vol. 53, iss. 3-4, p. p.~463–485, 2014.
@ARTICLE{HamkinsJohnstone2014:ResurrectionAxiomsAndUpliftingCardinals,
AUTHOR = "Joel David Hamkins and Thomas Johnstone",
TITLE = "Resurrection axioms and uplifting cardinals",
JOURNAL = "Archive for Mathematical Logic",
publisher= {Springer Berlin Heidelberg},
YEAR = "2014",
volume = "53",
number = "3-4",
pages = "p.~463--485",
month = "",
note = "",
url = "http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals",
eprint = "1307.3602",
archivePrefix = {arXiv},
primaryClass = {math.LO},
doi= "10.1007/s00153-014-0374-y",
issn= {0933-5846},
abstract = "",
keywords = "",
source = "",
file = F }
Abstract. We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of uplifting cardinal.
Many classical forcing axioms can be viewed, at least informally, as the claim that the universe is existentially closed in its forcing extensions, for the axioms generally assert that certain kinds of filters, which could exist in a forcing extension $V[G]$, exist already in $V$. In several instances this informal perspective is realized more formally: Martin’s axiom is equivalent to the assertion that $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions of the universe, meaning that $H_{\frak{c}}\prec_{\Sigma_1}V[G]$ for all such extensions; the bounded proper forcing axiom is equivalent to the assertion that $H_{\omega_2}$ is existentially closed in all proper forcing extensions, or $H_{\omega_2}\prec_{\Sigma_1}V[G]$; and there are other similar instances.
In model theory, a submodel $M\subset N$ is existentially closed in $N$ if existential assertions true in $N$ about parameters in $M$ are true already in $M$, that is, if $M$ is a $\Sigma_1$-elementary substructure of $N$, which we write as $M\prec_{\Sigma_1} N$. Furthermore, in a general model-theoretic setting, existential closure is tightly connected with resurrection, the theme of this article.
Elementary Fact. If $\mathcal{M}$ is a submodel of $\mathcal{N}$, then the following are equivalent.
1. The model $\mathcal{M}$ is existentially closed in $\mathcal{N}$.
2. $\mathcal{M}\subset \mathcal{N}$ has resurrection. That is, there is a further extension $\mathcal{M}\subset\mathcal{N}\subset\mathcal{M}^+$ for which $\mathcal{M}\prec\mathcal{M}^+$.
We call this resurrection because although certain truths in $\mathcal{M}$ may no longer hold in the extension $\mathcal{N}$, these truths are nevertheless revived in light of $\mathcal{M}\prec\mathcal{M}^+$ in the further extension to $\mathcal{M}^+$.
In the context of forcing axioms, we are more interested in the case of forcing extensions than in the kind of arbitrary extension $\mathcal{M}^+$ arising in the fact, and in this context the equivalence of (1) and (2) breaks own, although the converse implication $(2)\to(1)$ always holds, and every instance of resurrection implies the corresponding instance of existential closure. This key observation leads us to the main unifying theme of this article, the idea that
resurrection may allow us to formulate more robust forcing axioms
than existential closure or than combinatorial assertions about filters and dense sets. We therefore introduce in this paper a spectrum of new forcing axioms utilizing the resurrection concept.
Main Definition. Let $\Gamma$ be a fixed definable class of forcing notions.
1. The resurrection axiom $\text{RA}(\Gamma)$ is the assertion that for every forcing notion $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, with $\vdash_{\mathbb{Q}}\mathbb{R}\in\Gamma$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.
2. The weak resurrection axiom $\text{wRA}(\Gamma)$ is the assertion that for every $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.
The main result is to prove that various formulations of the resurrection axioms are equiconsistent with the existence of an uplifting cardinal, where an inaccessible cardinal $\kappa$ is uplifting, if there are arbitrarily large inaccessible cardinals $\gamma$ for which $H_\kappa\prec H_\gamma$. This is a rather weak large cardinal notion, having consistency strength strictly less than the existence of a Mahlo cardinal, which is traditionally considered to be very low in the large cardinal hierarchy. One highlight of the article is our development of “the world’s smallest Laver function,” the Laver function concept for uplifting cardinals, and we perform an analogue of the Laver preparation in order to achieve the resurrection axiom for c.c.c. forcing.
Main Theorem. The following theories are equiconsistent over ZFC:
1. There is an uplifting cardinal.
2. $\text{RA}(\text{all})$.
3. $\text{RA}(\text{ccc})$.
4. $\text{RA}(\text{semiproper})+\neg\text{CH}$.
5. $\text{RA}(\text{proper})+\neg\text{CH}$.
6. For some countable ordinal $\alpha$, the axiom $\text{RA}(\alpha\text{-proper})+\neg\text{CH}$.
7. $\text{RA}(\text{axiom-A})+\neg\text{CH}$.
8. $\text{wRA}(\text{semiproper})+\neg\text{CH}$.
9. $\text{wRA}(\text{proper})+\neg\text{CH}$.
10. For some countable ordinal $\alpha$, the axiom $\text{wRA}(\alpha\text{-proper})+\neg\text{CH}$.
11. $\text{wRA}(\text{axiom-A})+\neg\text{CH}$.
12. $\text{wRA}(\text{countably closed})+\neg\text{CH}$.
The proof outline proceeds in two directions: on the one hand, the resurrection axioms generally imply that the continuum $\frak{c}$ is uplifting in $L$; and conversely, given any uplifting cardinal $\kappa$, we may perform a suitable lottery iteration of $\Gamma$ forcing to obtain the resurrection axiom for $\Gamma$ in a forcing extension with $\kappa=\frak{c}$.
In a follow-up article, currently nearing completion, we treat the boldface resurrection axioms, which allow a predicate $A\subset\frak{c}$ and ask for extensions of the form $\langle H_{\frak{c}},{\in},A\rangle\prec\langle H_{\frak{c}}^{V[g\ast h]},{\in},A^\ast\rangle$, for some $A^\ast\subset\frak{c}^{V[g\ast h]}$ in the extension. In that article, we prove the equiconsistency of various formulations of boldface resurrection with the existence of a strongly uplifting cardinal, which we prove is the same as a superstrongly unfoldable cardinal. | 2017-07-29 11:36:12 | {"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": 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": 0.9504752159118652, "perplexity": 977.6218987245963}, "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-2017-30/segments/1500549427766.28/warc/CC-MAIN-20170729112938-20170729132938-00093.warc.gz"} |
https://www.physicsforums.com/threads/integrating-gaussians-with-complex-arguments.952436/ | # Integrating Gaussians with complex arguments
• A
The integral I'm looking at is of the form
$$\int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right)$$
Where $K \in \mathbb{R}$ and $J \in \mathbb{C}$
The book I am following (Kardar's Statistical Physics of Fields, Chapter 3 Problem 1) asserts that by completing the square this becomes $Z \exp\left( \frac{- |J|^2}{2K} \right)$ where $Z = \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 \right)$. I cant seem to reproduce this, and I think the trouble I'm running into arises from $|z|^2$ not being a square, but rather it involves conjugation as well. Therefore, I get the following
$$-\frac{1}{2}K|z|^2 + \bar{J}z = -\frac{1}{2}K\left( z\bar{z} -2 \frac{\bar{J}}{K}z \right)= -\frac{1}{2}K\left( z\bar{z} -2 \frac{\bar{J}}{K}z - 2 \frac{J}{K}\bar{z} +2 \frac{J}{K}\bar{z} + 4\frac{ |J|^2}{K^2} - 4 \frac{ |J|^2}{K^2} \right) =$$
$$-\frac{1}{2}K\left( z - 2 \frac{J}{K} \right) \left( \bar{z} -2 \frac{\bar{J}}{K} \right) -J\bar{z} +2 \frac{ |J|^2}{K}$$
Which means that I'm getting that
$$\int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right) = \left[ \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K \left| z-2 \frac{J}{K} \right|^2 - J\bar{z} \right) \right] \exp\left( 2 \frac{ |J|^2}{K} \right)$$
Which doesnt at all seem like
$$\left[ \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 \right) \right] \exp\left( \frac{- |J|^2}{2K} \right)$$
mathman
I haven't worked through all the details, but it looks like there was a shift in ##z##, i.e. ##z'=z-\frac{2J}{K^2}##.
I haven't worked through all the details, but it looks like there was a shift in ##z##, i.e. ##z'=z-\frac{2J}{K^2}##.
I dont think shifting $z$ by anything can help. Suppose you sent $z \mapsto z+a$ for any $a$ then I would get the following
$$-\frac{1}{2}K |z|^2+\bar{J}z \mapsto -\frac{1}{2}K |z+a|^2+\bar{J}(z + a) =$$
$$-\frac{1}{2}K \left( z\bar{z} + a\bar{z} + z\bar{a} +a\bar{a} -\frac{2\bar{J}}{K} z - \frac{2\bar{J}}{K}a \right) =$$
$$-\frac{1}{2}K \left( z\bar{z} + a\bar{z} + z \left( \bar{a} - \frac{2\bar{J}}{K} \right) +a\bar{a} - \frac{2\bar{J}}{K}a \right)$$
If I want this to look like $-\frac{1}{2}K|z+b|^2 + c$, then I need to add and subtract terms with $\bar{z}$ which means that I cant pull $e^c$ out of the integral.
mathman | 2022-05-28 14:09:19 | {"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": 0.9086538553237915, "perplexity": 298.56948384456445}, "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/1652663016853.88/warc/CC-MAIN-20220528123744-20220528153744-00438.warc.gz"} |
http://www.gradesaver.com/textbooks/science/physics/physics-principles-with-applications-7th-edition/chapter-6-work-and-energy-questions-page-161/5 | ## Physics: Principles with Applications (7th Edition)
a. Here, the force is the same. $F = k_{1}x_{1} = k_{2}x_{2}$. The work done on spring 1 is $$W_{1} = \frac{1}{2} k_{1}x_{1}^{2}$$ The work on spring 2 is $$W_{2} = \frac{1}{2} k_{2}x_{2}^{2} = \frac{1}{2} k_{2}\frac{k_{1}^{2} x_{1}^{2}}{ k_{2}^{2}}$$ $$W_{2} = W_{1} \frac{k_{1}}{k_{2}}$$ Because the first spring is stiffer, the work done on spring 2 is greater. b. Now, the distance stretched is the same. As before, the work done on spring 1 is $$W_{1} = \frac{1}{2} k_{1}x^{2}$$ The work on spring 2 is $$W_{2} = \frac{1}{2} k_{2}x^{2}$$ Because the first spring is stiffer, the work done on spring 1 is greater. | 2017-06-26 14:06:58 | {"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": 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": 0.748508632183075, "perplexity": 550.4066621652296}, "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-2017-26/segments/1498128320763.95/warc/CC-MAIN-20170626133830-20170626153830-00409.warc.gz"} |
https://biogeme.epfl.ch/sphinx/biogeme.html?highlight=split | Package description¶
Biogeme is a open source Python package designed for the maximum likelihood estimation of parametric models in general, with a special emphasis on discrete choice models.
biogeme.biogeme module¶
Implementation of the main Biogeme class that combines the database
and the model specification.
author
Michel Bierlaire
date
Tue Mar 26 16:45:15 2019
class biogeme.biogeme.BIOGEME(database, formulas, userNotes=None, numberOfThreads=None, numberOfDraws=1000, seed=None, skipAudit=False, removeUnusedVariables=True, suggestScales=True, missingData=99999)
Bases: object
Main class that combines the database and the model specification.
It works in two modes: estimation and simulation.
__init__(database, formulas, userNotes=None, numberOfThreads=None, numberOfDraws=1000, seed=None, skipAudit=False, removeUnusedVariables=True, suggestScales=True, missingData=99999)
Constructor
Parameters
• database (biogeme.database) – choice data.
• formulas (biogeme.expressions.Expression, or dict(biogeme.expressions.Expression)) – expression or dictionary of expressions that define the model specification. The concept is that each expression is applied to each entry of the database. The keys of the dictionary allow to provide a name to each formula. In the estimation mode, two formulas are needed, with the keys ‘loglike’ and ‘weight’. If only one formula is provided, it is associated with the label ‘loglike’. If no formula is labeled ‘weight’, the weight of each piece of data is supposed to be 1.0. In the simulation mode, the labels of each formula are used as labels of the resulting database.
• userNotes (str) – these notes will be included in the report file.
• numberOfThreads (int) – multi-threading can be used for estimation. This parameter defines the number of threads to be used. If the parameter is set to None, the number of available threads is calculated using cpu_count(). Ignored in simulation mode. Defaults: None.
• numberOfDraws (int) – number of draws used for Monte-Carlo integration. Default: 1000.
• seed (int) – seed used for the pseudo-random number generation. It is useful only when each run should generate the exact same result. If None, a new seed is used at each run. Default: None.
• skipAudit (bool) – if True, does not check the validity of the formulas. It may save significant amount of time for large models and large data sets. Default: False.
• removeUnusedVariables (bool) – if True, all variables not used in the expression are removed from the database. Default: True.
• suggestScales (bool.) – if True, Biogeme suggests the scaling of the variables in the database. Default: True. See also biogeme.database.Database.suggestScaling()
• missingData (float) – if one variable has this value, it is assumed that a data is missing and an exception will be triggered. Default: 99999.
calculateInitLikelihood()
Calculate the value of the log likelihood function
The default values of the parameters are used.
Returns
value of the log likelihood.
Return type
float.
calculateLikelihood(x, scaled, batch=None)
Calculates the value of the log likelihood function
Parameters
• x (list(float)) – vector of values for the parameters.
• scaled (bool) – if True, the value is diviced by the number of observations used to calculate it. In this case, the values with different sample sizes are comparable. Default: True
• batch (float) – if not None, calculates the likelihood on a random sample of the data. The value of the parameter must be strictly between 0 and 1, and represents the share of the data that will be used. Default: None
Returns
the calculated value of the log likelihood
Return type
float.
Raises
ValueError – if the length of the list x is incorrect.
calculateLikelihoodAndDerivatives(x, scaled, hessian=False, bhhh=False, batch=None)
Calculate the value of the log likelihood function and its derivatives.
Parameters
• x (list(float)) – vector of values for the parameters.
• hessian (bool) – if True, the hessian is calculated. Default: False.
• bhhh (bool) – if True, the BHHH matrix is calculated. Default: False.
• batch (float) – if not None, calculates the likelihood on a random sample of the data. The value of the parameter must be strictly between 0 and 1, and represents the share of the data that will be used. Default: None
Returns
f, g, h, bh where
• f is the value of the function (float)
• g is the gradient (numpy.array)
• h is the hessian (numpy.array)
• bh is the BHHH matrix (numpy.array)
Return type
tuple float, numpy.array, numpy.array, numpy.array
Raises
ValueError – if the length of the list x is incorrect
cfsqp(betas, bounds, parameters)
Invokes the CFSQP algorithm for estimation
Parameters
• betas (list of float) – initial values of the parameters to be estimated.
• bounds (list of tuple) – lower and upper bounds on each parameter.
• parameters (dict) –
user defined parameters for CFSQP
• mode = CBA: specifies job options as described below
• A = 0: ordinary minimax problems
• A = 1: ordinary minimax problems with each individual
function replaced by its absolute value, ie, an L_infty problem
• B = 0: monotone decrease of objective function after each iteration
• B = 1: monotone decrease of objective function after at most four iterations
• C = 1: default operation.
• C = 2: requires that constraints always be evaluated
before objectives during the line search.
• iprint: print level indicator with the following options
• iprint = 0: no normal output, only error
information (this option is imposed during phase 1)
• iprint = 1: a final printout at a local solution
• iprint = 2: a brief printout at the end of each iteration
• iprint = 3: detailed infomation is printed out
at the end of each iteration (for debugging purposes)
• miter: maximum number of iterations allowed by
the user to solve the problem
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing information about the lagorithm
Return type
numpay.array, dict(str:object)
changeInitValues(betas)
Modifies the initial values of the pameters in all formula
Parameters
betas (dict(string:float)) – dictionary where the keys are the names of the parameters, and the values are the new value for the parameters.
checkDerivatives(verbose=False)
Verifies the implementation of the derivatives.
It compares the analytical version with the finite differences approximation.
Parameters
verbose (bool) – if True, the comparisons are reported. Default: False.
Return type
tuple.
Returns
f, g, h, gdiff, hdiff where
• f is the value of the function,
• g is the analytical gradient,
• h is the analytical hessian,
• gdiff is the difference between the analytical and the finite differences gradient,
• hdiff is the difference between the analytical and the finite differences hessian,
confidenceIntervals(betaValues, intervalSize=0.9)
Calculate confidence intervals on the simulated quantities
Parameters
• betaValues (list(dict(str: float))) – array of parameters values to be used in the calculations. Typically, it is a sample drawn from a distribution.
• intervalSize (float) – size of the reported confidence interval, in percentage. If it is denoted by s, the interval is calculated for the quantiles (1-s)/2 and (1+s)/2. The default (0.9) corresponds to quantiles for the confidence interval [0.05, 0.95].
Returns
two pandas data frames ‘left’ and ‘right’ with the same dimensions. Each row corresponds to a row in the database, and each column to a formula. ‘left’ contains the left value of the confidence interval, and ‘right’ the right value
Example:
# Read the estimation results from a file
results = res.bioResults(pickleFile = 'myModel.pickle')
# Retrieve the names of the betas parameters that have been estimated
betas = biogeme.freeBetaNames
# Draw 100 realization of the distribution of the estimators
b = results.getBetasForSensitivityAnalysis(betas, size = 100)
# Simulate the formulas using the nominal values
simulatedValues = biogeme.simulate(betaValues)
# Calculate the confidence intervals for each formula
left, right = biogeme.confidenceIntervals(b, 0.9)
Return type
tuple of two Pandas dataframes.
createLogFile(verbosity=3)
Creates a log file with the messages produced by Biogeme.
The name of the file is the name of the model with an extension .log
Parameters
verbosity
types of messages to be captured
• 0: no output
• 1: warnings
• 2: only general information
• 3: more verbose
• 4: debug messages
Default: 3.
estimate(bootstrap=0, algorithm=<function simpleBoundsNewtonAlgorithmForBiogeme>, algoParameters=None, cfsqp_default_bounds=1000.0, saveIterations=False, file_iterations='__savedIterations.txt')
Estimate the parameters of the model.
Parameters
• bootstrap (int) – number of bootstrap resampling used to calculate the variance-covariance matrix using bootstrapping. If the number is 0, bootstrapping is not applied. Default: 0.
• algorithm (function) – optimization algorithm to use for the maximum likelihood estimation. If None, cfsqp is . Default: Biogeme’s Newton’s algorithm with simple bounds.
• algoParameters (dict) – parameters to transfer to the optimization algorithm
• cfsqp_default_bounds (float) – if the user does not provide bounds on the parameters, CFSQP assumes that the bounds are [-cfsqp_default_bounds, cfsqp_default_bounds]
• saveIterations (bool) – if True, the values of the parameters corresponding to the largest value of the likelihood function are saved in a pickle file at each iteration of the algorithm. Default: False.
• file_iterations (str) – name of the file where to save the values of the parameters. Default: ‘__savedIterations.txt’
Returns
object containing the estimation results.
Return type
biogeme.bioResults
Example:
# Create an instance of biogeme
biogeme = bio.BIOGEME(database, logprob)
# Gives a name to the model
biogeme.modelName = 'mymodel'
# Estimate the parameters
results = biogeme.estimate()
Raises
biogemeError – if no expression has been provided for the likelihood
getBoundsOnBeta(betaName)
Returns the bounds on the parameter as defined by the user.
Parameters
betaName (string) – name of the parameter
Returns
lower bound, upper bound
Return type
tuple
Raises
likelihoodFiniteDifferenceHessian(x)
Calculate the hessian of the log likelihood function using finite differences.
May be useful when the analytical hessian has numerical issues.
Parameters
x (list(float)) – vector of values for the parameters.
Returns
finite differences approximation of the hessian.
Return type
numpy.array
Raises
ValueError – if the length of the list x is incorrect
loadSavedIteration(filename='__savedIterations.txt')
Reads the values of the parameters from a text file where each line has the form name_of_beta = value_of_beta, and use these values in all formulas.
param filename
name of the text file to read. Default: ‘__savedIterations.txt’
type filename
str.
optimize(startingValues=None)
Calls the optimization algorithm.
The function self.algorithm is called. If None, CFSQP is invoked.
Parameters
startingValues – starting point for the algorithm
Type
list(float)
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing several information about the lagorithm
Return type
numpay.array, dict(str:object)
quickEstimate(algorithm=<function simpleBoundsNewtonAlgorithmForBiogeme>, algoParameters=None)
Estimate the parameters of the model. Same as estimate, where any extra
calculation is skipped (init loglikelihood, t-statistics, etc.)
Parameters
• algorithm (function) – optimization algorithm to use for the maximum likelihood estimation. If None, cfsqp is . Default: Biogeme’s Newton’s algorithm with simple bounds.
• algoParameters (dict) – parameters to transfer to the optimization algorithm
Returns
object containing the estimation results.
Return type
biogeme.results.bioResults
Example:
# Create an instance of biogeme
biogeme = bio.BIOGEME(database, logprob)
# Gives a name to the model
biogeme.modelName = 'mymodel'
# Estimate the parameters
results = biogeme.quickEstimate()
Raises
biogemeError – if no expression has been provided for the likelihood
simulate(theBetaValues=None)
Applies the formulas to each row of the database.
Parameters
theBetaValues (dict(str, float)) – values of the parameters to be used in the calculations. If None, the default values are used. Default: None.
Returns
a pandas data frame with the simulated value. Each row corresponds to a row in the database, and each column to a formula.
Return type
Pandas data frame
Example:
# Read the estimation results from a file
results = res.bioResults(pickleFile = 'myModel.pickle')
# Simulate the formulas using the nominal values
simulatedValues = biogeme.simulate(betaValues)
Raises
biogemeError – if the number of parameters is incorrect
validate(estimationResults, slices=5)
Perform out-of-sample validation.
The function performs the following tasks:
• it shuffles the data set,
• it splits the data set into slices of (approximatively) the same size,
• each slice defines a validation set (the slice itself) and an estimation set (the rest of the data),
• the model is re-estimated on the estimation set,
• the estimated model is applied on the validation set,
• the value of the log likelihood for each observation is reported.
Parameters
• estimationResults (biogeme.results.bioResults) – results of the model estimation based on the full data.
• slices (int) – number of slices.
Returns
a list containing as many items as slices. Each item is the result of the simulation on the validation set.
Return type
list(pandas.DataFrame)
class biogeme.biogeme.negLikelihood(like, like_deriv, scaled)
Provides the value of the function to be minimized, as well as its derivatives. To be used by the opimization package.
__init__(like, like_deriv, scaled)
Constructor
f(batch=None)
Calculate the value of the function
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for thre random batch. If None, the full data set is used. Default: None pass
Returns
value of the function
Return type
float
f_g(batch=None)
Calculate the value of the function and the gradient
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for the random batch. If None, the full data set is used. Default: None pass
Returns
value of the function and the gradient
Return type
tuple float, numpy.array
f_g_bhhh(batch=None)
Calculate the value of the function, the gradient and the BHHH matrix
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for the random batch. If None, the full data set is used. Default: None pass
Returns
value of the function, the gradient and the BHHH
Return type
tuple float, numpy.array, numpy.array
f_g_h(batch=None)
Calculate the value of the function, the gradient and the Hessian
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for the random batch. If None, the full data set is used. Default: None pass
Returns
value of the function, the gradient and the Hessian
Return type
tuple float, numpy.array, numpy.array
setVariables(x)
Set the values of the variables for which the function has to b calculated.
Parameters
x (numpy.array) – values
biogeme.database module¶
Implementation of the class Database, wrapping a pandas dataframe for specific services to Biogeme
author
Michel Bierlaire
date
Tue Mar 26 16:42:54 2019
class biogeme.database.Database(name, pandasDatabase)
Bases: object
Class that contains and prepare the database.
__init__(name, pandasDatabase)
Constructor
Parameters
• name (string) – name of the database.
• pandasDatabase (pandas.DataFrame) – data stored in a pandas data frame.
addColumn(expression, column)
Add a new column in the database, calculated from an expression.
Parameters
Returns
Return type
numpy.Series
Raises
ValueError – if the column name already exists.
buildPanelMap()
Sorts the data so that the observations for each individuals are contiguous, and builds a map that identifies the range of indices of the observations of each individuals.
checkAvailabilityOfChosenAlt(avail, choice)
Check if the chosen alternative is available for each entry in the database.
Parameters
• avail (list of biogeme.expressions.Expression) – list of expressions to evaluate the availability conditions for each alternative.
• choice (biogeme.expressions.Expression) – expression for the chosen alternative.
Returns
numpy series of bool, long as the number of entries in the database, containing True is the chosen alternative is available, False otherwise.
Return type
numpy.Series
count(columnName, value)
Counts the number of observations that have a specific value in a given column.
Parameters
• columnName (string) – name of the column.
• value (float) – value that is seeked.
Returns
Number of times that the value appears in the column.
Return type
int
descriptionOfNativeDraws()
Describe the draws available draws with Biogeme
Returns
dict, where the keys are the names of the draws, and the value their description
Example of output:
{'UNIFORM: Uniform U[0, 1]',
'UNIFORM_ANTI: Antithetic uniform U[0, 1]'],
'NORMAL: Normal N(0, 1) draws'}
Return type
dict
dumpOnFile()
Dumps the database in a CSV formatted file.
Returns
name of the file
Return type
string
generateDraws(types, names, numberOfDraws)
Generate draws for each variable.
Parameters
• types (dict) – A dict indexed by the names of the variables, describing the types of draws. Each of them can be a native type or any type defined by the function database.setRandomNumberGenerators
• names (list of strings) – the list of names of the variables that require draws to be generated.
• numberOfDraws (int) – number of draws to generate.
Returns
a 3-dimensional table with draws. The 3 dimensions are
1. number of individuals
2. number of draws
3. number of variables
Return type
numpy.array
Example:
types = {'randomDraws1': 'NORMAL_MLHS_ANTI',
'randomDraws2': 'UNIFORM_MLHS_ANTI',
'randomDraws3': 'UNIFORMSYM_MLHS_ANTI'}
theDrawsTable = myData.generateDraws(types,
['randomDraws1', 'randomDraws2', 'randomDraws3'], 10)
getNumberOfObservations()
Reports the number of observations in the database.
Note that it returns the same value, irrespectively if the database contains panel data or not.
Returns
Number of observations.
Return type
int
getSampleSize()
Reports the size of the sample.
If the data is cross-sectional, it is the number of observations in the database. If the data is panel, it is the number of individuals.
Returns
Sample size.
Return type
int
isPanel()
Tells if the data is panel or not.
Returns
True if the data is panel.
Return type
bool
panel(columnName)
Defines the data as panel data
Parameters
columnName (string) – name of the columns that identifies individuals.
remove(expression)
Removes from the database all entries such that the value of the expression is not 0.
Parameters
expression (biogeme.expressions.Expression) – expression to evaluate
sampleIndividualMapWithReplacement(size=None)
Extract a random sample of the individual map
from a panel data database, with replacement.
Useful for bootstrapping.
Parameters
size (int) – size of the sample. If None, a sample of the same size as the database will be generated. Default: None.
Returns
pandas dataframe with the sample.
Return type
pandas.DataFrame
sampleWithReplacement(size=None)
Extract a random sample from the database, with replacement.
Useful for bootstrapping.
Parameters
size (int) – size of the sample. If None, a sample of the same size as the database will be generated. Default: None.
Returns
pandas dataframe with the sample.
Return type
pandas.DataFrame
sampleWithoutReplacement(samplingRate, columnWithSamplingWeights=None)
Replace the data set by a sample for stochastic algorithms
Parameters
• samplingRate (float) – the proportion of data to include in the sample.
• columnWithSamplingWeights – name of the column with the sampling weights. If None, each row has equal probability.
• columnWithSamplingWeights – string
Returns
None
scaleColumn(column, scale)
Multiply an entire column by a scale value
Parameters
• column (string) – name of the column
• scale (float) – value of the scale. All values of the column will be multiplied by that scale.
setRandomNumberGenerators(rng)
Defines user-defined random numbers generators.
Parameters
rng (dict) – a dictionary of generators. The keys of the dictionary characterize the name of the generators, and must be different from the pre-defined generators in Biogeme: NORMAL, UNIFORM and UNIFORMSYM. The elements of the dictionary are functions that take two arguments: the number of series to generate (typically, the size of the database), and the number of draws per series.
Example:
def logNormalDraws(sampleSize, numberOfDraws):
return np.exp(np.random.randn(sampleSize, numberOfDraws))
def exponentialDraws(sampleSize, numberOfDraws):
return -1.0 * np.log(np.random.rand(sampleSize, numberOfDraws))
# We associate these functions with a name
dict = {'LOGNORMAL':(logNormalDraws,
'Draws from lognormal distribution'),
'EXP':(exponentialDraws,
'Draws from exponential distributions')}
myData.setRandomNumberGenerators(dict)
split(slices)
Prepare estimation and validation sets for validation.
Parameters
slices (int) – number of slices
Returns
list of estimation and validation data sets
Return type
tuple(pandas.DataFrame, pandas.DataFrame)
suggestScaling(columns=None)
Suggest a scaling of the variables in the database.
For each column, $$\delta$$ is the difference between the largest and the smallest value, or one if the difference is smaller than one. The level of magnitude is evaluated as a power of 10. The suggested scale is the inverse of this value.
$s = \frac{1}{10^{|\log_{10} \delta|}}$
where $$|x|$$ is the integer closest to $$x$$.
Parameters
columns (list(str)) – list of columns to be considered. If None, all of them will be considered.
Returns
A Pandas dataframe where each row contains the name of the variable and the suggested scale s. Ideally, the column should be multiplied by s.
Return type
pandas.DataFrame
sumFromDatabase(expression)
Calculates the value of an expression for each entry
in the database, and returns the sum.
Parameters
expression (biogeme.expressions.Expression) – expression to evaluate
Returns
sum of the expressions over the database.
Return type
float
useFullSample()
Re-establish the full sample for calculation of the likelihood
valuesFromDatabase(expression)
Evaluates an expression for each entry of the database.
Parameters
expression (biogeme.expressions.Expression.) – expression to evaluate
Returns
numpy series, long as the number of entries in the database, containing the calculated quantities.
Return type
numpy.Series
biogeme.distributions module¶
Implementation of the pdf and CDF of common distributions
:author:Michel Bierlaire
data
Thu Apr 23 12:01:49 2015
biogeme.distributions.logisticcdf(x, mu=0.0, s=1.0)
Logistic CDF
Cumulative distribution function of a logistic distribution
$f(x;\mu, \sigma) = \frac{1}{1+\exp\left(-\frac{x-\mu}{\sigma} \right)}$
Parameters
• x (float or biogeme.expression) – location parameter $$\mu$$ of the logistic distribution. Default: 0.
• x – scale parameter $$\sigma$$ of the logistic distribution. Default: 1.
Note
It is assumed that $$\sigma > 0$$, but it is not verified by the code.
Returns
value of the logistic CDF.
Return type
float or biogeme.expression
biogeme.distributions.lognormalpdf(x, mu=0.0, s=1.0)
Log normal pdf
Probability density function of a log normal distribution
$f(x;\mu, \sigma) = \frac{1}{x\sigma \sqrt{2\pi}} \exp{-\frac{(\ln x-\mu)^2}{2\sigma^2}}$
Parameters
• x (float or biogeme.expression) – location parameter $$\mu$$ of the lognormal distribution. Default: 0.
• s (float or biogeme.expression) – scale parameter $$\sigma$$ of the lognormal distribution. Default: 1.
Note
It is assumed that $$\sigma > 0$$, but it is not verified by the code.
Returns
value of the lognormal pdf.
Return type
float or biogeme.expression
biogeme.distributions.normalpdf(x, mu=0.0, s=1.0)
Normal pdf
Probability density function of a normal distribution
$f(x;\mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp{-\frac{(x-\mu)^2}{2\sigma^2}}$
Parameters
• x (float or biogeme.expression) – location parameter $$\mu$$ of the Normal distribution. Default: 0.
• s (float or biogeme.expression) – scale parameter $$\sigma$$ of the Normal distribution. Default: 1.
Note
It is assumed that $$\sigma > 0$$, but it is not verified by the code.
Returns
value of the Normal pdf.
Return type
float or biogeme.expression
biogeme.distributions.triangularpdf(x, a=-1.0, b=1.0, c=0.0)
Triangular pdf
Probability density function of a triangular distribution
$\begin{split}f(x;a, b, c) = \left\{ \begin{array}{ll} 0 & \text{if } x < a \\\frac{2(x-a)}{(b-a)(c-a)} & \text{if } a \leq x < c \\\frac{2(b-x)}{(b-a)(b-c)} & \text{if } c \leq x < b \\0 & \text{if } x \geq b. \end{array} \right.\end{split}$
Parameters
• x (float or biogeme.expression) – argument of the pdf
• a (float or biogeme.expression) – lower bound $$a$$ of the distribution. Default: -1.
• b (float or biogeme.expression) – upper bound $$b$$ of the distribution. Default: 1.
• c (float or biogeme.expression) – mode $$c$$ of the distribution. Default: 0.
Note
It is assumed that $$a < c < b$$, but it is not verified by the code.
Returns
value of the triangular pdf.
Return type
float or biogeme.expression
biogeme.distributions.uniformpdf(x, a=-1, b=1.0)
Uniform pdf
Probability density function of a uniform distribution.
$\begin{split}f(x;a, b) = \left\{ \begin{array}{ll} \frac{1}{b-a} & \text{for } x \in [a, b] \\ 0 & \text{otherwise}\end{array} \right.\end{split}$
Parameters
• x (float or biogeme.expression) – argument of the pdf
• a (float or biogeme.expression) – lower bound $$a$$ of the distribution. Default: -1.
• b (float or biogeme.expression) – upper bound $$b$$ of the distribution. Default: 1.
Note
It is assumed that $$a < b$$, but it is not verified by the code.
Returns
value of the uniform pdf.
Return type
float or biogeme.expression
biogeme.draws module¶
Generation of various types of draws.
author
Michel Bierlaire
date
Tue Jun 18 19:05:13 2019
biogeme.draws.getAntithetic(unif, sampleSize, numberOfDraws)
Returns antithetic uniform draws
Parameters
• unif (function) – function taking two arguments (sampleSize, numberOfDraws) and returning U[0, 1] draws
• sampleSize (int) – number of observations for which draws must be generated. If None, a one dimensional array will be generated. If it has a values k, then k series of draws will be generated
• numberOfDraws (int) – number of draws to generate.
Returns
numpy array with the antithetic draws
Return type
numpy.array
Example:
draws = dr.getAntithetic(dr.getUniform, sampleSize=3, numberOfDraws=10)
array([[0.48592363, 0.13648133, 0.35925946, 0.32431338, 0.32997936,
0.51407637, 0.86351867, 0.64074054, 0.67568662, 0.67002064],
[0.89261997, 0.0331808 , 0.30767182, 0.93433648, 0.17196124,
0.10738003, 0.9668192 , 0.69232818, 0.06566352, 0.82803876],
[0.81095587, 0.96171364, 0.40984817, 0.72177258, 0.16481096,
0.18904413, 0.03828636, 0.59015183, 0.27822742, 0.83518904]])
biogeme.draws.getHaltonDraws(sampleSize, numberOfDraws, symmetric=False, base=2, skip=0, shuffled=False)
Generate Halton draws
Parameters
• sampleSize (int) – number of observations for which draws must be generated. If None, a one dimensional array will be generated. If it has a values k, then k series of draws will be generated
• numberOfDraws (int) – number of draws to generate.
• symmetric (bool) – if True, draws from [-1: 1] are generated. If False, draws from [0: 1] are generated. Default: False
• base (int) – generate Halton draws for a given basis. Ideally, it should be a prime number. Default: 2.
• skip (int) – the number of elements of the sequence to be discarded.
• shuffled (bool) – if True, each series is shuffled
Returns
numpy array with the draws
Return type
numpy.array
Example:
halton = dr.getHaltonDraws(sampleSize=2, numberOfDraws=10, base=3)
array([[0.33333333, 0.66666667, 0.11111111, 0.44444444, 0.77777778,
0.22222222, 0.55555556, 0.88888889, 0.03703704, 0.37037037],
[0.7037037 , 0.14814815, 0.48148148, 0.81481481, 0.25925926,
0.59259259, 0.92592593, 0.07407407, 0.40740741, 0.74074074]])
biogeme.draws.getLatinHypercubeDraws(sampleSize, numberOfDraws, symmetric=False, uniformNumbers=None)
Implementation of the Modified Latin Hypercube Sampling proposed
by Hess et al, 2006.
Parameters
• sampleSize (int) – number of observations for which draws must be generated. If None, a one dimensional array will be generated. If it has a values k, then k series of draws will be generated
• numberOfDraws (int) – number of draws to generate.
• symmetric (bool) – if True, draws from [-1: 1] are generated. If False, draws from [0: 1] are generated. Default: False
• uniformNumbers (numpy.array) – numpy with uniformly distributed numbers. If None, the numpy uniform number generator is used.
Returns
numpy array with the draws
Return type
numpy.array
Example:
latinHypercube = dr.getLatinHypercubeDraws(sampleSize=3, numberOfDraws=10)
array([[0.43362897, 0.5275741 , 0.09215663, 0.94056236, 0.34376868,
0.87195551, 0.41495219, 0.71736691, 0.23198736, 0.145561 ],
[0.30520544, 0.78082964, 0.83591146, 0.2733167 , 0.53890906,
0.61607469, 0.00699715, 0.17179441, 0.7557228 , 0.39733102],
[0.49676864, 0.67073483, 0.9788854 , 0.5726069 , 0.11894558,
0.05515471, 0.2640275 , 0.82093696, 0.92034628, 0.64866597]])
biogeme.draws.getNormalWichuraDraws(sampleSize, numberOfDraws, uniformNumbers=None, antithetic=False)
Generate pseudo-random numbers from a normal distribution N(0, 1)
It uses the Algorithm AS241 Appl. Statist. (1988) Vol. 37, No. 3, which produces the normal deviate z corresponding to a given lower tail area of p; z is accurate to about 1 part in $$10^{16}$$.
Parameters
• sampleSize (int) – number of observations for which draws must be generated. If None, a one dimensional array will be generated. If it has a values k, then k series of draws will be generated
• numberOfDraws (int) – number of draws to generate.
• uniformNumbers (numpy.array) – numpy with uniformly distributed numbers. If None, the numpy uniform number generator is used.
• antithetic (bool) – if True, only half of the draws are actually generated, and the series are completed with their antithetic version.
Returns
numpy array with the draws
Return type
numpy.array
Example:
draws = dr.getNormalWichuraDraws(sampleSize=3, numberOfDraws=10)
array([[ 0.52418458, -1.04344204, -2.11642482, 0.48257162, -2.67188279,
-1.89993283, 0.28251041, -0.38424425, 1.53182226, 0.30651874],
[-0.7937038 , -0.07884121, -0.91005616, -0.98855175, 1.09405753,
-0.5997651 , -1.70785113, 1.57571384, -0.33208723, -1.03510102],
[-0.13853654, 0.92595498, -0.80136586, 1.68454196, 0.9955927 ,
-0.28615154, 2.10635541, 0.0436191 , -0.25417774, 0.01026933]])
biogeme.draws.getUniform(sampleSize, numberOfDraws, symmetric=False)
Uniform [0, 1] or [-1, 1] numbers
Parameters
• sampleSize (int) – number of observations for which draws must be generated. If None, a one dimensional array will be generated. If it has a values k, then k series of draws will be generated
• numberOfDraws (int) – number of draws to generate.
• symmetric (bool) – if True, draws from [-1: 1] are generated. If False, draws from [0: 1] are generated. Default: False
Returns
numpy array with the draws
Return type
numpy.array
Example:
draws = dr.getUniform(sampleSize=3, numberOfDraws=10, symmetric=False)
array([[0.13053817, 0.63892308, 0.55031567, 0.26347854, 0.16730932,
0.77745367, 0.48283887, 0.84247501, 0.20550219, 0.02373537],
[0.68935846, 0.03363595, 0.36006669, 0.26709364, 0.54907706,
0.22492104, 0.2494399 , 0.17323209, 0.52370401, 0.54091257],
[0.40310204, 0.89916711, 0.86065005, 0.94277699, 0.09077065,
0.40107731, 0.22554722, 0.47693135, 0.14058265, 0.17397031]])
draws = dr.getUniform(sampleSize=3, numberOfDraws=10, symmetric=True)
array([[ 0.74403237, -0.27995692, 0.33997421, -0.89405035, -0.129761 ,
0.86593325, 0.30657422, 0.82435619, 0.498482 , 0.24561616],
[-0.48239607, -0.29257815, -0.98342034, 0.68392813, -0.25379429,
0.49359859, -0.26459883, 0.14569724, -0.68860467, -0.40903446],
[ 0.93251627, -0.85166912, 0.58096917, 0.39289882, -0.65088635,
0.40114744, -0.61327161, 0.08900539, -0.20985417, 0.67542226]])
biogeme.exceptions module¶
Defines a generic exception for Biogeme
author
Michel Bierlaire
date
Tue Mar 26 16:47:11 2019
exception biogeme.exceptions.biogemeError
Bases: Exception
Defines a generic exception for Biogeme.
biogeme.expressions module¶
Defines the various arithmetic expressions accepted by Biogeme.
author
Michel Bierlaire
date
Tue Mar 26 16:47:49 2019
class biogeme.expressions.And(left, right)
Logical and
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Beta(name, value, lowerbound, upperbound, status)
Unknown parameters to be estimated from data.
__init__(name, value, lowerbound, upperbound, status)
Constructor
Parameters
• name (string) – name of the parameter.
• value (float) – default value.
• lowerbound (float) – if different from None, imposes a lower bound on the value of the parameter during the optimization.
• upperbound (float) – if different from None, imposes an upper bound on the value of the parameter during the optimization.
• status (int) – if different from 0, the parameter is fixed to its default value, and not modified by the optimization algorithm.
changeInitValues(betas)
Modifies the initial values of the Beta parameters.
The fact that the parameters are fixed or free is irrelevant here.
Parameters
betas (dict(string:float)) – dictionary where the keys are the names of the parameters, and the values are the new value for the parameters.
dictOfBetas(free=True, fixed=False)
Extract the set of parameters from the expression. Overload the generic function.
Parameters
• free (bool) – if True, the free parameters are included. Default: True.
• fixed (bool) – if True, the fixed parameters are included. Default: False.
Returns
a dict with the beta parameters appearing in the expression, the keys being the names of the parameters.
Return type
dict(string:biogeme.expressions.Expression)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the name of the expression between < >
2. the id of the expression between { }
3. the name of the parameter,
4. the status between [ ]
5. the unique ID, preceeded by a comma
6. the beta ID, preceeded by a comma
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
Raises
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
setOfBetas(free=True, fixed=False)
Extract the set of parameters from the expression. Overload the generic function.
Parameters
• free (bool) – if True, the free parameters are included. Default: True.
• fixed (bool) – if True, the fixed parameters are included. Default: False.
Returns
returns a set with the beta parameters appearing in the expression.
Return type
setSpecificIndices(indicesOfFreeBetas, indicesOfFixedBetas, indicesOfRandomVariables, indicesOfDraws)
Provide an index to all elementary expressions, specific to their type
Parameters
• indicesOfFreeBetas (dict(string:int)) – dictionary mapping the name of the free betas with their index
• indicesOfFixedBetas (dict(string:int)) – dictionary mapping the name of the fixed betas with their index
• indicesOfRandomVariables (dict(string:int)) – dictionary mapping the name of the random variables with their index
• indicesOfDraws (dict(string:int)) – dictionary mapping the name of the draws with their index
class biogeme.expressions.BinaryOperator(left, right)
Base class for arithmetic expressions that are binary operators. This expression is the result of the combination of two expressions, typically addition, substraction, multiplication or division.
__init__(left, right)
Constructor
Parameters
class biogeme.expressions.DefineVariable(name, expression, database)
Expression that defines a new variable and add a column in the database.
This expression allows the use to define a new variable that will be added to the database. It avoids that it is recalculated each time it is needed.
__init__(name, expression, database)
Constructor
Parameters
• name (string) – name of the variable.
• expression – formula that defines the variable
• type – biogeme.expressions.Expression
• database (biogeme.database.Database) – object identifying the database.
class biogeme.expressions.Derive(child, name)
Derivative with respect to an elementary expression
__init__(child, name)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the signatures of the child expression,
2. the name of the expression between < >
3. the id of the expression between { }
4. the id of the child, preceeded by a comma.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
setUniqueId(idsOfElementaryExpressions)
Provides a unique id to the elementary expressions.
Parameters
idsOfElementaryExpressions (dict(string:int)) – dictionary mapping the name of the elementary expression with their id.
class biogeme.expressions.Divide(left, right)
Division expression
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Elem(dictOfExpressions, keyExpression)
This returns the element of a dictionary. The key is evaluated from an expression.
__init__(dictOfExpressions, keyExpression)
Constructor
Parameters
• dictOfExpressions (dict(int: biogeme.expressions.Expression)) – dict of objects with numerical keys.
• keyExpression (biogeme.expressions.Expression) – object providing the key of the element to be evaluated.
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the signature of the expression defining the key
2. the signatures of all the children expressions,
3. the name of the expression between < >
4. the id of the expression between { }
5. the number of elements between ( )
6. the id of the expression defining the key
7. for each element: the value of the key and the id of the expression, separated by commas.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Elementary(name)
Elementary expression.
It is typically defined by a name appearing in an expression. It can be a variable (from the database), or a parameter (fixed or to be estimated using maximum likelihood), a random variable for numrerical integration, or Monte-Carlo integration.
__init__(name)
Constructor
Parameters
name (string) – name of the elementary experession.
getElementaryExpression(name)
Return: an elementary expression from its name if it appears in the expression. None otherwise.
setUniqueId(idsOfElementaryExpressions)
Provides a unique id to the elementary expressions. Overloads the generic function
Parameters
idsOfElementaryExpressions (dict(string:int)) – dictionary mapping the name of the elementary expression with their id.
class biogeme.expressions.Equal(left, right)
Logical equal
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Expression
Bases: object
This is the general arithmetic expression in biogeme. It serves as a base class for concrete expressions.
__add__(other)
Parameters
other (biogeme.expressions.Expression) – expression to be added
Returns
self + other
Return type
biogeme.expressions.Expression
__and__(other)
Parameters
other (biogeme.expressions.Expression) – expression for logical and
Returns
self and other
Return type
biogeme.expressions.Expression
__div__(other)
Parameters
other (biogeme.expressions.Expression) – expression for division
Returns
self / other
Return type
biogeme.expressions.Expression
__eq__(other)
Parameters
other (biogeme.expressions.Expression) – expression for equality
Returns
self == other
Return type
biogeme.expressions.Expression
__ge__(other)
Parameters
other (biogeme.expressions.Expression) – expression for greater or equal
Returns
self >= other
Return type
biogeme.expressions.Expression
__gt__(other)
Parameters
other (biogeme.expressions.Expression) – expression for greater than
Returns
self > other
Return type
biogeme.expressions.Expression
__init__()
Constructor
__le__(other)
Parameters
other (biogeme.expressions.Expression) – expression for less or equal
Returns
self <= other
Return type
biogeme.expressions.Expression
__lt__(other)
Parameters
other (biogeme.expressions.Expression) – expression for less than
Returns
self < other
Return type
biogeme.expressions.Expression
__mul__(other)
Parameters
other (biogeme.expressions.Expression) – expression to be multiplied
Returns
self * other
Return type
biogeme.expressions.Expression
__ne__(other)
Parameters
other (biogeme.expressions.Expression) – expression for difference
Returns
self != other
Return type
biogeme.expressions.Expression
__neg__()
Returns
-self
Return type
biogeme.expressions.Expression
__or__(other)
Parameters
other (biogeme.expressions.Expression) – expression for logical or
Returns
self or other
Return type
biogeme.expressions.Expression
__pow__(other)
Parameters
other (biogeme.expressions.Expression) – expression for power
Returns
self ^ other
Return type
biogeme.expressions.Expression
__radd__(other)
Parameters
other (biogeme.expressions.Expression) – expression to be added
Returns
other + self
Return type
biogeme.expressions.Expression
__rdiv__(other)
Parameters
other (biogeme.expressions.Expression) – expression for division
Returns
other / self
Return type
biogeme.expressions.Expression
__rmul__(other)
Parameters
other (biogeme.expressions.Expression) – expression to be multiplied
Returns
other * self
Return type
biogeme.expressions.Expression
__rpow__(other)
Parameters
other (biogeme.expressions.Expression) – expression for power
Returns
other ^ self
Return type
biogeme.expressions.Expression
__rsub__(other)
Parameters
other (biogeme.expressions.Expression) – expression to be substracted
Returns
other - self
Return type
biogeme.expressions.Expression
__rtruediv__(other)
Parameters
other (biogeme.expressions.Expression) – expression for division
Returns
other / self
Return type
biogeme.expressions.Expression
__sub__(other)
Parameters
other (biogeme.expressions.Expression) – expression to substract
Returns
self - other
Return type
biogeme.expressions.Expression
__truediv__(other)
Parameters
other (biogeme.expressions.Expression) – expression for division
Returns
self / other
Return type
biogeme.expressions.Expression
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
changeInitValues(betas)
Modifies the initial values of the Beta parameters.
The fact that the parameters are fixed or free is irrelevant here.
Parameters
betas (dict(string:float)) – dictionary where the keys are the names of the parameters, and the values are the new value for the parameters.
countPanelTrajectoryExpressions()
Count the number of times the PanelLikelihoodTrajectory is used in the formula. It should trigger an error if it is used more than once.
dictOfBetas(free=True, fixed=False)
Extract the set of parameters from the expression.
Parameters
• free (bool) – if True, the free parameters are included. Default: True.
• fixed (bool) – if True, the fixed parameters are included. Default: False.
Returns
a dict with the beta parameters appearing in the expression, the keys being the names of the parameters.
Return type
dict(string:biogeme.expressions.Expression)
dictOfDraws()
Recursively extract the random variables (draws for Monte-Carlo) appearing in the expression, and store them in a dictionary.
Returns
dict where the keys are the random variables and the elements the type of draws
Return type
dict(string:string)
dictOfRandomVariables()
Recursively extract the random variables appearing in the
expression, and store them in a dictionary.
Returns
returns a dict with the random variables appearing in the expression the keys being their names.
Return type
dict(string:biogeme.expressions.Expression)
dictOfVariables()
Recursively extract the variables appearing in the expression, and store them in a dictionary.
Returns
returns a dict with the variables appearing in the expression the keys being their names.
Return type
dict(string:biogeme.expressions.Expression)
embedExpression(t)
Check if the expression contains an expression of type t. Typically, this would be used to check that a MonteCarlo expression contains a bioDraws expression.
Return:
bool.
See: Expression.isContainedIn
getClassName()
Obtain the name of the top class of the expression structure
Returns
the name of the class
Return type
string
getElementaryExpression(name)
Return: an elementary expression from its name if it appears in the expression.
Parameters
name (string) – name of the elementary expression.
Returns
the expression if it exists. None otherwise.
Return type
biogeme.expressions.Expression
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the signatures of all the children expressions,
2. the name of the expression between < >
3. the id of the expression between { }
4. the number of children between ( )
5. the ids of each children, preceeded by a comma.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
getValue_c(database, numberOfDraws=1000)
Evaluation of the expression
In Biogeme the complexity of some expressions requires a specific implementation, in C++. This function invokes the C++ code to evaluate the value of the expression for a series of entries in a database. Note that this function will generate draws if needed.
Parameters
• database (biogeme.database.Database) – database
• numberOfDraws (int) – number of draws if needed by Monte-Carlo integration.
Returns
a list where each entry is the result of applying the expression on one entry of the dsatabase.
Return type
numpy.array
isContainedIn(t)
Check if the expression is contained in an expression of type t. Typically, this would be used to check that a bioDraws expression is contained in a MonteCarlo expression. If not, it cannot be evaluated.
Return:
bool.
See: Expression.embedExpression
setOfBetas(free=True, fixed=False)
Extract the set of parameters from the expression.
Parameters
• free (bool) – if True, the free parameters are included. Default: True.
• fixed (bool) – if True, the fixed parameters are included. Default: False.
Returns
returns a set with the beta parameters appearing in the expression.
Return type
setOfVariables()
Extract the set of variables used in the expression.
Returns
returns a set with the variables appearing in the expression.
Return type
setRow(row)
This function identifies the row of the database from which
the values of the variables must be obtained.
Parameters
row (int) – id of the row.
setSpecificIndices(indicesOfFreeBetas, indicesOfFixedBetas, indicesOfRandomVariables, indicesOfDraws)
Provides an index to all elementary expressions, specific to their type
Parameters
• indicesOfFreeBetas (dict(string:int)) – dictionary mapping the name of the free betas with their index
• indicesOfFixedBetas (dict(string:int)) – dictionary mapping the name of the fixed betas with their index
• indicesOfRandomVariables (dict(string:int)) – dictionary mapping the name of the random variables with their index
• indicesOfDraws (dict(string:int)) – dictionary mapping the name of the draws with their index
setUniqueId(idsOfElementaryExpressions)
Provides a unique id to the elementary expressions.
Parameters
idsOfElementaryExpressions (dict(string:int)) – dictionary mapping the name of the elementary expression with their id.
setVariableIndices(indicesOfVariables)
Provide an index to all variables
Parameters
indicesOfVariables (dict(string:int)) – dictionary mapping the name of the variables with their index
class biogeme.expressions.Greater(left, right)
Logical greater
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.GreaterOrEqual(left, right)
Logical greater or equal
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Integrate(child, name)
Numerical integration
__init__(child, name)
Constructor
Parameters
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the signatures of the child expression,
2. the name of the expression between < >
3. the id of the expression between { }, preceeded by a comma
4. the id of the children, preceeded by a comma
5. the index of the randon variable, preceeded by a comma
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
setSpecificIndices(indicesOfFreeBetas, indicesOfFixedBetas, indicesOfRandomVariables, indicesOfDraws)
Provide an index to all elementary expressions, specific to their type Overloads the generic function.
Parameters
• indicesOfFreeBetas (dict(string:int)) – dictionary mapping the name of the free betas with their index
• indicesOfFixedBetas (dict(string:int)) – dictionary mapping the name of the fixed betas with their index
• indicesOfRandomVariables (dict(string:int)) – dictionary mapping the name of the random variables with their index
• indicesOfDraws (dict(string:int)) – dictionary mapping the name of the draws with their index
setUniqueId(idsOfElementaryExpressions)
Provides a unique id to the elementary expressions. Overloads the generic function
Parameters
idsOfElementaryExpressions (dict(string:int)) – dictionary mapping the name of the elementary expression with their id.
class biogeme.expressions.Less(left, right)
Logical less
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.LessOrEqual(left, right)
Logical less or equal
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.LogLogit(util, av, choice)
Expression capturing the logit formula.
It contains one formula for the target alternative, a dict of formula for the availabilities and a dict of formulas for the utilities
__init__(util, av, choice)
Constructor
Parameters
• util (dict(int:biogeme.expressions.Expression)) – dictionary where the keys are the identifiers of the alternatives, and the elements are objects defining the utility functions.
• av (dict(int:biogeme.expressions.Expression)) – dictionary where the keys are the identifiers of the alternatives, and the elements are object of type biogeme.expressions.Expression defining the availability conditions. If av is None, all the alternatives are assumed to be always available
• choice (biogeme.expressions.Expression) – formula to obtain the alternative for which the logit probability must be calculated.
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the signatures of all the children expressions,
2. the name of the expression between < >
3. the id of the expression between { }
4. the number of alternatives between ( )
5. the id of the expression for the chosen alternative, preceeded by a comma.
6. for each alternative, separated by commas:
1. the number of the alternative, as defined by the user,
2. the id of the expression for the utility,
3. the id of the expression for the availability condition.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Minus(left, right)
Substraction expression
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.MonteCarlo(child)
Monte Carlo integration
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – arithmetic expression
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
class biogeme.expressions.NotEqual(left, right)
Logical not equal
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Numeric(value)
Numerical expression for a simple number
__init__(value)
Constructor
Parameters
value (float) – numerical value
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the name of the expression between < >
2. the id of the expression between { }
3. the value, preceeded by a comma.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Or(left, right)
Logical or
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.PanelLikelihoodTrajectory(child)
Likelihood of a sequences of observations for the same individual
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
countPanelTrajectoryExpressions()
Count the number of times the PanelLikelihoodTrajectory is used in the formula.
class biogeme.expressions.Plus(left, right)
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.Power(left, right)
Power expression
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.RandomVariable(name)
Random variable for numerical integration
__init__(name)
Constructor
Parameters
name (string.) – name of the random variable involved in the integration.
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
dictOfRandomVariables()
Recursively extract the random variables appearing in
the expression, and store them in a dictionary.
Returns
returns a dict with the random variables appearing in the expression the keys being their names.
Return type
dict(string:biogeme.expressions.Expression)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the name of the expression between < >
2. the id of the expression between { }
3. the name of the random variable,
4. the unique ID, preceeded by a comma,
5. the ID of the random variable.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
Raises
setSpecificIndices(indicesOfFreeBetas, indicesOfFixedBetas, indicesOfRandomVariables, indicesOfDraws)
Provide an index to all elementary expressions, specific to their type Overloads the generic function.
Parameters
• indicesOfFreeBetas (dict(string:int)) – dictionary mapping the name of the free betas with their index
• indicesOfFixedBetas (dict(string:int)) – dictionary mapping the name of the fixed betas with their index
• indicesOfRandomVariables (dict(string:int)) – dictionary mapping the name of the random variables with their index
• indicesOfDraws (dict(string:int)) – dictionary mapping the name of the draws with their index
class biogeme.expressions.Times(left, right)
Multiplication expression
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.UnaryMinus(child)
Unary minus expression
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.UnaryOperator(child)
Base class for arithmetic expressions that are unary operators.
Such an expression is the result of the modification of another expressions, typically changing its sign.
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
class biogeme.expressions.Variable(name)
Explanatory variable
This represents the explanatory variables of the choice model. Typically, they come from the data set.
__init__(name)
Constructor
Parameters
name (string) – name of the variable.
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
dictOfVariables()
Recursively extract the variables appearing in the expression, and store them in a dictionary.
Returns
returns a dict with the variables appearing in the expression the keys being their names. Here, it contains only one element.
Return type
dict(string:biogeme.expressions.Expression)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the name of the expression between < >
2. the id of the expression between { }
3. the name of the variable,
4. the unique ID, preceeded by a comma.
5. the variabvle ID, preceeded by a comma.
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
Raises
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
setVariableIndices(indicesOfVariables)
Provide an index to all variables
Parameters
indicesOfVariables (dict(string:int)) – dictionary mapping the name of the variables with their index
class biogeme.expressions.bioDraws(name, drawType)
Draws for Monte-Carlo integration
__init__(name, drawType)
Constructor
Parameters
• name (string) – name of the random variable with a series of draws.
• drawType (string) – type of draws.
audit(database=None)
Performs various checks on the expressions.
Parameters
database (biogeme.database.Database) – database object
Returns
tuple listOfErrors, listOfWarnings
Return type
list(string), list(string)
dictOfDraws()
Recursively extract the random variables (draws for Monte-Carlo). Overloads the generic function. appearing in the expression, and store them in a dictionary.
Returns
dict where the keys are the random variables and the elements the type of draws. Here, contains only one element.
Return type
dict(string:string)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the name of the expression between < >
2. the id of the expression between { }, preceeded by a comma
3. the name of the draws
4. the unique ID (preceeded by a comma),
5. the draw ID (preceeded by a comma).
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
Raises
setDrawIndex(idsOfDraws)
Provide an index to a series of draw for a random variable. Overload the generic function.
Parameters
idsOfDraws (dict(string:int)) – dictionary mapping the name of the draws with their id.
setSpecificIndices(indicesOfFreeBetas, indicesOfFixedBetas, indicesOfRandomVariables, indicesOfDraws)
Provide an index to all elementary expressions, specific to their type Overloads the generic function.
Parameters
• indicesOfFreeBetas (dict(string:int)) – dictionary mapping the name of the free betas with their index
• indicesOfFixedBetas (dict(string:int)) – dictionary mapping the name of the fixed betas with their index
• indicesOfRandomVariables (dict(string:int)) – dictionary mapping the name of the random variables with their index
• indicesOfDraws (dict(string:int)) – dictionary mapping the name of the draws with their index
class biogeme.expressions.bioLinearUtility(listOfTerms)
When the utility function is linear, it is expressed as a list of terms, where a parameter multiplies a variable.
__init__(listOfTerms)
Constructor
Parameters
listOfTerms (list(biogeme.expressions.Expression, biogeme.expressions.Expression)) – a list of tuple. Each tuple contains first a beta parameter, second the name of a variable.
Raises
biogeme.exceptions.biogemeError – if the object is not a list of tuples (parameter, variable)
dictOfBetas(free=True, fixed=False)
Extract the set of parameters from the expression.
Parameters
• free (bool) – if True, the free parameters are included. Default: True.
• fixed (bool) – if True, the fixed parameters are included. Default: False.
Returns
a dict with the beta parameters appearing in the expression, the keys being the names of the parameters.
Return type
dict(string:biogeme.expressions.Expression)
dictOfDraws()
Recursively extract the random variables (draws for Monte-Carlo). Overloads the generic function. appearing in the expression, and store them in a dictionary.
Returns
dict where the keys are the random variables and the elements the type of draws. Here, returns an empty dict.
Return type
dict(string:string)
dictOfRandomVariables()
Recursively extract the random variables appearing
in the expression, and store them in a dictionary.
Returns
returns a dict with the random variables appearing in the expression the keys being their names.
Return type
dict(string:biogeme.expressions.Expression)
dictOfVariables()
Recursively extract the variables appearing in the expression, and store them in a dictionary.
Returns
returns a dict with the variables appearing in the expression the keys being their names.
Return type
dict(string:biogeme.expressions.Expression)
getSignature()
The signature of a string characterizing an expression.
This is designed to be communicated to C++, so that the expression can be reconstructed in this environment.
The list contains the following elements:
1. the signatures of all the children expressions,
2. the name of the expression between < >
3. the id of the expression between { }
4. the number of terms in the utility ( )
5. for each term:
1. the id of the beta parameter
2. the unique id of the beta parameter
3. the name of the parameter
4. the id of the variable
5. the unique id of the variable
6. the name of the variable
Consider the following expression:
$2 \beta_1 V_1 - \frac{\exp(-\beta_2 V_2) }{ \beta_3 (\beta_2 \geq \beta_1)}.$
It is defined as:
2 * beta1 * Variable1 - expressions.exp(-beta2*Variable2) / (beta3 * (beta2 >= beta1))
And its signature is:
[b'<Numeric>{4780527008},2',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<Times>{4780526952}(2),4780527008,4780277152',
b'<Variable>{4511837152}"Variable1",5,2',
b'<Times>{4780527064}(2),4780526952,4511837152',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<UnaryMinus>{4780527120}(1),4780277656',
b'<Variable>{4511837712}"Variable2",6,3',
b'<Times>{4780527176}(2),4780527120,4511837712',
b'<exp>{4780527232}(1),4780527176',
b'<Beta>{4780277264}"beta3"[1],2,0',
b'<Beta>{4780277656}"beta2"[0],1,1',
b'<Beta>{4780277152}"beta1"[0],0,0',
b'<GreaterOrEqual>{4780527288}(2),4780277656,4780277152',
b'<Times>{4780527344}(2),4780277264,4780527288',
b'<Divide>{4780527400}(2),4780527232,4780527344',
b'<Minus>{4780527456}(2),4780527064,4780527400']
Returns
list of the signatures of an expression and its children.
Return type
list(string)
setOfBetas(free=True, fixed=False)
Extract the set of parameters from the expression.
Parameters
• free (bool) – if True, the free parameters are included. Default: True.
• fixed (bool) – if True, the fixed parameters are included. Default: False.
Returns
returns a set with the beta parameters appearing in the expression.
Return type
class biogeme.expressions.bioMax(left, right)
Maximum of two expressions
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.bioMin(left, right)
Minimum of two expressions
__init__(left, right)
Constructor
Parameters
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.bioMultSum(listOfExpressions)
This expression returns the sum of several other expressions.
It is a generalization of ‘Plus’ for more than two terms
__init__(listOfExpressions)
Constructor
Parameters
listOfExpressions (list(biogeme.expressions.Expression)) – list of objects representing the terms of the sum.
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
class biogeme.expressions.bioNormalCdf(child)
Cumulative Distribution Function of a normal random variable
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
biogeme.expressions.defineNumberingOfElementaryExpressions(collectionOfFormulas, variableNames)
Provides indices for elementary expressions
The numbering is done in the following order:
1. free betas,
2. fixed betas,
3. random variables for numrerical integration,
4. random variables for Monte-Carlo integration,
5. variables
The numbering convention will be performed for all expressions together, so that the same elementary expressions in several expressions will have the same index.
Parameters
• collectionOfFormula (list(biogeme.expressions.Expression)) – collection of Biogeme expressions.
• variableNames (list(string)) – list of the names of the variables
Returns
dict, free, freeNames, fixed, fixedNames, rv, rvNames, draws, drawsNames where
• dict is a dictionary mapping the names of the elementary expressions with their index,
• free is a dict with the free betas,
• freeNames is a list of the names of the free betas,
• fixed is a dict with the fixed betas,
• fixedNames is the list of the names of the fixed betas,
• rv is a dict with the random variables for numerical integration,
• rvNames is a list with their names,
• draws is a dict of the draws, and
• drawsNames is a list with their names.
class biogeme.expressions.exp(child)
exponential expression
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
biogeme.expressions.isNumeric(obj)
Identifies if an object is numeric, that is int, float or bool.
Parameters
obj (object) – any object
class biogeme.expressions.log(child)
logarithm expression
__init__(child)
Constructor
Parameters
child (biogeme.expressions.Expression) – first arithmetic expression
getValue()
Evaluates the value of the expression
Returns
value of the expression
Return type
float
biogeme.filenames module¶
Implements the function providing names for the output files.
author
Michel Bierlaire
date
Tue Mar 26 16:48:40 2019
biogeme.filenames.getNewFileName(name, ext)
Generate a file name that does not exist.
Parameters
• name (string) – name of the file.
• ext (string) – file extension.
Returns
name.ext if the file does not exists. If it does, returns name~xx.ext, where xx is the smallest integer such that the corresponding file does not exist. It is designed to avoid erasing output files inadvertently.
Return type
string
biogeme.loglikelihood module¶
Functions to calculate the log likelihood
author
Michel Bierlaire
date
Fri Mar 29 17:11:44 2019
biogeme.loglikelihood.likelihoodregression(meas, model, sigma)
Computes likelihood function of a regression model.
Parameters
Returns
The likelihood of the regression, assuming a normal distribution, that is
\begin{align}\begin{aligned}\frac{1}{\sigma} \phi\left( \frac{y-m}{\sigma} \right)\\Where :math:\phi(\cdot) is the pdf of the normal distribution.\end{aligned}\end{align}
Return type
biogeme.expressions.Expression
biogeme.loglikelihood.loglikelihood(prob)
Simply computes the log of the probability
Parameters
prob (biogeme.expressions.Expression) – An expression providing the value of the probability.
Returns
the logarithm of the probability.
Return type
biogeme.expressions.Expression
biogeme.loglikelihood.loglikelihoodregression(meas, model, sigma)
Computes log likelihood function of a regression model.
Parameters
Returns
the likelihood of the regression, assuming a normal distribution, that is
$-\left( \frac{(y-m)^2}{2\sigma^2} \right) - \log(\sigma) - \frac{1}{2}\log(2\pi)$
Return type
biogeme.expressions.Expression
biogeme.loglikelihood.mixedloglikelihood(prob)
Compute a simulated loglikelihood function
Parameters
prob – An expression providing the value of the probability. Although it is not formally necessary, the expression should contain one or more random variables of a given distribution, and therefore is defined as
$P(i|\xi_1,\ldots,\xi_L)$
Returns
the simulated loglikelihood, given by
$\ln\left(\sum_{r=1}^R P(i|\xi^r_1,\ldots,\xi^r_L) \right)$
where $$R$$ is the number of draws, and $$\xi_j^r$$
is the rth draw of the random variable $$\xi_j$$.
Return type
biogeme.expressions.Expression
biogeme.messaging module¶
Singleton managing the various levels of messages
author
Michel Bierlaire
date
Mon Jul 22 16:12:00 2019
class biogeme.messaging.Singleton
Bases: type
A singleton is a class with only one instance
class biogeme.messaging.bioMessage(screenLevel=0)
Bases: object
Manages the Biogeme messages
__init__(screenLevel=0)
Constructor
Parameters
screenLevel
level of message that must be displayed on the screen:
• 0: no output (default)
• 1: warnings only
• 2: only warnings and general information
• 3: more verbose
• 4: debug messages
addMessage(text, level)
Parameters
• text (string) – text of the message.
• level
level of the message
• 1: warning
• 2: general information
• 3: detailed information
• 4: debug message
Note
adding a message of level 0 is meaningless, as it correspond to silentmode.
allMessages(screenLevel=None)
Report all the messages up to a given level.
Parameters
fileLevel
level of message that must be reported in the file:
• 0: no output
• 1: warnings only
• 2: only warnings and general information
• 3: more verbose (default)
• 4: debug messages
If None (default), all messages are reported.
Returns
all messages.
Return type
str.
createLog(fileLevel=None, fileName='_biogeme')
Creates a log file
Parameters
• fileLevel
level of message that must be reported in the file:
• 0: no output
• 1: warnings only
• 2: only warnings and general information
• 3: more verbose (default)
• 4: debug messages
If None (default), all messages are reported.
• fileName (string) – name of the file (without extension). Default: ‘_biogeme’. A file called _biogeme.log will be created.
debug(text)
Parameters
text (string) – text of the message.
detailed(text)
Parameters
text (string) – text of the message.
general(text)
Parameters
text (string) – text of the message.
resetMessages()
resume()
Resume the regular operations of the logger after the use of temporarySilence
setDebug()
Set both screen and file levels to 4
setDetailed()
Set both screen and file levels to 3
setGeneral()
Set both screen and file levels to 2
setScreenLevel(level)
Change the level of messaging for the screen
Parameters
level (int) –
level of message that must be displayed on the screen:
• 0: no output
• 1: warnings only
• 2: only warnings and general information
• 3: more verbose
• 4: debug messages
setSilent()
Set both screen and file levels to 0
setWarning()
Set both screen and file levels to 1
temporarySilence()
Temporarily turns off the message, remembering the current screen level.
warning(text)
Parameters
text (string) – text of the message.
biogeme.models module¶
Implements various models.
author
Michel Bierlaire
date
Fri Mar 29 17:13:14 2019
biogeme.models.boxcox(x, ell)
Box-Cox transform
$B(x, \ell) = \frac{x^{\ell}-1}{\ell}.$
It has the property that
$\lim_{\ell \to 0} B(x,\ell)=\log(x).$
Parameters
Returns
the Box-Cox transform
Return type
biogeme.expressions.Expression
biogeme.models.checkValidityNestedLogit(V, nests)
Verifies if the nested logit model is indeed based on a partition
of the choice set.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• nests (tuple) –
A tuple containing as many items as nests. Each item is also
a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives belonging to the nest.
Example:
nesta = MUA, [1, 2, 3]
nestb = MUB, [4, 5, 6]
nests = nesta, nestb
Returns
a tuple ok, message, where the message explains the problem is the nested structure is not OK.
Return type
tuple(bool, str)
biogeme.models.cnl(V, availability, nests, choice)
Implements the cross-nested logit model as a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionaray, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA, alphaA
nestb = MUB, alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.cnl_avail(V, availability, nests, choice)
Same as cnl. Maintained for backward compatibility
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items
• an object of type biogeme.expressions.Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionary, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA, alphaA
nestb = MUB, alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.cnlmu(V, availability, nests, choice, mu)
Implements the cross-nested logit model as a MEV model with
the homogeneity parameters is explicitly involved
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionary, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA, alphaA
nestb = MUB, alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
• mu (biogeme.expressions.Expression) – Homogeneity parameter $$\mu$$.
Returns
choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.getMevForCrossNested(V, availability, nests)
Implements the MEV generating function for the cross-nested logit
model as a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionary, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA, alphaA
nestb = MUB, alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of the choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.getMevForCrossNestedMu(V, availability, nests, mu)
Implements the MEV generating function for the cross-nested logit
model as a MEV model with the homogeneity parameters is explicitly involved.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionary, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA, alphaA
nestb = MUB, alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
• mu (biogeme.expressions.Expression) – Homogeneity parameter $$\mu$$.
Returns
log of the choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.getMevForNested(V, availability, nests)
Implements the MEV generating function for the nested logit model
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
A tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions.Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives belonging to the nest.
Example:
nesta = MUA ,[1, 2, 3]
nestb = MUB ,[4, 5, 6]
nests = nesta, nestb
Returns
a dictionary mapping each alternative id with the function
$\ln \frac{\partial G}{\partial y_i}(e^{V_1}, \ldots,e^{V_J}) = e^{(\mu_m-1)V_i} \left(\sum_{i=1}^{J_m} e^{\mu_m V_i}\right)^ {\frac{1}{\mu_m}-1}$
where $$m$$ is the (only) nest containing alternative $$i$$,
and $$G$$ is the MEV generating function.
Return type
dict(int:biogeme.expressions.Expression)
biogeme.models.getMevForNestedMu(V, availability, nests, mu)
Implements the MEV generating function for the nested logit model, including the scale parameter
param V
dict of objects representing the utility functions of each alternative, indexed by numerical ids.
type V
dict(int:biogeme.expressions.Expression)
param availability
dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
type availability
dict(int:biogeme.expressions.Expression)
param nests
A tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives belonging to the nest.
Example:
nesta = MUA, [1, 2, 3]
nestb = MUB, [4, 5, 6]
nests = nesta, nestb
type nests
tuple
param mu
scale parameter
type mu
biogeme.expressions.Expression
return
a dictionary mapping each alternative id with the function
$\frac{\partial G}{\partial y_i}(e^{V_1},\ldots,e^{V_J}) = \mu e^{(\mu_m-1)V_i} \left(\sum_{i=1}^{J_m} e^{\mu_m V_i}\right)^{\frac{\mu}{\mu_m}-1}$
where $$m$$ is the (only) nest containing alternative $$i$$,
and $$G$$ is the MEV generating function.
rtype
dict(int:biogeme.expressions.Expression)
biogeme.models.logcnl(V, availability, nests, choice)
Implements the log of the cross-nested logit model as a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionaray, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA , alphaA
nestb = MUB , alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of the choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.logcnl_avail(V, availability, nests, choice)
Same as logcnl. Maintained for backward compatibility
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionary, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA, alphaA
nestb = MUB, alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.logcnlmu(V, availability, nests, choice, mu)
Implements the log of the cross-nested logit model as a MEV model
with the homogeneity parameters is explicitly involved.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
a tuple containing as many items as nests. Each item is
also a tuple containing two items
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a dictionary mapping the alternative ids with the cross-nested parameters for the corresponding nest. If an alternative is missing in the dictionary, the corresponding alpha is set to zero.
Example:
alphaA = {1: alpha1a,
2: alpha2a,
3: alpha3a,
4: alpha4a,
5: alpha5a,
6: alpha6a}
alphaB = {1: alpha1b,
2: alpha2b,
3: alpha3b,
4: alpha4b,
5: alpha5b,
6: alpha6b}
nesta = MUA , alphaA
nestb = MUB , alphaB
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
• mu (biogeme.expressions.Expression) – Homogeneity parameter $$\mu$$.
Returns
log of the choice probability for the cross-nested logit model.
Return type
biogeme.expressions.Expression
biogeme.models.logit(V, av, i)
The logit model
The model is defined as
$\frac{a_i e^{V_i}}{\sum_{i=1}^J a_j e^{V_j}}$
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• av (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• i (int) – id of the alternative for which the probability must be calculated.
Returns
choice probability of alternative number i.
Return type
biogeme.expressions.Expression
biogeme.models.loglogit(V, av, i)
The logarithm of the logit model
The model is defined as
$\frac{a_i e^{V_i}}{\sum_{i=1}^J a_j e^{V_j}}$
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• av (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• i (int) – id of the alternative for which the probability must be calculated.
Returns
choice probability of alternative number i.
Return type
biogeme.expressions.Expression
biogeme.models.logmev(V, logGi, av, choice)
Log of the choice probability for a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• logGi – a dictionary mapping each alternative id with the function
$\ln \frac{\partial G}{\partial y_i}(e^{V_1},\ldots,e^{V_J})$
where $$G$$ is the MEV generating function. If an alternative
$$i$$ is not available, then $$G_i = 0$$.
Parameters
• av (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of the choice probability of the MEV model, given by
$V_i + \ln G_i(e^{V_1},\ldots,e^{V_J}) - \ln\left(\sum_j e^{V_j + \ln G_j(e^{V_1}, \ldots,e^{V_J})}\right)$
biogeme.models.logmev_endogenousSampling(V, logGi, av, correction, choice)
Log of choice probability for a MEV model, including the
correction for endogenous sampling as proposed by Bierlaire, Bolduc and McFadden (2008).
Parameters
• V – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• logGi – a dictionary mapping each alternative id with the function
$\ln \frac{\partial G}{\partial y_i}(e^{V_1}, \ldots, e^{V_J})$
where $$G$$ is the MEV generating function. If an alternative
$$i$$ is not available, then $$G_i = 0$$.
Parameters
• av (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• correction (dict(int:biogeme.expressions.Expression)) – a dict of expressions for the correstion terms of each alternative.
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of the choice probability of the MEV model, given by
$V_i + \ln G_i(e^{V_1}, \ldots,e^{V_J}) + \omega_i - \ln\left(\sum_j e^{V_j + \ln G_j(e^{V_1}, \ldots, e^{V_J})+ \omega_j}\right)$
where $$\omega_i$$ is the correction term for alternative $$i$$.
biogeme.models.lognested(V, availability, nests, choice)
Implements the log of a nested logit model as a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
A tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives belonging to the nest.
Example:
nesta = MUA, [1, 2, 3]
nestb = MUB, [4, 5, 6]
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of choice probability for the nested logit model, based on the derivatives of the MEV generating function produced by the function getMevForNested
biogeme.models.lognestedMevMu(V, availability, nests, choice, mu)
Implements the log of the nested logit model as a MEV model, where
mu is also a parameter, if the user wants to test different normalization schemes.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
A tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions.Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives belonging to the nest.
Example:
nesta = MUA, [1, 2, 3]
nestb = MUB, [4, 5, 6]
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
• mu (biogeme.expressions.Expression) – expression producing the value of the top-level scale parameter.
Returns
the log of the nested logit choice probability based on the following derivatives of the MEV generating function:
$\frac{\partial G}{\partial y_i}(e^{V_1},\ldots,e^{V_J}) = \mu e^{(\mu_m-1)V_i} \left(\sum_{i=1}^{J_m} e^{\mu_m V_i}\right)^{\frac{\mu}{\mu_m}-1}$
where $$m$$ is the (only) nest containing alternative $$i$$, and $$G$$ is the MEV generating function.
Return type
biogeme.expressions.Expression
biogeme.models.mev(V, logGi, av, choice)
Choice probability for a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• logGi – a dictionary mapping each alternative id with the function
$\ln \frac{\partial G}{\partial y_i}(e^{V_1}, \ldots, e^{V_J})$
where $$G$$ is the MEV generating function. If an alternative
$$i$$ is not available, then $$G_i = 0$$.
Parameters
• av (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
Choice probability of the MEV model, given by
$\frac{e^{V_i + \ln G_i(e^{V_1}, \ldots,e^{V_J})}}{\sum_j e^{V_j + \ln G_j(e^{V_1},\ldots,e^{V_J})}}$
biogeme.models.mev_endogenousSampling(V, logGi, av, correction, choice)
Choice probability for a MEV model, including the correction
for endogenous sampling as proposed by Bierlaire, Bolduc and McFadden (2008).
Parameters
• V – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• logGi – a dictionary mapping each alternative id with the function
$\ln \frac{\partial G}{\partial y_i}(e^{V_1}, \ldots, e^{V_J})$
where $$G$$ is the MEV generating function. If an alternative
$$i$$ is not available, then $$G_i = 0$$.
Parameters
• av (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• correction (dict(int:biogeme.expressions.Expression)) – a dict of expressions for the correstion terms of each alternative.
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
log of the choice probability of the MEV model, given by
$V_i + \ln G_i(e^{V_1}, \ldots, e^{V_J}) + \omega_i - \ln\left(\sum_j e^{V_j + \ln G_j(e^{V_1},\ldots,e^{V_J})+ \omega_j}\right)$
where $$\omega_i$$ is the correction term for alternative $$i$$.
biogeme.models.nested(V, availability, nests, choice)
Implements the nested logit model as a MEV model.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative, indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
A tuple containing as many items as nests. Each item is also
a tuple containing two items:
• an object of type biogeme.expressions. Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives belonging to the nest.
Example:
nesta = MUA, [1, 2, 3]
nestb = MUB, [4, 5, 6]
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
Returns
choice probability for the nested logit model, based on the derivatives of the MEV generating function produced by the function getMevForNested
biogeme.models.nestedMevMu(V, availability, nests, choice, mu)
Implements the nested logit model as a MEV model, where mu is also a parameter, if the user wants to test different normalization schemes.
Parameters
• V (dict(int:biogeme.expressions.Expression)) – dict of objects representing the utility functions of each alternative, indexed by numerical ids.
• availability (dict(int:biogeme.expressions.Expression)) – dict of objects representing the availability of each alternative ($$a_i$$ in the above formula), indexed by numerical ids. Must be consistent with V, or None. In this case, all alternatives are supposed to be always available.
• nests (tuple) –
A tuple containing as many items as nests.
Each item is also a tuple containing two items:
• an object of type biogeme.expressions.Expression representing the nest parameter,
• a list containing the list of identifiers of the alternatives
belonging to the nest.
Example:
nesta = MUA ,[1, 2, 3]
nestb = MUB ,[4, 5, 6]
nests = nesta, nestb
• choice (biogeme.expressions.Expression) – id of the alternative for which the probability must be calculated.
• mu (biogeme.expressions.Expression) – expression producing the value of the top-level scale parameter.
Returns
the nested logit choice probability based on the following derivatives of the MEV generating function:
$\frac{\partial G}{\partial y_i}(e^{V_1},\ldots,e^{V_J}) = \mu e^{(\mu_m-1)V_i} \left(\sum_{i=1}^{J_m} e^{\mu_m V_i}\right)^{\frac{\mu}{\mu_m}-1}$
Where $$m$$ is the (only) nest containing alternative $$i$$, and $$G$$ is the MEV generating function.
Return type
biogeme.expressions.Expression
biogeme.models.piecewise(variable, thresholds)
Obsolete function. Present for compatibility only
biogeme.models.piecewiseFormula(variable, thresholds, initialBetas=None)
Generate the formula for a piecewise linear specification.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to $$-\infty$$ and $$+\infty$$, respectively. If $$t$$ is the variable of interest, for each interval $$[a:a+b[$$, we define a variable defined as:
$\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}$
New variables and new parameters are automatically created.
Parameters
• variable – variable for which we need the piecewise linear transform.
• thresholds (list(float)) – list of thresholds
• initialBetas (list(float)) – list of values to initialize the beta parameters. The number of entries should be the number of thresholds, plus one. If None, the value of zero is used. Default: none.
Returns
expression of the piecewise linear specification.
Return type
biogeme.expressions.Expression
biogeme.models.piecewiseFunction(x, thresholds, betas)
Plot a piecewise linear specification.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to $$-\infty$$ and $$+\infty$$, respectively. If $$t$$ is the variable of interest, for each interval $$[a:a+b[$$, we define a variable defined as:
$\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}$
Parameters
• x (float) – value at which the piecewise specification must be avaluated
• thresholds (list(float)) – list of thresholds
• betas (list(float)) – list of the beta parameters. The number of entries should be the number of thresholds, plus one.
Returns
value of the numpy function
Return type
float
biogeme.models.piecewiseVariables(variable, thresholds)
Generate the variables to include in a piecewise linear specification.
If there are K thresholds, K-1 variables are generated. The first and last thresholds can be defined as None, corresponding to $$-\infty$$ and $$+\infty$$,respectively. If $$t$$ is the variable of interest, for each interval $$[a:a+b[$$, we define a variable defined as:
$\begin{split}x_{Ti} =\left\{ \begin{array}{ll} 0 & \text{if } t < a \\ t-a & \text{if } a \leq t < a+b \\ b & \text{otherwise} \end{array}\right. \;\;\;x_{Ti} = \max(0, \min(t-a, b))\end{split}$
Parameters
• variable (biogeme.expressions.Expression or str) – variable for which we need the piecewise linear transform. The expression itself or the name of the variable can be given.
• thresholds (list(float)) – list of thresholds
Returns
list of variables to for the piecewise linear specification.
Return type
piecewiseFormula
biogeme.optimization module¶
Interface for the optimization algorithms.
author
Michel Bierlaire
date
Sun Apr 5 16:48:54 2020
biogeme.optimization.bfgs(H, d, y)
Update the BFGS matrix. Formula (13.12) of Bierlaire (2015)
where the method proposed by Powell (1977) is applied
Parameters
• H (numpy.array (2D)) – current approximation of the inverse of the Hessian
• d (numpy.array (1D)) – difference between two consecutive iterates.
• y (numpy.array (1D)) – difference between two consecutive gradients.
Returns
updated approximation of the inverse of the Hessian.
Return type
numpy.array (2D)
biogeme.optimization.bfgsLineSearch(fct, x0, initBfgs=None, eps=6.06273418136464e-06, maxiter=1000)
BFGS method with inexact line search (Wolfe conditions)
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• x0 (numpy.array) – starting point
• initBfgs (numpy.array) – matrix used to initialize BFGS. If None, the identity matrix is used. Default: None.
• eps (float) – the algorithm stops when this precision is reached. Default: $$\varepsilon^{\frac{1}{3}}$$
• maxiter (int) – the algorithm stops if this number of iterations is reached. Default: 1000
Returns
tuple x, messages, where
• x is the solution found,
• messages is a dictionary reporting various aspects related to the run of the
algorithm.
Return type
numpy.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if the dimensions of the matrix initBfgs do not match the length of x0.
biogeme.optimization.bfgsLineSearchForBiogeme(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on BFGS quasi-Newton method with LS.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the optimization routine:
• tolerance: when the relative gradient is below that threshold, the algorithm has
reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• maxiter: the maximum number of iterations (default: 100).
• initBfgs: the positive definite matrix that initalizes the BFGS updates. If None,
the identity matrix is used. Default: None.
Returns
tuple x, messages, where
• x is the solution found,
• messages is a dictionary reporting various aspects related to the run of the algorithm.
Return type
numpy.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if bounds are imposed on the variables.
biogeme.optimization.bfgsTrustRegion(fct, x0, initBfgs=None, delta0=1.0, eps=6.06273418136464e-06, dl=False, maxiter=1000, eta1=0.01, eta2=0.9)
BFGS method with trust region
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• x0 (numpy.array) – starting point
• initBfgs (numpy.array) – matrix used to initialize BFGS. If None, the identity matrix is used. Default: None.
• delta0 (float) – initial radius of the trust region. Default: 100.
• eps (float) – the algorithm stops when this precision is reached. Default: $$\varepsilon^{\frac{1}{3}}$$
• dl (bool) – If True, the Dogleg method is used to solve the trut region subproblem. If False, the truncated conjugate gradient is used. Default: False.
• maxiter (int) – the algorithm stops if this number of iterations is reached. Default: 1000.
• eta1 (float) – threshold for failed iterations. Default: 0.01.
• eta2 (float) – threshold for very successful iterations. Default 0.9.
Returns
tuple x, messages, where
• x is the solution found,
• messages is a dictionary reporting various aspects related to the run of the algorithm.
Return type
numpy.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if the dimensions of the matrix initBfgs do not match the length of x0.
biogeme.optimization.bfgsTrustRegionForBiogeme(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on Newton method with TR.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the optimization routine:
• tolerance: when the relative gradient is below that threshold, the algorithm has
reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• maxiter: the maximum number of iterations (default: 100).
• dogleg: if True, the trust region subproblem is solved using
the Dogleg method. If False, it is solved using the truncated conjugate gradient method (default: False).
• initBfgs: the positive definite matrix that initalizes the BFGS updates. If None,
the identity matrix is used. Default: None.
Returns
tuple x, messages, where
• x is the solution found,
• messages is a dictionary reporting various aspects
related to the run of the algorithm.
Return type
numpy.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if bounds are imposed on the variables.
biogeme.optimization.bioBfgs(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on BFGS quasi-Newton method with simple
bounds.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the
optimization routine:
• tolerance: when the relative gradient is below that threshold,
the algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• cgtolerance: when the norm of the residual is below that threshold, the conjugate gradient algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• infeasibleConjugateGradient: if True, the conjugate gradient algorithm may generate until termination. The result will then be projected on the feasible domain. If False, the algorithm stops as soon as an infeasible iterate is generated (default: False).
• maxiter: the maximum number of iterations (default: 1000).
• eta1: threshold for failed iterations (default: 0.01).
• eta2: threshold for very successful iteration (default 0.9).
• enlargingFactor: factor used to enlarge the trust region
during very successful iterations (default 10).
• hamabs: if True, a starting point is calculated using sotchastic Newton (default: False).
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing information about the lagorithm
Return type
numpay.array, dict(str:object)
class biogeme.optimization.bioBounds(b)
Bases: object
This class is designed for the management of simple bound constraints
__init__(b)
Parameters
b (list(tuple)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter.
Raises
biogeme.exceptions.biogemeError – if the bounds are incompatible
activity(x, epsilon=2.220446049250313e-16)
Determines the activity status of each variable.
Parameters
• x (numpy.array) – point for which the activity must be determined.
• epsilon (float) – a bound is considered active if the distance to it is less rhan epsilon.
Returns
a vector, same length as x, where each entry reports the activity of the
corresponding variable:
• 0 if no bound is active
• -1 if the lower bound is active
• 1 if the upper bound is active
Raises
breakpoints(x, d)
Projects the direction d, starting from x, on the intersection of the bound constraints
Parameters
• x (numpy.array) – current point
• d (numpy.array) – search direction
Returns
list of tuple (index, value), where index is the index of the variable, and value the value of the corresponding breakpoint.
Return type
list(tuple(int,float))
Raises
feasible(x)
Check if point verifies the bound constraints
Parameters
x (numpy.array) – point to project
Returns
True if x is feasible, False otherwise.
Return type
bool
Raises
biogeme.exceptions.biogemeError – if the dimensions are inconsistent
generalizedCauchyPoint(xk, gk, H, direction)
Implementation of Step 2 of the Specific Algorithm by Conn et al. (1988).
The quadratic model is defined as
$m(x) = f(x_k) + (x - x_k)^T g_k + \frac{1}{2} (x-x_k)^T H (x-x_k).$
Parameters
• xk (numpy.array. Dimension n.) – current point
• gk (numpy.array. Dimension n.) – vector g involved in the quadratic model definition.
• H (numpy.array. Dimension n x n.) – matrix H involved in the quadratic model definition.
Returns
generalized Cauchy point based on inexact line search.
Return type
numpy.array. Dimension n.
Raises
intersect(otherBounds)
Create a bounds object representing the intersection of two regions.
Parameters
otherBounds (class bioBounds) – other bound object that must be intersected.
Returns
bound object, intersection of the two.
Return type
class bioBounds
Raises
biogeme.exceptions.biogemeError – if the dimensions are inconsistent
intersectionWithTrustRegion(x, delta)
Create a bioBounds object representing the intersection between the feasible domain
and the trust region.
Parameters
• x (numpy.array) – center of the trust region
• delta (float) – radius of the tust region (infinity norm)
Raises
biogeme.exceptions.biogemeError – if the dimensions are inconsistent
maximumStep(x, d)
Calculates the maximum step thatcan be performed along a direction while staying feasible.
Parameters
• x (numpy.array) – reference point
• d (numpy.array) – direction
Returns
the largest alpha such that x + alpha * d is feasible and the list of indices achieving this value.
Return type
float, int
Raises
biogeme.exceptions.biogemeError – if the point is infeasible
project(x)
Project a point onto the feasible domain defined by the bounds.
Parameters
x (numpy.array) – point to project
Returns
projected point
Return type
numpy.array
Raises
biogeme.exceptions.biogemeError – if the dimensions are inconsistent
subspace(selectedVariables)
Generate a bioBounds object for selected variables
Parameters
• selectedVariables (numpy.array(bool)) – boolean vector. If an entry is True, the corresponding variables is considered.
• x – center of the trust region
• delta (float) – radius of the trust region (in infinity norm)
Type
numpy.array
Returns
bound object
Return type
class bioBounds
Raises
biogeme.exceptions.biogemeError – if the dimensions are inconsistent
biogeme.optimization.bioNewton(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on Newton’s method with simple
bounds.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the
optimization routine:
• tolerance: when the relative gradient is below that threshold,
the algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• cgtolerance: when the norm of the residual is below that threshold, the conjugate gradient algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• infeasibleConjugateGradient: if True, the conjugate gradient algorithm may generate until termination. The result will then be projected on the feasible domain. If False, the algorithm stops as soon as an infeasible iterate is generated (default: False).
• maxiter: the maximum number of iterations (default: 1000).
• eta1: threshold for failed iterations (default: 0.01).
• eta2: threshold for very successful iteration (default 0.9).
• enlargingFactor: factor used to enlarge the trust region
during very successful iterations (default 10).
• hamabs: if True, a starting point is calculated using sotchastic Newton (default: False).
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing information about the lagorithm
Return type
numpay.array, dict(str:object)
biogeme.optimization.cauchyNewtonDogleg(g, H)
Calculate the Cauchy, the Newton and the dogleg points.
The Cauchy point is defined as
$d_c = - \frac{\nabla f(x)^T \nabla f(x)}{\nabla f(x)^T \nabla^2 f(x) \nabla f(x)} \nabla f(x)$
The Newton point $$d_n$$ verifies Newton equation:
$H_s d_n = - \nabla f(x)$
where $$H_s$$ is a positive definite matrix generated with the method by
The Dogleg point is
$d_d = \eta d_n$
where
$\eta = 0.2 + 0.8 \frac{\alpha^2}{\beta |\nabla f(x)^T d_n|}$
and $$\alpha= \nabla f(x)^T \nabla f(x)$$, $$\beta=\nabla f(x)^T \nabla^2 f(x)\nabla f(x)$$
Parameters
• g (numpy.array) – gradient $$\nabla f(x)$$
• H (numpy.array) – hessian $$\nabla^2 f(x)$$
Returns
tuple with Cauchy point, Newton point, Dogleg point
Return type
numpy.array, numpy.array, numpy.array
Raises
biogeme.exceptions.biogemeError – if the quadratic model is not convex.
biogeme.optimization.dogleg(g, H, delta)
Find an approximation of the trust region subproblem using the dogleg method
Parameters
• H (numpy.array) – hessian of the quadratic model.
• delta (float) – radius of the trust region.
Returns
d, diagnostic where
• d is an approximate solution of the trust region subproblem
• diagnostic is the nature of the solution:
• -2 if negative curvature along Newton direction
• -1 if negative curvature along Cauchy direction (i.e. along the gradient)
• 1 if partial Cauchy step
• 2 if Newton step
• 3 if partial Newton step
• 4 if Dogleg
Return type
numpy.array, int
class biogeme.optimization.functionToMinimize
Bases: object
This is an abstract class. The actual function to minimize must be implemented in a concrete class deriving from this one.
abstract f(batch=None)
Calculate the value of the function
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for thre random batch. If None, the full data set is used. Default: None pass
Returns
value of the function
Return type
float
abstract f_g(batch=None)
Calculate the value of the function and the gradient
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for the random batch. If None, the full data set is used. Default: None pass
Returns
value of the function and the gradient
Return type
tuple float, numpy.array
abstract f_g_bhhh(batch=None)
Calculate the value of the function, the gradient and the BHHH matrix
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for the random batch. If None, the full data set is used. Default: None pass
Returns
value of the function, the gradient and the BHHH
Return type
tuple float, numpy.array, numpy.array
abstract f_g_h(batch=None)
Calculate the value of the function, the gradient and the Hessian
Parameters
batch (float) – for data driven functions (such as a log likelikood function), it is possible to approximate the value of the function using a sample of the data called a batch. This argument is a value between 0 and 1 representing the percentage of the data that should be used for the random batch. If None, the full data set is used. Default: None pass
Returns
value of the function, the gradient and the Hessian
Return type
tuple float, numpy.array, numpy.array
abstract setVariables(x)
Set the values of the variables for which the function has to b calculated.
Parameters
x (numpy.array) – values
biogeme.optimization.inverseBfgs(Hinv, d, y)
Update the inverse BFGS matrix. Formula (13.13) of Bierlaire (2015)
Parameters
• Hinv (numpy.array (2D)) – current approximation of the inverse of the Hessian
• d (numpy.array (1D)) – difference between two consecutive iterates.
• y (numpy.array (1D)) – difference between two consecutive gradients.
Returns
updated approximation of the inverse of the Hessian.
Return type
numpy.array (2D)
biogeme.optimization.lineSearch(fct, x, f, g, d, alpha0=1.0, beta1=0.0001, beta2=0.99, lbd=2.0)
Calculate a step along a direction that satisfies both Wolfe conditions
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• x (numpy.array) – current iterate.
• d (numpy.array) – descent direction.
• alpha0 (float) – first step to test.
• beta1 (float) – parameter of the first Wolfe condition.
• beta2 (float) – parameter of the second Wolfe condition.
• lbd (float) – expansion factor for a short step.
Returns
a step verifing both Wolfe conditions
Return type
float
Raises
biogeme.optimization.newtonLineSearch(fct, x0, eps=6.06273418136464e-06, maxiter=100)
Newton method with inexact line search (Wolfe conditions)
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• x0 (numpy.array) – starting point
• eps (float) – the algorithm stops when this precision is reached. Default: $$\varepsilon^{\frac{1}{3}}$$
• maxiter (int) – the algorithm stops if this number of iterations is reached. Defaut: 100
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing information about the lagorithm
Return type
numpay.array, dict(str:object)
biogeme.optimization.newtonLineSearchForBiogeme(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on Newton method.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the optimization routine:
• tolerance: when the relative gradient is below that threshold, the algorithm has
reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• maxiter: the maximum number of iterations (default: 100).
Returns
tuple x, nit, nfev, message, where
• x is the solution found,
• messages is a dictionary reporting various aspects related to the run of the
algorithm.
Return type
numpy.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if bounds are imposed on the variables.
biogeme.optimization.newtonTrustRegion(fct, x0, delta0=1.0, eps=6.06273418136464e-06, dl=False, maxiter=1000, eta1=0.01, eta2=0.9)
Newton method with trust region
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• x0 (numpy.array) – starting point
• delta0 (float) – initial radius of the trust region. Default: 100.
• eps (float) – the algorithm stops when this precision is reached. Default: $$\varepsilon^{\frac{1}{3}}$$
• dl (bool) – If True, the Dogleg method is used to solve the trut region subproblem. If False, the truncated conjugate gradient is used. Default: False.
• maxiter (int) – the algorithm stops if this number of iterations is reached. Default: 1000.
• eta1 (float) – threshold for failed iterations. Default: 0.01.
• eta2 (float) – threshold for very successful iterations. Default 0.9.
Returns
tuple x, messages, where
• x is the solution found,
• messages is a dictionary reporting various aspects related to the run of the algorithm.
Return type
numpy.array, dict(str:object)
biogeme.optimization.newtonTrustRegionForBiogeme(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on Newton method with TR.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell, u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the
optimization routine:
• tolerance: when the relative gradient is below that threshold, the algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• maxiter: the maximum number of iterations (default: 100).
• dogleg: if True, the trust region subproblem is solved using
the Dogleg method. If False, it is solved using the truncated conjugate gradient method (default: False).
Returns
tuple x, messages, where
• x is the solution found,
• messages is a dictionary reporting various aspects related to the run of the algorithm.
Return type
numpy.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if bounds are imposed on the variables.
biogeme.optimization.relativeGradient(x, f, g, typx, typf)
It is typically used for stopping criteria.
Parameters
• x (numpy.array) – current iterate.
• f (float) – value of f(x)
• g (numpy.array) – $$\nabla f(x)$$, gradient of f at x
• typx (numpy.array) – typical value for x.
• typf (float) – typical value for f.
Returns
$\max_{i=1,\ldots,n}\frac{(\nabla f(x))_i \max(x_i,\text{typx}_i)} {\max(|f(x)|, \text{typf})}$
Return type
float
biogeme.optimization.schnabelEskow(A, tau=6.06273418136464e-06, taubar=3.6756745753887175e-11, mu=0.1)
Modified Cholesky factorization by Schnabel and Eskow (1999).
If the matrix is ‘safely’ positive definite, the output is the classical Cholesky factor. If not, the diagonal elements are inflated in order to make it positive definite. The factor $$L$$ is such that $$A + E = PLL^TP^T$$, where $$E$$ is a diagonal matrix contaninig the terms added to the diagonal, $$P$$ is a permutation matrix, and $$L$$ is w lower triangular matrix.
Parameters
Returns
tuple $$L$$, $$E$$, $$P$$, where $$A + E = PLL^TP^T$$.
Return type
numpy.array, numpy.array, numpy.array
Raises
biogeme.optimization.scipy(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on the scipy minimize function.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the beta parameters
• bounds (list(tuple)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter
• parameters – dict of parameters to be transmitted to the optimization routine. See the scipy documentation.
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing several information about the lagorithm
Return type
numpay.array, dict(str:object)
biogeme.optimization.simpleBoundsNewtonAlgorithm(fct, bounds, x0, proportionTrueHessian=1.0, infeasibleConjugateGradient=False, delta0=1.0, tol=6.06273418136464e-06, cgtol=6.06273418136464e-06, maxiter=1000, eta1=0.01, eta2=0.9, enlargingFactor=10, hamabs=False)
Trust region algorithm for problems with simple bounds
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• bounds (class bounds) – bounds on the variables
• x0 (numpy.array) – starting point
• proportionTrueHessian (float) – proportion of the iterations where the true hessian is calculated. When not, the BFGS update is used. If 1.0, it is used for all iterations. If 0.0, it is not used at all.
• infeasibleConjugateGradient (bool) – if True, the conjugate gradient algorithm may generate until termination. The result will then be projected on the feasible domain. If False, the algorithm stops as soon as an infeasible iterate is generated. Default: False.
• delta0 (float) – initial radius of the trust region. Default: 100.
• tol (float) – the algorithm stops when this precision is reached. Default: $$\varepsilon^{\frac{1}{3}}$$
• cgtol (float) – the conjugate gradient algorithm stops when this precision is reached. Default: $$\varepsilon^{\frac{1}{3}}$$
• maxiter (int) – the algorithm stops if this number of iterations is reached. Default: 1000.
• eta1 (float) – threshold for failed iterations. Default: 0.01.
• eta2 (float) – threshold for very successful iterations. Default 0.9.
• enlargingFactor (float) – if an iteration is very successful, the radius of the trust region is multiplied by this factor. Default 10.
• hamabs – if True, a stochastic Newton algorithm is applied to find a starting point for the estimation process. This is particularly designed for estimation with large data sets. It is experimental, and inspired by the algorithm proposed by Lederrey et al. (2019)_.
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing information about the lagorithm
Return type
numpay.array, dict(str:object)
Raises
biogeme.exceptions.biogemeError – if the dimensions of the matrix initBfgs do not match the length of x0.
biogeme.optimization.simpleBoundsNewtonAlgorithmForBiogeme(fct, initBetas, bounds, parameters=None)
Optimization interface for Biogeme, based on variants of Newton method with simple
bounds.
Parameters
• fct (optimization.functionToMinimize) – object to calculate the objective function and its derivatives.
• initBetas (numpy.array) – initial value of the parameters.
• bounds (list(tuples)) – list of tuples (ell,u) containing the lower and upper bounds for each free parameter. Note that this algorithm does not support bound constraints. Therefore, all the bounds must be None.
• parameters (dict(string:float or int)) –
dict of parameters to be transmitted to the
optimization routine:
• tolerance: when the relative gradient is below that threshold,
the algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• cgtolerance: when the norm of the residual is below that threshold, the conjugate gradient algorithm has reached convergence (default: $$\varepsilon^{\frac{1}{3}}$$);
• proportionAnalyticalHessian: proportion of iterations when the analytical Hessian is calculated (default: 0).
• infeasibleConjugateGradient: if True, the conjugate gradient algorithm may generate until termination. The result will then be projected on the feasible domain. If False, the algorithm stops as soon as an infeasible iterate is generated (default: False).
• maxiter: the maximum number of iterations (default: 1000).
• eta1: threshold for failed iterations (default: 0.01).
• eta2: threshold for very successful iteration (default 0.9).
• enlargingFactor: factor used to enlarge the trust region
during very successful iterations (default 10).
• hamabs: if True, a starting point is calculated using sotchastic Newton (default: False).
Returns
x, messages
• x is the solution generated by the algorithm,
• messages is a dictionary describing information about the lagorithm
Return type
numpay.array, dict(str:object)
biogeme.optimization.truncatedConjugateGradient(g, H, delta)
Find an approximation of the trust region subproblem using the truncated conjugate gradient method
Parameters
• H (numpy.array) – hessian of the quadrartic model.
• delta (float) – radius of the trust region.
Returns
d, diagnostic, where
• d is the approximate solution of the trust region subproblem,
• diagnostic is the nature of the solution:
• 1 for convergence,
• 2 if out of the trust region,
• 3 if negative curvature detected.
• 4 if a numerical problem has been encountered
Return type
numpy.array, int
biogeme.optimization.truncatedConjugateGradientSubspace(xk, gk, Hk, delta, bounds, infeasibleIterate=False, tol=6.06273418136464e-06)
Find an approximation of the solution of the trust region subproblem using the truncated
conjugate gradient method within the subspace of free variables. Free variables are those corresponding to inactive constraints at the generalized Cauchy point.
Parameters
• H (numpy.array) – hessian of the quadrartic model.
• delta (float) – radius of the trust region.
• bounds (class bioBounds) – bounds on the variables.
• infeasibleIterate (bool) – if True, the algorithm may generate until termination. The result will then be projected on the feasible domain. If False, the algorithm stops as soon as an infeasible iterate is generated. Default: False.
Returns
d, diagnostic, where
• d is the approximate solution of the trust region subproblem,
• diagnostic is the nature of the solution:
• 1 for convergence,
• 2 if out of the trust region,
• 3 if negative curvature detected.
• 4 if a numerical problem has been encountered
Return type
numpy.array, int
Raises
biogeme.exceptions.biogemeError – if the dimensions are inconsistent
biogeme.optimization.trustRegionIntersection(dc, d, delta)
Calculates the intersection with the boundary of the trust region.
Consider a trust region of radius $$\delta$$, centered at $$\hat{x}$$. Let $$x_c$$ be in the trust region, and $$d_c = x_c - \hat{x}$$, so that $$\|d_c\| \leq \delta$$. Let $$x_d$$ be out of the trust region, and $$d_d = x_d - \hat{x}$$, so that $$\|d_d\| \geq \delta$$. We calculate $$\lambda$$ such that
$\| d_c + \lambda (d_d - d_c)\| = \delta$
Parameters
• dc (numpy.array) – xc - xhat.
• d (numpy.array) – dd - dc.
• delta (float) – radius of the trust region.
Returns
$$\lambda$$ such that $$\| d_c + \lambda (d_d - d_c)\| = \delta$$
Return type
float
biogeme.results module¶
Implementation of class contaning and processing the estimation results.
author
Michel Bierlaire
date
Tue Mar 26 16:50:01 2019
… todo:: rawResults should be a dict and not a class.
class biogeme.results.beta(name, value, bounds)
Bases: object
Class gathering the information related to the parameters of the model
__init__(name, value, bounds)
Constructor
Parameters
• name (string) – name of the parameter.
• value (float) – value of the parameter.
• bounds (float,float) – tuple (l,b) with lower and upper bounds
isBoundActive(threshold=1e-06)
Check if one of the two bound is ‘numerically’ active.
Being numerically active means that the distance between the value of the parameter and one of its bounds is below the threshold.
Parameters
threshold – distance below which the bound is considered to be active. Default: $$10^{-6}$$
Returns
True is one of the two bounds is numericall y active.
Return type
bool
setBootstrapStdErr(se)
Records the robust standard error calculated by bootstrap, and calculates and
records the corresponding t-statistic and p-value
Parameters
se (float) – standard error calculated by bootstrap.
setRobustStdErr(se)
Records the robust standard error, and calculates and records
the corresponding t-statistic and p-value
Parameters
se (float) – robust standard error
setStdErr(se)
Records the standard error, and calculates and records
the corresponding t-statistic and p-value
Parameters
se (float) – standard error.
class biogeme.results.bioResults(theRawResults=None, pickleFile=None)
Bases: object
Class managing the estimation results
__init__(theRawResults=None, pickleFile=None)
Constructor
Parameters
• theRawResults (biogeme.results.rawResults) – object with the results of the estimation. Default: None.
• pickleFile (string) – name of the file containing the raw results in pickle format. Default: None.
Raises
biogeme.exceptions.biogemeError – if no data is provided.
getBetaValues(myBetas=None)
Retrieve the values of the estimated parameters, by names.
Parameters
myBetas (list(string)) – names of the requested parameters. If None, all available parameters will be reported. Default: None.
Returns
dict containing the values, where the keys are the names.
Return type
dict(string:float)
Raises
biogeme.exceptions.biogemeError – if some requested parameters are not available.
getBetasForSensitivityAnalysis(myBetas, size=100, useBootstrap=False)
Generate draws from the distribution of the estimates, for sensitivity analysis.
Parameters
• myBetas (list(string)) – names of the parameters for which draws are requested.
• size (int) – number of draws. Default: 100.
• useBootstrap (bool) – if True, the variance-covariance matrix generated by the bootstrapping is used for simulation. If False, the robust variance-covariance matrix is used. Default: False.
Raises
biogeme.exceptions.biogemeError – if useBootstrap is True and the bootstrap matrix is not available.
Returns
numpy table with as many rows as draws, and as many columns as parameters.
Return type
numpy.array
getBootstrapVarCovar()
Obtain the bootstrap variance covariance matrix as a Pandas data frame.
Returns
bootstrap variance covariance matrix, or None if not available
Return type
pandas.DataFrame
getCorrelationResults()
Get the statistics about pairs of coefficients as a Pandas dataframe
Returns
Pandas data frame with the correlation results
Rtpye
pandas.DataFrame
getEstimatedParameters()
Gather the estimated parameters and the corresponding statistics in a Pandas dataframe.
return
Pandas dataframe with the results
rtype
pandas.DataFrame
getGeneralStatistics()
Format the results in a dict
Returns
dict with the results. The keys describe each content. Each element is a tuple, with the value and its preferred formatting.
Example:
'Init log likelihood': (-115.30029248549191, '.7g')
Return type
dict(string:float,string)
getHtml()
Get the results coded in HTML
Returns
HTML code
Rtpye
string
getLaTeX()
Get the results coded in LaTeX
Returns
LaTeX code
Return type
string
getRobustVarCovar()
Obtain the robust variance covariance matrix as a Pandas data frame.
Returns
robust variance covariance matrix
Return type
pandas.DataFrame
getVarCovar()
Obtain the Rao-Cramer variance covariance matrix as a Pandas data frame.
Returns
Rao-Cramer variance covariance matrix
Return type
pandas.DataFrame
writeHtml()
Write the results in an HTML file.
writeLaTeX()
Write the results in a LaTeX file.
writePickle()
Dump the data in a file in pickle format.
Returns
name of the file.
Return type
string
biogeme.results.calcPValue(t)
Calculates the p value of a parameter from its t-statistic.
The formula is
$2(1-\Phi(|t|)$
where $$\Phi(\cdot)$$ is the CDF of a normal distribution.
Parameters
t – t-statistics
Type
float
Returns
p-value
Return type
float
class biogeme.results.rawResults(theModel, betaValues, fgHb, bootstrap=None)
Bases: object
Class containing the raw results from the estimation
__init__(theModel, betaValues, fgHb, bootstrap=None)
Constructor
Parameters
• theModel (biogeme.BIOGEME) – object with the model
• betaValues (list(float)) – list containing the estimated values of the parameters
• fgHb (float,numpy.array, numpy.array, numpy.array) –
tuple f,g,H,bhhh containing
• f: the value of the function,
• H: the second derivative matrix,
• bhhh: the BHHH matrix.
• bootstrap (numpy.array) –
output of the bootstrapping. numpy array, of size B x K, where
• B is the number of bootstrap iterations
• K is the number of parameters to estimate
Default: None.
biogeme.tools module¶
Implements some useful functions
author
Michel Bierlaire
date
Sun Apr 14 10:46:10 2019
biogeme.tools.calculatePrimeNumbers(upperBound)
Calculate prime numbers
Parameters
upperBound (int) – prime numbers up to this value will be computed
Returns
array with prime numbers
Return type
list(int)
biogeme.tools.checkDerivatives(theFunction, x, names=None, logg=False)
Verifies the analytical derivatives of a function by comparing them with finite
difference approximations.
Parameters
• theFunction (function) –
A function object that takes a vector as an argument, and
returns a tuple.
• The first element of the tuple is the value of the function $$f$$,
• the second is the gradient of the function,
• the third is the hessian.
• x (numpy.array) – arguments of the function
• names (list(string)) – the names of the entries of x (for reporting).
• logg (bool) – if True, messages will be displayed.
Returns
tuple f, g, h, gdiff, hdiff where
• f is the value of the function at x,
• g is the analytical gradient,
• h is the analytical hessian,
• gdiff is the difference between the analytical gradient and the finite
difference approximation
• hdiff is the difference between the analytical hessian and the finite
difference approximation
Return type
float, numpy.array,numpy.array, numpy.array,numpy.array
biogeme.tools.countNumberOfGroups(df, column)
This function counts the number of groups of same value in a column. For instance: 1,2,2,3,3,3,4,1,1 would give 5
biogeme.tools.findiff_H(theFunction, x)
Calculates the hessian of a function $$f$$ using finite differences
Parameters
• theFunction (function) – A function object that takes a vector as an argument, and returns a tuple. The first element of the tuple is the value of the function $$f$$, and the second is the gradient of the function. The other elements are not used.
• x (numpy.array) – argument of the function
Returns
numpy matrix containing the hessian calculated by finite differences.
Return type
numpy.array
biogeme.tools.findiff_g(theFunction, x)
Calculates the gradient of a function :mathf using finite differences
Parameters
• theFunction (function) – A function object that takes a vector as an argument, and returns a tuple. The first element of the tuple is the value of the function $$f$$. The other elements are not used.
• x (numpy.array) – argument of the function
Returns
numpy vector, same dimension as x, containing the gradient calculated by finite differences.
Return type
numpy.array
biogeme.tools.getPrimeNumbers(n)
Get a given number of prime numbers
Parameters
n (int) – number of primes that are requested
Returns
array with prime numbers
Return type
list(int)
biogeme.version module¶
Information about the version of Biogeme
author
Michel Bierlaire
date
Tue Mar 26 16:45:15 2019
biogeme.version.getHtml()
Package information in HTML format
Returns
HTML code.
Return type
string
biogeme.version.getLaTeX()
Package information in LaTeX format
Returns
Return type
string
biogeme.version.getText()
Package information in text format
Returns
package information
Return type
string
biogeme.version.getVersion()
Version of the software
Returns
version number, and the release.
Return type
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http://pld.fk.ui.ac.id/mqpbr/decay-constant-symbol-2481e0 | We can find the decay constant directly from Equation \ref{eq8}. Solution. 1,000,000 times stronger than those of the electronic and molecular forces. The radioactive decay constant is usually represented by the symbol λ. To determine the activity, we first need to find the number of nuclei present. The decay constant l is the probability that a nucleus will decay per second so its unit is s-1. The decay constant has dimensions of inverse time, and the SI unit of time is the second, so the units of the decay constant are inverse seconds (1/s). * mean lifetime - symbol τ - the average lifetime of any given particle. • mean lifetime — symbol τ — … Find (a) its decay constant and (b) the initial activity of 1.00 g of the material. This method will work with any of the symbols above, substituting the appropriate code before typing ALT+X. Potassium-40 (40 K) is a radioactive isotope of potassium which has a long half-life of 1.251 × 10 9 years. The probability to decay/time is termed the ”decay constant”, and is given the symbol λ. P = λ Δt If you have the Lucida Sans Unicode font available, this will type the equilibrium symbol without going to the insert symbol menu. The decay constant (symbol: λ and units: s −1 or a −1) of a radioactive nuclide is its probability of decay per unit time.The number of parent nuclides P therefore decreases with time t as dP/P dt = −λ. The Lambda logo (λ) is a symbol found frequently in the Half-Life universe. Strategy. For a particular decay mechanism, the radioactive decay constant for a nuclide is defined as the probability per unit time that a given nucleus of that nuclide will decay by that mechanism. A half-life is the time it takes for half of the nuclei to disappear. The decay rate, or activity, of a radioactive substance are characterized by: Constant quantities: * half life - symbol t1 / 2 - the time for half of a substance to decay. A = activity in becquerel (Bq) N = the number of undecayed nuclei l = decay constant (s-1) Radioactive decay law. The decay constant is found to be The half-life of strontium-90, $$\ce{_{38}^{90}Sr}$$, is 28.8 y. "Λ" is the 11th letter in the Greek alphabet. The symbol l = 1/t is known as the decay constant. It represents the Greek letter "Λ" (lowercase "λ"), and is a radioactive decay constant used in the half-life equation. λ ≡ the probabilty to decay per unit time (units of 1/time) From this assumption, one can ”derive” the half-life decay … activity = decay constant x the number of undecayed nuclei. The value of the decay constant depends on the nature of the particular decay process. * decay constant - symbol λ - … The definition may be expressed by the equation. 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Radioactive decay constant is usually represented by the symbol λ the appropriate code before ALT+X... | 2021-08-04 05:10:05 | {"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": 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": 0.9204402565956116, "perplexity": 998.1749878121295}, "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-2021-31/segments/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00632.warc.gz"} |
https://tex.meta.stackexchange.com/questions/4076/who-wants-to-win-a-duck-wait-what/4118 | # Introduction
Hello there, fellow TeX.sx users! I'm organizing a lottery. The idea is very similar to Who Wants to Win a LaTeX Book?, but both the lottery algorithm and the prize are different. Let me explain some things first.
## Who are you?
I'm some random dude on the Internet who happens to hang out a lot in this very community. :)
## What is the prize?
The prize will be a lovely duck.
## Wait a minute, did you say duck?
Yes! :) But not a real duck, it will be a hand puppet duck:
Lovely, isn't it?
## Why a duck? Why not a lion?
Good question. :) I always like to provide examples with ducks when I write answers. To my surprise, the theme got a very positive feedback from the community and it spread to everybody. Of course, a lion would be better, but hey, it's free! And it's a duck! :)
## Why a hand puppet?
I have absolutely no idea. But since I got this duck, I thought it would be a good idea to start a contest! :)
# Rules
• To win the duck you must guess a number in the range 1-200. The first person who guesses the right number gets the duck.
• Contestants may submit 2 valid guesses.
• Contestants must have been a member of TeX.sx before 21 December, 2013.
# The lottery
I'll use the following code to pick the winner:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{expl3}
\usepackage{xparse}
\usepackage{pgf}
\pgfmathsetseed{\number\pdfrandomseed}
\ExplSyntaxOn
\prop_new:N \g_duck_contest_prop
\int_new:N \g_duck_contest_upper_limit_int
\bool_new:N \l_duck_contest_winner_bool
\bool_set_false:N \l_duck_contest_winner_bool
\tl_new:N \l_duck_contest_winner_tl
\int_new:N \l_duck_contest_winning_number_int
\NewDocumentCommand { \person } { m m }
{
\clist_map_inline:nn { #1 }
{
\prop_if_in:NnTF \g_duck_contest_prop { ##1 }
{
Hey! ~ #2 ~ wants ~ to ~ be ~ naughty! ~
The ~ number ~ ##1 ~ was ~ already ~ chosen! \par
}
{
\prop_gput:Nnn \g_duck_contest_prop { ##1 } { #2 }
\int_gset:Nn \g_duck_contest_upper_limit_int
{
\int_max:nn { \g_duck_contest_upper_limit_int } { ##1 }
}
}
}
}
\NewDocumentCommand{ \winner } { }
{
\bool_do_until:Nn \l_duck_contest_winner_bool
{
\pgfmathrandom{1, \g_duck_contest_upper_limit_int}
\prop_get:NoN \g_duck_contest_prop { \pgfmathresult } \l_duck_contest_winner_tl
\quark_if_no_value:NF \l_duck_contest_winner_tl
{
\bool_set_true:N \l_duck_contest_winner_bool
}
}
\int_set_eq:NN \l_duck_contest_winning_number_int \pgfmathresult
\l_duck_contest_winner_tl
}
\cs_generate_variant:Nn \prop_get:NnN { No }
\NewDocumentCommand{ \winningnumber } { }
{
\int_to_arabic:n \l_duck_contest_winning_number_int
}
\ExplSyntaxOff
\begin{document}
% ==============================
% For example, in the following line
% I picked 1 and 27, but I'm not
% participating in the contest :)
% \person{1,27}{Paulo Cereda}
\person{137,143}{Marco Daniel}
\person{65,105}{Ethan Bolker}
\person{68,130}{giordano}
\person{123,77}{Werner}
\person{124,100}{Stiff Jokes}
\person{15,97}{David Carlisle}
\person{22,44}{egreg}
\person{42,37}{tohecz}
\person{108,69}{Count Zero}
\person{47,167}{ppr}
\person{26,2}{marczellm}
\person{3,30}{Papiro}
\person{131,144}{percusse}
\person{33,66}{Harish Kumar}
\person{1,200}{Przemysław Scherwentke}
\person{4,5}{Frank Mittelbach}
\person{23,177}{cgnieder}
\person{11,121}{texenthusiast}
\person{50,52}{Peter LeFanu Lumsdaine}
\person{14,29}{ricmarques}
\person{117,183}{Newb}
\person{82,88}{Francesco Endrici}
\person{122,171}{Claudio Fiandrino}
\person{17,21}{Dror}
\person{153,154}{topskip}
\person{74,147}{Sigur}
\person{193,38}{ComFreek}
\person{113,31}{Wayne Werner}
\person{6,7}{Andrew Stacey}
\person{12,171}{azetina}
\person{8,63}{Ignasi}
\person{28,99}{Andrea L.}
\person{25,116}{Benedikt Bauer}
\person{126,129}{Alan Munn}
\person{99,199}{lvaneesbeeck}
\person{197,53}{Pouya}
\person{20,125}{afrendeiro}
\person{45,152}{kan}
\person{114,115}{Philip}
\person{24,133}{doncherry}
\person{27,35}{Yori}
\person{79,186}{knut}
\person{12,173}{fifaltra}
\person{28,196}{Michael Hoppe}
\person{94,81}{XZS}
\person{10,172}{OSjerick}
\person{56,163}{laxxy}
% there was an entry to the code, but not as
% a proper answer, I'm sorry
% \person{8,9}{Joe Corneli}
% ==============================
% the announcement
The winner is \winner, with \winningnumber! Congratulations!
\end{document}
I'll run this code with all the contestants, post a video of it and announce the winner.
# The date
Hopefully, I'll get the code running on January 1st, 2014, and announce the winner. :)
Good luck! :) And by the way, welcome to TeX.sx, the friendliest and most awesome community in the whole StackExchange network! They give us hats, right? We give ducks!
----------------------------------------------------------------------------------
|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|009|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|016|◖■◗|018|019|◖■◗|
----------------------------------------------------------------------------------
|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|032|◖■◗|034|◖■◗|036|◖■◗|◖■◗|039|040|
----------------------------------------------------------------------------------
|041|◖■◗|043|◖■◗|◖■◗|046|◖■◗|048|049|◖■◗|051|◖■◗|◖■◗|054|055|◖■◗|057|058|059|060|
----------------------------------------------------------------------------------
|061|062|◖■◗|064|◖■◗|◖■◗|067|◖■◗|◖■◗|070|071|072|073|◖■◗|075|076|◖■◗|078|◖■◗|080|
----------------------------------------------------------------------------------
|◖■◗|◖■◗|083|084|085|086|087|◖■◗|089|090|091|092|◖■◗|◖■◗|095|096|◖■◗|098|◖■◗|◖■◗|
----------------------------------------------------------------------------------
|101|102|103|104|◖■◗|106|107|◖■◗|109|110|111|112|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|118|119|120|
----------------------------------------------------------------------------------
|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|◖■◗|127|128|◖■◗|◖■◗|◖■◗|132|◖■◗|134|135|136|◖■◗|138|139|140|
----------------------------------------------------------------------------------
|141|142|◖■◗|◖■◗|145|146|◖■◗|148|149|150|151|◖■◗|◖■◗|◖■◗|155|156|157|158|159|160|
----------------------------------------------------------------------------------
|161|162|◖■◗|164|165|166|◖■◗|168|169|170|◖■◗|◖■◗|◖■◗|174|175|176|◖■◗|178|179|180|
----------------------------------------------------------------------------------
|181|182|◖■◗|184|185|◖■◗|187|188|189|190|191|192|◖■◗|194|195|◖■◗|◖■◗|198|◖■◗|◖■◗|
----------------------------------------------------------------------------------
◖■◗
# Update
Congrats to Dror for picking 17, the winning number! And thanks to all that joined the contest! Happy New Year! Here's the video with me running the code. :)
• Gambling is not good for children. – kiss my armpit Dec 22 '13 at 18:48
• @StiffJokes: maybe it teaches a valuable lesson: do not gamble. Not even once. :) – Paulo Cereda Dec 22 '13 at 18:56
• Nine nine nine nine nine nine. – Qrrbrbirlbel Dec 22 '13 at 23:42
• @Qrrbrbirlbel: LOL if you check some of the previous reviews of the code, you'll see David's optimization, which employs quite the same implementation of Dilbert's. :P – Paulo Cereda Dec 23 '13 at 10:46
• 4 - the IEEE vetted random number (according to XKCD's hovertext) – Wayne Werner Dec 23 '13 at 20:15
• $$\pi$$ – Kartik Dec 27 '13 at 8:02
• 4 numbers per person might be useful to finalize this game much faster. – kiss my armpit Dec 28 '13 at 20:19
• Am I supposed to edit the code above so that I'm in it? Because I can't. Does that mean I can't take part in the lottery? – fifaltra Dec 29 '13 at 0:28
• With no better way to close competitions, 'off-topic' it is: the competition is now closed to new entries, and a winner will be announced (edited in/accepted) soon. – Joseph Wright Dec 31 '13 at 23:01
• Interesting things about the video: (1) You failed to highlight your code from top to bottom so you needed to do it from bottom to top. (2) Sadly, even you did not use arara to compile the code. :-) – kiss my armpit Jan 1 '14 at 23:12
• @JosephWright: How long do we have to wait for closing this question? Or should it be in "on hold" status forever? – kiss my armpit Jan 5 '14 at 19:17
• @StiffJokes The 'on hold' business lasts a week: it's there because there was a feeling that 'closed' was too 'final' in general. It's not something that's selectable for individual questions: closing puts them on hold for one week then they show closed. – Joseph Wright Jan 5 '14 at 22:02
Mmh... I hope that FakeNameGenerator will grant me some luck!
Number One:28
Number Two:99
Happy Christmas Eve!
\bye
25, because it's today's date. And 116 because I need another number!
My numbers:
Number 1: 126
Number 2: 129
Not random, not chosen by Emacs.
Number 1: 1
Number 2: 200
Neither random nor probable.
## Edit
So my new choices:
Number 1: 197
Number 2: 53
P.S. Have you noticed how every one is biased toward prime numbers!?
1. 20
2. 125
This is a long answer now :)
• Number 45
• Number 152
I was the winner last time. And, I want ducks too :-)
Juhu! My first answer in the tex community, 114, 115.
Number one: 24
Number two: 133
27 and 35 because my current reputation is 2735.
• Darn, my reputation changed. Can I still change my choice of numbers ? :) – yori Dec 31 '13 at 11:57
Number 1: 82
Number 2: 88
Very nice idea!
Number 1: 122.
Number 2: 171.
I wanted to take 3 and 6 (the age of my children who would love the duck). But the numbers are already away.
So I take:
79 and 186
I used a random function with check for already used numbers.
This sounds fun! I will go for 12 and 173.
Well, my lucky numbers are 28 and 196.
As I mostly deal with TikZ, I let PGF handle the job for me.
\documentclass{standalone}
\usepackage{pgf}
\begin{document}
\pgfmathtruncatemacro{\first}{random(1, 200)}
\pgfmathtruncatemacro{\second}{random(1, 200)}
\textbackslash{}person\{\first,\second\}\{xzs\}
\end{document}
My fantastic lucky numbers are: 010 and 172.
56, 163. Quack, quack, quack, quack!! | 2020-04-06 22:34:23 | {"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.3965412974357605, "perplexity": 6477.25571451223}, "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-2020-16/segments/1585371660550.75/warc/CC-MAIN-20200406200320-20200406230820-00484.warc.gz"} |
https://math.stackexchange.com/questions/436016/solution-of-sde-ds-t-alpha-s-tftdw-t | # solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$
does someone know how to solve the following SDE
$$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$
where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution for this SDE? Many thanks!
Note that the SDE $$dS_t = (\alpha \cdot S_t+f(t)) \, dW_t \qquad S_0 = s$$ is a linear SDE and therefore there is indeed an explicit formula for the solution of the SDE. One approach is the following:
1. Solve the homogeneous SDE $$dS_t = \alpha \cdot S_t \qquad X_0 = x$$ There are several possibilities to do so: Either you have some intuition how the solution might look like or you apply Itô's formula to $X_t := g(W_t)$ to obtain conditions on (the derivatives of) $g$ allowing you to determine $g$.
2. Solve the inhomogeneous SDE $$dS_t = (\alpha \cdot S_t+f(t)) \, dW_t$$ One approach is the so-called variation of constants: Let $S_t^0$ such that $\frac{1}{S_t^0}$ solves the homogeneoues SDE, $S_0^0 = 1$ (we solved this SDE in the first step, so there is an explicit formula). Now we apply Itô's formula to $$Z_t := S_t \cdot S_t^0$$ which works fine since we know $dS_t$, $dS_t^0$. Therefore, we can determine $Z_t$ and consequently also $S_t = \frac{Z_t}{S_t^0}$.
Literature: e.g. René L. Schilling/Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes
It is easy to check that \begin{align*} d\left(e^{-\frac{\alpha^2}{2}t - \alpha W_t } S_t \right) = e^{-\frac{\alpha^2}{2}t - \alpha W_t } f(t) dW_t. \end{align*} Then, $S_t$ can be solved subsequently.
• What is a motivation behind considering the discounting factor? – Idonknow Dec 11 '19 at 15:39
• It is the integration factor, and is motivated by the usual treatment for mean-reverting processes. – Gordon Dec 11 '19 at 16:24
• Thanks for your reply. Is the integrating factor same as in first order ordinary differential equation? – Idonknow Dec 11 '19 at 23:39
• Yes. Very similar idea. – Gordon Dec 12 '19 at 12:19 | 2020-04-05 05:02:54 | {"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": 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": 0.9615051746368408, "perplexity": 231.1933767276695}, "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-2020-16/segments/1585370528224.61/warc/CC-MAIN-20200405022138-20200405052138-00175.warc.gz"} |
https://homework.zookal.com/questions-and-answers/give-an-example-of-a-function-that-tends-to-a-462784015 | 1. Math
2. Calculus
3. give an example of a function that tends to a...
# Question: give an example of a function that tends to a...
###### Question details
Give an example of a function that tends to a finite limit even though it is not defined at a point. | 2021-02-24 22:43:12 | {"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": 0.931032121181488, "perplexity": 157.3744567336436}, "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/1614178349708.2/warc/CC-MAIN-20210224223004-20210225013004-00094.warc.gz"} |
https://www.physicsforums.com/threads/prove-the-distribution.413277/ | # Homework Help: Prove the distribution
1. Jun 29, 2010
### physics.alex
1. The problem statement, all variables and given/known data
I am asked to prove the following distribution equal to 1. The distribution is obtained by using classic Binomial distribution and apply stirling approximation.
2. Relevant equations
P(X) = $$\frac{1}{\sqrt{Nx(1-x)}}$$exp(-NI(x))
where I(x) = x ln (x/p) + (1-x) ln (1-x/1-p)
This form seems complicated to me. Any suggest for the first step to simplify the proof??
2. Jun 30, 2010
### Staff: Mentor
I would say substitute the expression you have for I(x) into the expression you have for P(X), but what you have for I(x) is ambiguous, especially this part:
ln(1 - x/1 - p)
If I take this at face value, it is
$$ln(1 - \frac{x}{1} - p)$$
I don't think that's what you meant, though, so I will have to interpret what you have written.
Is it
$$ln(\frac{1 - x}{1 - p})$$?
Or is it
$$ln(1 - \frac{x}{1 - p})$$?
3. Jun 30, 2010
### physics.alex
sorry for unclear formula.
it should be
ln{ (1-x) / (1-p) }
*I am not familiar with the latex format sorry..
Alex
4. Jun 30, 2010
### Staff: Mentor
Whether you know LaTeX or not you should be aware of how to write rational expressions so that they are not ambiguous; that is, by using enough parentheses in the right places so that their meaning is clear.
For your problem, work with I(x) using the properties of logs. From this, you should get the log of the product of two expessions involving the fractions x/p and (1 - x)/(1 - p).
From there, replace the two factors by their Stirling approximations.
I haven't worked this all the way through, but that's the direction I would take.
5. Jul 1, 2010
### physics.alex
Thanks and will try it.
Alex | 2018-06-25 15:46:47 | {"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": 0.6200743317604065, "perplexity": 1638.8759654523042}, "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-2018-26/segments/1529267868135.87/warc/CC-MAIN-20180625150537-20180625170537-00262.warc.gz"} |
https://uwestroinski.wordpress.com/category/microeconomics/page/2/ | ## Prices and Demands (Part II)
October 19, 2009
In part one we have seen that in microeconomics if we treat demand and price as observables on a Hilbert space, then not both of them can be bounded linear operators. Especially, since all linear operators on a finite dimensional Hilbert space are bounded, the state space for our market cannot be finite dimensional. All this is a consequence of a single assertion, namely demand invariance under price-scaling and somehow resembles the situation in quantum theory.
So far our considerations concerning prices and demands were quite abstract. To do some real computations we need a representation of these concepts. As promised last week, I now provide such a representation for the observables price ${p_i}$, demand ${d_i}$ and excess demand ${z_i}$ on an appropriate Hilbert space.
The results so far, lead to the following approach: The Hilbert space is given as ${X = L^2(\mathbb{R}^n)}$. This, in a way, is the simplest non-finite dimensional Hilbert space and therefore an unsurprising first choice. Thus, the state of the market with ${n\in\mathbb{N}}$ goods is described by a function ${\xi\in X}$. Let the coordinates of ${\xi}$ be denoted as ${(x_1,\ldots,x_n)\in \mathbb{R}^n}$, then the demand ${d_i: D(d_i)\rightarrow X}$ is given as a differential operator
$\displaystyle d_i \xi = -i \mu_i \frac{d}{d x_i} \xi$
with domain
$\displaystyle D(d_i) = \{ \xi\in X: \xi \textnormal{ absolutely continuous and } \xi' \in X \}.$
The excess demand operator ${z_i=d_i-\omega_i \textnormal{id}_X}$ has the same domain as ${d_i}$.
Define the function ${e_i:\mathbb{R}^n\rightarrow \mathbb{R}}$ as ${e_i(x)=e^{x_i}}$. Then, the price operator ${p_i:D(p_i)\rightarrow X}$ is given as a multiplication operator
$\displaystyle p_i \xi = e_i \cdot \xi$
with domain
$\displaystyle D(p_i) = \{ \xi\in X: e_i \cdot \xi \in X \}.$
All operators ${p_i, d_i, z_i}$ are self-adjoint, ${p_i}$ is positive, the commutator satisfies
$\displaystyle \left[p_i,z_i\right]\xi = \left[p_i,d_i\right]\xi = -i \mu_i \left(e_i \cdot \frac{d}{d x_i}\xi - e_i \cdot \xi - e_i \cdot \frac{d}{d x_i} \xi\right)= i \mu_i p_i \xi$
and thus the market axioms are fulfilled.
That still might look a little abstract if one is not used to Functional Analysis. The corresponding representation in quantum mechanics however lead to major new insights into the field.
Next time I shall give a formal comparison of microeconomics and quantum mechanics. As you can imagine by now, they are similar on some abstract level. However, there are also some striking discrepancies like the different commutation relations and thus the step towards the desired Schrödinger type equation for markets is not straight forward.
Stay tuned …
## Prices and Demands (Part I)
October 5, 2009
You might remember, that we are looking for general laws describing the behavior of markets. To that purpose, at first glance, it is not self-evident to describe such markets, i.e. prices ${p}$ of and demands ${d}$ for goods, by observables acting on some Hilbert space ${X}$. However, as we have seen in Commutation Relations in Markets and in Scientific Laws there might be good reasons for doing this and if we do it, the observables ${p}$ and ${d}$ have to satisfy the following axioms:
• (MA1) The price ${p_i}$ of good ${i}$ is a positive observable on ${X}$ for all goods ${1\leq i\leq n}$.
• (MA2) The demand ${d_i}$ of good ${i}$ is an observable on ${X}$ for all goods ${1\leq i\leq n}$.
• (MA3) The endowment ${\omega_i}$ of good ${i}$ is a real number ${\omega_i \in\mathbb{R}}$ for all goods ${1\leq i\leq n}$.
• (MA4) Prices ${p_i}$ and demands ${d_j}$ interact according to
$\displaystyle \left[p_i, d_j\right]=i \mu_i p_i \delta_{i,j}$
for a fixed real ${\mu_i\in\mathbb{R}}$.
Up to now, we have also learned that the demand ${d}$ essentially is the generator of the price-scaling group and that mathematicians often represent groups as linear operators acting on vector spaces to get an idea of what is going on. Let us just do that and find some ‘matrices’ to represent price-scaling. Maybe we then get a better understanding of such market descriptions.
Unfortunately, there is a famous result of H. Wielandt stating that if linear operators ${A,B}$ satisfy a commutation relation ${[A,B]=\textnormal{id}_X}$, then not both can be bounded simultaneously. Since there are no unbounded linear operators on finite dimensional vector spaces, the Hilbert space then must be infinite dimensional.
Just as a side note, you might notice that (scalar multiples of) momentum and position in quantum dynamics satisfy the above commutation relation and this is the reason why we cannot represent these observables as matrices. On our hunt for market laws that might be a little setback. Certainly not for mathematicians or physicists, but probably for economists. As far as i know, infinite dimensional Hilbert spaces are up to now not part of their syllabus.
Today we are not doing in quantum mechanics and thus there is still hope. If we divide market axiom 4 by ${i \mu}$ we observe that our commutation relation is ${[A,B]=A}$ which is certainly different from the quantum situation.
Let us do some mathematics.
Assume ${A,B}$ to be bounded linear operators (acting on some Banach space) with ${A^n\neq 0}$ for all ${n\in \mathbb{N}}$. Assume furthermore ${[A,B]= A}$ and as induction hypothesis ${[A^n,B]= nA^n}$. Then ${[A^{n+1},B]= A[A^n,B]+[A,B]A^n=nA^{n+1}+A^{n+1}=(n+1)A^{n+1}}$. The norm estimate ${n \|A^n\| = \|[A^n,B]\|\leq 2 \|A^n\| \|B\|}$ yields a contradiction since ${A^n\neq 0}$ for all ${n\in\mathbb{N}}$. Therefore, either at least one of the operators ${A}$ and ${B}$ is unbounded and/or the commutation relation ${[A,B]=A}$ does not hold.
In our situation the commutation relation holds as stated in market axiom 4. Since, by market axiom 1, the observable ${p_i}$ is positive we obtain ${p_i^n\neq 0}$ for ${n\in\mathbb{N}}$ and hence, at least one of the operators ${p_i}$ and ${d_i}$ is unbounded.
Oh, oh … not nice. But, it could have been worse. Next week I choose a Hilbert space and give you a representation of price and demand as unbounded operators on this Hilbert space. Things will look much more down-to-earth then. Promised …
• ## Commutation Relations in Markets
September 8, 2009
To derive commutation relations in microeconomics we first have to reach sure ground. What is a minimal set of assumptions we need to derive something interesting, but still comprehensive enough to describe something meaningful?
In a market, it is definitely safe to assume that we have ${n}$ goods for some number ${1\leq n \in \mathbb{N}}$. This goods are being traded and therefore we need to talk about prices and demand. Call ${p_i}$ the price of good ${i}$ and ${d_i}$ the demand for good ${i}$ and let ${1\leq i \leq n}$. What else do we need?
Sure, we need a lot more, but not now! As we have seen in Scientifc Laws all we need now is a symmetry between price and demand. The key to this symmetry is found in any basic text book like e.g. Microeconomic Theory by A. Mas-Colell, M.D. Whinston and is called invariance of demand under price-scaling. What is meant by that? Let me just give you an example. When continental europe introduced the Euro currency, many nations swapped their national currency for the new Euro. In Germany, 1 Euro was worth 1.95583 Deutsche Mark. All prices, wages, debts aso. where scaled by ${\frac{1}{1.95583}}$. The day after, no increase of demand for fridges, cars, credits aso. was observed. That was no surprise for economists. Where should a change of demand come from? A redefinition of the currency is not enough to generate demand. That is generally believed and a pillar in the following argumentation.
Just for the sake of completeness let me emphasize that price scaling, as introduced above form a group. Whenever we scale by a factor ${\alpha\in\mathbb{R}_{>0}}$ and then scale by a factor ${\beta\in\mathbb{R}_{>0}}$ we obtain a scaling by the factor ${\alpha\beta}$. Scaling with 1 is the neutral element and for each scale factor ${\alpha\in\mathbb{R}_{>0}}$ we can go back by scaling with ${\frac{1}{\alpha}}$.
As mathematicians, we often represent abstract groups (like the above price-scaling) as linear operators acting on some vector space. To that purpose, we choose the state of the market to be given by a non-zero vector ${\xi}$ in a Hilbert space ${X}$ with inner product denoted by ${\left\langle \cdot | \cdot \right\rangle}$. Of course, in the moment you can think of ${X}$ as a finite dimensional Hilbert space ${\mathbb{R}^n}$ or ${\mathbb{C}^n}$. On the other hand, it is always good to be suspicious and fixing the dimension to be finite might be premature. Observables are self-adjoint operators on this Hilbert space and satisfy the following axioms:
• (MA1) The price ${p_i}$ of good ${i}$ is a positive observable on ${X}$ for all goods ${1\leq i\leq n}$.
• (MA2) The demand ${d_i}$ of good ${i}$ is an observable on ${X}$ for all goods ${1\leq i\leq n}$.
A positive observable ${a}$ on ${X}$ is an observable with ${\langle a\xi | \xi \rangle>0}$ for all ${0\neq\xi}$ in the domain ${D(a)}$ of ${a}$.
By a famous result of E. Noether, symmetries and invariants are closely tied together. What are the market invariants of the asymmetric market under the price-scaling symmetry? To see this, let ${\left(U_i(\alpha)\right)_{0 < \alpha\in \mathbb{R}}}$ be a strongly continuous family of unitary operators on ${X}$ such that
$\displaystyle U_i^{-1}(\alpha)p_i U_i(\alpha)=\alpha p_i.$
The family ${U_i(\cdot)}$ satisfies the following properties for all ${\alpha>0}$ and ${\beta>0}$:
• ${U_i(1)= \textnormal{id}_X}$
• ${U_i(\alpha)U_i(\beta)=U_i(\alpha\beta)=U_i(\beta)U_i(\alpha)}$
• ${U_i^{-1}(\alpha) = U_i\left(\frac{1}{\alpha}\right)}$
Define ${T_i(t):=U_i(e^t)}$ and observe
• ${T_i(0)= \textnormal{id}_X}$
• ${T_i(t)T_i(s)=T_i(t+s)=T_i(s)T_i(t)}$
• ${T_i^{-1}(t) = T_i(-t)}$
This yields ${T_i}$ to be a strongly continuous group of unitary operators acting on ${X}$. Thus, the theorem of Stone ensures the existence of a skew-adjoint generator ${A_i}$. Set ${\alpha = e^t}$ and with ${U(\alpha)=T(\ln \alpha)}$ it follows that
$\displaystyle \begin{array}{rcl} p_i & = & \frac{d}{d\alpha}\left(U_i^{-1}(\alpha)p_i U_i(\alpha)\right) \\ & = & \frac{d}{d\alpha}\left(T_i(-\ln \alpha)p_i T_i(\ln \alpha)\right) \\ & = & - \frac{1}{\alpha} T_i(-\ln \alpha) A_i p_i T_i(\ln \alpha) + \frac{1}{\alpha} T_i(-\ln \alpha)p_i A_i T_i(\ln \alpha). \end{array}$
Evaluation at ${\alpha=1}$ yields
$\displaystyle \left[p_i, A_i\right] = p_i. \ \ \ \ \ (1)$
Since a generator commutes with the strongly continuous group it generates it is easily seen that ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ also commutes with ${U_i(\alpha)}$ for any ${\beta_i,\gamma_i\in\mathbb{C}}$. Hence ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ represents a market invariant under price-scaling.
Now we derive an economic interpretation of ${A_i}$. We know already that ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ represents a market invariant under price-scaling for any ${\beta_i,\gamma_i\in\mathbb{C}}$. Since ${A_i}$ is skew-adjoint and ${\beta_i A_i + \gamma_i\textnormal{id}_X}$ needs to be an observable, we get that ${\beta_i = i \mu_i }$ and ${\gamma_i = \omega_i}$ for some ${\mu_i, \omega_i \in\mathbb{R}}$. Furthermore, since scaling of one price does not influence scaling of the others (i.e., ${\left[p_i, U_j(\alpha)\right]=0}$ for ${i\neq j}$) we can use (1) and obtain
$\displaystyle \left[p_i, i \mu_i A_j - \omega_i \textnormal{id}_X\right] = i \mu_i p_i \delta_{i,j}.$
The operator ${i \mu_i A_i + \omega_i \textnormal{id}_X}$ is an observable and is invariant under price-scaling. Economic intuition therefore leads us to identify this operator with the demand respectively excess demand for good ${i}$ if ${\mu_i\neq 0}$. The real parameter ${\omega_i}$ is identified as endowment. The other real parameter ${\mu_i}$ represents a new feature. Intuitively it measures the difference of first selling and then buying a good versus first buying and then selling that good.
The observations in the last paragraph yield the final axioms.
• (MA3) The endowment ${\omega_i}$ of good ${i}$ is a real number ${\omega_i \in\mathbb{R}}$ for all goods ${1\leq i\leq n}$.
• (MA4) Prices ${p_i}$ and demands ${d_j}$ interact according to
$\displaystyle \left[p_i, d_j\right]=i \mu_i p_i \delta_{i,j}$
for a fixed real ${\mu_i\in\mathbb{R}}$.
There are still a lot of things to say, e.g. on how measurements are done, on the dimension of the Hilbert space ${X}$, on representations of demand ${d_i}$ and price ${p_i}$ as operators and on a comparison to the commutation relations of quantum mechanics. Stay tuned … | 2019-08-22 23:03:44 | {"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": 0, "img_math": 149, "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": 0.9059715867042542, "perplexity": 309.1415549510654}, "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/1566027317516.88/warc/CC-MAIN-20190822215308-20190823001308-00163.warc.gz"} |
http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html | ### 1. SPECIAL RELATIVITY AND FLAT SPACETIME
We will begin with a whirlwind tour of special relativity (SR) and life in flat spacetime. The point will be both to recall what SR is all about, and to introduce tensors and related concepts that will be crucial later on, without the extra complications of curvature on top of everything else. Therefore, for this section we will always be working in flat spacetime, and furthermore we will only use orthonormal (Cartesian-like) coordinates. Needless to say it is possible to do SR in any coordinate system you like, but it turns out that introducing the necessary tools for doing so would take us halfway to curved spaces anyway, so we will put that off for a while.
It is often said that special relativity is a theory of 4-dimensional spacetime: three of space, one of time. But of course, the pre-SR world of Newtonian mechanics featured three spatial dimensions and a time parameter. Nevertheless, there was not much temptation to consider these as different aspects of a single 4-dimensional spacetime. Why not?
Consider a garden-variety 2-dimensional plane. It is typically convenient to label the points on such a plane by introducing coordinates, for example by defining orthogonal x and y axes and projecting each point onto these axes in the usual way. However, it is clear that most of the interesting geometrical facts about the plane are independent of our choice of coordinates. As a simple example, we can consider the distance between two points, given by
(1.1)
In a different Cartesian coordinate system, defined by x' and y' axes which are rotated with respect to the originals, the formula for the distance is unaltered:
(1.2)
We therefore say that the distance is invariant under such changes of coordinates.
This is why it is useful to think of the plane as 2-dimensional: although we use two distinct numbers to label each point, the numbers are not the essence of the geometry, since we can rotate axes into each other while leaving distances and so forth unchanged. In Newtonian physics this is not the case with space and time; there is no useful notion of rotating space and time into each other. Rather, the notion of "all of space at a single moment in time" has a meaning independent of coordinates.
Such is not the case in SR. Let us consider coordinates (t, x, y, z) on spacetime, set up in the following way. The spatial coordinates (x, y, z) comprise a standard Cartesian system, constructed for example by welding together rigid rods which meet at right angles. The rods must be moving freely, unaccelerated. The time coordinate is defined by a set of clocks which are not moving with respect to the spatial coordinates. (Since this is a thought experiment, we imagine that the rods are infinitely long and there is one clock at every point in space.) The clocks are synchronized in the following sense: if you travel from one point in space to any other in a straight line at constant speed, the time difference between the clocks at the ends of your journey is the same as if you had made the same trip, at the same speed, in the other direction. The coordinate system thus constructed is an inertial frame.
An event is defined as a single moment in space and time, characterized uniquely by (t, x, y, z). Then, without any motivation for the moment, let us introduce the spacetime interval between two events:
(1.3)
(Notice that it can be positive, negative, or zero even for two nonidentical points.) Here, c is some fixed conversion factor between space and time; that is, a fixed velocity. Of course it will turn out to be the speed of light; the important thing, however, is not that photons happen to travel at that speed, but that there exists a c such that the spacetime interval is invariant under changes of coordinates. In other words, if we set up a new inertial frame (t', x', y', z') by repeating our earlier procedure, but allowing for an offset in initial position, angle, and velocity between the new rods and the old, the interval is unchanged:
(1.4)
This is why it makes sense to think of SR as a theory of 4-dimensional spacetime, known as Minkowski space. (This is a special case of a 4-dimensional manifold, which we will deal with in detail later.) As we shall see, the coordinate transformations which we have implicitly defined do, in a sense, rotate space and time into each other. There is no absolute notion of "simultaneous events"; whether two things occur at the same time depends on the coordinates used. Therefore the division of Minkowski space into space and time is a choice we make for our own purposes, not something intrinsic to the situation.
Almost all of the "paradoxes" associated with SR result from a stubborn persistence of the Newtonian notions of a unique time coordinate and the existence of "space at a single moment in time." By thinking in terms of spacetime rather than space and time together, these paradoxes tend to disappear.
Let's introduce some convenient notation. Coordinates on spacetime will be denoted by letters with Greek superscript indices running from 0 to 3, with 0 generally denoting the time coordinate. Thus,
(1.5)
(Don't start thinking of the superscripts as exponents.) Furthermore, for the sake of simplicity we will choose units in which
(1.6)
we will therefore leave out factors of c in all subsequent formulae. Empirically we know that c is the speed of light, 3 × 108 meters per second; thus, we are working in units where 1 second equals 3 × 108 meters. Sometimes it will be useful to refer to the space and time components of x separately, so we will use Latin superscripts to stand for the space components alone:
(1.7)
It is also convenient to write the spacetime interval in a more compact form. We therefore introduce a 4 × 4 matrix, the metric, which we write using two lower indices:
(1.8)
(Some references, especially field theory books, define the metric with the opposite sign, so be careful.) We then have the nice formula
(1.9)
Notice that we use the summation convention, in which indices which appear both as superscripts and subscripts are summed over. The content of (1.9) is therefore just the same as (1.3).
Now we can consider coordinate transformations in spacetime at a somewhat more abstract level than before. What kind of transformations leave the interval (1.9) invariant? One simple variety are the translations, which merely shift the coordinates:
(1.10)
where a is a set of four fixed numbers. (Notice that we put the prime on the index, not on the x.) Translations leave the differences x unchanged, so it is not remarkable that the interval is unchanged. The only other kind of linear transformation is to multiply x by a (spacetime-independent) matrix:
(1.11)
or, in more conventional matrix notation,
(1.12)
These transformations do not leave the differences x unchanged, but multiply them also by the matrix . What kind of matrices will leave the interval invariant? Sticking with the matrix notation, what we would like is
(1.13)
and therefore
(1.14)
or
(1.15)
We want to find the matrices such that the components of the matrix are the same as those of ; that is what it means for the interval to be invariant under these transformations.
The matrices which satisfy (1.14) are known as the Lorentz transformations; the set of them forms a group under matrix multiplication, known as the Lorentz group. There is a close analogy between this group and O(3), the rotation group in three-dimensional space. The rotation group can be thought of as 3 × 3 matrices R which satisfy
(1.16)
where 1 is the 3 × 3 identity matrix. The similarity with (1.14) should be clear; the only difference is the minus sign in the first term of the metric , signifying the timelike direction. The Lorentz group is therefore often referred to as O(3,1). (The 3 × 3 identity matrix is simply the metric for ordinary flat space. Such a metric, in which all of the eigenvalues are positive, is called Euclidean, while those such as (1.8) which feature a single minus sign are called Lorentzian.)
Lorentz transformations fall into a number of categories. First there are the conventional rotations, such as a rotation in the x-y plane:
(1.17)
The rotation angle is a periodic variable with period 2. There are also boosts, which may be thought of as "rotations between space and time directions." An example is given by
(1.18)
The boost parameter , unlike the rotation angle, is defined from - to . There are also discrete transformations which reverse the time direction or one or more of the spatial directions. (When these are excluded we have the proper Lorentz group, SO(3,1).) A general transformation can be obtained by multiplying the individual transformations; the explicit expression for this six-parameter matrix (three boosts, three rotations) is not sufficiently pretty or useful to bother writing down. In general Lorentz transformations will not commute, so the Lorentz group is non-abelian. The set of both translations and Lorentz transformations is a ten-parameter non-abelian group, the Poincaré group.
You should not be surprised to learn that the boosts correspond to changing coordinates by moving to a frame which travels at a constant velocity, but let's see it more explicitly. For the transformation given by (1.18), the transformed coordinates t' and x' will be given by
(1.19)
From this we see that the point defined by x' = 0 is moving; it has a velocity
(1.20)
To translate into more pedestrian notation, we can replace = tanh-1v to obtain
(1.21)
where = 1/. So indeed, our abstract approach has recovered the conventional expressions for Lorentz transformations. Applying these formulae leads to time dilation, length contraction, and so forth.
An extremely useful tool is the spacetime diagram, so let's consider Minkowski space from this point of view. We can begin by portraying the initial t and x axes at (what are conventionally thought of as) right angles, and suppressing the y and z axes. Then according to (1.19), under a boost in the x-t plane the x' axis (t' = 0) is given by t = xtanh, while the t' axis (x' = 0) is given by t = x/tanh. We therefore see that the space and time axes are rotated into each other, although they scissor together instead of remaining orthogonal in the traditional Euclidean sense. (As we shall see, the axes do in fact remain orthogonal in the Lorentzian sense.) This should come as no surprise, since if spacetime behaved just like a four-dimensional version of space the world would be a very different place.
It is also enlightening to consider the paths corresponding to travel at the speed c = 1. These are given in the original coordinate system by x = ±t. In the new system, a moment's thought reveals that the paths defined by x' = ±t' are precisely the same as those defined by x = ±t; these trajectories are left invariant under Lorentz transformations. Of course we know that light travels at this speed; we have therefore found that the speed of light is the same in any inertial frame. A set of points which are all connected to a single event by straight lines moving at the speed of light is called a light cone; this entire set is invariant under Lorentz transformations. Light cones are naturally divided into future and past; the set of all points inside the future and past light cones of a point p are called timelike separated from p, while those outside the light cones are spacelike separated and those on the cones are lightlike or null separated from p. Referring back to (1.3), we see that the interval between timelike separated points is negative, between spacelike separated points is positive, and between null separated points is zero. (The interval is defined to be s2, not the square root of this quantity.) Notice the distinction between this situation and that in the Newtonian world; here, it is impossible to say (in a coordinate-independent way) whether a point that is spacelike separated from p is in the future of p, the past of p, or "at the same time".
To probe the structure of Minkowski space in more detail, it is necessary to introduce the concepts of vectors and tensors. We will start with vectors, which should be familiar. Of course, in spacetime vectors are four-dimensional, and are often referred to as four-vectors. This turns out to make quite a bit of difference; for example, there is no such thing as a cross product between two four-vectors.
Beyond the simple fact of dimensionality, the most important thing to emphasize is that each vector is located at a given point in spacetime. You may be used to thinking of vectors as stretching from one point to another in space, and even of "free" vectors which you can slide carelessly from point to point. These are not useful concepts in relativity. Rather, to each point p in spacetime we associate the set of all possible vectors located at that point; this set is known as the tangent space at p, or Tp. The name is inspired by thinking of the set of vectors attached to a point on a simple curved two-dimensional space as comprising a plane which is tangent to the point. But inspiration aside, it is important to think of these vectors as being located at a single point, rather than stretching from one point to another. (Although this won't stop us from drawing them as arrows on spacetime diagrams.)
Later we will relate the tangent space at each point to things we can construct from the spacetime itself. For right now, just think of Tp as an abstract vector space for each point in spacetime. A (real) vector space is a collection of objects ("vectors") which, roughly speaking, can be added together and multiplied by real numbers in a linear way. Thus, for any two vectors V and W and real numbers a and b, we have
(1.22)
Every vector space has an origin, i.e. a zero vector which functions as an identity element under vector addition. In many vector spaces there are additional operations such as taking an inner (dot) product, but this is extra structure over and above the elementary concept of a vector space.
A vector is a perfectly well-defined geometric object, as is a vector field, defined as a set of vectors with exactly one at each point in spacetime. (The set of all the tangent spaces of a manifold M is called the tangent bundle, T(M).) Nevertheless it is often useful for concrete purposes to decompose vectors into components with respect to some set of basis vectors. A basis is any set of vectors which both spans the vector space (any vector is a linear combination of basis vectors) and is linearly independent (no vector in the basis is a linear combination of other basis vectors). For any given vector space, there will be an infinite number of legitimate bases, but each basis will consist of the same number of vectors, known as the dimension of the space. (For a tangent space associated with a point in Minkowski space, the dimension is of course four.)
Let us imagine that at each tangent space we set up a basis of four vectors , with {0, 1, 2, 3} as usual. In fact let us say that each basis is adapted to the coordinates x; that is, the basis vector is what we would normally think of pointing along the x-axis, etc. It is by no means necessary that we choose a basis which is adapted to any coordinate system at all, although it is often convenient. (We really could be more precise here, but later on we will repeat the discussion at an excruciating level of precision, so some sloppiness now is forgivable.) Then any abstract vector A can be written as a linear combination of basis vectors:
(1.23)
The coefficients A are the components of the vector A. More often than not we will forget the basis entirely and refer somewhat loosely to "the vector A", but keep in mind that this is shorthand. The real vector is an abstract geometrical entity, while the components are just the coefficients of the basis vectors in some convenient basis. (Since we will usually suppress the explicit basis vectors, the indices will usually label components of vectors and tensors. This is why there are parentheses around the indices on the basis vectors, to remind us that this is a collection of vectors, not components of a single vector.)
A standard example of a vector in spacetime is the tangent vector to a curve. A parameterized curve or path through spacetime is specified by the coordinates as a function of the parameter, e.g. x(). The tangent vector V() has components
(1.24)
The entire vector is thus V = V. Under a Lorentz transformation the coordinates x change according to (1.11), while the parameterization is unaltered; we can therefore deduce that the components of the tangent vector must change as
(1.25)
However, the vector itself (as opposed to its components in some coordinate system) is invariant under Lorentz transformations. We can use this fact to derive the transformation properties of the basis vectors. Let us refer to the set of basis vectors in the transformed coordinate system as . Since the vector is invariant, we have
(1.26)
But this relation must hold no matter what the numerical values of the components V are. Therefore we can say
(1.27)
To get the new basis in terms of the old one we should multiply by the inverse of the Lorentz transformation . But the inverse of a Lorentz transformation from the unprimed to the primed coordinates is also a Lorentz transformation, this time from the primed to the unprimed systems. We will therefore introduce a somewhat subtle notation, by writing using the same symbol for both matrices, just with primed and unprimed indices adjusted. That is,
(1.28)
or
(1.29)
where is the traditional Kronecker delta symbol in four dimensions. (Note that Schutz uses a different convention, always arranging the two indices northwest/southeast; the important thing is where the primes go.) From (1.27) we then obtain the transformation rule for basis vectors:
(1.30)
Therefore the set of basis vectors transforms via the inverse Lorentz transformation of the coordinates or vector components.
It is worth pausing a moment to take all this in. We introduced coordinates labeled by upper indices, which transformed in a certain way under Lorentz transformations. We then considered vector components which also were written with upper indices, which made sense since they transformed in the same way as the coordinate functions. (In a fixed coordinate system, each of the four coordinates x can be thought of as a function on spacetime, as can each of the four components of a vector field.) The basis vectors associated with the coordinate system transformed via the inverse matrix, and were labeled by a lower index. This notation ensured that the invariant object constructed by summing over the components and basis vectors was left unchanged by the transformation, just as we would wish. It's probably not giving too much away to say that this will continue to be the case for more complicated objects with multiple indices (tensors).
Once we have set up a vector space, there is an associated vector space (of equal dimension) which we can immediately define, known as the dual vector space. The dual space is usually denoted by an asterisk, so that the dual space to the tangent space Tp is called the cotangent space and denoted T*p. The dual space is the space of all linear maps from the original vector space to the real numbers; in math lingo, if Tp* is a dual vector, then it acts as a map such that:
(1.31)
where V, W are vectors and a, b are real numbers. The nice thing about these maps is that they form a vector space themselves; thus, if and are dual vectors, we have
(1.32)
To make this construction somewhat more concrete, we can introduce a set of basis dual vectors by demanding
(1.33)
Then every dual vector can be written in terms of its components, which we label with lower indices:
(1.34)
In perfect analogy with vectors, we will usually simply write to stand for the entire dual vector. In fact, you will sometime see elements of Tp (what we have called vectors) referred to as contravariant vectors, and elements of Tp* (what we have called dual vectors) referred to as covariant vectors. Actually, if you just refer to ordinary vectors as vectors with upper indices and dual vectors as vectors with lower indices, nobody should be offended. Another name for dual vectors is one-forms, a somewhat mysterious designation which will become clearer soon.
The component notation leads to a simple way of writing the action of a dual vector on a vector:
(1.35)
This is why it is rarely necessary to write the basis vectors (and dual vectors) explicitly; the components do all of the work. The form of (1.35) also suggests that we can think of vectors as linear maps on dual vectors, by defining
(1.36)
Therefore, the dual space to the dual vector space is the original vector space itself.
Of course in spacetime we will be interested not in a single vector space, but in fields of vectors and dual vectors. (The set of all cotangent spaces over M is the cotangent bundle, T*(M).) In that case the action of a dual vector field on a vector field is not a single number, but a scalar (or just "function") on spacetime. A scalar is a quantity without indices, which is unchanged under Lorentz transformations.
We can use the same arguments that we earlier used for vectors to derive the transformation properties of dual vectors. The answers are, for the components,
(1.37)
and for basis dual vectors,
(1.38)
This is just what we would expect from index placement; the components of a dual vector transform under the inverse transformation of those of a vector. Note that this ensures that the scalar (1.35) is invariant under Lorentz transformations, just as it should be.
Let's consider some examples of dual vectors, first in other contexts and then in Minkowski space. Imagine the space of n-component column vectors, for some integer n. Then the dual space is that of n-component row vectors, and the action is ordinary matrix multiplication:
(1.39)
Another familiar example occurs in quantum mechanics, where vectors in the Hilbert space are represented by kets, |. In this case the dual space is the space of bras, |, and the action gives the number |. (This is a complex number in quantum mechanics, but the idea is precisely the same.)
In spacetime the simplest example of a dual vector is the gradient of a scalar function, the set of partial derivatives with respect to the spacetime coordinates, which we denote by "d":
(1.40)
The conventional chain rule used to transform partial derivatives amounts in this case to the transformation rule of components of dual vectors:
(1.41)
where we have used (1.11) and (1.28) to relate the Lorentz transformation to the coordinates. The fact that the gradient is a dual vector leads to the following shorthand notations for partial derivatives:
(1.42)
(Very roughly speaking, "x has an upper index, but when it is in the denominator of a derivative it implies a lower index on the resulting object.") I'm not a big fan of the comma notation, but we will use all the time. Note that the gradient does in fact act in a natural way on the example we gave above of a vector, the tangent vector to a curve. The result is ordinary derivative of the function along the curve:
(1.43)
As a final note on dual vectors, there is a way to represent them as pictures which is consistent with the picture of vectors as arrows. See the discussion in Schutz, or in MTW (where it is taken to dizzying extremes).
A straightforward generalization of vectors and dual vectors is the notion of a tensor. Just as a dual vector is a linear map from vectors to R, a tensor T of type (or rank) (k, l ) is a multilinear map from a collection of dual vectors and vectors to R:
(1.44)
Here, "×" denotes the Cartesian product, so that for example Tp × Tp is the space of ordered pairs of vectors. Multilinearity means that the tensor acts linearly in each of its arguments; for instance, for a tensor of type (1, 1), we have
(1.45)
From this point of view, a scalar is a type (0, 0) tensor, a vector is a type (1, 0) tensor, and a dual vector is a type (0, 1) tensor.
The space of all tensors of a fixed type (k, l ) forms a vector space; they can be added together and multiplied by real numbers. To construct a basis for this space, we need to define a new operation known as the tensor product, denoted by . If T is a (k, l ) tensor and S is a (m, n) tensor, we define a (k + m, l + n) tensor T S by
(1.46)
(Note that the and V(i) are distinct dual vectors and vectors, not components thereof.) In other words, first act T on the appropriate set of dual vectors and vectors, and then act S on the remainder, and then multiply the answers. Note that, in general, T S S T.
It is now straightforward to construct a basis for the space of all (k, l ) tensors, by taking tensor products of basis vectors and dual vectors; this basis will consist of all tensors of the form
(1.47)
In a 4-dimensional spacetime there will be 4k + l basis tensors in all. In component notation we then write our arbitrary tensor as
(1.48)
Alternatively, we could define the components by acting the tensor on basis vectors and dual vectors:
(1.49)
You can check for yourself, using (1.33) and so forth, that these equations all hang together properly.
As with vectors, we will usually take the shortcut of denoting the tensor T by its components T ... ... . The action of the tensors on a set of vectors and dual vectors follows the pattern established in (1.35):
(1.50)
The order of the indices is obviously important, since the tensor need not act in the same way on its various arguments. Finally, the transformation of tensor components under Lorentz transformations can be derived by applying what we already know about the transformation of basis vectors and dual vectors. The answer is just what you would expect from index placement,
(1.51)
Thus, each upper index gets transformed like a vector, and each lower index gets transformed like a dual vector.
Although we have defined tensors as linear maps from sets of vectors and tangent vectors to R, there is nothing that forces us to act on a full collection of arguments. Thus, a (1, 1) tensor also acts as a map from vectors to vectors:
(1.52)
You can check for yourself that TV is a vector ( i.e. obeys the vector transformation law). Similarly, we can act one tensor on (all or part of) another tensor to obtain a third tensor. For example,
(1.53)
is a perfectly good (1, 1) tensor.
You may be concerned that this introduction to tensors has been somewhat too brief, given the esoteric nature of the material. In fact, the notion of tensors does not require a great deal of effort to master; it's just a matter of keeping the indices straight, and the rules for manipulating them are very natural. Indeed, a number of books like to define tensors as collections of numbers transforming according to (1.51). While this is operationally useful, it tends to obscure the deeper meaning of tensors as geometrical entities with a life independent of any chosen coordinate system. There is, however, one subtlety which we have glossed over. The notions of dual vectors and tensors and bases and linear maps belong to the realm of linear algebra, and are appropriate whenever we have an abstract vector space at hand. In the case of interest to us we have not just a vector space, but a vector space at each point in spacetime. More often than not we are interested in tensor fields, which can be thought of as tensor-valued functions on spacetime. Fortunately, none of the manipulations we defined above really care whether we are dealing with a single vector space or a collection of vector spaces, one for each event. We will be able to get away with simply calling things functions of x when appropriate. However, you should keep straight the logical independence of the notions we have introduced and their specific application to spacetime and relativity.
Now let's turn to some examples of tensors. First we consider the previous example of column vectors and their duals, row vectors. In this system a (1, 1) tensor is simply a matrix, Mij. Its action on a pair (, V) is given by usual matrix multiplication:
(1.54)
If you like, feel free to think of tensors as "matrices with an arbitrary number of indices."
In spacetime, we have already seen some examples of tensors without calling them that. The most familiar example of a (0, 2) tensor is the metric, . The action of the metric on two vectors is so useful that it gets its own name, the inner product (or dot product):
(1.55)
Just as with the conventional Euclidean dot product, we will refer to two vectors whose dot product vanishes as orthogonal. Since the dot product is a scalar, it is left invariant under Lorentz transformations; therefore the basis vectors of any Cartesian inertial frame, which are chosen to be orthogonal by definition, are still orthogonal after a Lorentz transformation (despite the "scissoring together" we noticed earlier). The norm of a vector is defined to be inner product of the vector with itself; unlike in Euclidean space, this number is not positive definite:
(A vector can have zero norm without being the zero vector.) You will notice that the terminology is the same as that which we earlier used to classify the relationship between two points in spacetime; it's no accident, of course, and we will go into more detail later.
Another tensor is the Kronecker delta , of type (1, 1), which you already know the components of. Related to this and the metric is the inverse metric , a type (2, 0) tensor defined as the inverse of the metric:
(1.56)
In fact, as you can check, the inverse metric has exactly the same components as the metric itself. (This is only true in flat space in Cartesian coordinates, and will fail to hold in more general situations.) There is also the Levi-Civita tensor, a (0, 4) tensor:
(1.57)
Here, a "permutation of 0123" is an ordering of the numbers 0, 1, 2, 3 which can be obtained by starting with 0123 and exchanging two of the digits; an even permutation is obtained by an even number of such exchanges, and an odd permutation is obtained by an odd number. Thus, for example, = - 1.
It is a remarkable property of the above tensors - the metric, the inverse metric, the Kronecker delta, and the Levi-Civita tensor - that, even though they all transform according to the tensor transformation law (1.51), their components remain unchanged in any Cartesian coordinate system in flat spacetime. In some sense this makes them bad examples of tensors, since most tensors do not have this property. In fact, even these tensors do not have this property once we go to more general coordinate systems, with the single exception of the Kronecker delta. This tensor has exactly the same components in any coordinate system in any spacetime. This makes sense from the definition of a tensor as a linear map; the Kronecker tensor can be thought of as the identity map from vectors to vectors (or from dual vectors to dual vectors), which clearly must have the same components regardless of coordinate system. The other tensors (the metric, its inverse, and the Levi-Civita tensor) characterize the structure of spacetime, and all depend on the metric. We shall therefore have to treat them more carefully when we drop our assumption of flat spacetime.
A more typical example of a tensor is the electromagnetic field strength tensor. We all know that the electromagnetic fields are made up of the electric field vector Ei and the magnetic field vector Bi. (Remember that we use Latin indices for spacelike components 1,2,3.) Actually these are only "vectors" under rotations in space, not under the full Lorentz group. In fact they are components of a (0, 2) tensor F, defined by
(1.58)
From this point of view it is easy to transform the electromagnetic fields in one reference frame to those in another, by application of (1.51). The unifying power of the tensor formalism is evident: rather than a collection of two vectors whose relationship and transformation properties are rather mysterious, we have a single tensor field to describe all of electromagnetism. (On the other hand, don't get carried away; sometimes it's more convenient to work in a single coordinate system using the electric and magnetic field vectors.)
With some examples in hand we can now be a little more systematic about some properties of tensors. First consider the operation of contraction, which turns a (k, l ) tensor into a (k - 1, l - 1) tensor. Contraction proceeds by summing over one upper and one lower index:
(1.59)
You can check that the result is a well-defined tensor. Of course it is only permissible to contract an upper index with a lower index (as opposed to two indices of the same type). Note also that the order of the indices matters, so that you can get different tensors by contracting in different ways; thus,
(1.60)
in general.
The metric and inverse metric can be used to raise and lower indices on tensors. That is, given a tensor T, we can use the metric to define new tensors which we choose to denote by the same letter T:
(1.61)
and so forth. Notice that raising and lowering does not change the position of an index relative to other indices, and also that "free" indices (which are not summed over) must be the same on both sides of an equation, while "dummy" indices (which are summed over) only appear on one side. As an example, we can turn vectors and dual vectors into each other by raising and lowering indices:
(1.62)
This explains why the gradient in three-dimensional flat Euclidean space is usually thought of as an ordinary vector, even though we have seen that it arises as a dual vector; in Euclidean space (where the metric is diagonal with all entries +1) a dual vector is turned into a vector with precisely the same components when we raise its index. You may then wonder why we have belabored the distinction at all. One simple reason, of course, is that in a Lorentzian spacetime the components are not equal:
(1.63)
In a curved spacetime, where the form of the metric is generally more complicated, the difference is rather more dramatic. But there is a deeper reason, namely that tensors generally have a "natural" definition which is independent of the metric. Even though we will always have a metric available, it is helpful to be aware of the logical status of each mathematical object we introduce. The gradient, and its action on vectors, is perfectly well defined regardless of any metric, whereas the "gradient with upper indices" is not. (As an example, we will eventually want to take variations of functionals with respect to the metric, and will therefore have to know exactly how the functional depends on the metric, something that is easily obscured by the index notation.)
Continuing our compilation of tensor jargon, we refer to a tensor as symmetric in any of its indices if it is unchanged under exchange of those indices. Thus, if
(1.64)
we say that S is symmetric in its first two indices, while if
(1.65)
we say that S is symmetric in all three of its indices. Similarly, a tensor is antisymmetric (or "skew-symmetric") in any of its indices if it changes sign when those indices are exchanged; thus,
(1.66)
means that A is antisymmetric in its first and third indices (or just "antisymmetric in and "). If a tensor is (anti-) symmetric in all of its indices, we refer to it as simply (anti-) symmetric (sometimes with the redundant modifier "completely"). As examples, the metric and the inverse metric are symmetric, while the Levi-Civita tensor and the electromagnetic field strength tensor F are antisymmetric. (Check for yourself that if you raise or lower a set of indices which are symmetric or antisymmetric, they remain that way.) Notice that it makes no sense to exchange upper and lower indices with each other, so don't succumb to the temptation to think of the Kronecker delta as symmetric. On the other hand, the fact that lowering an index on gives a symmetric tensor (in fact, the metric) means that the order of indices doesn't really matter, which is why we don't keep track index placement for this one tensor.
Given any tensor, we can symmetrize (or antisymmetrize) any number of its upper or lower indices. To symmetrize, we take the sum of all permutations of the relevant indices and divide by the number of terms:
(1.67)
while antisymmetrization comes from the alternating sum:
(1.68)
By "alternating sum" we mean that permutations which are the result of an odd number of exchanges are given a minus sign, thus:
(1.69)
Notice that round/square brackets denote symmetrization/antisymmetrization. Furthermore, we may sometimes want to (anti-) symmetrize indices which are not next to each other, in which case we use vertical bars to denote indices not included in the sum:
(1.70)
Finally, some people use a convention in which the factor of 1/n! is omitted. The one used here is a good one, since (for example) a symmetric tensor satisfies
(1.71)
and likewise for antisymmetric tensors.
We have been very careful so far to distinguish clearly between things that are always true (on a manifold with arbitrary metric) and things which are only true in Minkowski space in Cartesian coordinates. One of the most important distinctions arises with partial derivatives. If we are working in flat spacetime with Cartesian coordinates, then the partial derivative of a (k, l ) tensor is a (k, l + 1) tensor; that is,
(1.72)
transforms properly under Lorentz transformations. However, this will no longer be true in more general spacetimes, and we will have to define a "covariant derivative" to take the place of the partial derivative. Nevertheless, we can still use the fact that partial derivatives give us tensor in this special case, as long as we keep our wits about us. (The one exception to this warning is the partial derivative of a scalar, , which is a perfectly good tensor [the gradient] in any spacetime.)
We have now accumulated enough tensor know-how to illustrate some of these concepts using actual physics. Specifically, we will examine Maxwell's equations of electrodynamics. In 19th-century notation, these are
(1.73)
Here, E and B are the electric and magnetic field 3-vectors, J is the current, is the charge density, and × and . are the conventional curl and divergence. These equations are invariant under Lorentz transformations, of course; that's how the whole business got started. But they don't look obviously invariant; our tensor notation can fix that. Let's begin by writing these equations in just a slightly different notation,
(1.74)
In these expressions, spatial indices have been raised and lowered with abandon, without any attempt to keep straight where the metric appears. This is because is the metric on flat 3-space, with its inverse (they are equal as matrices). We can therefore raise and lower indices at will, since the components don't change. Meanwhile, the three-dimensional Levi-Civita tensor is defined just as the four-dimensional one, although with one fewer index. We have replaced the charge density by J0; this is legitimate because the density and current together form the current 4-vector, J = (, J1, J2, J3).
From these expressions, and the definition (1.58) of the field strength tensor F, it is easy to get a completely tensorial 20th-century version of Maxwell's equations. Begin by noting that we can express the field strength with upper indices as
(1.75)
(To check this, note for example that F01 = F01 and F12 = B3.) Then the first two equations in (1.74) become
(1.76)
Using the antisymmetry of F, we see that these may be combined into the single tensor equation
(1.77)
A similar line of reasoning, which is left as an exercise to you, reveals that the third and fourth equations in (1.74) can be written
(1.78)
The four traditional Maxwell equations are thus replaced by two, thus demonstrating the economy of tensor notation. More importantly, however, both sides of equations (1.77) and (1.78) manifestly transform as tensors; therefore, if they are true in one inertial frame, they must be true in any Lorentz-transformed frame. This is why tensors are so useful in relativity - we often want to express relationships without recourse to any reference frame, and it is necessary that the quantities on each side of an equation transform in the same way under change of coordinates. As a matter of jargon, we will sometimes refer to quantities which are written in terms of tensors as covariant (which has nothing to do with "covariant" as opposed to "contravariant"). Thus, we say that (1.77) and (1.78) together serve as the covariant form of Maxwell's equations, while (1.73) or (1.74) are non-covariant.
Let us now introduce a special class of tensors, known as differential forms (or just "forms"). A differential p-form is a (0, p) tensor which is completely antisymmetric. Thus, scalars are automatically 0-forms, and dual vectors are automatically one-forms (thus explaining this terminology from a while back). We also have the 2-form F and the 4-form . The space of all p-forms is denoted , and the space of all p-form fields over a manifold M is denoted (M). A semi-straightforward exercise in combinatorics reveals that the number of linearly independent p-forms on an n-dimensional vector space is n!/(p!(n - p)!). So at a point on a 4-dimensional spacetime there is one linearly independent 0-form, four 1-forms, six 2-forms, four 3-forms, and one 4-form. There are no p-forms for p > n, since all of the components will automatically be zero by antisymmetry.
Why should we care about differential forms? This is a hard question to answer without some more work, but the basic idea is that forms can be both differentiated and integrated, without the help of any additional geometric structure. We will delay integration theory until later, but see how to differentiate forms shortly.
Given a p-form A and a q-form B, we can form a (p + q)-form known as the wedge product A B by taking the antisymmetrized tensor product:
(1.79)
Thus, for example, the wedge product of two 1-forms is
(1.80)
Note that
(1.81)
so you can alter the order of a wedge product if you are careful with signs.
The exterior derivative "d" allows us to differentiate p-form fields to obtain (p + 1)-form fields. It is defined as an appropriately normalized antisymmetric partial derivative:
(1.82)
The simplest example is the gradient, which is the exterior derivative of a 1-form:
(1.83)
The reason why the exterior derivative deserves special attention is that it is a tensor, even in curved spacetimes, unlike its cousin the partial derivative. Since we haven't studied curved spaces yet, we cannot prove this, but (1.82) defines an honest tensor no matter what the metric and coordinates are.
Another interesting fact about exterior differentiation is that, for any form A,
(1.84)
which is often written d2 = 0. This identity is a consequence of the definition of d and the fact that partial derivatives commute, = (acting on anything). This leads us to the following mathematical aside, just for fun. We define a p-form A to be closed if dA = 0, and exact if A = dB for some (p - 1)-form B. Obviously, all exact forms are closed, but the converse is not necessarily true. On a manifold M, closed p-forms comprise a vector space Zp(M), and exact forms comprise a vector space Bp(M). Define a new vector space as the closed forms modulo the exact forms:
(1.85)
This is known as the pth de Rham cohomology vector space, and depends only on the topology of the manifold M. (Minkowski space is topologically equivalent to R4, which is uninteresting, so that all of the Hp(M) vanish for p > 0; for p = 0 we have H0(M) = . Therefore in Minkowski space all closed forms are exact except for zero-forms; zero-forms can't be exact since there are no -1-forms for them to be the exterior derivative of.) It is striking that information about the topology can be extracted in this way, which essentially involves the solutions to differential equations. The dimension bp of the space Hp(M) is called the pth Betti number of M, and the Euler characteristic is given by the alternating sum
(1.86)
Cohomology theory is the basis for much of modern differential topology.
Moving back to reality, the final operation on differential forms we will introduce is Hodge duality. We define the "Hodge star operator" on an n-dimensional manifold as a map from p-forms to (n - p)-forms,
(1.87)
mapping A to "A dual". Unlike our other operations on forms, the Hodge dual does depend on the metric of the manifold (which should be obvious, since we had to raise some indices on the Levi-Civita tensor in order to define (1.87)). Applying the Hodge star twice returns either plus or minus the original form:
(1.88)
where s is the number of minus signs in the eigenvalues of the metric (for Minkowski space, s = 1).
Two facts on the Hodge dual: First, "duality" in the sense of Hodge is different than the relationship between vectors and dual vectors, although both can be thought of as the space of linear maps from the original space to R. Notice that the dimensionality of the space of (n - p)-forms is equal to that of the space of p-forms, so this has at least a chance of being true. In the case of forms, the linear map defined by an (n - p)-form acting on a p-form is given by the dual of the wedge product of the two forms. Thus, if A(n - p) is an (n - p)-form and B(p) is a p-form at some point in spacetime, we have
(1.89)
The second fact concerns differential forms in 3-dimensional Euclidean space. The Hodge dual of the wedge product of two 1-forms gives another 1-form:
(1.90)
(All of the prefactors cancel.) Since 1-forms in Euclidean space are just like vectors, we have a map from two vectors to a single vector. You should convince yourself that this is just the conventional cross product, and that the appearance of the Levi-Civita tensor explains why the cross product changes sign under parity (interchange of two coordinates, or equivalently basis vectors). This is why the cross product only exists in three dimensions - because only in three dimensions do we have an interesting map from two dual vectors to a third dual vector. If you wanted to you could define a map from n - 1 one-forms to a single one-form, but I'm not sure it would be of any use.
Electrodynamics provides an especially compelling example of the use of differential forms. From the definition of the exterior derivative, it is clear that equation (1.78) can be concisely expressed as closure of the two-form F:
(1.91)
Does this mean that F is also exact? Yes; as we've noted, Minkowski space is topologically trivial, so all closed forms are exact. There must therefore be a one-form A such that
(1.92)
This one-form is the familiar vector potential of electromagnetism, with the 0 component given by the scalar potential, A0 = . If one starts from the view that the A is the fundamental field of electromagnetism, then (1.91) follows as an identity (as opposed to a dynamical law, an equation of motion). Gauge invariance is expressed by the observation that the theory is invariant under A A + d for some scalar (zero-form) , and this is also immediate from the relation (1.92). The other one of Maxwell's equations, (1.77), can be expressed as an equation between three-forms:
(1.93)
where the current one-form J is just the current four-vector with index lowered. Filling in the details is left for you to do.
As an intriguing aside, Hodge duality is the basis for one of the hottest topics in theoretical physics today. It's hard not to notice that the equations (1.91) and (1.93) look very similar. Indeed, if we set J = 0, the equations are invariant under the "duality transformations"
(1.94)
We therefore say that the vacuum Maxwell's equations are duality invariant, while the invariance is spoiled in the presence of charges. We might imagine that magnetic as well as electric monopoles existed in nature; then we could add a magnetic current term 4(*JM) to the right hand side of (1.91), and the equations would be invariant under duality transformations plus the additional replacement J JM. (Of course a nonzero right hand side to (1.91) is inconsistent with F = dA, so this idea only works if A is not a fundamental variable.) Long ago Dirac considered the idea of magnetic monopoles and showed that a necessary condition for their existence is that the fundamental monopole charge be inversely proportional to the fundamental electric charge. Now, the fundamental electric charge is a small number; electrodynamics is "weakly coupled", which is why perturbation theory is so remarkably successful in quantum electrodynamics (QED). But Dirac's condition on magnetic charges implies that a duality transformation takes a theory of weakly coupled electric charges to a theory of strongly coupled magnetic monopoles (and vice-versa). Unfortunately monopoles don't exist (as far as we know), so these ideas aren't directly applicable to electromagnetism; but there are some theories (such as supersymmetric non-abelian gauge theories) for which it has been long conjectured that some sort of duality symmetry may exist. If it did, we would have the opportunity to analyze a theory which looked strongly coupled (and therefore hard to solve) by looking at the weakly coupled dual version. Recently work by Seiberg and Witten and others has provided very strong evidence that this is exactly what happens in certain theories. The hope is that these techniques will allow us to explore various phenomena which we know exist in strongly coupled quantum field theories, such as confinement of quarks in hadrons.
We've now gone over essentially everything there is to know about the care and feeding of tensors. In the next section we will look more carefully at the rigorous definitions of manifolds and tensors, but the basic mechanics have been pretty well covered. Before jumping to more abstract mathematics, let's review how physics works in Minkowski spacetime.
Start with the worldline of a single particle. This is specified by a map M, where M is the manifold representing spacetime; we usually think of the path as a parameterized curve x(). As mentioned earlier, the tangent vector to this path is dx/d (note that it depends on the parameterization). An object of primary interest is the norm of the tangent vector, which serves to characterize the path; if the tangent vector is timelike/null/spacelike at some parameter value , we say that the path is timelike/null/spacelike at that point. This explains why the same words are used to classify vectors in the tangent space and intervals between two points - because a straight line connecting, say, two timelike separated points will itself be timelike at every point along the path.
Nevertheless, it's important to be aware of the sleight of hand which is being pulled here. The metric, as a (0, 2) tensor, is a machine which acts on two vectors (or two copies of the same vector) to produce a number. It is therefore very natural to classify tangent vectors according to the sign of their norm. But the interval between two points isn't something quite so natural; it depends on a specific choice of path (a "straight line") which connects the points, and this choice in turn depends on the fact that spacetime is flat (which allows a unique choice of straight line between the points). A more natural object is the line element, or infinitesimal interval:
(1.95)
From this definition it is tempting to take the square root and integrate along a path to obtain a finite interval. But since ds2 need not be positive, we define different procedures for different cases. For spacelike paths we define the path length
(1.96)
where the integral is taken over the path. For null paths the interval is zero, so no extra formula is required. For timelike paths we define the proper time
(1.97)
which will be positive. Of course we may consider paths that are timelike in some places and spacelike in others, but fortunately it is seldom necessary since the paths of physical particles never change their character (massive particles move on timelike paths, massless particles move on null paths). Furthermore, the phrase "proper time" is especially appropriate, since actually measures the time elapsed on a physical clock carried along the path. This point of view makes the "twin paradox" and similar puzzles very clear; two worldlines, not necessarily straight, which intersect at two different events in spacetime will have proper times measured by the integral (1.97) along the appropriate paths, and these two numbers will in general be different even if the people travelling along them were born at the same time.
Let's move from the consideration of paths in general to the paths of massive particles (which will always be timelike). Since the proper time is measured by a clock travelling on a timelike worldline, it is convenient to use as the parameter along the path. That is, we use (1.97) to compute (), which (if is a good parameter in the first place) we can invert to obtain (), after which we can think of the path as x(). The tangent vector in this parameterization is known as the four-velocity, U:
(1.98)
Since d = - dxdx, the four-velocity is automatically normalized:
(1.99)
(It will always be negative, since we are only defining it for timelike trajectories. You could define an analogous vector for spacelike paths as well; null paths give some extra problems since the norm is zero.) In the rest frame of a particle, its four-velocity has components U = (1, 0, 0, 0).
A related vector is the energy-momentum four-vector, defined by
(1.100)
where m is the mass of the particle. The mass is a fixed quantity independent of inertial frame; what you may be used to thinking of as the "rest mass." It turns out to be much more convenient to take this as the mass once and for all, rather than thinking of mass as depending on velocity. The energy of a particle is simply p0, the timelike component of its energy-momentum vector. Since it's only one component of a four-vector, it is not invariant under Lorentz transformations; that's to be expected, however, since the energy of a particle at rest is not the same as that of the same particle in motion. In the particle's rest frame we have p0 = m; recalling that we have set c = 1, we find that we have found the equation that made Einstein a celebrity, E = mc2. (The field equations of general relativity are actually much more important than this one, but " R - Rg = 8GT" doesn't elicit the visceral reaction that you get from "E = mc2".) In a moving frame we can find the components of p by performing a Lorentz transformation; for a particle moving with (three-) velocity v along the x axis we have
(1.101)
where = 1/. For small v, this gives p0 = m + mv2 (what we usually think of as rest energy plus kinetic energy) and p1 = mv (what we usually think of as [Newtonian] momentum). So the energy-momentum vector lives up to its name.
The centerpiece of pre-relativity physics is Newton's 2nd Law, or = m = d/dt. An analogous equation should hold in SR, and the requirement that it be tensorial leads us directly to introduce a force four-vector f satisfying
(1.102)
The simplest example of a force in Newtonian physics is the force due to gravity. In relativity, however, gravity is not described by a force, but rather by the curvature of spacetime itself. Instead, let us consider electromagnetism. The three-dimensional Lorentz force is given by = q( + × ), where q is the charge on the particle. We would like a tensorial generalization of this equation. There turns out to be a unique answer:
(1.103)
You can check for yourself that this reduces to the Newtonian version in the limit of small velocities. Notice how the requirement that the equation be tensorial, which is one way of guaranteeing Lorentz invariance, severely restricted the possible expressions we could get. This is an example of a very general phenomenon, in which a small number of an apparently endless variety of possible physical laws are picked out by the demands of symmetry.
Although p provides a complete description of the energy and momentum of a particle, for extended systems it is necessary to go further and define the energy-momentum tensor (sometimes called the stress-energy tensor), T. This is a symmetric (2, 0) tensor which tells us all we need to know about the energy-like aspects of a system: energy density, pressure, stress, and so forth. A general definition of T is "the flux of four-momentum p across a surface of constant x". To make this more concrete, let's consider the very general category of matter which may be characterized as a fluid - a continuum of matter described by macroscopic quantities such as temperature, pressure, entropy, viscosity, etc. In fact this definition is so general that it is of little use. In general relativity essentially all interesting types of matter can be thought of as perfect fluids, from stars to electromagnetic fields to the entire universe. Schutz defines a perfect fluid to be one with no heat conduction and no viscosity, while Weinberg defines it as a fluid which looks isotropic in its rest frame; these two viewpoints turn out to be equivalent. Operationally, you should think of a perfect fluid as one which may be completely characterized by its pressure and density.
To understand perfect fluids, let's start with the even simpler example of dust. Dust is defined as a collection of particles at rest with respect to each other, or alternatively as a perfect fluid with zero pressure. Since the particles all have an equal velocity in any fixed inertial frame, we can imagine a "four-velocity field" U(x) defined all over spacetime. (Indeed, its components are the same at each point.) Define the number-flux four-vector to be
(1.104)
where n is the number density of the particles as measured in their rest frame. Then N0 is the number density of particles as measured in any other frame, while Ni is the flux of particles in the xi direction. Let's now imagine that each of the particles have the same mass m. Then in the rest frame the energy density of the dust is given by
(1.105)
By definition, the energy density completely specifies the dust. But only measures the energy density in the rest frame; what about other frames? We notice that both n and m are 0-components of four-vectors in their rest frame; specifically, N = (n, 0, 0, 0) and p = (m, 0, 0, 0). Therefore is the = 0, = 0 component of the tensor p N as measured in its rest frame. We are therefore led to define the energy-momentum tensor for dust:
(1.106)
where is defined as the energy density in the rest frame.
Having mastered dust, more general perfect fluids are not much more complicated. Remember that "perfect" can be taken to mean "isotropic in its rest frame." This in turn means that T is diagonal - there is no net flux of any component of momentum in an orthogonal direction. Furthermore, the nonzero spacelike components must all be equal, T11 = T22 = T33. The only two independent numbers are therefore T00 and one of the Tii; we can choose to call the first of these the energy density , and the second the pressure p. (Sorry that it's the same letter as the momentum.) The energy-momentum tensor of a perfect fluid therefore takes the following form in its rest frame:
(1.107)
We would like, of course, a formula which is good in any frame. For dust we had T = UU, so we might begin by guessing ( + p)UU, which gives
(1.108)
(1.109)
Fortunately, this has an obvious covariant generalization, namely p. Thus, the general form of the energy-momentum tensor for a perfect fluid is
(1.110)
This is an important formula for applications such as stellar structure and cosmology.
As further examples, let's consider the energy-momentum tensors of electromagnetism and scalar field theory. Without any explanation at all, these are given by
(1.111)
and
(1.112)
You can check for yourself that, for example, T00 in each case is equal to what you would expect the energy density to be.
Besides being symmetric, T has the even more important property of being conserved. In this context, conservation is expressed as the vanishing of the "divergence":
(1.113)
This is a set of four equations, one for each value of . The = 0 equation corresponds to conservation of energy, while Tk = 0 expresses conservation of the kth component of the momentum. We are not going to prove this in general; the proof follows for any individual source of matter from the equations of motion obeyed by that kind of matter. In fact, one way to define T would be "a (2, 0) tensor with units of energy per volume, which is conserved." You can prove conservation of the energy-momentum tensor for electromagnetism, for example, by taking the divergence of (1.111) and using Maxwell's equations as previously discussed.
A final aside: we have already mentioned that in general relativity gravitation does not count as a "force." As a related point, the gravitational field also does not have an energy-momentum tensor. In fact it is very hard to come up with a sensible local expression for the energy of a gravitational field; a number of suggestions have been made, but they all have their drawbacks. Although there is no "correct" answer, it is an important issue from the point of view of asking seemingly reasonable questions such as "What is the energy emitted per second from a binary pulsar as the result of gravitational radiation?" | 2016-05-03 08:50:35 | {"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": 0.8851256966590881, "perplexity": 270.8818950470579}, "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-2016-18/segments/1461860121090.75/warc/CC-MAIN-20160428161521-00025-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://theambientproject.com/6g00aaq/numerical-questions-on-gravitation-for-class-9-1b71ce | From quality notes to books and from exercises to past papers. So, have a look at physics 9th class notes containing numerical, short questions, long questions and multiple choice question. PDF download of these motion class 9 numericals is also available. Question numerical. (d) thrust. A body of 90 kg f on the surface of earth. The number of oxygen atoms in 4.4 g of CO 2 is approx. Therefore I encourage you to practice these mcqs set like a quiz. Galileo Galilei. March 14, 2018 at 5:24 pm Reply . (d) 9.8 m/s 2 Question 2. Lessons. The value of acceleration due to gravity on the surface of the earth at sea level is (a) 4.9 m/s 2 (b) 6 m/s 2 (c) 8 m/s 2 (d) 9.8 m/s 2 Answer. (c) 10 kg. Universal law of gravity. 0. These 9th class physics notes of 5th Chapter are according to the new syllabus (2013) of Punjab Boards including Lahore Board, Gujranwala Board, Sahiwal Board, Rawalpindi Board, Faisalabad Board, Multan Board, Bahawalpur Board, Federal Board, … Page No: 100. 2. Revise the difference between weight and mass. 3. Who formulated the universal law of gravitation? Answer. H C Verma class 9 Numerical Problems. Active; Voted; Newest; Oldest; 0. doubts Posted May 21, 2018 . Question from very important topics are covered by NCERT Exemplar Class 9.You also get idea about the type of questions and method to answer in your Class 9th examination. Learn. Contains solved exercises, review questions, MCQs, important board questions and chapter overview. 7. This document is highly rated by Class 9 students and has been viewed 61836 times. A body of 90 kg f on the surface of earth. We provide you with everything that is related to Physics. Q1. 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Law of Gravitation was coined by Sir Issac Newton known as pressure of the... # 5 for 9th Class notes containing numerical, short questions, mcqs, important Board questions and Chapter.... | 2021-10-16 22:07:18 | {"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": 0.3606748580932617, "perplexity": 2578.269773116231}, "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/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00221.warc.gz"} |
https://scicomp.stackexchange.com/questions/35610/openmp-inconsistently-segfaulting | # OpenMP inconsistently segfaulting
I parallelized my code with openMP, and now have this bug in my code that's really odd. If it's the first time I run on a computer, it segfaults, but if I run the program again it runs fine. I've never seen anything like this and was curious if anyone has seen this before or has good resources or hints on how to fix this. It's a very simple loop that it segfaults on, and I parallelized every loop with default none. The variable declaration followed by the loop is below. I had previously posted a stripped down version of the loop for readability, but it was requested that I fill it in for the sake of more clarity :
real(dp) :: q, oneOverq, L, D, Pressure, u, v, En, rho, oneOverrho
real(dp) :: normal(Mesh%numDims), wL(Mesh%numFields), cd_dql(Mesh%numFields, Mesh%numEdges)
real(dp) :: norm_dx1(Mesh%numDims, Mesh%numDims), norm_dx2(Mesh%numDims, Mesh%numDims)
real(dp) :: L_dx(Mesh%numDims, Mesh%numNodes), D_dx(Mesh%numDims, Mesh%numNodes)
real(dp) :: p_dql(Mesh%numFields), L_dql(Mesh%numFields, Mesh%numEdges), Res(Mesh%numFields, Mesh%numTri)
INTEGER(i4) :: i, j, k, m, nodecount, cellNum, edgeNum, Node1, Node2, node, cell, cells(3)
!$$OMP PARALLEL DO Reduction(+:L, D, cL_dql, L_dx, cD_dql, D_Dx) default(none) shared(Mesh, W, grad, gamma, dRdX, Jac_2nd) & !$$OMP private(cellnum, edgenum, node1, node2, rho, oneoverrho, u, v, En, pressure, normal, norm_dx1, norm_dx2, p_dql, wL, cell, node, cells)
!DIR$IVDEP:LOOP do cellnum = 1, Mesh%numTri edgeNum = Mesh%airfoilEdges(i) Node1 = Mesh%edgelist(edgenum,1) Node2 = Mesh%edgelist(edgenum,2) cellNum = Mesh%edgeList(edgenum,3) call cell_stencil(Mesh, cellnum, cells) wL = W(:, cellnum) call linear_limited_reconstruction(cellnum, edgenum, Mesh, grad(:,:,cellNum), wL) rho = wL(1) oneOverrho = 1.0_dp/rho u = wL(2)*oneOverrho v = wL(3)*oneOverrho En = wL(4)*oneOverrho call get_pressure(rho, En, u, v, pressure) p_dql(1) = (gamma - 1._dp)*half*( u**2 + v**2 ) p_dql(2) = (gamma - 1._dp)*-u p_dql(3) = (gamma - 1._dp)*-v p_dql(4) = gamma-1._dp call norm_dx(node1, node2, Mesh, normal, norm_dx1, norm_dx2) L = L + Pressure*normal(2) D = D + Pressure*normal(1) do j = 1, 4 if (j == 1) then cell = cellnum else cell = cells(j-1) end if if (cell /= 0) then cL_dql(:,cell) = cL_dql(:,cell) + matmul(P_dql(:),Jac_2nd%wL_rc_dq(:,:,j,edgenum))*normal(2) cD_dql(:,cell) = cD_dql(:,cell) + matmul(P_dql(:),Jac_2nd%wL_rc_dq(:,:,j,edgenum))*normal(1) end if end do L_dx(:, node1) = L_dx(:, node1) + Pressure*norm_dx1(2, :) L_dx(:, node2) = L_dx(:, node2) + Pressure*norm_dx2(2, :) D_dx(:, node1) = D_dx(:, node1) + Pressure*norm_dx1(1, :) D_dx(:, node2) = D_dx(:, node2) + Pressure*norm_dx2(1, :) do j = 1, 6 node = Mesh%NodeStencil(j, cellnum) if (node /= 0) then do k = 1, Mesh%numFields L_dx(:,node) = L_dx(:,node) + P_dql(k)*dRdX%wL_rc_dx(k,:,j,edgenum)*normal(2) D_dx(:,node) = D_dx(:,node) + P_dql(k)*dRdX%wL_rc_dx(k,:,j,edgenum)*normal(1) end do end if end do end do !$OMP END PARALLEL DO
I can put in the full code up in the code block, but for the sake of readability I thought it was best not to. If anyone has any advice I'd greatly appreciate it.
Thanks!
• Often these sorts of heisenbugs happens because of an out-of-bounds memory access. Depending on the operating system's whim of where arrays get allocated into heap memory, that memory access can either segfault or run over into already allocated data, the first of which is obvious and the second of which is not. You can try compiling with -fsanitize=address and see what happens; there are also thread, leak, memory sanitizers, etc. Jul 24 '20 at 15:20
• As with all things, I posted and then found the answer (I think). It looks like default none didn't properly work. I hadn't specified two indices, i and j, that I was using to assign the data to the appropriate arrays, so I guess sometimes it corrected had them as private, and other times not so. It looks like your hunch was right, and I'll definintely use that flag in the future for my memory debugging. Do you have any other suggestions as to what sanitizers to use, etc? thanks!
– EMP
Jul 24 '20 at 15:30
• The address and leak sanitizers have been the most useful to me, but I code more in C/C++ than Fortran and there you have to be more careful about memory leaks. What I would definitely recommend is that you put your project is on github or gitlab, set up an automated test suite for it, and enable address sanitizer for the testing build. Once that's set up, it's easy to add more sanitizers later. Jul 24 '20 at 17:16
• If default (none) is not working properly that is a VERY broken compiler - what is it? To be honest that would be so broken that I can't believe what you attest to be the problem. But to say more we really need to see at the very least "Computational things" and the declaration of the variables. Jul 24 '20 at 18:32
• It's the intel ifort compiler. I used the i and j variables inside the computational section. From my understanding of default(none) it should have given an error since I hadn't specified either public or private. Since I made this change, I haven't had a problem.
– EMP
Jul 24 '20 at 19:07 | 2021-10-27 17:39:25 | {"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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30950117111206055, "perplexity": 1630.5762019766416}, "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/1634323588216.48/warc/CC-MAIN-20211027150823-20211027180823-00519.warc.gz"} |
https://www.cuemath.com/questions/how-to-find-the-radius-and-diameter-of-a-circle/ | # The radius and diameter of a circle
## Question: How to find the radius and diameter of a circle?
Diameter is the longest chord of the circle and radius is half of the diameter.
## Answer: Using the formula d = C/π and d = 2r we can find diameter and radius of a circle
Let's compute the diameter and radius of the circle
## Explanation:
Circumference of a circle (C) = 2πr or πd
where, r = radius of the circle, d = diameter of the circle and d = 2r
=> d = C/π ----------------------------- (1)
Let's take an example to understand this
If the circumference of a circle is 20π units then,
Diameter(d) = 20π / π = 20 units (from equation (1))
Radius = d/2= 20/2= 10 units (since, d = 2r) | 2021-05-08 17:26:19 | {"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": 0.9193443655967712, "perplexity": 1605.9388480030366}, "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/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00433.warc.gz"} |
https://www.jeremyinstem.com/puzzle/consecutive-integers/ | Consecutive Integers
Three consecutive integers are multiplied together, and the middle number is added to the total.
E.g. $(4 \times 5 \times 6)+5=125=5^3$
Prove that this is always true, with any set of three consecutive integers.
Source | 2019-06-16 01:33:53 | {"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": 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": 0.34537258744239807, "perplexity": 529.1940138562193}, "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-2019-26/segments/1560627997508.21/warc/CC-MAIN-20190616002634-20190616024634-00515.warc.gz"} |
https://en.m.wikipedia.org/wiki/Pareto_principle | Pareto principle
The Pareto Principle is derived from Vilfredo Pareto's observation that only a "vital few" of the peapods in his garden produced the majority of peas.
The Pareto principle (also known as the 80/20 rule, the law of the vital few, or the principle of factor sparsity)[1] states that, for many events, roughly 80% of the effects come from 20% of the causes.[2] Management consultant Joseph M. Juran suggested the principle and named it after Italian economist Vilfredo Pareto, who noted the 80/20 connection while at the University of Lausanne in 1896, as published in his first paper, "Cours d'économie politique". Essentially, Pareto showed that approximately 80% of the land in Italy was owned by 20% of the population; Pareto developed the principle by observing that about 20% of the peapods in his garden contained 80% of the peas.[3]
It is a common rule of thumb in business; e.g., "80% of your sales come from 20% of your clients." Mathematically, the 80/20 rule is roughly followed by a power law distribution (also known as a Pareto distribution) for a particular set of parameters, and many natural phenomena have been shown empirically to exhibit such a distribution.[4]
The Pareto principle is only tangentially related to Pareto efficiency. Pareto developed both concepts in the context of the distribution of income and wealth among the population.
In economicsEdit
The original observation was in connection with population and wealth. Pareto noticed that 80% of Italy's land was owned by 20% of the population.[5] He then carried out surveys on a variety of other countries and found to his surprise that a similar distribution applied.
A chart that gave the inequality a very visible and comprehensible form, the so-called 'champagne glass' effect,[6] was contained in the 1992 United Nations Development Program Report, which showed that distribution of global income is very uneven, with the richest 20% of the world's population controlling 82.7% of the world's income.[7]
Distribution of world GDP, 1989[8]
Quintile of population Income
Richest 20% 82.70%
Second 20% 11.75%
Third 20% 2.30%
Fourth 20% 1.85%
Poorest 20% 1.40%
In scienceEdit
The more predictions a theory makes, the greater the chance is of some of them being cheaply testable. Modifications of existing theories make many fewer new unique predictions, increasing the risk that the few predictions remaining will be very expensive to test.[9]
In softwareEdit
In computer science and engineering control theory, such as for electromechanical energy converters, the Pareto principle can be applied to optimization efforts.[10]
For example, Microsoft noted that by fixing the top 20% of the most-reported bugs, 80% of the related errors and crashes in a given system would be eliminated.[11]
In load testing, it is common practice to estimate that 80% of the traffic occurs during 20% of the time period.[citation needed]
In software engineering, Lowell Arthur expressed a corollary principle: "20 percent of the code has 80 percent of the errors. Find them, fix them!"[12]
In sportsEdit
It is said that about 20% of sportsmen participate in 80% of big competitions and out of them, 20% win 80% of the awards. This could also be applied to teams in many popular games.
The Pareto principle has also been applied to training, where roughly 20% of the exercises and habits have 80% of the impact and the trainee should not focus so much on a varied training.[13] This does not necessarily mean eating heartily or going to the gym are not important, just that they are not as significant as the key activities.
The law of the few can be also seen in betting, where it is said that with 20% effort you can match the accuracy of 80% of the bettors.[14]
Occupational health and safetyEdit
Occupational health and safety professionals use the Pareto principle to underline the importance of hazard prioritization. Assuming 20% of the hazards account for 80% of the injuries, and by categorizing hazards, safety professionals can target those 20% of the hazards that cause 80% of the injuries or accidents. Alternatively, if hazards are addressed in random order, a safety professional is more likely to fix one of the 80% of hazards that account only for some fraction of the remaining 20% of injuries.[15]
Aside from ensuring efficient accident prevention practices, the Pareto principle also ensures hazards are addressed in an economical order as the technique ensures the resources used are best used to prevent the most accidents.[16]
Other applicationsEdit
In the systems science discipline, Epstein and Axtell created an agent-based simulation model called SugarScape, from a decentralized modeling approach, based on individual behavior rules defined for each agent in the economy. Wealth distribution and Pareto's 80/20 principle became emergent in their results, which suggests the principle is a collective consequence of these individual rules.[17]
The Pareto principle has many applications in quality control.[citation needed] It is the basis for the Pareto chart, one of the key tools used in total quality control and Six Sigma techniques. The Pareto principle serves as a baseline for ABC-analysis and XYZ-analysis, widely used in logistics and procurement for the purpose of optimizing stock of goods, as well as costs of keeping and replenishing that stock.[18]
The Pareto principle was also mentioned in the book 24/8 - The Secret for being Mega-Effective by Achieving More in Less Time by Amit Offir. Offir claims that if you want to function as a one-stop shop, focus on the 20% of a project that is important and you will save a lot of time and energy.
In health care in the United States, 20% of patients have been found to use 80% of health care resources.[19]
The Dunedin Study has found 80% of crimes are committed by 20% of criminals.[20] This statistic is used to support both stop-and-frisk policies and broken windows policing, as catching those criminals committing minor crimes will likely net many criminals wanted for (or who would normally commit) larger ones.
Mathematical notesEdit
The idea has a rule of thumb application in many places, but it is commonly misused. For example, it is a misuse to state a solution to a problem "fits the 80/20 rule" just because it fits 80% of the cases; it must also be that the solution requires only 20% of the resources that would be needed to solve all cases. Additionally, it is a misuse of the 80/20 rule to interpret data with a small number of categories or observations.
This is a special case of the wider phenomenon of Pareto distributions. If the Pareto index α, which is one of the parameters characterizing a Pareto distribution, is chosen as α = log45 ≈ 1.16, then one has 80% of effects coming from 20% of causes.
It follows that one also has 80% of that top 80% of effects coming from 20% of that top 20% of causes, and so on. Eighty percent of 80% is 64%; 20% of 20% is 4%, so this implies a "64/4" law; and similarly implies a "51.2/0.8" law. Similarly for the bottom 80% of causes and bottom 20% of effects, the bottom 80% of the bottom 80% only cause 20% of the remaining 20%. This is broadly in line with the world population/wealth table above, where the bottom 60% of the people own 5.5% of the wealth, as near a 64/4 connection.
The 64/4 correlation also implies a 32% 'fair' area between the 4% and 64%, where the lower 80% of the top 20% (16%) and upper 20% of the bottom 80% (also 16%) relates to the corresponding lower top and upper bottom of effects (32%). This is also broadly in line with the world population table above, where the second 20% control 12% of the wealth, and the bottom of the top 20% (presumably) control 16% of the wealth.
The term 80/20 is only a shorthand for the general principle at work. In individual cases, the distribution could just as well be, say, nearer to 80/10 or 80/30. There is no need for the two numbers to add up to the number 100, as they are measures of different things, (e.g., 'number of customers' vs 'amount spent'). However, each case in which they do not add up to 100%, is equivalent to one in which they do. For example, as noted above, the "64/4 law" (in which the two numbers do not add up to 100%) is equivalent to the "80/20 law" (in which they do add up to 100%). Thus, specifying two percentages independently does not lead to a broader class of distributions than what one gets by specifying the larger one and letting the smaller one be its complement relative to 100%. Thus, there is only one degree of freedom in the choice of that parameter.
Adding up to 100 leads to a nice symmetry. For example, if 80% of effects come from the top 20% of sources, then the remaining 20% of effects come from the lower 80% of sources. This is called the "joint ratio", and can be used to measure the degree of imbalance: a joint ratio of 96:4 is very imbalanced, 80:20 is significantly imbalanced (Gini index: 60%), 70:30 is moderately imbalanced (Gini index: 40%), and 55:45 is just slightly imbalanced.
The Pareto principle is an illustration of a "power law" relationship, which also occurs in phenomena such as brush fires and earthquakes.[21] Because it is self-similar over a wide range of magnitudes, it produces outcomes completely different from Gaussian distribution phenomena. This fact explains the frequent breakdowns of sophisticated financial instruments, which are modeled on the assumption that a Gaussian relationship is appropriate to, for example, stock price movements.[22]
Equality measuresEdit
Gini coefficient and Hoover indexEdit
Using the "A : B" notation (for example, 0.8:0.2) and with A + B = 1, inequality measures like the Gini index (G) and the Hoover index (H) can be computed. In this case both are the same.
${\displaystyle H=G=|2A-1|=|1-2B|\,}$
${\displaystyle A:B=\left({\frac {1+H}{2}}\right):\left({\frac {1-H}{2}}\right)}$
Theil indexEdit
The Theil index is an entropy measure used to quantify inequalities. The measure is 0 for 50:50 distributions and reaches 1 at a Pareto distribution of 82:18. Higher inequalities yield Theil indices above 1.[23]
${\displaystyle T_{T}=T_{L}=T_{s}=2H\,\operatorname {artanh} (H).\,}$
ReferencesEdit
1. ^ THE APPLICATION OF THE PARETO PRINCIPLE IN SOFTWARE ENGINEERING. Ankunda R. Kiremire 19th October, 2011
2. ^ Bunkley, Nick (March 3, 2008), "Joseph Juran, 103, Pioneer in Quality Control, Dies", New York Times
3. ^ "What is 80/20 rule?". 80/20 Rule of Presenting Ideas. Archived from the original on January 28, 2013. Retrieved October 4, 2015.
4. ^ Newman, MEJ. "Power laws, Pareto Distributions, and Zipf's law" (PDF). p. 11. Retrieved 10 April 2011.
5. ^ Pareto, Vilfredo; Page, Alfred N. (1971), Translation of Manuale di economia politica ("Manual of political economy"), A.M. Kelley, ISBN 978-0-678-00881-2
6. ^ Gorostiaga, Xabier (January 27, 1995), "World has become a 'champagne glass' globalisation will fill it fuller for a wealthy few", National Catholic Reporter
7. ^ United Nations Development Program (1992), 1992 Human Development Report, New York: Oxford University Press
8. ^ Human Development Report 1992, Chapter 3, retrieved 2007-07-08
9. ^ "Illustrations of the Logic of Science" 1877–1878, Charles Sanders Peirce
10. ^ Gen, M.; Cheng, R. (2002), Genetic Algorithms and Engineering Optimization, New York: Wiley
11. ^ Rooney, Paula (October 3, 2002), Microsoft's CEO: 80–20 Rule Applies To Bugs, Not Just Features, ChannelWeb
12. ^ Pressman, Roger S. (2010). Software Engineering: A Practitioner's Approach (7th ed.). Boston, Mass: McGraw-Hill, 2010. ISBN 978-0-07-337597-7.
13. ^
14. ^
15. ^ Woodcock, Kathryn (2010). Safety Evaluation Techniques. Toronto, ON: Ryerson University. p. 86.
16. ^ "Introduction to Risk-based Decision-Making" (PDF). USCG Safety Program. United States Coast Guard. Retrieved 14 January 2012.
17. ^ Epstein, Joshua; Axtell, Robert (1996), Growing Artificial Societies: Social Science from the Bottom-Up, MIT Press, p. 208, ISBN 0-262-55025-3
18. ^ Rushton, Oxley & Croucher (2000), pp. 107–108.
19. ^ Myrl Weinberg: In health-care reform, the 20-80 solution | Contributors | projo.com | The Providence Journal
20. ^ Nicola, Davis (2016), 'High social cost' adults can be predicted from as young as three, says study, The Guardian
21. ^ Bak, Per (1999), How Nature Works: the science of self-organized criticality, Springer, p. 89, ISBN 0-387-94791-4
22. ^ Taleb, Nassim (2007), The Black Swan, pp. 229–252, 274–285
23. ^ On Line Calculator: Inequality | 2017-06-23 17:28:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.6743828058242798, "perplexity": 2216.9547353034436}, "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-2017-26/segments/1498128320077.32/warc/CC-MAIN-20170623170148-20170623190148-00499.warc.gz"} |
https://dsp.stackexchange.com/questions/71288/why-this-system-is-linear | # Why this system is linear?
Hi guys i'm studying signals and systems, and my professor told us that
$$y(t) = \int\limits_{ t+T }^{t-T/2} {x(a+T/2)}\mathrm{d} a$$
is a linear system.
But a primitive of $$x$$ isn't $$x^2$$ ? How it's possible that's linear ?
That's not an integral of variable $$x$$. The notation $$x(a+T/2)$$ stands for a function $$x(\cdot)$$ of variable $$a$$.
So applying the fundamental theorem of calculus, and assuming there exists a function $$G(a)$$ such that $$G'(a) = x(a)$$, then you will have :
$$\int x(a+T/2) da = \int G'(a+T/2)da = G(a+T/2) + C$$
where the constant of integration, $$C$$, will be omitted in the definite integral :
$$\int_{t+T}^{t-T/2} x(a+T/2) da = \int_{t+T}^{t-T/2} G'(a+T/2)da = G(a+T/2)|_{t+T}^{t-T/2}$$
So the system has nothing with a square function.
Coming to its linearity, you can show this in line with the linearity of the integral operator...
• Even if was the integral of $x$ times $(a + T/2)$, the integration is in $a$, so it's still linear in $x$. – TimWescott Nov 7 '20 at 0:15
• @TimWescott Good catch! – Fat32 Nov 7 '20 at 0:29
The apparently complicated integral bounds $$t+T$$ and $$t-T/2$$, or the shift $$(a+T/2)$$ with independent variable $$a$$ inside the integral, obfuscate the simplicity of the system. It computes a signed area, on a constantly moving window $$[T,-T/2]$$ that moves around $$t$$, for an input that has a constant shift. All those ingredients suggest that the system could be linear.
To see that in a clearer fashion, it could be useful to simplify it a bit. By a variable change $$u=a+T/2$$, the system $$S$$ becomes:
$$y(t) = S(x(t))=-\int_t^{t+3T/2}x(u)\mathrm{d}u\,.$$
Then one can verify whether $$S(\lambda x_1(t)+\mu x_2(t))$$ is equal to $$\lambda S( x_1(t))+\mu S(x_2(t))$$. It was possible to check already on the original formula, maybe it is simpler with the simplified form. | 2021-04-22 10:46:39 | {"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": 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": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.897511899471283, "perplexity": 269.48183552191256}, "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/1618039603582.93/warc/CC-MAIN-20210422100106-20210422130106-00141.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/35324-linear-transformation-question-print.html | # Linear Transformation Question
• Apr 20th 2008, 09:51 PM
lllll
Linear Transformation Question
let $B$ be an $n \times p$ matrix. For each $j \ (1 \leq j \leq p)$ let $v_j$ denote the jth column of B. Prove that:
$v_j =Be_j$, where $e_j$ is the jth standard vector of $F^p$
so far all I have is:
$v_j = \left( \begin{array}{c}
B_{1j} \\
B_{2j} \\
\vdots \\
B_{mj}
\end{array} \right) = Be_{j} \therefore v_j = Be_j$
is this correct? | 2017-07-25 07:20:05 | {"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": 8, "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": 0.9923959970474243, "perplexity": 397.269278179646}, "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-2017-30/segments/1500549425082.56/warc/CC-MAIN-20170725062346-20170725082346-00560.warc.gz"} |
https://astronomy.stackexchange.com/questions/48247/what-is-precision-of-planet-periods-data | # What is precision of planet periods data?
Wikipedia gives $$10759.22$$ days for sidereal period of Saturn. I have calculated a period from de441_part-1.bsp and obtained $$10736.247\bar{2}$$ days. Why such a big difference? Which is more accurate? What accuracy in percentage I can expect from periods of Jupiter, Saturn, Uranus and Neptune?
EDIT: Code in Python 2.7:
from datetime import datetime, timedelta
from skyfield.framelib import ecliptic_frame
def f(t):
sun_eph = eph['sun']
s_eph = eph['saturn barycenter']
_, s_lon, _ = sun_eph.at(t).observe(s_eph).frame_latlon(ecliptic_frame)
s_angle = s_lon._degrees
print("s_angle = ", s_angle)
t0 = ts.utc(-13188, 8, 11, 14, 4)
f(t0)
# ('s_angle = ', 1.7441721183417348e-05)
t1 = ts.utc(1878, 10, 8, 3, 55)
f(t1)
# ('s_angle = ', 1.4075963901504881e-05)
(t1-t0)/512.0
# 10747.657377115886
Indeed there was error somewhere, now I got another number, but it still is not equal to that of Wikipedia.
• Not only Wikipedia, Nasa's website also states the same number, check here nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html, Even tropical year of saturn is 10,746.94 earth days only Jan 20 at 14:29
• I think you are looking for accuracy, not precision . Jan 20 at 15:22
• It would be better if you had shown your calculation. You might well have made a mistake, and there's no way to tell. Jan 20 at 20:11
• This may be helpful: Nuances of the terms (mean / osculating / Keplerian / orbital) elements Jan 21 at 21:13
The planetary orbits can not strictly be described by the usual Kepler elements anymore as they are disturbed by the other planets (Saturn in particular will for instance be heavily affected by Jupiter). The Kepler elements can therefore not be accurately defined but are only used as 'osculating elements' that approximate the actual orbit but vary from point to point. If you go to NASA's Horizons website and get the ephemeris for the last Saturn year, you can see that the sidereal period (which is calculated for each point via Kepler's law from the semimajor axis A) varies from PR=1.072402392741634E+04 d to PR=1.083416198008542E+04 d over one Saturn orbit, whereas in the header data it gives 10755.698 d for the sidereal period (which is some average over recent data for a not further specified period).
So as the orbital period (and the other Kepler elements) fluctuate in this sense with time, they are not really suitable to use as fixed values for applications where high accuracy is needed. And for uncritical applications any value that falls within the range of the fluctuations should do.
• We should also remember that the Sun itself doesn't even "stay in one place" but dances around with Jupiter, Saturn, Uranus and Neptune; the last of which is surprising because of its low relative mass until one considers that it makes up for that with distance.
– uhoh
Jan 21 at 23:21
• @uhoh If you select 'osculating orbital elements' as the ephemeris type in the |Horizons app, these will refer to the center of the sun anyway (as they are essentially Kepler elements, albeit time dependent). It is only if you select 'Vector table' (i.e. x,y,z coordinates) that they are referred to the barycenter of the solar system. Jan 22 at 9:47
• @uhoh You can not refer the osculating orbital elements to any other than the center of the sun. It will result in an error message in the Horizons app if you try to do that. You can only do it for if you select ''Vector Table' (which gives you x,y,z coordinates) or 'Observer Table' (which gives you RA,DEC) Jan 22 at 12:27
• @uhoh I was referring to the osculating orbital elements of planets above. In general, they are only defined with regard to the primary body of the orbit. So you can not refer the orbital elements of a moon of Saturn to Jupiter for instance. You will get the error message "Cannot output osculating elements of satellite wrt non-primary center body". In this case you can select only Saturn or the Sun (in the latter case it gives some non-sensical elements though like a negative semi-major axis and and infinite orbital period) Jan 22 at 13:11
• @uhoh This error message when trying to use a reference point other than the primary body has been appearing at lest for the least 2 years or so (I have not used Horizons much before that, so I could not tell how it used to be). You can of course choose the solar system barycenter as reference point for the osculating orbital elements, but this is not how they are normally defined and used. There would be a high risk of misinterpreting your data if you define them this way. Anyway, this option has disappeared from the dropdown menu. You have to explicitly type 500@0 for this. Jan 23 at 19:07 | 2022-05-16 05:24:09 | {"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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6102641224861145, "perplexity": 1118.5644107812225}, "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-2022-21/segments/1652662509990.19/warc/CC-MAIN-20220516041337-20220516071337-00464.warc.gz"} |
https://www.physicsforums.com/threads/root-test-and-integral-test-question.781999/ | # Root Test and Integral Test Question
1. Nov 15, 2014
### RJLiberator
1. The problem statement, all variables and given/known data
From K=4 to infinity the Σ (-1)^k (k/e^k)
Converge or diverge?
Use:
a) Ratio Test
b) Root Test
c) Integral Test
d) Alternating series test
2. Relevant equations
3. The attempt at a solution
For the alternating series test and ratio test I have the correct answer that it converges. These were fairly simple for me to proceed with.
However, I am stuck on the Root test and Integral test.
For the root test I DID get an answer, but it seems corrupt:
Lim as n approaches infinity of (|(-1)^k (k/e^k|))^(1/k)
With some simplification I narrowed it down to
The lim as n-->infinity of (|n^(1/n)|/e)
Which doesn't seem solvable ?
And for the Integral test, I am seeing the answer requires imaginary numbers, etc. which we do not use in this class. Is it possible that the instructor did not realize this? Does this problem demand the use of imaginary numbers, etc? If so, I imagine I would be able to pass on this part.
Thanks for any guidance.
2. Nov 15, 2014
### Staff: Mentor
Root test: it is solvable. The numerator is a well-known limit problem with a standard answer, but here you do not need the exact limit - it is sufficient to find some upper bound, and that is easier to find.
You can show absolute convergence instead of the (weaker) convergence. There, you don't get issues with complex numbers.
3. Nov 15, 2014
### RJLiberator
Ah, music to my ears. I see exactly what to do with the absolute convergence of the integral test. I took out the (-1)^k and then integrated to get a result o 5/e^4 which concludes absolute convergence.
Now, I will try to work on the root test.
4. Nov 17, 2014
### RJLiberator
For the Root Test:
I took the limit of the numerator and denominator. For the common limit of n^(1/n) the limit is 1. For e, the limit is the constant --> e.
Thus, answer is 1/e and the limit is less then 1 meaning absolute convergence. | 2017-12-17 10:52:29 | {"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": 0.9069150686264038, "perplexity": 685.6615240198181}, "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-2017-51/segments/1512948595342.71/warc/CC-MAIN-20171217093816-20171217115816-00686.warc.gz"} |
https://ohhaskme.com/3878/examples-differential-equations-surprisingly-solutions | What are some examples of "ugly" differential equations with surprisingly simple solutions?
I'm not sure this quite fits here but I was very surprised by how the KdV- and KP hierachies are "solved by moduli theory". Perhaps these equations are not ugly and perhaps Weierstrass and theta functions are not simple though.
To be honest, even "nice" differential equations can have very ugly solutions, and most equations don't even have solutions, so I wouldn't bet on an ugly equation to have a "nice" solution.
Quite interested in what other responses you might get though
This might not be "ugly" but it's an infinite-order ODE:
y - y' - y'' - y''' - ... = 0
Every (differentiable) function satisfies many differential equations. Shouldn't be too hard too cook up an "ugly" equatipn satisfied by, say, e^x.
IDK if this qualifies but the Lindblad equation is pretty ugly when you write it all the way out.
A permitted solution to general relativity field equations is empty space, so most terms are zero, although there is still inertia.
take any nice equation of two variables and convert it from polar form into cartesian coordinates.
0 like 0 dislike | 2023-01-30 10:25:33 | {"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": 0.8413887023925781, "perplexity": 1113.3469692627464}, "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/1674764499816.79/warc/CC-MAIN-20230130101912-20230130131912-00467.warc.gz"} |
http://crypto.stackexchange.com/tags?page=8&tab=popular | # Tags
A tag is a keyword or label that categorizes your question with other, similar questions. Using the right tags makes it easier for others to find and answer your question.
Type to find tags:
wpa2-psk× 5 Wi-Fi Protected Access 2 - Pre-Shared Key (also called WPA or WPA2 Personal) is a method of securing your network using WPA2 with the use of the optional Pre-Shared Key (PSK) authentication, which was… universal-hash× 5 untagged× 4 simon× 4 A family of lightweight symmetric block-ciphers designed for good performance in hardware with block sizes ranging from 32 to 128 bits and key sizes ranging from 64 to 256 bits. tea× 4 a block cipher by David Wheeler and Roger Needham of the Cambridge Computer Laboratory; notable for its simplicity of description and implementation. avalanche× 4 evident if, when an input is changed slightly (for example, flipping a single bit) the output changes significantly (e.g., half the output bits flip). decryption-oracle× 4 cfb× 4 an encryption mode, that builds a self-synchronizing stream-cipher from a block-cipher. camellia× 4 a 128-bit, symmetrical block cipher jointly developed by Mitsubishi and NTT of Japan. dual-ec-drgb× 4 short for Dual Elliptic Curve Deterministic Random Bit Generator; a pseudorandom number generator based on the elliptic curve discrete logarithm problem. information-theory× 4 concerned with sending messages via electronic signals in the most efficient and error-free way. schnorr-identification× 4 The Schnorr Identification Protocol relies upon the security of the Discrete Logarithm Problem. Schnorr's protocol was introduced after, and is comparable to, the identification protocol of Fiat and S… luby-rackoff× 4 a Feistel cipher where in each round the nonlinear function used is assumed to be chosen uniformly at random from the set of all such functions. These ciphers are mainly of th… key-recovery× 4 A means of recovering cryptographic keys when the usual means for obtaining them is unavailable.; the ability to uncover the secret key to a cryptographic message. knapsack× 4 the problem of determining which numbers from a given collection of numbers have been added together to yield a specific sum: used in cryptography to encipher (and sometimes decipher) mess… key-check-value× 4 needham-schroeder× 4 Needham–Schroeder refers to both, a symmetric key or a public key authentication protocol. mixnets× 4 routing protocols that create hard-to-trace communications by using a chain of proxy servers known as mixes which take in messages from multiple senders, shuffle them, and send them b… ocb× 4 The Offset CodeBook Mode, an authenticated encryption mode of operation for a block cipher. onion-routing× 3 A method for anonymous communications over a wide area network such as the Internet. mixing-function× 3 to "scramble" or mix the internal state of a hash (or cipher) function. The input to the function is the current internal state and the output of the function bec… kbkdf× 3 a key derivation function ([tag:kdf]) that uses a key in the computation. key-generation× 3 the process of generating keys for cryptographic purposes. searchable-encryption× 3 predicate-encryption× 3 an encryption paradigm which gives a master secret key owner fine-grained control over access to encrypted data. plausible-deniability× 3 Plausible deniability may refer $1)$ to deniable encryption schemes allowing to decrypt a ciphertext for a message $m$ to some distinct message $m'$ or $2)$ to a feature provided by deniable file syst… pir× 3 A PIR (Private Information Retrieval) protocol allows a user to retrieve an item from a server in possession of a database without revealing which item is retrieved. ipsec× 3 frequency-analysis× 3 the study of letters or groups of letters contained in a ciphertext in an attempt to partially reveal the message. ecies× 3 Elliptic Curve Integrated Encryption Scheme (ECIES) is a public key encryption system proposed by Victor Shoup in 2001. feal× 3 a block cipher designed by NTT. It is known to be insecure. embedded× 3 a computer system with a dedicated function within a larger mechanical or electrical system, often with real-time computing constraints. congruence× 3 If two numbers $b$ and $c$ have the property that their difference $b-c$ is integrally divisible by a number $m$ (i.e., $(b-c)/m$ is an integer), then $b$ and $c$ are said to be "congruent modulo $m$.… alternating-step× 3 a cryptographic pseudorandom number generator intended to be used in a stream cipher. biclique-attack× 3 bijection× 3 a function $f$ from a set $X$ to a set $Y$ with the property that, for every $y$ in $Y$, there is exactly one $x$ in $X$ such that $f(x) = y$. It follows from … | 2014-10-25 20:43:28 | {"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": 0.5256319046020508, "perplexity": 1052.1068527595892}, "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-2014-42/segments/1414119650193.38/warc/CC-MAIN-20141024030050-00113-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.zbmath.org/?q=ai%3Abergamasco.adalberto-panobianco+ai%3Ahounie.j | # zbMATH — the first resource for mathematics
Global properties of a class of vector fields in the plane. (English) Zbl 0662.58021
The paper is concerned with the global solvability of the problem $$Ln=0$$, dn$$\neq 0$$ on $${\mathbb{R}}^ 2$$, where L is a complex vector field without singularities. First it is shown that for a suitable class of vector fields L the Mizohata operator $$\partial_ t-it\partial y$$ is a model operator in a neighborhood of the characteristic set of L. Then several integrability conditions are discussed and some global range theorems for the Mizohata operator are given. In an appendix relations to hyperelliptic vector fields are considered.
Reviewer: N.Jacob
##### MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text:
##### References:
[1] Baouendi, S; Treves, F, A local constancy principle for the solutions of certain overdetermined systems of first order linear partial differential equations, Math. anal. appl. stud., 7A, 245-262, (1981) [2] Hörmander, L, Pseudodifferential operators of principal type, (), 69-96 [3] Kamke, E; Kamke, E, Über die partielle differentialgleichung f(x, y)zx + g(x, y)zy = h(x, y), II, Math. Z., Math. Z., 42, 287-300, (1936) · Zbl 0015.34804 [4] Nehari, Z, Conformal mapping, (1952), McGraw-Hill New York · Zbl 0048.31503 [5] Nirenberg, L, Lectures on linear partial differential equations, () · Zbl 0267.35001 [6] Springer, G, Introduction to Riemann surfaces, (1957), Addison-Wesley Reading, MA · Zbl 0078.06602 [7] Sjöstrand, Note on a paper of F. treves concerning mizohata type operators, Duke math. J., 41, 3, 601-608, (1980) · Zbl 0471.35076 [8] Treves, F, Remarks about certain first-order linear PDE in two variables, Comm. PDE, 5, 381-425, (1980) · Zbl 0519.35008 [9] Treves, F, Hypoelliptic PDE’s of principal type, sufficient conditions and necessary conditions, Comm. pure appl. math., 24, 631-670, (1971) · Zbl 0234.35019 [10] Treves, F, Approximation and representations of functions and distributions annihilated by a system of complex vector fields, (1981), École Polytech, Centre de Math Palaiseau, France · Zbl 0515.58030 [11] Ważewski, T, Sur un problème de caractère intégral relatif à l’équation Zx + Q(x, y)zy = 0, Mathematica cluj, 8, 103-116, (1934) · Zbl 0008.39403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-01-25 18:36:53 | {"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": 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": 0.7025043368339539, "perplexity": 2990.4681906331307}, "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-2021-04/segments/1610703587074.70/warc/CC-MAIN-20210125154534-20210125184534-00421.warc.gz"} |
http://mail.scipy.org/pipermail/numpy-discussion/2009-January/039484.html | # [Numpy-discussion] Zoom fft code
Stéfan van der Walt stefan@sun.ac...
Mon Jan 5 07:58:11 CST 2009
2009/1/5 Neal Becker <ndbecker2@gmail.com>:
> I was not aware that chirp-z transform can be used to efficiently compute DFT over a limited part of the spectrum. I could use this. Any references on this technique?
The only reference I have is the one mentioned in the source:
Rabiner, L.R., R.W. Schafer and C.M. Rader.
The Chirp z-Transform Algorithm.
IEEE Transactions on Audio and Electroacoustics, AU-17(2):86--92, 1969
The discrete z-transform,
X(z_k) = \sum_{n=0}^{N-1} x_n z^{-n}
is calculated at M points,
z_k = AW^-k, k = 0,1,...,M-1.
You can think of the z_k's as a spiral, where A controls the outside
radius (starting frequency) and W the rate of inward spiralling.
Regards
Stéfan | 2014-08-23 09:38:38 | {"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": 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": 0.6874953508377075, "perplexity": 6447.014010119099}, "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-2014-35/segments/1408500825567.38/warc/CC-MAIN-20140820021345-00080-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://kops.uni-konstanz.de/handle/123456789/6109 | ## Efficient Topology-Aware Overlay Network
2003
Rinaldi, Roberto
Journal article
##### Published in
Computer Communication Review ; 33 (2003), 1
##### Abstract
Peer-to-peer (P2P) networking has become a household word in the past few years, being marketed as a work-around for server scalability problems and wonder drug'' to achieve resilience. Current widely-used P2P networks rely on central directory servers or massive message flooding, clearly not scalable solutions. Distributed Hash Tables (DHT) are expected to eliminate flooding and central servers, but can require many long-haul message deliveries. We introduce Mithos, an content-addressable overlay network that only uses minimal routing information and is directly suitable as an underlay network for P2P systems, both using traditional and DHT addressing. Unlike other schemes, it also efficiently provides locality-aware connectivity, thereby ensuring that a message reaches its destination with minimal overhead. Mithos provides for highly efficient forwarding, making it suitable for use in high-throughput applications. Paired with its ability to have addresses directly mapped into a subspace of the IPv6 address space, it provides a potential candidate for native deployment. Additionally, Mithos can be used to support third-party triangulation to quickly select a close-by replica of data or services.
##### Subject (DDC)
004 Computer Science
##### Cite This
ISO 690WALDVOGEL, Marcel, Roberto RINALDI, 2003. Efficient Topology-Aware Overlay Network. In: Computer Communication Review. 33(1). Available under: doi: 10.1145/774763.774779
BibTex
@article{Waldvogel2003Effic-6109,
year={2003},
doi={10.1145/774763.774779},
title={Efficient Topology-Aware Overlay Network},
number={1},
volume={33},
journal={Computer Communication Review},
author={Waldvogel, Marcel and Rinaldi, Roberto}
}
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<dcterms:abstract xml:lang="eng">Peer-to-peer (P2P) networking has become a household word in the past few years, being marketed as a work-around for server scalability problems and wonder drug'' to achieve resilience. Current widely-used P2P networks rely on central directory servers or massive message flooding, clearly not scalable solutions. Distributed Hash Tables (DHT) are expected to eliminate flooding and central servers, but can require many long-haul message deliveries. We introduce Mithos, an content-addressable overlay network that only uses minimal routing information and is directly suitable as an underlay network for P2P systems, both using traditional and DHT addressing. Unlike other schemes, it also efficiently provides locality-aware connectivity, thereby ensuring that a message reaches its destination with minimal overhead. Mithos provides for highly efficient forwarding, making it suitable for use in high-throughput applications. Paired with its ability to have addresses directly mapped into a subspace of the IPv6 address space, it provides a potential candidate for native deployment. Additionally, Mithos can be used to support third-party triangulation to quickly select a close-by replica of data or services.</dcterms:abstract>
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<dcterms:bibliographicCitation>First publ. in: Computer Communication Review 33 (2003), 1</dcterms:bibliographicCitation>
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https://en.wikiversity.org/wiki/Primary_mathematics:Boolean_logic | # Primary mathematics:Boolean logic
Boolean logic (also called Boolean algebra) is a complete system for logical operations, used often since popularization of mathematical logic and discussions concerning the foundations of mathematics. It was named after George Boole, who first defined an algebraic system of logic in the mid 19th century. Boolean logic has many applications in electronics, computer hardware and software, and is the basis of all modern digital electronics. In 1938, Claude Shannon showed how electric circuits with relays could be modeled with Boolean logic. This fact soon proved enormously consequential with the emergence of the electronic computer.
Using the algebra of sets, this article contains a basic introduction to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications.
## Set logic vs. Boolean logic
Sets can contain any elements. We will first start out by discussing general set logic, then restrict ourselves to Boolean logic, where elements (or "bits") each contain only two possible values, called various names, such as "true" and "false", "yes" and "no", "on" and "off", or "1" and "0".
## Terms
Venn diagram showing the intersection of sets "A AND B" (in violet/dark shading), the union of sets "A OR B" (all the colored regions), and the exclusive OR case "set A XOR B" (all the colored regions except the violet). The "universe" is represented by all the area within the rectangular frame.
Let X be a set:
• An element is one member of a set and is denoted by ${\displaystyle \in }$ . If the element is not a member of a set it is denoted by ${\displaystyle \notin }$ .
• The universe, sometimes denoted by 1, is "all elements being considered", which is not necessarily the same as "all elements there are".
• The empty set or null set is the set of no elements, denoted by ${\displaystyle \varnothing }$ and sometimes 0.
• A unary operator applies to a single set. There is only one unary operator, called logical NOT. It works by taking the complement (opposite) of a set.
• A binary operator applies to two sets. The basic binary operators are logical OR and logical AND. They perform the union and intersection of sets.
There are also other derived binary operators, such as XOR (exclusive OR, i.e., "one or the other, but not both"), and set difference, A−B.
• The identity or equivalence of two sets is denoted by ${\displaystyle A\equiv B}$ and means that every element in set A is also in set B and every element in set B is also in set A.
• A subset is denoted by ${\displaystyle A\subseteq B}$ and means every element in set A is also in set B.
• A superset is denoted by ${\displaystyle A\supseteq B}$ and means every element in set B is also in set A.
• A proper subset is denoted by ${\displaystyle A\subset B}$ and means every element in set A is also in set B and the two sets are not identical.
• A proper superset is denoted by ${\displaystyle A\supset B}$ and means every element in set B is also in set A and the two sets are not identical.
a = a and ~a = ~a and a =/= ~a
a + 0 = a and a - 0 = a and ~a + 0 = ~a and ~a - 0 = ~a
a.1 = a and a/1 = a and ~a.1 = ~a and ~a/1 = ~a
a.0 = 0 and a/0 = 0 and ~a.0 = 0 and ~a/0 = 0
a + 1 = 1 + a and a - 1 = -1 + a and ~a + 1 = 1 + ~a and ~a - 1 = -1 + ~a
0 + a = a and 0 - a = -a and 0 + ~a = ~a and 0 - ~a = - ~a
a + a = a + a and a . a = a and a - a = 0 and a / a = 1
a + ~a = 0 and a . ~a = 0 and a - ~a = 1 and a / ~a = -1
a . b = b . a and a + b = b + a
(a + b) + c = a + (b + c) and (a . b) . c = a . (b . c)
a + (b . c) = (a + b) . (a + c) and a . (b + c) = (a . b) + (a . c)
~(a + b) = ~a + ~b and ~(a . b) = ~a . ~b
a / b = a / b and a - b = -b + a
(a - b) - c = a - (b - c) and (a / b) / c = a / (b / c)
a - (b / c) = (a - b) / (a - c) and a / (b - c) = (a / b) - (a / c)
~(a - b) = ~a - ~b and ~(a / b) = ~a / ~b
## Example
Imagine that set A contains all even numbers (multiples of two) in "the universe" (defined in the example below as all integers between 0 and 30 inclusive) and set B contains all multiples of three in "the universe". Then the intersection of the two sets (all elements in sets A AND B) would be all multiples of six in "the universe". The complement of set A (all elements NOT in set A) would be all odd numbers in "the universe".
### Chaining operations together
While at most two sets are joined in any Boolean operation, the new set formed by that operation can then be joined with other sets utilizing additional Boolean operations. Using the previous example, we can define a new set C as the set of all multiples of five in "the universe". Thus "sets A AND B AND C" would be all multiples of 30 in "the universe". If more convenient, we may consider set AB to be the intersection of sets A and B, or the set of all multiples of six in "the universe". Then we can say "sets AB AND C" are the set of all multiples of 30 in "the universe". We could then take it a step further, and call this result set ABC.
### Use of parentheses
While any number of logical ANDs (or any number of logical ORs) may be chained together without ambiguity, the combination of ANDs and ORs and NOTs can lead to ambiguous cases. In such cases, parentheses may be used to clarify the order of operations. As always, the operations within the innermost pair is performed first, followed by the next pair out, etc., until all operations within parentheses have been completed. Then any operations outside the parentheses are performed.
### Application to binary values
In this example we have used natural numbers, while in Boolean logic binary numbers are often used. The universe, for example, could contain just two elements, "1" and "0" (or "true" and "false", "yes" and "no", "on" or "off", etc.). We could also combine binary values together to get binary words, such as, in the case of two digits, "00", "01", "10", and "11". Applying set logic to those values, we could have a set of all values where the first digit is "0" ("00" and "01") and the set of all values where the first and second digits are different ("01" and "10"). The intersection of the two sets would then be the single element, "01". This could be shown by the following Boolean expression, where "1st" is the first digit and "2nd" is the second digit:
(NOT 1st) AND (1st XOR 2nd)
## Properties
We define symbols for the two primary binary operations as ${\displaystyle \land /\cap }$ (logical AND/set intersection) and ${\displaystyle \lor /\cup }$ (logical OR/set union), and for the single unary operation ${\displaystyle \lnot /\sim }$ (logical NOT/set complement). We will also use the values 0 (logical FALSE/the empty set) and 1 (logical TRUE/the universe). The following properties apply to both Boolean logic and set logic (although only the notation for Boolean logic is displayed here):
${\displaystyle a\lor \lnot a=1}$ ${\displaystyle a\land \lnot a=0}$ complements ${\displaystyle a\lor a=a}$ ${\displaystyle a\land a=a}$ idempotency ${\displaystyle a\lor 0=a}$ ${\displaystyle a\land 1=a}$ boundedness ${\displaystyle a\lor 1=1}$ ${\displaystyle a\land 0=0}$ ${\displaystyle \lnot 0=1}$ ${\displaystyle \lnot 1=0}$ 0 and 1 are complements ${\displaystyle \lnot (\lnot a)=a}$ involution ${\displaystyle a\lor (b\lor c)=(a\lor b)\lor c}$ ${\displaystyle a\land (b\land c)=(a\land b)\land c}$ associativity ${\displaystyle a\lor b=b\lor a}$ ${\displaystyle a\land b=b\land a}$ commutativity ${\displaystyle a\lor (a\land b)=a}$ ${\displaystyle a\land (a\lor b)=a}$ absorption ${\displaystyle a\lor (b\land c)=(a\lor b)\land (a\lor c)}$ ${\displaystyle a\land (b\lor c)=(a\land b)\lor (a\land c)}$ distributivity ${\displaystyle \lnot (a\lor b)=\lnot a\land \lnot b}$ ${\displaystyle \lnot (a\land b)=\lnot a\lor \lnot b}$ de Morgan's laws
## Truth tables
For Boolean logic using only two values, 0 and 1, the INTERSECTION and UNION of those values may be defined using truth tables such as these:
${\displaystyle \cap }$ 0 1
0 0 0
1 0 1
${\displaystyle \cup }$ 0 1
0 0 1
1 1 1
• More complex truth tables involving multiple inputs, and other Boolean operations, may also be created.
• Truth tables have applications in logic, interpreting 0 as FALSE, 1 as TRUE, ${\displaystyle \cap }$ as AND, ${\displaystyle \cup }$ as OR, and ¬ as NOT.
## Other notations
Mathematicians and engineers often use plus (+) for OR and a product sign (${\displaystyle \cdot }$) for AND. OR and AND are somewhat analogous to addition and multiplication in other algebraic structures, and this notation makes it very easy to get sum of products form for normal algebra. NOT may be represented by a line drawn above the expression being negated (${\displaystyle {\overline {x}}}$). It also commonly leads to giving ${\displaystyle \cdot }$ a higher precedence than +, removing the need for parenthesis in some cases.
Programming languages often use a pipe symbol (|) for OR, an ampersand (&) for AND, and a tilde (~) for NOT. In many programming languages, these symbols stand for bitwise operations, however, and "||", "&&", and "!" may be used for the logical OR, AND, and NOT operations.
Another notation uses "meet" for AND and "join" for OR. However, this can lead to confusion, as the term "join" is also commonly used for any Boolean operation which combines sets together, which includes both AND and OR.
## Basic mathematics use of Boolean terms
Underlying the "language of mathematics" are boolean assumptions that are seldom explicitly stated. The following examples show the unstated boolean relationship.
• In the case of simultaneous equations, they are connected with an implied logical AND:
(4x + y = 2) and (2x − y = 2)
• Similarly, for simultaneous inequalities:
(x + y < 2) and (x − y < 7)
• Both the greater-than-or-equals and less-than-or-equals inequalities most often implicitly have an OR boolean joining them:
(x ≤ 2) means
(x < 2) or (x = 2)
• The plus/minus sign (${\displaystyle \pm }$), as in the case of the solution to a square root problem, may be taken as logical OR:
(Width ± 3) means
(Width = 3) or (Width = −3)
## English language use of Boolean terms
Care should be taken when converting an English sentence into a formal Boolean statement. Many English sentences have imprecise meanings, for example "all that glitters is not gold", which could mean that "nothing that glitters is gold" or "some things which glitter are not gold".
AND and OR can also be used interchangeably in English, in certain cases:
• "I always carry an umbrella for when it rains and snows."
• "I always carry an umbrella for when it rains or snows."
Sometimes the English words AND and OR have the opposite meaning in Boolean logic:
• "Give me all the red and blue berries" usually means "Give me all berries that are red or blue". An alternative phrasing for standard written English: "Give me all berries that are red as well as all berries that are blue".
Also note that the word OR in English may correspond with either logical OR or logical XOR, depending on the context:
• "I start to sweat when the humidity or temperature is high." (logical OR)
• "You want ice cream and candy? You may have ice cream or candy." (logical XOR)
The combination AND/OR is sometimes used in English to specify a logical OR, when just using the word OR alone might have been mistaken as meaning logical XOR:
• "I'm having chicken and/or beef for dinner." (logical OR). An alternative phrasing for standard written English: "I'm having chicken, or beef, or both, for dinner."
A case where this is an issue is when specifications for a computer program or electronic circuit are supplied as an English paragraph describing their function. For example, the statement: "the program should verify that the applicant has checked the male or female box", should be taken as an XOR, and a check added to ensure that one, and only one, box is selected. In other cases, the interpretation of English may be less certain, and the author of the specification may need to be consulted to determine their true intent.
## Applications
### Digital electronic circuit design
Boolean logic is also used for circuit design in electrical engineering; here 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.
Basic logic gates such as AND, OR, and NOT gates may be used alone, or in conjunction with NAND, NOR, and XOR gates, to control digital electronics and circuitry. Whether these gates are wired in series or parallel controls the precedence of the operations.
### Database applications
For this application, each record in a table may be considered to be a row "element" of a data "set". There are Boolean data types, and then there is Boolean logic used for the selection of elements from any set of data types. Examples of both are shown.
Some Relational databases (DB Management Systems, DBMS) can query a set of data having Boolean values. The standardized query language, SQL, supports a three-valued logic: true, false, or unknown. This Boolean data type was introduced in the ISO SQL:1999 standard.
CREATE TABLE test1
(
a int,
b boolean
);
INSERT INTO test1
VALUES (1, true);
INSERT INTO test1
VALUES (2, false);
INSERT INTO test1
VALUES (3, null);
INSERT INTO test1
VALUES (4, unknown);
SELECT *
FROM test1;
a b
------------- ----------------
1 TRUE
2 FALSE
3 NULL
4 UNKNOWN
The SQL Boolean data type did not gain widespread adoption, owing to the following inconsistency: SQL data types can have the special null value as well. The standard says that the NULL BOOLEAN and UNKNOWN "may be used interchangeably to mean exactly the same thing". This identification creates the possibility that UNKNOWN = UNKNOWN is not TRUE but UNKNOWN/NULL. Most SQL DBMSs use other data types like bit, byte, and char to simulate the behavior of Boolean data types. PostgreSQL does support the standard SQL Boolean data type.
Here are some examples using Boolean logic "NOT", "AND", and "OR":
SELECT * FROM employees WHERE last_name = 'Dean' AND first_name = 'James' ;
That example will produce a list of all employees, and only those employees, named James Dean.
SELECT * FROM employees WHERE last_name = 'Dean' OR first_name = 'James' ;
That example will produce a list of all employees whose first name is James OR whose last name is Dean. Any and all employees named James Dean (from the first example) would also appear in this list.
SELECT * FROM employees WHERE NOT last_name = 'Dean' ;
That example will produce a list of all employees whose last name is not Dean. All employees named James from the second example would appear on this list, except for those employees named James Dean.
In the field of Electronic Medical Records, the Boolean logic is called a Concept Processing technology.
### Search engine queries
Search engine queries also employ Boolean logic. For this application, each web page on the Internet may be considered to be an "element" of a "set". The following examples use a syntax supported by Google.[1]
• Doublequotes are used to combine whitespace-separated words into a single search term.[2]
• Whitespace is used to specify logical AND, as it is the default operator for joining search terms:
"Search term 1" "Search term 2"
• The OR keyword is used for logical OR:
"Search term 1" OR "Search term 2"
• The minus sign (approximated as a hyphen) is used for logical NOT (AND NOT):
"Search term 1" -"Search term 2"
## Notes and references
1. Not all search engines support the same query syntax. Additionally, some organizations provide "specialized" search engines that support alternate or extended syntax. (See e.g.,Syntax cheatsheet, Google codesearch supports regular expressions).
2. Doublequote-delimited search terms are called "exact phrase" searches in the Google documentation. | 2019-10-17 19:03:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 40, "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": 0.6321381330490112, "perplexity": 1055.1968480021155}, "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-2019-43/segments/1570986675598.53/warc/CC-MAIN-20191017172920-20191017200420-00519.warc.gz"} |
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14 1
oh look a I doodled x heh,,,, #xxxtentacion
4 0 | 2017-12-17 08:01:59 | {"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": 0.8068647980690002, "perplexity": 1618.4473119309953}, "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-2017-51/segments/1512948594665.87/warc/CC-MAIN-20171217074303-20171217100303-00042.warc.gz"} |
https://mail.python.org/pipermail/pypy-commit/2009-March/031767.html | cfbolz at codespeak.net cfbolz at codespeak.net
Tue Mar 31 16:15:28 CEST 2009
Author: cfbolz
Date: Tue Mar 31 16:15:25 2009
New Revision: 63449
Modified:
Log:
accepted most of anto's changes
==============================================================================
+++ pypy/extradoc/talk/icooolps2009/paper.tex Tue Mar 31 16:15:25 2009
@@ -153,49 +153,14 @@
environments, such as C/Posix, the CLI and the JVM. This is done by a component
of PyPy called the \emph{translation toolchain}.
-
-\anto{XXX: are the following paragraphs really needed? I don't think this
- details are needed in Section \ref{sect:implementation}. I would say
- something much shorter, see below.}
-
-The central idea of this way to implement VMs is that the interpreter
-implementation in RPython should be as free as possible of low-level
-implementation details, such as memory management strategy, threading model or
-object layout. Instead, these details are inserted into the VM during the
-translation process by the translation toolchain. This makes it possible to
-change these details later, if that becomes necessary. This is something that is
-hard to do with a traditional VM written in a low-level language such as C,
-since the low-level details need to be fixed early in the development-process.
-XXX is this paragraph really needed?
-
-In the following we will describe some details of the translation process, since
-they are needed in Section \ref{sect:implementation}. The first step is to produce
-control flow graphs of all functions of the RPython program. Afterwards, type
-inference is performed to gain type information about all the variables in the
-flow graphs. Afterwards, the abstraction level of the operations in the graphs
-is lowered in a stepwise fashion. At the end of this process, all operations in
-the graphs correspond rather directly to a simple operation in C. The variables
-are annotated with a C-like type system containing primitive types (like
-\texttt{Signed}, \texttt{Bool}, etc.) or pointer types pointing to Structs,
-Arrays or Functions.
-
-XXX example. reuse the one of the tracing jit?
-
-The translation process usually just turns these graphs into C code so that they
-can be compiled into an executable. However, they can also be interpreted in
-various ways. This is useful for testing and debugging the translation toolchain
-because the produced error messages in case of a crash are a lot more helpful
-than what would be produced after compilation to C. These low-level graphs are
-also what the tracing JIT takes as input, as we will see later.
-
-\anto{By writing VMs in a high-level language, we keep the implementation of
- the language free of low-level details such as memory management strategy,
- during the translation process which consists in a series of steps, each
- step transforming the representation of the program produced by the previous
- one until we get the final executable. As we will see later, this internal
- low-level representation of the program is also used as an input for the
- tracing JIT.}
+By writing VMs in a high-level language, we keep the implementation of the
+language free of low-level details such as memory management strategy,
+during the translation process which consists in a series of steps, each step
+transforming the representation of the program produced by the previous one
+until we get the final executable. As we will see later, this internal
+low-level representation of the program is also used as an input for the
+tracing JIT.
%- original goal: Python interpreter in Python
@@ -233,15 +198,14 @@
The code for those common loops however should be highly optimized, including
aggressive inlining.
-\sout{The generation of loops works as follows: at first, everything is interpreted.}
-\anto{At first, when the program starts, everything is interpreted.}
+At first, when the program starts, everything is interpreted.
The interpreter does a bit of lightweight profiling to figure out which loops
are run often. This lightweight profiling is usually done by having a counter on
each backward jump instruction that counts how often this particular backward jump
was executed. Since loops need a backward jump somewhere, this method finds
loops in the user program.
-When a \sout{common}\anto{hot} loop is identified, the interpreter enters a
+When a hot loop is identified, the interpreter enters a
special mode, called \emph{tracing mode}. When in tracing mode, the interpreter
records a history (the \emph{trace}) of all the operations it executes, in addition
to actually performing the operations. During tracing, the trace is repeatedly | 2016-09-27 19:34:23 | {"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": 0.5826185345649719, "perplexity": 3477.076722389206}, "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-2016-40/segments/1474738661155.8/warc/CC-MAIN-20160924173741-00028-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.openfoam.com/documentation/guides/latest/api/doubleScalar_8C_source.html | The open source CFD toolbox
doubleScalar.C
Go to the documentation of this file.
1/*---------------------------------------------------------------------------*\
2 ========= |
3 \\ / F ield | OpenFOAM: The Open Source CFD Toolbox
4 \\ / O peration |
5 \\ / A nd | www.openfoam.com
6 \\/ M anipulation |
7-------------------------------------------------------------------------------
8 Copyright (C) 2011-2013 OpenFOAM Foundation
9 Copyright (C) 2017 OpenCFD Ltd.
10-------------------------------------------------------------------------------
11License
12 This file is part of OpenFOAM.
13
14 OpenFOAM is free software: you can redistribute it and/or modify it
15 under the terms of the GNU General Public License as published by
16 the Free Software Foundation, either version 3 of the License, or
17 (at your option) any later version.
18
19 OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
20 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
21 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
22 for more details.
23
24 You should have received a copy of the GNU General Public License
25 along with OpenFOAM. If not, see <http://www.gnu.org/licenses/>.
26
27\*---------------------------------------------------------------------------*/
28
29#include "doubleScalar.H"
30#include "error.H"
31#include "parsing.H"
32#include "IOstreams.H"
33
34#include <cstdlib>
35#include <sstream>
36
37// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
38
39// Scalar.C is used for template-like substitution
40
41#define Scalar doubleScalar
42#define ScalarVGREAT doubleScalarVGREAT
43#define ScalarVSMALL doubleScalarVSMALL
44#define ScalarROOTVGREAT doubleScalarROOTVGREAT
45#define ScalarROOTVSMALL doubleScalarROOTVSMALL
46#define ScalarRead readDouble
47// Convert using larger representation to properly capture underflow
48#define ScalarConvert ::strtold
49
50#include "Scalar.C"
51
52#undef Scalar
53#undef ScalarVGREAT
54#undef ScalarVSMALL
55#undef ScalarROOTVGREAT
56#undef ScalarROOTVSMALL
57#undef ScalarRead
58#undef ScalarConvert
59
60// ************************************************************************* //
Useful combination of include files which define Sin, Sout and Serr and the use of IO streams general... | 2023-01-31 13:47:33 | {"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": 0.979034960269928, "perplexity": 4856.484662241521}, "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-2023-06/segments/1674764499871.68/warc/CC-MAIN-20230131122916-20230131152916-00828.warc.gz"} |
https://mathhelpforum.com/threads/log-system.146718/ | # log-system
#### dhiab
Solve the system
$$\displaystyle log_{a^{2}}(x)-log_{a^{4}}(y)=3$$
$$\displaystyle log_{a^{6}}(x)+log_{a^{8}}(y)=4$$
#### undefined
MHF Hall of Honor
Solve the system
$$\displaystyle log_{a^{2}}(x)-log_{a^{4}}(y)=3$$
$$\displaystyle log_{a^{6}}(x)+log_{a^{8}}(y)=4$$
Is $$\displaystyle a$$ a constant?
Here are some things you can do to the first equation.
$$\displaystyle log_{a^{2}}(x)-log_{a^{4}}(y)=3$$
$$\displaystyle \Longrightarrow \frac{log_a(x)}{log_a(a^2)}-\frac{log_a(y)}{log_a(a^4)}=3$$
$$\displaystyle \Longrightarrow \frac{log_a(x)}{2}-\frac{log_a(y)}{4}=3$$
$$\displaystyle \Longrightarrow 2log_a(x)-log_a(y)=12$$
$$\displaystyle \Longrightarrow log_a(x^2)-log_a(y)=12$$
$$\displaystyle \Longrightarrow log_a\left(\frac{x^2}{y}\right)=12$$
Keep in mind that $$\displaystyle x>0$$ and $$\displaystyle y>0$$.
dhiab
#### Soroban
MHF Hall of Honor
Hello, dhiab!
Edit: I've corrected my blunder . . .
Solve the system:
. . $$\displaystyle \log_{a^2}(x)-\log_{a^4}(y) \;=\; 3 \;\;[1]$$
. . $$\displaystyle \log_{a^6}(x)+\log_{a^8}(y)\:=\:4\;\;[2]$$
We have:
. . $$\displaystyle \frac{\ln x}{\ln a^2} \;{\color{red}-}\; \frac{\ln y}{\ln a^4} \;=\;3 \quad\Rightarrow\quad \frac{\ln x}{2\ln a} \;-\; \frac{\ln y}{4\ln a} \;=\;3 \quad\Rightarrow\quad 2\ln x \;-\; \ln y \;=\;12\ln a \;\;[1]$$
. . $$\displaystyle \frac{\ln x}{\ln a^6} \;+\; \frac{\ln y}{\ln a^8} \;=\;4 \quad\Rightarrow\quad \frac{\ln x}{6\ln a} \;+\; \frac{\ln y}{8\ln a} \;=\;4 \quad\Rightarrow\quad 4\ln x \;+\; 3\ln y \;=\;96\ln a \;\;[2]$$
$$\displaystyle \begin{array}{ccccccc}3\times [1]: & 6\ln x -3\ln y &=& 36\ln a \\ \text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}$$
And we have: .$$\displaystyle 10\ln x \:=\:132\ln a \quad\Rightarrow\quad \ln x \:=\:\tfrac{66}{5}\ln a \:=\:\ln\left(a^{\frac{66}{5}}\right)$$
. . Therefore: .$$\displaystyle \boxed{x \;=\;a^{\frac{66}{5}}}$$
$$\displaystyle \begin{array}{ccccc}\text{-}2\times [1]\!: & \text{-}4\ln x + 2\ln y &=& \text{-}24\ln a \\ \text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}$$
And we have: .$$\displaystyle 5\ln y \:=\:72\ln a \quad\Rightarrow\quad \ln y \:=\:\tfrac{72}{5}\ln a \:=\: \ln\left(a^{\frac{72}{5}}\right)$$
. . Therefore: .$$\displaystyle \boxed{y \;=\;a^{\frac{72}{5}}}$$
Last edited:
bjhopper
#### dhiab
Hello, dhiab!
It may be neater with natural logs . . . or maybe not.
We have:
. . $$\displaystyle \frac{\ln x}{\ln a^2} + \frac{\ln y}{\ln a^4} \;=\;3 \quad\Rightarrow\quad \frac{\ln x}{2\ln a} + \frac{\ln y}{4\ln a} \;=\;3 \quad\Rightarrow\quad 2\ln x + \ln y \;=\;12\ln a \;\;[1]$$
. . $$\displaystyle \frac{\ln x}{\ln a^6} + \frac{\ln y}{\ln a^8} \:=\:4 \quad\Rightarrow\quad \frac{\ln x}{6\ln a} + \frac{\ln y}{8\ln a} \:=\:4 \quad\Rightarrow\quad 4\ln x + 3\ln y \;=\;96\ln a \;\;[2]$$
$$\displaystyle \begin{array}{ccccccc}\text{-}3\times [1]: & \text{-}6\ln x -3\ln y &=& \text{-}36\ln a \\$$$$\displaystyle \text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}$$
And we have: .$$\displaystyle -2\ln x \:=\:60\ln a \quad\Rightarrow\quad \ln x \:=\:-30\ln a \:=\:\ln\left(a^{-30}\right)$$
. . Therefore: .$$\displaystyle \boxed{x \;=\;a^{-30}}$$
$$\displaystyle \begin{array}{ccccc}\text{-}2\times [1]\!: & \text{-}4\ln x - 2\ln y &=& \text{-}24\ln a \\$$$$\displaystyle \text{Add [2]:} & 4\ln x + 3\ln y &=& 96\ln a \end{array}$$
And we have: .$$\displaystyle \ln y \:=\:72\ln a \quad\Rightarrow\quad \ln y \:=\:\ln\left(a^{72}\right)$$
. . Therefore: .$$\displaystyle \boxed{y \;=\;a^{72}}$$
HELLO : thank you but in first equation you have - not +(Evilgrin) | 2019-11-22 20:08:47 | {"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": 0.9894300699234009, "perplexity": 13817.485437822932}, "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-2019-47/segments/1573496671548.98/warc/CC-MAIN-20191122194802-20191122223802-00072.warc.gz"} |
https://testbook.com/question-answer/a-sinusoidal-voltage-v-50-sin-t-is-applie--5f15b773c337a40d111c054c | # A sinusoidal voltage V = 50 sin ωt is applied to a series RL circuit. The current in the circuit is given by I = 25 sin (ωt – 53°). The apparent power consumed by the load is
This question was previously asked in
ISRO Scientist Electrical 2015 Paper
View all ISRO Scientist EE Papers >
1. 375 VA
2. 625 VA
3. 2500 VA
4. 750 VA
Option 2 : 625 VA
## Detailed Solution
Concept:
The power triangle is as shown below.
P = Active power (or) Real power in W = Vrms Irms cos ϕ
Q = Reactive power in VAR = Vrms Irms sin ϕ
S = Apparent power in VA = Vrms Irms
S = P + jQ
$$S = \sqrt {{P^2} + {Q^2}}$$
ϕ is the phase difference between the voltage and current
Power factor $$\cos \phi = \frac{P}{S}$$
Calculation:
Given that, V = 50 sin ωt
I = 25 sin (ωt – 53°)
Apparent power, $$P = \frac{{50}}{{\sqrt 2 }} \times \frac{{25}}{{\sqrt 2 }} = 625\;VA$$
Free
ME Subject Test 1: Strength of Materials
9318
20 Questions 20 Marks 18 Mins | 2021-12-07 18:21:35 | {"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": 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": 0.4571365416049957, "perplexity": 4442.407218055534}, "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/1637964363405.77/warc/CC-MAIN-20211207170825-20211207200825-00094.warc.gz"} |
http://www.math.utah.edu/ugrad/research/symps/13Fall.html | ## Monday December 16 Session - 2pm-4pm - LCB 222
02:00-02:20pm, Brady Thompson, The Discriminant Relation Formula
Mentor: Gordan Savin
Abstract: One of the most significant invariants of an algebraic number field is the discriminant. The discriminant is an idea we're all familiar with since basic algebra, but idea can be generalized for number fields and becomes an ubiquitous tool in number theory. I will present general definitions and properties of the discriminant of an algebraic number field. I will also discuss the Discriminant Relation Formula, which is a tool that we can use to determine certain properties about a tower of fields. These properties include finding the ring of integers in a number field and verifying whether a field is a Hilbert class field. I will provide an example to illustrate it's usefulness.
02:20-02:40pm, Drew Ellingson, Tropical Analogues of Classical Theorems
Mentor: Steffen Marcus
Abstract: Tropical Geometry is a field derived from the study of the worst possible degenerations of classical Algebraic Geometry. This talk will develop the bare-bones concepts necessary to start thinking about Tropical Geometry. We then build intuition into the subject by stating and investigating a few tropical analogues of famous theorems in Algebraic Geometry. We will talk about Bézout's Theorem, and then move on to the more challenging group law on cubic curves
02:40-03:00pm, Drew Ellingson, Computation of Top Intersections on the Moduli Space of Curves
Mentors: Steffen Marcus and Drew Johnson
Abstract: Current algorithms for computing Top Intersections on the Moduli Space of Curves do not satisfactorily handle boundary classes. The goal of the research I have conducted with Steffen Marcus and Drew Johnson is to implement algorithms in the mathematics software SAGE to compute intersection numbers for arbitrary boundary strata. In this presentation, I will introduce the Moduli Space of Curves, its compactification, and the dual graph of a nodal curve. I will then talk about some combinatorial and graph-theoretic problems that arise in computation.
03:00-03:20pm, Jonathan Race, Explorations in GARCH(1,1) Processes
Mentor: Lajos Horváth
Abstract: It is often the case in financial and economic data that we need models which account for dynamic volatility, or variance. GARCH processes are a relatively recent development in such non-linear modeling. In this presentation I will review the application of GARCH processes and some necessary conditions for their existence.
03:20-03:40pm, Nathan Briggs, Optimal three material design on the microstructure and macrostructure scale
Mentor: Andrej Cherkaev
Abstract: The problem of optimal three material composite as formulated and solved by Cherkaev and Dzierzanowski is investigated [1]. Namely stress energy plus cost is minimized for an elastic body loaded on the boundary consisting of a strong and expensive material, a cheap but weak material, and a void. This minimization finds the optimal microstructures by solving a multivariable nonconvex minimization problem which is reduced to determination of the quasiconvex envelope of a multiwell Lagrangian, where the wells represent materials' energies plus their costs; the quasiconvex envelope represents the energy and the cost of an optimal composite [2]. After finding the microstructures the problem of optimal design of a body with these microstructures is investigated. The main focus is on a special case corresponding to a specific cost of the weak material. Finally the roll of each material in the design is investigated and applications are discussed. This work is in collaboration with Grzegorz Dzierzanowski.
[1] A. Cherkaev and G. Dzierzanowski. Three-phase plane composites of minimal elastic stress energy: High-porosity structures. International Journal of Solids and Structures, 50, 25-26, pp. 4145-4160, 2013.
[2] A. Cherkaev. Variational Methods for Structural Optimization. Springer
03:40-04:00pm, Sophia Hudson, Exploring 2D Truss Structures Through Finite Element Simulation
Mentor: Andrej Cherkaev
Abstract: Lattices in the Euclidean plane can be modeled as a collection of nodes and edges, forming a graph, with nodes corresponding to intersections between the trusses modeled by the edges. The problem of our particular interest is that of understanding what happens to these structures when physical properties are applied to the lattice. In this project, we apply forces to the boundary nodes of n x n truss structures, modeled as connected collections of equilateral triangles. We use a finite element model to understand the stresses and strains on the trusses and to visualize the displacement of nodes and edges from their initial conditions.
## Tuesday December 17 Session - 2pm-4pm - LCB 222
02:00-02:20pm, Logan Calder, Fractal Models of Finance
Mentor: Jingyi Zhu
Abstract: Fractals have already been used to solve technological problems in communication, and the fractal power of modeling natural features is widely known. Fractals have been used to create realistic animated landscapes and special effects in movies. The complex features found in living organisms can be recreated by repeating a simple pattern. Even in aspects of more modern systems, such as the risk in financial markets, fractals provide more understanding of reality than standard models.
Typically, models of finance have been based on the normal probability distribution. The data though, doesn't fit the model. There are two many big changes in market prices to allow for an easy model to come from the normal distribution. The normal distribution also doesn't allow for dependent events. Instead, the fractal dimension of market graphs may better help us understand the dependence of price changes on each other and therefore allow us to predict more accurately the variance of price changes over time.
The short term and long term dependence of price changes can be determined in one of two ways: Finding the fractal dimension of the graph of market prices over time, or plotting the log of variance of price changes versus the log of different time intervals.
We found that price changes in gold have short term dependence over a period of about 100 days. This allows for easier determination of variance of changes over different time intervals.
02:20-02:40pm, Michael Senter, Random Motion in Media with Memory
Mentor: Christel Hohenegger
Abstract: Robert Brown discovered random particle motion in the 19th century. We will discuss the model developed by Langevin to describe this motion, as well as the results of Ornstein and Uhlenbeck. We will then proceed to look at random motion in media with memory.
02:40-03:00pm, Camille Humphries, Numerical Methods for the Advection Equation: Comparison of Lax-Friedrichs and Central Schemes
Mentor: Yekaterina Epshteyn
Abstract: This presentation will include an introduction to the advection equation and will focus on two numerical approximation schemes. The structure of the Lax-Friedrichs and Central Schemes will be presented and explained. A comparison of the accuracy and error ratios in both schemes will be shown for a test function.
03:20-03:40pm, Ryan Durr, The Quantification of Exit Times for Varying Fluid Models
Mentor: Christel Hohenegger
Abstract: This project begins with a classic theoretical approach to modeling using the Langevin equation. This model assumes that there is no lasting effect from the fluid on the kinematics of a particle. The advantages of this approach is that there is a known analytic solution that can verify the mathematical simulation. The concluding portion of this project is to model the kinematics of a particle that is traversing a fluid with lasting effects and to quantify its exit times. This model does not have analytic solution. The goal of this research is to quantify the exit time for the different fluid models.
03:40-04:00pm, Wyatt Mackey, On the McKay Graphs of the Projective Representations of $S_n$
Mentor: Dan Ciubotaru
Abstract: The McKay correspondence details a specific connection between the finite automorphisms of $\mathbb{R}^3$ and the Dynkin diagrams of certain Lie algebras. In particular, one creates the Mckay Graphs" by observing multiplicities of representations in the tensor products of the representations of a group with its spin representation, then using this to create a weighted, directed graph. Applying this process on the projective covers of the finite automorphisms of $\mathbb{R}^3$, we achieve equivalent graphs to certain interesting Dynkin diagrams.
The finite groups of automorphisms of $\mathbb{R}^3$ correspond to the cyclic groups, the dihedral groups, and $A_3, \ A_4, \ S_3,$ and $S_4$. This paper is interested in examining possible patterns in the McKay graphs of the projective covers of the symmetric group $S_n$ for larger $n$; in particular, we examine the graphs of $n=5$ and $n=6$. To this end, we wrote a program capable of doing all necessary computations to draw the McKay graphs, given the character table of the group. We did not find similar results to McKay's correspondence, however. Interestingly, we find that the McKay graph of the projective cover of $S_5$ is non-planar, quite different from the cases of $n \le 4$, wherein all of the graphs were trees. Surprisingly, this does not hold for the projective cover of $S_6$, which is again planar. | 2018-12-16 00:57:39 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5307254791259766, "perplexity": 610.1298416895404}, "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-2018-51/segments/1544376827175.38/warc/CC-MAIN-20181216003916-20181216025916-00050.warc.gz"} |
https://bio.libretexts.org/Bookshelves/Ecology/Book%3A_Quantitative_Ecology_-_A_New_Unified_Approach_(Lehman_Loberg_and_Clark)/14%3A_Humans_as_Prey/14.05%3A_The_Strange_Case_of_Polio | 14.5: The Strange Case of Polio
Polio had long been a relatively rare disease of infants, called “infantile paralysis.” In the middle of the twentieth century, however, it became more common and started affecting older children and adults. A new form of the disease seemed to be emerging.
Human Health and Hygiene
Which of these people, do you think, performed the greatest service to human health and hygiene in the twentieth century, but inadvertently triggered this mid-century polio epidemic?
1. Louis Pasteur, discoverer of pasteurization
2. Alexander Fleming, discoverer of penicillin
3. Jonas Salk, creator of the polio vaccine
4. Franklin Delano Roosevelt, President of the United States and polio victim
5. Henry Ford, creator of the production line.
This seems a strange question, with industrialist Henry Ford under consideration. But indeed, the answer is Henry Ford!
At the beginning of the twentieth century, most local transportation was by horse and powered, of course, almost entirely by the biofuel hay. While it has now largely left social memory, in the early decades of the twentieth century the streets were a slurry of gravel and horse manure. Flies were everywhere, and caused little concern paid, for this was the norm. People’s outhouses were ventilated to the open air, and flies laid eggs there and in the streets, then freely entered houses and landed on food. A number of diseases take advantage of the fecal–oral pathway, and polio is one of them.
But automobiles and tractors intervened. As the horse-drawn era closed, manure generally vanished, running water arrived and flush toilets arrived, hygiene improved, sealed screen doors became common, and flies died in vast numbers. Without intending it, Henry Ford became the greatest fly killer of all time. The availability of the fecal–oral pathway diminished, and the infectivity of related diseases fell.
Figure $$PageIndex{1}$$ shows the horse population declining slowly until World War I, then falling rather steadily as the number of cars increased in stages. The first increase in the number of cars ended around 1930 with the Great Depression, when many people could not afford cars. The end of World War II in 1945 brought another boom in car purchases, and by 1975 society had replaced almost every horse per capita with a car.
As the chance of catching polio fell, the average age of catching it increased. To understand this, consider residents of the northern hemisphere living at various latitudes. Because residents of the High Arctic have a chance to see the northern lights—the aurora borealis—every week, children living there will likely see the aurora before their first birthday. Farther south, at 50 degrees north latitude, the aurora may appear only once every few years, especially near the lights of cities, so a child could be 5 or 10 years old before ever seeing it. And finally, say at 35 degrees north latitude, the aurora may appear but once or twice in a lifetime, so many people could be in middle age before viewing them, and others might go an entire lifetime without being touched by their hypnotic display.
So it is with disease. The number of opportunities for catching a highly infectious disease, naturally, is high. If the quantity of pathogens in the environment is such that all individuals encounter them on average once a year, only about one-third of infants will avoid infection in their first year. (Actually the number is 1/e = 0.367..., if the chance of infection is completely random.) The same fraction of the remaining infants will catch the disease during their first year, and the rest will be age two or older when they catch the disease. Therefore, as the pathways for transmitting polio diminished during the twentieth century, the chances of catching it in any year decreased and the age of onset correspondingly increased.
Polio is like some other diseases that are not usually virulent in infants and young children. A baby infected with polio might have a cold and a runny nose, and the infection might go without particular notice. In an older child, however, it can stop bone growth and muscle development, crippling the child. The polio epidemic of mid-century America was thus not a new disease emerging, but an ancient disease dying out.
Albert Sabin, of polio vaccine fame, suspected a connection with flies. In 1941 he and his colleagues reported in Science on a study they performed in areas of the United States where polio had struck. They captured flies, pureed them in sterile fluid, and gave them to monkeys in feedings, nosedrops, or injections. As they put it, “Down came the monkeys with polio.”
With further improvements in hygiene and broad use of vaccines, rates of polio have dropped to nearly zero. Figure 14.8 shows a moderate number of cases of polio before the late 1940s, an outbreak lasting until the early 1960s, nearly nothing in the years following.
This page titled 14.5: The Strange Case of Polio is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Clarence Lehman, Shelby Loberg, & Adam Clark (University of Minnesota Libraries Publishing) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | 2022-08-12 20:41:36 | {"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": 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": 0.4637599289417267, "perplexity": 3269.6199628021604}, "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-2022-33/segments/1659882571758.42/warc/CC-MAIN-20220812200804-20220812230804-00687.warc.gz"} |
http://crowlspace.com/?paged=3 | ## More MathJax Testing
This one uses a Javascript to load MathJax direct. Seems easier than the plug-in.
In equation 1, we find the value of an
interesting integral:
$$\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}$$
or this:
$$\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}$$
or this:
$$\mathcal{\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}}$$
## MathJax on Crowlspace
I’ve just downloaded the MathJax Plugin for WordPress and this is a test equation:
$$E=mc^2$$
So what do you think?
## Mission to Ceres
Ceres is in the news, thanks to the marvellous “Dawn” mission, which has seen a plucky little solar-powered ion-drive achieve orbit around two heavenly bodies on one tank of propellant. However the low power-to-mass ratio of the ion-drive means a multi-year journey, which is punishing for human crew and would-be colonists. A more reasonable design was proposed by James Longuski and his team at Purdue:
Abstract
A low-thrust trajectory design study is performed for a mission to send humans to Ceres and back. The flight times are constrained to 270 days for each leg, and a grid search is performed over propulsion system power, ranging from 6 to 14 MW, and departure V?V?, ranging from 0 to 3 km/s. A propulsion system specific mass of 5 kg/kW is assumed. Each mission delivers a 75 Mg payload to Ceres, not including propulsion system mass. An elliptical spiral method for transferring from low Earth orbit to an interplanetary trajectory is described and used for the mission design. A mission with a power of 11.7 MW and departure V?V? of 3 km/s is found to offer a minimum initial mass in low Earth orbit of 289 Mg. A preliminary supply mission delivering 80 Mg of supplies to Ceres is also designed with an initial mass in low Earth orbit of 127 Mg. Based on these results, it appears that a human mission to Ceres is not significantly more difficult than current plans to send humans to Mars.
I believe the basis for the above paper is the 2011 Student Project Vision here:
Project Vision
…which has this rather elaborate Crew Transfer Vehicle doing the heavy-lifting of carrying a crew to Ceres:
…which requires a bit of explanation:
Getting to Ceres is not easy. The major delta-vee budget is due to the plane change (Ceres is inclined to the ecliptic by 10.6 degrees) and the lack of high energy capture orbits, aerocapture or aerobraking at such a small object. Yet it’s not much more difficult than getting to Mars in some respects – if you include the landing delta-vee budget. The major enticement is the chance of abundant water ice and, perhaps, some sort of easy access to liquid water from cryovolcanic vents. “Dawn” has given us the mysterious White Spot, which is at least a kilometre above the crater floor it is in the middle of. Could it be a protusion of the water ice from below the asphalt black crust? Or something more exotic – an icy fumerole? There’s water vapour around Ceres, which hopefully “Dawn” will study in more detail.
The real crying need for such missions is multi-megawatt space-power supplies. Until that’s developed, such missions will remain paper studies.
## Exotic Biochemistries
Check out Paul Gilster’s discussion of azotosome-based life in the methane lakes of Titan
[Ref: “Membrane alternatives in worlds without oxygen: Creation of an azotosome” Science Advances Vol. 1, No. 1 (27 February 2015), e1400067.]
His essay prompted this quick discussion.
In 1961 Isaac Asimov, who was a research Chemist as well as uber-writing machine, wrote a highly influential essay (for “Fantasy & Science-Fiction” magazine) on exotic biochemistries. For those who want to read what the Good Doctor had to say it was reprinted in the old “Cosmic Search” newsletter and is available online here:
Not as We Know it – The Chemistry of Life
Asimov suggested the following options, in order of decreasing temperature:
There, then, is my list of life chemistries, spanning the temperature range from near red heat down to near absolute zero:
1. fluorosilicone in fluorosilicone
2. fluorocarbon in sulfur
3.*nucleic acid/protein (O) in water
4. nucleic acid/protein (N) in ammonia
5. lipid in methane
6. lipid in hydrogen
Of this half dozen, the third only is life-as-we-know-it. Lest you miss it, I’ve marked it with an asterisk.
I originally read about Asimov’s typology in “Man and the Stars”, the collection of discussions on Extraterrestrial contact by the ASTRA group in Scotland published by Duncan Lunan.
A more recent discussion of exotic biochemistry, which inspired Stephen Baxter’s recent depiction of Titanian life in his novel “Ultima”, is found in William Bains’ essay here:
In turn Bains’ work led to a collaboration with Sara Seager which provocatively argues for a hydrogen-based photosynthetic life:
Photosynthesis in Hydrogen-Dominated Atmospheres [Open Accesss]
[Ref: Life 2014, 4(4), 716-744; doi:10.3390/life4040716]
…the full implications of which are yet to be explored – the essay was published late last year. One irritating conclusion is that such H2 based biospheres might be very hard to detect remotely.
Another exotic option is the possibility of chlorinic photosynthesis, making chlorine based compounds instead of oxygen as a by-product:
[Ref: Astrobiology. 2010 Nov;10(9):953-63. doi: 10.1089/ast.2009.0364]
…though chlorine compounds do tend to be very opaque and may make the surface too dark to sustain life. In his “Manifold” trilogy, book 2 “Space”, Stephen Baxter imagined a world poisoned by the deliberate seeding of its oceans with chlorine producing organisms. If such a photosynthetic pathway is possible, then its spontaneous evolution in our own oceans is a possibility that we might’ve be lucky enough to avoid thus far. Other worlds, maybe not.
## Ceres Entices…
The Bright Spots of Ceres entice with their “sudden” appearance in sunlight and their delayed fade as they cross the terminator. Is it some bright projection of salty-ice, freshly exuding into a crater floor, or something more dramatic like a salt diapir that has pushed through from below?
## Ceres: Its Origin and Predicted Bulk Chemical Composition
Andrew Prentice’s Modern Laplacian Theory (MLT) has made definite predictions, with a reasonable success rate, for the bodies of the Solar System for the last ~40 years. The latest new body in view of our probes is Ceres, which the MLT predicts is a metal/silicate core wrapped in water ice and salt.
Note Prentice’s statement:
Perhaps Dawn will find the surface of Ceres to be very flat, though roughened through aeons of impacts, with fresh craters having bright floors and ejecta.
…in light of this enigma:
## O’Neill Cylinder from 1931
In its early days, the “Buck Rogers” comic-strip kept an eye on new developments in astronomy and tried for scientific plausibility based on the (admittedly shaky) facts of the day. The asteroid Eros was discovered in 1931 and appeared in a story that same year. A particularly prescient tale it featured a cylindrical habitat 20 miles long and 5 wide, with gravity produced by centrifugal force.
## Centauri Dreams: What Comets Are Made Of
Informative musings from Paul Gilster. Dirty Snowball or Dusty, Greasy Snowball? NASA has tried to replicate what happens to water-ice and hydrocarbons in cryogenic conditions – with unexpected results.
## Next Big Future – Where the Next Trend is Found
Brian Wang’s future-news blog is the Source for the scape of the coming-soon. | 2016-06-25 14:00:54 | {"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": 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": 0.24269582331180573, "perplexity": 5251.195871112794}, "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/1466783393332.57/warc/CC-MAIN-20160624154953-00200-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/29206/quantifying-how-good-im-producing-a-spectral-density | # Quantifying how good I'm producing a spectral density
I recently asked a question here on how to create noise with a specific spectral density $S_{xx}(f)$. The helpful people of this stack exchange told me one way to do it was filtering white noise with a unitary spectral density using a filter with impulse response $H(f) = \sqrt{S_{xx}(f)}$ and then suggested how to do this. The way that was proposed was using a frequency sampling based FIR filter. As I use Mathematica the idea was to use FrequencySamplingFilterKernel.
Now, this seems to work reasonably well qualitatively, but I am trying to find a way of quantifying this. So my question comes down to how I quantitaviely test how well the noise produced with my filter resembles the desired spectral density.
To take a specific example, I am trying to produce (not exclusively, but to begin with) the Ornstein Uhlenbeck spectral density given by $S_{xx}(f) = \frac{2c\tau^2}{1+(2 \pi f)^2}$. So I suppose it is the easiest if I just show my work and then get to my specific question. What I did was generate the amplitudes
Γ = 10; (* bandwidth of Lorentzian*)
P = 2; (* std dev of noise *)
τn = 1/Γ;(* relaxation time of OU process *)
c = 2 P^2 Γ;(* diffusion constant of OU process *)
S[f_] := (2 c τn^2)/(
1 + (2 Pi f τn)^2); (* power spectral density *)
K = 100; (* number of frequency components taken into account *)
fmax = 5; (*max frequency*)
Δf = fmax/(K - 1);(* freq step *)
fk = Range[0, fmax, Δf]; (* sampling frequencies *)
FilterAmps = Sqrt[S[fk]]; (* Amplitudes for the FIR filter *)
The parameter choices are arbitrary, and definitely not necessarily what I want to do in practice, but that shouldn't matter too much for now. I then generate some white noise (the times list is not needed at this point, but I'm just doing it to keep track of what I'm doing for now):
Ns = 1000; (* number of time points generated; output time points \
after filter is n = Ns - 2K + 2*)
Δt = 1/(2 fmax); (*Nyquist rate*)
Tmax = (Ns - 1) Δt;
times = Range[0,
Tmax, Δt]; (*Using Nyquist rate as step size*)
UnitDataWhite =
RandomVariate[NormalDistribution[0, 1], Length[times]];
Okay, great, so then we filter
FilteredNoise =
ListConvolve[FrequencySamplingFilterKernel[FilterAmps],
UnitDataWhite];
FilteredTimes =
Range[0, (Length[FilteredNoise] -
1) Δt, Δt];
Giving me a dataset of $n = N - K + 2$ noise datapoints which should behave according to the spectral density of the Ornstein Uhlenbeck process. The problem is, how do I quantify how well this works? The first thing that came to mind was just constructing the power spectral density and seeing how well it agreed with theory. Well, there I ran into my first real meeting with the fact that one does not simply find the power spectral density from a dataset to some arbitrary degree of precision. Mathematica has a built in function periodogram which is supposed to do this, but while the shape is definitely as desired, is not exactly.
Okay, so then I thought perhaps it is too much to ask to reconstruct the spectral density in this way. Should I look instead at other quantities that carry the same information, such as the autocovariance? For the OU process this is given by $C_{x}(t) = \frac{c\tau}{2} e^{-t/\tau}$ so that's also relatively nice. But calculating the autocovariance with CovarianceFunction again runs into amplitude trouble, so that doesn't seem like a good method either.
So in the end my question boils down to this. Creating noise with a spectral density using the way I described here, how do I reliably quantify how well I am succeeding?
*Note that I know that there are nicer ways of creating OU noise, but I am trying to construct a method for general spectral densities
• You are getting the correct shape even by using FilterAmps = S[fk] instead of FilterAmps = Sqrt[S[fk]]? – SleuthEye Mar 4 '16 at 3:23
• That's an embarrassing mistake. The question still holds, but that does already fix it a little. – user129412 Mar 4 '16 at 14:40
• I can probably add to this in the following way. Although perhaps with carefully picking the specific parameters of periodogram (averaging over partitions, averaging, smoothing functions) I can eventually get close to the correct power spectral density, but this is only because I know where I want to go. I haven't really been able to find a reliable guide for how to pick your overlap/averaging sizes depending on the data size, most that I can find tells you that you should adjust this according to your specific situation, which doesn't say that much. – user129412 Mar 4 '16 at 15:04
• Keep in mind that a periodogram is only an estimation method, and you are generating colored noise which is by nature unknown (you could generate something that has exactly the spectrum you want in a deterministic way, but then it would no longer be 'noise'). If that periodogram converges to what you expect as you increase the number of samples (so you can get a better estimate) then your signal generation is probably fine. – SleuthEye Mar 4 '16 at 15:16
• That does make sense, but I'm aiming to use this for an experiment where it will be quite crucial to accurately control the parameters of the noise. In the end I'll be moving on to a physical spectrum analyser so perhaps that will help. Still a little confused on why the autocorrelations are misbehaving however, but perhaps it is simply some vertical offset that I need to understand. – user129412 Mar 4 '16 at 15:20 | 2020-05-29 23:49:11 | {"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": 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": 0.7081028819084167, "perplexity": 555.9912108915004}, "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/1590347406785.66/warc/CC-MAIN-20200529214634-20200530004634-00419.warc.gz"} |
http://ptsymmetry.net/?m=201207 | July 2012
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## Reciprocity and unitarity in scattering from a non-Hermitian complex PT-symmetric potential
Zafar Ahmed
In non-relativistic quantum scattering, Hermiticity is necessary for both reciprocity and unitarity. Reciprocity means that both reflectivity (R) and transmitivity (T) are insensitive to the direction of incidence of a wave (particle) at a scatterer from left/right. Unitarity means that R+T=1. In scattering from non-Hermitian PT-symmetric structures the (left/right) handedness (non-reciprocity) of reflectivity is known to be essential and unitarity remains elusive so far. Here we present a surprising occurrence of both reciprocity and unitarity in scattering from a complex PT-symmetric potential. In special cases, we show that this potential can even become invisible (R=0, T=1) from both left and right sides. We also find that this optical potential can give rise to a perfect transmission (T=1) this time without both unitarity and reciprocity (of reflectivity).
http://arxiv.org/abs/1207.6896
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
## PT-Symmetric Pseudo-Hermitian Relativistic Quantum Mechanics With a Maximal Mass
V. N. Rodionov
The quantum-field model described by non-Hermitian, but a $${\cal PT}$$-symmetric Hamiltonian is considered. It is shown by the algebraic way that the limiting of the physical mass value $$m \leq m_{max}= {m_1}^2/2m_2$$ takes place for the case of a fermion field with a $$\gamma_5$$-dependent mass term ($$m\rightarrow m_1 +\gamma_5 m_2$$). In the regions of unbroken $$\cal PT$$ symmetry the Hamiltonian $$H$$ has another symmetry represented by a linear operator $$\cal C$$. We exactly construct this operator by using a non-perturbative method. In terms of $$\cal C$$ operator we calculate a time-independent inner product with a positive-defined norm. As a consequence of finiteness mass spectrum we have the $$\cal PT$$-symmetric Hamiltonian in the areas $$(m\leq m_{max})$$, but beyond this limits $$\cal PT$$-symmetry is broken. Thus, we obtain that the basic results of the fermion field model with a $$\gamma_5$$-dependent mass term is equivalent to the Model with a Maximal Mass which for decades has been developed by V.Kadyshevsky and his colleagues. In their numerous papers the condition of finiteness of elementary particle mass spectrum was introduced in a purely geometric way, just as the velocity of light is a maximal velocity in the special relativity. The adequate geometrical realization of the limiting mass hypothesis is added up to the choice of (anti) de Sitter momentum space of the constant curvature.
http://arxiv.org/abs/1207.5463
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
## Time-dependent Hamiltonians with 100% evolution speed efficiency
Raam Uzdin, Uwe Guenther, Saar Rahav, Nimrod Moiseyev
The evolution speed in projective Hilbert space is considered for Hermitian Hamiltonians and for non-Hermitian (NH) ones. Based on the Hilbert-Schmidt norm and the spectral norm of a Hamiltonian, resource-related upper bounds on the evolution speed are constructed. These bounds are valid also for NH Hamiltonians and they are illustrated for an optical NH Hamiltonian and for a non-Hermitian $$\mathcal{PT}$$-symmetric matrix Hamiltonian. Furthermore, the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective Hilbert space. A recipe for the construction of time-dependent Hamiltonians which ensure 100% speed efficiency is given. Generally these efficient Hamiltonians are NH but there is a Hermitian efficient Hamiltonian as well. Finally, the extremal case of a non-Hermitian non-diagonalizable Hamiltonian with vanishing energy difference is shown to produce a 100% efficient evolution with minimal resources consumption.
http://arxiv.org/abs/1207.5373
Quantum Physics (quant-ph)
## Breakdown of adiabatic transfer schemes in the presence of decay
Eva-Maria Graefe, Alexei A. Mailybaev, Nimrod Moiseyev
In atomic physics, adiabatic evolution is often used to achieve a robust and efficient population transfer. Many adiabatic schemes have also been implemented in optical waveguide structures. Recently there has been increasing interests in the influence of decay and absorption, and their engineering applications. Here it is shown that contrary to what is often assumed, even a small decay can significantly influence the dynamical behaviour of a system, above and beyond a mere change of the overall norm. In particular, a small decay can lead to a breakdown of adiabatic transfer schemes, even when both the spectrum and the eigenfunctions are only sightly modified. This is demonstrated for the decaying version of a STIRAP scheme that has recently been implemented in optical waveguide structures. It is found that the transfer property of the scheme breaks down at a sharp threshold, which can be estimated by simple analytical arguments.
http://arxiv.org/abs/1207.5235
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)
## Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments
Alessandro Sergi, Konstantin G. Zloshchastiev
We consider a non-Hermitian Hamiltonian in order to effectively describe a two-level system coupled to a dissipative environment. The total Hamiltonian of the model is obtained by adding a general anti-Hermitian part, depending on four parameters, to the Hermitian Hamiltonian of a tunneling two-level system. The time evolution is formulated and derived in terms of the density matrix of the model, different types of decays are revealed and analyzed. In particular, the population difference and coherence are defined and calculated analytically. We have been able to mimic various physical situations with different properties, such as dephasing and vanishing population difference.
http://arxiv.org/abs/1207.4877
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
## PT-symmetric quantum Liouvillean dynamics
Tomaz Prosen
We discuss a combination of unitary and anti-unitary symmetry of quantum Liouvillian dynamics, in the context of open quantum systems, which implies a D2 symmetry of the complex Liovillean spectrum. For sufficiently weak system-bath coupling it implies a uniform decay rate for all coherences, i.e. off-diagonal elements of the system’s density matrix taken in the eigenbasis of the Hamiltonian. As an example we discuss symmetrically boundary driven open XXZ spin 1/2 chains.
http://arxiv.org/abs/1207.4395
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
## Dynamics of higher-order solitons in regular and PT-symmetric nonlinear couplers
R. Driben, B. A. Malomed
Dynamics of symmetric and antisymmetric 2-solitons and 3-solitons is studied in the model of the nonlinear dual-core coupler and its PT-symmetric version. Regions of the convergence of the injected perturbed symmetric and antisymmetric N-solitons into symmetric and asymmetric quasi-solitons are found. In the PT-symmetric system, with the balanced gain and loss acting in the two cores, borders of the stability against the blowup are identified. Notably, in all the cases the stability regions are larger for antisymmetric 2-soliton inputs than for their symmetric counterparts, on the contrary to previously known results for fundamental solitons (N=1). Dynamical regimes (switching) are also studied for the 2-soliton injected into a single core of the coupler. In particular, a region of splitting of the input into a pair of symmetric solitons is found, which is explained as a manifestation of the resonance between the vibrations of the 2-soliton and oscillations of energy between the two cores in the coupler.
http://arxiv.org/abs/1207.3917
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
## The two dimensional harmonic oscillator on a noncommutative space with minimal uncertainties
Sanjib Dey, Andreas Fring
The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.
http://arxiv.org/abs/1207.3303
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
## Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations
Sanjib Dey, Andreas Fring
We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets are clearly identified.
http://arxiv.org/abs/1207.3297
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph) | 2018-04-20 02:47:24 | {"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": 0.6439723968505859, "perplexity": 1122.7284264623022}, "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-2018-17/segments/1524125937113.3/warc/CC-MAIN-20180420022906-20180420042906-00294.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/proc.2013.2013.183 | Article Contents
Article Contents
# Small data solutions for semilinear wave equations with effective damping
• We consider the Cauchy problem for the semi-linear damped wave equation
$u_{tt} - \Delta u + b(t)u_t = f(t,u),\qquad u(0,x) = u_0(x),\qquad u_t(0,x) = u_1(x).$
We prove the global existence of small data solution in low space dimension, and we derive $(L^m\cap L^2)-L^2$ decay estimates, for $m\in[1,2)$. We assume that the time-dependent damping term $b(t)>0$ is effective, that is, the equation inherits some properties of the parabolic equation $b(t)u_t - \Delta u = f(t,u)$.
Mathematics Subject Classification: 35L71.
Citation:
• [1] M. D'Abbicco, M.R. Ebert, Hyperbolic-like estimates for higher order equations, J. Math. Anal. Appl. 395 (2012), 747-765, doi:10.1016/j.jmaa.2012.05.070. [2] M. D'Abbicco, M.R. Ebert, A class of dissipative wave equations with time-dependent speed and damping J. Math. Anal. Appl. 399 (2013), 315-332, doi:10.1016/j.jmaa.2012.10.017. [3] M. D'Abbicco, S. Lucente, A modified test function method for damped wave equations Adv. Nonlinear Studies 13 (2013), 867-892. [4] M. D'Abbicco, S. Lucente, M. Reissig, Semilinear wave equations with effective damping, Chinese Ann. Math. 34B (2013), 3, 345-380, doi:10.1007/s11401-013-0773-0. [5] H. Fujita, On the blowing up of solutions of the Cauchy Problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac.Sci. Univ. Tokyo 13 (1966), 109-124. [6] R. Ikehata, Y. Mayaoka, T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^N$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373. [7] R. Ikehata, M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbbR^N$, J. Math. Anal. Appl., 269 (2002), 87-97. [8] R. Ikehata, K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $R^N$ with noncompactly supported initial data, Nonlinear Analysis 61 (2005), 1189-1208. [9] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with Space-Dependent Potential, Funkcial. Ekvac. 52 (2009), 411-435. [10] J. Lin, K. Nishihara, J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discrete and Continuous Dynamical Systems, 32 (2012), 4307-4320, http://www.aimsciences.org/journals/pdfs.jsp?paperID=7562&mode=full [11] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS. 12 (1976), 169-189. [12] G. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. of Differential Equations 174 (2001), 464-489. [13] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations 232 (2007), 74-103. [14] Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 109-114.
Open Access Under a Creative Commons license | 2023-03-23 04:23:45 | {"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.5761892795562744, "perplexity": 1485.8308944388082}, "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-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00771.warc.gz"} |
https://koreascience.kr/article/JAKO199127053856972.page?&lang=ko | # SCANNING ELECTRON MICROSCOPIC STUDY OF THE JUNCTION BETWEEN GOLD INLAYS AND GINGIVAL CAVOSURFACE MARGINS
• 박준일 (서울대학교 치과대학 치과보존학교실) ;
• 권혁춘 (서울대학교 치과대학 치과보존학교실)
• Park, Joon-Il (Department of Conservative Dentistry, College of Dentistry, Seoul National University) ;
• Kown, Hyuk-Choon (Department of Conservative Dentistry, College of Dentistry, Seoul National University)
• 발행 : 1991.10.31
#### 초록
Present - day inlay casting procedures have been developed for more than 100 years and experimentation has focused on the perfect adaptation to the cavity preparation. Marginal adaptation is considered to be an important indicator of the acceptability of the cast restotration, especially on the gingival margin. The purpose of this study was to evaluate the effects of a dissecting microscope and burnishing on vertical discrepancies, horizontal discrepancies, and cement thicknesson master die. Extracted premolars were prepared for class II gold inlays and master dies were made with conventional techniques. The experiments consisted of 4 groups. Group 1 : unaided eye, no burnishing on master die. Group 2 : unaided eye, burnishing on master die. Group 3 : microscope, no burnishing on master die. Group 4 : microscope, burnishing on master die. Cemented inlays were embedded in the hard resin and sectioned with microcutter through the gingival margins. The sectioned surfaces were polished with emery paper and finally with aluminum oxide powders. The results of the experiments were measured for vertical discrepancies, horizontal discrepancied and cement thickness under the scanning electron microscpe at the beveled gingival margin. The results of the study were summarized as follows. 1. Group 1 showed the vertical discrepancies of $81.6{\mu}m({\pm}48.6{\mu}m)$, horizontal discrepancies of $60.1{\mu}m({\pm}41.1{\mu}m)$, and cement thickness of $59.6{\mu}m({\pm}24.6{\mu}m)$. 2. Group 2 showed the vertical discrepancies of $78.6{\mu}m({\pm}30.9{\mu}m)$, horizontal discrepancies of $36.9{\mu}m({\pm}20.7{\mu}m)$, and cement thickness of $54.0{\mu}m({\pm}21.6{\mu}m)$. 3. Group 3 showed the vertical discrepancies of $57.5{\mu}m({\pm}26.4{\mu}m)$, horizontal discrepancies of $28.4{\mu}m({\pm}17.5{\mu}m)$, and cement thickness of $37.2{\mu}m({\pm}17.4{\mu}m)$. 4. Group 4 showed the vertical discrepancies of $56.7{\mu}m({\pm}35.0{\mu}m)$, horizontal discrepancies of $31.8{\mu}m({\pm}24.2{\mu}m)$, and cement thickness of $45.6{\mu}m({\pm}19.8{\mu}m)$. 5. Vertical discrepancies were not significantly different at any groups(p>.050). 6. Microscope groups(Group 3, 4) showed significantly improved horizontal marginal adaptation (p<.050). 7. Although cement thickness showed the subset of Group 3. 4, 2 and Group 4, 2, 1. Group 3 showed significantly smaller thickness than Group l(p<.050). 8. Finishing and polishing by means of a microscope produced significantly smaller discrepancies than doing so with the unaided eye(p<.050).
#### 과제정보
연구 과제 주관 기관 : 서울대학교병원 | 2022-08-10 13:29:33 | {"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": 0.7207033634185791, "perplexity": 9674.541754088576}, "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/1659882571190.0/warc/CC-MAIN-20220810131127-20220810161127-00123.warc.gz"} |
https://rosettacode.org/wiki/Null | Null object
(Redirected from Null)
Null object
You are encouraged to solve this task according to the task description, using any language you may know.
Null (or nil) is the computer science concept of an undefined or unbound object. Some languages have an explicit way to access the null object, and some don't. Some languages distinguish the null object from undefined values, and some don't.
Show how to access null in your language by checking to see if an object is equivalent to the null object.
This task is not about whether a variable is defined. The task is about "null"-like values in various languages, which may or may not be related to the defined-ness of variables in your language.
11l
```F f([Int]? &a)
I a != N
a.append(1)
f(N)
[Int] arr
f(&arr)
print(arr)```
Output:
```[1]
```
6502 Assembly
Translation of: Z80 Assembly
Technically there is no such thing as a null pointer; all pointers point to something. It's a matter of what you're willing to give up. Often the null pointer is thought of as memory address 0, and on many CPUs this is the case - however this is most assuredly not the case on the 6502. Reason being, zero page RAM is limited and quite valuable, given that there are only 256 bytes of it and it is much more efficient to access than regular memory. As such, declaring `\$0000` to be the null pointer would be a very poor choice. Ideally, the null pointer on a 6502 should:
• Be somewhere that isn't zero page RAM
• Be somewhere that we cannot change at runtime (e.g. read-only memory) or doing so would cause major problems (the vector table)
• Point to something that has no value to the programmer.
We can choose quite a few places, the easiest one I can think of is `\$FFFF`. Although it sort of breaks our first rule, as when dereferenced as a 16-bit value, you get the value stored at `\$0000` as the high byte, we still can access `\$0000` normally anyway. Since it points to the high byte of the interrupt request vector, it's something we don't want to (or most likely can't) modify at runtime, and is of no use to us (if we really wanted the IRQ handler's address we'd dereference `\$FFFE` instead.)
How a null pointer is implemented is very simple. You decide beforehand what your null pointer will be, and before you dereference a pointer variable, compare it to the null pointer, and if they're equal, don't dereference it. That's all there is to it.
```lda pointer ;a zero-page address that holds the low byte of a pointer variable.
CMP #\$FF
BNE .continue
lda pointer+1
CMP #\$FF
BNE .continue
RTS ;return without doing anything
.continue```
8th
```null? if "item was null" . then
```
AArch64 Assembly
Works with: as version Raspberry Pi 3B version Buster 64 bits
```/* ARM assembly AARCH64 Raspberry PI 3B */
/* program nullobj64.s */
/*******************************************/
/* Constantes file */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
/*******************************************/
/* Initialized data */
/*******************************************/
.data
szCarriageReturn: .asciz "\n"
szMessResult: .asciz "Value is null.\n" // message result
qPtrObjet: .quad 0 // objet pointer
/*******************************************/
/* UnInitialized data */
/*******************************************/
.bss
/*******************************************/
/* code section */
/*******************************************/
.text
.global main
main: // entry of program
ldr x0,[x0] // load pointer value
cbnz x0,100f // is null ?
ldr x0,qAdrszMessResult // yes -> display message
bl affichageMess
100: // standard end of the program
mov x0,0 // return code
mov x8,EXIT // request to exit program
svc 0 // perform the system call
/********************************************************/
/* File Include fonctions */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"```
Action!
```TYPE Object=[
BYTE byteData
INT intData
CARD cardData]
PROC IsNull(Object POINTER ptr)
IF ptr=0 THEN
PrintE("Object is null")
ELSE
PrintE("Object is not null")
FI
RETURN
PROC Main()
Object a
Object POINTER ptr1=a,ptr2=0
IsNull(ptr1)
IsNull(ptr2)
RETURN```
Output:
```Object is not null
Object is null
```
ActionScript
```if (object == null)
trace("object is null");
```
ActionScript also has an undefined value: see Undefined values#ActionScript.
```with Ada.Text_Io;
if Object = null then
end if;
```
ALGOL 68
In ALGOL 68 the NIL yields a name that does not refer to any value. NIL can never be naturally coerced and can only appear where the context is strong.
Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d
```REF STRING no result = NIL;
STRING result := "";
IF no result :=: NIL THEN print(("no result :=: NIL", new line)) FI;
IF result :/=: NIL THEN print(("result :/=: NIL", new line)) FI;
IF no result IS NIL THEN print(("no result IS NIL", new line)) FI;
IF result ISNT NIL THEN print(("result ISNT NIL", new line)) FI;
COMMENT using the UNESCO/IFIP/WG2.1 ALGOL 68 character set
result := °;
IF REF STRING(result) :≠: ° THEN print(("result ≠ °", new line)) FI;
END COMMENT
# Note the following gotcha: #
REF STRING var := NIL;
IF var ISNT NIL THEN print(("The address of var ISNT NIL",new line)) FI;
IF var IS REF STRING(NIL) THEN print(("The address of var IS REF STRING(NIL)",new line)) FI```
Output:
```no result :=: NIL
result :/=: NIL
no result IS NIL
result ISNT NIL
The address of var ISNT NIL
The address of var IS REF STRING(NIL)
```
NIL basically is an untyped ref (pointer) that does not refer anywhere.
ALGOL 68 also has empty. This is a "constant" of size 0 and type void. c.f. Roots of a function for two different examples of usage.
• empty as an undefined argument to a routine.
• empty as a routine return if no result is found.
empty is typically used to refer to am empty leaf in a tree structure.
Basically:
• ALGOL 68's empty is python's `None`,
• ALGOL 68's void is python's `NoneType`, and
• ALGOL 68's nil is python's `hash(None)`
ALGOL W
```begin
% declare a record type - will be accessed via references %
record R( integer f1, f2, f3 );
% declare a reference to a R instance %
reference(R) refR;
% assign null to the reference %
refR := null;
% test for a null reference - will write "refR is null" %
if refR = null then write( "refR is null" ) else write( "not null" );
end.```
AmigaE
```DEF x : PTR TO object
-> ...
IF object <> NIL
-> ...
ENDIF```
APL
APL is a vector/array-based language, so rather than a 'null pointer' or 'null value' there is the 'null vector'.
``` ⍝⍝ GNU APL
]help ⍬
niladic function: Z ← ⍬ (Zilde)
Zilde is the empty numeric vector (aka. ⍳0)
Not a function but rather an alias for the empty
vector:
⍬≡⍳0
1
```
AppleScript
Many applications will return `missing value`, but `null` is also available.
```if x is missing value then
display dialog "x is missing value"
end if
if x is null then
display dialog "x is null"
end if
```
ARM Assembly
Works with: as version Raspberry Pi
```/* ARM assembly Raspberry PI */
/* program nullobj.s */
/* Constantes */
.equ STDIN, 0 @ Linux input console
.equ STDOUT, 1 @ Linux output console
.equ EXIT, 1 @ Linux syscall
.equ READ, 3 @ Linux syscall
.equ WRITE, 4 @ Linux syscall
/* Initialized data */
.data
szCarriageReturn: .asciz "\n"
szMessResult: .asciz "Value is null.\n" @ message result
iPtrObjet: .int 0 @ objet pointer
/* UnInitialized data */
.bss
/* code section */
.text
.global main
main: @ entry of program
ldr r0,[r0] @ load pointer value
cmp r0,#0 @ is null ?
ldreq r0,iAdrszMessResult @ yes -> display message
bleq affichageMess
100: @ standard end of the program
mov r0, #0 @ return code
pop {fp,lr} @ restaur 2 registers
mov r7, #EXIT @ request to exit program
svc 0 @ perform the system call
/******************************************************************/
/* display text with size calculation */
/******************************************************************/
/* r0 contains the address of the message */
affichageMess:
push {r0,r1,r2,r7,lr} @ save registres
mov r2,#0 @ counter length
1: @ loop length calculation
ldrb r1,[r0,r2] @ read octet start position + index
cmp r1,#0 @ if 0 its over
bne 1b @ and loop
@ so here r2 contains the length of the message
mov r1,r0 @ address message in r1
mov r0,#STDOUT @ code to write to the standard output Linux
mov r7, #WRITE @ code call system "write"
svc #0 @ call systeme
pop {r0,r1,r2,r7,lr} @ restaur des 2 registres */
bx lr @ return```
Arturo
```v: null
if v=null -> print "got NULL!"
```
Output:
`got NULL!`
AutoHotkey
```If (object == null)
MsgBox, object is null
```
AutoIt
```Local \$object = Null
If \$object = Null Then MsgBox(0, "NULL", "Object is null")
```
AWK
Undefined elements correspond to an empty string; when converted to a numerical value, it evaluates to 0. In order to distinguish a undefined value from a value of 0, length(var) need to be used.
```#!/usr/bin/awk -f
BEGIN {
b=0;
print "<"b,length(b)">"
print "<"u,length(u)">"
print "<"u+0,length(u+0)">";
}
```
Output
```<0 1>
< 0>
<0 1>```
Axe
Null pointers can be checked by simply comparing the pointer with 0.
```If P=0
Disp "NULL PTR",i
End```
Babel
In this example, we place nil on the stack, then perform an if-then-else (ifte) based on the value returned by the 'nil?' operator which returns true if top-of-stack (TOS) is nil. If TOS is nil, then we can be relieved, otherwise, the interpreter has gone absolutely haywire. The '<<' operator prints the selected string to STDOUT.
`{ nil { nil? } { "Whew!\n" } { "Something is terribly wrong!\n" } ifte << }`
BASIC
Applesoft BASIC
Applesoft has no built-in object system. The closest values to NULL or nil for each of the types are 0 for integers and floating point numbers, and "" for strings. There is also the NUL character: CHR\$(0). One could create an object system using global variables and include a special value for NULL, but this is probably a mistake.
```TRUE = 1 : FALSE = 0
NULL = TRUE
IF NULL THEN PRINT "NULL"
NULL = FALSE
IF NOT NULL THEN PRINT "NOT NULL"```
Output:
```NULL
NOT NULL```
BBC BASIC
A null object has a pointer with a value of zero or one.
``` PROCtestobjects
END
DEF PROCtestobjects
PRIVATE a(), b(), s{}, t{}
DIM a(123)
DIM s{a%, b#, c\$}
IF !^a() <= 1 PRINT "a() is null" ELSE PRINT "a() is not null"
IF !^b() <= 1 PRINT "b() is null" ELSE PRINT "b() is not null"
IF !^s{} <= 1 PRINT "s{} is null" ELSE PRINT "s{} is not null"
IF !^t{} <= 1 PRINT "t{} is null" ELSE PRINT "t{} is not null"
ENDPROC
```
Output:
```a() is not null
b() is null
s{} is not null
t{} is null
```
GWBASIC/QBasic/QB/VBDOS
These dialects of BASIC have no built-in object system. One STRING variable can have a default empty ("") value and a numeric one a default zero (0) value. A STRING variable can be assigned with the NULL (Chr\$(0)) value if needed and can be assesed with the instruction.
```IF VAR\$ = CHR\$(0) THEN PRINT "Variable has a null value."
```
Bracmat
Bracmat has no null objects.
The operators for multiplication, addition and concatenation have neutral elements, which are `1`, `0` and the empty string, respectively, but these are values like any other string.
```a:?x*a*?z {assigns 1 to x and to z}
a:?x+a+?z {assigns 0 to x and to z}
a:?x a ?z {assigns "" (or (), which is equivalent) to x and to z}```
C
C has the null pointer, written as "0", whose internal representation is often, though not always, the same as integer zero. It is (supposedly) garanteed to be pointing to nothing, so receiving one of those likely means you are not looking at an object--but, there are occasions where changing content of a null pointer actually does something (say, on DOS); and a function that's supposed to return a pointer on success doesn't always return a 0 otherwise (e.g. mmap returns -1 for failure).
There is a very common macro, `NULL`, which evaluates to `(void*) 0` or an equivalent value. NULL is compatible with all pointer types, including both data pointers and function pointers.
The standard library defines NULL in locale.h, stddef.h, stdio.h, stdlib.h, string.h, time.h and wchar.h. POSIX systems also define NULL in dirent.h and unistd.h. Many C files include at least one of these headers, so NULL is almost always available.
```#include <stdio.h>
int main()
{
char *object = 0;
if (object == NULL) {
puts("object is null");
}
return 0;
}
```
C#
As with Java, any reference type may be null, and testing for nullity uses ordinary boolean operators.
```if (foo == null)
Console.WriteLine("foo is null");
```
C# 2.0 introduced nullable types for situations in which even primitive value types may have undefined or unknown values (for example, when reading from a database). Prior to the introduction of nullable types, these situations would require writing wrapper classes or casting to a reference type (e.g., object), incurring the penalties of boxing and reduced type safety. A variable with nullable type can be declared simply by adding the '?' operator after the type.
Works with: C# version 2.0+
```int? x = 12;
x = null;
```
Also new in C# 2.0 was the null coalescing operator, '??', which is simply syntactic sugar allowing a default value to replace an operand if the operand is null:
Works with: C# version 2.0+
```Console.WriteLine(name ?? "Name not specified");
//Without the null coalescing operator, this would instead be written as:
//if(name == null){
// Console.WriteLine("Name not specified");
//}else{
// Console.WriteLine(name);
//}
```
C++
In C++ non-pointer types do not support null. (C++ provides value semantics rather than reference semantics). When using pointers C++ permits checking for null by comparing the pointer to a literal of 0, or (as in C) by way of a macro (NULL) which simply expands to 0.
```#include <iostream>
#include <cstdlib>
if (object == 0) {
std::cout << "object is null";
}
```
std::optional is available since C++17 (or Boost's boost::optional via boost/optional.hpp for earlier standards) for cases where the programmer wishes to pass by value, but still support a null value.
```#include <iostream>
#include <optional>
std::optional<int> maybeInt()
int main()
{
std::optional<int> maybe = maybeInt();
if(!maybe)
std::cout << "object is null\n";
}
```
C++11
In C++11 there is `nullptr` of type `nullptr_t` which represents a pointer to an invalid place. You can use it like
```int *p = nullptr;
...
if (p == nullptr){
// do some thing
}
//or just
if (p){
// do some thing
}
```
Chapel
Objects variables without an initializer expression will be initiallized to nil:
```class C { };
var c:C; // is nil
writeln(if c == nil then "nil" else "something");
```
Clojure
Clojure's `nil` is equivalent to Java's `null`.
```(let [x nil]
(println "Object is" (if (nil? x) "nil" "not nil")))
```
Test wether symbol `foo` is defined:
```(find (ns-interns *ns*) 'foo)
```
Undefining `foo`:
```(ns-unmap *ns* 'foo)
```
COBOL
Works with GnuCOBOL 2.0
``` identification division.
program-id. null-objects.
remarks. test with cobc -x -j null-objects.cob
data division.
working-storage section.
01 thing-not-thing usage pointer.
*> call a subprogram
*> with one null pointer
*> an omitted parameter
*> and expect void return (callee returning omitted)
*> and do not touch default return-code (returning nothing)
procedure division.
call "test-null" using thing-not-thing omitted returning nothing
goback.
end program null-objects.
*> Test for pointer to null (still a real thing that takes space)
*> and an omitted parameter, (call frame has placeholder)
*> and finally, return void, (omitted)
identification division.
program-id. test-null.
data division.
01 thing-one usage pointer.
01 thing-two pic x.
procedure division using
thing-one
optional thing-two
returning omitted.
if thing-one equal null then
display "thing-one pointer to null" upon syserr
end-if
if thing-two omitted then
display "no thing-two was passed" upon syserr
end-if
goback.
end program test-null.
```
Output:
```prompt\$ cobc -x -j null-objects.cob
thing-one pointer to null
no thing-two was passed
```
Common Lisp
Basics
Common Lisp has an object denoted by the symbol `nil`. When the symbol `nil` is evaluated as an expression, it evaluates to itself.
`nil` uniquely represents boolean false, and so code like
```(if (condition) (do-this))
```
is actually testing whether `(condition)` returns the value `nil`. The object `nil` is also used to denote the empty list which also terminates other lists. The value is also used as a default when some function returns fewer values than expected. `(list (values))` produces `(nil)` (list containing one element, which is the empty list), because `(values)` produces no value, but the function call `(list ...)` needs to reduce the expression to a single argument value, and so `nil` is supplied.
Beginnings of Null Object
The idea of making functions accept `nil` without failing did not appear in early Lisps. For instance `(car nil)` was erroneous: it was incorrect to try to access the first element of a non-list.
The defaulting behavior `(car nil)` which Common Lisp programmers take for granted was introduced in InterLisp, and then copied into MacLisp. (InterLisp had other liberties that do not survive into Common Lisp: it was possible to call a function with insufficient arguments, and the missing ones defaulted to `nil`. Likewise, excess arguments were ignored. CL has a disciplined syntax and semantics for default and variable arguments.)
This `(car nil) -> nil` behavior shows `nil` in an kind of new role: the role of a null object which takes methods that apply to other objects and provides some default non-failing behavior. It is the beginnings of the [null object design pattern].
Object-Oriented Null Object
In Common Lisp, in fact, there is a class called `null`, of which the object `nil` is understood to be the only instance. Furthermore, the `null` class is at the bottom of the type spindle: it is a subclass of every class. This is in contrast with the type `T` which is a superclass of every class.
Since `null` is at the bottom of the class hierarchy, it is possible to write methods specialized to parameters of class `null` which will only be applicable if the argument is the object `nil`. No other object is a subtype of `null`.
Some traditional Lisp functions could be expressed using the object system like this. Suppose that the `car` function did not have a safe defaulting behavior for `nil`. We could use the methods of the object system to define a `car*` which does have the safe behavior:
```(defmethod car* ((arg cons))
(car arg))
(defmethod car* ((arg null))
nil)
```
Now if we invoke `car*` on something which is neither a cons, nor `nil`, we get an error about no applicable method being found.
We can handle that ourselves by writing a method specialized to the master supertype `t`:
```(defmethod car* ((arg t)) ;; can just be written (defmethod car* (arg) ...)
(error "CAR*: ~s is neither a cons nor nil" arg))
```
The classes `t` and `null` are widely exploited in Lisp OO programming.
Component Pascal
```MODULE ObjectNil;
IMPORT StdLog;
TYPE
Object = POINTER TO ObjectDesc;
ObjectDesc = RECORD
END;
VAR
x: Object; (* default initialization to NIL *)
PROCEDURE DoIt*;
BEGIN
IF x = NIL THEN
StdLog.String("x is NIL");StdLog.Ln
END
END DoIt;
END ObjectNil.
```
Crystal
In Crystal, nil is represented by an instance of the Nil type, accessed by the identifier `nil`. A variable can only become nil if Nil is one of its possible types. All objects inheriting from the base Object class implement the method `.nil?` which returns true if the object is nil and false if it isn't. The equality and case equality operators can also be used to check for nil. The compiler returns an error if an object may be nil but is not treated as such. This can be suppressed with the `.not_nil!` method, which throws an exception at runtime if the object is in fact nil.
```foo : Int32 | Nil = 5 # this variable's type can be Int32 or Nil
bar : Int32? = nil # equivalent type to above, but shorter syntax
baz : Int32 = 5 # this variable can never be nil
foo.not_nil! # nothing happens, since 5 is not nil
puts "Is foo nil? #{foo.nil?}"
foo = nil
puts "Now is foo nil? #{foo.nil?}"
puts "Does bar equal nil? #{bar == nil}"
puts "Is bar equivalent to nil? #{bar === nil}"
bar.not_nil! # bar is nil, so an exception is thrown
```
Output:
```Is foo nil? false
Now is foo nil? true
Does bar equal nil? true
Is bar equivalent to nil? true
Unhandled exception: Nil assertion failed (NilAssertionError)
...```
D
In D is is used to perform bitwise identity, like to compare an object reference against null.
```import std.stdio;
class K {}
void main() {
K k;
if (k is null)
writeln("k is null");
k = new K;
if (k !is null)
writeln("Now k is not null");
}
```
Output:
```k is null
Now k is not null```
Delphi
``` // the following are equivalent
if lObject = nil then
...
if not Assigned(lObject) then
...
```
See Delphi
Dyalect
Dyalect has a notion of `nil` - a special sigleton value which can be used in the cases when no other meaningful value can be provided.
```var x = nil
if x == nil {
//Do something
}```
Déjà Vu
There isn't an actual null object, so generally falsy objects are used to indicate a missing value, or when that's impractical a specific ident:
```if not obj:
pass #obj is seen as null
if = :nil obj:
pass #obj is seen as null```
E
`object == null`
EchoLisp
The null object - null - is the same as the empty list (). It may be tested with the null? or !null? predicates. NB : null is not the same as the boolean #f (false). null evaluates to #t (true) in logical operations.
```null → null
() → null
(null? 3) → #f
(!null? 4) → #t
(null? null) → #t
;; careful - null is not false :
(if null 'OUI 'NON) → OUI
;; usual usage : recursion on lists until (null? list)
(define (f list)
(when (!null? list)
(write (first list)) (f (rest list))))
(f '( a b c)) → a b c
```
Eiffel
Any reference type variable can be Void. In the following example, STRING is a reference type, while INTEGER is an expanded type. The keyword "detachable" (as opposed to "attached") is used to indicate that the variable "s" may be Void. The default interpretation when neither of these two keywords is used depends on a compiler option. The first if statement will cause a compiler warning because an expanded type variable such as i will never be Void.
```class
APPLICATION
inherit
ARGUMENTS
create
make
feature {NONE} -- Initialization
make
local
i: INTEGER
s: detachable STRING
do
if i = Void then
print("i = Void")
end
if s = Void then
print("s = Void")
end
end
end
```
Output:
`s = Void`
Elixir
`nil` is atom in fact:
```iex(1)> nil == :nil
true
iex(2)> is_nil(nil)
true
```
`nil` is thought of as being `false` in the conditional expression.
If the condition given to `if/2` returns `false` or `nil`, the body given between `do`/`end` is not executed and it simply returns `nil`.
```iex(3)> if nil, do: "not execute"
nil
```
Erlang
Erlang does not have an null object.
As an alternative, many applications tend to pick a convention for returning an empty condition and use that.
Example alternatives:
1. Something like
`{ok, 3} % normal case`
or
`{err, no_more} % error case`
on error.
2. Don't ever allow an undefined return value, and throw an exception instead.
3. Return an atom:
1. undefined*
2. undef
3. null
4. nil
5. none
undefined is often used by records as an initial value and the stdlib module.
Atoms are erlang's user-defined constants that always evaluates to is itself. It is also equal to no other value else but itself.
F#
As a .Net languages F# inherits the null as a potential value for object variables. Other than in interfacing assemblies written in other .Net languages, null rarely serves a purpose in F# code. Contrived code, to show using null, as per task description:
```let sl : string list = [null; "abc"]
let f s =
match s with
| null -> "It is null!"
| _ -> "It's non-null: " + s
for s in sl do printfn "%s" (f s)
```
Factor
```: is-f? ( obj -- ? ) f = ;
```
Fantom
Test for equality with 'null', which is the null value.
```fansh> x := null
fansh> x == null
true
fansh> x = 1
1
fansh> x == null
false```
Note, nullable objects have a type ending in a question mark, for example:
`Int? y := null` is valid, but
`Int y := null` is not.
Forth
Standard ANS Forth does not distinguish a particular invalid memory value like NULL. Instead, ALLOCATE returns an out-of-band success code to indicate a failed allocation. Dictionary words have the option of throwing an exception on a dictionary space overrun. Forth lacks a NULL symbol because it has such a wide variety of target platforms. On some embedded targets, the memory space may be as small as 64 direct-mapped addresses, where eliminating a valid zero address would have a high price.
In practice, all notable hosted implementations follow the C practice of being able to treat a zero address (i.e. FALSE) as a null address for the purpose of list termination.
FreeBASIC
```'FB 1.05.0 Win64
' FreeBASIC does not have a NULL keyword but it's possible to create one using a macro
#Define NULL CPtr(Any Ptr, 0) '' Any Ptr is implicitly convertible to pointers of other types
Type Dog
name As String
age As Integer
End Type
Dim d As Dog Ptr = New Dog
d->Name = "Rover"
d->Age = 5
Print d->Name, d->Age
Delete d
d = NULL '' guard against 'd' being used accidentally in future
' in practice many FB developers would simply have written: d = 0 above
Sleep
```
Output:
```Rover 5
```
FutureBasic
While objects such as strings can be NULL in FB, arrays, dictionaries and other collections cannot contain NULL objects.
```// Object dimensioned, but not assigned
CFStringRef object
if ( object == NULL )
print "object is NULL"
end if
HandleEvents```
Output:
```object is NULL
```
Go
Nil is a predefined identifier, defined for six types in Go. In each case, it represents the zero value for the type, that is, the memory representation of all zero bytes. This is the value of a newly created object. In the cases of these six types, an object must be subsequently initialized in some way before it has much use. Examples of initialization are given in the Go solution of task Undefined values.
```package main
import "fmt"
var (
s []int // slice type
p *int // pointer type
f func() // function type
i interface{} // interface type
m map[int]int // map type
c chan int // channel type
)
func main() {
fmt.Println(s == nil)
fmt.Println(p == nil)
fmt.Println(f == nil)
fmt.Println(i == nil)
fmt.Println(m == nil)
fmt.Println(c == nil)
}
```
Output is "true" in each case.
Haskell does not have a universal null value. There is a 'value of every type', the undefined value (sometimes written ⊥, 'bottom'), but it is essentially a sort of exception — any attempt to use it is an error.
```undefined -- undefined value provided by the standard library
error "oops" -- another undefined value
head [] -- undefined, you can't take the head of an empty list
```
When one would use "null" as a marker for "there is no normal value here" (e.g. a field which is either an integer or null), one uses the Maybe type instead. The definition of Maybe is:
``` data Maybe a = Nothing | Just a
```
That is, a Maybe Integer is either Nothing or Just <some integer>.
There are many ways to work with Maybe, but here's a basic case expression:
```case thing of
Nothing -> "It's Nothing. Or null, whatever."
Just v -> "It's not Nothing; it is " ++ show v ++ "."
```
It is easy to work with Maybe type using do-notation (since Maybe is a monad):
```add_two_maybe_numbers x y do
a <- x
b <- y
return (a+b)
```
Then
```*Main> add_two_maybe_numbers (Just 2) (Just 3)
Just 5
Nothing
```
Icon and Unicon
Icon/Unicon have a null value/datatype. It isn't possible to undefine a variable.
```procedure main()
nulltest("a",a) # unassigned variables are null by default
nulltest("b",b := &null) # explicit assignment is possible
nulltest("c",c := "anything")
nulltest("c",c := &null) # varibables can't be undefined
end
procedure nulltest(name,var)
return write(name, if /var then " is" else " is not"," null.")
end
```
Io
```if(object == nil, "object is nil" println)
```
J
J doesn't have an untyped NULL. Instead, it has a concept of "fill". Numeric fill is 0, character fill is the space character, and boxed fill is the ace (a:) which is an empty box. Fill is what is used to pad an array structure when that is needed. (And some operations support using a user specified value in place of the default fill.)
To indicate "missing data", "normal" data is usually pressed into service (e.g. 0 or _1 in a numeric context, ' ' in a literal context, a: in a boxed context, etc). Frequently, missing data is represented by the empty vector '', or other arrays without any elements.
That said, undefined names in J are not associated with any data of any type. Furthermore, any attempt to use the value of an undefined is treated as an error (this is distinct from the concept of an empty array, which contains no data but which is not an error to use). However, it is possible to check if a name is defined before attempting to use it:
```isUndefined=: _1 = nc@boxxopen
```
Example use:
``` isUndefined 'foo'
1
foo=:9
isUndefined 'foo'
0
```
Note, of course, that this "name is not defined" state is not a first class value in J -- you can not create a list of "undefineds".
Finally, note: the concept of an empty array can be natural in J (and APL) for representing data which is not there -- it is the structural equivalent of the number zero. That said, its implications can sometimes be non-obvious for people coming from a languages which requires that arrays have content. As a result, you will sometimes encounter empty array jokes...
Marie Pennysworth, having spent a productive day shopping, stopped by Robert Cuttingham's butcher shop.
"Eleven dollars per pound," he responded.
"Sirloin is thirteen dollars per pound today," he answered.
"But Harkin's Grocery down the street is selling sirloin for nine dollars per pound!" she exclaimed.
"Well, they're out," she sighed.
He smiled, "When I am out, I only charge seven dollars a pound."
That said, note that a typical way to indicate missing or invalid data, in J, is to have a parallel array which is a bit mask (which selects the desired or valid values and, by implication, does not select the invalid values). Or, as a logical equivalent: a list of indices which select the desired and/or valid values. Alternatively, you can have an array without the invalid values and a bit mask which demonstrates how the data would be populated on a larger array -- in other words instead of 3,4,null,5 you could have (3 4 5) and (1 1 0 1). And you can transform between some of these representations:
``` 1 1 0 1#3 4 _ 5 NB. use bitmask to select numbers
3 4 5
I.1 1 0 1 NB. get indices for bitmask
0 1 3
0 1 3 { 3 4 _ 5 NB. use indices to select numbers
3 4 5
1 1 0 1 #inv 3 4 5 NB. use bitmask to restore original positions
3 4 0 5
1 1 0 1 #!._ inv 3 4 5 NB. specify different fill element
3 4 _ 5
3 4 5 (0 1 3}) _ _ _ _ NB. use indices to restore original positions
3 4 _ 5
```
Java
In Java, "null" is a value of every reference type.
```// here "object" is a reference
if (object == null) {
System.out.println("object is null");
}
```
JavaScript
In Javascript null is the value that isn't anything. null is not an object, but because of a bug typeof null will return "object".
```if (object === null) {
// The object is nothing
}
typeof null === "object"; // This stands since the beginning of JavaScript
```
jq
jq has null as a value, but while on the subject of nothing, it may be worth mentioning that jq also has a filter, empty, for producing an empty sequence, a.k.a. nothing.
null is distinct from false. Here are some examples:
```null|type # => "null"
null == false # => false
null == null # => true
empty|type # => # i.e. nothing (as in, nada)
empty == empty # => # niente
empty == "black hole" # => # Ничего```
Jsish
Like Javascript, Jsish has undefined and null. Unlike Javascript, null is not typed as object, but null.
Jsish, with parameter typed functions, also allows void as a type spec, to indicate the parameter (of whatever type) may be omitted by a caller.
```/* null non value */
if (thing == null) { puts("thing tests as null"); }
if (thing === undefined) { puts("thing strictly tests as undefined"); }
puts(typeof thing);
puts(typeof null);
puts(typeof undefined);
```
Output:
```prompt\$ jsish nulling.jsi
thing tests as null
thing strictly tests as undefined
undefined
null
undefined
```
Julia
See language reference: https://docs.julialang.org/en/v1/manual/faq/#Nothingness-and-missing-values
K
K has well developed notions of data null :
The special numeric atoms 0I and 0N refer to integer infinity and “not-a-number” (or “null” in database parlance) concepts, and similarly 0i and 0n for floating-point.
eg : ( 1 2 3 0N 6 7 )
and and missing value nil :
Empty expressions in both list expressions and function expressions actually represent a special atomic value called nil. ... A list may contain one or more empty items (i.e. the nil value _n), which are typically indicated by omission:
``` (1;;2) ~ (1 ; _n ; 2) / ~ is ''identical to'' or ''match'' .
1
_n ~' ( 1 ; ; 2 ) / ''match each''
0 1 0
additional properties : _n@i and _n?i are i; _n`v is _n```
For more detail on K's concept of typed nulls, see http://code.kx.com/wiki/Reference/Datatypes#Primitive_Types
Klingphix
```%t nan !t
\$t nan == ?```
Output:
`1`
Kotlin
Kotlin distinguishes between non-nullable types and nullable types. The latter are distinguished from the former by a '?' suffix. Only nullable types have a 'null' value indicating that they don't currently refer to an object of their non-nullable equivalent.
In addition, Kotlin has a Nothing type which has no instances and is a sub-type of every other type. There is also a nullable Nothing? type whose only value is 'null' and so, technically, this is the type of 'null' itself.
Here are some examples:
```// version 1.1.0
fun main(args: Array<String>) {
val i: Int = 3 // non-nullable Int type - can't be assigned null
println(i)
val j: Int? = null // nullable Int type - can be assigned null
println(j)
println(null is Nothing?) // test that null is indeed of type Nothing?
}
```
Output:
```3
null
true
```
langur
Null can be compared for directly, using equality operators, or can be checked with the isNull() function. Operators ending with a ? mark propagate null. A null in an expression test is a non-truthy result.
Works with: langur version 0.10
Prior to 0.10, multi-variable declaration/assignment would use parentheses around variable names and values.
```val .x, .y = true, null
writeln .x == null
writeln .y == null
writeln .x ==? null
writeln .y ==? null
# null not a "truthy" result
writeln if(null: 0; 1)```
Output:
```false
true
null
null
1```
Lasso
```local(x = string, y = null)
#x->isA(::null)
// 0 (false)
#y->isA(::null)
// 1 (true)
#x == null
// false
#y == null
//true
#x->type == 'null'
// false
#y->type == 'null'
//true
```
Latitude
Nil is an object in Latitude, like any other.
```foo := Nil.
if { foo nil?. } then {
putln: "Foo is nil".
} else {
putln: "Foo is not nil".
}.```
In particular, Nil satisfies the Collection mixin, so it can be treated as an (immutable) collection.
`Nil to (Array). ;; []`
Nil is the default value returned if a method body is empty.
```func := {}.
func. ;; Nil```
Lily
Lily doesn't provide a built-in nothing type, but allows one to be created using enum class:
```enum class Option[A] {
Some(A)
None
}
# Only variables of class Option can be assigned to None.
# Type: Option[integer]
var v = Some(10)
# Valid: v is an Option, and any Option can be assigned to None
v = None
# Invalid! v is an Option[integer], not just a plain integer.
v = 10
# Type: integer
var w = 10
# Invalid! Likewise, w is an integer, not an Option.
w = None```
Lingo
Null/nil is called "<Void>" in Lingo. Lingo doesn't distinguish undefined variables from <Void> objects, and by using the constant VOID you can even assign <Void> to variables. Functions that don't return anything, return <Void>. Checking for <Void> (e.g. by using built-in function voidP) can be used to implement optional function arguments: if voidP() returns TRUE (1) for some argument, a default value can be assigned in the function body.
```put _global.doesNotExist
-- <Void>
put voidP(_global.doesNotExist)
-- 1
x = VOID
put x
-- <Void>
put voidP(x)
-- 1```
Logo
```to test :thing
if empty? :thing [print [list or word is empty]]
end
print empty? [] ; true
print empty? "|| ; true```
Lua
```isnil = (object == nil)
print(isnil)
```
M2000 Interpreter
For Com Objects
There is a Nothing to assign to a COM object to released (but time to actually released depends from system). A com pointer can't get another value (only the first value, and the Nothing at the end).
```Module CheckWord {
Declare Alfa "WORD.APPLICATION"
Declare Alfa Nothing
Print Type\$(Alfa)="Nothing"
Try ok {
Declare Alfa "WORD.APPLICATION"
\\ we can't declare again Alfa
}
If Not Ok Then Print Error\$ ' return Bad Object declaration
}
CheckWord```
For Containers
Container's pointers (for arrays, inventories, stack) we have to assign an empty container, there is not a null one.
```Module CheckContainers {
\\ Arrays (A() and B() are value types)
Dim A(10)=1, B()
\\ B() get a copy of A(), is not a reference type
B()=A()
\\ we make a pointer to Array
B=A()
\\ now B is a reference type object
Print Len(B)=10 ' ten items
B+=10
Print A(3)=11, A(7)=11
\\ we can change pointer using a pointer to an empty array
B=(,)
\\ we can erase A() and B()
Dim A(0), B(0)
Print Len(A())=0, Len(B())=0
Print Len(B)=0
B=(123,)
\\ B() is a value type so get a copy
B()=B
Print Len(B)=1, B(0)=123
\\ Using Clear we pass a new empty array
Clear B
Print Type\$(B)="mArray"
Print Len(B)=1, B(0)=123
\\ Inventrories. Keys must be unique (for normal inventories)
Inventory M=1,2,3,4:=400,5
Print M
Clear M
Inventory M=1,2,3,4,5
Print M
\\ Inventory Queue can have same keys.
Inventory Queue N=1,1,2:="old",2:="ok",3
If Exist(N,2) Then Print Eval\$(N)="ok", Eval(N!)=3 ' 4th item
Clear N
Print Len(N)=0, Type\$(N)="Inventory"
\\ Stack Object
Z=Stack:=1,2,3
Stack Z {
While not empty {Print Number}
}
Print Len(Z)=0
Z=Stack((Stack:=1,2,3,4),Stack:=20,30,40 )
Print Len(Z)=7
Print Z ' 1 2 3 4 20 30 49
Z=Stack ' This is an empty stacl
Print Len(Z)=0
Print Type\$(Z)="mStiva"
}
CheckContainers```
For Groups
Groups are value types, but we can make reference to them,or pointer to them
A Named referenced can't get a new reference
A pointer to a named group is actual a reference, and can change type and reference
A pointer to a copy of group (as float group) is actually a pointer to group.
Pointers to groups can be point to an Null Group, assigning a 0& value (a long type)
```class something {
}
class alfa as something {
x=10, y=20
}
a->alfa()
Print a is type alfa = true
Print a is type something = true
a->0&
Print a is type null = true
\\ beta is a named object, is static
group beta {
type: something, alfa
x=10, y=20
}
Print beta is type alfa = true
Print beta is type something = true
\\ now a is a pointer as a weak reference to beta
a->beta
print a is type alfa = true
print a is type something = true
a=pointer() ' same as a->0&
Print a is type null = true
\\ now a is a pointer of a copy of beta
a->(beta)
print a is type alfa = true
print a is type something = true
a=pointer() ' same as a->0&
Print a is type null = true```
Maple
In Maple, NULL and () represent the null object.
```a := NULL;
a :=
is (NULL = ());
true
if a = NULL then
print (NULL);
end if;```
A null object is different from an undefined value.
```b := Array([1, 2, 3, Integer(undefined), 5]);
b := [ 1 2 3 undefined 5 ]
numelems(b);
5
b := Array([1, 2, 3, Float(undefined), 5]);
b := [ 1 2 3 Float(undefined) 5 ]
numelems(b);
5
b := Array([1, 2, 3, NULL, 5]);
b := [ 1 2 3 5 ]
numelems(b);
4```
Mathematica/Wolfram Language
Mathematica can assign a Null value to a symbol, two examples:
```x=Null;
```
```x =.
x = (1 + 2;)
FullForm[x]
```
Both set x to be Null. To specifically test is something is Null one can use the SameQ function (with infix operator: ===):
```SameQ[x,Null]
```
Or equivalent:
```x===Null
```
will give back True if and only if x is assigned to be Null. If x is empty (nothing assigned) this will return False. To test if an object has something assigned (number, list, graphics, null, infinity, symbol, equation, pattern, whatever) one uses ValueQ:
```x =.;
ValueQ[x]
x = 3;
ValueQ[x]
```
gives:
```False
True
```
MATLAB / Octave
The closest think to a NULL element in Matlab/Octave is an empty field or empty string; empty fields in a conditional expression evaluate to false.
```a = []; b='';
isempty(a)
isempty(b)
if (a)
1,
else,
0
end;
```
```octave:4> a = []; b='';
octave:5> isempty(a)
ans = 1
octave:6> isempty(b)
ans = 1
octave:7> if (a) 1, else, 0, end;
ans = 0```
Maxima
There is no null object in Maxima. Usually, a function that returns nothing (as the builtin "disp") returns in fact the symbol 'done.
MAXScript
`if obj == undefined then print "Obj is undefined"`
Modula-3
In Modula-3, `NIL` is a value, and `NULL` is a type. The `NULL` type contains only one value, `NIL`. `NULL` is a subtype of all reference types, which allows all reference types to have the value `NIL`.
This can lead to errors, if for example you write:
```VAR foo := NIL
```
This (most likely incorrectly) gives foo the type `NULL`, which can only have the value `NIL`, so trying to assign it anything else will not work. To overcome this problem, you must specify the reference type when declaring foo:
```VAR foo: REF INTEGER := NIL;
```
```IF foo = NIL THEN
IO.Put("Object is nil.\n");
END;
```
MUMPS
A variable can be declared implicitly by using it as on the left side in a SET, or by making a new version for the current scope with a NEW statement. A variable can have descendants without having a value set.
The \$DATA (or \$D) function will return a number:
\$DATA returns: Variable is defined
Variable has children No Yes
No 0 1
Yes 10 11
Or, by examples (in immediate mode):
```CACHE>WRITE \$DATA(VARI)
0
CACHE>SET VARI="HELLO" WRITE \$DATA(VARI)
1
CACHE>NEW VARI WRITE \$DATA(VARI) ;Change to a new scope
0
CACHE 1S1>SET VARI(1,2)="DOWN" WRITE \$DATA(VARI)
10
CACHE 1S1>WRITE \$DATA(VARI(1))
10
CACHE 1S1>WRITE \$D(VARI(1,2))
1
CACHE 1S1>SET VARI(1)="UP" WRITE \$DATA(VARI(1))
11
<CACHE 1S1>QUIT ;Leave the scope
<CACHE>W \$DATA(VARI)," ",VARI
1 HELLO```
Nanoquery
Translation of: Ursa
```\$x = \$null
if (\$x = \$null)
println "x is null"
else
println "x is not null"
end if```
Neko
```/**
<doc>
<p>Neko uses <i>null</i> for undefined variables,
and also as a programmer accessible value.</p>
<p>The <i>null</i> value can be treated as a boolean value with the
builtin \$istrue, and tests as false.</p>
</doc>
*/
var n = null
if n == null \$print("n is null\n")
if \$not(\$istrue(n)) \$print("and tests as boolean false\n")
```
NetRexx
In NetRexx as in Java, "null" is a value of every reference type.
```/* NetRexx */
options replace format comments java crossref symbols binary
robject = Rexx -- create an object for which the value is undefined
say String.valueOf(robject) -- will report the text "null"
if robject = null then say 'Really, it''s "null"!'
```
Output:
```null
Really, it's "null"!
```
NewLISP
```#! /usr/local/bin/newlisp
(setq myobject nil)
(println (nil? myobject))
(exit)
```
```true
```
Nim
There is a `nil` value in Nim, which is the same as a 0. It can be explicitly forbidden as a value:
```let s: pointer = nil
{.experimental: "notnil".}
let ns: pointer not nil = nil # Compile time error
```
The value "nil" can be used for pointers, references (i.e. pointers managed by the garbage collector) and procedures. It was also used for strings and sequences, but this is no longer the case (option `--nilseqs:on` allows to retrieve the old behavior).
Testing if a pointer “p” is `nil` can be done either by using `==` or using the procedure `isNil`.
```var p: ptr int
if p == nil: echo "it is nil"
if p != nil: echo "it is not nil"
if p.isNil: echo "it is nil"
```
Oberon-2
Works with: oo2c
```MODULE Null;
IMPORT
Out;
TYPE
Object = POINTER TO ObjectDesc;
ObjectDesc = RECORD
END;
VAR
o: Object; (* default initialization to NIL *)
BEGIN
IF o = NIL THEN Out.String("o is NIL"); Out.Ln END
END Null.
```
Output:
```o is NIL
```
Objeck
In Objeck, "Nil" is a value of every reference type.
```# here "object" is a reference
if(object = Nil) {
"object is null"->PrintLine();
};```
Objective-C
The value `nil` is used to indicate that an object pointer (variable of type `id`) doesn't point to a valid object.
```// here "object" is an object pointer
if (object == nil) {
NSLog("object is nil");
}
```
An interesting thing is that in Objective-C, it is possible to send a message to `nil`, and the program will not crash or raise an exception (nothing will be executed and `nil` will be returned in place of the usual return value).
```[nil fooBar];
```
Note that `nil` is distinct from `NULL`, which is only used for regular C pointers.
For class pointers (values of type `Class`), they have a separate null pointer value called `Nil`.
Confusingly, there is also `NSNull`, a singleton class with one value, `[NSNull null]`, used as a dummy object to represent the lack of a useful object. This is needed in collections like arrays and dictionaries, etc., because they do not allow `nil` elements, so if you want to represent some "empty" slots in the array you would use this.
OCaml
Maybe the closest type of OCaml would be the type option, which is defined like this in the standard library:
```type 'a option = None | Some of 'a
```
```match v with
| None -> "unbound value"
| Some _ -> "bounded value"
```
Oforth
null is an object, the only instance of Null class. When an object is created, all attributes are initiallized to null value. When a method or function is called, all local variables begin with null value.
```null isNull
"abcd" isNull
: testNull { | a | a ifNull: [ "Variable value is null" println ] ;```
Ol
Ol has no null object in task meaning sense. To indicate the "unassigned" variable state typically used #false because this is only value that triggers the 'unless' and 'if not'.
The builtin null and #null (that a same) means "an empty list". Important note: in contrast with CL the null means #true in 'if' statement!
ooRexx
ooRexx has a special singleton object called .nil that is used to indicate the absence of values in some situations (such as the default values returned from collection objects).
```
if a[i] == .nil then say "Item" i "is missing"
```
Uninitialized ooRexx variables do not evaluate to .nil, but rather the character string name of the variable (all uppercase). The var() built-in function allows variable validity to be tested:
```a=.array~of('A','B')
i=3
if a[i] == .nil then say "Item" i "of array A is missing"
if \var("INPUT") then say "Variable INPUT is not assigned"
if \var("var") then say "Variable" var "is not assigned"
```
Output:
```Item 3 of array A is missing
Variable INPUT is not assigned
Variable VAR is not assigned```
Oz
There is no explicit null in Oz.
Unbound variables
If an unbound variable is accessed, the current thread will be suspended:
```declare
X
in
{Show X+2} %% blocks```
If you later assign a value to X in another thread, the original thread will resume and print the result of the addition. This is the basic building block of Oz' declarative concurrency.
Undefined values
Access to undefined values (like using an out-of-range array index or a non-existing record feature) will usually provoke an exception in Oz.
It is also possible to assign a unique "failed" value to a variable. Such a failed value encapsulates an exception. This can be useful in concurrent programming to propagate exceptions across thread boundaries.
```declare
X = {Value.failed dontTouchMe}
in
{Wait X} %% throws dontTouchMe```
Sometimes algebraic data types like Haskell's Maybe are simulated using records.
```declare
X = just("Data")
in
case X of nothing then skip
[] just(Result) then {Show Result}
end```
PARI/GP
GP does not have good facilities for this, but this test suffices for most purposes:
`foo!='foo`
See Delphi
Perl
In Perl, `undef` is a special scalar value, kind of like null in other languages. A scalar variable that has been declared but has not been assigned a value will be initialized to `undef`. (Array and hash variables are initialized to empty arrays or hashes.)
If `strict` mode is not on, you may start using a variable without declaring it; it will "spring" into existence, with value `undef`. In `strict` mode, you must declare a variable before using it. Indexing an array or hash with an index or key that does not exist, will return `undef` (however, this is not an indication that the index or key does not exist; rather, it could be that it does exist, and the value is `undef` itself). If `warnings` is on, most of the time, if you use the `undef` value in a calculation, it will produce a warning. `undef` is considered false in boolean contexts.
It is possible to use `undef` like most other scalar values: you can assign it to a variable (either by doing `\$var = undef;` or `undef(\$var);`), return it from a function, assign it to an array element, assign it to a hash element, etc. When you do list assignment (i.e. assign a list to a list of variables on the left side), you can use `undef` to "skip" over some elements of the list that you don't want to keep.
You can check to see if a value is `undef` by using the `defined` operator:
```print defined(\$x) ? 'Defined' : 'Undefined', ".\n";
```
From the above discussion, it should be clear that if `defined` returns false, it does not mean that the variable has not been set; rather, it could be that it was explicitly set to `undef`.
Starting in Perl 5.10, there is also a defined-or operator in Perl. For example:
```say \$number // "unknown";
```
prints \$number if it is defined (even if it is false) or the string "unknown" otherwise.
Phix
There is a builtin NULL, however it is equivalent to the integer 0 and will trigger a type check if assigned to a variable declared as string or sequence. In most programs the zero-length string/sequence (""/{}) suffices, but if you want a variable that can be a string/sequence or NULL, but not other arbitrary integer/float values, use something like the following user-defined types:
```type nullableString(object o)
return string(o) or o=NULL
end type
nullableString s
s = "hello"
s = NULL
--s = 1 -- error
--s = {1,2,3} -- error
type nullableSequence(object o)
return sequence(o) or o=NULL
end type
nullableSequence q
q = {1,2,3}
q = "string" -- fine (strings are a subset of sequences)
q = NULL
--q = 1 -- error
```
PHL
`if (obj == null) printf("obj is null!\n");`
PHP
There is a special value NULL. You can test for it using is_null() or !isset()
```\$x = NULL;
if (is_null(\$x))
echo "\\$x is null\n";
```
PicoLisp
New internal symbols are initialized with the value NIL. NIL is also the value for "false", so there is never really an "undefined value". 'not' is the predicate to check for NIL, but many other (typically flow control) functions can be used.
```(if (not MyNewVariable)
(handle value-is-NIL) )```
or
```(unless MyNewVariable
(handle value-is-NIL) )```
Pike
In Pike all variables are initialized to ${\displaystyle 0}$, regardless of their type. thus ${\displaystyle 0}$ functions as a `Null` value for all types except integer.
${\displaystyle 0}$ is also used to indicate the absence of a key or object member.
to tell the difference between a value ${\displaystyle 0}$ and absence of a key, `zero_type()` is used:
```> mapping bar;
> bar;
Result: 0
> bar = ([ "foo":0 ]);
> bar->foo;
Result 0;
> zero_type(bar->foo);
Result: 0
> bar->baz;
Result: 0
> zero_type(bar->baz);
Result: 1
```
PL/I
```declare x fixed decimal (10);
...
if ^valid(x) then signal error;
declare y picture 'A9XAAA9';
...
if ^valid(y) then signal error;```
Comment:- In the picture specification, the content of variable y must consist of letters where the letter 'A' is given, digits or space where the digit '9' appears, and the letter X signfies that any character is acceptable.
PowerShell
In PowerShell the automatic variable `\$null` represents a null value. Comparisons are not left/right symmetrical which means placing `\$null` on the left side greatly assists when comparing to an array.
```if (\$null -eq \$object) {
...
}
```
PureBasic
All variables that has not yet been given any other value will be initiated to #Null
```If variable = #Null
Debug "Variable has no value"
EndIf
```
Python
```x = None
if x is None:
print "x is None"
else:
print "x is not None"
```
Output:
```x is None
```
R
R has the special value NULL to represent a null object. You can test for it using the function is.null. Note that R also has a special value NA to represent missing or unknown values.
```is.null(NULL) # TRUE
is.null(123) # FALSE
is.null(NA) # FALSE
123==NULL # Empty logical value, with a warning
foo <- function(){} # function that does nothing
foo() # returns NULL
```
Racket
"null", or its literal form "'()", is used to denote empty lists and sometimes it is used as a generic null value.
```-> null
'()
-> (null? null)
#t
-> (null? 3)
#f
```
But a value that is more used as a generic "nothing" value is "#f", false. Racket also has a void value, mostly the result of side-effect functions. (And an undefined value.)
Raku
(formerly Perl 6)
(as it were...)
In Raku you can name the concept of Nil, but it not considered an object, but rather the absence of an object, more of a "bottom" type. The closest analog in real objects is an empty list, but an empty list is considered defined, while Nil.defined always returns false. Nil is what you get if you try to read off the end of a list, and () is just very easy to read off the end of... :-)
If you try to put Nil into a container, you don't end up with a container that has Nil in it. Instead the container reverts to an uninitialized state that is consistent with the declared type. Hence, Raku has the notion of typed undefined values, that are real objects in the sense of "being there", but are generic in the sense of representing type information without being instantiated as a real object. We call these type objects since they can stand in for real objects when one reasons about the types of objects. So type objects fit into the type hierarchy just as normal objects do. In physics terms, think of them as "type charge carriers" that are there for bookkeeping between the "real" particles.
All type objects derive from Mu, the most-undefined type object, and the object most like "null" in many languages. All other object types derive from Mu, so it is like Object in other languages as well, except Mu also encompasses various objects that are not discrete, such as junctions. So Raku distinguishes Mu from Any, which is the type that functions the most like a discrete, mundane object.
Mostly the user doesn't have to think about it. All object containers behave like "Maybe" types in Haskell terms; they may either hold a valid value or a "nothing" of an appropriate type. Most containers default to an object of type Any so you don't accidentally send quantum superpositions (junctions) around in your program.
```my \$var;
say \$var.WHAT; # Any()
\$var = 42;
say \$var.WHAT; # Int()
say \$var.defined; # True
\$var = Nil;
say \$var.WHAT; # Any()
say \$var.defined # False
```
You can declare a variable of type Mu if you wish to propagate superpositional types:
```my Mu \$junction;
say \$junction.WHAT; # Mu()
\$junction = 1 | 2 | 3;
say \$junction.WHAT; # Junction()
```
Or you can declare a more restricted type than Any
```my Str \$str;
say \$str.WHAT; # Str()
\$str = "I am a string.";
say \$str.WHAT; # Str()
\$str = 42; # (fails)
```
But in the Raku view of reality, it's completely bogus to ask for a way "to see if an object is equivalent to the null object." The whole point of such a non-object object is that it doesn't exist, and can't participate in computations. If you think you mean the null object in Raku, you really mean some kind of generic object that is uninstantiated, and hence undefined. One of those is your "null object", except there are many of them, so you can't just check for equivalence. Use the defined predicate (or match on a subclass of your type that forces recognition of abstraction or definedness).
Raku also has Failure objects that, in addition to being undefined carriers of type, are also carriers of the reason for the value's undefinedness. We tend view them as lazily thrown exceptions, at least until you try to use them as defined values, in which case they're thrown for real.
Raven
This example is in need of improvement: Add NULL handling with MySQL data.
```NULL as \$v
\$v NULL = # TRUE
\$v NULL != # FALSE
1 NULL = # FALSE
1.1 NULL = # FALSE
NULL as \$v2
\$v2 \$v = # TRUE```
REBOL
```x: none
print ["x" either none? x ["is"]["isn't"] "none."]
```
Output:
`x is none.`
REBOL also has the concept of `unset` values, testable with `get/any`
```unset? get/any 'some-var
unset? get 'some-var
```
Output:
```true
** Script Error: some-var has no value
** Near: unset? get 'some-var```
REXX
REXX can have variables with a null value.
With the symbol built-in function, it can be determined if a variable is defined (or not).
The length built-in function can be used to see what the length of the value of a defined variable.
A variable with a null value has a length of 0 (zero).
The drop statement can be used to "undefine" a REXX variable.
```/*REXX program demonstrates null strings, and also undefined values. */
if symbol('ABC')=="VAR" then say 'variable ABC is defined, value='abc"<<<"
else say "variable ABC isn't defined."
xyz=47
if symbol('XYZ')=="VAR" then say 'variable XYZ is defined, value='xyz"<<<"
else say "variable XYZ isn't defined."
drop xyz
if symbol('XYZ')=="VAR" then say 'variable XYZ is defined, value='xyz"<<<"
else say "variable XYZ isn't defined."
cat=''
if symbol('CAT')=="VAR" then say 'variable CAT is defined, value='cat"<<<"
else say "variable CAT isn't defined."
```
output
```variable ABC isn't defined.
variable XYZ is defined, value=47<<<
variable XYZ isn't defined.
variable CAT is defined, value=<<<
```
Ring
```see isnull(5) + nl + # print 0
isnull("hello") + nl + # print 0
isnull([1,3,5]) + nl + # print 0
isnull("") + nl + # print 1
isnull("NULL") # print 1```
Ruby
The value when referring to the instance variable which isn't initialized is nil.
```puts "@object is nil" if @object.nil? # instance variable
puts "\$object is nil" if \$object.nil? # global variable, too
# It recognizes as the local variable even if it isn't executed.
object = 1 if false
puts "object is nil" if object.nil?
# nil itself is an object:
puts nil.class # => NilClass
```
Output:
```@object is nil
\$object is nil
object is nil
NilClass
```
Rust
```// If an option may return null - or nothing - in Rust, it's wrapped
// in an Optional which may return either the type of object specified
// in <> or None. We can check this using .is_some() and .is_none() on
// the Option.
fn check_number(num: &Option<u8>) {
if num.is_none() {
println!("Number is: None");
} else {
println!("Number is: {}", num.unwrap());
}
}
fn main() {
let mut possible_number: Option<u8> = None;
check_number(&possible_number);
possible_number = Some(31);
check_number(&possible_number);
}
```
S-lang
S-Lang uses NULL; it is the only object of type Null_Type:
```variable foo = NULL;
print(foo);
if (foo == NULL)
print(typeof(foo));```
Output:
```NULL
Null_Type
```
Scala
This blog post has a good explanations of the different types of null-like values.
```scala> Nil
res0: scala.collection.immutable.Nil.type = List()
scala> Nil == List()
res1: Boolean = true
scala> Null
Null
^
scala> null
res3: Null = null
scala> None
res4: None.type = None
scala> Unit
res5: Unit.type = object scala.Unit
scala> val a = println()
a: Unit = ()
```
Scheme
```(null? object)
```
Note: "null?" here tests whether a value is the empty list.
Sidef
The absence of a value is represented by nil
```var undefined; # initialized with an implicit nil
say undefined==nil; # true
say defined(nil) # false
```
However, nil is not an object, so we can't call methods on it. Alternatively, Sidef provides the null object:
```var null_obj = null; # initialize with a null value
say null_obj.is_a(null); # true
say defined(null_obj); # true
```
Slate
`Nil isNil = True.`
Smalltalk
```object isNil ifTrue: [ "true block" ]
ifFalse: [ "false block" ].
nil isNil ifTrue: [ 'true!' displayNl ]. "output: true!"
foo isNil ifTrue: [ 'ouch' displayNl ].
x := (foo == nil).
x := foo isNil
```
notice that nil is the singleton instance of the UndefinedObject class; i.e. it is a first class object. Thus we can do:
```foo := nil.
foo class. "-> UndefinedObject"
foo respondsTo: #'bar'. "asking if a message is implemented"
foo class compile:'fancyOperation ^ 123'.
foo fancyOperation "->123"
```
the last example being for demonstration only - it is not considered well behaved to add arbitrary code that way, except for framework support, such as encoding, decoding marshalling etc.)
Standard ML
Maybe the closest type of Standard ML would be the type option, which is defined like this in the standard library:
```datatype 'a option = NONE | SOME of 'a
```
```case v of NONE => "unbound value"
| SOME _ => "bounded value"
```
Swift
Swift has `Optional<T>` type, where `nil` means a lack of value. `T?` is syntactic sugar for `Optional<T>`.
```let maybeInt: Int? = nil
```
To just check if variable is nil, you can use `==` operator.
```if maybeInt == nil {
print("variable is nil")
} else {
print("variable has some value")
}
```
Usually you want to access the value after checking if it's nil. To do that you use `if let`
```if let certainlyInt = maybeInt {
print("variable has value \(certainlyInt)")
} else {
print("variable is nil")
}
```
Tailspin
Tailspin does not have a null value, but a transform is allowed to produce nothing at all, in which case that chain of computation simply stops. A templates transform can explicitly label cases as producing nothing by !VOID but also input values for which there is no matching branch will produce nothing. If you need computation to continue even in the event of nothing being produced, you can wrap the transform in an array/list to get an empty list.
```templates mightBeNothing
when <=0> do !VOID
when <=1> do 'something' !
end mightBeNothing
1 -> mightBeNothing -> 'Produced \$;. ' -> !OUT::write
0 -> mightBeNothing -> 'Won''t execute this' -> !OUT::write
2 -> mightBeNothing -> 'Won''t execute this' -> !OUT::write
// capture the transform in a list to continue computation when no result is emitted
[1 -> mightBeNothing] -> \(
when <=[]> 'Produced nothing. ' !
otherwise 'Produced \$(1);. ' !
\) -> !OUT::write
[0 -> mightBeNothing] -> \(
when <=[]> 'Produced nothing. ' !
otherwise 'Produced \$(1);. ' !
\) -> !OUT::write
[2 -> mightBeNothing] -> \(
when <=[]> 'Produced nothing. ' !
otherwise 'Produced \$(1);. ' !
\) -> !OUT::write```
Output:
`Produced something. Produced something. Produced nothing. Produced nothing. `
It is an error to try to assign nothing to a symbol or field.
```// throws an error
def nothing: 0 -> mightBeNothing;
// throws an error
{ nothing: 0 -> mightBeNothing }
// OK, simply results in the empty structure without a field called 'nothing'
{ 0 -> mightBeNothing -> (nothing: \$) }```
Tcl
In Tcl, where every value is a string, there is no out-of band value corresponding to NULL. In many cases, using the empty string is sufficient:
```if {\$value eq ""} ...
```
A stricter approximation to NULL can be had with non-existing variables or elements of a dict or array:
```if {![info exist nullvar]} ...
if {![info exists arr(nullval)]} ...
if {![dict exists \$dic nullval]} ...
```
Note that lists do not support anything like nulls, since they are strictly sequences of values.
Ursa
```# the type at declaration doesn't matter
decl int x
set x null
if (= x null)
out "x is null" endl console
else
out "x is not null" endl console
end if```
VBA
```Public Sub Main()
Dim c As VBA.Collection
' initial state: Nothing
Debug.Print c Is Nothing
' create an instance
Set c = New VBA.Collection
Debug.Print Not c Is Nothing
' release the instance
Set c = Nothing
Debug.Print c Is Nothing
End Sub
```
Visual Basic
Works with: VB6
Null by the definition of this task is called "Nothing" in VB6:
```Public Sub Main()
Dim c As VBA.Collection
' initial state: Nothing
Debug.Assert c Is Nothing
' create an instance
Set c = New VBA.Collection
Debug.Assert Not c Is Nothing
' release the instance
Set c = Nothing
Debug.Assert c Is Nothing
End Sub
```
The Null keyword has a different meaning in VB6: it's one of the states that a Variant type variable can be in. Null means that a Variant doesn't hold valid (i.e.: defined) data.
```Public Sub Main()
Dim v As Variant
' initial state: Empty
Debug.Assert IsEmpty(v)
Debug.Assert VarType(v) = vbEmpty
v = 1&
Debug.Assert VarType(v) = vbLong
' assigning the Null state
v = Null
' checking for Null state
Debug.Assert IsNull(v)
Debug.Assert VarType(v) = vbNull
End Sub
```
Wart
The null value `nil` is also the only false value.
`(not nil)`
Wren
In Wren, it is not technically possible for a variable to have no value at all.
If you define a variable without giving it a value, then it is given the special value null which is the only instance of the Null class and also a keyword.
Similarly, if you define a function but don't give it a return value, then it returns null. In this particular respect, null is similar to void in C.
In boolean expressions, null is considered to be false whereas (apart from false itself) all other values (even 0) are considered to be true. For consistency the Null class therefore overrides the ! operator, which it inherits from the Object class, so that !null returns true.
It is always easy to test for nullness either by querying a variable's type or checking the value of a boolean expression involving a potentially null variable.
```// Declare a variable without giving it an explicit value.
var s
// We can now check for nullness in one of these ways.
System.print(s)
System.print(s == null)
System.print(s is Null)
System.print(s.type == Null)
// Similarly, if we define this function without giving it a return value.
var f = Fn.new {
System.print("I'm a function with no explicit return value.")
}
// And now call it.
var g = f.call()
// We find that the return value is null.
System.print(g)
```
Output:
```null
true
true
true
I'm a function with no explicit return value.
null
```
zkl
In zkl, there isn't a C like 0/NULL, a value that, if referenced, causes bad things to happen. There is an Object, Void, that is used as generic NULL like thing but it is just another object.
```if(Void == n) ...
return(Void)```
Z80 Assembly
Translation of: ARM Assembly
Even though this doesn't really apply to assembly, as there is no "null pointer" per se (that is, all data is a number), it would be interesting to demystify what `NULL` really is. At the lowest level, the null pointer is just a pointer to a memory location that is of no use to the programmer. The Z80 does not segfault, so any memory address we read is fair game. So without the hardware enforcing `NULL`, we need to pick an address we don't mind losing. Typically, using `&0000` to equal C's `NULL` is acceptable, as address `&0000` on any randomly chosen Z80-based hardware is typically a jump to the kernel's entry point (or the main program's entry point, depending on the implementation.)
```ld a,(&0000) ;dereference the null pointer as a uint8
ld hl,(&0000) ;dereference the null pointer as a uint16
```
Typically, you would get 0xC3 when dereferencing as an 8-bit value and 0xC3nn when dereferencing as a 16-bit value, where nn is the low byte of the address of the aforementioned entry point. Neither of these are particularly useful, so having `&0000` as the null pointer is perfectly fine. Although there are technically other options, most CPUs can compare with zero more efficiently than most other numbers, and the Z80 is no exception. Of course, the Z80 will not check if a pointer is NULL for you, so you have to do it yourself:
```LD HL,myPointers
;there is no LD BC,(HL) so we have to do this:
LD c,(hl)
inc hl
LD b,(hl)
;and compare to null
LD a,b
or c ;compare BC to zero
JR z,isNull
``` | 2022-10-02 09:25:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.2741208076477051, "perplexity": 8505.258154439962}, "config": {"markdown_headings": false, "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-2022-40/segments/1664030337307.81/warc/CC-MAIN-20221002083954-20221002113954-00636.warc.gz"} |
https://www.physicsforums.com/threads/electric-field-outside-a-solenoid.744191/ | # Electric Field Outside a Solenoid
1. Mar 19, 2014
### sashab
1. The problem statement, all variables and given/known data
A solenoid has a radius of 1.85 cm and 1110 turns per meter. Over a certain time interval the current varies with time according to the expression I = 2.50t, where I is in amperes and t is in seconds. Calculate the electric field 5.47 cm from the axis of the solenoid.
2. Relevant equations
E = (μ$_{0}$n/2)(di/dt)(R$^{2}$/r), r>R
Where μ$_{0}$ = 4∏x10$^{-7}$, n = 1110 turns/meter, di/dt = 2.50, R = 0.0185m, r=0.0547m
3. The attempt at a solution
After plugging in all the numbers and putting things in the right units, I got 1.09x10$^{-4}$ V/m, but this is incorrect. If someone could tell me where I went wrong, I would really appreciate it! Thanks
2. Mar 19, 2014
### sashab
Nevermind, I found my mistake! | 2018-02-20 22:17:04 | {"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": 0.5706859230995178, "perplexity": 817.9291524060485}, "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-2018-09/segments/1518891813109.36/warc/CC-MAIN-20180220204917-20180220224917-00487.warc.gz"} |
http://rstudio-pubs-static.s3.amazonaws.com/253773_a38d1747e5884be7b189020a0c72e848.html | # Language Detection Shiny App
26th February 2017
### Overview
• This app determines the language(s) that a user-supplied text is possibly written in, and then enables the user to select one of the top-scoring languages to visualise word frequencies in the text and in the language's corpus
• All the plots in the app are interactive
• Supported languages (and corresponding two-letter ISO 639-1 code)
Afrikaans (af), Breton (br), Bosnian (bs), Catalan (ca), Czech (cs), Danish (da), German (de), English (en), Esperanto (eo), Spanish (es), Estonian (et), Basque (eu), Finnish (fi), French (fr), Galician (gl), Croatian (hr), Hungarian (hu), Indonesian (id), Icelandic (is), Italian (it), Lithuanian (lt), Latvian (lv), Malay (ms), Dutch (nl), Norwegian (no), Polish (pl), Portuguese (pt), Romanian (ro), Slovak (sk), Slovene (sl), Albanian (sq), Serbian (sr), Swedish (sv), Tagalog (tl), Turkish (tr)
• Data source for the language corpora
2016 OpenSubtitles Frequency Word Lists
• Source code
### Scoring the text
• For each supported language, a score between 0 and 1 is assigned using a simple algorithm: the frequency of all the words in the text is calculated, and then the frequencies of the words that also appear in the top n (default: 500) most frequent words of the supported languages are added together
• For Bosnian, Croatian, and Serbian, the number of words used to detect the language should be increased as the default 500 words may not be enough to discriminate between these three (similar) languages
• The highest scores determine the languages that the text is probably written in
• The top 10 scores are shown as a bar graph in the app
Example for the text Oom Gert Vertel en Ander Gedigte:
language score langName
1 af 0.69075677 Afrikaans (Afrikaans)
24 nl 0.48370466 Dutch (Nederlands, Vlaams)
8 en 0.17943289 English
7 de 0.16350580 German (Deutsch)
4 ca 0.12437857 Catalan (català)
2 br 0.12378015 Breton (brezhoneg)
15 gl 0.10688639 Galician (galego)
33 sv 0.10127048 Swedish (svenska)
6 da 0.09514822 Danish (dansk)
14 fr 0.08833548 French (français)
The highest scoring language is Afrikaans, and the text is indeed actually written in Afrikaans.
### Properties of the word frequencies in the corpus and in the text
• In the language that the text is written in, the frequencies of the words in the corpus and in the text tend to be similar, especially for the most frequent words, as can be seen in the word frequencies plot
• In the actual language of the example text (leftmost plot), there are many points, and they tend to be grouped around the line of equal frequencies, especially at higher frequencies
• In the two next best candidate languages, there are less points and they are more scattered.
• Zipf's law states that in a natural language, the frequency of any word is inversely proportional to its rank in the frequency table
• The leftmost part of the plot below show the word frequencies in the previous example text and in the Afrikaans corpus, according to their rank: the distributions of the frequencies are similar
• The rightmost part of the plot replaces the example text with an artificial text consisting of the 500 most frequent words in Afrikaans, each used exactly once: the distributions are completely different
### Going further
• Scoring algorithm
The algorithm used in this app obtains accurate results, the scoring function1 could however be improved to:
• Further discriminate between similar languages (e.g. Bosnian, Croatian, and Serbian)
• Support languages that use non-Latin scripts (e.g. Chinese, Arabic, Hindi)
• Corpora
The corpora for the supported languages are based on subtitles of TV series and films, and are therefore biased towards the spoken form of the languages.
For instance, first and second person pronouns such as “I” or “you” appear more frequently than in traditional written works
Additional or alternative corpora could be created2 to handle other forms of the languages (e.g. non-fiction and fiction written works, online forums and chats) and obtain better results depending on the type of input that is fed to the app. | 2018-02-18 08:23:29 | {"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": 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": 0.3005973696708679, "perplexity": 7340.103477211508}, "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-2018-09/segments/1518891811795.10/warc/CC-MAIN-20180218081112-20180218101112-00720.warc.gz"} |
https://www.physicsforums.com/threads/help-with-dielectrics.282755/ | # Help with Dielectrics
1. Jan 4, 2009
### rbtqwt
Hi. I have a problem in trying to find the field $$\vec E$$ in the following situation:
I have an infinite charged plane, with charge density $$\sigma$$, and two dielectrics, like in picture:
http://img53.imageshack.us/img53/2301/testrb0.jpg [Broken]
Now, if i think of $$\vec D$$ being orthogonal to the charged plane, using Gauss law i get $$\vec D = \frac{\sigma}{2} \vec k$$, then i get the fields $$\vec E$$in the dielectrics: $$\vec E_1 = \frac{\sigma}{2\varepsilon_0 k_1} \vec k$$ and $$\vec E_2 = \frac{\sigma}{2\varepsilon_0 k_2} \vec k$$.. but, because of $$\oint \vec E \cdot d\vec x = 0$$, I obtain $$E_{t_1} = E_{t_2}$$ , where $$E_{t_i}$$ is the tangential (to the contact surface of dielectrics) component of $$\vec E$$ in dielectric $$i$$. But $$E_{t_1} = \|\vec E_1}\| \ne \|\vec E_2\| = E_{t_2}$$.
What is wrong?
Last edited by a moderator: May 3, 2017
2. Jan 4, 2009
### clem
Re: Dielectrics
The basic equation is E1=E2. Then find D1 and D2. There will be a sigma1 and silgma2.
3. Jan 4, 2009
### rbtqwt
Re: Dielectrics
In the problem $$\sigma$$ is fixed :shy: | 2017-11-24 17:41:27 | {"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": 0.8623379468917847, "perplexity": 1372.056678971619}, "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-2017-47/segments/1510934808260.61/warc/CC-MAIN-20171124161303-20171124181303-00412.warc.gz"} |
http://mathoverflow.net/revisions/23642/list | MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
"the quadratic variation of a Brownian motion between $0$ and $T$ is equal to $T$"
this is only true that if $\mathcal{D}^N$ is a nested sequence of partitions of $[0,T]$ (with mesh size going to $0$) then the quadratic variation of a Brownian motion along these partitions converges towards $T$, almost surely. If we define the quadratic variation of a continuous function $f$ as we would like to, $$Q(f,[0,T]) = \sup_{0=t_0<\ldots, t_n=T } \sum |f(t_k)-f(t_{k+1})|^2,$$ then the Brownian paths have almost surely infinite quadratic variation. | 2013-06-20 04:47:04 | {"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": 0.9038848280906677, "perplexity": 225.39221163994253}, "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/1368710299158/warc/CC-MAIN-20130516131819-00038-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://gamedev.stackexchange.com/questions/172157/roguelike-map-and-entities-how-to-store-it | # RogueLike - Map and Entities - how to store it?
Started to write my own RL, and already fell into analysis (paralysis?) when implementing map class.
My TILE class looks like:
class CMapTile
{
public:
enum eMapTileType
{
E_UNKNOWN = 0,
E_FREE,
E_WALL,
E_ALL
};
CMapTile();
virtual ~CMapTile();
uint8_t GetType();
uint8_t IsBlocking();
uint8_t GetSymbol();
protected:
uint8_t m_uiType;
uint8_t m_uiBlocking;
uint8_t m_uiSymbol;
};
Then i wrote a class for the map itself
#include "CMapTile.hpp"
class CMap
{
public:
CMap();
void Init(const uint8_t& ruiWidth, const uint8_t& ruiHeight);
void Generate();
void Print();
CMapTile GetTile(const uint8_t& ruiX, const uint8_t& ruiY) const;
private:
uint8_t m_uiWidth;
uint8_t m_uiHeight;
std::vector<CMapTile> m_vTiles;
};
And here is me thinking. With the design above, Map has no idea about entities. Entities have no idea about map.
So when i want to move Player RIGHT given above i will have to make two checks (totally separated). A check for map if tile is blocking and a check in a vector of entities if there is any entity on that position (X+1).
// Obtain tile to the right
CMapTile *pTile = m_Map->GetTile(pPlayer->GetPosX()+1, pPlayer->GetPosY());
// Obtain entity to the right
CEntity *pEntityToRight = m_EntitiesManager.GetEntity(pPlayer->GetPosX()+1, pPlayer->GetPosY());
// Check if blocking
if(pTile != NULL && pTile->IsBlocking() == false)
{
// Check if any entity there and blocking
if(pEntityToRight != NULL && pEntityToRight->IsBlocking == false)
{
// Update Player Pos to Tile position
pPlayer->SetPosition(pTile->GetPosX(), pTile->GetPosY());
// Update Map Entity Id for this Tile
_Tile->SetEntityId(pPlayer->GetID());
}
}
Maybe instead i should have an integer in the CMapTile storing an id of an entity, like below:
class CMapTile
{
public:
enum eMapTileType
{
E_UNKNOWN = 0,
E_FREE,
E_WALL,
E_ALL
};
CMapTile();
virtual ~CMapTile();
uint8_t GetType();
uint8_t IsBlocking();
uint8_t GetSymbol();
int GetEntityID();
protected:
uint8_t m_uiType;
uint8_t m_uiBlocking;
uint8_t m_uiSymbol;
int m_iEntityID; // <<<<<<<-----
};
Thanks to which i can make a simple call to CMap like this to check if i can move player Right:
// Obtain tile to the right
CMapTile *pTile = m_Map->GetTile(pPlayer->GetPosX()+1, pPlayer->GetPosY());
// Check if blocking
if(pTile->IsBlocking() == false)
{
// Update Player Pos to Tile position
pPlayer->SetPosition(pTile->GetPosX(), pTile->GetPosY());
// Update Map Entity Id for this Tile
_Tile->SetEntityId(pPlayer->GetID());
}
With the above CMapTile::IsBlocking would have to know about Entity class, to figure out if entity is a blocking type or not.
Questions This sounds simple but that initial design has an effect on later development. Examples:
What happens later when for an example NPC would have to attack player?
How from the NPC class the logic knows where is the player located or if he (NPC) can move right ? - having no idea about the map.
Should i pass the MAP pointer to the AI logic ?
Which is the proper way to go to separate the logic and make all of the above simple in future stages of development ?
• Edit title to make it into a proper question. And edit last paragraph to remove "opinion/suggestion", as that will result in your question tagged as "opinion based" and be locked. Recommendation for title: "How to render this/a tilemap?", "In which order to render this tilemap?". – Hatoru Hansou May 21 '19 at 20:54
• @HatoruHansou - thank you for the hints, i have updated both title / question. – PeeS May 21 '19 at 20:56
• Good call @HatoruHansou, and thank you for making these updates, PeeS. – DMGregory May 21 '19 at 21:00
I don't see a problem with the representation of your tilemap in memory. I think I can simplify it by make a tilemap a simple vector<uint8_t> but as we don't know what features you need let's not touch that topic now, as there isn't really a specific way of doing this.
About tile walkability: different entities may interact with tiles in different ways. Example: a ghost may pass through walls. Consider having a single uint8_t to represent a tile_type, and let logic for different entities to decide if they can walk or no walk a tile. For example, birds can pass over a tree tile, the player's character can't. This way you can have ghosts and birds. Do not ask the tile if it is blocking, ask the entity if it can walk a tile.
If you don't like logic in entities, ask the engine if a given entity can walk a tile. If you are using Entity/Component/System approach, ask the relevant system if a given entity can walk a tile.
Should i pass the MAP pointer to the AI logic? I would have a globals::MAP, but this depends on which design patterns you want to follow. You can, it isn't either correct or incorrect. Tip: I found myself requiring map at different functions in graphical RPGs, that's why I prefer globals. If you are working alone no need to overprotect your map, you know where you may modify it or not.
About storing entities IDs in tilemap: again, it isn't incorrect, and for text-based probably not overkill performance wise, as long as you are aware that this makes your tilemap also an scene grid. An scene grid is a form of scene partitioning, either for physics or for rendering. In this case, you are using the grid to easy check if near squares contain an entity.
One question you must ask yourself: Do you need more than a single entity in an square at a given time? Most graphical RPGs don't allow it and for the rare "cutscenes" where there may be special cases they just code them as scripted animations (suspending game "physics" temporarily), so you may very well be fine by having your tilemap being your scene grid with a single field for an entity ID. Problems start if the answer is "yes, I need more than a single entity".
Possible solutions are:
• Replace the entity ID field for a vector (required memory will skyrocket, but depending on your bigger map worst case you may not care).
• Don't store entities IDs in the tilemap, just store them in a separate vector/list and when you want two entities to interact search for suitable entities by iterating through the whole list. This forces you to think in advance in your worst possible case, the problem is that you may change your mind later, so this doesn't scale. Take this as a premature optimization for the specific scenario of allowing a very small number of entities alive at the same time.
If you already know that you only need 100 entities at the time, and that your maps will be big, 1000x1000. You can estimate like this:
• a list with 100 entities ids, with 32 bit per element, is only 400 Bytes. I calling it an entity list, but you probably want to use a vector<>. For a linked list there is extra data.
• a 1000x1000 tilemap, with a 32 bit field for entities ids is 4 MB only for the entities references, doesn't matter if their are set to 0 or reference an actual alive entity, I didn't add the other fields the tile structure contains. Yes, bigger than the list approach, but still a joke for today's hardware.
You must also think in list traversal time vs scene grid near square checking. Of-course the grid beats the list, but at what memory cost. All depends on the number of entities you expect and the size of maps you may want to have.
Some notes: I said "very small number of entities" but for today machines it may be thousand of them (the simpler their interactions are the more you may have). The list strategy plays well with an entity spawn strategy where you load a bunch of entities when entering a certain room/going near a certain zone, and unload them when they no more pursuit you. But if you want an honest open world simulation, and load/unload entities are a no go for you, entity lists aren't your friend (except that your big open world map only requires 100 entities in total, then any strategy would do).
If you can't decide on an strategy, just test them with a fabricated worst possible case (a really big map with thousands entities). Maybe you don't know yet how many you will need for the worst map your final game will have, but you may have an idea, just use bigger numbers than in your worst possible imagined case.
The only value you obtain from this answer is to help you think in some problems now. Problems you will hit eventually anyway while advancing the project.
Need help deciding? Go for the scene grid and single entity per square. Because it's simpler to code. Also, force yourself to think in things like max possible entities at a given time smells to premature optimization. The bigger possible map is not only limited by their size in memory, but also by the time you need to expend in creating them, so I don't expect them to be very big (of-course, there is the case of procedural generation, that makes big maps cheaper to create).
Tip: you are already using getters, and that's OK. By having a proper wrapping interface, if you switch strategies at some point, while your getters prototype doesn't require to change, the rest of the code doesn't require any change either, you will only have to rewrite the getters inner code.
In brief:
• Ask the map the type of tile of a given coordinate.
• Ask the entity/system if a given entity can walk that type of tile.
• Use scene grid because is simpler to code, until you hit the horizon where it won't work for you anymore (and probably you won't hit it)
• Be smart with wrapper functions prototypes in case you decide to switch strategies in the future.
Possible optimization: for most possible scenarios, you only need a tile to be a uint8_t value, and nothing more. The value determines everything, including the symbol used to render it (if that is what the symbol member is for in your code). But I can think of a case where this is not valid, if you want to have secret passages, then you may disguise a passable tile with the symbol of a wall, but note that that can be solved too by having another tile value for the passable wall tile.
tile 23, renders as ¤ and not passable by player.
tile 24, renders as ¤ but passable by player.
¤¤¤¤¤¤¤¤¤¤
¤¤¤T¤¤¤¤¤¤
¤¤¤¤¤¤¤¤¤¤
¤¤¤¤¤¤¤ ££
¤¤ ¤¤ £££
£E
P E £
Above, the two ¤ are tile 24, while all the others are tile 23, the player can move to T by walking over those two tiles. The presence of T gives the hint that there is a secret passage.
• Thank You, i really like the answers and suggestions here. Especially v.good idea to ask the entity if can walk the tile. Your answer allows me to move forward and do some test/try approaches - instead of being looped in thinking. Again, appreciated! Hope there will be more chances to discuss if i face a blocker in the future. – PeeS May 22 '19 at 15:12
• @PeeS Glad it helped. Remember, divide complex topic in smaller chunks, so the question is answerable. This one was a bit open, but still answerable. Also, single topic questions have more viewers. – Hatoru Hansou May 25 '19 at 2:05 | 2020-08-03 23:51:54 | {"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": 0.27931883931159973, "perplexity": 2108.734971893116}, "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/1596439735836.89/warc/CC-MAIN-20200803224907-20200804014907-00413.warc.gz"} |
https://community.wolfram.com/groups/-/m/t/1432072 | # [CALL] For Curious Cases of Word Histories
Posted 8 months ago
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NOTE: This is a long page with many images. Scroll through to find some gems.
WordFrequencyData is a nifty instrument for mining oceans of texts and discovering wonderful historical semantic curiosities. This post is a call for you to share your discoveries of interesting word histories. Rules are very simple.
• Post you discovery as a comment on this thread
• Your discovery should be curious histories of some words that can be seen in their WordFrequencyData
• Start your comment with a title clearly indicating the meaning of your discovery (use # as the first character to make a title)
• Your comment must contain a plot WordFrequencyData of your terms. You can use the function I provide below. Alternatively you can use your own what to visualize WordFrequencyData.
• Your comment must contain Wolfram Language code you use to make the plot
• Your comment must contain some text explaining why you think the words you found are curious and interesting in your opinion
• If you want to comment on someone's work please click REPLY to his/her specific post so it is clear to what you refer and nested structure of comments is preserved.
Please see comment below for good examples.
### FUNCTION for PLOTs
Feel free to use this function for your visualizations and change or improve it if you wish. Note what kind of options you can provide to this plot. I tried to limit those options to only very important once, fixing other options to make a nice plot.
ClearAll@WordFrequencyPlot;
Options[WordFrequencyPlot]=
{"YearStart"->1800,"YearEnd"->Now,"Case"->True,
"Smooth"->3,"Scaling"->None,"Style"->Automatic};
WordFrequencyPlot[words_,OptionsPattern[]]:=
With[{
$data=WordFrequencyData[words,"TimeSeries", {OptionValue["YearStart"],OptionValue["YearEnd"]}, IgnoreCase->OptionValue["Case"]]}, DateListPlot[ MapThread[Callout, {MeanFilter[#,Quantity[OptionValue["Smooth"],"Years"]]&/@ Values[$data],words}],
ScalingFunctions->OptionValue["Scaling"],
PlotRange->All,
PlotTheme->"Detailed",
PlotStyle->OptionValue["Style"],
FrameTicks->{Automatic,None},
ImageSize->Large,
FrameLabel->{"YEAR","FREQUENCY in TEXT"}]
]
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Posted 8 months ago
# Benford's Law
I wanted to start from something spectacular in its simplicity - demonstration of Benford's Law. From MathWorld:
Benford's Law is a phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability ∼30%, much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages.
Surprisingly, this law holds not only for the digits usage in texts, but also for the word-names of the digits, - see plots below. It would be nice to hear any explanations of this.
WordFrequencyPlot[ToString /@ Range[0, 9]]
## Log scaling of vertical axis
WordFrequencyPlot[ToString /@ Range[0, 9], "Scaling" -> "Log"]
## Direct plot of digit "names"
WordFrequencyPlot[IntegerName[Range[0, 9]]]
## Log scaling of vertical axis for digit "names"
WordFrequencyPlot[IntegerName[Range[0, 9]], "Scaling" -> "Log"]
Posted 8 months ago
Surprisingly, this law holds not only for the digits usage in texts, but also for the word-names of the digits, - see plots below. It would be nice to hear any explanations of this. "One" is also an indefinite pronoun ("no one", "one of the group", "if one wishes"), so appears a lot more often in English than just as a spelled-out number. For the cases of actual numeric representation, nearly any numbered list will include one (such as this brief statement, for one); those that go to two will also include one (but not three), those that go to three will also include two (but not four), and so on.
Posted 8 months ago
# The Five Basic Senses
There are five basic senses: touch, sight, hearing, smell and taste. I remember reading somewhere that the maximal information flow human experience normally is due to the vision. The diagram below reflects on that, - note it is a logarithmic scale, - word "see" is much more frequent than others (Although it can have other meanings too, besides the direct act of vision itself, like "understand" etc. But so can other words too.). Note curious fall of "taste" below "hear" and "touch".
WordFrequencyPlot[{"see", "hear", "touch", "taste", "smell"}, "Scaling" -> "Log"]
Posted 8 months ago
# The Five Ws and H
WordFrequencyPlot[{"how", "why", "what", "where", "when", "who"}]
According to Wikipedia, The Five Ws and H are questions whose answers are considered basic in information gathering or problem solving. They are often mentioned in journalism, research, and police investigations. They constitute a formula for getting the complete story on a subject. According to the principle of the Six Ws, a report can only be considered complete if it answers these questions starting with an interrogative word:
• Who was involved?
• What happened?
• Where did it take place?
• When did it take place?
• Why did that happen?
• How did it happen?
Rudyard Kipling in his "Just So Stories" (1902) writes:
I keep six honest serving-men
(They taught me all I knew);
Their names are What and Why and When
And How and Where and Who.
It is quite remarkable to observe that these questions have various degrees of usage perhaps reflecting on their relevant importance, with "when" being the key question nowadays, which was not always the case. "Who" lead in past but became less prominent.
Posted 8 months ago
There is one more with wh:and the popular names show the importance of doing:
Posted 8 months ago
# Freedom, Liberty, Justice, Equality
WordFrequencyPlot[{"freedom", "liberty", "justice", "equality"}]
Political theorists dating back to the Hellenic period have examined the tensions between these concepts, especially between equality and freedom. It's interesting to see how "equality" stays fairly flat while "justice" and "liberty" have decreased (though "justice" seems to be on a relatively recent uptick). Interestingly, "freedom" has increased over time, especially around the time of World War II and the following decades.
Posted 8 months ago
# Religion and Politics
WordFrequencyPlot[{"religion", "politics"}]
It's a common adage that discussing politics and religion won't make you any friends, especially at social gatherings such as dinner parties. But which of these terms has been mentioned more frequently over time? Not terribly surprising that the term "religion" has decreased over time, with general trends of people becoming more secular, but the fact the terms "politics" and "religion" seem to meet in our current period is somewhat telling. Karl Marx once quipped religion is the opium of the people. Perhaps politics and political discourse are shaping up to take its place, for better or worse.
Posted 8 months ago
# Trigonometry and Calculator
WordFrequencyPlot[{"trigonometry", "calculator"}]
It's a pretty well-know fact and self-evident that calculators have ruined trigonometry.
# Smartphone, iPhone, Apple, Samsung, Orange
WordFrequencyPlot[{"iPhone", "Smartphone", "Samsung", "Apple",
"Orange"}, "Scaling" -> "Log"]
People back in 1900 didn't think iPhone was the best phone ever. But if you think Apple is the best company ever, have you ever tried Orange? Way more stable and reliable.
I didn't resist in a humorist take in this thread.
Posted 8 months ago
Somewhat related to health: WordFrequencyPlot[{"homeopathy", "acupuncture", "chemotherapy", "fasting", "antibiotic"}]
Posted 8 months ago
## Four letter words from f to k
These are more than one and the one does not come up (OMG):
flak was important during WW II - even here it throws its shadows ....
Posted 8 months ago
## Western Thinking
Posted 8 months ago
## Social Occupation
The King is still the king, unbelievable
may be the King of Rock n'Roll and the King of Pop are included. Without the king
the servant declines, the employee raises, not much gain in it, isn't it? The abolition of slavery seems to be reflected.
Posted 8 months ago
## Political Systems
WordFrequencyPlot[{"capitalism", "nationalism", "socialism",
"communism", "fascism", "populism"}]
Posted 8 months ago
## World Powers
Posted 8 months ago
## Recession word frequency vs actual recessions
Let's plot the frequency of the word "recession".
WordFrequencyPlot[{"recession"}, "Scaling" -> "Log"]
The federal reserve bank in St. Louis keeps a set of indicators which includes the recession periods since 1854.
fred = ServiceConnect["FederalReserveEconomicData"];
usrec = fred["SeriesData", "ID" -> "USREC"];
recWord = WordFrequencyData["recession", "TimeSeries", {1854, Now}];
recLogWord = TimeSeriesMap[Log, recWord];
{min, max} = {Min[#], Max[#]} &@recLogWord["Values"];
recScaled =
MovingAverage[TimeSeriesMap[Rescale[#, {min, max}] &, recLogWord], 3];
DateListPlot[{recScaled, usrec}, Filling -> Axis, ImageSize -> Large,
FrameTicks -> {Automatic, None},
PlotLegends -> {"recession word freq(Log)", "Recessions"},
PlotRange -> {{DateObject[{1854}], DateObject[{2010}]}, {0, 1}}]
Posted 8 months ago
# Colors and the "rise" of blue
I think colors are quite interesting. In the plots below note the "rise of blue" in texts. There is a research field that relates language and perception of color. See for example "Russian blues reveal effects of language on color discrimination". Color blue takes a special place, in some opinions less frequent in ancient literature. Also many languages do not distinguish between what in English are described as "blue" and "green" and instead use a cover term spanning both; this might have an effect on English translations. Please respond to this comment if you have any thoughts about this phenomenon.
color={"white","black","red","yellow","green","blue",
"orange","purple","gray","indigo","pink"};
Interpreter["Color"][color]
Show [WordFrequencyPlot[color[[;; 6]], "Case" -> False, "Scaling" -> "Log",
"Style" -> Interpreter["Color"][color[[;; 6]]]],Background -> GrayLevel[.8]]
A bit more of colors in regular non-Log scaling:
Show [WordFrequencyPlot[color, "Case" -> False,
"Style" -> Interpreter["Color"][color]], Background -> GrayLevel[.8]]
Posted 8 months ago
# Is money the root of all evil?
I just wondered if money was the root of all evil. I'm not great with math, so I wasn't sure how to get the square or cube root of the values for money to see if they aligned with the value for "evil" at a certain point in history. However, "evil" is not mentioned as frequently as "money", so I don't think "money" is any root of "evil". It might be the other way around though.
WordFrequencyPlot[{"money", "evil"}]
Posted 8 months ago
Navigator's tools. Notice the blip during WWII: WordFrequencyPlot[{"sextant", "chronometer", "compass", "pelorus", "almanac"}, "Scaling" -> "Log"]
Posted 8 months ago
# MRB
Here are my initials's (MRB) occurrence since I was born. (I discovered the MRB constant in 1999 -- any connection between that and the MRB uptick in the graph after 2000?)
ClearAll@WordFrequencyPlot;
Options[WordFrequencyPlot] = {"YearStart" -> 1995, "YearEnd" -> Now,
"Case" -> True, "Smooth" -> 3, "Scaling" -> None,
"Style" -> Automatic};
WordFrequencyPlot[words_, OptionsPattern[]] :=
With[{$data = WordFrequencyData[words, "TimeSeries", {OptionValue["YearStart"], OptionValue["YearEnd"]}, IgnoreCase -> OptionValue["Case"]]}, DateListPlot[ MapThread[ Callout, {MeanFilter[#, Quantity[OptionValue["Smooth"], "Years"]] & /@ Values[$data],
words}], ScalingFunctions -> OptionValue["Scaling"],
PlotRange -> All, PlotTheme -> "Detailed",
PlotStyle -> OptionValue["Style"], FrameTicks -> {Automatic, None},
ImageSize -> Large, FrameLabel -> {"YEAR", "FREQUENCY in TEXT"}]]
Posted 8 months ago
# War and Peace
Apparently "Peace" is not as talked about (or written about) as "War"...
WordFrequencyPlot[{"war", "peace"}]
# Earth and Space
Or perhaps running out of options for peace on earth, space becomes the next frontier...
WordFrequencyPlot[{"earth", "space"}]
Posted 8 months ago
I have looked at a number of example some similar to the ones above. Some word histories tell nice stories, for example about medicine. Here are three words for malaria: Options[WordFrequencyPlot] = {"YearStart" -> 1800, "YearEnd" -> Now, "Case" -> True, "Smooth" -> 3, "Scaling" -> None, "Style" -> Automatic}; WordFrequencyPlot[words_, OptionsPattern[]] := With[{$data = WordFrequencyData[words, "TimeSeries", {OptionValue["YearStart"], OptionValue["YearEnd"]}, IgnoreCase -> OptionValue["Case"]]}, DateListPlot[ MapThread[ Callout, {MeanFilter[#, Quantity[OptionValue["Smooth"], "Years"]] & /@ Values[$data], words}], ScalingFunctions -> OptionValue["Scaling"], PlotRange -> All, PlotTheme -> "Detailed", PlotStyle -> OptionValue["Style"], FrameTicks -> {Automatic, None}, ImageSize -> Large, FrameLabel -> {"YEAR", "FREQUENCY in TEXT"}]] WordFrequencyPlot[{"ague", "malaria", "paludism"}] Until 1880, when Laveran first discovered the parasite, ague and malaria have basically the same frequency. Malaria is derived from malaria aria "bad air", whereas ague comes from acute febris "acute fever".Sometimes we can also observe a shift in the frequency of words reflecting meaning the same thing Options[WordFrequencyPlot] = {"YearStart" -> 1800, "YearEnd" -> Now, "Case" -> True, "Smooth" -> 3, "Scaling" -> None, "Style" -> Automatic}; WordFrequencyPlot[{"Moslem", "Muslim"}] In those cases a relative frequency plot, i.e. displaying quantiles could be interesting: StackedDateListPlot[ MapThread[ Callout, {Values[ WordFrequencyData[{"Moslem", "Muslim"}, "TimeSeries"]], {"Moslem", "Muslim"}}], PlotRange -> All, PlotLayout -> "Percentile", ImageSize -> Large, PlotStyle -> {Red, Green}, LabelStyle -> Directive[Bold, 16], PlotTheme -> "Detailed"] Such a plot is also useful to compare opposites like the words peace and war, which are also studied in an earlier post: StackedDateListPlot[ MapThread[ Callout, {Values[ WordFrequencyData[{"war", "peace"}, "TimeSeries"]], {"war", "peace"}}], PlotRange -> All, PlotLayout -> "Percentile", ImageSize -> Large, PlotStyle -> {Red, Green}, LabelStyle -> Directive[Bold, 16], PlotTheme -> "Detailed"] It is interesting to see that since about 1850 the word war is more frequent than peace.These plot also reflect use of words such as bike and bicycle StackedDateListPlot[ MapThread[ Callout, {Values[ WordFrequencyData[{"bicycle", "bike"}, "TimeSeries"]], {"bicycle", "bike"}}], PlotRange -> All, PlotLayout -> "Percentile", ImageSize -> Large, PlotStyle -> {Red, Green}, LabelStyle -> Directive[Bold, 16], PlotTheme -> "Detailed"] I would have expected that bike becomes more prominent during the 20th century. Between 1800 and 1880 is is also surprisingly common. I am not sure why, but this could be due to the other meaning of bike which is something like "nest or swarm of bees". It would be interesting to consider the change of meaning of words. I tried to look at the word "gay" which has changed meaning over the years from lighthearted (13th century), bright and showy (14th century) and happy. It could also imply morality and mean gay women (prostitute) or gay man (womaniser), gay house (brothel). around 1900 it was something like "cheerful"; in the 1980 young users would use it to mean "lame, stupid" around 1990 it got to mean homosexual. I tried to use google n-grams to figure that out, but it didn't really work well. Here are words that are used close to gay over the years: Table[{k, StringSplit[ StringSplit[ StringSplit[ StringSplit[ URLExecute[ "https://books.google.com/ngrams/graph?content=gay+*_ADJ&\ year_start=" <> ToString[k] <> "&year_end=" <> ToString[k + 20] <> "&corpus=15&smoothing=3"], "direct_url="][[2]], " width"][[1]], "gay%20"][[3 ;;]], "_"][[All, 1]]}, {k, 1800, 2000, 20}] which givesThe frequency plot is: Options[WordFrequencyPlot] = {"YearStart" -> 1800, "YearEnd" -> Now, "Case" -> True, "Smooth" -> 3, "Scaling" -> None, "Style" -> Automatic}; WordFrequencyPlot[{"gay"}] or over longer times: Options[WordFrequencyPlot] = {"YearStart" -> 1500, "YearEnd" -> Now, "Case" -> True, "Smooth" -> 3, "Scaling" -> None, "Style" -> Automatic}; WordFrequencyPlot[{"gay"}] Also plastic has changed meaning from the the characteristic of being plastic to the material plastic: WordFrequencyPlot[{"plastic"}] In general we can see when different products have been developed: WordFrequencyPlot[{"radio", "telephone", "computer", "car", "watch", "electricity"}, "YearStart" -> 1700, "YearEnd" -> Now] Of course, words can come out of fashion, too. For example: WordFrequencyPlot[{"Pence", "Dollar", "Shilling", "Euro", "Sterling", "Farthing", "Florin", "Dime", "Yen", "Yuan"}, "YearStart" -> 1700, "YearEnd" -> Now] In fact we can study this more systematically, by looking at the correlations between frequency curves: words = {"war", "peace", "communism", "capitalism", "socialism", "democracy", "unemployment", "conflict", "crisis", "terrorism", "military", "welfare", "bomb", "weapons", "combat"} worddata = (WordFrequencyData[#, "TimeSeries"])["Values"] & /@ words; cm = Correlation[ Transpose@ worddata[[All, 1 ;; Min[ Table[Length[worddata[[i, ;;]]], {i, 1, Length[words] - 1}]]]]]; Column[{GraphicsRow[words[[1 ;;]], ImageSize -> 1000, Frame -> All], Row[{GraphicsColumn[words[[1 ;;]], ImageSize -> 67, Frame -> All], Overlay[{ArrayPlot[cm, ColorFunction -> (ColorData["TemperatureMap"][(1 + #)/2] &), Frame -> None, Mesh -> True, PlotRangePadding -> 0, ImageSize -> 1000, ColorFunctionScaling -> False], GraphicsGrid[Map[NumberForm[#, 2] &, cm, {2}], ImageSize -> 1000]}]}]}, Alignment -> Right, Spacings -> 0] We can use a BandwidthOrdering Needs["GraphUtilities"] {r, c} = MinimumBandwidthOrdering[cm, Method -> "RCMD"] cm2 = Correlation[ Transpose@ worddata[[r]][[All, 1 ;; Min[ Table[Length[worddata[[r]][[i, ;;]]], {i, 1, Length[words]}]]]]]; Column[{GraphicsRow[words[[r]], ImageSize -> 1000, Frame -> All], Row[{GraphicsColumn[words[[r]], ImageSize -> 67, Frame -> All], Overlay[{ArrayPlot[cm2, ColorFunction -> (ColorData["TemperatureMap"][(1 + #)/2] &), Frame -> None, Mesh -> True, PlotRangePadding -> 0, ImageSize -> 1000, ColorFunctionScaling -> False], GraphicsGrid[Map[NumberForm[#, 2] &, cm2, {2}], ImageSize -> 1000]}]}]}, Alignment -> Right, Spacings -> 0] Using that we can try to find words with a similar behaviour: StackedDateListPlot[ MapThread[Callout, Log /@ {Values[ WordFrequencyData[{"Democracy", "War", "Peace"}, "TimeSeries"]], {"Democracy", "War", "Peace"}}], PlotRange -> All, PlotLayout -> "Percentile", ImageSize -> Large, PlotStyle -> {Red, Green, Blue}, LabelStyle -> Directive[Bold, 16], PlotTheme -> "Detailed"] which indicates near constant ratios over a long time. This is not that easy to see in the FrequencyPlot WordFrequencyPlot[{"Democracy", "War", "Peace"}, "YearStart" -> 1900, "YearEnd" -> Now, "Scaling" -> {None, "Log"}] .Finally, it is interesting to look at other languages such as tu vs usted in Spanish Options[WordFrequencyPlotSpanish] = {"YearStart" -> 1800, "YearEnd" -> Now, "Case" -> True, "Smooth" -> 3, "Scaling" -> None, "Style" -> Automatic}; WordFrequencyPlotSpanish[words_, OptionsPattern[]] := With[{$data = WordFrequencyData[words, "TimeSeries", {OptionValue["YearStart"], OptionValue["YearEnd"]}, IgnoreCase -> OptionValue["Case"], Language -> "Spanish"]}, DateListPlot[ MapThread[ Callout, {MeanFilter[#, Quantity[OptionValue["Smooth"], "Years"]] & /@ Values[$data], words}], ScalingFunctions -> OptionValue["Scaling"], PlotRange -> All, PlotTheme -> "Detailed", PlotStyle -> OptionValue["Style"], FrameTicks -> {Automatic, None}, ImageSize -> Large, FrameLabel -> {"YEAR", "FREQUENCY in TEXT"}]] WordFrequencyPlotSpanish[{"vosotros", "ustedes"}] or Du and Sie in German Options[WordFrequencyPlotGerman] = {"YearStart" -> 1800, "YearEnd" -> Now, "Case" -> True, "Smooth" -> 3, "Scaling" -> None, "Style" -> Automatic}; WordFrequencyPlotGerman[words_, OptionsPattern[]] := With[{$data = WordFrequencyData[words, "TimeSeries", {OptionValue["YearStart"], OptionValue["YearEnd"]},(*IgnoreCase\[Rule]OptionValue["Case"],*) IgnoreCase -> False, Language -> "German"]}, DateListPlot[ MapThread[ Callout, {MeanFilter[#, Quantity[OptionValue["Smooth"], "Years"]] & /@ Values[$data], words}], ScalingFunctions -> OptionValue["Scaling"], PlotRange -> All, PlotTheme -> "Detailed", PlotStyle -> OptionValue["Style"], FrameTicks -> {Automatic, None}, ImageSize -> Large, FrameLabel -> {"YEAR", "FREQUENCY in TEXT"}]] WordFrequencyPlotGerman[{"Du", "Sie"}] I suppose that there are interesting mechanisms working here. It definitely feels that "Du" becomes more prevalent as opposed to the more formal "Sie". But there might be an affect due to (social?) media etc. in the opposite direction. Here are a couple of pronouns in English: WordFrequencyPlot[{"you", "thou", "ye", "thee", "thy"}, "YearStart" -> 1200, "YearEnd" -> Now] which might look better on a percentile plot: StackedDateListPlot[ MapThread[ Callout, {Values[ WordFrequencyData[{"you", "thou", "ye", "thee", "thy"}, "TimeSeries"]], {"you", "thou", "ye", "thee", "thy"}}], PlotRange -> All, PlotLayout -> "Percentile", ImageSize -> Large, PlotStyle -> RandomColor[5], LabelStyle -> Directive[Bold, 16], PlotTheme -> "Detailed"] Logarithmically, this becomes: StackedDateListPlot[ MapThread[Callout, Log@{Values[ WordFrequencyData[{"you", "thou", "ye", "thee", "thy"}, "TimeSeries"]], {"you", "thou", "ye", "thee", "thy"}}], PlotRange -> All, PlotLayout -> "Percentile", ImageSize -> Large, PlotStyle -> RandomColor[5], LabelStyle -> Directive[Bold, 16], PlotTheme -> "Detailed"] Cheers,Marco
Posted 8 months ago
Curiosity killed the cat: WordFrequencyPlot[{"curiosity", "cat"}]
Posted 8 months ago
# Transportation
WordFrequencyPlot[{"airplane", "car", "train", "speed"}]
Posted 8 months ago
# Programming Languages
WordFrequencyPlot[{"Wolfram", "Mathematica", "Fortran", "Cobol",
"HTML", "CSS", "Ruby", "JavaScript", "PHP", "Matlab", "LabVIEW",
"Python", "Java", "Swift"}, "Scaling" -> "Log"]
Posted 8 months ago
- A man who writen "Philosophiæ Naturalis Principia Mathematica". WordFrequencyPlot[{"Pythagoras", "Archimedes", "Euclid", "Fibonacci", "Descartes", "Newton", "Leibniz", "Gauss", "Euler", "Fermat", "Turing"}, "YearStart" -> 1800] WordFrequencyPlot[{"Newton", "Copernicus", "Gauss", "Einstein", "Hawking"}, "YearStart" -> 18
Posted 7 months ago
# North, South, East, West
WordFrequencyPlot[{"north", "south", "east", "west"}, "Scaling" -> "Log"]
`
I was a bit surprised to see south is dominating north considering that the later is a standard and "the fundamental direction" in various geography, cartography, GIS, etc. applications and conventions. South surpassed north around 1900. | 2019-04-25 11:59:04 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17080064117908478, "perplexity": 10896.924432361826}, "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-2019-18/segments/1555578721441.77/warc/CC-MAIN-20190425114058-20190425135626-00004.warc.gz"} |
https://twitwi.github.io/Presentation-2016-12-13-Journee-Ml-Opt/ | Please wait, while our marmots are preparing the hot chocolate…
## {image-full top-left darkened /black-bg /no-status /first-slide title-slide fancy-slide bot}
- - - - 5 points {notes} - who - worked on unsup learning for 5-6 y - you'll get an overview of Graphical models vs NN, for unsupervised - you'll get a taste of interesting recent works - NIPS 2016 last week ## Disclaimer {infobox image-full top-right darkened /black-bg /no-status}
- Some notations are atypical. // due to the mix of domains - I will, almost surely, skip sections. - Don't hesitate to ask questions lives. ## TITLE {#plan plan overview /with-ujm} - Unsupervised Representation Learning {intro} - Notations and problem formulation {setup} - Probabilistic (graphical) models {probmod} - Auto-encoders {autoenc} - Generative Adversarial Networks {gan} - adversarial examples and training - GANs - Focus on … {focuses} - optimization {focus optim} - space and time convolutions {focus conv} - *depth {focus depth}*, *breadth/width {focus width}* - semantics {focus semantics} - sequential/temporal aspects {focus temporal} - recent GAN[s](#recentgans) {focus recentgans} - Wrap up {conclusion} # @copy:#plan: %+class:highlight: .intro ## Representation Learning at Hubert Curien // supervised at Data Intelligence, mostly equivalent to learning @svg: repr/representation-di.svg 800 500 - @anim: #tasks |#data-low |#data-mid |#data-high |#methods-title |#methods-mining |#methods-metric |#methods-local |#methods-grammar |#methods-deep ## Unsupervised (Representation) Learning - No labels available - Learning intermediate features or representations - Task agnostic - Related to (data) density estimation - Related to compression ## Example: motif mining in videos / temporal data @svg: motif-mining/motif-mining-task.svg 750 400 - @anim: #layer1 + -#init | #layer2 | #layer3 | #layer7 | #layer4 | #layer6 | #layer5 - Key points: structure? compression? density estimation? {slide} // st {hard, related, that I know} well-enough - {notes} - n. relation to compression, ... - n. notion of need to have a some structure/assumptions/priors - n. structured data probability density estimation - n. something that is {hard, related, that I know} well-enough # @copy:#plan: %+class:highlight: .setup # Notations and problem formulation {#setup} ## Notations and Problem Formulation - Notations - $x$ : data (observations) - $y$ : value to predict (for supervised cases) - $z$ : unknown, unobserved latent information - $\theta$ (or $W$) : model parameters // will come back on the differences z vs θ - Unsupervised learning - only $x$ is given - need to find the parameters ($\theta$, $W$) - may want to further infer the latent variables ($z$) # @copy:#plan: %+class:highlight: .probmod # Probabilistic (graphical) models {#probmod} ## Generative Model, Parameters, Latent Vars… - Observations / Data - Supposition, we have a mixture of 3 gaussians {slide} - Challenge {slide} - gaussians have unknown *parameters* - which point belongs to which component is *not observable* - @anim: .first ## Probabilistic Modeling: principle {libyli} - Adopting a generative approach - think about how the world generated the data - describe it in a “generative model” - Formalize your assumptions about the observations (data) - choose/design a model - a model formulates how *some unknown variables* that are “responsible” for the *observations* (data) - set some priors on the unknown variables - Naming convention: different types of unknowns - parameters: unknown global parameters of the model - latent variables: unknown observation-specific variables // usually unknown - With a mixture of Gaussians - parameters: mean and covariances (and weight) of all Gaussians - latent variables: which Gaussian each data points comes from ## Probabilistic Model Learning {libyli} - The model is generative - describes how the data ($x$) gets generated - “forward model” - the probability of the observations: $p(x | \theta)$ - Finding the unknowns (parameters, latent var.) is challenging - reversing the generative process - finding (or maximizing) $p(\theta | x)$ or $p(\theta, z | x)$ or $p(z | x, \theta)$ - high dimensional parameter/latent spaces - highly non-convex functions ## M1 − PCA: intuition @svg: media/wikipedia-pca.svg 800 500 - @anim: #patch_3 | #patch_4 ## M1: PCA @svg: media/wikipedia-pca.svg 200 200 {model} @svg: graphs/theta-x.svg 100 250 {model m11 clearright} @svg: graphs/x-f-theta.svg 100 250 {model minv} // @svg: media/factor.svg 40 300 {model m12} - Principle Component Analysis (eigen-*) - dimensionality reduction - capture the maximum amount of data variance - PCA probabilistic view {libyli} - observations come from a single low-dimensional gaussian distribution - ... and are transformed with a linear transformation (rotation + scale), - ... and have added noise noisy - @anim: .m11 - Over-generic graphical representation {slide} - $\theta$ is linear transformation - data points $x$ depend on $\theta$ - no *explicit* latent variables *{ico-pencil}* - @anim: .minv - Inference problem: $f$ {slide} - dedicated algorithms
(covariance matrix eigenvalues, iterative methods, …) ## M2 − Topic Modeling: matrix factorization - Probabilistic Latent Semantic Analysis (PLSA) - matrix decomposition {step2} - non-negative {step2} - probabilistic formulation {step2} - $p(w|d) = \sum_z p(w|z) \times p(z|d)$ {step2} - or $x^i = \theta^T \cdot z^i$ (for a document $i$) {step3} - @anim: .svg1 | #documents | #topics - @anim: .step2 | .step3 @svg: media/matrix-decomposition.svg 700 200 {svg1} ## M2: Topic Models {libyli} // our notation in this pres is highly confusion with standards of this domain @svg: graphs/theta-x.svg 50 200 {model m21} @svg: graphs/thetaz-x.svg 130 200 {model m22 clearright} @svg: graphs/x-f-thetaz.svg 130 200 {model m23} - LDA, topic models […](file:///home/twilight/doc/PublicationsAndPresentations/2012-cpms/day-11/cpms-lecture-11-topic-models.html#slide-4) - Latent Dirichlet Allocation - mixture of discrete distributions (categorical/multinomial) - Bayesian formulation of // won't go into details about bayesian - LSA, LSI (Latent Semantic Indexing) // we don't distinguish pLSA/LDA here - Probabilistic formulation of - NMF (non-negative matrix factorization) - @anim: .m21 - $x^i = \theta^T \cdot z^i$ (for a document $i$) *{ico-pencil}* - @anim: .m22 | .m23 - Learning/Inference, $f$ - Gibbs sampling - EM: expectation maximization - variational inference # @copy:#plan: %+class:highlight: .autoenc # Auto-encoders (not yet) {autoenc} ## Feed-forward Neural Networks (supervised) {libyli} @svg: graphs/ffnet.svg 120 400 {model} - Supervised learning (regression, classification, …) - the $x$ are given - the corresponding labels $y$ are given - Building blocks of “neural nets” - a neuron computes a weighted sum of its inputs - the sum is followed by an “activation” $\sigma$ - weights are learned ($W$) - $f^o(x^i, W) = \sigma\left( \sum_d W_{o,d} \times x^i_d \right) = \sigma\left( W_{o,.} ^T \cdot x^i\right)$ - Define a network architecture (class of functions) - number and dimension of layers - activation functions (sigmoid, tanh, ReLU, …) - … actually any composition of differentiable functions - Learning with stochastic gradient descent (SGD) and variants ## M3: Autoencoders {libyli} @svg: graphs/autoenc.svg 120 500 {model m31} - Idea: use a feed-forward approach - … for unsupervised learning (no labels) - to learn a compact data representation - Principle *{ico-pencil}* - try to predict the input form the input - have a latent **bottleneck**: limited model capacity - **encoder** $f$: from the input $x$ to the latent $z$ - **decoder** $g$: from the latent $z$ to the input $x$ - @anim: .m31, .cup - Learning principles of $f$ and $g$ - mean square reconstruction error: $\left\| g(f(x)) - x \right\|^2$ - SGD (like any neural net) - sparsifying regularization: sparse activations ($z$, $f(x)$) - add noise to the input (denoizing autoencoders) n. RBM ? {notes} n. VAE ? {notes} # @copy:#plan: %+class:highlight: .gan # Generative Adversarial Networks {#gan} ## Adversarial Examples (Goodfellow, 2014) {libyli} // ostrich - @anim: %attr:.hasFS:height:200 - In high dimensional spaces - a huge part of the input space is never seen / irrelevant - models are easy to fool - models are wrongly calibrated (bad confidence estimation) - Goal - build machine learning methods robust to adversarial examples - (relation to anomaly detection) - Idea of adversarial training - generate adversarial examples automatically - train also using these examples ## GAN Intuition {infobox image-full top-left darkened /black-bg /no-status}
- Ongoing struggle between two players: - one that makes fake samples, - one that tries to detect them. ## M4: Generative Adversarial Networks {libyli} @svg: graphs/ganright.svg 90 500 {model heightauto} @svg: graphs/ganleft.svg 90 482 {model heightauto alignbottom} - Principle: train two networks - $G$: to generate samples from noise - $D$: to discriminate between true and generated samples - NB: $G$ will try to fool $D$ - Elements *{ico-pencil}* - $x$: a training sample (real) - $z$: a random point in a latent space - $\tilde{x}{}$: a generated sample (fake) - $y$: a binary “fake” ($0$) or “real” ($1$) value - GAN is a minimax game - $\min_G \max_D V(D, G)$ - $V(D, G) = \; \mathbb{E}_{x} [log( D(x) ) ] + \mathbb{E}_z [log(1 - D(G(z))) ]$ ## GAN Target {libyli} @svg: graphs/ganright.svg 90 500 {model heightauto} - GAN optimization is a minimax - $\min_G \max_D V(D, G)$ - $V(D, G) = \; \mathbb{E}_{x} [log( D(x) ) ] + \mathbb{E}_z [log(1 - D(G(z))) ]$ - find a $G$ that minimizes the accuracy of the **best** $D$ - Equilibrium and best strategies - $D$ ideally computes $D(x) = \frac{p_{data}(x)}{p_{data}(x) + p_{gen}(x)}$ - thus $G$ should ideally fit $p_{data}(x)$ - … $G$ samples for $p_{data}(x)$ - Optimization in practice - alternate optimization of $G$ and $D$ - warning: $\min \max$ is not $\max \min$ - saddle point finding (hot topic) ## Example of GAN-generated Digits - DCGAN, Radford et al., 2015/2016 ## Example of GAN-generated Images @svg: media/dcgan-faces.svg 800 500 @anim: div.hasSVG | %viewbox:#zzz | %viewbox:#zzz2 # @copy:#plan: %+class:highlight: .focuses, .focus.optim ## How is all This Optimized {libyli} - Probabilistic models *{ico-pencil}* - Gibbs sampling - Expectation Maximization - Variational Inference - Black-box variational inference (e.g., [Edward](https://github.com/blei-lab/edward)) - Deep models (composition of differentiable function) - … using “back-propagation” (chain rule) - (S)GD - SGD with momentum - SGD with adaptation: RMSProp, ADAM, … - batch normalization trick - link: other tricks for [learning GANs](https://github.com/soumith/ganhacks) - are local minima any good? - link: [which optimizer?](http://sebastianruder.com/optimizing-gradient-descent/index.html#whichoptimizertouse) ## An overview of gradient descent optimization algorithms {no-print} [which optimizer to use?](http://sebastianruder.com/optimizing-gradient-descent/index.html#whichoptimizertouse) ## An overview of gradient descent optimization algorithms {no-print} [which optimizer to use?](http://sebastianruder.com/optimizing-gradient-descent/index.html#whichoptimizertouse) # @copy:#plan: %+class:highlight: .focuses, .focus.conv ## Convolution Models - Extensions of topic models - replace topic with motifs (with temporal structure) - PLSM, HDLSM (Emonet et al., 2014) - Convolutional Neural Networks - most of Christian's talk (ConvNets) - pixelRNN, … # @copy:#plan: %+class:highlight: .focuses, .focus.depth ## Depth in Unsupervised Learning {libyli} - Neural Network depth = Hierarchical probabilistic models - Neural Networks - “deep learning” - adding layers - handling depth with ReLU - handling depth with “ResNets”, Residual Networks (Deep residual learning for image recognition, He et al. 2015) - Hierarchical probabilistic models - Topic Models (LDA, Blei, Ng, Jordan, 2003) - Deep exponential families (Ranganath et al., 2015) *{ico-pencil}* // blei AISTATS - Deep Gaussian Processes (Damianou, Lawrence, 2013) // AISTATS # @copy:#plan: %+class:highlight: .focuses, .focus.width ## Width in Unsupervised Learning {libyli} - Width - Topic model: number of topics - Autoencoder: number of neurons in the hidden layer - GAN: size of $z$ - Non-parametric approaches, HDP, HDLSM (Emonet et al., 2014) - Gaussian process as an infinitely wide NN layer (Damianou, Lawrence, 2013) - universal function approximator - Autoencoders with group sparsity (Bascol et al., 2016) - allow for many hidden units - penalize the use of too many of them # @copy:#plan: %+class:highlight: .focuses, .focus.semantics ## Semantics in Unsupervised Learning {libyli} - Probabilistic models - inference is difficult - consider the “explains away” principle - lead to better interpretability (meaningful $z$) // rain, sun allergy, umbrella - Simpler feed-forward model - independent processing - inhibitory feedback is difficult - Bascol et al, 2016 - group-sparsity on filters - local activation inhibition - global activation entropy maximization - AdaReLU: activation function that zeroes low-energy points # @copy:#plan: %+class:highlight: .focuses, .focus.temporal ## Sequential and Temporal Modeling - cf. Christian Wolf's talk - HMM, CRF =?= RNN - LSTM =?= HSMM *{ico-hugepencil}* # @copy:#plan: %+class:highlight: .focuses, .focus.recentgans # Recent GAN Works {#recentgans} ## M5: BiGAN, ALI (2016) @svg: graphs/biganright.svg 200 500 {model} @svg: graphs/biganleft.svg 200 500 {model} *{ico-hugepencil}* ## M6: InfoGAN (2016) {libyli} @svg: graphs/infoganright.svg 250 480 {model} // @svg: graphs/ganleft.svg 93 480 {model heightauto alignbottom} - GAN noise ($z$) - is unstructured - can be partly ignored by $G$ - InfoGAN idea and principle - part of the noise is a code $c$ - enforce high mutual information
between $c$ and $\tilde{x}{}$ - in practice, predict $c$ from $\tilde{x}{}$ - use a coder $Q$ - @anim: .model - Structure in the code *{ico-pencil}* - Cartesian product of anything - (categorical, continuous, ...) - $\min_{G,Q} \max_D V_{InfoGAN}(D, G, Q) = V_{InfoGAN}(D, G) - \lambda L_I(G, Q)$ {denser} ## InfoGAN: some results ## *GAN as a Modeling Tool {libyli} - Conditional GANs and variants (2016) - the GAN process is conditioned on some data - e.g., image generation condition on a semantic mask - e.g., image conditioned on a text sentence - e.g., audio conditioned on a text sentence - e.g., image conditioned on class and keypoints - … - Very complex (and operational) setups ## Ex: Learning What and Where to Draw ## Ex: Learning What and Where to Draw - Scott Reed et al. # @copy:#plan: %+class:highlight: .conclusion ## Take-home Message {infobox takehome image-fit top-left darkened /black-bg /no-status /fancy-slide}
- Return of the generative approaches. - Two ways of estimating densities - generative models, - generative networks. - The golden age of Variational Inference. // (and black-box VI) - The golden age of SGD. // is the Gibbs sampling of continuous spaces - Saddle points! // active and will progress a lot ## Thank You! | 2020-11-27 20:59:23 | {"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": 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": 0.48916172981262207, "perplexity": 10962.838405477598}, "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-2020-50/segments/1606141194171.48/warc/CC-MAIN-20201127191451-20201127221451-00391.warc.gz"} |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=DBSHCJ_2012_v27n2_411 | OPTIMALITY CONDITIONS AND DUALITY FOR SEMI-INFINITE PROGRAMMING INVOLVING SEMILOCALLY TYPE I-PREINVEX AND RELATED FUNCTIONS
Title & Authors
OPTIMALITY CONDITIONS AND DUALITY FOR SEMI-INFINITE PROGRAMMING INVOLVING SEMILOCALLY TYPE I-PREINVEX AND RELATED FUNCTIONS
Jaiswal, Monika; Mishra, Shashi Kant; Al Shamary, Bader;
Abstract
A nondifferentiable nonlinear semi-infinite programming problem is considered, where the functions involved are $\small{{\eta}}$-semidifferentiable type I-preinvex and related functions. Necessary and sufficient optimality conditions are obtained for a nondifferentiable nonlinear semi-in nite programming problem. Also, a Mond-Weir type dual and a general Mond-Weir type dual are formulated for the nondifferentiable semi-infinite programming problem and usual duality results are proved using the concepts of generalized semilocally type I-preinvex and related functions.
Keywords
multiobjective programming;semi-infinite programming;optimality;duality;
Language
English
Cited by
References
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S. K. Mishra, S. Y. Wang, and K. K. Lai,Generalized Convexity and Vector Optimization, Springer-Verlag, Berlin Heidelberg, 2009.
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X. M. Yang and D. Li, Semistrictly preinvex functions, J. Math. Anal. Appl. 258 (2001), no. 1, 287- 308. | 2018-10-19 15:07:06 | {"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": 1, "/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": 0.3191607594490051, "perplexity": 4044.2901352835624}, "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-2018-43/segments/1539583512411.13/warc/CC-MAIN-20181019145850-20181019171350-00489.warc.gz"} |
https://stats.stackexchange.com/questions/225357/how-do-i-calculate-sample-size-and-what-is-the-is-the-meaning-of-power-in-a-cros | How do I calculate sample size and What is the is the meaning of power in a cross-sectional study?
Here is an example:
Previous research reported that the prevalence of HIV testing among gays is 19.2% . Sample size required to detect this percentage with 95% confidence and .05 precision is 273 (using n = (Z^2 × P(1 – P))/e^2 ). e is precision and Z is the Z score corresponding to 95% i.e. 1.96
Have I done it correctly?
Now what's the importance of having such power: Is it to claim that the sample is representative of that population regarding HIV testing and hence HIV prevalence in my study should be trusted even if it was different from previous research?
1 Answer
Firstly: My calculations yield a required sample size of $239$ (see below for more details).
Secondly: In this context, the concept of power is not needed at all. Power is only needed in the context of statistical testing, where you need to know the distribution of your test statistic under $H_1$. In this context, you are simply interested in "sharpening" your inference made from a sample and don't actually test any hypothesis. The result that you will get can be interpreted as:
"If I gather a sample of size 239, I will be able to conclude with at least 95% probability that the true prevalence of HIV testing amoung homosexuals is within 19.2% $\pm$ 5%." (This holds only if your study also finds a prevalence of 19.2% - see below for more details)
Without going too much into the details (I am sure this is a topic well covered online), we obtain a CI with 1-$\alpha$ probability of covering the proportion $p$ using: $\hat{p} \pm Q^{\mathcal{N} (0,1)}(1-\frac{\alpha}{2})*\sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}$.
If we now decide on a desired "precision" (as you called it) $\delta$, which will reflect half of the width of the obtained CI, we can calculate the required sample size with respect to the chosen $\alpha, \delta$ and $\hat{p}$ as:
$Q^{\mathcal{N} (0,1)}(1-\frac{\alpha}{2})*\sqrt{\frac{\hat{p}*(1-\hat{p})}{n}} \leq \delta$ $\Leftrightarrow$ $n \geq (\frac{Q^{\mathcal{N} (0,1)}(1-\frac{\alpha}{2})}{\delta})^2 * \hat{p}*(1-\hat{p})$.
$Q^{\mathcal{N} (0,1)} (x)$ refers to the $x$ quantile of the $\mathcal{N} (0,1)$ distribution.
In reality, $\hat{p}$ might deviate from the estimated proportion in previous studies (let me call that $\hat{\hat{p}}$). If this is the case, you might not be able to achieve your desired "precision" (In fact you will not be able to achieve your desired precision, if $|\hat{\hat{p}}-0.5| > |\hat{p}-0.5|$) and you will be even more precise if $|\hat{\hat{p}}-0.5| < |\hat{p}-0.5|$.
Now, using the above formula and $\alpha = 0.05$, $\delta = 0.05$ and $p=0.192$, I obtain a required sample size of $239$. | 2019-11-19 09:47:54 | {"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": 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": 0.9406294822692871, "perplexity": 376.7573524596366}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2019-47/segments/1573496670135.29/warc/CC-MAIN-20191119093744-20191119121744-00544.warc.gz"} |
http://physics.stackexchange.com/tags/pressure/hot | # Tag Info
47
Atmospheric pressure is equivalent to supporting a weight of 10 tonnes (about 10 average cars) per metre squared. Put like that, it's not surprising that those metal tanks crumple. However, in the comments you raise the point that you pump your bike tyres to 40 psi (about 3 atm) and yet they don't explode. I think this gets to the crux of your confusion. ...
19
First of all, as mentioned, atmospheric pressure can exert very high loads when integrated over significant areas. As an example, an overpressure of just 2psi is sufficient to destroy many houses and can kill people. That's about 13% of atmospheric pressure. Secondly there is an important scale question. You give an example of a bike tyre: a road bike ...
6
Here's some tennis racket physics from Rod Cross, including links to several Am. J. Phys articles (the physics educators' journal, thus excellent for learning from) and this excellent diagram: There are at least three "sweet spots": The node, at the center of the strings, is a point where the natural standing waves in a vibrating racket don't have any ...
6
Gases in containers at high pressures have those pressures because there are more molecules in them than in the same container at atmospheric pressure, not because there is a difference between the molecular energies. At the same temperature, two containers with different numbers of molecules in them have the same probability distribution of energies. The ...
5
Will a tennis ball go further if i hit it with the side of the racket? No. You want the racket to deform, not the ball. This means using the strings to elastically store energy and return it to the ball. The Ball The ball's deformation upon impact is undesirable because "a tennis ball is required by the rules of tennis to dissipate a fraction of ...
4
The micron used in this way is a unit of pressure. It's short hand for "micron of mercury". It's the pressure that causes the column of mercury in a mercury manometer (pressure gauge) to rise one micro meter. One Torr is one millimeter of mercury, and atmospheric pressure is 760 Torr. 1 $\mu$ = 0.133 Pa.
4
A tank is shaped for pressure from the inside, not the outside. The hull of the tank is convex. Pressure on the inside will cause the hull to assume a shape maximizing the volume per surface which leads to spherical or cylindrical shapes. This does not need much rigidity: balloons come in similar shapes. Pressure on the outside instead will maximize ...
4
Drawing a vacuum in the tank puts the tank walls under a compressive load. The ability of a structure to take compressive load depends on its stability. For a tank car, if we ignore the end caps, compressive loads are acting in two directions - lengthwise and radial. The cylindrical tank will be very stable in lengthwise compression - any buckling forces are ...
3
If you look at the tank from its circular side you could see how it has to perform like an arch to support the load of atmospheric pressure. Let's imagine we cut a section 1 meter long of this cylinder and cut the bottom part off to have a nice round arch and inspect how it works. It is roughly 3 meters diameter so it has to support a load of 3 x 1 meters x ...
3
It's simply inherent to the definition of polytropic processes that they don't allow the system to increase both its pressure and volume at the same time. That doesn't mean you can't increase a system's pressure and volume. You just need a non-polytropic process to do so. For example, it could be a compound process consisting of two polytropic processes with ...
3
First, the reason why the finger becomes more wind-sensitive with some saliva isn't that the saliva evaporates but because the saliva, or water, is a good thermal conductor. The finger has to be warmer than the air so the heat flows from the finger to the air and a good thermal conductor such as saliva helps this flux to take place. Second, because it's the ...
3
This is really three separate questions. Static and dynamic pressure: It is the static pressure that really matters in practical situations. The dynamic pressure is related to the kinetic energy of the fluid which, when it changes, causes a corresponding change in the static pressure. Condenser/evaporator application: The basic Bernoulli equation ...
2
Interactions between the molecules of the gas are not required. In fact ideal gases are modeled as if the molecules have zero interaction. They do however move and interact with the container. That is sufficient to explain the behavior. Imagine that you have a vessel with two identical halves that are connected by a small portal that can be opened and ...
2
Assuming an incompressible liquid, Bernoulli for instationary flow (neglecting friction) is $$\int_1^2 \frac{\partial c}{\partial t} \, \mathrm{d}s + \tfrac12 (c_2^2-c_1^2) + g(z_2-z_1) + \frac1{\rho}(p_2-p_1)=0$$ with velocity $c$, gravitation accceleration $g$, height $z$, density $\rho$ and pressure $p$ and $1$ and $2$ denoting the two positions ...
2
Suppose you do a force balance on the portion of the fluid situated between elevations z and $z +\Delta z$ in the left column. You get: $$p(z+\Delta z)S-p(z)S+\rho g S\Delta z=\rho S\Delta z \frac{dv}{dt}\tag{1}$$where $v$ is the downward velocity in the left column:$$v=-\frac{dx}{dt}\tag{2}$$ The latter equation is correct because the fluid is ...
2
A deodorant can contains a liquid hydrocarbon, typically a propane/butane mixture, and the pressure inside the can is due to the vapour pressure of this hydrocarbon. The pressure can be set to any desired value by varying the composition of the propellant - more propane makes a higher pressure while more butane makes a lower pressure. For a deodorant the ...
2
Gold will compress to about half of its volume at atmospheric pressure if you compress it to 2 million atmospheres at room temperature, which is something that I'm sure has been done with diamond anvil cells. For many metals, the atomic lattice will also undergo structural phase transitions from one lattice type to another at certain pressures, but I don't ...
2
There are two questions: "Why does vaccum crush the steel tank?" and Why the tank implode?" lemon's answered the first question perfectly - multiply the 1 atm pressure by the surface area of the tank and you will get the force, that crushed it. The second answer is not that simple. The tank walls are designed to transform the pressure forces (perpendicular ...
2
There are two ways you can change the internal energy of a gas, one is macroscopic, that is, performing work on or by the gas, if the gas either expands or contracts. The other is microscopically through heat. If the compressed gas is at the same temperature than the outside gas, these microscopic collisions will not result in an exchange of energy, because ...
2
In 3000atm it's speed of decompression will be slower because it is facing greater air density and the expanding spring has to move it. There will also be less "resonance" as the denser air damps the spring movement.
1
In a polytropic process other than adiabatic, you are controlling the temperature in tandem with P and V in such a way that n is constant. You can certainly achieve negative values of n by controlling the temperature appropriately. From the ideal gas law, if T and P are expressed parametrically in terms of V, then:$$\frac{P}{P_0}=\left(\frac{V_0}{V}\right)^... 1 The critical pressure is given by$$P_c=\frac{a}{27b^2},$$while the critical temperature is$$T_c=\frac{8a}{27bR}=\frac{8bP_c}{R}.$$The parameter b is related to to the effective volume occupied by the molecules,$$b=4N_0V_0,$$where V_0 is the volume of the molecule and N_0 is the Avogadro number. So at least theoretically you can chose P_c=1\, \... 1 Whether an answer exists depends on your definition of "near" compared to STP. There are a few fluids that have their critical point at a temperature close to STP, but higher pressure. For example, (see http://www.engineeringtoolbox.com/critical-point-d_997.html) material Tc(K) Pc(atm) acetylene 309.5 61.6 ethylene 283.1 50.5 ethane ... 1 I tried to look around but I couldn't find anything. It does seem like supercritical CO2 is very popular in applications because the critical temperature is just a little over 30 Celsius, but it still requires 73 atmospheres of pressure. An interesting thing mentioned on the Wiki page is that Venus may have had supercritical CO2 oceans many years ago. 1 A thermodynamic process is called reversible if an infinitesimal change of the external condition reverses the process. Consider a gas enclosed by a freely moving piston in a cylinder. Let us say it is in mechanical equilibrium with the atmosphere, that is, the pressures on the piston match. If you increase the external pressure infinitesimally the piston ... 1 Since the pressure p is intensive and the volume V and enthalpy H are extensive variables the function p=p(V,H) is homogeneous degree 0 so you always have$$H \frac{\partial p}{\partial H} + V \frac{\partial p}{\partial V} = 0
1
re: "Why don't high pressure gases stored in containers lose energy?" They can gain & lose energy: Energy (heat) is lost from a gas as the gas is compressed (whether thru mechanical compression or thru cooling compression (e.g. passing a gas thru a tube that is immersed in a very cold liquid -- like liquid nitrogen). Energy (heat) is gained by a gas ...
1
No, a feather dropped in a vacuum jar will not drop at the same speed as in air. There must be a thousand video's out there of this very topic but my favourite is this one. An object moving through air, or any fluid, will experience a drag force resisting its motion. This force increases as the speed of the object through the fluid increases. It also ...
1
Depends on the wall thickness, for example you can collapse a plastic bottle sucking with your mouth but you can't with a glass bottle. There is an Asme code to calculate the minimum wall thickness of a steel tank. The code for external pressure is diferent for internal pressure because geometry of the vessel is very important. Flat and convexe geometry ...
1
According to the second law of thermodynamics,entropy of an isolated system tends to increase. Considering the high pressure region and low pressure region as an isolated system, its total entropy goes up, making fluid flow from high pressure to low pressure to increase the disorder(entropy of the system).This behavior follows from statistical models of ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2016-07-26 20:02:59 | {"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": 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": 0.5756582617759705, "perplexity": 628.8218425445542}, "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-2016-30/segments/1469257825124.22/warc/CC-MAIN-20160723071025-00306-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/please-help-conceptual-motion-of-a-transverse-wave-on-a-string-question.775852/ | 1. Oct 13, 2014
### agonydrum
I am losing my mind over this, it seems the longer i think about it the further i get from a definitive answer. It started with me trying to understand the variables the would result in a standing wave, ie: what needs to occur, why, and how it occurs.
At first i was confused because it seemed that the normal force at the fixed end should just halt all motion since it is an application of dampening but i was told that something happens which distorts the reflected wave in some way? Im not sure what could be happening to distort the wave.
Second, I dont understand how exactly the reflected wave is created in the first place, like i said above it seems to me that the energy should just be dissipated at the fixed end by the normal force not reflected upside down.
The last concept im just unsure about, I did some quick math with a simplified wave force diagram and it seems that the nodes could only be located were the two opposing waves first intersect, but that would mean you could generate nodes even if the reflected wave didnt math the energy of the incident wave?
Any help would be greatly appreciated I've spent days obsessing over this and i get too fixated to just move past it and finish the chapters.
2. Oct 13, 2014
### Andrew Mason
In order for a string to vibrate, it has to have tension so it is either a string fixed at both ends (like a guitar string) or something equivalent (eg. a string with a weight hanging from it). Both ends are effectively fixed. If they are fixed, they can't move so the ends are nodes. The only wave that can exist on such a string is one that has nodes at both ends.
There is no need to think of a reflected wave. Just think of it as a vibration of the whole string. It vibrates at a particular frequency because of the tension, length and mass per unit length of the string. See: http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html
AM
3. Oct 13, 2014
### Staff: Mentor
Consider the mechanism that leads to waves in a string in the first place: We model the string as a spring; when we displace a segment the string stretches increasing tension; this tension accelerates the displaced segment towards the centerline; but by the time it gets there the segment has picked up some velocity so overshoots and is displaced to the other side; and the cycle repeats.
This is basically the interaction that you're describing when you write down with the differential equation for the wave (and it's also very similar to the dynamics of a harmonic oscillator). Viewed this way, there's no energy dissipation at the fixed end; the end is tugged way and then the other but it doesn't move so $d$ is zero in $W=Fd$ and no work is done.
That's for an ideal spring/string described by the classical wave equation and with no damping. In any real physical device, some energy will be dissipated in friction at the fixed point (very similar to the dynamics of a damped harmonic oscillator) so the reflected wave will have slightly less energy and amplitude than the incident wave. Thus, to set up the standing wave and keep it going we have to drive the system, adding energy to replace that lost by damping. Otherwise....
Saying that the energies don't match is equivalent to saying that the amplitudes are not the same. Thus, the two waves won't be able to exactly cancel at the points where you expect the nodes to appear, and a standing wave won't form.
4. Oct 15, 2014
### olivermsun
It isn't so surprising. Think of what happens when an elastic ball strikes a wall. It bounces; the energy doesn't just dissipate. Similarly, think of what happens when waves in a swimming pool hit the end (the wall): they also reflect, they don't just dissipate on the wall. Ocean waves encountering a cliff or sea wall reflect as well (alhough waves that impinge on beaches do tend to dissipate—for more complicated reasons).
5. Oct 16, 2014
### Philip Wood
If the string is fixed to a fixed anchorage, as the wave arrives at the end, it can't move the anchorage and can't do work on it, so can't lose energy to the anchorage. But the anchorage will exert a force on the string to the force equal and opposite to the force which the string exerts on the anchorage, and it is that force on the string which sends a wave back down the string, reversed in phase at the anchorage.
As Andrew Mason remarked, there is no necessity to regard a stationary or standing wave as a combination of progressive waves travelling in opposite directions. It is just one way of thinking about what is going on.
6. Nov 12, 2014
### agonydrum
I really appreciate all of you posting on here to try and help me understand this, some of your explanations have helped me see areas where i was modeling the system incorrectly but I am still having trouble understanding the reflected wave. I understand generally that the reflection is generated by reflecting the force in the X and Y at the fixed end, what i don't understand is how the forces in the Y don't cancel immediately. It seems that if you look at whats happening in a second by second shot than the normal Y force would have to be double the incident force since the new amplitude is a change of 200%. Once to cancel out the incident wave and a second to generate the reflected wave.
7. Nov 12, 2014
### olivermsun
One thing that may be helpful to realize is that, in the idealized string problem, the anchor can exert as much force as is necessary to enforce the boundary conditions of zero displacement and zero velocity (and hence zero energy transmission) at the end of the string.
8. Nov 12, 2014
### Staff: Mentor
It's a bit like bouncing a ball off a wall - there's no problem getting enough force to change the velocity of the ball by 200% with an idealized perfectly rigid wall.
9. Nov 13, 2014
### Philip Wood
The forces I was talking about in #5 were on different bodies (the wall and the string) and therefore don't cancel in a physical sense.
One of the things that makes it difficult to get one's head round what's happening is that it's easy to flip inadvertently between two valid approaches: (1) considering the string's motion in terms of net forces on the parts of the string, (2) considering the motion to be the resultant of superposed progressive waves travelling in opposite directions.
In approach (2) the string actually moves up and down at the end attached to the wall – for each of the progressive waves. But the resultant displacement here is zero. Another thing that you have to watch with approach (2) is that the angle at which the string meets the wall for the individual waves, and therefore the Y component of the force between string and wall due to individual waves, isn't the same as for the resultant wave. | 2017-08-17 10:49:24 | {"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": 0.4726501703262329, "perplexity": 365.8366222229951}, "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/1502886103167.97/warc/CC-MAIN-20170817092444-20170817112444-00532.warc.gz"} |
https://codegolf.stackexchange.com/questions/170676/divide-the-work | # Divide the work [duplicate]
There is a job which can be decomposed into x equally-sized smaller tasks. You have a team of size y <= x, where every member works equally fast on any task. The goal for this challenge is to divide the work as evenly as possible such that every member of your team has at least 1 task to perform. As evenly as possible means that given any member a, the number of tasks it must perform may be at most one more than any other member b. A single task cannot be further divided or worked on simultaneously by two members.
# Input
Your program/function will take as input two positive integers x and y. y is guaranteed to be less than or equal to x. You are free to decide in what order your program will take these inputs. You may take the inputs from any input source desired.
# Output
Your program/function will output a list of positive integers of length y representing the number of tasks each member must perform. The list may be in any order. For example, the following outputs are identical:
2 1 2 1
1 1 2 2
2 2 1 1
Outputs may be to any output sink desired.
# Examples
Each line pair denotes inputs and one possible output. These example inputs specify x first.
1 1
1
4 1
4
4 2
2 2
4 3
2 1 1
4 4
1 1 1 1
10 3
3 4 3
10 7
1 2 2 2 1 1 1
# Scoring
This is code golf; shortest code wins. Standard loopholes apply.
# R, 26 bytes
function(x,y)table(1:x%%y)
Try it online!
Counts the number of occurrence of 1,2,...,y in [1...x] modulus y. 12 bytes golfed by @mnel, and than an additional 6 by @digEmAll.
• I think this can be reduced to 32 bytes with function(v,w)table(rep(1:w,l=v)) – mnel Aug 15 '18 at 22:54
• @mnel indeed! We have a number of active golfeRs these days... join the fun :) – JayCe Aug 15 '18 at 23:20
• Nice approach! I guess this is also valid : 26 bytes – digEmAll Aug 19 '18 at 8:33
• @digEmAll indeed! – JayCe Aug 19 '18 at 20:09
# JavaScript (ES6), 34 bytes
Takes input as (y)(x).
y=>g=x=>y?[k=x/y--|0,...g(x-k)]:[]
Try it online!
### Example for x = 10, y = 3
Remaining tasks | # of tasks for next worker | Workers
---------------------+----------------------------+-------------------------------------
O O O O O O O O O O | 10 / 3 = 3.333... -> 3 | [ O O O ] [ pending ] [ pending ]
O O O O O O O - - - | 7 / 2 = 3.5 -> 3 | [ O O O ] [ O O O ] [ pending ]
O O O O - - - - - - | 4 / 1 = 4 -> 4 | [ O O O ] [ O O O ] [ O O O O ]
x#y=map(divy)[x..x+y-1]
Try it online!
# Jelly, 3 bytes
sZẈ
Try it online!
### How?
sZẈ - Link: integer tasks, integer workers
s - split into chunks of length workers
Z - transpose
Ẉ - length of each
# Python 2, 4038 36 Bytes
-2 bytes thanks to Mr. Xcoder
lambda x,y:x%y*[x/y+1]+[x/y]*(y-x%y)
Try it Online!
• 36: lambda x,y:x%y*[x/y+1]+[x/y]*(y-x%y) – Mr. Xcoder Aug 15 '18 at 21:26
# Jelly, 3 bytes
œsẈ
Try it online!
-1 byte thanks to Jonathan Allan, showing me a built-in I haven't seen before. œs splits the range $[1 \dots\:x]$ in $y$ similarly sized pieces, then Ẉ retrieves the length of each chunk.
• On mobile didn't see other answers already... Ẉ takes this to 3 too – Jonathan Allan Aug 15 '18 at 21:22
• @JonathanAllan Thanks, I wonder how I never saw that built-in. – Mr. Xcoder Aug 15 '18 at 21:25
• Only been around for a couple of months I think – Jonathan Allan Aug 15 '18 at 21:26
• Hmm, I've actually used it once but have since forgotten about it. Just edited about 3 answers to with this built-in. Thanks for the heads-up! – Mr. Xcoder Aug 15 '18 at 21:34
# C (gcc), 48 bytes
Takes i task size, j team size and returns tasks to array k.
l;f(i,j,k)int*k;{for(l=j;l--;)k[l]=i/j+(i%j>l);}
Try it online!
# Brain-Flak, 68 bytes
({}([{}])<{({}(()))}{}>){({}[()]<({}<{({}<>)<>}>()){<>({}<>)}{}>)}{}
Try it online!
Initializes the stack with y ones, then rolls the stack x-y times, each time adding 1 to the former top of the stack.
# 05AB1E, 7 bytes
LôζðK€g
L # Create a list in the range [1, first (implicit) input]
# i.e. 10 → [1,2,3,4,5,6,7,8,9,10]
ô # Split it in chunks of size second (implicit) input
# i.e. [1,2,3,4,5,6,7,8,9,10] and 3 → [[1,2,3],[4,5,6],[7,8,9],[10]]
ζ # Zip, swapping rows and columns (with space as filling character by default)
# i.e. [[1,2,3],[4,5,6],[7,8,9],[10]] → [[1,4,7,10],[2,5,8,' '],[3,6,9,' ']]
ðK # Remove all those spaces
# i.e. [[1,4,7,10],[2,5,8,' '],[3,6,9,' ']]
# → [['1','4','7','10'],['2','5','8'],['3','6','9']]
€g # And then take the length of each inner list as result
# i.e. [['1','4','7','10'],['2','5','8'],['3','6','9']] → [4,3,3]
# MATL, 6 bytes
:gie!s
Try it online!
(implicit input, y)
: % range, push 1...x
g % convert to logical (all ones)
i % push y
e % reshape to matrix of y rows, padding with 0s
!s % row sums; as a row vector
(implicit output)
• I have created [this meta answer] (codegolf.meta.stackexchange.com/a/16802/80010) as a Community wiki so you can modify it. – JayCe Aug 16 '18 at 13:58
• ^ I hope it will let you upvote it after you modify it... otherwise making this a community wiki was a really bad idea! – JayCe Aug 16 '18 at 14:00
x#y=[div x y+sum[1|i<=mod x y]|i<-[1..y]]
Try it online!
# Perl 5-a, 31 bytes
$a[$_--%$F[1]]++while$_;say"@a"
Try it online!
# Java, 55 bytes
By making use of var in Java 10 we can reduce by 1 byte.
y->x->{var a=new int[y];for(;0<x;a[x--%y]++);return a;}
Try it online!
Java 8 Version with 56 bytes
y->x->{int[]a=new int[y];for(;0<x;a[x--%y]++);return a;}
Works by just iterating over the array representing the workers until no tasks are left.
• Unless it's explicitly stated otherwise in the question (which it isn't here), you'll need to include private static int[] g(int x, int y) { and the closing brace in your byte count. – Οurous Aug 16 '18 at 0:24
• Hi, welcome to PPCG! Although @Οurous is indeed right that snippets aren't allowed and it should be either a function or full program, in Java 8+ just y->x->{/*code here*/} would be enough in this case. So it's 55 bytes: y->x->{var a=new int[y];for(;0<x;a[x--%y]++);return a;}. Nice answer though! After you've changed it to comply to the default rules I will upvote it. Enjoy your stay! :) – Kevin Cruijssen Aug 16 '18 at 7:42
• Oh, and if you haven't done so yet, Tips for golfing in Java and Tips for golving in <all languages> might be interesting to read through. – Kevin Cruijssen Aug 16 '18 at 7:44
• @Οurous Yeah, I missed that, was already late :/ – cryoW0lf Aug 16 '18 at 9:03
• @KevinCruijssen Thanks, I changed that. But with using Lambda it will increase to 62, due to the requirements that lambda parameters have to be effectively final. But thanks :) – cryoW0lf Aug 16 '18 at 9:03
# Groovy, 35 bytes
{x,y->o=[0]*y;while(x--)o[x%y]++;o}
Try it online!
# Brachylog, 14 bytes
⟨{h⟦₁}ġ₎t⟩z₁lᵐ
Try it online!
Port of Jonathan Allan's method - range, split, transpose/zip, length - via the explanation in Kevin Cruijssen's answer. Can also be tT&h⟦₁;Tġ₎z₁lᵐ equivalently.
### 19 bytes
h~+.ℕ₁ᵐ≜⟨⌉-⌋⟩<2&t~l
Try it online!
# UGL 1.0.0, 20 Bytes
CÑORCPFÐWC_+_WNINI
Port of the Haskell answer. (even though the language transpiles to Python)
Takes 2 inputs via stdin.
## Explanation (Warning: This may get out of hand)(update: it did not)
C takes two arguments, and apart from some other functions, it is (lambda x,y:map(x,y)).
Ñ is the alias of the int() function. (When you pass it as parameter instead of feeding arguments to it)
O takes one function with two arguments, and two anything, and applies the function to them.
R defines a lambda with two arguments: lambda _,W: <stuff>
C, again is map.
P is functools.partial
F is flip (lambda f: lambda x,y: f(y,x))
Ð is the alias of Division (/)
W is the second argument of the lambda we defined with R
C is exclusive range this time
_ is the first argument of the lambda we defined with R
+_W is literally _+W
NINI is two inputs taken from stdin and converted to integers.
So, this thing is (functions redefined in order to make the last line a bit short):
import functools as ft
apply2 = lambda x,y,z: x(y,z)
flip = lambda f: lambda x,y: f(y,x)
div = lambda x,y: x/y
rg = range
p = print
ip = input()
p(map(int,apply2(lambda _,W: map(ft.partial(flip(div),W),rg(_,(_+W))),int(ip()),int(ip()))))
# Lua, 50 49 bytes
1 byte saved thanks to DLosc :)
x,y=...repeat print(x//y)x=x-x//y y=y-1until y<=0
Try it online!
The output is decimal and in different lines because of the way Lua implements the function print(), if this is a problem I can edit the submission to use io.write() instead.
• 1 byte savings is possible by eliminating the space between 1 and until. :) – DLosc Sep 21 '18 at 21:01
# Clean, 39 bytes
Not much room for originality in this one.
import StdEnv
$x y=[i/y\\i<-[x..x+y-1]] Try it online! Defines the function $ :: Int Int -> [Int], always returning higher numbers towards the end.
# Charcoal, 19 bytes
ＮθＮηＩＥθ⁻÷×⊕ιηθ÷×ιηθ
Try it online! Link is to verbose version of code. Based on Bresenham's line algorithm. Explanation:
Ｎθ Input team size
θ Team size
Ｅ Map over implicit range
ι ι Current value
⊕ Incremented
× × Multiply
θ θ Team size
÷ ÷ Integer divide
⁻ Subtract
Ｉ Cast to string
Implicitly print
# APL (Dyalog), 19 bytes
4 bytes saved thanks to @Adám
{×⍵:(⍺-1)∇⍵-⎕←⌊⍵÷⍺}
Try it online!
• 0=⍵:⋄×⍵: – Adám Aug 16 '18 at 14:52
• Can't you remove k←? – Adám Aug 16 '18 at 14:58
• @Adám thanks, gone a bit rusty :P – Uriel Aug 16 '18 at 15:04
# Forth (gforth), 47 bytes
: f tuck /mod swap rot 0 do 2dup i > - . loop ;
Try it online!
### Explanation
Divides the work by the number of people, add 1 to all workers whose index is less than (or equal) to the amount of excess work (the remainder/modulus)
### Code Explanation
: f \ start a new word definition
tuck \ stick a copy of the number of workers in the "back" of the stack
/mod swap \ get the quotient and remainder, move remainder to the top
rot \ grab the copy of the number of workers from earlier and move it to the top
0 do \ start counted loop from 0 to #workers - 1
2dup \ duplicate the modulus and minimum work
i > \ check if the index is greater than the remainder
- \ subtract from minimum work (forth uses -1 for "true")
. \ output the result
loop \ end the counted loop
; \ end the word definition | 2021-05-06 04:27:11 | {"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": 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": 0.3206058740615845, "perplexity": 4532.549962029303}, "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-2021-21/segments/1620243988725.79/warc/CC-MAIN-20210506023918-20210506053918-00159.warc.gz"} |
https://techfi.tech/what-is-ve-3-3-andre-cronjes-latest-defi-project-caused-fantom-to-take-off/ | TLDR: A new protocol, being built by two of crypto's most well-known developers, aims to change the mold for how DeFi works.
Fantom has grown to become the fifth-largest DeFi chain by TVL on the back of the hype about "ve (3,3)." Here's what the investors need to know about Andre Cronje's latest DeFi experiment.
Andre Cronje and Daniele Sestagalli - Who are they?
Andre Cronje
Andre has been at the forefront of DeFi since it really took off in the summer of 2020. His most successful project, yearn finance, has been a staple of DeFi since its launch in July 2020. Boasting over $4 billion in total value locked and a market cap of over$800 million, the yield optimizer is truly one of the blue-chip DeFi products. Another one of Andre's projects is Keep3r. While Keep3r is a bit more complex than Yearn, it is essentially a protocol that fosters collaboration between protocols and developers. Between Andre's two main projects, Yearn and Keep3r, he has a proven track record of creating successful protocols.
Daniele Sestagalli
If Andre's specialty is coding, then Daniele's specialty is community building. Over the past year, Daniele has built an impressive crypto community known as Frog Nation. It is a group of DeFi projects and protocols that Dani is involved in. The core Frog Nation projects are Abracadabra, Popsicle Finance, and Wonderland. Those are all successful projects.
What is ve(3,3)?
What are VE (Vote Escrowed) tokenomics?
VE just stands for "Vote escrowed," and it was introduced with Curve Finance. The Curve is a DEX that is primarily used for stablecoin swaps. The token behind Curve Finance is CRV.
When you have CRV, you have the option to vote to lock it. By vote-locking CRV, you receive veCRV in return. veCRV gains you two things:
1. Fee revenues generated by the protocol.
2. Governance over Curve Finance.
The longer you lock your CRV, the more veCRV you receive. Once you lock your CRV for veCRV, it cannot be reversed before the expiration date, and it cannot be transferred.
So using ve-tokenomics is a mechanism to better incentivize long-term believers in a project.
It does this by assigning more rights and benefits for those who lock their tokens for longer periods of time and taking tokens temporarily out of the circulating supply, which effectively reduces selling pressure. It also lowers the liquidity of tokens.
What is 3,3?
(3,3) was introduced by the protocol OlympusDAO. Basically, (3,3) is the way in which OlympusDAO baked in an element of game theory into the tokenomics of their protocol.
For example, there are only 2 participants in this game involving OHM, the native token of Olympus. They have the option of staking, bonding, or selling:
• In order to stake, the participant increases the price of OHM because they must purchase the OHM off the market. By staking, the participant passively increases their OHM exposure through auto-compounding. That is beneficial for the protocol and the participants and assigned a value of (+2).
• By bonding, a participant exchanges a certain amount of assets, typically stablecoins or LP tokens, for discounted OHM from OlympusDAO that vests over time. In theory, this shouldn't affect the market price of OHM. However, it is beneficial to the protocol, so it is assigned a value of (+1).
• Lastly, by selling, the participant decreases the price of OHM. This is negative for both the participant and the protocol, as the value of OHM is now decreased. We assign this a value of (-2).
Based on this, both participants are the best off if they both stake, while they are the worst off if they both sell. This game with only 2 participants involved is obviously a gross oversimplification of how the OHM market works practically. Although in theory (3,3)/staking is the best option for participants, that is not always the case for a game that involves thousands, millions of participants.
And the combination of them - ve(3,3)
As a result, ve(3,3) combines ve and (3,3). But more than that, Ve(3,3) hopes to pass them and solve the problem of token distribution and management of DeFi applications.
Users will be able to deposit a base token in return for a non-transferable token, which will be locked in the protocol. In return, they will receive transferable incentive tokens as a reward.
Current problems and new solutions for Defi
First problem: DAO attack
On November 11, 2021, Mochi formally announced themselves as a new player in the Curve Wars, writing that "Curve is the backbone of DeFi, and Convex is the kingmaker of Curve."
First, Mochi launched its governance token, MOCHI INU, and incentivized liquidity for its USDM stablecoin. As is customary, DeFi users aped in, and liquidity quickly grew above $100M. A Mochi team member swapped$46 million in USDM for DAI using the Mochi Curve pool, swapped the DAI for ETH, and used a large portion of that ETH to purchase massive quantities of CVX, which they then locked.
That would have allowed them to vote on additional CRV rewards for the Mochi pool, attracting additional liquidity and allowing them to swap even more USDM for stablecoins to buy more CVX. That ultimately creates a flywheel, tilty CRV rewards in their favor, and attracts huge sums of liquidity to their platform.
Andre Cronje called it "amazingly scammy."
The Emergency DAO ultimately elected to cut off the Mochi pool's rewards. However, this attack brought up an issue: KeeperDAO, FRAX, Olympus, CREAM, and other DAO communities are voting or have voted to pursue similar strategies (if at a smaller scale) and is DAO decentralized?
So here is Andre's solution:
Emission rates' flexibility
Emission rates—or the amount of newly created tokens—will be determined by the circulating supply, and rewards will be greater if fewer tokens are locked across the entire protocol.
On the contrary, if anyone wants to abuse token staking to gain control, they will face the problem that the amount of returned tokens will be fewer, thereby preventing the risks that the current DAO model faces.
Second problem: Incentives for lockers
After LPs receive native tokens from liquidity mining, most of them will immediately sell for profits. Their behavior will affect the development of the project.
Emission rates' flexibility
With the above emission rate's flexibility, weekly emissions are adjusted as a percentage of the circulating supply.
If 0%, 50%, and 100% of the token are locked for ve, the weekly emission would be 2,000,000, 1,000,000, and 0, respectively. This solution will make tokens more and more scarce and incentivize lockers to lock their tokens.
ve lockers increase their holdings proportional to the weekly emission
Assume 1,000,000 weekly emissions, a total supply of 20,000,000, and a locked supply of 10,000,000. It would mean that 1,000,000 new tokens are minted and provided as incentives, a 5% supply increase. Our goal is to ensure that ve lockers are never diluted; as such, ve lockers have their holdings increased by 5%.
Ve(3,3) is integrated into NFT
It casts the lock-up certificate into VeNFT, which can be circulated in the secondary market, and users can use it at any time. In addition, it supports VeNFT mortgage, lending, etc., and enjoys voting governance rights.
With those deviations from the standard, tokens will be distributed more efficiently by user activities.
Third problem: Fee Distribution
In the old AMM model, LPs will receive token rewards issued by the application, not generated fees from protocols.
For Curve, veCRV holders receive 50% of fees no matter what gauges they vote for. It can be problematic as someone who owns veCRV could vote for a pool that generates no fees for Curve, yet they still receive 50% of the protocol's aggregate fee revenues. And there is a probability that they will vote for low-fee earning pools. Thus, it is hard for the project to incentivize the high-earning pools.
Ve-lockers have the right to decide whether the pool will be boosted
This approach is more decentralized and can create an AMM war, like Convex and Curve.
Ve-lockers can also earn 100% of all fees generated on only pools they vote for
Participants are incentivized to vote for the pools with the highest fees as that is the only way to receive cash flows from the protocol.
New AMM - The first product of ve(3,3)
Ve(3,3) introduced the first AMM designed with a P2P (protocol-to-protocol) model - Solidly. Solidly will be able to integrate the AMM, and the AMM will also have a Uniswap v2 compatible interface. Therefore, the veNFTs (which represent 25% of the Total Supply) distribution is based on the Total Value Locked (TVL) of the top 25 protocols on Fantom.
Conclusion
Ve(3,3) is an attempt to allow DeFi applications to adjust the inflation rate of native tokens according to the way of community co-governance rather than purely market action. What Andre and Dani are building over at Solidly seems special. There are tons of innovative ideas going into this protocol, and it could truly be a game-changer by being a P2P AMM. I believe that Solidly Exchange could bolster the whole Fantom ecosystem with it. Next time, we will talk about Solidly Exchange in more detail.
Resources
[1] Andre Cronje - DeFi Architect: Creator of YFI & Keep3rV1 , Alexandria (coinmarketcap.com), accessed 5th December 2022.
[2[ Who Is Daniele Sestagalli?, Crypto Briefing, accessed 5th December 2022.
[3] ve(3,3), medium.com, accessed 6th December 2022.
[4] ve(3,3) Ouroboros: Part 1 - Fee Distribution, medium.com. accessed 6th December 2022.
[5] ve(3,3) Ouroboros: Part 2 - Fees explored, medium.com, accessed 6th December 2022.
[6] ve(3,3): Curves, Initial distribution, Competition, & Building a protocol for protocols, medium.com, accessed 6th December 2022.
[7] ve(3,3): An Introduction into the New and Ambitious Solidly Exchange, substack.com, accessed 6th December 2022.
[8] FAQ - Olympus, Olympusdao.finance, accessed 7th December 2022.
[9] Understanding Voting - Curve Finance, resources.curve, accessed 7th December 2022.
[10] Curve Turns Off Mochi's Rewards After 'Amazingly Scammy' Tactics Trigger Outcry, finance.yahoo.com, accessed 7th December 2022. | 2023-02-05 01:16:00 | {"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": 0.2688722312450409, "perplexity": 6710.6217036175885}, "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-2023-06/segments/1674764500158.5/warc/CC-MAIN-20230205000727-20230205030727-00705.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/physics-for-scientists-and-engineers-a-strategic-approach-with-modern-physics-4th-edition/chapter-27-current-and-resistance-exercises-and-problems-page-763/36 | ## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)
The required diameter of the aluminum wire is $0.64~mm$
We can write a general expression for the resistance: $R = \frac{\rho~L}{A} = \frac{\rho~L}{\pi~r^2}$ We can write an expression for the resistance of the copper wire: $R_c = \frac{\rho_c~L}{\pi~r_c^2}$ We can write an expression for the resistance of the aluminum wire: $R_a = \frac{\rho_a~L}{\pi~r_a^2}$ Since the resistance must be equal, we can equate both expressions for the resistance: $R_c = R_a$ $\frac{\rho_c~L}{\pi~r_c^2} = \frac{\rho_a~L}{\pi~r_a^2}$ $r_a^2 = \frac{\rho_a}{\rho_c}~r_c^2$ $r_a = \sqrt{\frac{\rho_a}{\rho_c}}~r_c$ $r_a = \sqrt{\frac{2.8\times 10^{-8}~\Omega~m}{1.7\times 10^{-8}~\Omega~m}}~(0.25~mm)$ $r_a = 0.32~mm$ Since the diameter is twice the radius, the required diameter of the aluminum wire is $0.64~mm$ | 2020-03-31 23:06:15 | {"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": 0.7698096036911011, "perplexity": 183.63018515005288}, "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-2020-16/segments/1585370504930.16/warc/CC-MAIN-20200331212647-20200401002647-00273.warc.gz"} |
http://www.journalofinequalitiesandapplications.com/content/2012/1/128 | Research
Jensen's inequality for monetary utility functions
Jing Liu* and Long Jiang
Author Affiliations
College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu, 221116, People's Republic of China
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Journal of Inequalities and Applications 2012, 2012:128 doi:10.1186/1029-242X-2012-128
Received: 5 July 2011 Accepted: 7 June 2012 Published: 7 June 2012
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we prove that Jensen's inequality holds true for all monetary utility functions with respect to certain convex or concave functions by studying the properties of monetary utility functions, convex functions and concave functions.
Keywords:
monetary utility functions; Jensen's inequality; convex functions; concave functions
1 Introduction and preliminaries
1.1 Introduction
Monetary utility functions have recently attracted much attention in the mathematical finance community, see e.g. [1-4]. According to [2], a monetary utility function U can be identified with a convex risk measure ρ by the formula U(ξ) = - ρ(ξ); convex risk measure, introduced in [5,6], is a popular notion in particular since the Basel II accord. Some times we want to not only measure the utility of an uncertain random variable but also estimate the utility of its function (see e.g. 2.2). Jensen's inequality will be a useful tool solving this problem.
It is well-known that Jensen's inequality holds true for classical expectation, which, in terms of operator, can be seen as a particular type of monetary utility functions. But with respect to some convex or concave functions, it does not hold true for all monetary utility functions, as stated in our Example 2.1. This suggests a natural question: with respect to which kind of convex or concave functions does it hold true? In this paper, we study this question and give a sufficient and reasonable condition under which Jensen's inequality holds for all monetary utility functions.
1.2 Notations and assumptions
Let be a probability space. Denote by the collection of all real-valued essentially bounded -measurable random variables in . The Definition 1.1 and Remark 1.1 are cited from [2].
Definition 1.1. A function is called a monetary utility function if it is concave, non-decreasing with respect to the order of , satisfies the normalization condition U(0) = 0 and has the cash-invariance property (U(X + b) = U(X) + b for every and b ∈ ℝ).
Remark 1.1. The normalization U(0) = 0 does not restrict the generality as it may be obtained by adding a constant to U.
2 Jensen's inequality for monetary utility functions
In this section, we will show our main results and give two examples. For proving the results, we need Proposition 2.1 on monetary utility functions.
Proposition 2.1. Let be a monetary utility function. Then for any k ∈ ℝ, , we have
(2.1)
(2.2)
Proof.
(i) If 0 ≤ k ≤ 1, for the concavity of U, we have
Take Y = 0 and consider U(0) = 0, then we have
(ii) If k ≥ 1, then . By (2.1) we have
It follows that
(2.3)
If -1 ≤ k ≤ 0, then 0 ≤ -k ≤ 1. By (2.1) we have
Since
therefore
(2.4)
Hence
If k ≤ -1, then -k ≥ 1. By (2.3) we have
Combining the above inequality with (2.4) which is actually true for all k in R, we have
The proof of Proposition 2.1 is complete.
□
Now let us introduce the main results of this paper, i.e. Jensen's inequality for monetary utility functions.
Theorem 2.1. Let φ be a convex function on ℝ. Suppose that for any and . Then for any and any monetary utility function , we have φ(U(X)) ≤ U(φ(X)).
Proof. As stated in Definition 1.1, for every , U(X) is finite, i.e. , so we have
For any
based on the subgradient inequality in [7], we have
For the arbitrariness of x, we have
Consider the monotonicity and cash-invariance property of U and (2.1) in Proposition 2.1, then we have
Hence
□
It is also possible to obtain the Jensen inequality for monetary utility functions with respect to certain concave functions (Theorem 2.2) and prove it by the subgradient inequality in [7] and (2.2) in Proposition 2.1. As the proof is very similar with the proof of Theorem 2.1, we omit it and just give the result.
Theorem 2.2. Let ψ be a concave function onℝ. Suppose that for any or . Then for any and any monetary utility function , we have ψ(U(X)) ≥ U(ψ(X)).
Then let us illustrate the reasonableness of the conditions on convex and concave functions in Theorems 2.1 and 2.2 through an example. Actually, Jensen's inequality usually is not true for all monetary utility functions even when the related convex or concave function is a linear function.
Example 2.1. Let φ(x) = kx + a (k < 0 or k > 1) and ψ(x) = hx + b (0 < h < 1), obviously φ (respectively ψ) is a convex (respectively concave) function on ℝ that do not satisfy the condition in Theorem 2.1 (respectively Theorem 2.2). We consider a particular type of monetary utility function U, the entropic utility function in [4], which is defined as
(2.5)
We choose a such that . It is easy to check that U(kX) < kU(X) and U(hX) > hU(X). Then we have
Thus Jensen's inequality does not holds. So the conditions in Theorems 2.1 and 2.2 are reasonable.
At the end of this paper, let us discuss an application of Jensen's inequality for monetary utility functions.
Example 2.2. We still consider the entropic utility function. Sometimes, we want to estimate the entropic utility of X+ or -X- using the entropic utility of the future outcome X. For this kind of problem, Jensen's inequality will be a useful tool. Let φ(x) = x+, ψ(x) = -x-, then φ is a convex function satisfying the condition in Theorem 2.1 and ψ is a concave function satisfying the condition in Theorem 2.2 By Theorems 2.1 and 2.2 we have
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
JL study the properties of the functions, carried out the proof of the results and drafted the manuscript. LJ conceived of the study, designed the research method, and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their detailed comments and valuable suggestions. This research was supported by the National Natural Science Foundation of China (No. 10971220), the FANEDD (No. 200919), and the National Undergraduate Innovation Experiment Project of China (No. 091029030).
References
1. Delbaen, F, Peng, S, Rosazza Gianin, E: Representation of the penalty term of dynamic concave utilities. Finance and Stochastics. 14(3), 449–472 (2010). Publisher Full Text
2. Jouini, E, Schachermayer, W, Touzi, N: Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance. 18(2), 269–292 (2008). Publisher Full Text
3. Filipović, D, Kupper, M: Equilibrium Prices for Monetary Utility Functions. International Journal of Theoretical and Applied Finance(IJTAF). 11(3), 325–343 (2008). Publisher Full Text
4. Acciaio, B: Optimal risk sharing with non-monotone monetary functionals. Finance and Stochastics. 11, 267–289 (2007). Publisher Full Text
5. Föllmer, H, Schied, A: Convex measures of risk and trading constraints. Finance and Stochastics. 6(4), 429–447 (2002). Publisher Full Text
6. Frittelli, M, Rosazza Gianin, E: Putting order in risk measures. Journal of Banking & Finance. [http://www.geocities.ws/smhurtado/RiskMeasures.pdf] 26(7), 1473–1486 (2002). . PubMed Abstract | Publisher Full Text
7. Rockafellar, RT: Convex Analysis, Princeton: Princeton University Press (1970) | 2013-05-24 19:03:12 | {"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": 0.8424028754234314, "perplexity": 766.849564768447}, "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-2013-20/segments/1368704986352/warc/CC-MAIN-20130516114946-00032-ip-10-60-113-184.ec2.internal.warc.gz"} |
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# The first trenches that were cut into a 500-acre site at
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The first trenches that were cut into a 500-acre site at [#permalink]
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07 Aug 2010, 16:04
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The first trenches that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but independently of the more celebrated city-states of southern Mesopotamia, in what is now southern Iraq.
A. that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but
B. that were cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously with but also
C. having been cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously but
D. cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence of centrally administered complex societies in northern regions of the Middle East arising simultaneously but also
E. cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East arose simultaneously with but
[Reveal] Spoiler: OA
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Trying hard to achieve something unachievable now....
Last edited by dkverma on 08 Aug 2010, 03:42, edited 1 time in total.
If you have any questions
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07 Aug 2010, 17:32
10
KUDOS
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This post was
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Ok,I did POE and reached at E
First subject is trenches --- anywhere you see yields is out , B and D - out
left with A, C and E - having is almost always wrong on gmat - quickly eliminated C
A has got evidence for and E has got evidence that
evidence that is correct - So,E
Also, not sure whether were arising is correct in A , since the question talks about past - arose is better
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07 Aug 2010, 23:48
Yep it should be E.
Other options have glaring errors.
A is poorly constructed and doesn't convey the right meaning
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08 Aug 2010, 02:11
E is for me
trenches is plural, leave A,C,E option.
"cut into...." E is more clear than A.C
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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04 Mar 2012, 14:03
In learning of foundation of prounouns, such as that, I probably have lost the track here!
Isn't "that" suppose to refer to a singular idea/thing?
Q1) Is usage of that appropriate here ? And why?
The first trenches--- that ----were cut into a 500-acre site at Tell Hamoukar,
Shouldn't it be "those" instead of "that", since trenches - the subject is plural? That & were - how do they go along?!!!!!
Q2) Shouldn't it be "those" here, instead of "that"?
If it represents a feeling /noun, then that should follow singular very - is/was ----
"Egypt, have yielded strong evidence for blah blah societies in blah blah ?"that"? were arising simultaneously ....
Q3)
What does prounoun "that" refer to?
Soceities right? (not the middle east)
Then shouldn't it be plural - those??
"Egypt, have yielded strong evidence for blah blah societies in blah blah ?"that"? were arising simultaneously ....
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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04 Mar 2012, 21:28
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In learning of foundation of prounouns, such as that, I probably have lost the track here!
Isn't "that" suppose to refer to a singular idea/thing?
Q1) Is usage of that appropriate here ? And why?
Q2) Shouldn't it be "those" here, instead of "that"?
Q3)
What does prounoun "that" refer to?
Yes That is supposed to refer to the singular idea/thing.
A1: Usage of that is inappropriate because you already have a main subject "the trenches". Why would you want to reiterate a subject which already exists immediately after you have introduced it, right? Later on in the sentence it makes sense to refer to it but not immediately after introduction.
A2: No. We should not use either that or those so early on. This is again drawing on answer to point 1.
A3: Correct. That here refers to the complex societies. But is wrongly used. This is the reason why the correct answer eliminates all usage of that completely. There is no ambiguity of that or those at all.
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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04 Mar 2012, 22:40
Much thanks !
That was interesting !
However, I am still trying to buzz my mind around that/those.
i see some grammer pundits saying, "that" goes even with plural subject.
items produced by this machine are guaranteed to be accurately sized
vs.
items that are produced by this machine are guaranteed to be accurately sized
(there's no real difference here. so, the first one is marginally better, if only because it's more concise)
well, i understand, you dont want "that", immediately after subject.
but lets say if that's not the case....then
Would you justify use of pronoun "that", in such plural verb occasions?
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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04 Mar 2012, 22:50
OK, gotcha!!
There is a second point I forgot to mention and one that is tested on the GMAT quite often. The usage of "that" vs."which" as restrictive/non-restrictive pronouns.
-------------------------------------------
Source: GMAT Hacks website.
Here's a handy way to remember the rule: "That" is restrictive, while "which" is non-restrictive. The rule will make more sense after some further discussion.
Consider two similar sentences:
I'm staying at the hotel in Chicago that the Andersons operate.
I'm staying at the hotel in Chicago, which the Andersons operate.
------------------------------------------------
So, the next question: Do that and which refer to plurals or singulars?
They can refer to the whole sentence/noun phrase/noun/pronoun.
So, in summary, that is a pronoun, a restrictive prounoun and pronoun used for comparisons. You should be familiar with each of the usages for GMAT SC.
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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07 Mar 2012, 19:09
First of, thanks for the research, mourinhogmat1 !
Regarding
mourinhogmat1 wrote:
Consider two similar sentences:
I'm staying at the hotel in Chicago that the Andersons operate.
I'm staying at the hotel in Chicago, which the Andersons operate.
------------------------------------------------
So, the next question: Do that and which refer to plurals or singulars?
They can refer to the whole sentence/noun phrase/noun/pronoun.
So, in summary, that is a pronoun, a restrictive prounoun and pronoun used for comparisons. You should be familiar with each of the usages for GMAT SC.
Sorry, but I think "that" in above sentence is not used as pronoun. It's used as a starter for a modifier.
The which doesnt really care about the "the andersons operate" (modifier), but on it's antecedant hotel
Take a look at this -
[Incorrect] I am staying at various hotels, which Paris Hilton operates.
[Correct] I am staying at various hotels, that paris hilton operate.
I think instead of studying business, I am doing research on English grammer now
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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07 Mar 2012, 19:13
so I researched more (on english grammer, not in business), and this is what I think -
mourinhogmat1 wrote:
Yes That is supposed to refer to the singular idea/thing.
A1: Usage of that is inappropriate because you already have a main subject "the trenches". Why would you want to reiterate a subject which already exists immediately after you have introduced it, right? Later on in the sentence it makes sense to refer to it but not immediately after introduction.
It's perfectly fine to use that, immediately after the subject. Let it re-iterate itself
It's little idomatic-modifier start-up usage.
Welcoming suggestion from all gmat pundits here (including mourinhogmat1)
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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05 Feb 2013, 03:48
Hi guy
In all fairness, I think is wrong to attack such complex question in the way you did, with the use of that as pivot point.
Moreover, I'm honest to say that me too had problems simply because
- this kind of question is difficult only for the reason that when you read it from the beginning, when you are in the end of the phrase you already forgot where you stand: lost
- is important to understand the exact time line, without this process you always will pick such question wrong or at least you pick right but after five minute (during the exam the pressure blow your mind for sure) that is the same to pick it wrong.
Now back to the question: the acheologists do something NOW (cut a site into pieces) and discover something else (in that place complex societies took place) and the societies AROSE, in the past.
If you use arise or arising the societies seem still there. as an ongoing situation
cut (now) into a 500-acre site at Tell Hamoukar, Syria,have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East arose simultaneously (one time) with but and so on
If you do not understand clearly thi first split (the land were cut not in the past but NOW) by someone. They cut the land into acres $$NOW$$
Focus on the whole picture. Grammar is important but try to understand a macro vision of the sentence
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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05 Feb 2013, 06:52
Why don't we need "that" after trenches ?
We can use THAT after trenches (as in the case of option A & B) . We can use either relative clause or participle (among others) to modify a noun. It's not specific modifier that matters but the meaning of the sentence.
that were cut into a 500-acre site at Tell Hamoukar, Syria,
cut into a 500-acre site at Tell Hamoukar, Syria,
Both of the above colored portion are CORRECTLY modifying noun TRENCHES. But there are other error present in option A & B. The Errors are as follows:-
(A) that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but -
Evidence prove that societies AROSE (at one point of time in the past) but the evidences do not prove in any way the period/ process of evolution.
(In simple words, if we want to indicate a simple action in the past, use simple past tense as GMAT prefers simplicity & concision)
(B) that were cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously with but also -
Evidence prove that societies AROSE (at one point of time in the past) but the evidences do not prove in any way the period/ process of evolution. (In simple words, if we want to indicate a simple action in the past, use simple past tense as GMAT prefers simplicity & concision)
Simple present is generally used for facts, truth etc, thus use of YIELDS is incorrect over here
Fame
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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05 Feb 2013, 12:06
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targetgmatchotu wrote:
The first trenches that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but independently of the more celebrated city-states of southern Mesopotamia, in what is now southern Iraq.
(A) that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but
(B) that were cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously with but also
(C) having been cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously but
(D) cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence of centrally administered complex societies in northern regions of the Middle East arising simultaneously but also
(E) cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East arose simultaneously with but
Although I accept that "evidence for" is wrongly used and "evidence that" is the right usage, some discussions said about the use of "that"
Why don't we need "that" after trenches ?
How do we know that author is talking about "first trenches" or he is talking about the "first trenches that were cut into"?
And for the second "that" used after "Middle East" is there any reason why it is wrongly used other than that it modifies "middle east" ,although it should have modified "societies"
I'm happy to help with this. This is SC#70 from the OG13.
The opening choices ------
"The first trenches that were cut ...." ----- this modifies "trenches" with a subordinate clause, a clause beginning with "that". This is perfectly correct.
"The first trenches having been cut ...." --- participle with a strange tense, not correct
"The first trenches cut ...." ---- as fameatop pointed out above, this is participial phrase, also 100% correct. For more on participial phrases, see:
http://magoosh.com/gmat/2012/participle ... -the-gmat/
The difference between this would be like the difference between
(a) The horse that was traded for an electric guitar was now .....
(b) The horse, traded for an electric guitar, was now ....
(a) is a "that" clause construction, (b) is a participial construction, and both are correct.
You see, grammar is complex. You can just memorize a simple rule like don't drop the word "that" ----- There are two very different "that" clauses to consider.
Category #1: relative clauses
This is what appears in this sentence. Here, the word "that" acting as a relative pronoun -- others include who, whom, whoever, etc. Within the relative clause, the relative pronoun acts as a pronoun within the clause, often the subject of the clause. Let's look at (A) from the prompt ---- the relative clause is in green.
(1) The first trenches that were cut into a 500-acre site at Tell Hamoukar, Syria have yielded ...
Within that clause, the pronoun "that" is the subject of the clause, the subject of the verb "were cut."
Other examples includes
(2) The horse that was traded for an electric guitar was now ....
(3) The regions of Europe that Julius Caesar conquered were not .....
In #2, the word "that" is also the subject of the clause, now the subject of the verb "was traded." In #3, the word "that" is the direct object of the verb "conquered."
Nobody drops the "that" from a relative clause ----- since "that" is acts as a pronoun in the clause, it always sound terribly awkward to drop a pronoun. Pick any sentence with a pronoun, and say the sentence without the pronoun --- it will sound bizarre and incomplete Nobody makes this mistake. The dropping the "that" mistake is never a concern with relative clauses.
Category #2: substantive clauses
For more on this structure, read these two posts:
http://magoosh.com/gmat/2012/substantiv ... -the-gmat/
http://magoosh.com/gmat/2012/gmat-idiom ... ieve-that/
This is what we have following the word "evidence" in the SC sentence above ---- evidence that ..., know that ...., hope that ...., wish that ...., believe that ..... hypothesis that .... etc. etc. etc.
Here, the word "that" is followed by a full [noun] + [verb] clause. Examples, with substantive clause in green ----
(4) .... evidence that centrally administered complex societies in northern regions of the Middle East arose simultaneously with but independently of the more celebrated city-states of southern Mesopotamia, in what is now southern Iraq.
(5) The Declaration of Independence states that all men are created equal.
(6) The senator said that he will not seek reelection.
In all three cases, what follows "that" is a full clause --- in each case, we could extract the green section, throw away the word "that", and the rest of the green part could stand on its own as a full complete sentence. Here, the word "that" is NOT acting as a pronoun --- rather, it is serving to introduce a full clause. Because the word "that" plays no essential role within the clause, it is very tempting to drop it --- in fact, people do all the time in casual conversation, and the GMAT always tests this. This is where one has to have one's antennae up, looking for this very predictable mistake.
Does all this make sense?
Mike
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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05 Feb 2013, 14:56
Your explantion Mike is outstanding but in my opinion who study Sentence Correction more clinically than logically tend to struggle because their view becomes too technical.
I admit the importance and the distinction among the rules to clarify and consolidate the concepts.
however, on the upper level question to catch the gist of the sentence is more important.
Grammar rules lead you to the next level but after some point if you rely too much on rules, you lose the compass.
I didn't ask to myself where "that" standed for and the significance of it.
I understood that "cut" was the key and D was ackward and wrong because if you read the entire sentence it unfolds not so clearly.
Otherwise A student could be stuck in a limbo for endless time. my personal opinion, acceptable or not
For the rest thanks for the super super super amazing explanation
To completely understand what I mean here an article (it is like a windfall in this situation) from brian galvin - veritas prep
Fraud or Phenom In Sentence Correction
Best Regards
carcass
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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29 Aug 2013, 23:42
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2013gmat wrote:
1)that were cut(verb)
2)cut into(modifier)
then which one should I choose? and are there any specific rules for that???
First, meaning is key to solve this question
The trenches cut into something (Active voice)
--OR--
The trenches were cut into something? (Passive voice)
Clearly, the trenches only cut into something (how the trenches were cut (passive voice) into something? )
The first trenches that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but independently of the more celebrated city-states of southern Mesopotamia, in what is now southern Iraq.
A. that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but
Wrong. Passive voice is wrong.
B. that were cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously with but also
Wrong.
Same as in A. --> Passive voice is wrong.
Trenches is plural --> "yields" is wrong.
Wrong idiom: but also (the correct one is: not only... but also)
C. having been cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously but
Wrong. "having been cut" is ungrammatical.
D. cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence of centrally administered complex societies in northern regions of the Middle East arising simultaneously but also
Wrong idiom: but also (the correct one is: not only... but also)
E. cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East arose simultaneously with but
Correct.
Active voice "the trenches cut into something" <-- correct.
Contrast meaning: arose simultaneously with but independently.... <-- correct.
Hope it helps.
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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10 Sep 2013, 21:37
The first trenches ...
that were cut into a 500-acre site at Tell Hamoukar, Syria,
Contd...have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East
that( Refers to societies) were arising simultaneously with but independently of the more celebrated city-states of southern Mesopotamia, in what is now southern Iraq.
I have a question regarding the idiom "Evidence that", "Evidence for" and "Evidence of". Evidence that is the right idiom? It was between A and E ( I eliminated A cos it was passive voice) Others had subject-verb agreement issues ( yields)
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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11 Sep 2013, 08:50
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Let me give a different perspective about the use of active and passive voices here.
Quote:
The trenches that were cut
The trenches cut
IMO, both the above are passive voices. One may say that first one is wordier by two words. You might see that a trench has to be cut by somebody. It cannot cut itself or cut another. The cut is used in the sense of a past participle and not past tense at best; one may say that the first is wordier by two words
So whenever you use trench with the verb cut, it will always be in passive. However a trench can run along some route, when it will be in active voice. I hope this difference is realized
Therefore the reason that A is wrong is because of that idiom, evidence for; evidence that is the accepted idiom, So E wins.
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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12 Sep 2013, 00:57
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nelz007 wrote:
The first trenches ...
that were cut into a 500-acre site at Tell Hamoukar, Syria,
Contd...have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East
that( Refers to societies) were arising simultaneously with but independently of the more celebrated city-states of southern Mesopotamia, in what is now southern Iraq.
I have a question regarding the idiom "Evidence that", "Evidence for" and "Evidence of". Evidence that is the right idiom? It was between A and E ( I eliminated A cos it was passive voice) Others had subject-verb agreement issues ( yields)
Hi Nelson,
Let me address your confusion regarding the above mentioned usages of “evidence” with simple examples.
a. The police found evidence that Syrio was present at the crime scene when the crime happened.
b. The police found evidence for Syrio was present at the crime scene… (Police found evidence because Syrio was present at the crime scene. Changes the meaning).
c. The police found evidence of Syrio was present at the crime scene…(Completely incorrect. It doesn’t make sense to use a clause after “evidence of”)
d. There was ample evidence for the police to file a case against Syrio.
e. The police found no evidence of gun at the crime scene.
Notice how when “evidence” is followed by that, the “that clause” describes what that evidence in fact is. This is absolutely in line with how a typically noun modifier works. You have a noun that is followed by a that clause that explains this noun.
Likewise, when evidence is followed by “for” or “on”, you can see that it is followed by a noun.
So really speaking you do not need to think of “evidence” in terms of an idiom. It works in the same way as any other noun entity would work. But yes, whether evidence should be followed by “that” or by a preposition “for” or “on” depends on what you want to communicate through the sentence.
Hope this helps!
Regards,
Krishna
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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03 Dec 2013, 09:58
Clause breakup
1. the first trenches - NOUN
2. that were cut into a 500 acre site at ..syria , have yielded strong evidence for centrally administered complex societies in northern regions of the middle east –
Error -1 fragment error – two verb in one clause for one subject – S- V error
That correctly refers to the trenches but need to change that clause modifier to verb ed modifier to adjust S-V pair .
3.that were arising simultaneously with but independently of the more celebrated
city states of southern Mesopotamia, in
Error 2– that is used to modify preceding clause that incorrectly modifies the preceding noun Middle east only where it should modify the whole preceding clause
Error 3 – As per OG rule for greater clarity and concision, 2 DC [Subordinate] structure should be condensed into one
4. what is now southern Iraq.- Non underlined portion.
1. plural subject – trenches and have yielded ok - S-V pair ok
2. verb tense ok .
3. S-V pair error
4. Modifier placement error
Is the analysis ok ? Want to know except A why B,C,D are incorrect ?
A. that were cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence for centrally administered complex societies in northern regions of the Middle East that were arising simultaneously with but
B. that were cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously with but also
C. having been cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East were arising simultaneously but
D. cut into a 500-acre site at Tell Hamoukar, Syria, yields strong evidence of centrally administered complex societies in northern regions of the Middle East arising simultaneously but also
E. cut into a 500-acre site at Tell Hamoukar, Syria, have yielded strong evidence that centrally administered complex societies in northern regions of the Middle East arose simultaneously with but[/quote]
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Re: The first trenches that were cut into a 500-acre site at [#permalink]
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04 Dec 2013, 04:14
not only... but also
to say about 2 thing similar
...but ...
to say about two things different, a contrast.
try to understand meaning to use words, a game on sc.
noun+having+done, never exist in english grammar and never exist on gmat land. this is a hard and fast rule which we can use to eliminate an answer choice without reading/understanding the meaning.
if we see a split
noun +(noun+that clause)
vs
noun +that clause.
then the focus of meaning is changed. normally the under pattern is correct.
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Re: The first trenches that were cut into a 500-acre site at [#permalink] 04 Dec 2013, 04:14
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https://www.taho.cc/c/progressive/science | ## 数学：K函数
In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function.
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It can also be given in closed form as
where ζ'(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and
Another expression using polygamma function is[1]
where A is Glaisher constant.
The K-function is closely related to the gamma function and the Barnes G-function; for natural numbers n, we have
More prosaically, one may write
The first values are
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While Einstein’s general relativity has been well tested by various observations, it is possible that at the cosmological scale it might require some modification or extension, which could, in turn, explain dark matter or dark energy. There exist a plethora of modified theories of gravity, from simple straightforward generalizations of the Hilbert-Einstein action (e.g. f(R) gravity), to utilizing other geometric structures (e.g. f(T) gravity that models gravity as spacetime torsion instead of curvature), it is important to constrain these theories from both observations and theoretical grounds (a good theory should be mathematically self-consistent and free of pathologies). [Figure source: Tessa Baker]
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iData 是第三方交流学术成果的公益互联网项目，所有信息均来自公开、透明的互联网查询网站，号称是全球最大的知网镜像网站。它从一开始便提供免费的学术文献浏览和下载。
## 物理学：如何理解黑洞的图像
4月10日晚（今晚），第一幅真实的黑洞图像就会被公布。这个视频是最近发布的（发布于昨天），讲解了如何理解黑洞的图像。 | 2019-11-17 22:25:52 | {"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": 0.930414617061615, "perplexity": 2233.236641540118}, "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-2019-47/segments/1573496669352.5/warc/CC-MAIN-20191117215823-20191118003823-00552.warc.gz"} |
https://www.freemathhelp.com/forum/threads/averaging-percentages-whats-the-trend.109594/ | # Averaging Percentages: What's the trend?
#### cmbarona
##### New member
I'm trying to find the best way to present average giving to an organization based on how much had been given in previous years, but I realized that different methods of averaging give me different percentages. I understand why I get different figures, but I'm having a hard time wrapping my head around why that is and what those different figures mean. To give context, I'm comparing year-to-date giving across the five previous years to see where we should be by this point in our current year. So, for example, I should be able to say we should have received x% of our giving by this point in the current year, and therefore make a recommendation of how we should focus future efforts for the rest of the year. I'm applying this problem across a number of different groups of donors, but the core of the problem remains the same.
I'll refer to the raw data in the following way: C=Current (Year-to-date) giving. C1-C5 refer to the five previous years, and CN refers to that giving in the current year (N=Now). T=Total giving, so T1-T5 refer to the total giving in those previous years. What I'm trying to find is PN, or the percentage of total giving I should have in the current year. I'm using the percentage I'll generate to compare to two different figures: the average of total giving in the five previous years, and the goal set at the beginning of the current year. Which of the following methods would be best, or should I use a different method? What are the implications for the different methods?
1. Method 1: Average the percentages. Find P1-P5. That is, C1/T1=P1. After this is done for all five previous years, (P1+P2+P3+P4+P5)/5=PN
2. Method 2: Percent of the averages. ((C1+C2+C3+C4+C5)/5)/((T1+T2+T3+T4+T5)/5)=PN
3. Method 3: Percent of raw figures (no averaging). (C1+C2+C3+C4+C5)/(T1+T2+T3+T4+T5)=PN
#### j-astron
##### Junior Member
Hi cmbarona,
Methods 2 and 3 should give you the exact same answer, because the factor of 1/5 cancels from both the numerator and denominator of Method 2, and you just end up with the ratio from Method 3.
Neither of these is what you want. If the question you're trying to answer is "by this point in the year, what percentage of our annual earnings have we typically received?", then Method 1 provides the answer to that question. You're averaging the percentages, which gives you what fraction of the total annual earnings you've earned YTD (year-to-date), on average, over the last 5 years.
To illustrate why Method 2 is not what you want, it's because the average of the absolute YTD earnings in two different years is misleading, since YTD earnings would be very different in different years if the total donation amount were very different in those two years. I think this is best illustrated with a very simplistic example, let's say you got only $1000 in year 1, but$5000 in year two. Furthermore, let's say that by the end of June, you had $750 in year 1 and$2500 in year 2. So, to summarize:
C1 = $750 C2 =$2500
T1 = $1000 T2 =$5000
Using Method 1: P1 = 750/1000 = 0.75, and P2 = 2500/5000 = 0.5, and therefore P_avg = P1+P2/2 = (0.75+0.5)/2 = 1.25/2 = 0.625
Here we got an answer that is representative what's actually happening. By this point in year1 you had 75% of your donations, and by this point in year 2, you had 50% of your donations, so on average, you get about 62.5% of your donations by mid-year
Using Method 2: C_avg = (C1 + C2)/2 = (750 + 2500)/2 = 3250/2 = 1625
T_avg = (T1 + T2)/2 = (1000 + 5000)/2 = 6000/2 = 3000
P_avg = C_avg / T_avg = 1625/3000 = 0.54
So, despite the fact that in year 1, you got 75% of your donations by mid-year (which is a lot more than half) you're reporting an average mid-year percentage earnings of only 54%. That's clearly not right. Your average percentage value is low because the absolute earnings in year 1 were low, illustrating that this is not the right statistic.
One final suggestion: you could considering quantifying this using more than just a single number. If you have YTD earnings for every single month of the year, for several years, you could make a plot (a graph) of the cumulative percentage earnings vs. time for each year, and compare them. Maybe in some years it increases towards 100% very steeply at first, because most donations came in early in the year and then it tapers off. Or maybe, every year, it's close to a straight line (with constant slope), because the rate of donations is steady over the year: you get about the same amount coming in every month). That graph, I think, would be very informative to look at.
#### cmbarona
##### New member
Thanks, that makes a lot of sense now.
This brings me to part 2 of my question. Is there a meaningful percentage I can present to show variance from goal, and if so, how do I calculate it? Keep in mind, I'll project this based on two figures, a 5-year average of total yearly giving and goals we set at the beginning of the year.
So, let's say by this point in the year we expect to have received 60% of total yearly donations. And let's say the average giving over the last 5 years is $750k, but we set an optimistic goal for this year of$1M.
Adjusted goal for 5-year average is therefore .6 * 750000 = 450,000. And the adjusted goal for the goal we set is .6 * 1000000 = 600,000. Easy enough, we've got our benchmarks now.
Now, let's compare actual current giving to those benchmarks. Let's say we've raised $500k thus far this year. We're$50k ahead of the average, but $100k behind our goal. Here's the question. Is there any meaningful way to represent that +$50k and -$100k in their own percentages? How would I calculate it? #### j-astron ##### Junior Member Now, let's compare actual current giving to those benchmarks. Let's say we've raised$500k thus far this year. We're $50k ahead of the average, but$100k behind our goal.
Here's the question. Is there any meaningful way to represent that +$50k and -$100k in their own percentages? How would I calculate it?
I don't know if I fully understand what you're trying to do, but I could take a stab at it:
500k/450k = 1.111... or 111% of the 5-year YTD giving average. So you could say you're up 11% from the average.
In contrast, 500k/600k is only 0.8333... = 83.3% of your YTD goal for this year. So you're up 11% from the average, but down 16.67% from this year's goal.
Is that the sort of thing you'd like to be able to report? 'Cause it could make equal sense to just report the surplus or shortfall in absolute dollar amounts, depending on the point of all this.
#### cmbarona
##### New member
I am reporting absolute dollar amounts, but bear in mind this report is done across different donor groups, and the people who see it - who are not knee-deep in data all day - are more likely to be able to wrap their heads around what the different figures mean if I've got a meaningful percentage I can attach to them. For example: "overall donations are behind 10%, and most donor groups are close to that figure, but donations from graduates are down 30%; therefore, we should refocus efforts to reach graduates and investigate what campaigns have been successful in the past." Using absolute dollar figures is useful, but it's hard to show meaningful deviation from expectations if we just use dollar figures. They vary enough that some other statistical figure would be more useful, I think. What I'm doing now is to compare the average percentage expected by this point in the year to the percent of either the projected 5-year average or the yearly goal.
Examples: We've received $500k. That's 66% of the$750k average year-end giving, and 50% of the $1M goal. Now, we expect to have 60% of all giving in by this point of the year, so that means we're (66%-60%) = 6% ahead of average giving, but (50%-60%) = -10% behind our yearly goals. Again, it's not so much the math that's tripping me up, but what the different methods imply about the data. This method obviously differs substantially from the method you suggested, but what do the different figures mean? #### j-astron ##### Junior Member I am reporting absolute dollar amounts, but bear in mind this report is done across different donor groups, and the people who see it - who are not knee-deep in data all day - are more likely to be able to wrap their heads around what the different figures mean if I've got a meaningful percentage I can attach to them. For example: "overall donations are behind 10%, and most donor groups are close to that figure, but donations from graduates are down 30%; therefore, we should refocus efforts to reach graduates and investigate what campaigns have been successful in the past." Using absolute dollar figures is useful, but it's hard to show meaningful deviation from expectations if we just use dollar figures. They vary enough that some other statistical figure would be more useful, I think. What I'm doing now is to compare the average percentage expected by this point in the year to the percent of either the projected 5-year average or the yearly goal. Examples: We've received$500k. That's 66% of the $750k average year-end giving, and 50% of the$1M goal. Now, we expect to have 60% of all giving in by this point of the year, so that means we're (66%-60%) = 6% ahead of average giving, but (50%-60%) = -10% behind our yearly goals. Again, it's not so much the math that's tripping me up, but what the different methods imply about the data. This method obviously differs substantially from the method you suggested, but what do the different figures mean?
Yup, the challenging part of stats is often not doing the calculations themselves, but figuring out which calculations to do in the first place in order to accurately represent what you're trying to show.
Your percentages make perfect sense and are meaningful:
You're at 50% of your yearly goal
Normally by this point in the year, you're at 60% of your yearly goal, so you're 10% behind in terms of proportion of received donations YTD
However, your yearly goal is optimistic. You're at 67% of the 5-year average yearly earnings, so you're actually outpacing the average rate of earnings by a bit (66.667% / 60% = 11% higher rate of earnings).
I'm gonna think out loud and make some assumptions here. Let's assume the rate of earning is constant throughout the year. So normally you make $750k/12 =$62,500 monthly. If you like, this is the monthly average. However, to meet your new goal, you'd have to make a monthly average of $1M/12 =$83,333.33. That's 20,833 more per month. It's 33% more per month. It's 33% more in any time interval.
So...in this case, YTD, you normally make $450k, but you actually have made$500k. That's $50k more, which is only 11% more. You need to have made 33% more. So your surplus over the past year's average YTD earnings is only 11/33 = 1/3 of what it needs to be. It needed to be$150k to get you from 450k to 600k. Instead it's only 50k.
So yeah, this is an interesting way of looking at it. Your rate of earnings is so far 1.11x past years, but needs to be 1.33x past years to meet your optimistic goal. If you project this rate of earnings to year end, you'll predict that you'll make 1.11 * 750k = \$833k. You'll be at 83.3% of your optimistic goal, and thus fall short of it by 16.7%. (Note that 1.11/1.33 = 0.83). So all the numbers we've computed have come up and mean something. Which ones you use depend on what question you are trying to answer
Last edited: | 2019-08-24 12:04:04 | {"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": 0.6474083662033081, "perplexity": 1036.6221368655065}, "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/1566027320734.85/warc/CC-MAIN-20190824105853-20190824131853-00019.warc.gz"} |
http://andresmendes.github.io/Vehicle-Dynamics-Lateral/html/DocTireLinear.html | # Linear tire model
Linear relationship between tire lateral force and slip angle.
The code of this class can be found in TireLinear. It inherits methods from abstract class Tire. The complete list of class codes is in API.
## Sintax
The class LinearTire has two methods
Fy = TireModel.Characteristic(alpha)
TireModel.PlotTire()
## Arguments
The following table describes the input arguments
## Outputs
Fy Tire lateral force [N]
## Description
The lateral force of the tire can be calculated as
$F_y = K \alpha$
where $$F_y$$ is the lateral force, $$K$$ is the cornering stiffness and $$\alpha$$ is the tire slip angle.
Hypothesis
• Linear tire model
• Valid only for small values of slip angle | 2018-07-16 00:36:44 | {"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": 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": 0.36322712898254395, "perplexity": 3304.29800657242}, "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-2018-30/segments/1531676589029.26/warc/CC-MAIN-20180716002413-20180716022334-00030.warc.gz"} |
http://c-s-a.org.cn/html/2021/1/7742.html | 计算机系统应用 2021, Vol. 30 Issue (1): 122-128 PDF
Image Inpainting Based on New Encoder and Similarity Constraint
LIN Zhu, WANG Min
College of Computer and Information, Hohai University, Nanjing 211100, China
Abstract: The existing image repair methods have some problems such as obvious trace, semantic discontinuity, unclear, etc. To solve these problems, this study proposes an image repair method based on a new encoder and context-aware loss. In this paper, the generative adversarial network is adopted as the basic network architecture. In order to fully learn the image features and get clearer repair results, SE-ResNet is introduced to extract the effective features of the image. At the same time, the joint context-aware loss training generating network is proposed to constrain the similarity of local features, so that the repaired image is closer to the original and more real and natural. Experiments on multiple public datasets in this paper prove that the proposed method can repair the damaged images better.
Key words: generative adversarial network image inpainting residual network contextual loss
(1)在生成网络和全局上下文以及局部上下文判别网络部分添加了基于SE-ResNet的残差块更好的提取特征.
(2)增加了上下文感知损失网络以辅助约束局部高频特征的相似性来修复图像.
1 相关内容
Yu等人[10]提出一种端到端的图像修复模型, 通过采用一种堆叠型的生成网络确保与周边颜色以及纹理的连贯性, 同时引入了注意力模块从距离较远的区域提取近似待修复区域的特征.
Liu等人[11]提出通过在卷积过程中更新掩膜并使用更新的掩膜值归一化卷积核的权重值, 保证卷积核能够专注于有效的像素值.
Yu等人[12]通过引入门控卷积, 学习一种特征通道的动态选择机制, 以提高色彩的一致性, 同时提出一种高效的判别器SN-PatchGAN用于辅助修复随机缺失的图像.
2 网络结构 2.1 SE-ResNet
2.2 生成网络
${L_{\rm {adv}}} = - {E_{x \sim {p_r}(x)}}D(G(M \odot x))$ (1)
图 1 生成网络结构图
${L_{\rm {res}}} = - {E_{x \sim {p_r}(x)}}[{\left\| {M \odot (x - G(M \odot x))} \right\|_2}]$ (2)
${L_{ {CX}}} = - \log [CX(\Phi (x),\Phi (G(M \odot x)))]$ (3)
${L_{\rm {res}}} + {\lambda _1}{L_{\rm {adv}}} + {\lambda _2}{L_{{CX}}}$ (4)
2.3 判别网络
图 2 判别网络结构图
${L_{\rm {dis}}} = - {E_{x \sim {p_r}}}[\log (D(x)) + \log (1 - D(G(x)))]$ (5)
2.4 上下文感知损失
${L_{ {CX}}} = - \log [CX(\Phi (x),\Phi (G(M \odot x)))]$ (6)
$CX(x,y) = CX(X,Y) = \frac{1}{N}\sum\limits_j {\mathop {\max }\limits_i } C{X_{ij}}$ (7)
$C{X_{ij}} = {w_{ij}}\Bigg/\sum\limits_k {{w_{ik}}}$ (8)
${w_{ij}} = \exp \left( {\dfrac{{1 - {d_{\rm {similar}}}}}{h}} \right)$ (9)
${d_{\rm {similar}}} = \frac{{{d_{ij}}}}{{{{\min }_k}{d_{ik}} + \varepsilon }}$ (10)
${d}_{ij}$ 归一化, 其中 ${d}_{ij}$ ${x}_{i}$ ${y}_{j}$ 的余弦距离. 上述 ${d}_{ij}$ 计算公式为:
${d_{ij}} = \left( {1 - \frac{{({x_i} - {\mu _y}) \cdot ({y_j} - {\mu _y})}}{{{{\left\| {{x_i} - {\mu _y}} \right\|}_2}{{\left\| {{y_j} - {\mu _y}} \right\|}_2}}}} \right)$ (11)
${\mu _y} = \frac{1}{N}\sum\limits_j {{y_j}}$ (12)
3 实验 3.1 数据集
3.2 训练过程
3.3 SE-ResNet的效果分析
图 3 添加SE-ResNet残差块与否对比图
3.4 上下文感知损失的效果分析
图 4 采用上下文感知损失与否对比图
3.5 与现有方法的比较
图 5 与文献[3]方法的中心缺失修复效果对比图
图 6 与文献[3]方法的随机缺失修复效果对比图
图 7 边缘检测图对比
4 结束语
[1] Darabi S, Shechtman E, Barnes C, et al. Image melding: Combining inconsistent images using patch-based synthesis. ACM Transactions on Graphics, 2012, 31(4): 82. [2] Pathak D, Krahenbühl P, Donahue J, et al. Context encoders: Feature learning by inpainting. Proceedings of 2016 IEEE Conference on Computer Vision and Pattern Recognition. Las Vegas, NV, USA. 2016. 2536–2544. [3] Iizuka S, Simo-Serra E, Ishikawa H, et al. Globally and locally consistent image completion. ACM Transactions on Graphics, 2017, 36(4): 107. [4] Hu J, Shen L, Sun G. Squeeze-and-excitation networks. Proceedings of 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. Salt Lake City, UT, USA. 2018. 7132–7141. [5] He KM, Zhang XY, Ren SQ, et al. Deep residual learning for image recognition. Proceedings of 2016 IEEE Conference on Computer Vision and Pattern Recognition. Las Vegas, NV, USA. 2016. 770–778. [6] Mechrez R, Talmi I, Zelnik-Manor L, et al. The contextual loss for image transformation with non-aligned data. Proceedings of the 15th European Conference on Computer Vision. Munich, Germany. 2018. 800–815. [7] Simonyan K, Zisserman A. Very deep convolutional networks for large-scale image recognition. arXiv: 1409.1556, 2014. [8] Goodfellow I, Pouget-Abadie J, Mirza M, et al. Generative adversarial nets. Proceedings of the 27th International Conference on Neural Information Processing Systems. Cambridge, MA, USA. 2014. 2672–2680. [9] Song YH, Yang C, Shen YJ, et al. SPG-Net: Segmentation prediction and guidance network for image inpainting. arXiv: 1805.03356, 2018. [10] Yu JH, Lin Z, Yang JM, et al. Generative image inpainting with contextual attention. Proceedings of 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. Salt Lake City, UT, USA. 2018. 5505–5514. [11] Liu HY, Jiang B, Xiao Y, et al. Coherent semantic attention for image inpainting. Proceedings of 2019 IEEE/CVF International Conference on Computer Vision. Seoul, Republic of Korea. 2019. 4169–4178. [12] Yu JH, Lin Z, Yang JM, et al. Free-form image inpainting with gated convolution. Proceedings of 2019 IEEE/CVF International Conference on Computer Vision. Seoul, Republic of Korea. 2019. 4471–4480. [13] Yu F, Koltun V. Multi-scale context aggregation by dilated convolutions. arXiv: 1511.07122, 2015. [14] Liu ZW, Luo P, Wang XG, et al. Large-scale celebfaces attributes (CelebA) dataset. http://mmlab.ie.cuhk.edu.hk/projects/CelebA.html, 2018. [15] Huang GB, Ramesh M, Berg T, et al. Labeled faces in the wild: A database for studying face recognition in unconstrained environments. Amherst: University of Massachusetts, 2007. [16] 佟雨兵, 张其善, 祁云平. 基于PSNR与SSIM联合的图像质量评价模型. 中国图象图形学报, 2006, 11(12): 1758-1763. DOI:10.11834/jig.2006012307 [17] Horé A, Ziou D. Image quality metrics: PSNR vs. SSIM. Proceedings of the 20th International Conference on Pattern Recognition. Istanbul, Turkey. 2010. 2366–2369. | 2021-03-08 03:02:47 | {"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": 0.7680411338806152, "perplexity": 11912.493682272145}, "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/1614178381803.98/warc/CC-MAIN-20210308021603-20210308051603-00183.warc.gz"} |
http://www.math.nus.edu.sg/~mattohkc/SNLSDP.html | Kim-Chuan Toh , Pratik Biswas, and Yinyu Ye
The software was last updated in 21 Oct 2008.
It implemented an SDP based approach with regularization for solving sensor network localization problems. The algorithm first solves an SDP relaxation (with regularization) of the non-convex minimization problem (1), and use the SDP computed solution as the starting point for a gradient descent method with backtracking line search to solve the smooth unconstrained problem (2).
This software package is designed for solving small size senor network localization problems with up to 200 sensors and a few thousands given distances.
$$(1)\quad \min \big\{ \sum_{(i,j)\in {\cal E}} | \|x_i-x_j\|^2-d_{ij}^2| + \sum_{(k,j)\in {\cal F}} | \|a_k-x_j\|^2-d_{kj}^2 | \big\} \\[10pt] (2)\quad \min \big\{ \sum_{(i,j)\in {\cal E}} ( \|x_i-x_j\|-d_{ij})^2 + \sum_{(k,j)\in {\cal F}} (\|a_k-x_j\|-d_{kj})^2 \big\}$$
where $$d_{ij}, d_{kj}$$ are distance data, $$x_j$$ is the position of the jth sensor, and $$a_k$$ is the position of the kth anchor.
If you find SNLSDP useful in your work, please cite the following paper:
[1] P. Biswas, T.-C. Liang, K.-C. Toh, T.-C. Wang, and Y. Ye, Semidefinite programming approaches for sensor network localization with noisy distance measurements, IEEE Transactions on Automation Science and Engineering, 3 (2006), pp. 360--371. | 2016-02-13 21:15:40 | {"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": 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": 0.5692810416221619, "perplexity": 1453.8215221969624}, "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-2016-07/segments/1454701168011.94/warc/CC-MAIN-20160205193928-00014-ip-10-236-182-209.ec2.internal.warc.gz"} |
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Fluorine is one of the most reactive elements. In nature, halogens always exist as F 2, Cl 2, Br 2, I 2 and At 2. Properties of the Halogens. It is a corrosive and highly toxic gas. Trend of change in the physical properties However, some of the physical properties mentioned above vary gradually when going down Group 17, … CC BY-SA 3.0. http://en.wikibooks.org/wiki/A-level_Chemistry/AQA/Module_2/Group_VII:_The_Halogens%23Physical_Properties ThoughtCo uses cookies to provide you with a great user experience. The halogens include fluorine (F), chlorine (Cl), bromine (Br), iodine (I), and astatine (At). Wikipedia Liquid bromine has a high vapor pressure, and the reddish vapor is readily visible in Figure 18.60. Halogens react with metals to form salts. These reactive nonmetals have seven valence electrons. Halogens share many similar properties including: They all form acids when combined with hydrogen. So group seven, aka the halogens. They gain electrons very fast making them most reactive of all chemical elements. Therefore, the physical state of the elements down the group changes from gaseous fluorine to solid iodine. The interhalogens of form XY have physical properties intermediate between those of the two parent halogens. The boiling point increases moving down the group because the Van der Waals force is greater with increases size and atomic mass. Thus the colour of the Astatine … Common properties of Halogens The elements classed as Halogens have the following properties in common: They are non-metals; Low melting and boiling points; Brittle when solid; Poor conductors of heat and electricity; Have coloured vapours; Their molecules … % Progress . Halogens can gain an electron by reacting with atoms of other elements. Fluorine is pale yellow, chlorine is green, bromine is orange and iodine is grey. Iodine crystals have a … Fluorine, chlorine, bromine, iodine, and astatine definitely are halogens. This particular resource used the following sources: http://www.boundless.com/ Boundless vets and curates high-quality, openly licensed content from around the Internet. They have relatively low melting and boiling points that increase steadily down the group. They readily combine with metals to form salts. General properties of halogens Physical properties. Even so, it will share some common properties with the other elements in its group. The covalent bond between the two atoms has some ionic character, the less electronegative halogen, X, being oxidised and having a partial positive charge. They have a valence of 1 and form covalent bonds with non-metals atoms, or ionic bonds with metal atoms. Therefore, most of the chemical reactions that involve halogens are oxidation-reduction reactions in aqueous solution. The halogens have the following properties: They are non-metals stable as diatomic molecules (this means at room temperature and pressure, they exist as molecules made of two atoms , e.g. Elements typically become more metallic or basic on descending a main group. Wiktionary Types of Halogens . CC BY-SA 3.0. http://en.wiktionary.org/wiki/halogen Fluorine and chlorine are in the gaseous state, bromine in liquid and iodine in the solid state. They can be found toward the right-hand side of the table, in a vertical line. Halogen, any of the six nonmetallic elements that constitute Group 17 (Group VIIa) of the periodic table. Chemical Properties of Halogens: They exist in all three classical states of matter – solid, liquid and gas. Properties of the Halogens Halogens (fluorine, chlorine, bromine, iodine, astatine) are nonmetal elements that are highly electronegative and reactive. CC BY-SA 3.0. http://en.wiktionary.org/wiki/electronegativity It slowly reacts to form hydrogen bromide (HBr) and hypobromous acid (HBrO): $Br_2 (g) + H_2O (l) \rightarrow HBr (aq) + HBrO (aq)$. The group of halogen Liquid bromine has a high vapor pressure, and the reddish vapor is readily visible in the figure below. Properties of the Halogens Fluorine is a pale yellow gas, chlorine is a greenish-yellow gas, bromine is a deep reddish-brown liquid, and iodine is a grayish-black crystalline solid. Oxidizing power: An important feature of the halogen is their oxidizing property which is due to high electron affinity of halogen atoms. Chlorine has maximum solubility of 7.1 g per kg of water at ambient temperature (21 °C). The elements in group 7 are called the halogens. Fluorine is a pale yellow gas, chlorine is a greenish-yellow gas, bromine is a deep reddish-brown liquid, and iodine is a grayish-black crystalline solid. Note: It is not easy for non-metals like halogens to form cations. It oxidizes other halide ions to halogens in solution or when dry. The Halogens. In these compounds, halogens are present in the form of halide anions with a charge of -1 (for example, Cl -, Br -).The ending -id indicates the presence of halide anions; for example, Cl is called “chloride”.. Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a science writer, educator, and consultant. All halogens form salts of group I with similar properties. Fluorine has the highest electronegativity of all elements. She has taught science courses at the high school, college, and graduate levels. The halogens have very high electronegativities. As a group, halogens exhibit highly variable physical properties. Halogen, any of the six nonmetallic elements that constitute Group 17 (Group VIIa) of the periodic table. From the table of physical properties it can be inferred that the depth of colour of the halogens increases in atomic number. Halogens are very reactive because they have seven valence electrons and need one more to have eight valence electrons (an octet). The chlorine ion, usually obtained from table salt (NaCl) is essential for human life. … They share similar chemical properties. Properties of the Halogens. Fluorine is the most active halogen, and astatine is the least. What Element in the Halide Family is a Liquid? CC BY-SA 3.0. http://en.wikipedia.org/wiki/Halogen Astatine is the rarest naturally-occurring element. As a diatomic molecule, fluorine has the weakest bond due to repulsion between electrons of the small atoms. Electronegativity is the ability of an atom to attract electrons or electron density towards itself within a covalent bond. As a group, halogens exhibit highly variable physical properties. electronegativityThe tendency of an atom to attract electrons to itself. Iodine is minimally soluble in water, with a solubility of 0.03 g per 100 g water. Therefore, they are highly reactive and can gain an electron through reaction with other elements. Learn more about the properties of halogens in this article. Liquid bromine has a high vapor pressure, and the reddish vapor is readily visible in Figure $$\PageIndex{1}$$. Near room temperature, the halogens span all of the physical states: Fluorine and chlorine are gases, bromine is a liquid, and iodine is a solid. Wikipedia Physical Properties Electronegativity depends upon the attraction between the nucleus and bonding electrons in the outer shell. This, in turn, depends on the balance between the number of protons in the nucleus, the distance between the nucleus and bonding electrons, and the shielding effect of inner electrons. Fluorine is the strongest oxidizing agent. Cl 2 ). Chemical properties of Halogens. Halogens display physical and chemical properties typical of nonmetals. Iodine crystals have a noticeable vapor pressure. The Periodic Table - the Halogens. When halogens combine or react with metals, they form ionic bonds. They are highly reactive, especially with alkali metals and alkaline earths. Toxicity decreases with heavier halogens until you get to astatine, which is dangerous because of its radioactivity. Chemical Properties of HALOGEN. They have seven valence electrons (one short of a stable octet). Reason: the ionization energy (amount of energy required to lose an electron(s) from the outermost energy level of a gaseous atom) is very large. The Halogens exhibit some very interesting properties in the periodic table. In addition, the chemical properties of halogens allow them to act as oxidizing agents - to oxidize metals. It is the only element group that includes elements capable of existing in three of the four main states of matter at room temperature: solid, liquid, and gas. CC BY-SA 3.0. http://en.wikipedia.org/wiki/File:Halogens.jpg The halogens are the only periodic table group containing elements in all three familiar states of matter (solid, liquid, and gas) at standard temperature and pressure. The halogens are a group of elements on the periodic table. Today the two in between: bromine and iodine. ALFRED PASIEKA / SCIENCE PHOTO LIBRARY / Getty Images. Group 7 is also known by its more modern name of Group 17. What Are the Properties of the Alkaline Earth Metals? Halogens range from solid (I 2) to liquid (Br 2) to gaseous (F 2 and Cl 2) at room temperature. The halogens all have a strong and often nasty smell; The halogen elements are extremely toxic; Poor conductors of heat and electricity; Low melting and boiling points; Chemical Properties . When this happens, the atoms become stable and have noble gas configurations. Halogens. When halogens react with metals, they produce a wide range of salts, including calcium fluoride, sodium chloride, silver bromide and potassium iodide. The melting and boiling point of halogens increases with increase in the atomic number of the element. What this means is that their molecules exist with two atoms each. The element group is a particular class of nonmetals. The halogens are particularly reactive with the alkali metals and alkaline earths, forming stable ionic crystals. Element 117, which has the placeholder name of ununseptium, might have some properties in common with the other elements. Halogens range from solid (I2) to liquid (Br2) to gaseous (F2 and Cl2) at room temperature. The halogens are a series of non-metal elements from group 17 of the periodic table (formerly VII). It reacts with otherwise inert materials such as glass, and it forms compounds with the heavier noble gases. The chemical properties of halogens allow them to easily join with most of the elements, so they are never found unbound in nature. The artificially created element 117 (ununseptium) may also be considered a halogen. CC BY-SA 3.0. http://en.wikipedia.org/wiki/Halogens This means the shared electrons are further from the halogen nucleus, which increases the shielding of inner electrons. This means electronegativity decreases down the group. Fluorine and chlorine are gases, while bromine is a liquid and iodine and astatine are solids. CHEM - Properties and Reactions of Halogens Halogens are Group 7 non-metals, including fluorine (F), chlorine (Cl), bromine (Br), iodine (I) and astatine (At). When fluorine exists as a diatomic molecule, the F–F bond is unexpectedly weak. Progress % … The name "halogen" means "salt-producing". Halogens are the most reactive nonmetals. This is because fluorine atoms are the smallest of the halogens—the atoms are bonded close together, which leads to repulsion between free electrons in the two fluorine atoms. Properties and Trends of Halogens Colour and state of halogens at room temperature : As halogens go down the group, melting point and boiling point increases. Atoms get bigger down the group as additional electron shells are filled. These patterns result from their physical properties and give me the rare opportunity to incorporate some organic chemistry. There is a trend in state from gas to liquid to solid as you go down the group . they exist naturally in various mineral salts in […] This is of course a typical property of non-metals. The artificially created element 117, tennessine, may also be a halogen. Owing to their high reactivity, these are never found in a pure form in the nature. Describe the physical and chemical properties of halogens. Halogens can gain an electron by reacting with atoms of other elements. The halogen elements are: Although element 117 is in Group VIIA, scientists predict it may behave more like a metalloid than a halogen. Practice. Bromine has a solubility of 3.41 g per 100 g of water. Halogens are nonmetals in group 17 (or VII) of the periodic table. Elements in group seven have a number of similar properties, most importantly they have low melting and boiling points. Liquid bromine has a high vapor pressure, and the reddish vapor is readily visible in . 3. Boundless Learning The artificially created element 117 (ununseptium) may also be considered a halogen. The halogens are a group in the periodic table consisting of five chemically related elements: fluorine, chlorine, bromine, iodine, and astatine. Thus fluorine must be handled with substances like the inert organofluorine compound Teflon. This occurs with the addition of potassium iodide (KI), forming a triiodide ion. Iodine crystals have a noticeable vapor pressure. This oxidizing ability decreases down the group as the electron affinity decreases. All halogens form salts of group I with similar properties. Halogens are diatomic when kept under room temperature. Chlorine bleach and iodine tincture are two well-known examples. Liquid bromine has a high vapor pressure, and the reddish vapor is readily visible in (Figure 3.12.1). From the lowest boiling and melting point to the highest, the group in order is fluorine, chlorine, bromine, iodine and astatine. Wiktionary They gain electrons very fast making them most reactive of all chemical elements. Halogens are diatomic, which means they form molecules of two atoms. There are either five or six halogen elements, depending on how strictly you define the group. They react with metals and other halogens to get an octet. (a) Halogen is a Greek word which means salt-former’. 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Per kg of water at ambient temperature ( 21 °C ) - to oxidize metals: //www.boundless.com/chemistry/textbooks/boundless-chemistry-textbook/ anions are... | 2023-03-30 11:49:11 | {"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": 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": 0.4800979793071747, "perplexity": 3355.8529168615323}, "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/1679296949181.44/warc/CC-MAIN-20230330101355-20230330131355-00725.warc.gz"} |
https://math.stackexchange.com/questions/3589437/if-any-infinite-numerable-subset-of-the-t-1-space-x-have-a-limit-point-then | # If any infinite numerable subset of the $T_1$ space $X$ have a limit point then $X$ is countably compact
Following a reference from "Elementos de Topología General" by Angel Tamariz and Fidel Casarrubias.
Theorem Let be $$X$$ a $$T_1$$ space such that every infinite and numerable subset have a limit point: so $$X$$ is countably compact.
proof. Well we suppose that $$X$$ is not countably compact and so there exist a open numerable cover $$\mathcal{U}=\{U_n:n\in\mathbb{N}\}$$ such that it have not a finite subcover.
So we choose $$x_1\in X$$ and $$U_{n_1}\in\mathcal{U}$$ such that $$x\in U_{n_1}$$. So since $$\mathcal{U}$$ have not a finite subcover it is $$X\setminus(\bigcup_{i=1}^{n_1}U_i)\neq\varnothing$$ and so we choose $$x_2\in X\setminus(\bigcup_{i=1}^{n_1}U_i)$$. Then since $$\mathcal{U}$$ is an open cover there exist $$n_2\in\mathbb{N}$$ such that $$x_2\in U_{n_2}$$ (observe that $$n_1). So we suppose that we made $$n_1,...,n_k\in\mathbb{N}$$ and $$x_1,...,x_k\in X$$ such thaat $$n_1<... and $$x_1\in U_{n_1}$$ and $$x_j\in U_{n_j}\setminus(\bigcup_{i=1}^{n_j-1}U_i)$$ for any $$j\in\{2,...,k\}$$. Since $$\bigcup_{i=1}^{n_k}U_i\neq X$$ there are exist $$x_{n_{k+1}}\in X\setminus(\bigcup_{i=1}^{n_k}U_i)$$ and $$n_{k+1}\in\mathbb{N}\setminus\{1,2,...,k\}$$ such that $$x_{n_{k+1}}\in U_{n_{k+1}}$$. So in this way we recursively made the point $$x_{k+1}\in U_{n_{k+1}}\setminus(\bigcup_{i=1}^{n_k}U_i)$$.
So using the previous recursive process we can define the infinite numerable set $$F=\{x_k:k\in\mathbb{N}\}$$ and the sequence $$\{U_k:k\in\mathbb{N}\}$$ in $$\mathcal{U}$$. So we prove that $$F$$ have not limit point in $$X$$. Indeed for any $$x\in X$$ there exist $$n\in\mathbb{N}$$ such that $$x\in U_n$$ and $$U_n$$ contains at most a finite collection $$G$$ of points of $$F$$. So $$(U_n\setminus G)\cup\{x\}$$ is a neighborhood of $$x$$ that not contains point of $$X$$ that are different from $$x$$.
Well I don't understand why $$U_n$$ contains at most a finite collection $$G$$ of points of $$F$$. Could someone help me, please?
Here the original proof in Spanish (I hope mine was a good translation).
• As per the comments on my answer: the sequence of elements from $\mathcal{U}$ should probably say $\{U_{n_k}: k \in \Bbb N\}$ to make it clearer. Mar 21 '20 at 22:54
• Hi professor Brandsma, could I ask your assistance to solve the problem that I explain here? Mar 22 '20 at 18:25
• I’ll type something later. Busy now. Mar 22 '20 at 18:28
• Okay, don't worry; thanks!!! Mar 22 '20 at 18:29
So having $$x \in X$$ and some $$U_n$$ ($$n$$ fixed for this argument) containing it, let $$n_k$$ be the first of the indices used in the recursive construction that is larger or equal to $$n$$. By construction then $$x_{k+1}$$ and higher indexed ones, like $$x_{k+2}$$ etc. ) are not in $$U_n$$ as $$n \le n_k < n_{k+1}$$, and $$x_{k+1}$$ was chosen to lie outside $$\bigcup_{i=1}^{n_k} U_i$$ in the recursive step, so only those $$x_l$$ with $$l \le k$$ can be in $$U_n$$, so at most finitely many, as claimed.
• Okay, this it is clear if $U_n$is an element of the sequence $\{U_k:k\in\mathbb{N}\}$; but if $U_n$ is an element of $\mathcal{U}$ such that is not in $\{U_k:k\in\mathbb{N}\}$, why the result holds? Mar 21 '20 at 18:15
• @AntonioMariaDiMauro you can always choose a $U_n$ because it’s a cover. Mar 21 '20 at 18:17
• So the made sequence $\{U_k:\in\mathbb{N}\}$ of element of $\mathcal{U}$ is a permutation of the element of $\mathcal{U}$? Mar 21 '20 at 18:18
• Sorry, but I don't understan what you wrote. Perhaps do you want say that $\{U_k:k\in\mathbb{N}\}$ is a subsequence of $\mathcal{U}$ such that is a cover of $X$? Mar 21 '20 at 18:22 | 2021-12-03 06:55:36 | {"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": 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": 58, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9213662147521973, "perplexity": 154.44540198936204}, "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/1637964362605.52/warc/CC-MAIN-20211203060849-20211203090849-00250.warc.gz"} |
http://archive.numdam.org/item/AIHPA_1981__34_1_85_0/ | The geometrical and gauge structure of a generalized theory of gravitation
Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1, p. 85-94
@article{AIHPA_1981__34_1_85_0,
author = {Moffat, J. W.},
title = {The geometrical and gauge structure of a generalized theory of gravitation},
journal = {Annales de l'I.H.P. Physique th\'eorique},
publisher = {Gauthier-Villars},
volume = {34},
number = {1},
year = {1981},
pages = {85-94},
mrnumber = {605358},
language = {en},
url = {http://www.numdam.org/item/AIHPA_1981__34_1_85_0}
}
Moffat, J. W. The geometrical and gauge structure of a generalized theory of gravitation. Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1, pp. 85-94. http://www.numdam.org/item/AIHPA_1981__34_1_85_0/
[1] J.W. Moffat, Phys. Rev., t. D19, 1979, p. 3554. | MR 538568
[2] J.W. Moffat, Ibid., t. D19, 1979, p. 3562. | MR 538570
[3] J.W. Moffat, J. Math. Phys., t. 21, 1980, p. 1798. | Zbl 0451.35091
[4] R.B. Mann and J.W. Moffat, University of Toronto, preprint, 1980.
[5] G. Kunstatter, J.W. Moffat and P. Savaria, Can. J. Phys., t. 58, 1980, p. 729. | MR 586630 | Zbl 1043.83554
[6] Cf. Y.M. Cho, Phys. Rev., t. D14, 1976, p. 2421.
[7] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience Pub. John Wiley, New York, t. II, 1969). | MR 238225 | Zbl 0175.48504
[8] Here the superspace is defined for eight bose coordinates and not for four bose and four fermi coordinates as is done in superspace versions of supersymmetry theories (cf. P. Nath and R. Arnowitt, Phys. Lett., t. 56B, 1975, p. 177). The present work can readily be extended to the case of superspace supersymmetry by defining the N manifold in terms of four fermi coordinates with the basis vectors ξm satisfying { ξm, ξn } = { ∂m, ∂n } = 0 (see: J.W. Moffat, to be published in Lett. in Math. Physics).
[9] Cf. Supergravity, eds. D. Z. FREEDMAN and P. VAN NIEUWENHUIZEN, North Holland, Pub. 1979. | MR 592447
[10] Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin, Math.-Phys., K1, 1921, p. 966. | JFM 48.1327.01 | 2020-10-30 02:13:03 | {"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": 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": 0.5032685399055481, "perplexity": 6883.164996160075}, "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/1603107906872.85/warc/CC-MAIN-20201030003928-20201030033928-00558.warc.gz"} |
https://lisnbaa.wordpress.com/2016/03/20/algebraic-versus-symplectic-toric-manifolds/ | # Algebraic versus symplectic… toric manifolds
As the second and probably final (for the moment at least) comparative post between algebraic and symplectic geometry, I thought I’d compare their respective toric geometries and related combinatorics. ‘Symplectic’ as terminology was proposed by Weyl to reflect the similarities with complex geometry by replacing the Latin word ‘complex’ with its Greek equivalent. As this anecdote suggests, there is much shared between the two!
In the toric world, there are at least three ways to construct a toric manifold from some combinatorial data; for instance a polytope or a fan. One of them is by using lattice points (= sections for an algebraic geometer) to embed the relevant torus into some big projective space and take the closure. If you like, this is just applying Proj to the toric ample divisor corresponding to the polytope. Given a Delzant polytope $\Delta$ with lattice points as vertices – notice that this can sometimes (always?) be achieved by suitable scaling of the symplectic form – one can perform this process to obtain a toric variety, which will be a smooth subvariety of some projective space by the Delzant condition. In particular, it inherits a symplectic form from the restriction of the Fubini-Study form as a complex submanifold of a projective space.
The torus action it also carries turns out to be Hamiltonian with respect to this symplectic structure as witnessed by its moment map, which has moment polytope… $\Delta$. That is, by Delzant’s theorem (which says that there’s a unique way to reconstruct a symplectic toric manifold from a polytope) the algebraic and symplectic constructions of a toric manifold from a lattice Delzant polytope agree. | 2017-09-23 14:42:45 | {"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": 0, "img_math": 2, "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": 0.8028391003608704, "perplexity": 476.097828103633}, "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/1505818689686.40/warc/CC-MAIN-20170923141947-20170923161947-00366.warc.gz"} |
https://www.bunnypub.net/en/technology/topics/107553 | ## World's first holeless phone
Phones have been shedding ports and buttons for a while now. Headphone jacks are going extinct on most devices, and home and back buttons have been replaced on nearly every iOS and Android phone with software UIs. The new Meizu Zero is the first to take that trend to its natural endpoint: it has no ports, buttons, or holes that mar the exterior of the device.
According to Meizu, the Zero has "no buttons, no speakers holes, no SIM card slot, and no charging port." Most of the exterior of the device is a seamless slab of ceramic (an effect that's only slightly ruined by a protruding camera bump on the back and what appear to be microphone holes on the bottom).
The Zero is proudly advertised as "the world's first holeless phone," so Meizu has had to figure out several workarounds for the usual functions on a phone. Hence, the 5.99-inch AMOLED display, which, notably for a 2019 smartphone, has a Galaxy S9 chin and top bezel instead of a notch. It features an under-glass fingerprint sensor as well as Meizu's "mSound 2.0" technology that allows it to function as a speaker.
Charging is completely wireless, too. Meizu is promising speeds of up to 18W through its own "Super mCharge Wireless" technology, which would far surpass the 7.5W spec Apple uses and the 9W speeds offered by Samsung.
There are no buttons, but Meizu is using a haptic feedback system to offer virtual buttons on the side to turn the phone on and off and to adjust the volume. And eSIM technology replaces the SIM card slot. Naturally, due to the fact that there are essentially no holes on the outside of the device, it also has an IP68 rating against water and dust.
There's still a lot Meizu hasn't announced about the Zero, including what the specs will look like outside of the fact that the Zero will use last year's Snapdragon 845 processor. We'll have to wait to try out the phone in person before any judgment can be made about how well these new technologies actually work.
• 22
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Blagoje
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Blagoje
5.0 BRB (≈ $2.5) expand Don't underestimate human beings wisdom, they might be able to change the technology beyond imagination. Blagoje 5.0 BRB (≈$ 2.5)
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expand | 2019-09-15 20:56:44 | {"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": 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": 0.29754090309143066, "perplexity": 7960.124988317398}, "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-2019-39/segments/1568514572289.5/warc/CC-MAIN-20190915195146-20190915221146-00103.warc.gz"} |
https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Bilinear_transform.html | # Bilinear transform
related topics {math, number, function} {system, computer, user} {math, energy, light} {style, bgcolor, rowspan}
The bilinear transform (also known as Tustin's method) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear transform is a special case of a conformal mapping (namely, the Möbius transformation), often used to convert a transfer function $H_a(s) \$ of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function $H_d(z) \$ of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the $j \omega \$ axis, $Re[s]=0 \$, in the s-plane to the unit circle, $|z| = 1 \$, in the z-plane. Other bilinear transforms can be used to warp the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays $\left( z^{-1} \right) \$ with first order all-pass filters.
The transform preserves stability and maps every point of the frequency response of the continuous-time filter, $H_a(j \omega_a) \$ to a corresponding point in the frequency response of the discrete-time filter, $H_d(e^{j \omega_d T}) \$ although to a somewhat different frequency, as shown in the Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency. | 2017-11-23 13:38:33 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "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": 0.9198439717292786, "perplexity": 613.6468661443922}, "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-2017-47/segments/1510934806832.87/warc/CC-MAIN-20171123123458-20171123143458-00236.warc.gz"} |
https://www.semanticscholar.org/paper/Curvit%3A-An-open-source-Python-package-to-generate-Joseph-Stalin/4f8df51c5af076242ea0745c809bbbeda9024456 | Curvit: An open-source Python package to generate light curves from UVIT data
@article{Joseph2021CurvitAO,
title={Curvit: An open-source Python package to generate light curves from UVIT data},
author={Prajwel Joseph and C. S. Stalin and Shyam N. Tandon and S. K. Ghosh},
journal={Journal of Astrophysics and Astronomy},
year={2021},
volume={42},
pages={1-10}
}
• Published 16 January 2021
• Physics
• Journal of Astrophysics and Astronomy
Curvit is an open-source Python package that facilitates the creation of light curves from the data collected by the Ultra-Violet Imaging Telescope (UVIT) onboard AstroSat, India’s first multi-wavelength astronomical satellite. The input to Curvit is the calibrated events list generated by the UVIT-Payload Operation Center (UVIT-POC) and made available to the principal investigators through the Indian Space Science Data Center. The features of Curvit include: (i) automatically detecting sources…
1 Citations
AstroSat observation of the HBL 1ES 1959+650 during its October 2017 flaring
• Physics
• 2021
We present the results of the X-ray flaring activity of 1ES 1959+650 during October 25-26, 2017 using AstroSat observations. The source was variable in the X-ray. We investigated the evolution of the
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CCDLAB: A Graphical User Interface FITS Image Data Reducer, Viewer, and Canadian UVIT Data Pipeline
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CCDLAB was originally developed as a FITS image data reducer and viewer, and development was then continued to provide ground support for the development of the UVIT detector system provided by the | 2022-11-27 15:52:58 | {"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": 0.3714299499988556, "perplexity": 7142.491774087853}, "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-2022-49/segments/1669446710409.16/warc/CC-MAIN-20221127141808-20221127171808-00715.warc.gz"} |
https://www.nature.com/articles/s41598-017-12239-0?error=cookies_not_supported&code=9cebd327-6133-4704-8733-36214730d92d | Article | Open
Allele Age Under Non-Classical Assumptions is Clarified by an Exact Computational Markov Chain Approach
• Scientific Reports 7, Article number: 11869 (2017)
• doi:10.1038/s41598-017-12239-0
Accepted:
Published online:
Abstract
Determination of the age of an allele based on its population frequency is a well-studied problem in population genetics, for which a variety of approximations have been proposed. We present a new result that, surprisingly, allows the expectation and variance of allele age to be computed exactly (within machine precision) for any finite absorbing Markov chain model in a matter of seconds. This approach makes none of the classical assumptions (e.g., weak selection, reversibility, infinite sites), exploits modern sparse linear algebra techniques, integrates over all sample paths, and is rapidly computable for Wright-Fisher populations up to N e = 100,000. With this approach, we study the joint effect of recurrent mutation, dominance, and selection, and demonstrate new examples of “selective strolls” where the classical symmetry of allele age with respect to selection is violated by weakly selected alleles that are older than neutral alleles at the same frequency. We also show evidence for a strong age imbalance, where rare deleterious alleles are expected to be substantially older than advantageous alleles observed at the same frequency when population-scaled mutation rates are large. These results highlight the under-appreciated utility of computational methods for the direct analysis of Markov chain models in population genetics.
Introduction
Allele age is generally defined as the duration of time a mutant allele has been segregating in a population. The problem of calculating the expected age of an allele given its current population frequency is an important problem in population genomics (e.g., ref.1) with a long history of theoretical investigations (e.g., refs2,3,4,5,6; reviewed in ref.7). One reason that allele age remains an important problem is that the effects of selection and age can be highly confounded in terms of their influence on population frequency. That is, an allele may be at low frequency because it is deleterious or simply because it is young. Inferences about the fitness effects of segregating polymorphisms must therefore make some consideration of allele age, either explicitly or implicitly, and methods for inferring fitness impacts based on allele ages have even been proposed8.
The first theoretical analysis of allele age was developed by Kimura and Ohta3 using a continuous-time diffusion approximation to the age of a neutral allele in a finite population. Later work added consideration of selection9, yielding the well-known result that allele age is expected to be symmetric with respect to the direction of selection, and that neutral alleles are expected to be older than selected alleles observed at the same frequency (the “Maruyama effect” hereafter). Recently, an interesting exception to these classical results has been pointed out10,11,12. Mafessoni et al.12 showed that weakly selected rare alleles are expected to be about 5% older than neutral alleles observed at the same frequency, when heterozygote fitness is non-additive. This phenomenon appears to be an example of a more general behaviour recently termed ‘stochastic slowdown’10, where weak selection counter-intuitively prolongs, rather than shortens, the average time to absorption. It is important to understand the generality of these findings, since, as Mafessoni et al.12 point out, many new mutations arising in a population are expected to be recessive and weakly deleterious, and it is conceivable that this slowdown effect could thereby mislead attempts to make inferences about natural selection.
Previous investigations of allele age, and classical approaches in population genetics more generally, have required that mutation rates are assumed to be so slow that no additional mutations can occur during the segregation of an initial variant (implying that the population-scaled mutation rate, θ, is very small or ≈0). However, cases where this assumption is violated in nature are increasingly being reported, and it is likely in such cases that classical population genetic theory will be unreliable at best13. While in most eukaryotes, θ is estimated to be $≪$0.05, several examples of so-called hyperdiverse eukaryotes are known with $θ ˆ$ between 0.05 and 0.1514. In bacteria, it is not uncommon for estimates of θ to be at the high end of this range or significantly larger. For example, Sung et al.15 reported average estimates taken from the literature of θ = 0.15 for Helicobacter pylori and 0.12 for Salmonella enterica, both of significant biomedical interest. Hughes et al.16 also reported $θ ˆ$ in Pseudomonas syringae to be 0.55. In addition, θ in some organisms including viruses and pathogens has been estimated to be much larger, by a variety of analytical methods, with estimates often exceeding 1. For example, θ in HIV-1 has been estimated to be between 10 and 36917 in one study, and >1 using the effective population size estimated by Pennings et al.18,19 together with mutation rates from other studies; similarly, θ in macaque monkeys infected with RT-SHIV (an engineered simian immundeficiency virus encoding human HIV-1 reverse transcriptase) has been estimated to be greater than one20. Other arguments that classical assumptions about θ may be violated in nature have also been recently put forward. For example, Messer and Petrov21 have highlighted that most known cases of molecular adaptation across diverse organisms show signatures of soft selective sweeps (but see ref.22), where adaptive alleles have multiple origins either by recurrent mutation or migration. These findings are potentially unexpected if evolution is strongly mutation-limited and may indicate that the effective population-scaled mutation rate is underestimated in many cases and/or that adaptation may tend to occur during periods of episodically large population size (and thus, high θ)23. We therefore decided to revisit the problem of calculating allele age based on population frequency under non-classical assumptions, and in particular to examine the impact of large values of θ on the expected age of an allele. For beneficial variants, the values of θ that we consider are expected to produce adaptive fixations that may have either multiple mutational origins or single origins24.
To study the effects of non-classical parameter ranges on allele age, we develop a new exact approach capable of rapidly computing moments of the allele age distribution under any absorbing discrete-time Markov chain model of population genetics. This approach exploits sparsity, parallelism, and modern computational architectures25, and is completely general with respect to the underlying model. It therefore requires none of the classical simplifying assumptions (e.g., weak selection, weak mutation, infinite sites, etc). For the purposes of the present study, we assumed a biallelic diploid Wright-Fisher model26 including bidirectional mutation, selection and dominance. Computationally, our solution mainly relies on back-substitutions using an LU decomposition of a sparse matrix derived from the model’s transition matrix, and does not use any matrix-matrix multiplications, which are computationally expensive. This computational implementation is similar to that in ref.25, where we applied sparse matrix techniques to the calculation of population genetic quantities such as the probability of fixation and sojourn times (but not allele age). To the best of our knowledge, this is the first computationally feasible, exact approach for computing allele age (or its moments) to be proposed. Calculation of the expected value and variance of allele age is fast, exact and scales easily to realistic population sizes (N e ≈ 105 for Wright-Fisher type models, and much larger for Moran models due to their greater sparsity; see Discussion). We have implemented this method in our software package Wright-Fisher Exact Solver, WFES25 (available at https://github.com/dekoning-lab/wfes/).
Results
Using the approach outlined above and described fully in the Methods, we considered allele age and related quantities in a biallelic Wright-Fisher model including bidirectional mutation, selection, and dominance. For selection coefficient s and dominance coefficient h, the homozygous wildtype fitness was defined as 1, heterozygote fitness as 1 + sh, and homozygous mutant fitness as 1 + s (following standard definitions26). Bi-directional mutation was modelled in the Wright-Fisher transition matrix26, with extinction and fixation states assumed to be absorbing. This assumption implies a return process such that when a mutant frequency of 1 is attained, the population is returned to a frequency of 0 (equivalent to swapping the labels for the wild-type and mutant states); this allows properties of average trajectories to be easily calculated based on their starting or ending states.
In a biallelic diploid model, each individual may be either wild-type or mutant at each locus and chromosome. We define an “allele” here explicitly as the mutant genotype. Thus, allele age refers to how long the mutant state has been segregating in the population, starting from a population that was monomorphic for the wild-type state. By allowing mutation, we assume that an arbitrary number of new mutations could potentially arise in the population while an initial mutant is segregating, and thus the assumption of shared ancestry of all segregating mutants is not necessarily made. In the context of classical theory it may seem unnatural to consider allele age while including mutation. However, this is because classical theory makes the assumption that mutation cannot be recurrent, while there is no such prohibition in nature. Furthermore, even when θ is large, allele trajectories include long periods of time spent at the boundaries, and it therefore remains reasonable to demarcate the behaviour of such trajectories based on their visits to the boundaries. This may no longer be true when θ is so large that a population always contains all possible alleles ($θ≫1$).
Except where otherwise specified, all results that follow are for a rare allele observed in x = 10 copies, sampled from an effective population size of N e = 10,000 diploids. Forward and backward mutation rates were assumed equal. We consider a range of population-scaled mutation rates, θ = 4N e μ, between θ = 0.0048 and θ = 0.96, where μ is the mutation rate per site per chromosome, and N e the effective population size. Results obtained using values of θ that were two orders of magnitude smaller than θ = 0.0048 were largely similar (not shown).
Validation by comparison to other methods
We first examined the correspondence between expected allele age determined by exact computation with the Wright-Fisher Markov model and the expected allele age approximated using Kimura and Ohta’s diffusion approach3. Since Kimura and Ohta’s method assumes no selection and no mutation, we ran our computations on a Wright-Fisher model having these same assumptions. Across a range of effective populations sizes and observed allele counts, the methods exhibited close correspondence (Table 1), where Kimura and Ohta’s method consistently overestimated allele age by a few generations.
We next validated our method and its implementation by comparing results to allele age simulations that included selection, dominance, and mutation (Table 2). Allele age probability distributions can be approximated by simulation by reversing the direction of time in a Wright-Fisher model that is modified to have the same stationary distribution as the original (forward-time) transition matrix27. “Forward time” simulations of this reversed model can then be performed starting at the observed frequency, x/(2N e ), and running until the beginning of the sample path (p/(2N e ); see Methods for details). Simulations performed in this manner agreed well with the model-based computations across the entire parameter range. Allele-frequency probability distributions approximated by simulation are shown for a subset of cases in Fig. 1.
Computational advantages of the exact approach
Allele age simulations were implemented in C++ and parallelized, so that their runtimes would be reasonably fast (see https://github.com/dekoning-lab/allele_age_simulator/). Simulations were much more time consuming than the direct computation of the moments using our approach (e.g., 15 minutes versus 0.6 seconds for θ = 0.01; Table 3). As θ was increased, the simulations took increasingly more time both because the allele trajectories grew longer on average and because higher mutation rates also increased the variance in the duration of allele age trajectories. For θ = 0.96, running a 10 million replicate simulation over 32 cores took approximately 13 hours. On the other hand, the runtime for the exact matrix method was constant across different mutation rates and averaged about 0.5 seconds. Thus, when moments provide sufficient information, they can be obtained much more efficiently using our exact approach.
Direct demonstration of classical results
Several classical results pertaining to allele age can be directly obtained by examining expected allele age and variance as a function of selection (Fig. 2). It should be emphasized that these plots are neither probability distributions nor estimates. Rather, they are the exact moments of allele age derived directly from the Wright-Fisher model, as explained in the Methods section.
For rare alleles, the expected allele age has a large variance relative to the mean and the mean age is roughly symmetric with respect to the sign of the selection coefficient, with neutral alleles expected to be older than selected alleles (Fig. 2B, leftmost column; the Maruyama effect9). The symmetry of allele age with respect to the direction of selection is among the most conspicuous classical findings on allele age, and has been the subject of recent study, where different authors have both supported it using population genomic data8 and argued against it using simulations that included linkage28.
Selective strolls and stochastic slowdowns
Recent work10,11,12 has convincingly demonstrated, using primarily simulation and diffusion theory methods, that weakly selected alleles are sometimes expected to be older than neutral alleles observed at the same frequency when fitness in heterozygotes is non-additive. This idea was termed “selective strolls” by Mafessoni et al.12, referring to the observation that selected variants may sometimes persist in a population slightly longer than neutral ones. Here we directly reproduce this effect for rare recessive alleles (h = 0), where it can be seen that weakly deleterious alleles are expected to be older than neutral alleles at the same frequency (Fig. 2A, leftmost column), and for dominant alleles (h = 1), where it can be seen that weakly advantageous alleles are expected to be older than neutral alleles at the same frequency (Fig. 2C, leftmost column). Consistent with the findings of Mafessoni et al.12, it is apparent that the selective stroll effect size is not very large and is on the order of about 5%.
Recurrent mutation and age imbalance
Contrary to the Maruyama effect, for population-scaled mutation rates approaching θ ≈ 1 the mean allele age becomes strongly asymmetric around s = 0 (Fig. 2, c.f. left to right) such that weakly to moderately deleterious alleles can on average be substantially older than advantageous alleles at the same frequency. We refer to this previously unobserved phenomenon as “age imbalance”.
Under age imbalance, slightly deleterious alleles are also expected to generally be older than neutral alleles at the same frequency. This new example of stochastic slowdown is observed even when heterozygote fitness is additive (i.e., with h = 0.5). The effect size in this case is substantially larger than for the previously noted slowdowns with small θ (or θ = 0; ref.12). For example, expected extinction times for the oldest alleles with h = 0.5 are approximately 22.7% longer than for neutral alleles.
Rare recessive alleles (h = 0) under recurrent mutation and large θ (Fig. 2, right) experience the same effect but to an even greater degree. Recessivity and fast mutation appear to have a similar and mutually reinforcing effect on both age imbalance and the stochastic slowdown under weak selection. Both selective stroll and age imbalance results appear to be explained primarily by the average time to extinction (Fig. 3, left), which indicates that when mutation rates are bidirectionally fast, weakly deleterious alleles counter-intuitively take longer to go extinct than do advantageous (or neutral) alleles. For h = 0 extinction times are even longer for deleterious recessive alleles than for those with h = 0.5, but now the expected fixation times also show a similar imbalance with respect to the direction of selection (Fig. 3, c.f. A and B), which accentuates the stochastic slowdown further. Remarkably, the expected time to extinction for the oldest, weakly selected recessive alleles is about 66.9% longer than for neutral alleles (Fig. 2A, left). The same results for h = 1 are shown in Fig. 3C, where fixation times are shifted to the right rather than the left, which seems to largely cancel out the stochastic slowdown caused by the left-shifted extinction times.
To help explain Fig. 3, we also calculated the conditional sojourn times for mutants that go to extinction, and compared these to sojourn times for neutral variants (Fig. 4). For deleterious alleles, we see that the time spent at low frequencies increases as we move away from 2N e s = 0 until 2N e s = −2.53 is reached; the stronger selection is within this parameter range, the more extinction sojourns are dominated by residency at lower frequencies compared to neutral. While this trend is expected since negative selection opposes increases in allele frequency, it is surprising that the net change in non-neutral sojourn times is positive. That is, the increased time at low frequencies surpasses the decreased time spent at high frequencies, resulting in longer sojourns overall. This phenomenon has also been reported for previously noted stochastic slowdowns11.
Allowing the starting number of copies to vary
When population-scaled mutation rates are very high it can become plausible that an originating mutation enters the population in several copies (i.e., that it simultaneously occurs in several individuals). For example, when θ = 0.96, the average number of mutations entering the population per generation is 0.96/2 = 0.48, so on average there will be a new mutation every two generations. The probability of a population generating multiple copies of the mutant allele in a single generation, assuming mutations are Poisson distributed, is ≈0.38. This may pose problems for any method for calculating allele age, since when the likelihood of a population simultaneously generating more than one mutant becomes non-negligible, the starting number of copies, p, should be integrated out.
To integrate over p we consider the probability of starting in p copies, given that $p∼Poisson(λ=θ/2)$. This can be easily implemented in our computational procedure starting at Equation 8 by reusing the LU decomposition of (I − Q)T, which does not depend on p (see Methods). Since this decomposition is by far the most computationally expensive operation, the integrated solution is trivially harder than when assuming a single starting copy. In addition, since the probability of large numbers of mutations occurring in the same generation will typically be negligible, we define a threshold ε such that only starting configurations with a probability greater than ε are considered. Below, we assumed ε = 10−5.
In Fig. 5 we show the effect of numerically integrating over p when θ = 0.96 for the range of mutation rates, selection coefficients, and dominance coefficients considered throughout the manuscript. In most cases, the results were identical at better than three to four decimal places, and only began to diverge slightly when θ was very large (i.e., θ = 0.96). It is possible that other statistics of the Markov process might change more than this as a function of p, and thus to be conservative one may choose to always integrate over p (particularly since this adds only seconds to the compute time). However, we conclude that assuming p = 1 (as is done by convention in all previous studies of allele age that we are aware of) is likely to introduce no bias unless θ is quite large (i.e., $≫$1).
Discussion
Computational population genetics approaches offer the relatively straightforward ability to explore parameter ranges or assumptions that may be inaccessible to classical theory. Usually simulations are used to address scenarios where the assumptions of classical theory may be violated. However, simulations can often be slow, require long runtimes to obtain precise estimates for rare events, and can scale poorly to large populations. An alternative computational approach is to find a class of models whose properties can be interrogated directly, without the need for simulation. For example, Steinruecken et al.29 recently showed how the transition density function of biallelic Wright-Fisher diffusions30 could be approximately computed, eliminating the need for a variety of simulations (although allele age has not been considered in this framework). Here we have shown that even the exact computational analysis of biallelic Markov models (including Wright-Fisher models) can be made efficient enough to often eliminate the need for either simulations or diffusion approximations in the first place. Markov chain models are typically discounted early in the lifecycle of a population genetic investigation in favour of diffusion approximations, since they are widely viewed as impractical to work with due to their large and potentially unwieldy state spaces. Contrariwise, here and elsewhere25, we have shown that judicious computation, sparsity, and parallelism can be together exploited to rather surprising effect, making exact computation under general Markov models not only tractable but capable of generating new insights with ease. Working directly with the underlying Markov models of population genetics has a number of advantages. For example, when strong mutation is included, absorbing boundaries can artificially become inaccessible in a diffusion. There is no corresponding problem when studying the unapproximated Markov chain. In addition, diffusion approaches cannot easily describe behaviours at the absorbing boundaries (but see ref.31).
One of the most appealing aspects of this computational population genetics approach is that it is general with respect to underlying modelling assumptions, as long as they can be expressed as a finite absorbing Markov chain. This approach also has several advantages over simulations, including fast runtimes that are relatively insensitive to modelling assumptions (Table 3), and exact results (within machine precision) even for small effects or rare events that would otherwise require long-run, high replicate simulations to study. For a population size of N e = 10,000, exact calculation of the expected allele age and variance, absorption probabilities and times, and conditional sojourn times, takes only about 6.5 seconds using 16 Intel E5-2670 cores (2.60 GHz) in our reference implementation25. Models with greater sparsity are even faster and can scale much better. For example, the same analysis under a comparable Moran model takes only about 0.25 seconds25.
The method proposed for calculating allele age is based on the efficient computation of the moments of the probability distribution of allele ages. It is therefore appropriate to view these quantities not as estimates, but as exact results for a given model. An advantage of this approach is that the expected value of the allele age probability distribution will more often be much closer to the true allele age than would a maximum likelihood estimator, since the age distributions are both highly skewed and very long tailed (see Fig. 1). A potential disadvantage is that we must assume that the true population frequency is known without error. In cases where it is not, error in the observed frequency could be accounted for by computing allele age for a range of population frequencies centred on the observed value.
As shown in Fig. 2, classical allele age results3,9,32 can be easily obtained for general population genetic models with our approach. We also reproduced exact representations of recently discovered effects, such as “selective strolls”, which have a smaller effect on expected allele age when mutation rates are low (also see ref.12). By exploiting the generality of our approach, we discovered new evidence for a stochastic slowdown that occurs when bidirectional mutation is fast, such that rare, weakly deleterious alleles are expected to be substantially older than neutral alleles. In the most extreme case, average extinction times for the oldest alleles were 22% and 68% longer than for neutral alleles (for h = 0.5 and h = 0, respectively). Finally, we found that when relaxing the assumption of weak mutation, a large age imbalance arises with respect to selection, such that rare deleterious alleles are expected to be old and rare advantageous alleles very young. This may be explained in part by the expectation that with strong mutation pressure and positive selection, allele frequencies will rise rapidly following origination. When this is true, the best explanation for a beneficial allele being rare is that it only arose quite recently. This expected rapid rise in mutant frequency under strong mutation and positive selection may also be responsible for the much faster extinction times for beneficial alleles compared to deleterious ones (Fig. 3: left), since the longer beneficial alleles persist, the more likely their frequencies are to be pushed upwards towards fixation. Consequently, the mutants that go to extinction are most likely to do so quickly.
A potential limitation of our approach to calculating allele age is that we have assumed equilibrium demography with constant population size. However, this is a limitation of our implementation rather than of the method itself. One solution to this problem is to consider instantaneous switches among different population sizes under a Markov-modulated model. By virtue of our sparse linear algebra approach, this would only be linearly more difficult than the constant population size approach. It could also have advantages over existing diffusion theory methods33, for example, by faithfully modelling an increase in the population mutation rate during population growth that includes the effect of recurrent mutation. Such considerations may be important for understanding adaptation in organisms with “boom and bust” population dynamics23. We leave exploration of these ideas for future work.
Methods
Theory
Let X(t) be an absorbing discrete-time Markov chain with known transition matrix P and state-space defined by the number of copies of a mutant allele in a population of N e effective diploid individuals. Let Q be the submatrix of P that contains only transient-to-transient state transitions. Assume that the current number of mutant alleles x is a transient state, so the allele in question is neither extinct nor fixed. We also assume that the allele entered the population at a specific frequency p/(2N e ), where p is a transient state (we later show how this assumption can be relaxed). In practice, we consider p = 1 unless stated otherwise.
The probability of transitioning from state p to state x in time t is simply $P p , x t$, or equivalently $Q p , x t$ since both p and x are transient states. Since the Markov chain is absorbing,
$∑ t = 0 ∞ Q p , x t = ( I − Q ) p , x − 1$
(1)
is finite34, where I is the identity matrix. This finiteness allows us to fix x and p and specify a probability distribution of the allele age.
$f p , x (t)= Q p , x t ∑ t = 0 ∞ Q p , x t = Q p , x t ( I − Q ) p , x − 1$
(2)
A complete measure theoretic construction of this distribution can be found in the supplementary material S1 Appendix. The exact moments of this distribution can be written in terms of the matrix Q by using matrix sum identities. We show the first three below using [A] b,c to denote the entry in the b-th row and c-th column of matrix A.
$μ 1 = ∑ t = 0 ∞ t f p , x (t)= ∑ t =0 ∞ t Q p , x t ∑ t = 0 ∞ Q p , x t = [ Q ( I − Q ) − 2 ] p , x [ ( I − Q ) − 1 ] p , x$
(3)
$μ 2 = ∑ t = 0 ∞ t 2 f p , x (t)= ∑ t = 0 ∞ t 2 Q p , x t ∑ t = 0 ∞ Q p , x t = [ Q ( I + Q ) ( I − Q ) − 3 ] p , x [ ( I − Q ) − 1 ] p , x$
(4)
$μ 3 = ∑ t =0 ∞ t 3 f p , x (t)= ∑ t = 0 ∞ t 3 Q p , x t ∑ t = 0 ∞ Q p , x t = [ Q ( Q 2 + 4 Q + 1)( I − Q ) − 4 ] p , x [ ( I − Q ) − 1 ] p , x$
(5)
The expected allele age is given by μ1, and the variance is given by $μ 2 − μ 1 2$.
It is interesting, and relevant if the reader wishes to compute higher moments than those listed above, to notice that the k-th moment μ k is closely linked to the matrix polylogarithm function Lik (Q) by the following equation.
$μ k = [ L i − k ( Q ) ] p , x [ ( I − Q ) − 1 ] p , x$
(6)
where
$L i − s (z)= ∑ k =1 ∞ z k k s = ( z ∂ ∂ z ) s (z (1 − z ) − 1 )$
(7)
Combining equations 6 and 7 therefore allows for the rapid symbolic computation of the closed-form expressions for any moment μ k .
Implementation
Computation of the moments in Equations 3, 4 and 5 can be greatly simplified. This simplification requires obtaining a single LU decomposition of a sparse matrix and using it to solve multiple linear systems by back-substitution. This computational approach is similar to our approach in ref.25, where it was applied to the calculation of quantities such as the probability of fixation and sojourn times.
The first step is to calculate the LU decomposition of (I − Q)T, where T denotes transpose. LU decomposition has a theoretical time complexity on the same order as matrix multiplication, and thus can be as large as O(n3) floating point operations for a dense n × n matrix. However, much faster solutions are possible for sparse matrices, which scale in terms of the number of non-zero entries (e.g., refs35,36). For Wright-Fisher models, Q and hence (I − Q)T, are typically very sparse (at machine precision), and thus a potentially large time savings can be obtained by exploiting this sparsity. Computation of the LU decomposition is by far the most time-intensive step, but we find it is still feasible for population sizes around 105 on typical workstation computers as of the time of writing25. As noted earlier, much larger effective population sizes can be easily considered with the more sparse Moran model.
The second step is to use forward and back substitution to solve multiple linear systems. Given the LU decomposition, this is quite fast and typically requires only a few seconds. First we solve for M1 in
$( I − Q ) T M 1 = e p$
(8)
where e p is the p-th column of the identity matrix. Note that $M 1 T$ is the p-th row of (I − Q)−1, so that the x-th entry of M1 is in fact $( I − Q ) p , x − 1$ as required in the denominator of Equations 3 and 4.
Next, we use the same LU decomposition to solve for M2 in
$( I − Q ) T M 2 = M 1$
(9)
Notice that
$( ( I − Q ) 2 ) T M 2 = ( I − Q ) T ( I − Q ) T M 2 = ( I − Q ) T M 1 = e p$
(10)
so that $M 2 T$ is actually the p-th row of (I − Q)−2. We next take the dot product of $M 2 T$ with the x-th column of Q, which we call Q x .
$M 2 T ⋅ Q x = [ ( I − Q ) − 2 Q ] p , x = [ Q ( I − Q ) − 2 ] p , x$
(11)
which is what was required in the numerator of Equation 3.
We repeat the procedure and solve for M3 in
$( I − Q ) T M 3 = M 2$
(12)
Again, we have
$( ( I − Q ) 3 ) T M 3 = ( I − Q ) T ( I − Q ) T M 2 = ( I − Q ) T M 1 = e p$
(13)
so that $M 3 T$ is the p-th row of (I − Q)−3. In order to compute the numerator of the second moment, we also need the x-th column of Q(I + Q), which we call A x . Note this does not in any way necessitate a full matrix multiplication, as we require only the x-th column. Although this is potentially an expensive O(n2) computation, in practice, sparsity makes it trivially easy. Now we have
$M 3 T ⋅ A x = [ ( I − Q ) − 3 Q ( I + Q ) ] p , x = [ Q ( I + Q ) ( I − Q ) − 3 ] p , x$
(14)
as required in the numerator of Equation 4.
Hence we have calculated all necessary components of the expected value and variance as given in Equations 3 and 4.
The computation of higher moments can be easily implemented as well. To do this, one would first use equations 6 and 7 to obtain closed-form expressions for the needed moments. We recommend using a factored form of the expression so that matrix multiplication is never required in the implementation (it is a convenient property of the polylogarithm that all closed-form expressions of Lis (z) factor completely over the reals). The implementation would then require extending the above algorithm as needed, i.e. iteratively solving
$( I − Q ) T M k + 1 = M k$
(15)
for M k+1, where $M k T$ is the p-th row of (I − Q)k.
We have implemented this approach for the first two moments in our software package Wright-Fisher Exact Solver, WFES25 (available at https://github.com/dekoning-lab/wfes/). In practice it takes only seconds to minutes to calculate the relevant quantities for population sizes under N e = 100,000.
As an aside, we note that the full probability distribution can also be feasibly approximated for small N e to an arbitrary degree of precision by taking the summation in equation 2 to some large finite value.
Simulations
In order to simulate a distribution of allele ages, we must reverse the process, i.e. use the reversed absorbing Markov chain. Specifically, the simulation will start at state x and essentially run backwards in time until it hits state p. It will then either keep going, or stop with a probability equal to the probability that the current visit to state p is the beginning of the chain (when the mutation first entered the population). This backwards simulation can be done by creating a reversed transition matrix and running it in a forwards simulation.
We use the method presented in Chae et al.27, which is as follows. The states of the reversed absorbing Markov chain are {1, 2, …, 2N e − 2, 2N e − 1, stop}, where the stop state is absorbing and all others are transient. The reversed Markov chain does not regard fixation or extinction as absorbing states, and in fact does not allow transition to these states at all.
Let P′ be the matrix of transition probabilities of the reversed absorbing Markov chain. In its canonical form,
$P= ( Q ′ R ′ 0 I )$
(16)
We have
$Q j , k ′ = Q k , j N p , k N p , j and R j , i ′ = ( N p , p − 1 if j = p , i = stop 0 otherwise$
(17)
where Q and N are the transient-to-transient state transition matrix and the fundamental matrix, respectively, of the original Markov chain. (Note that N here is used by convention to represent the fundamental matrix and has no relationship to N e defined above).
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Acknowledgements
We thank Nathan Bryans for helpful comments on the manuscript and German Luna Patiarroy for mathematical insight regarding measure theory. This work was supported by a Discovery Grant to APJdK from the Natural Sciences and Engineering Research Council of Canada (NSERC DG 03651), by an NSERC USRA award (BDS), and by an Alberta Innovates doctoral fellowship to IK The authors gratefully acknowledge infrastructure support from the Canada Foundation for Innovation (CFI LOF #31908, APJdK) and the Alberta Children’s Hospital Research Institute.
Author information
Affiliations
1. University of Calgary, Cumming School of Medicine, Dept. of Biochemistry and Molecular Biology, Calgary, Alberta, Canada
• Bianca De Sanctis
• & A. P. Jason de Koning
2. University of Calgary, Cumming School of Medicine, Department of Biochemistry and Molecular Biology Graduate Program (Bioinformatics stream), Calgary, Alberta, Canada
• Ivan Krukov
• & A. P. Jason de Koning
3. University of Calgary, Cumming School of Medicine, Department of Medical Genetics, and Alberta Children’s Hospital Research Institute, Calgary, Alberta, Canada
• A. P. Jason de Koning
Contributions
A.P.J.d.K. designed the research, B.D.S. formulated the methodology, A.P.J.d.K. implemented the methodology and performed the analyses, I.K. implemented the simulation methodology and performed analyses, and A.P.J.d.K. and B.D.S. wrote the manuscript. All authors reviewed the manuscript.
Competing Interests
The authors declare that they have no competing interests.
Corresponding author
Correspondence to A. P. Jason de Koning. | 2017-12-16 14:43:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 32, "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": 0.789271891117096, "perplexity": 1642.6966150714234}, "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-2017-51/segments/1512948588251.76/warc/CC-MAIN-20171216143011-20171216165011-00055.warc.gz"} |
http://codeforces.com/problemset/problem/804/B | B. Minimum number of steps
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
We have a string of letters 'a' and 'b'. We want to perform some operations on it. On each step we choose one of substrings "ab" in the string and replace it with the string "bba". If we have no "ab" as a substring, our job is done. Print the minimum number of steps we should perform to make our job done modulo 109 + 7.
The string "ab" appears as a substring if there is a letter 'b' right after the letter 'a' somewhere in the string.
Input
The first line contains the initial string consisting of letters 'a' and 'b' only with length from 1 to 106.
Output
Print the minimum number of steps modulo 109 + 7.
Examples
Input
ab
Output
1
Input
aab
Output
3
Note
The first example: "ab" → "bba".
The second example: "aab" → "abba" → "bbaba" → "bbbbaa". | 2020-01-28 02:17:09 | {"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": 0.2560245096683502, "perplexity": 1618.8910859657315}, "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-2020-05/segments/1579251737572.61/warc/CC-MAIN-20200127235617-20200128025617-00169.warc.gz"} |
http://crypto.stackexchange.com/tags/linear-cryptanalysis/new | # Tag Info
We define $T$ as: $T(B) = (b_0,b_1,b_2,b_3)$ We use $D$ to represent the difference of $X$ and $Y$: $D = X \oplus Y$ Compute $T(D)$: $T(D) = (d_0,d_1,d_2,d_3)$ $= (x_0 \oplus y_0, x_1 \oplus y_1, x_2 \oplus y_2, x_3 \oplus y_3)$ $=(x_0,x_1,x_2,x_3) \oplus (y_0,y_1,y_2,y_3)$ which is by definition of $T$: $T(X) \oplus T(Y)$ | 2014-11-28 12:20:22 | {"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": 0.923124372959137, "perplexity": 113.20278893707196}, "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-2014-49/segments/1416931010166.36/warc/CC-MAIN-20141125155650-00068-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/tags/thermal-field-theory/hot | # Tag Info
## Hot answers tagged thermal-field-theory
9
Here is a proof following Ojima, "Lorentz Invariance vs. Temperature in QFT", Letters in Mathematical Physics (1986) Vol. 11, Issue 1 (1986) 73-80. The first two pages of the paper are available for free here, but the website wants money for more of the paper. (Click the orange "Look Inside" button if the paper doesn't open automatically.) Fortunately, the ...
6
We calculate the free energy (density) for the Higgs field $\phi$ at finite temperature. In the Standard Model, this looks like $\mathcal{F}_{SM}(\phi,T) = -\frac{\pi^2}{90}g_* T^4+V_{SM}(\phi, T) \ ,$ where $g_*$ is the number of degrees of freedom in the SM ($g_*=106.75$). The potential has the form $V_{SM}(\phi,T) = D(T^2-T_0^2)\phi^2 - ET\phi^3+\frac{... 5 An enclosed cavity near thermal equilibrium at some temperature will be filled with blackbody radiation, which has a well-defined spectrum that depends only (!) on the temperature of the cavity. If you poke a small hole in the cavity, some of those photons will leak out of the hole as thermal radiation. For instance, the pupils of your eyes are dark ... 4 There are two particle candidates for dark matter, and both are still in the realm of hypothesis. These particles are candidates exactly because they cannot decay to something lighter, they are stable. Neutrinos, which also cannot decay to something lighter, has such a small mass that they cannot model the way dark matter is attracted to gravitational ... 3 You're using different boundary conditions around the thermal circle, so there's no reason to expect the results to be related. When you use periodic boundary conditions around the thermal circle, it's equivalent to inserting a factor$(-1)^F$in the trace $$Z = {\rm Tr\ } (-1)^F e^{-\beta H}$$ and because of supersymmetry, only the groundstates can ... 3 That trace is nothing but the sum of the non-vanishing$\rho$eigenvalues to the 1/r power and taking their multiplicities into account. Since those eigenvalues$\lambda$belong to$(0,1]$, then$\lambda^{1/r}\to 1$as$r\to +\infty$. In summary$tr(\rho^{1/r})$tends to the number of non-vanishing eigenvalues of$\rho$taking multiplicities into account. In ... 3 That is the same situation as in QFT, as soon as you drop Poincaré invariance. I mean in curved spacetime. There you may have uncountably many inequivalent representations of the same algebra of observables just by varying some continuous parameter (curvatures). If you put all these reps in orthogonal sectors in a common Hilbert space, it must be non.... 3 Roughly speaking solid matter is on a lattice form, A three-dimensional lattice filled with two molecules A and B, here shown as black and white spheres. The molecules fit like LEGO , the forces tying them together are mainly the spill over electric field forces , attractive and repulsive forming the patterns of the lattice. In a single crystal one ... 2 1) What you call$\rho$should really be called$\epsilon$(this is the energy density, not the particle density). 2) The thermodynamic variables$\epsilon$and$P$are expectation values of certain operators in a thermal ensemble. You should not confuse equations for the operators with equations for thermodynamic quantities. 3) The operator that ... 2 There seems to be some confusion in your question between thermal fluctuations and quantum fluctuations, so I will try to address both of them in my answer. Spontaneous symmetry breaking occurs when the ground state of the Hamiltonian does not exhibit the full symmetries of the Hamiltonian. In other words, there are multiple degenerate configurations with ... 2 The point is that to achieve the sum over Matsubara frequencies $$\sum_{n} g(i\omega_n)$$ we can use a contour integral $$\oint_C g(z) f(z)$$ with the contour described in fig 1 here, so long as we choose an$f(z)$with simple poles exactly at the Matsubara frequencies$\omega_n$. This determines$f(z)$to be proportional to the Bose-Einstein distribution. ... 2 A Euclidean correlation function may be interpreted as a Lorentzian expectation value by "cutting" the path integral and continuing the time coordinate. Let me review how this procedure relates Euclidean correlation functions on a closed manifold$M\times S^1_\beta$to thermal expectation values in a Lorentzian quantum field theory on$M \times \mathbb{R}...
2
Two remarks: Essentially every function can be regarded as a distribution, although the converse is not true. In this sense, you can indeed regard $(e^{\beta |p_0|} - 1)^{-1}$ as a distributions. Distributions of this kind are known as regular distributions. Non-regular distributions are known as singular distributions. It is not true that you cannot ...
2
First write $\rho, y$ in the eigenbasis of the Hamiltonian, $$\rho = \sum_n p_n |n\rangle \langle n| \\ y = \sum_n q_n |n\rangle \langle n|$$ with $q_n^4 = p_n$, in particular $0\le q_n \le 1$. In this basis the inequality becomes $$\sum_{n,m} |V_{n,m}|^2 q_n^{1+\eta} q_m^{3-\eta} \le \sum_{n,m} |V_{n,m}|^2 q_n q_m^2$$ At this point the ...
1
I am not going to go into the black hole stuff, but thermalization is never loss of matter and/or probability, but - so to say - only a "rearrangement" of density matrix eigenstates and corresponding probabilities. In other words, it is an evolution from an arbitrary initial state $\rho$, which may be pure ($\;\rho = \rho^2 = |\Psi\rangle\langle \Psi |\;$) ...
1
The effective action is defined so that $\frac{\delta \Gamma \left[ \phi \right]}{\delta \phi^i \left( x \right)}=J_{\phi}^i \left( x \right)$, where $J$ is a classical current, so we see that this definition coincides with the usual definition of the action in classical field theory. Moreover, the procedure of evaluation of this action involves evaluation ...
1
Concerning OP's explicit example $$\frac{\delta(p^2 + m^2 )}{e^{\beta |p_0|} - 1} , \qquad\beta\neq 0. \tag{D}$$ The massive case $m\neq 0$. Then the singularity of $\frac{1}{e^{\beta |p_0|} - 1}$ does not overlap with the support of $\delta(p^2 + m^2 )$, so the product distribution is mathematically well-defined. The massless case $m=0$. Then the ...
1
No. Consider for example observables of the form $\mathcal O = e^{\beta H/2}$; the value exponentially increases as the probability exponentially decreases and you can get what's in principle an infinite sum of constant terms.
1
In general, the expectation value $tr(A e^{-\beta H})$ is not defined for a generic selfadjoint operator $A$ (of the form ${\cal O}^2$ or not) if it is unbounded as it is the standard situation in QFT. $tr(A e^{-\beta H})$ however converges if (with the written order!) the range of $e^{-\beta H}$ belongs to the domain of $A$ and the composition is trace ...
1
I think the second formula is also not very profound. This is just based on the fact that the chemical potential $$\Delta S = \int dt \, \mu Q$$ enters the action like the zeroth component of an (imaginary) $U(1)$ gauge field $A_\beta=(i\mu,\vec{0})$. Note that this is the Polyakov line for a $U(1)$ background field, not the Polyakov line of (for example)...
1
Temperature is a macroscopic measure, it is statistical , a thermodynamic variable. At best, an ensemble of particles with a statistical distribution of energy in a thermodynamic equilibrium will have a kinetic temperature defined as connected with the average kinetic energy of the ensemble of particles. On the other hand, the scattering cross section ...
1
You can always go straightforward and use that you know how $\hat{a}$ and $\hat{a}^\dagger$ act on the basis $|n\rangle$. Let's write down straightforwadly what $\mathrm{Tr}$ is $$\mathrm{Tr}\Big(\hat{O}e^{-\beta\hat{H}}\Big)=\sum_n\langle n|\hat{O}e^{-\beta\hat{H}}|n\rangle=\sum_n e^{-\beta(n+1/2)}\langle n|\hat{O}|n\rangle$$ Now,...
1
Yes emissivity depends on temperature: $$\epsilon(T)= \frac{E(T)}{E_b (T)}$$ $\epsilon$ is total hemespherical emissivity. $E$ is the emissive power of the actual body which depend on temperature and $E_b$ is the emissive power of a blackbody: $E_b(T)=\sigma T^4$
1
Collisions of a state with other particles, present at finite density, influence the life-time of the state as energy can be transferred to those other particles during inelastic collisions (i.e. they can change state). This change in life-time is related to a change in the width via the uncertainty relation and gives the thermal width. There is a physical ...
1
Finite temperature Feynman rules are simply zero temperature Feynman rules for Euclidean ($t\to i\tau$) QFT in periodic imaginary time. So instead of continuous values for the momenta, you will have a discrete spectrum for the timelike moments (such as in the infinite potential well in basic quantum mechanics). It's called the Matsubara formalism, if you ...
1
Thermal expansion from an atomistic perspective: The energetic potential between two atoms can be approximated by two exponential functions, one for the attractive force between the atoms, one for the repulsive force. The superposition of these two force fields has a minimum at a certain distance. Examples for such empirical potentials are Stillinger-Weber, ...
1
Let's define $T_{EW}$ the temperature where the coefficient $m^2_H(T)$ of the operator $H^2$ in the SM lagrangian vanishes: $$m_H^2(T=T_{EW})=0\,.$$ For $T>T_{EW}$ the Higgs vev is vanishing, the EW symmetry in unbroken, and the elementary particles are massless. For $T<T_{EW}$ the the vev is non-vanishing, $v_T\propto -m_H^2/\lambda\neq 0$, the EW ...
1
A simple (and quite accurate) answer is that quantum fluctuations are the fluctuations that exist at zero temperature. What it means is that even at zero temperature, there might be fluctuations in the measurements of observables, which does not happen for classical systems at zero temperature, due to the non-commutativity of the dynamical and potential ...
1
Found a sketch of a proof on a referee's report on a paper RELATIVISTIC INVARIANCE OF THE VACUUM by Adam Bednorz. The referee's sketch is: Comment Hundreds of calculations in Fnite temperature Feld theory have been published. To my knowledge, none of these calculations have ever conflicted with Lorentz invariance in the limit $\beta \to \infty$ ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2019-08-26 00:18:54 | {"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": 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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9296967387199402, "perplexity": 321.21766664464394}, "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-2019-35/segments/1566027330913.72/warc/CC-MAIN-20190826000512-20190826022512-00458.warc.gz"} |
https://proofwiki.org/wiki/Mathematician:Eratosthenes_of_Cyrene | # Mathematician:Eratosthenes of Cyrene
Jump to navigation Jump to search
## Mathematician
Greek geometer and astronomer best known for his estimate of the size of the Earth.
Also famous for his Sieve of Eratosthenes.
Famously a correspondent with Archimedes of Syracuse, who addressed the preamble of his The Method to him.
Custodian of the Library of Alexandria.
Went blind in his old age, and it is said that he committed suicide by starving himself to death.
Greek
## History
• Born: c. 276 BCE, Cyrene, North Africa (now Shahhat, Libya)
• Died: c. 194 BCE, Alexandria, Egypt
## Theorems and Topics
Results named for Eratosthenes of Cyrene can be found here.
## Publications
• Platonicus (on the philosophy of Plato)
• Hermes
### Other works
These works are referred to by other ancient writers, but are now believed lost:
• On the measurement of the Earth
• On means (now lost: recognised by Pappus as one of the great works of geometry)
## Critical View
... a good mathematician but quite a fop.
-- 1937: Eric Temple Bell: Men of Mathematics: Chapter $\text{II}$: Modern Minds in Ancient Bodies
## Also known as
Ancient Greek: Ἐρατοσθένης. | 2021-10-25 07:06:25 | {"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": 0.8156476020812988, "perplexity": 13949.796465865527}, "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-2021-43/segments/1634323587655.10/warc/CC-MAIN-20211025061300-20211025091300-00195.warc.gz"} |
https://mmediting.readthedocs.io/en/v0.12.0/inpainting_models.html | # Inpainting Models¶
## DeepFillv1 (CVPR’2018)¶
### Abstract¶
Recent deep learning based approaches have shown promising results for the challenging task of inpainting large missing regions in an image. These methods can generate visually plausible image structures and textures, but often create distorted structures or blurry textures inconsistent with surrounding areas. This is mainly due to ineffectiveness of convolutional neural networks in explicitly borrowing or copying information from distant spatial locations. On the other hand, traditional texture and patch synthesis approaches are particularly suitable when it needs to borrow textures from the surrounding regions. Motivated by these observations, we propose a new deep generative model-based approach which can not only synthesize novel image structures but also explicitly utilize surrounding image features as references during network training to make better predictions. The model is a feed-forward, fully convolutional neural network which can process images with multiple holes at arbitrary locations and with variable sizes during the test time. Experiments on multiple datasets including faces (CelebA, CelebA-HQ), textures (DTD) and natural images (ImageNet, Places2) demonstrate that our proposed approach generates higher-quality inpainting results than existing ones.
### Citation¶
@inproceedings{yu2018generative,
title={Generative image inpainting with contextual attention},
author={Yu, Jiahui and Lin, Zhe and Yang, Jimei and Shen, Xiaohui and Lu, Xin and Huang, Thomas S},
booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
pages={5505--5514},
year={2018}
}
### Results and models¶
Places365-Challenge
DeepFillv1 square bbox 256x256 3500k Places365-val 11.019 23.429 0.862 model | log
CelebA-HQ
DeepFillv1 square bbox 256x256 1500k CelebA-val 6.677 26.878 0.911 model | log
## DeepFillv2 (CVPR’2019)¶
### Abstract¶
We present a generative image inpainting system to complete images with free-form mask and guidance. The system is based on gated convolutions learned from millions of images without additional labelling efforts. The proposed gated convolution solves the issue of vanilla convolution that treats all input pixels as valid ones, generalizes partial convolution by providing a learnable dynamic feature selection mechanism for each channel at each spatial location across all layers. Moreover, as free-form masks may appear anywhere in images with any shape, global and local GANs designed for a single rectangular mask are not applicable. Thus, we also present a patch-based GAN loss, named SN-PatchGAN, by applying spectral-normalized discriminator on dense image patches. SN-PatchGAN is simple in formulation, fast and stable in training. Results on automatic image inpainting and user-guided extension demonstrate that our system generates higher-quality and more flexible results than previous methods. Our system helps user quickly remove distracting objects, modify image layouts, clear watermarks and edit faces.
### Citation¶
@inproceedings{yu2019free,
title={Free-form image inpainting with gated convolution},
author={Yu, Jiahui and Lin, Zhe and Yang, Jimei and Shen, Xiaohui and Lu, Xin and Huang, Thomas S},
booktitle={Proceedings of the IEEE International Conference on Computer Vision},
pages={4471--4480},
year={2019}
}
### Results and models¶
Places365-Challenge
DeepFillv2 free-form 256x256 100k Places365-val 8.635 22.398 0.815 model | log
CelebA-HQ
DeepFillv2 free-form 256x256 20k CelebA-val 5.411 25.721 0.871 model | log
## Global&Local (ToG’2017)¶
### Abstract¶
We present a novel approach for image completion that results in images that are both locally and globally consistent. With a fully-convolutional neural network, we can complete images of arbitrary resolutions by flling in missing regions of any shape. To train this image completion network to be consistent, we use global and local context discriminators that are trained to distinguish real images from completed ones. The global discriminator looks at the entire image to assess if it is coherent as a whole, while the local discriminator looks only at a small area centered at the completed region to ensure the local consistency of the generated patches. The image completion network is then trained to fool the both context discriminator networks, which requires it to generate images that are indistinguishable from real ones with regard to overall consistency as well as in details. We show that our approach can be used to complete a wide variety of scenes. Furthermore, in contrast with the patch-based approaches such as PatchMatch, our approach can generate fragments that do not appear elsewhere in the image, which allows us to naturally complete the image.
### Citation¶
@article{iizuka2017globally,
title={Globally and locally consistent image completion},
author={Iizuka, Satoshi and Simo-Serra, Edgar and Ishikawa, Hiroshi},
journal={ACM Transactions on Graphics (ToG)},
volume={36},
number={4},
pages={1--14},
year={2017},
publisher={ACM New York, NY, USA}
}
### Results and models¶
Note that we do not apply the post-processing module in Global&Local for a fair comparison with current deep inpainting methods.
Places365-Challenge
Global&Local square bbox 256x256 500k Places365-val 11.164 23.152 0.862 model | log
CelebA-HQ
Global&Local square bbox 256x256 500k CelebA-val 6.678 26.780 0.904 model | log
## PConv (ECCV’2018)¶
### Abstract¶
Existing deep learning based image inpainting methods use a standard convolutional network over the corrupted image, using convolutional filter responses conditioned on both valid pixels as well as the substitute values in the masked holes (typically the mean value). This often leads to artifacts such as color discrepancy and blurriness. Post-processing is usually used to reduce such artifacts, but are expensive and may fail. We propose the use of partial convolutions, where the convolution is masked and renormalized to be conditioned on only valid pixels. We further include a mechanism to automatically generate an updated mask for the next layer as part of the forward pass. Our model outperforms other methods for irregular masks. We show qualitative and quantitative comparisons with other methods to validate our approach.
### Citation¶
@inproceedings{liu2018image,
title={Image inpainting for irregular holes using partial convolutions},
author={Liu, Guilin and Reda, Fitsum A and Shih, Kevin J and Wang, Ting-Chun and Tao, Andrew and Catanzaro, Bryan},
booktitle={Proceedings of the European Conference on Computer Vision (ECCV)},
pages={85--100},
year={2018}
}
### Results and models¶
Places365-Challenge | 2022-10-05 18:40:11 | {"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": 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": 0.31933170557022095, "perplexity": 3377.87956478415}, "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/1664030337663.75/warc/CC-MAIN-20221005172112-20221005202112-00204.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-11-sequences-induction-the-binomial-theorem-chapter-test-page-860/2 | ## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)
$$\displaystyle a_1=4\\ a_2=14 \\ a_3=44 \\a_4=134 \\ a_5= 404$$
We are given that $a_n=3a_{n-1}+2$ and $a_1=4$. In order to determine the remaining values, we will have to substitute $n=2,3,4,5$ into the given sequence: $$\displaystyle a_2=3a_1+2=(3)(4)+2=14 \\ a_3=3 a_2+3=(3) (14)+2=44 \\a_4=3 a_3+2=(3)(44)+2=134 \\ a_5=3a_4+2=(3)(134)+2=404$$ | 2021-10-17 07:30:21 | {"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": 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": 0.8865649104118347, "perplexity": 454.5883925022471}, "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/1634323585121.30/warc/CC-MAIN-20211017052025-20211017082025-00453.warc.gz"} |
https://math.stackexchange.com/questions/673526/how-to-change-variables-in-a-surface-integral-without-parametrizing/673942 | How to change variables in a surface integral without parametrizing
This is a doubt that I carry since my PDE classes.
Some background (skippable):
In the multivariable calculus course at my university we made all sorts of standard calculations involving surface and volume integrals in $$R^3$$, jacobians and the generalizations of the fundamental theorem of calculus. In order to make those calculations we had to parametrize domains and calculate differentials.
A couple of years later I took a PDE course. We worked with Evans' Partial differential equations book. This was my first experience with calculus in $$\mathbb R^n$$ and manipulations like $$\text{average}\int_{B(x,r)}f(y)\,dy= \text{average}\int_{B(0,1)}f(x+rz)\,dz.$$ This was an ordinary change of variables. $$y=x+rz,\,\,dy=r^n\,dz$$ and the mystery was solved. Like in that case, I was able to justify most of these formal manipulations after disentangling definitions.
That aside, I found these quick formal calculations to be very powerful.
However,
I realized that I wasn't able to justify this: $$\text{average} \int_{\partial B(x,r)}f(y)dS(y)= \text{average}\int_{\partial B(0,1)}f(x+rz)\,dS(z).$$ I have some vague idea of what's happening: the same substitution as before, but this time the jacobian is $$r^{n-1}$$ because the transformation is actually happening between regions which "lack one dimension". Also, I see some kind of pattern: a piece of arc-length in the plane is $$r\,d\theta$$, a piece of sphere-area is $$r^2 \sin\theta \, d\phi \,d\theta$$, "and so on". Maybe some measure-theoretic argument can help me: I know, roughly speaking, that for any measure $$\mu$$, $$\int_\Omega f\circ \phi \,d\mu=\int_{\phi(\Omega)} f \, d(\mu\circ\phi^{-1}).$$ I'd say $$\phi(z)=(z-x)/r$$ and $$\phi^{-1}(y)=ry+x$$, but I actually don't know how $$dS(y)$$ looks like "as a measure" (It's not a product measure or a restriction of one, but it somehow relates to Lebesgue's in $$\mathbb R^n$$...). Why would I conclude that $$dS(y)\circ \phi^{-1}=r^{n-1}dS(z)$$? I have an intuition, but either I lack the mathematical concepts and definitions to express it or I'm just too confused. Is there some theory that I could learn in order to understand? Maybe something about the measure $$dS$$. Is it expressible in terms of the Lebesgue measure in some way? Or set-theoretically, maybe, without having to resort to $$n-1$$ parameters and complicated relations?
Maybe all of this would not have been a problem if I had ever mastered n-dimensional spherical coordinates. But even so, more generally, is there a way of changing variables when I'm integrating over a subregion of "dimension$$" without necessarily parametrizing?
Sorry for the vagueness, but I don't really know what to ask for exactly.
Note: I saw some of the answers to this post, but none of them were deep enough in the direction I intend.
Note II: If there are no general methods or theories, maybe restricting to linear transformations, to Lebesgue measure exclusively, or to subregions defined by simple expressions like $$g(x)=C$$ or $$g(|x|)=C$$ could get me somewhere.
Edit: I have not yet studied differential geometry, which has been mentioned in a comment. I added it to the tags.
• There is a unifying framework for the question that's bothering you, differential-geometry. In this case, we think of $dS$ as an "$(n-1)$-differential form" in $\Bbb R^n$, and integrate it along the "$(n-1)$-chain" which is your sphere. What you require is called a "pull-back" of $dS$ via the differentiable mapping $y\mapsto x+ry$. Hopefully, this might point you in the right direction. – Jonathan Y. Feb 12 '14 at 9:20
• @JonathanY. Do you know any good books on the subject? Rigorous, with good prose, motivations, nice examples, maybe even illustrations (not a requirement, though). Also, is this some specific subtopic? The Wikipedia article mentions "Riemannian manifolds", "tensors", the Theorema Egregium, and "symplectic geometry". I wouldn't know exactly where to put my attention. – dafinguzman Feb 13 '14 at 8:03
• While I haven't formally studied differential geometry in its broadest setting, I have used parts of it and in this aspect the books Introduction to Smooth Manifolds and Riemannian Manifolds by John Lee were very...smooth. =) – Mark Fantini Feb 13 '14 at 8:14
• @dafinguzman myself, I like Spivak's "calculus on manifolds" or "comprehensive introduction to differential geometry". You're looking for integration on chains/manifolds and the (proper instance of the) change of variable theorem. – Jonathan Y. Feb 13 '14 at 12:20
No parametrization is needed, just some careful uncovering of definitions. Maybe the following can help:
Let $S^n(r)=\{x\in\mathbb R^{n+1}|\|x\|=r\}\subset\mathbb R^{n+1}$ (the sphere of radius $r>0$ centered at the origin).
There is a natural volume form $\mu_{r}$ on $S^n(r)$, i.e. a nowhere-vanishing $n$-form, induced by the volume form $\mu=dx_1\wedge\ldots \wedge dx_{n+1}$ on $\mathbb R^{n+1}$ and the "outward" unit normal $N:S^n(r)\to\mathbb R^{n+1}$, $N(x)=x/r$, and is defined as follows:
$$\mu_r(x)(v_1,\ldots,v_n):=\mu(x)(N(x), v_1, \ldots, v_n)$$ for all $v_1, \ldots, v_n\in T_x S^n(r).$
(More informally, we write $\mu_r=\mu/dr$. Note that $dr(N)=1$).
Next let $\Phi:S^n(1)\to S^n(r)$ be defined by $\Phi(x)=rx$. Then $$\Phi^*\mu_r=r^n\mu_1.$$ This you can prove without any parametrization, just from the definitions above and generalities like the chain rule, pull back, etc.
It then follows, by the change of variables formula, that for any continuous function $f:S^n(r)\to\mathbb R$, $$\int_{S^n(r)}f\mu_r=\int_{S^n(1)} \Phi^*(f\mu_r)=r^n\int_{S^n(1)} (f\circ\Phi) \mu_1.$$
Makes sense?
• Could you maybe elaborate some definitions? I don't know what $n$-forms, $T_x$ or "pull back" are. As you present it, it seems that $\mu$ is a product measure, somehow. Also, I don't know what $dr(N)$ is supposed to mean. I can guess it means (by the notation) that "surface infinitesimal elements" get amplified by $1$ near the points which are on the sphere. – dafinguzman Feb 13 '14 at 8:20
• @dafinguzman: sorry, cannot elaborate, it's too long. I am using standard language taught in most differential geometry courses. You need to take such a course or pick a textbook (popular ones are Spivak, Lee, Guillemin+Pollack. My personal favorite is Arnold's mathematical methods of classical mechanics, intuitive and rigorous at the same time; but it's not easy as a first textbook). Now I am sure it is possible to translate my answer to a more elementary language (although less rigorous). Perhaps another user of the site will try to do it (or even myself if I find the time and energy). – Gil Bor Feb 13 '14 at 16:08
• OK, after a long time I read through some chapters of Spivak's Calculus with Manifolds and now I get what you are trying to say. Trying to adapt to the definitions in Spivak, though, I find that $(\Phi ^ * \mu_r)(x)((v_1)_x,..., (v_n)_x) = r^n \mu (rx) (x, (v_1)_{rx}, ..., (v_n)_{rx})$. Then I'm tempted to say that equals $r^n \mu (x) (x, (v_1)_{x}, ..., (v_n)_{x})$ by "invariance of $\mu$ under translations", which is known for Lebesgue measure in $R^n$. Is that correct? Thanks for your answer: once the definition of "$dS$" is made explicit everything becomes clearer. – dafinguzman Jun 15 '15 at 22:48
I know this is an old question, but I thought this explanation might be helpful to some.
By definition (in $\mathbb R^3$):
$$\int_{\partial B(\pmb x,r)}f(\pmb y)dS(\pmb y)= \int_U f(\pmb y(s,t))\left\|\frac{\partial\pmb y}{\partial s}\times\frac{\partial\pmb y}{\partial t}\right\|dsdt$$
Now, observe that $f(\pmb y)=f(\pmb x+r(\frac{\pmb y-\pmb x}{r}))$, and that if $\pmb y(s,t)$ is a parametrization of $\partial B(\pmb x,r)$ for $(s,t)\in U$, then $\frac{\pmb y(s,t)-\pmb x}{r}$ is a parametrization of $\partial B(\pmb 0,1)$ for $(s,t)\in U$. Finally we observe that
$$\left\|\frac{\partial\pmb y}{\partial s}\times\frac{\partial\pmb y}{\partial t}\right\|= r^2\left\|\frac{\partial}{\partial s} \left (\frac{\pmb y-\pmb x}{r} \right )\times\frac{\partial }{\partial t} \left (\frac{\pmb y-\pmb x}{r} \right )\right\|$$
So if we let $\pmb z(s,t)=\frac{\pmb y(s,t)-\pmb x}{r}$, then we have
$$\int_U f(\pmb y(s,t))\left\|\frac{\partial\pmb y}{\partial s}\times\frac{\partial\pmb y}{\partial t}\right\|dsdt= r^2\int_U f(\pmb x +r\pmb z(s,t))\left\|\frac{\partial\pmb z}{\partial s}\times\frac{\partial\pmb z}{\partial t}\right\|dsdt\\= r^2\int_{\partial B(\pmb 0,1)}f(\pmb x+r\pmb z)dS(\pmb z)$$
• This is great! I see that the cross product gets in the way of immediately generalising this reasoning to $\mathbb R^n$. I'm guessing this is more or less the role of the wedge products in Gil Bor's answer. – dafinguzman Sep 25 '17 at 6:10
• @dafinguzman I think you can generalize if you use the fact that in $\mathbb R^n$ $\int_{\partial B(\pmb x,r)}f(\pmb y)dS(\pmb y)= \int_U f(\pmb y(\pmb z))\left|\det\left(\frac{\partial\pmb y}{\partial z_1}, \dots,\frac{\partial\pmb y}{\partial z_{n-1}},\pmb n\right)\right|d^{n-1}\pmb z$, where $\pmb n$ is the normal vector to the surface, and that $\pmb n$ does not change when the surface is scaled and translated. – user5753974 Sep 26 '17 at 10:08
• @user5753974 Hi, is $\pmb z$ a $(n-1)$-tuple? And does $d^{n-1}\pmb z$ mean $dz_1 dz_2 ... dz_{n-1}$? Thanks! – Sam Wong Oct 11 '18 at 5:48
I think the measure theoretic approach works fine, note that the surface measure is n-1 dimensional Hausdorff measure, in general for the s-dimensional Hausdorff measure we have $$H^{s} (rA)=r^{s} H^{s}(A)$$ and this measure is translation invariant. Now use the measure theoretic change of variable formula.
• Does this mean that the measure $dS$ in the integral over $\partial B$ is the $n-1$ dimensional Hausdorff measure? I hadn't thought of it that way. – dafinguzman Feb 19 at 21:34
• Yes, the surface measure $dS$ which is used on the boundary of n-dimensional objects is exactly the $n-1$ Hausdorff measure on the boundary. – Arya Jamshidi Feb 20 at 5:10 | 2019-07-21 01:35:29 | {"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": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9304451942443848, "perplexity": 3708.237839381101}, "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-2019-30/segments/1563195526799.4/warc/CC-MAIN-20190720235054-20190721021054-00408.warc.gz"} |
https://adrian-007.eu/2019/02/01/raspberry-pi-cctv/ | # Surveillance camera based on Raspberry Pi
Back in 2018, when construction of my new home began, I ended up driving to construction site almost every day - that was wasting approximately 1 hour every day just to drive back and forth. Because of that huge time waster and obvious reason that “things” gets “displaced” on construction sites, I’ve decided that I need some kind of surveillance on-site.
First and obvious choice would be some cheap IP camera, but given conditions under which camera would work, I needed to consider some prerequisites:
• power delivery: someone could plug out power cable and that would be pretty much it for surveillance, hence a requirement was backup battery power supply
• ability to store live feed in remote location: since there would be no way of accessing camera remotely (dynamic IP, firewall), camera would have to store live feed in remote location - i.e. private, publicly accessible server
• Internet connection: since I don’t have wired connection on-site, my only choice would be to use 3G - fortunately I already had a USB dongle and a card with 100 GB of mobile transfer per month already in place
• given that 3G connection can give low upload bandwidth, camera had to have option to customize resolution, frame rate, codec options (preferably variable bit rate)
• ability to capture in poor lighting / at night
After doing a little bit of digging I realized that already available solutions might cost quite a bit and not necessarily meet all the requirements / expected quality of service. Since I already had some parts lying around and to be honest - because it would be fun - I’ve decided that I will build my own camera. And so it began…
## Hardware
Raspberry Pi Zero was my first choice, but due to some unexpected problems with power supply, which fried my board, I ended up using regular Raspberry Pi 2 I had in stock. I was already familiar with this platform, it runs under vanilla GNU/Linux distribution, has decent performance, hardware-accelerated H.264 encoding and most importantly - dedicated camera port.
Parts listing:
1. Raspberry Pi Zero ($5) replaced with Raspberry Pi 2 Model B (in stock,$25)
2. Waveshare Camera HD Night Vision H (~$40) 3. ZTE MF820D 3G USB modem (~$20) replaced with TP-Link TL-WN722N Wi-Fi card (in stock, ~$10) and Huawei E5372 Wireless 3G router (~$13)
4. Battery - AKUELL BT-02-108 12V 12Ah (~$30) 5. Prototype circuit boards (in stock, ~$0.30)
6. Main power supply with voltage regulator based on LM317T (~$0.25/piece) acting as regular power supply and a buffered battery charger 7. Secondary power supply with voltage regulator based on LM2576T for 5V line (~$1/piece)
8. Secondary power supply with voltage regulator based on LM2576T for 3.3V line (~$1/piece) 9. 230/2x9V transformer (~$15)
10. PCF8563 RTC clock (~$0.55) 11. Fans, resistors, capacitors, coils, diodes, wires, connectors, LEDs (all in stock, around$20 in total)
12. Case (~$7) In total (discarding replaced parts - I had to pay for my mistakes ;) it’s around$165. Is it or isn’t it much for a camera? It depends - you probably could find a solution that would meet most of the requirements (though as far as I know, most products doesn’t provide sufficient backup power supply and means for providing Internet connection).
### Raspberry Pi with camera module
Main device of the project. It’s small, supports I2C (so I could attach RTC), has little appetite for power, decent performance, dedicated camera port and a powerful support from software side - excellent choice for a home-made project, as thousands have already proven. And I already had a history with it…
Camera module - night vision version. It provides decent video quality at day and night, considering poor lighting and only 9W for artificial light source. This is a version with wide-angle lense, so there’s more area covered by it. Initial idea was to use 3 separate camera with dedicated multiplexer, but after some testing, I concluded that:
• hardware multiplexer has poor quality if you want to capture live video feeds from 3 cameras (frames were interlacing with each other) - I’ve written whole capture software basing on raspivid and I’ve managed to actually capture two simultaneous streams, but every few seconds, for a second, I got capture from wrong camera. I wasn’t able to make muxing synchronization to work.
• single video stream, H.264 hardware-encoded, 1280x720 @ 5 FPS, grayscale with quantization set to 20 and variable bit rate produced constant data flow with rates up to 1 Mb/s. For 3 cameras that would be 3 Mb/s at peak - that would create quite a data backlog (though that would be extremely rare case). Due to mobile internet connection constraints (when I exceed data limit, my transfer rate goes down to 1 Mb/s), this was another deal breaker for 3 cameras setup.
Having those two problems that couldn’t be simply solved, I abandoned this approach and went with a single camera setup (but I still feel a little bit unsatisfied).
### Internet connection
My first approach to Internet connection was to use 3G dongle, in that case ZTE MF820D. As expected, it turned out that Raspberry Pi cannot provide enough power to the device - it was shutting down as soon as it tried to connect to cellular network. That forced me to provide means for powering dongle externally. Since I’ve been building my own voltage regulator specifically for 5V line, it seemed easy enough, right? Just put a 5V line into the USB cable, solder USB port onto the board, and voila - modem should be stable… Not quite. Because I’ve been using prototype board, connections were rather crude. Elements placement, along with loose wires didn’t help either - there was just too much noise around the device. At first modem was pretty stable so I considered that this part is done, but once I put everything together and ran stress tests, it turned out that modem couldn’t last 2 days without locking itself up.
Simplest solution? Just reset the device! That should work, except that modem had external power supply, so I couldn’t just power-cycle it. Resetting USB hub inside Raspberry Pi? The device didn’t show up after reset. So I thought that since I have a power signal from Raspberry Pi over USB port, I could put external power for modem through MOSFET transistor and if there’s no power on USB port, the device would reset. And although this could solve the power-cycle problem, it didn’t help the modem - it was still locking up, only physical reconnect was helping out. Because it’s a no-go for a surveillance device to have such instability issues, I’ve decided to ditch USB dongle and buy Wi-Fi router with 3G capabilities. I found used Huawei E5372 on auction and bought it for 13 bucks. It has an internal backup battery, supports 2.4 and 5 GHz networks and pretty much just works out of the box. Although it may seem like a workaround, it actually turned out to be a better solution - placement of the camera would determine where 3G antennas would be. With mobile router, I could put it in a place where I have best signal and still be able to serve Internet connection to camera wherever it would be placed, within Wi-Fi range.
### Power supply
One of the most important features of the camera was to have a good backup power supply that will be able to work outdoors. I could consider an existing UPS solutions, but they weren’t meant to be used outdoors. Another reason for my own supply was that I could enclose it in one case with the rest of the hardware.
I started by estimating how much power whole hardware will draw and how long power outages I could expect - it’s a construction site, everything is out in the open, someone could damage power lines or just plug the camera out. I estimated (very roughly) current draw for each component (assuming maximal load):
• Raspberry Pi 2 Model B: ~400mA @ 5V (2W).
• Camera: (~ 200mA @ 3.3V - 0.66W) with 3 IR diodes, each rated at 3W (9.66W in total).
• Wi-Fi card: ~40mA (0.2W).
• PCF8563 RTC clock: ~10mA (0.05W).
• Fan for LM317T voltage regulator: 100mA (0.5W).
That gives 12.5 Watts. Knowing that (and current ratings) I estimated how much power will be lost on voltage regulators:
• First LM2576T has efficiency around 77% (per datasheet). For 5V line I estimated that current draw will be roughly at 700mA (3.5W). Given that, regulator will draw 4.55 Watts of power, and that means it should dissipate 1.05 Watts of power as heat.
• Second LM2576T that will regulate 3.3V line have lower efficiency, but it’s not given in datasheet, I assumed it was at 72%. Current draw for this line was estimated as 2.74A (9W). If regulator outputs that much current, with efficiency at 72%, it will draw 12.5W and would dissipate 3.5 Watts as heat.
After taking into account power lost as heat (4.55W), I end up with 17.05 Watts of power. Having a battery with 12Ah we can calculate:
12V x 12Ah = 144Wh
144Wh / 17.05W = 8.45h
That gives us more than 8 hours of constant work on battery. Bear in mind that this is just theoretical and very rough estimation! On one hand, most components won’t work at it’s peak current rating, on the other hand, battery most probably will be either degraded or little bit discharged and won’t have perfect characteristics that manufacturer declares, not to mention battery’s own efficiency. Even so, we can safely assume that picked battery will withstand at least a couple of hours of video streaming.
Last component is the buffered power supply circuit based on LM317T. Given that input voltage is 20V, output power is 17.05W and output voltage is set to 13.65V, I can calculate that output current for this regulator would be 1.2A - almost at device’s limit, which is 1.5A. Just for curiosity I’ve also calculated how much power will be wasted as heat on this last circuit:
V(diff) = 20V - 13.65V = 6.35V
P(loss) = 1.2A x V(diff) = 7.62W
So, in total, power consumption should be 24.67W. Below are results from live readings with IR diodes off and on:
Without IR diodes, power consumption is a little bit above 18W.
With IR diodes on, power consumption is around 26W. Given estimates that didn’t take into account loses at passive components (resistors, diodes) and transformer (plus dissipated heat by it), live readings goes quite close to estimated value.
As for schematics, 5V and 3.3V voltage regulators circuits are based on standard application schematic from datasheet. As for the buffered power supply that is used to either supply power or charge the battery I used article from "Elektronika dla wszystkich" 10/98
### Case
There are two fans in the case - one on LM317T’s heatsink, as this component will dissipate most power as heat and if being too hot, could limit maximal rating of output current (it has thermal protection built-in) and on case itself, as inside there’s a battery that cannot be sealed completely off - case must be ventilated. Smart routing could probably save design from having two fans, but I decided to be lazy in that matter.
In theory, case should be hermetic or at least water and splash resistant - but because battery is inside the case, it must have some sort of ventilation. Due to that reason and prices of professional outdoor cases, I went with a standard indoor-use plastic case that I’ve reinforced with metal plates that holds the battery (it weight quite a bit) and closed most holes with hot glue - it should protect the case from rain and snow.
During most of my tests I was struggling with random resets, messages from kernel about under-voltage being detected, etc. I always thought that I was just physically messing with some wires and due to rather poor wire connections, I just shorted or disconnected power lines for a second and that’s that. Test was running at night? Cat yet again wanted to chew on power cable. But as I was getting closer to the end of the work on hardware, I still couldn’t get resets to go away - that was the moment when I realized there’s something really wrong, so I started searching. What I did prior to finding the right solution:
• checked pins connections,
• checked if pins are soldered correctly,
• re-soldered pins anyway,
• checked wires and theirs connectors,
• checked filtering capacitors at power supply inputs and outputs,
• replaced those capacitor with new ones,
• thickened power lines on circuit boards,
• re-routed wires to minimize interference.
And nothing worked - I still had random resets. So I gave up for a while until anything new would pop up into my mind to try out. And I waited… One day I was watching this guy on YouTube talking about twisting power lines and why you should do it and then it hit me - all my power lines were happily dangling around - and in close proximity I had Wi-Fi card, Wi-Fi router, AC power lines, transformer and bunch of invisible noise that could affect my circuits. So I went with his advice and twisted all power lines - and that was it, I got one week uptime without any more efforts.
Lesson learned - always twist your wires in order to eliminate external noise.
## Software
Requirement was simple - capture video from camera, encode it, encapsulate it in some sort of container and send over the network. All that could be done with stock Raspbian image, but I decided to choose another path - build my own Linux distribution!
### Operating System
Building Linux distribution by hand is hard, really, really hard. Tons of packages that would have to be built is just enormous. Luckily there are tools that help with that process. One notable example is Yocto. Unfortunately I don’t know this framework and since time was a pressing matter for this project, among other things, I decided to go with what I already knew - Buildroot.
• picking right settings for your platform: compiler, Linux kernel, init system, filesystem layout, target packages,
• compiling every aspect of OS in the right order,
• packaging result OS in suitable format (in this case, SD card image).
Raspberry Pi as a platform has very good support in Buildroot, so there was no problem with starting off. Once I built my first image and tested that it’s running good on target device, I started customizing.
First thing I did was to build my own toolchain. Although Buildroot provides it’s own toolchain for given platform, once you clean your build, it also cleans the toolchain - that’s a serious time waste. For building toolchain I’ve used a tool similar to Buildroot itself called crosstool-NG - it helps with building GCC compiler (8.2), binutils, standard C library (uClibC-ng) and other required tools. With easy to use ncurses Linux-like menuconfig, you can customize most of toolchain aspects, pick binaries version, minimal Linux kernel support, supported languages, features, etc. I went with this config - based mostly on defaults, with options that were mostly required by Buildroot.
Once I had a working toolchain, I’ve configured Buildroot to use it, rebuilt system image, ran stress tests and after everything seemed to be working stable, I started customizing packages. First, I wanted to have the most recent versions of packages - most importantly - FFmpeg, rpi-userland (contains raspivid binary) and rpi-firmware. As for former packages there was no problem - it just needs an updated git commit values, FFmpeg was harder, because Buildroot was sticking to 3.x release, where latest one as of time of writing this is 4.1. After patching Buildroot patches and adding some of my own, I ended up with the most recent FFmpeg package I could get.
After selecting needed packages, I focused on system’s init system. By default Buildroot uses Busybox as it’s init system - and that’s a good choice most of the time. But as I was experimenting with different approaches, I quickly decided to give systemd a go - mainly due to it’s powerful unit system, it’s really neat and easy to write services in systemd. And so I went with that choice for a while… Until I started actually measuring how much time systemd is wasting on, well, itself. Obvious choice would be to minimize the number of services that device wasn’t gonna use anyway, but default systemd setup in Buildroot already did a good job with that. From my measurements systemd was taking more than 30 seconds to spawn login prompt. It was using NetworkManager to establish network connection. I thought that I could live with that, but it just couldn’t let me go. My inner frustration with that time waste was bigger than reluctance to rewrite all services to another init system (and write new ones that currently were covered by systemd). I went back to Busybox - and, coupled with connman, from a 30+ seconds startup it went down to whooping less than 5 seconds for login prompt. Last time I measured Raspberry Pi’s performance was back in 2013 when I was still at university. Back then systemd wasn’t a viable choice for embedded systems, it seems it’s still not in 2019.
My Buildroot fork with raspberry-cctv_defconfig and other customization is available on github.com.
### Capture software
One may say - obviously, motion! Well, I didn’t. My initial concept with 3 cameras basically excluded it from the start. Once I realized that only one camera will be in place, I already had a working prototype based on raspivid + FFmpeg - that’s one. Second reason is that although it makes sense to use motion to detect movement, this detection can be faulty and CPU consuming - since I was going for power efficiency, post-processing should rather be done somewhere else. When Raspberry Pi 2 Model B is idling, it consumes roughly 220mA, but it goes up to 400mA when it’s at full load. On the other hand, Wi-Fi card is consuming 40mA - given that H.264 encoding is hardware-accelerated, I would argue that constant data transfer will consume less power than constant movement detection - but to be honest, that’s just my estimation, not actual measurement.
If motion is out of the question, what can be done with raspivid and FFmpeg? Plenty. Raspivid is probably most advanced software that can set various camera parameters - I was going for Variable Bit Rate (VBR) and this application is able to do exactly that. With VBR and 4 FPS I get funny 30 KB worth of video stream per second for mostly still images and up to 70 KB for all-over-the-place movement. That’s under my minimal upload bandwidth constraint of 128 KB/s - all good so far.
Sending a video over the network is a problem in itself. FFmpeg supports HLS standard and it supports writing output files in remote location - just what this project needs. Except the fact that once there’s a network outage, FFmpeg doesn’t buffer much of that video, instead it just discards backlog data and gets only newest frames - that’s unacceptable. If I cannot rely on FFmpeg capabilities to send video to remote location, then I had to come up with something else - so I wrote a simple C++ application that does number of things:
• sends video to remote location,
• keeps persistent connection to the server (it’s important because we don’t waste bandwidth and resources for establishing connection, HTTPS handshake, etc.),
• keeps a backlog of video material - even hours of material can be kept on persistent storage and send later, if we have enough throughput.
With this application I’ve solved FFmpeg’s problem of sending files over the network, but I still needed FFmpeg package. Raspivid produces raw H.264 video stream, but that data cannot be safely sent over the network. Since I decided to use HLS protocol, files had to be segmented into smaller fragments (10 seconds) and encapsulated in some kind of container, in this case MPEG-TS (Transport Stream) - it’s a widely used container, I guess mostly in DVB-T/S. Container maintains essential information about the stream and contains data that makes segments error prone. In addition, after encapsulation, segment files can be played by ordinary web browser.
In summary, capturing software contains three components:
• raspivid for video capture,
• FFmpeg for stream encapsulation,
• camera_streamer for sending video segments over the network.
### Server-side support
Server that will receive segment files must be properly configured and be publicly available. I used vanilla nginx with PHP-FPM - tandem that I’ve been using for more than 5 years and it didn’t let me down once.
Camera Streamer project expects server to support PUT method, we can get it done by enabling WebDAV module inside nginx:
location /camera {
# ...
client_max_body_size 10m;
create_full_put_path on;
dav_methods PUT DELETE;
dav_access group:rw all:r;
}
And that should be it for the web server. But putting segment files on the server isn’t exactly usable - we still need to do some stuff before we can use them.
First is viewing - I could get each segment and play it i.e. in VLC, but that’s just too much overhead. Instead, I’ve written a PHP script that will glob video’s root directory, group results, glob once more, this time for video files for given date and produce m3u8 playlist from found files. Having a playlist, we can simply point any player to that location and enjoy live stream from our camera!
But here arises another problem - storage. TS files for 24 hours of capture can take as much as 4 GB of space - that’s a lot. What could be done with that? Each segment contains I-frame and each segment is 10 seconds long - that’s just not efficient. I could use FFmpeg one more time to simply concatenate segment files - but putting them all into one file won’t reduce overall size. I cannot get away with re-encoding the material. And if I’m forced to re-encode it, I might just as well use more efficient codec - here H.265 comes into place. 24 hours of video that takes ~3 GB as segment files after re-encoding with H.265 takes just 400 MB, which is far more storage space friendly. Encoding on 3 threads is done with x2 speed (i.e. a 4 FPS video is processed with 8 FPS speed) - enough to encode one day’s worth of video in less than actual day and to keep encoding machine responsive.
All server-side scripts are available on github.com.
## Results
I’ve been testing camera for more than two weeks now and I’m more than happy with the results. Video quality is even better than I’ve expected, I’ve managed to keep power and bandwidth constraints, all the features I wanted are up and running - I consider it a success.
Below is a sample video fragment from tests (re-encoded for story purposes):
… and some photos of the device itself:
I started working on this project around the summer and ended it at the beginning of 2019 - I’m certainly not happy how long this project took, but would I do it again? Certainly! I’ve learned a lot about video processing, circuits (power supplies especially) and I had tons of fun while doing it! | 2021-07-25 18:54:52 | {"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": 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": 0.29217347502708435, "perplexity": 2707.5905573953733}, "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-2021-31/segments/1627046151760.94/warc/CC-MAIN-20210725174608-20210725204608-00532.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-6-rational-expressions-and-equations-6-1-rational-expressions-6-1-exercise-set-page-379/9 | ## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$x=0$.
Let us solve the given problem in the following way: $-11x=0$ Divide by $-11$ throughout. $\dfrac{-11x}{-11}=\dfrac{0}{-11}$ This implies that $x=0$ Hence, the final answer will be $x=0$. | 2019-12-10 23:53:13 | {"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": 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": 0.9214194416999817, "perplexity": 386.41722544922317}, "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-2019-51/segments/1575540529516.84/warc/CC-MAIN-20191210233444-20191211021444-00249.warc.gz"} |
https://socratic.org/questions/how-do-you-simplify-2-19-using-the-distributive-property | # How do you simplify 2(19) using the distributive property?
Jul 2, 2016
$2 \left(19\right) = 38$
#### Explanation:
$2 \left(19\right) = 2 \times 19$
= $2 \times \left(20 - 1\right)$ now using distributive property
= $2 \times 20 - 2 \times 1$
= $40 - 2$
= $38$ | 2019-06-16 17:36:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.8993343710899353, "perplexity": 7367.030960280947}, "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-2019-26/segments/1560627998288.34/warc/CC-MAIN-20190616162745-20190616184745-00227.warc.gz"} |
https://www.zbmath.org/?q=ai%3Aernvall-hytonen.anne-maria+ai%3Alepisto.arto | # zbMATH — the first resource for mathematics
Bounds and computational results for exponential sums related to cusp forms. (English) Zbl 1246.11138
The holomorphic cusp forms are defined by the Fourier series $F(z)=\sum_{n=1}^{\infty}a(n)n^{\frac{\kappa-1}{2}}\text{e}(nz),$ where $$\text{Re} \;z>0$$, $$\text{e}(x)=\text{e}^{2\pi i x}$$, $$\kappa$$ is the weight of the form, and the numbers $$a(n)$$ are called normalized Fourier coefficients.
This paper presents some computer data suggesting the size of bounds for exponential sums $\sum_{M\leq n\leq M+\Delta}a(n)\text{e}(n\alpha),$ where $$\Delta$$ is considerably smaller than $$M$$.
##### MSC:
11L07 Estimates on exponential sums 11Y35 Analytic computations
Full Text:
##### References:
[1] PARI/GP, Version @vers. 2006. available from · pari.math.u-bordeaux.fr [2] Apostol, T. M.: Modular functions and Dirichlet series in number theory. volume 41 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990 · Zbl 0697.10023 [3] Ernvall-Hytönen, A.-M.: A relation between Fourier coefficients of holomorphic cusp forms and exponential sums. to appear in Publications de l’Institut Mathematique · Zbl 1279.11083 · doi:10.2298/PIM0900097E [4] Ernvall-Hytönen, A.-M., Karppinen, K.: On short exponential sums involving Fourier coefficients of holomorphic cusp forms. Int\?ath. Res. Not. IMRN, (10) : Art. ID. rnn022, 44, 2008 · Zbl 1247.11106 · doi:10.1093/imrn/rnn022 [5] Ernvall-Hytönen, A.-M.: An improvement on the upper bound of exponential sums of holomorphic cusp forms. submitted [6] Ivić, A.: Large values of certain number-theoretic error terms. Acta Arith., 56(2) : 135-159, 1990 · Zbl 0659.10053 · eudml:206303 [7] Jutila, M.: On exponential sums involving the Ramanujan function. Proc. Indian Acad. Sci. Math. Sci., 97(1-3) : 157-166 (1988), 1987 · Zbl 0658.10043 · doi:10.1007/BF02837820 [8] Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen. Springer-Verlag, Berlin, 1998 · Zbl 0895.11001 [9] Rankin, R. A.: Contributions to the theory of Ramanujan’s function $$\tau (n)$$ and similar arithmetical functions ii. The order of Fourier coefficients of integral modular forms. Proc. Cambridge Philos. Soc., 35 : 357-372, 1939 · Zbl 0021.39202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-01-25 18:09:45 | {"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": 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": 0.8796509504318237, "perplexity": 2070.437561724812}, "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-2021-04/segments/1610703587074.70/warc/CC-MAIN-20210125154534-20210125184534-00593.warc.gz"} |
https://mathoverflow.net/questions/338026/intermediate-extension-and-perverse-cohomologies | Intermediate extension and perverse cohomologies
Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A of U be a local system S on A. Is it true that the intermediate extension IC(U, S) is one of the perverse cohomologies of R (I.e. the cohomologies of R with respect to the perverse t-structure)?
No. Let $$X=U=\Bbb C$$, $$V=\emptyset$$, $$A=X\setminus\{0\}$$, $$S=\Bbb C_A[1]$$, and $$R=Rj_*S$$, where $$j\colon A\to X$$ is inclusion. One can show that $$R$$ is perverse. However, $$j_{!*}S=\Bbb C_X[1] \neq R$$. | 2019-08-18 00:12:46 | {"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9417611956596375, "perplexity": 103.06576249329405}, "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-2019-35/segments/1566027313501.0/warc/CC-MAIN-20190817222907-20190818004907-00400.warc.gz"} |
https://answers.ros.org/question/40941/how-to-label-2d-laser-based-map-with-some-objects-door-bed-etc/ | # How to label 2D laser based map with some objects (door, bed, .... etc)
Hello
Im trying to put some labels of object like doors or beds in my yaml map. I have the map,its already created. Just want to define a object like for example door or bed in the map. The door should be label with some frame and coordinate, that later my laser scan can find them and give me some parameters like distance to this door or bed.
The “GMapping” package, which implements a simultaneous localization and mapping algorithm, has been employed to effectively learn occupancy grid maps from the 2D laser range data. The generated map can then be re-used to localize the platform in the learned environment producing the trajectories followed by the user.
Two dimensional map of the environment is already built from the laser range finder and IMU data. So I have that map.
This is my object node. So I put the object labels , coordinates here yes?
# include <vector>
sensor_msgs::LaserScan laser_scan; float min_range;
void scanCallback(const sensor_msgs::LaserScan::ConstPtr& msg) {
std::vector<float> laser;
laser = msg->ranges;
int size_laser = laser.size();
for (int i=0;i<size_laser;i++){
if (laser[i] < 0.01){
laser[i] = 99999;
}
if (laser[i] > 45){
laser[i] = 99999;
}
}
min_range = 2;
int index_min;
for (int i=0;i<size_laser;i++){
if (laser[i] < min_range){
min_range = laser[i];
index_min = i;
ROS_INFO("Minimum Range = %f", min_range);
}
}
for (int j=0;j<size_laser;j++){
if (laser[j] > min_range + 0.5){
laser[j] = 0;
}
}
laser_scan = *msg;
laser_scan.ranges.clear();
laser_scan.ranges = laser;
}
int main(int argc, char **argv) {
ros::init(argc, argv, "object_node");
ros::NodeHandle n;
ROS_INFO("Minimum Range = %f", min_range);
ros::Subscriber sub = n.subscribe("scan", 1000, scanCallback);
ros::Rate loop_rate(10);
while (ros::ok())
{
laser_pub.publish(laser_scan);
ros::spinOnce();
loop_rate.sleep();
}
return 0;
}
And this is the yaml map file
image: pow_real_time.pgm resolution: 0.050000 origin: [-100.000000, -100.000000, 0.000000] negate: 0 occupied_thresh: 0.65 free_thresh: 0.196
and this is my launch file
Perform AMCL localisation: runs several nodes to generate odometry from laser scans (ICP) & IMU, loads a map of the POW assessment area, runs AMCL, plays back a dataset for localisation, and runs the visualiser with the correct visualisation parameters configured. -->
<launch>
<node name="rosplay" pkg="rosbag" type="play" args="/home/Data/13-48-20.bag --clock"/>
<node pkg="tf" type="static_transform_publisher" name="baselink_laser" args="0 0 0 0 0 0 /base_link /laser 10"/>
<node pkg="tf" type="static_transform_publisher" name="laser_imu" args="0 0 0 0 0 0 /laser /base_imu 10"/>
<node pkg="tf" type="static_transform_publisher" name="baselink_camera" args="0 0 0 0 0 0 /base_link /camera 10"/>
<!-- Start the map server node and specify the map file (*.pgm) and the map resolution in metres/pixel -->
<node name="map_server" pkg="map_server" type="map_server" args="\$(find amcl_listener)/maps/pow_real_time.yaml" output="screen"/>
<!--Start the Laser_scan_matcher package, to provide odometry from laser data (ICP)-->
<node pkg="laser_scan_matcher" type ...
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@Astronaut Hi my name is Cris and im working on labeling obstacles and add semantic value to the map, could you finally manage to ad the labels to the map? And if you did how did you do it?? any help would be really helpful and appreciated
( 2013-08-22 09:33:01 -0600 )edit
Sort by » oldest newest most voted
I think this strongly depends on what you are going to do. If you plan to use more knowledge about you environment and you also want to to reasoning about it, you could think of using a knowledge base like KnowRob ( http://www.ros.org/wiki/knowrob ) to store e.g. semantic environment information (in a so called semantic map) and also any other kind of information you want to use. A big advantage of this approach is that you can do reasoning about objects and the environment with ot like e.g. finding the most likely storage location of an object or similar (check the Knowrob Wiki: http://ias.cs.tum.edu/kb/wiki/index.php/Main_Page )
But if you just want to have some locations of objects in your environment, this might be a bit of an overkill, so in this case you might just stick with a list of objects and coordinates...
more
At the beginning I just want to have some locations of objects in my environment. So very very basic stuff. So how to put one object for beginning and label it with coordinate? Any tutorials about that? So just label an object with coordinates and frame in my environment.
( 2012-08-08 22:24:51 -0600 )edit
I guess in this case guess I would just write a ROS node that takes a list ob objects with coordinates and posts a TF transform for every object... You can check the TF tutorials about how to add transforms: http://www.ros.org/wiki/tf/Tutorials
( 2012-08-09 00:24:27 -0600 )edit
ok. So in that case i will have this object on the laser scan map and than can calculate some parameters like minimal distance to these objects. Wright?
( 2012-08-09 02:44:07 -0600 )edit
You will have tf transforms which can be in relation to every coordinate system known to tf (like e.g. the /map-frame). Thus you can easily calculate distances from the object poses to other coordinate systems (like e.g. that of the current robot pose) by just using the functions offered by TF
( 2012-08-09 03:11:52 -0600 )edit
Plus you can visualize your map and object positions using RVIZ
( 2012-08-09 03:21:35 -0600 )edit
ok. also can visualize single beams, like minimal distance to that object?
( 2012-08-09 03:32:28 -0600 )edit
So I can visualize the objects of my yaml map , right?
( 2012-08-09 03:35:40 -0600 )edit
And these object have to be put in the map. So have to fill my yaml map file with the object parameters?Or how it is?
( 2012-08-09 16:55:23 -0600 )edit | 2021-03-04 16:48:44 | {"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": 0.21121031045913696, "perplexity": 4847.445761941797}, "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/1614178369420.71/warc/CC-MAIN-20210304143817-20210304173817-00431.warc.gz"} |
https://zbmath.org/?q=an%3A0983.06002 | # zbMATH — the first resource for mathematics
Relatively complemented ordered sets. (English) Zbl 0983.06002
The authors investigate conditions for the existence of relative complements in ordered sets. For relatively complemented ordered sets with $$0$$ they show that each element $$b\neq 0$$ is the least one of the set of all upper bounds of all atoms contained in $$b$$.
##### MSC:
06A06 Partial orders, general 06C15 Complemented lattices, orthocomplemented lattices and posets
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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.4 pp.758-773
DOI : https://doi.org/10.7232/iems.2020.19.4.758
# Radiotherapy Service Improvement: Simulation Study
Chawis Boonmee, Auttharat Kosayanon, Imjai Chitapanarux, Chompoonoot Kasemset*
Center of Healthcare Engineering System, Department of Industrial Engineering, Faculty of Engineering, Chiang Mai University Chiang Mai 50200, Thailand
Department of Industrial Engineering, Faculty of Engineering, Chiang Mai University Chiang Mai 50200, Thailand
Division of Radiation Oncology, Department of Radiology, Faculty of Medicine, Chiang Mai University Chiang Mai 50200, Thailand
Center of Healthcare Engineering System, Department of Industrial Engineering, Faculty of Engineering, Chiang Mai University Chiang Mai 50200, Thailand
*Corresponding Author, E-mail: chompoonoot.kasemset@cmu.ac.th
February 5, 2020 July 1, 2020 December 15, 2020
## ABSTRACT
Radiotherapy service has seen a spur of interest in the health care system. Many cancer centers aim to develop and enhance radiotherapy service to support the increasing demand. Because Thailand is a developing country, some cancer centers still have limited resources, both staff and machinery, while the number of cancer patients in Thailand has increased every year. To enhance the operational efficiency of radiotherapy service, this research aims to propose an integrated simulation approach and theory of constraints (TOC) approach for increasing radiotherapy service capability in Thailand. As simulation and TOC were applied in the case study, the real bottlenecks were identified in each treatment room. Considering four treatment rooms, only two rooms were selected for improvement (Room 2 and Room 3) after the simulation. The eight proposed solutions concentrated on improving both human- and machine-related bottlenecks. The simulation experiments were conducted to evaluate each solution. The results presented the best solution as adding one technician to Room 3 and replacing the radiation machine in Room 2 with the same machine as in Room 3. With this solution, the number of patients served was increased by 12.32% from the current system. As for the investment needed, the benefit-cost ratio and payback period were calculated as 1.89 and 2.80 years, respectively, for this solution.
## 1. INTRODUCTION
Healthcare management or healthcare administration is the management or administration of healthcare systems, public health systems, hospitals, whole hospital networks, and other medical facilities. Healthcare management aims to ensure that individual sections run smoothly and efficiently, qualified employees are hired, information is disseminated efficiently throughout the organization or network, specific outcomes are reached, and resources are used efficiently—among many other responsibilities. Recently, healthcare management has experienced enormous demand since it can assist in managing a hospital and related medical facilities. The scope of the healthcare management system is steadily increasing worldwide. The solutions are not only related to management in the healthcare system but also related to healthcare services, health policies and enhancement in the demand for world-class healthcare facilities in healthcare management as well. Nowadays, healthcare services still suffer from inefficient operation. Many hospitals and other medical facilities urgently need to improve their processes and services. However, redesign projects in healthcare service systems have often been unsuccessful (Boonmee and Kasemset, 2019).
Since 2003, the number of cancer patients in Thailand has been continuously increasing, with the number of dead having increased in proportion (about 60,000 people per annum on average since 2003). Most of the males (16.2%) have died from rectum and colon cancer while most of the females (37.5%) have died from breast cancer (Social and Quality of life database system, 2018). Radiotherapy is one of the major methods of cancer treatment; it utilizes curative therapy and palliative therapy. Curative therapy is treatment that aims to completely cure patients; palliative therapy is treatment for relieving the symptoms and reducing the suffering caused by cancer. As the number of cancer patients has increased, the demand for radiotherapy has continually increased in kind. Therefore, the operational efficiency of radiotherapy service is an important issue at present.
Currently, Thai cancer centers are endeavoring to support the enhancement of radiotherapy service capability. However, redesign projects in radiotherapy service systems have often been unsuccessful due to complicated systems, misidentification of the real problems, and the limitation of resources—both of staff and machinery. Hence, this research aims to propose solutions for radiotherapy service improvement at the Thai cancer center case study using simulation-based procedures and the concept of Theory of Constraints (TOC). There are three contributions of this research. Firstly, this research presents the application of simulation-based procedures for TOC implementation in healthcare system improvement. Secondly, this research considers the complicated healthcare system including multiple patient types, multiple machines with different capabilities, and multiple other resources. The third contribution is to present several solutions based on multicharacteristic parameters and evaluate solutions under uncertain situations using simulations.
The remainder of this paper is organized as follows: Section 2 presents related literature review in the health care system. Section 3 presents the research methodology of this research. Section 4 presents a case study in the Thai cancer center. Section 5 presents the results of research, including data collection, simulation models, bottleneck identification, implemented design, and improvement plans evaluation. Finally, a conclusion and discussion are given in Section 6.
## 2. LITERATURE REVIEW
This section presents an overview of the relevant literature. Healthcare management has become an interesting topic due to the increased demand for service. Many researchers aim to develop and enhance all systems in the health care supply chain in order to support the increasing demand in the future by employing a variety of methods for improvement in the health care system. The simulation approach is one popular technique that can be applied in this field (Boonmee and Kasemset, 2019;Monks et al., 2015). The simulation approach can deal with more complicated systems and requires fewer simplification assumptions. Moreover, it is able to be used to study non-existing systems, conduct experiments that are expensive to perform in reality, and predict complicated outputs of actions and developments (Lagergren, 1998). Aboueljinane et al. (2014) proposed a discrete event simulation model for analyzing the performance of SAMU (the acronym of Urgent Medical Aid Services in French) regarding 94 processes. The model determined the possible changes in the SAMU which led to enhanced operational efficiency for coverage performance in the simulation. Five strategies of scenarios were proposed for the improvement related to the location of rescue teams and the level of resources used throughout the service area. The results suggested that repositioning a portion of the existing teams into potential bases increased the 20- minute coverage performance up to 4.5% on average. Furthermore, this improvement in coverage reached 7.3% when the whole fleet was relocated based on the multiperiod redeployment plan obtained from simulation optimization. Lu et al. (2014) proposed a genuine agentbased adoptive scheduling model for service that aims to decrease outpatient waiting time in the Hand and Foot Team Clinic of the orthopedic surgery department at a hospital in Taiwan. The results showed improvement: The agent-based collaborative control system was able to decrease waiting time between 29% and 36% for walk-in patients as well as 61% to 63% for scheduled patients. Similarly, Kaushal et al. (2015) presented an agent-based simulation model in order to evaluate fast-track treatment in the hospital emergency department. The objective of this research aimed to reduce patient waiting time and to study the behavior change of entities and resources in a complex emergency department system including static and dynamic fast-track treatment processes. Implementation of this research reduced the patient waiting time by 50%. Bhattacharjee and Ray (2016) proposed a simulation and analysis of appointment systems for scheduling patients. This paper applied the simulation approach in order to control and synchronize the arrival of patients with resource availability, thereby reducing the patient waiting time and increasing the utilization of resources. The paper determined the appointment system with multiple classes of patients where different classes of patients might vary in punctuality, no-show probabilities, mean service times and service time variability. Other related papers that represented the simulation approach in the health care system were Abuhay et al. (2016), Ahmed and Amagoh (2008), Andreev et al. (2013), Brenner et al. (2010), Jones and Evans (2008), Lee et al. (2010), Ngowtanasuwan and Ruengtam (2013), and Monks et al. (2012).
In order to provide more efficiency in the implementation of health care service, several papers integrated the simulation model with other techniques, such as plant layout Kritchanchai and Hoeur (2018), optimization (Cabrera et al., 2012), six-sigma (Venkatadri et al. 2011), data envelopment analysis (Weng et al., 2011), what-if analysis Wong et al. (2011), pull system concept (Tiwari and Sandberg, 2016), and Theory of Constraints (TOC) (Somayeh, 2009).
Theory of Constraints (TOC) also known as constraint( s) management, is a management theory which emphasizes the importance of enhancing system performance by including a smarter utilization of existing resources, especially by considering the bottleneck, before increasing the system’s capacity. The principal of TOC focuses on the weakest point of any system (i.e., “constraint” or “bottleneck”) and emphasizes the improvement of that constraint’s performance—directly resulting in improving the overall performance of the system (Somayeh, 2019). The method of TOC comprises five steps: 1) identifying the bottleneck process, 2) bottleneck exploitation, 3) bottleneck subordination, 4) system evaluation, and 5) looping back to the first step to find any additional bottleneck in the system (Kasemset and Kachitvichyanukul, 2007;Kasemset, 2011). To enhance healthcare service, some papers applied the simulation approach and TOC in healthcare system improvement. Tiwari and Sandberg (2016) proposed perioperative bed capacity planning by using the TOC and simulation approach with the logic of the simulation model being guided by the TOC concept. The paper focused on space planning and bed capacity decisions under the various stages of the perioperative system: preoperative, intraoperative, and postoperative capacity. Then, Grida and Zeid (2019) proposed a system-dynamics-based model to implement the TOC in a healthcare system. The system dynamic model simulated a typical medium-sized hospital where different types of patients are served using the same limited resources based on the TOC philosophy. The results demonstrated that the number of served patients increased by 6% without any resource elevation.
Few papers presented application of the simulation approach and the TOC approach in the radiotherapy sector. Mohammadi and Eneyo (2012) presented an application of TOC’s drum-buffer-rope approach in scheduling the radiotherapy system of a hospital. This research aimed to reduce the waiting time and waiting lists for radiotherapy treatments. The same paper was proposed by Somayeh (2009), who proposed the adoption of a scheduling policy based on TOC’s drum-buffer-rope tool for an outpatient cancer facility. All above-mentioned papers are concluded in Table 1. From the literature reviews, when resource planning was considered, TOC is one concept that help in this kind of problem.
Based on the above-mention review, few papers have focused on the radiotherapy service system, so there is a lack of knowledge in this area includes multiple patient types, multiple machines, multiple rooms, and multiple radiation therapy techniques. Hence, this paper aims to propose an application of the simulation approach and the TOC approach for enhancement of the radiotherapy service system. The paper determines not only multiple patient types, multiple machines, and multiple rooms but also multiple radiation therapy techniques in the system. Furthermore, this paper aims to propose several solutions that could be effective even with the limited resources in the case study in order to enhance the radiotherapy service system, in which economic analysis is applied to evaluate all solution candidates as well.
## 3. METHODOLOGY
The methodology of this research consisted of six steps as shown in Figure 1. Detailed descriptions of each process are presented as follows:
### 3.1 Data Collection
The existing process of radiation therapy service of the case study was studied and the case study’s data were collected including the proportion of patients, treatment site type, radiotherapy technique, and processing time in each activity.
### 3.2 Simulation Model Formulation
This step was to formulate the simulation model. For data analysis, the number of patients and the processing time of each operation were fitted to appropriate statistical distribution using the Input Analyzer from ARENA software. Then the simulation model was formulated to present the existing process of radiation therapy service via ARENA software. Finally, model verification and validation were conducted to test whether the model was well-structured and a correct representative or not. To calculate the number of replications in simulation experiments, Equation (1) was conducted in this research (Kelton et al., 2009):
$n ≅ n 0 h 0 2 h 2$
(1)
where
• n = the number of replications,
• n0 = the number of initial replications (10 to 15 replications are usually considered),
• h0 = the half-width obtained from the initial set of replications, and
• h = the desired half-width from decision makers.
### 3.3 Bottleneck Identification
The simulation model of the existing process was run to collect statistical data. The results from the simulation model were used to identify the system bottleneck (constraints) using TOC technique based on the proposed method of Kasemset and Kachitvichyanukul (2007).
### 3.4 Proposing Improvements
After the system bottlenecks were identified, improvement plans were proposed. During this step, the concept of TOC was considered to improve the system until the system bottlenecks were removed.
### 3.5 Evaluation Improvement Plans
This step was to evaluate each improvement plan using the simulation technique. The results of each solution were compared with the current system. All comparisons were conducted based on statistical comparison using 95% CI (significance level [α] = 0.05) and economic perspectives including payback period and benefit-cost ratio. The payback period and benefit-cost ratios were calculated as Equation (2) and Equation (3) (Blank and Tarquin, 2012):
$0 = − P + N C F ( P / A , i , n p )$
(2)
(3)
where
• i = interest rate,
• P = investment cost,
• NCF = average net cash flow (per year), and
• np = payback period.
### 3.6 Discussion and Conclusion
Finally, the discussion and conclusion were conducted at the end of this research.
## 4. CASE STUDY
This section presents the case study in which we applied our approach: a Thai cancer center. This cancer center is located in the northern region of Thailand. Supertertiary medical care is provided for residential patients from 8 provinces. The center provides 4 techniques for radiotherapy services: Conventional Radiotherapy (2D), Three-Dimensional Conformal Radiotherapy (3D-CRT), Intensity Modulated Radiation Therapy (IMRT), and Image Guided Radiation Therapy (IGRT). More details of the radiotherapy techniques can be found in studies by American Cancer Society (2019) and Rehman et al. (2018). Currently, there are four treatment rooms in which different radiation machines are used for different treatment techniques. The details of techniques in each room are presented in Table 2.
The general process of radiotherapy service is presented in Figure 2. When patients arrive at the division, they are classified as two categories: (1) ambulant or (2) non-ambulant patients. Ambulant patients are able to move to other operations without assistance from the division’s staff, while non-ambulant patients need assistance. The proportions of ambulant and non-ambulant patients are presented in Table 3. The preparation step is started when patients arrive to radiotherapy rooms. Then, their postures are adjusted in the immobilization device based on treatment locations. For the preparation, there are two technicians working at each room to support patients. There are five treatment sites: the central nervous system (CNS); head and neck (H&N); thoracic; abdomen; and pelvis. The proportions of patients for each treatment site are presented in Table 4. The different treatment sites and types of patients are affected by different operations and movement times. Before a patient is treated with each technique (the proportion of patients with each technique can be found in Table 5), simulation and verification should be carried out to confirm the treatment plan. After the radiotherapy is finished, patients are discharged and move out, and the room is cleared to serve the next patient.
## 5. RESULTS
### 5.1 Current System
Following the exiting process of radiation therapy addressed in Section 4, data including the arrival pattern of patients, operation time of each process, and transfer time were randomly collected every working day during the official working hours (50 values for each data item).
### 5.2 Building the Simulation Model for the Existing System
There were three main tasks presented as follows.
#### 5.2.1 Data Analysis
Collected data were fitted to an appropriate basic statistical distribution using the Input Analyzer of Arena software. When the number of each item was 50 values, the chi-square test was employed with a significance level (α) of 0.50. The results of the data fitting are presented in Table 6 - Table 10.
#### 5.2.2 Model Building
The simulation model of the existing system was developed via Arena software (as presented in Figure 3). From Figure 3, the model consisted of 4 parts represented in each treatment room with similar structures. Starting from creating entities to represent patients, entities were designated as ambulant or non-ambulant patients based on the proportions presented in Table 3. Then entities were sent along the flow modules based on their different assigned treatment sites and radiation techniques. The operation time of each type is different. At the end, entities were counted separately for each room before they were disposed of.
#### 5.2.3 Model Verification and Validation
After the simulation model was developed, the model was verified and validated to confirm the appropriateness of the model. The flows of the entities were compared with the flow process of the existing system by the model developer. Test runs were conducted initially using 10 replications. The outputs from the simulation tests were investigated and used to find the appropriate number of replications for the simulation experiments. Initial results of 10 replications presented the average system output as 117.20 persons per day with the initial half-width (95% confidence interval, CI) as 6.66 or 5.68% error in comparison with the point estimator (average value). To calculate the appropriate number of replications as addressed as Equation (1), the desired half-width smaller than the initial half-width was provided from the decision-maker’s preference as 3 or 2.56% error from the point estimator. Calculations following Equation (1) for 49.28 replications (or approximately 50 replications) were set to perform the validation test. Therefore, the parameter settings for the simulation experiments were determined to be 1 day (8 hours) for run length and 50 replications for each scenario. Then the simulation test was conducted again to verify the model.
A comparison of the results is presented in Table 11. The total output and the output of each room are presented and were compared with the real data from the existing system using a 95% confidence interval on mean (average) of patients served per day. When, the two real numbers are (lower bound, upper bound) of 95% confidence interval on mean calculated based on equation (4) (Montgomery and Runger, 2014).
$x ¯ − t α 2 , n − 1 S n ≤ μ ≤ x ¯ + t α 2 , n − 1 S n$
(4)
when α = 0.05 and n = 50, the lower bound of mean is $x ¯ − t α 2 , n − 1 S n$ and the upper bound of mean is $x ¯ + t α 2 , n − 1 S n$.
The results presented no significant difference between the output ranges of the two systems, when the intervals of both systems overlap, so we concluded that the developed simulation model can be used to represent the real system.
### 5.3 Bottleneck Identification
To improve the system, bottleneck identification was firstly carried out following the 5 steps of TOC implementation. The simulation-based procedure proposed by Kasemset and Kachitvichyanukul (2007) was employed to identify the system bottlenecks of the existing system. The results from the simulation test previously conducted in the validation step were analyzed. Operation time of 5 processes referred to Figure 2 are presented in Table 12.
There are 2 resources used during the process of radiotherapy: man (technicians) and machine (radiation machines). There are 3 operations mainly worked by technicians while 2 operations are mainly conducted by a radiation machine. However, all operations occupy both man’s and machine’s time for each treatment.
To identify the system bottleneck (Kasemset and Kachitvichyanukul, 2007), the first task was to select the bottleneck candidates. Then all candidates were tested until the real system bottleneck could be identified using the simulation-based method. Considering Table 12, Figure 4 is presented to classify the operation time mainly used by man and machines.
From Figure 4, the bottleneck candidates can be selected based on longer operation time occupied by either man or machine. Bottleneck candidates were listed as follows: (i) for Room 1 was man and machine, (ii) for Room 2 was machine, and (iii) for Room 3 and Room 4 were man.
To identify the real system bottleneck, simulation tests were conducted by increasing the capacity of each bottleneck candidate and observing whether the significant improvement existed or not (Kasemset and Kachitvichyanukul, 2007). The simulation experiments were designed based on the bottleneck candidates as follows: (i) man: increasing the number of technicians; and (ii) machine: reducing 50% of the processing time at the treatment operation. The results from simulation tests are presented in Table 13.
The results presented in Table 13 were used in identifying the system bottleneck for each room when the number of patients treated per day was significantly increased and different compared to the existing results. Either adding a technician or decreasing treatment time is designed for increasing the resource’s capacity to identify the bottleneck when output can be improved. The intervals of outputs of three situations with (*) did not overlap. Finally, the conclusions of this step were as follows:
• - The real bottleneck was the radiation machine for Rooms 1, 2, and 4; and
• - The real bottleneck was technicians for Room 3.
### 5.4 Improvement Plans
Following the 5 steps of TOC implementation, after bottlenecks were identified, exploitation was considered. The key point of the bottleneck exploitation is to maximize the bottleneck’s utilization. Figure 5 presented the utilization of each treatment room.
From Figure 5, the utilization of each room is quite high (approximately 90% or over) excluding Room 3, which is the lowest at 83.68%. Interviews of the center’s experts were conducted; these experts reported that the number of appointed patients per day is fixed and cannot be increased because the variation of treatment time for this service is quite large. The policy is to make the number of appointments that can be completely serviced within the office hours. Thus, the cancer center sets a buffer time that cannot be reduced to maximize the utilization of each room to be close to 100%. Then the improvements were planned based on the bottleneck subordination and system elevation concepts from TOC. The concept of subordination is to support the bottleneck to work continuously, while the concept of elevation is to enhance the system performance by adding new resources.
From Table 13, there were three rooms that machines were identified as bottlenecks, whereas only the room no. 3 that the bottleneck was identified as man. Therefore, the improvement for the room no. 3 was set as adding one technician as scenario 1.
For the treatment rooms at which machines were identified as bottlenecks, the process time of treatment step was considered. From Table 12, considering total processing time of all treatment rooms, the room no. 2 has the longest total processing time following by the room no. 1, 4, and 3, respectively. Also, the treatment time of the room no. 2 was the longest among other rooms. Then the department’s experts (doctors and staffs) recommended to improve this treatment room by replacing the current machine (using for a long time and outdated technology) by new machine models that can work faster with updated technology. Thus scenario 2 was proposed for the improvement of the room no. 2.
For the room no. 1 and 4 having machines as bottlenecks, they were not considered any improvement because (i) the machine at the room no. 1 is the newest with highest technology reserved for complicated cases that resulted in long treatment time and (ii) the number of output patients of the room no. 4 is the highest comparing with other rooms as shown in Table 11. Thus, solutions were proposed only for the room no. 2 and 3 as follows:
• - Scenario 1: Adding one technician to Room 3; and
• - Scenario 2: Replacing the radiation machine in Room 2.
For scenario 2, to replace the radiation machine in Room 2, there were 3 sub-scenarios mentioned as follows:
• - Scenario 2.1: Replacing with the same radiation machine as in Room1;
• - Scenario 2.2: Replacing with the same radiation machine as in Room 3; and
• - Scenario 2.3: Replacing with the same radiation machine as in Room 4.
As previously noted in Table 2, the radiation machine in Room 3 can serve the same treatment techniques as those served in Room 2. Although the radiation machine in Room 1 is used for IMRT and IGRT techniques for complicated cases, this machine can be used for all techniques covering the same techniques as in Room 3. Consequently, the simulation tests for scenario 2.1 and 2.3 could be conducted without changing the proportion of patients for each technique in these scenarios. Conversely, when the radiation machine in Room 4 was considered to replace the one in Room 3, the proportion of patients for each technique had to change among all treatment rooms because this machine can serve only 2D and 3D techniques. Thus, the patients assigned to IMRT and IGRT in Room 3 were moved to other rooms as they would be affected by the proportion of patients assigned to each room and each technique. Consequently, for scenario 2.3, there were three sub-cases proposed (Table 14).
The scenarios presented in Table 14 were formulated by balancing the proportion of each technique when changing the combination of techniques in each room.
Finally, seven simulation tests were conducted follows:
• - Scenario 0: Current system;
• - Scenario 1: Adding a technician to Room 3;
• - Scenario 2.1: Replacing with the same radiation machine as in Room 1;
• - Scenario 2.2: Replacing with the same radiation machine as in Room 3;
• - Scenario 2.3.1: Replacing with the same radiation machine as in Room 4 and switching 2D-patients in Room 3 with IMRT-patients in Room 2;
• - Scenario 2.3.2: Replacing with the same radiation machine as in Room 4, switching 3D-patients from Room 3 with IMRT-patients from Room 2, and moving IGRT-patients from Room 3 to Room 1; and
• - Scenario 2.3.3: Replacing with the same radiation machine as in Room 4, assigning 2D-patients to Room 2, 3D-patients to Room 4, and IMRTpatients to Room 3.
### 5.5 Evaluation of the Improvements
All scenarios mentioned previously were evaluated by simulation tests. All results—the average outputs per day—were compared with the existing system and are shown in Table 15.
As presented in Table 15 and Figure 6, there were three scenarios from the first tests (Scenarios 1, 2.1, and 2.2) in which the average numbers of patients served of the overall system was significantly improved (without overlapping with scenario 0 (current)). Then, Scenarios 3 and 4 were proposed from the combination of Scenarios 1 and 2.1 and the combination 1 and 2.2, respectively. The results from Scenarios 3 and 4 also presented non-overlap interval comparing with Scenario 0, so the system outputs of both scenarios showed significant improvement comparing with the current situation, as well.
## 6. DISCUSSION
The results from the simulation tests can be addressed as follows:
• - Adding one additional technician to Room 3 can help to increase the number of patients served for the system.
• - For replacement of the radiation machine in Room 2, only the machine that works similarly to the one in Room 1 and 3 can help to increase the number of patients served for the system; machines that work similarly to one in Room 4 should not be considered.
• - The combination between adding one technician to Room 3 and replacing the machinein Room 2 with a machine like the one in Room 1 or 3 can also help to increase the number of patients served for the system.
All scenarios excluding 2.3.1 to 2.3.3 can significantly increase the number of patients served for the system in comparison with the existing system. However, some discussion can be drawn as follows:
• (i) Scenarios 1, 2.1, 2.2, and 3 were not significantly different. Thus, any one of them can be applied with similar results.
• (ii) Scenarios 1 and 2.1 were dominated by Scenario 4, so both solutions in scenario 1 and 2.1 can be eliminated.
• (iii) From (ii), alternatives for the system improvement were reduced to Scenarios 2.2, 3 and 4. These three scenarios can help in improving the system output similarly when considering only simulation results.
As more than one of the solutions are acceptable, each scenario was also evaluated based on economic perspectives. This research conducted economic assessment using payback period and benefit-cost ratio following Equations (2) and (3). The calculation for Scenarios 1, 2.1, 2.2, 3 and 4 are presented in Table 16.
As in the discussion point (iii), Scenarios 2.2, 3 and 4 can increase the system output similarly. In addition, these three solutions need investment. Considering payback period and benefit-cost ratio, Scenario 4 should be prefered with a maximum benefit-cost ratio of 1.89 and minimum payback period of 2.80 years.
Scenario 1 should be recommended if the cancer center cannot accept the investment; also, this solution is the easiest for real implementation.
## 7. CONCLUSION AND RECOMMENDATION
This research presented a simulation study of the radiotherapy service of a cancer center case study. The current system was studied and the simulation model was developed to represent the system. Based on TOC, bottleneck identification was carried out using simulation as a tool. Resource capacity constraints (or bottlenecks) of the system were identified and evaluated. Then solutions were provided for two out of four treatment rooms as follows: add an additional technician to Room 3, and replace the radiation machine in Room 2. Based on the simulation experiments, the number of patients served by the system can be increased when adding one technician to Room 3. Moreover, this solution can be practically implemented with no investment.
To replace the machine in Room 2, three different machines can be considered. Based on the simulation tests, either machine as in Rooms 1 or 3 can help to increase the number of patients served from the system. Furthermore, for the combination scenarios, adding one technician to Room 3 together with replacing the machine in Room 2 with machines as in either Room 1 or 3 can also help to improve in the number of patients served.
The recommendation when considering economic assessments is to add one technician to Room 3 and replace the radiation machine in Room 2 with the model as used in Room 3 because the benefit-cost ratio is the highest and payback period is minimum of all proposed solutions.
This research presents the application of simulation and TOC concepts in improving the healthcare service of the case study. Radiotherapy service has multiplicity in types of patients and symptoms, treatment techniques and detail in each operation. Simulation is a useful tool for representing the system and providing information leading to system improvement, and TOC is a concept that concentrates on improving the system bottleneck to improve the overall system’s output. Simulation-based procedure to identify bottleneck following TOC concept help in reducing the number of simulation tests when only the bottleneck candidate(s) are considered in simulation tests (Kasemset and Kachitvichyanukul, 2010). Implementing this procedure can help in identifying the bottleneck, as well as providing and evaluating the solutions. Finally, practical solutions can be proposed to decision-makers for further real implementation.
Although, this paper is mainly presented the application of simulation in healthcare real case study, the new general academic insight is to apply the concept of TOC in healthcare in systematic procedure. As we know TOC is widely applied in manufacturing systems for identifying system’s bottlenecks and improving system’s throughput, but we found few research works applied this concept in healthcare research. Among few researches, TOC was mentioned as in idea of bottleneck identification and improvement without systematic approach (Grida and Zeid, 2019;Somayeh, 2009), whereas, TOC with systematic approach based on simulation was presented in this research with the real case of healthcare system. When TOC was applied to improve any system, the step of bottleneck identification is to consider basic performance measurements (i.e. resource utilization, processing time, number in queue), whereas, this paper applied simulation experiments to identify real system bottleneck. The advantage of this procedure is to identify the system bottleneck correctly at the beginning, so the solution can be proposed at the most effectiveness. Moreover, simulation experiment for bottleneck identification is useful when key system parameters are high variability as in healthcare system.
Further study is recommended for bottleneck exploitation. Currently, there are fixed numbers of appointed patients per day that cannot be changed. The recommendation is to analyze the current number of appointments per day. To analyze this issue, simulation and scheduling techniques can be employed. In addition, a stochastics model should be considered due to the variation of the problem as mentioned previously.
## ACKNOWLEDGMENTS
This research was supported by the Murata Science Foundation under the Memorandum of Understanding between Chiang Mai University and The Murata Science Foundation.
## Figure
The methodology of the research.
Process of radiotherapy service in each room.
The simulation model of the Division of Radiation Oncology at the Thai cancer center case study.
Operation times classified based on main resources (man or machine).
The average utilization of each treatment room (man and machine are the same).
95% CI of the average number of patients served per day (lower bound, upper bound).
## Table
List of studies in the literature review
Techniques of each treatment room at the case study division.
The proportion of ambulant and non-ambulant patients of each room
The proportion of treatment sites of each room.
The proportion of radiotherapy service to patients with respect to technique
The number of patients (persons/day)
Operation time of the verification (minutes)
Operation time of preparation for each treatment site (minutes)
Transferring time of each patient type (minutes)
Operation time for radiotherapy in each room with different techniques (minutes)
Results comparison between real system and simulation
Average time of each process in each radiotherapy room
Comparison of results for the bottleneck identification using 95% CI of the average number of patients per day presented as the interval as (lower bound, upper bound)
The new service schedule of each scenario for adjusting the proportion of patients
Results comparison for additional tests using 95% CI of the average number of patients served per day presented as the interval as (lower bound, upper bound).
Economic assessment results
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12. Charfeddine, M. and Montreuil, B. (2010), Integrated agent-oriented modeling and simulation of population and healthcare delivery network: Application to COPD chronic disease in a Canadian region, Proceedings of the 2010 Winter Simulation Winter Conference, IEEE, 2327-2339
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14. Jones, S. S. and Evans, R. S. (2008), An agent based simulation tool for scheduling emergency department physicians, AMIA Annual Symposium Proceedings, American Medical Informatics Association, 338-342.
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17. Kasemset, C. and Kachitvichyanukul, V. (2010), Effect of Confidence Interval on Bottleneck Identification via Simulation, Proceedings of the 2010 IEEE International Conference on Industrial Engineering and Engineering Management, Macao, China, 1592-1595.
18. Kaushal, A. , Zhao, Y. , Peng, Q. , Strome, T. , Weldon, E. , Zhang, M. , and Chochinov, A. (2015), Evaluation of fast track strategies using agent-based simulation modeling to reduce waiting time in a hospital emergency department, Socio-Economic Planning Sciences, 50, 18-31.
19. Kelton, W. D. , Sadowski, R. P. , Swets, N. B. (2009), Simulation with Arena (5th ed.), McGraw-Hill, New York.
20. Kritchanchai, D. and Hoeur, S. (2018), Simulation modeling for facility allocation of outpatient department, International Journal of Healthcare Management, 11(3), 193-201.
21. Lagergren, M. (1998), What is the role and contribution of models to management and research in the health services? A view from Europe, European Journal of Operational Research, 105(2), 257-266.
22. Lee, B. Y. , Brown, S. T. , Korch, G. W. , Cooley, P. C. , Zimmerman, R. K. , Wheaton, W. D. , Zimmer, S. M. , Grefenstette, J. J. , Bailey, R. R. , Assi, T. , and Burkea, D. S. (2010), A computer simulation of vaccine prioritization, allocation, and rationing during the 2009 H1N1 influenza pandemic, Vaccine, 28(31), 4875-4879.
23. Lu, T. P. , Tsai, P. F. , and Chu, Y. C. (2014), An agent-based collaborative model for orthopedic outpatient scheduling, Proceedings of the 2014 IEEE 18th International Conference on Computer Supported Cooperative Work in Design (CSCWD), Hsinchu, Taiwan, 621-626.
24. Mohammadi, A. and Eneyo, E. S. (2012), Application of drum-buffer-rope methodology in scheduling of healthcare system, Architectural Engineering - Faculty Publications, 92, 1-18.
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26. Monks, T. , Pitt, M. , Stein, K. , and James, M. (2012), Maximizing the population benefit from thrombolysis in acute ischemic stroke: A modeling study of in-hospital delays, Stroke, 43(10), 2706-2711.
27. Montgomery, D. C. and Runger, G. C. (2014), Applied Statistics and Probability for Engineers (6th ed.), Wiley, Singapore.
28. Ngowtanasuwan, G. and Ruengtam, P. (2013), Applied simulation model for design of improving medical record area in out-patient department (OPD) of a governmental Hospital, Procedia-Social and Behavioral Sciences, 101, 147-158.
29. Rehman, J. , Zahra, Ahmad, N. , Khalid, M. , Noor ul Huda Khan Asghar, H. M. , Gilani, Z. A. , Ullah, I. , Nasar, G. , Akhtar, M. M. , and Usmanie, M. N. (2018), Intensity modulated radiation therapy: A review of current practice and future outlooks, Journal of Radiation Research and Applied Sciences, 11, 361-367.
30. Shi, J. , Peng, Y. , and Erdem, E. (2014), Simulation analysis on patient visit efficiency of a typical VA primary care clinic with complex characteristics, Simulation Modelling Practice and Theory, 47, 165-181.
31. Social and Quality of life database system (2018), Reported a record number of cancer patients in 2003-2015, cited 2018 Nov. 13, Available from: http://soci al.nesdc.go.th/social/Default.aspx?tabid=131.
32. Somayeh, S. (2009), Theory of constraints for publicly funded health systems, Doctor thesis, University of Toronto, Ontario, Canada.
33. Tiwari, V. and Sandberg, W. S. (2016), Perioperative bed capacity planning guided by theory of constraints, In Proceedings of the 2016 Winter Simulation Conference, Washington, DC, 1894-1903.
34. Venkatadri, V. , Raghavan, V. A. , Kesavakumaran, V. , Lam, S. S. , and Srihari, K. (2011), Simulation based alternatives for overall process improvement at the cardiac catheterization lab, Simulation Modelling Practice and Theory, 19(7), 1544-1557.
35. Weng, S. J. , Tsai, B. S. , Wang, L. M. , Chang, C. Y. , and Gotcher, D. (2011), Using simulation and data envelopment analysis in optimal healthcare efficiency allocations, Proceedings of the 2011 Winter Simulation Conference (WSC), Phoenix, AZ, 1295-1305.
36. Wong, S. Y. , Tsui, K. L. , Chin, K. S. , and Xu, M. (2011), A simulation study to achieve healthcare service quality improvement in accident and emergency department (AED), Proceedings of the 2011 IEEE International Conference onQuality and Reliability (ICQR), Bangkok, Thailand, 259-263.
37. Zeinali, F. , Mahootchi, M. , and Sepehri, M. M. (2015), Resource planning in the emergency departments: A simulation-based metamodeling approach, Simulation Modelling Practice and Theory, 53, 123-138. | 2021-03-06 17:32:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "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": 0.4987654685974121, "perplexity": 2496.434856350153}, "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/1614178375274.88/warc/CC-MAIN-20210306162308-20210306192308-00630.warc.gz"} |
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The sentence: > Miguel le dio a su novia un anillo. Translates into: > Miguel gave a ring to his girlfriend. I would think that there would be no need for the "le", since the direct... more
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Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. | 2021-06-14 16:33:52 | {"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": 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": 0.33237454295158386, "perplexity": 6549.371807380571}, "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/1623487612537.23/warc/CC-MAIN-20210614135913-20210614165913-00593.warc.gz"} |
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# XVI International Symposium on Very High Energy Cosmic Ray Interactions (ISVHECRI 2010)
US/Central
One West (Fermilab)
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
,
Description
The 16th meeting of the biennial conference series, the International Symposium on Very High Energy Cosmic Ray Interactions (ISVHECRI 2010), was held at Fermilab June 28 to July 2, 2010.
The URL for the Proceedings is http://www.slac.stanford.edu/econf/C1006284.
Topics coverd in this Symposium:
• Recent accelerator data and results
• Sensitivity of Monte Carlo models to data
• Extensive air shower experiments, E > 100 TeV
• Experiments above the Ankle
• Emulsion chambers
• Anisotropy
• Muons
• Balloon and satellite experiments
Support
• Monday, 28 June
• 8:45 AM 9:00 AM
Welcome One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 8:45 AM
Symposium Opening Remarks 5m
Speaker: Local Organizing Committee (ISVHECRI 2010)
• 8:50 AM
Welcome from the Fermilab Directorate 10m
Speaker: Dr Young-Kee Kim (Deputy Director, Fermilab)
• 9:00 AM 10:40 AM
Introductory presentations One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 9:00 AM
Accelerator Data 50m
I shall present selected examples of accelerator data, mainly from hadron colliders, that are relevant for understanding cosmic ray showers. I focus on the forward region, x(Feynman) > 0.05, where high energy data are scarce, since the emphasis in collider physics became high-pT phenomena. I discuss whether that situation can be improved.
Speaker: Dr Michael Albrow (Fermi National Accelerator Laboratory)
• 9:50 AM
Cosmic rays: current status 50m
Important new results in four areas of particle astrophysics are on the agenda of this conference: atmospheric leptons; direct measurements of composition and spectrum to 100 TeV; air shower measurements from the knee to the ankle; and the upper end of the cosmic-ray spectrum. Each of these topics has a long history, with the techniques and the basic questions being established early on. What is relative contribution of pions, kaons and charm to leptons in the atmosphere? Do all species of primary cosmic rays have the same source spectra and propagation history? Where is the transition from galactic cosmic rays to a higher energy population of particles from extra-galactic sources? Is there a suppression of the highest energy particles due to energy loss during propagation through the cosmic background radiation? In this introductory talk I will comment on the current status of each topic in its historical context.
Speaker: Prof. Thomas Gaisser (University of Delaware)
• 10:40 AM 11:10 AM
coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 11:10 AM 12:00 PM
Introductory presentations: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 11:10 AM
Relating accelerator data and models 50m
The study of high energy cosmic rays requires a good understanding of the properties of hadronic interactions. Information on the strong interactions can be obtained in experimental studies at accelerators, however the modeling of cosmic rays showers requires an extrapolation of the observations made at accelators with some guidance from theoretical ideas. This talk will review some of the key problems for these extrapolations and the resulting systematic uncertainties. The possibility to obtain information on the hadronic nteractions from cosmic ray observations will also be considered.
Speaker: Dr Paolo Lipari (INFN Roma 1)
• 12:00 PM 12:30 PM
Recent relevant accelerator data and results: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Dr Bryan Pattison (CERN)
• 12:00 PM
Particle production Experiments and their relevance to understanding Extensive Air showers 30m
Calculations of fluxes of atmospheric neutrinos and muons from extensive air showers suffer from our lack of knowledge of hadronic production processes. We are dependent of particle production models which suffer from systematics from both model dependent assumptions as well as the data used to tune them. We will present recent published data from NA49, and NA61 experiments as well as present analysis from the MIPP experiment relevant to particle production and air showers. Prospects of getting higher quality data using the MIPP upgrade will be discussed.
Speaker: Dr Rajendran Raja (Fermilab)
• 12:30 PM 1:30 PM
lunch 1h Cafeteria
### Cafeteria
#### Fermilab
PO Box 500 Batavia, IL 60510
• 1:30 PM 3:30 PM
Recent relevant accelerator data and results: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Dr Bryan Pattison (CERN)
• 1:30 PM
Perspectives on Nuclear Physics Input into High-Energy Cosmic Ray Interactions 30m
Recent ultra high-energy cosmic ray data hints an increase of heavier nuclei in the composition of the cosmic ray flux, accentuating the importance of more precise nuclear physics input. In this talk recent results from relativistic heavy ion and other nuclear experiments will be summarized and the possible impact of these results on understanding cosmic ray interactions will be discussed.
Speaker: Prof. Baha Balantekin (University of Wisconsin)
• 2:00 PM
Recent accelerator data and results from the Tevatron 30m
We present relevant results from CDF and D0, including diffractive and elastic scattering, and other inclusive measurements.
Speaker: Dr Mary Convery (Fermilab)
• 2:30 PM
Status and Prospects from the ATLAS Detector 30m
Since the startup of the LHC in December 2009, the ATLAS detector has been accumulating data from collisions at center of mass energies of 900 GeV and 7 TeV. Although the integrated luminosity is still low, it is increasing at an accelerated pace. The data have already made it possible to commission and calibrate the various subdetectors, understand their performance in detail and refine the trigger and software reconstruction algorithms. Initial measurements on charged particle multiplicities at \sqrt{s} = 900 GeV and 7 TeV as a function of pseudorapidity and transverse momentum have allowed comparisons to results from other experiments at the lower center of mass energy and to various Monte Carlo models of minimum bias events. Standard Model electroweak processes are also being used as benchmarks for validating the analysis and simulation tools. With the higher luminosity expected in the coming year, stringent tests of higher order QCD processes could be achieved. Various models of new physics could be probed and significant constraints obtained. The status of the detector will be summarized, and a brief review of physics results and expectations from early analyses will be given.
Speaker: Prof. Georges Azuelos (Univ. de Montreal)
• 3:00 PM
Recent Results from CMS 30m
The status of CMS concerning the 2009 run and the first data recorded at 7 TeV in 2010 will be reported. After a summary of the LHC and detector performance, including some example of interesting events, the talk will focus to the first results obtained. In particular, emphasis will be given to low-pT QCD physics including charged hadron spectra, the measurement of Bose-Einstein correlations (BEC) and of underlying event properties.
Speaker: Dr Ambra Gresele (Trento University)
• 3:30 PM 4:00 PM
coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:00 PM 5:35 PM
Recent relevant accelerator data and results: 3 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Oscar Saavedra (Università di Torino / INFN, Torino)
• 4:00 PM
Status and prospects from TOTEM 30m
Totem is exploring the forward region at pseudorapidity larger than 3.1; its main goal is the measurement of the total and elastic cross-section at 14 TeV and the study of diffractive physics in the forward region. The experiment is now built and almost completely commissioned; data taking started in December 2009. TOTEM aims at measuring the total cross section beyond 1 TeV/c with the unprecedented precision of 1 % by using the luminosity independent method, based on the simultaneous detection of elastic scattering at low momentum transfer and of the inelastic interactions. To achieve this, protons scattered at very small angles in elastic or quasi-elastic reactions will be measured in telescopes of silicon detectors enclosed in Roman Pots, placed on both sides of the intersection regions; inelastically produced secondaries will be measured by a forward inelastic detector covering the region 3 < eta <7 with full azimutal acceptance. The TOTEM physics program includes the measurement of forward charged multiplicity distributions at the TEV scale, important for the understanding of the cosmic ray events. TOTEM will take data under all LHC beam conditions including standard high luminosity runs to maximize its physics goals.
• 4:30 PM
LHCf measurements of very forward particles at LHC 30m
LHCf (Large Hadron Collider forward) is a dedicated experiment to measure the neutral particles emitted around zero degree of LHC interactions. Energy and Pt spectra of photons, pi-zero and neutral hadrons at such forward region are crucial to qualify the existing interaction models and to improve them for cosmic-ray physics. From the end of 2009, LHCf has successfully taken data at LHC collisions at sqrt(s)= 0.9 and 7TeV. In this presentation, the first results of LHCf mainly obtained since April 2010 will be presented together with the prediction of various interaction models.
Speaker: Dr Takashi SAKO (Solar-Terrestrial Environment laboratory, Nagoya University)
• 5:00 PM
CASTOR LHC and cosmic rays 15m
CASTOR, a very forward (5.2<η<6.6) Čerenkov-light, tungsten/quartz calorimeter was installed and commissioned at CMS (LHC) in 2009. The calorimeter, with 16-fold φ-segmentation, 14-fold z-segmentation (224 channels) and 10λ(int), has been obtaining data since November 2009. The physics to be addressed with CASTOR include forward energy flow in pp, AA and pA, critical for the screening of EAS MC codes, as well as “exotic” topics, such as “Centauro” and “long penetrating” events, observed in VHE cosmic-ray data. The later constitute the reason for the novel design of the calorimeter. The first operational experience with CASTOR at CMS and the possibility of identifying “long penetrating” events will be presented and discussed.
Speaker: Prof. Edwin Norbeck (University of Iowa)
• 5:15 PM
First physics results at LHCb 20m
First pp collisions at sqrt(s) = 0.9 and 7 TeV have been recorded by the LHCb detector using a minimum bias trigger. These data are very valuable to commission the detector and trigger algorithms, but will also be used to perform a number of interesting minimum bias physics measurements, in the forward region covered by the LHCb detector (polar angles between 15 and 300 mrad), amongst which measurements of the prompt Kshort, Lambda, anti-Lambda, proton, anti-proton production cross sections, as well as of the Lambda transverse polarization. The motivations, ingredients and status of such measurements will be discussed, and preliminary results shown where available.
Speaker: Mr Christian Linn (University Heidelberg)
• 5:35 PM 7:30 PM
Reception 1h 55m Wilson Hall 15 North Crossover
### Wilson Hall 15 North Crossover
#### Fermilab
PO Box 500 Batavia, IL 60510
The reception is scheduled for two hours and will end at 7:30 pm
Buses will leave at 7:30 pm
• Tuesday, 29 June
• 8:30 AM 8:50 AM
Recent relevant accelerator data and results: 4 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Oscar Saavedra (Università di Torino / INFN, Torino)
• 8:30 AM
First Results from the ALICE Experiment at the LHC 20m
The Large Hadron Collider (LHC) at CERN (Geneva, Switzerland) has successfully started operation in 2009. Collisions of protons at energies of 7 TeV are being provided to the experiments, the highest center-of-mass energy ever achieved in accelerators. The ALICE experiment at the LHC is designed for the investigation of heavy-ion collisions, but it is also well suited for studies of pp collisions. In this talk, first results of the ALICE experiment from pp collisions at the LHC will be presented.
Speaker: Dr Henner Buesching (University of Frankfurt)
• 8:50 AM 10:30 AM
Balloon and Satellite Experiments: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 8:50 AM
Balloon-borne and Space-based Particle Measurements with Magnetic Spectrometers 50m
Using high-performance superconducting or permanent magnets coupled with precision detector systems, magnetic-rigidity spectrometers have the unique ability to completely identify incident particles by charge, charge-sign, mass, and energy. Magnetic spectrometers are central to measurements of cosmic antiparticles and the spectra of light isotopes and elements. Positron and antiproton spectra measured by magnetic spectrometers are important in constraining dark-matter models as well as models for the origin, acceleration, and transport of cosmic rays in the Galaxy and Heliosphere. Searches for heavier antinuclei probe symmetry-breaking processes in the early Universe. Measurements of light-isotope spectra to relativistic velocities constrain models for cosmic-ray transport and storage in the Galaxy. Instrumental techniques used in modern magnetic-rigidity spectrometers and results from recent experiments will be reviewed. Prospects for future magnetic spectrometer instruments will be discussed.
Speaker: Dr John Mitchell (NASA Goddard Space Flight Center)
• 9:40 AM
Balloon-borne and Space-based Experiments with Non-magnetic Detectors 50m
Direct measurements of cosmic rays with satellite or balloon-borne detectors are used for understanding cosmic ray origin, acceleration and propagation, exploring the supernova acceleration limit, and searching for exotic sources such as dark matter. Their energy reach is currently limited to ~10^15 eV by the detector size and exposure time, but incident particles are identified element-by-element with excellent charge resolution. A challenge of balloon-borne and space-based experiments is that the detectors must be large enough to collect adequate statistics, yet stay within the weight limit for available space flight. Innovative approaches now promise high quality measurements over an energy range that was not previously possible. Recent measurement results will be reviewed and their implications will be discussed. The outlook for existing and future experiments with non-magnetic detectors will also be discussed.
Speaker: Prof. Eun-Suk Seo (University of Maryland)
• 10:30 AM 11:00 AM
Conference Photo & coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
All ISVHECRI 2010 participants are invited to be a part of the Conference Photo. The location will be in the atrium or on the front steps, depending on the weather.
• 11:00 AM 12:20 PM
Balloon and Satellite Experiments: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 11:00 AM
Status of AMS 20m
The Alpha Magnetic Spectrometer (AMS) is a major particle physics experiment on the International Space Station (ISS). AMS is a general purpose particle physics spectrometer using the technologies commonly employed at CERN and Fermilab and upgraded for space applications. The properties of the AMS detector are that it will provide a coordinate resolution of 10 microns, a timing resolution of 150 ps and a velocity resolution of 1 part in 1000. It will simultaneously measure e+, e-, p, p-bar and nuclei up to the TeV region. For its 20 year stay on the ISS it will provide a sensitive search for the origins of Dark Matter, the existence of antimatter, the existence of strangelets and so forth. AMS is a DOE sponsored international collaboration involving 600 scientists from 16 countries. It is schedule to be transported by the Space Shuttle to ISS in November 2010.
Speakers: Prof. Andrei Kounine (MIT), Prof. Samuel C.C. Ting (MIT)
• 11:20 AM
Balloon-borne gamma-ray telescope with nuclear emulsion 15m
We are planning to observe cosmic gamma-ray in the energy range 10MeV to 100GeV by balloon-borne gamma-ray telescope with nuclear emulsion. Nuclear emulsion is a precise tracker. By detecting starting point of electron pair, gamma-ray direction can be determined precisely (1.4mrad@1-2GeV). This is much better than Fermi Gamma-ray Space Telescope launched June 2008. Now we are developing the gamma-ray telescope with nuclear emulsion and are planning to observe by balloon flight. Overview and status of our telescope is talked in this presentation.
Speaker: Dr Satoru Takahashi (Nagoya University)
• 11:35 AM
The JEM-EUSO Mission to Explore the Extreme Universe 15m
The JEM-EUSO mission explores the origin of the extreme energy comic-rays (EECRs) above 10^20 eV and challenges to the limit of the basic physics, through the observations, of their arrival directions and energies. It is designed to observe more than 1,000 events of EECRs above 7x10^19 eV in its five-year operation with an exposure larger than 1 million km^2 /sr/year. The super-wide-field (60 degrees) telescope with a diameter of about 2.5m looks down the atmosphere of the night-side of the earth to detect near UV photons (330-400nm, both fluorescent and Cherenkov photons) emitted from the giant air-shower produced by an EECR. The arrival direction map with 1,000 events naturally tells us the origin of the EECRs and allows us to identify the EECR sources to known astronomical objects. The comparison among the energy spectra of the spatially resolved individual sources will clarify the acceleration/emission mechanism, and also finally confirm the Greisen-Zatse'pin-Kuzmin process for the validation of Lorentz invariance up to ~10^11. Neutral components (neutrinos and gamma rays) can also be detected as well, if their fluxes are high enough. The JEM-EUSO mission is planned to be launched by a H2B rocket about 2015 and transferred to ISS by H2 Transfer Vehicle (HTV). It will be attached to the external experiment platform of “KIBO” which completed July 2009 by STS-127 mission of the space shuttle.
Speaker: Dr James H. Adams, Jr. (NASA/MSFC)
• 11:50 AM
On the electron/positron excesses and the knee of cosmic ray spectra 15m
Based on the cosmic rays acceleration in the young supernova remnant like environment, electron and positron pair production through the interactions between high energy cosmic rays and radiation background photons is studied. It is found that both the electron/positron excesses and the knee structure of the cosmic ray spectra can be explained with one set of the source parameters.
Speaker: Prof. Yuqian Ma (IHEP)
• 12:05 PM
Atmospheric Effects of High Energy Cosmic Rays 15m
It has been suggested that events such as supernovae, gamma ray bursts (GRBs) and motion of the Sun perpendicular to the galactic plane may expose the Earth to an enhanced flux of high energy Cosmic Rays (HECRs). The electromagnetic component of the resulting air showers leads to an increase in ionization and dissociation in the atmosphere which results in a series of chemical reactions. These reactions occurring in the stratosphere deplete the ozone, resulting in an increase in the solar UVB flux at the ground level. This could be harmful to a variety of organisms such as phytoplanktons which form the base of the food chain. Enhanced ionization could also result in an increase in the low altitude cloud cover, thereby increasing the albedo and cooling the planet. Magnitude of these effects depend on the flux of cosmic rays hitting the atmosphere. Using CORSIKA and NASA GSFC 2D photochemical code, we perform detailed computer simulations of 10 GeV – 1 PeV range primaries interacting with the Earth's atmosphere and construct a model to quantify these effects for an arbitrary astrophysical source. Data up to PeV primaries is freely available and is being extended for EeV primaries.
Speaker: Mr Dimitra Atri (University of Kansas)
• 12:20 PM 1:20 PM
lunch 1h Cafeteria
### Cafeteria
#### Fermilab
PO Box 500 Batavia, IL 60510
• 1:20 PM 2:25 PM
Hadronic cross sections: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Thomas Gaisser (University of Delaware)
• 1:20 PM
"Hadron cross sections: from cyclotrons to colliders to cosmic rays" 50m
Using the Froissart bound as a unifying theme, I will show that the experimental data for hadronic crosssections, from nucleon-nucleon, pion-proton, gamma-p and gamma*-p, are all consistent with a high energy behavior saturating the Froissart bound, all rising with energy as log^2(s). Using analyticity constraints that tie in very accurate low-energy total cross section measurements for pp and pbar-p scattering, we make very precise predictions for both LHC and cosmic ray energy cross sections.
Speaker: Prof. Martin Block (Northwestern University)
• 2:10 PM
The proton-air inelastic cross-section measurement at sqrt(s) ~ 2 TeV from EAS-TOP experiment. 15m
The proton-air inelastic cross section measurement at sqrt(s) ~2 TeV from the EAS-TOP Extensive Air Shower experiment is reported. The technique exploits cosmic ray proton primaries, in the energy region $E_0 = 1.5- 2.5 x 10^15 eV, studying the absorption length of their cascades when detected at maximum development. Primary energies are selected through the EAS muon number, and proton originated cascades at maximum development by means of the shower size. The shower and detector fluctuations are obtained by means of simulations performed using the CORSIKA code and the QGSJET II and SIBYLL interaction models. The statistical and systematic uncertainties, as well as the relationships with the pp total cross section measurements are discussed. Speaker: Dr Gian Carlo TRINCHERO (INAF-IFSI and INFN Torino) • 2:25 PM 3:25 PM Sensitivity of Monte Carlo models to data One West ### One West #### Fermilab PO Box 500 Batavia, IL 60510 Convener: Dr Paolo Lipari (INFN Roma 1) • 2:25 PM Modeling Hadronic Multiparticle Production at Very High Energy 1h After introducing the general structure of event generators used for simulating cosmic ray interactions we describe the underlying philosophy of the Monte Carlo models EPOS, QGSJET, SIBYLL, and DPMJET. Some of the important assumptions of the models are reviewed in detail and the prediction obtained with the models are discussed. The reliability of the predictions is one of the key questions for which the new LHC data give valuable input. The relation of model predictions to general air shower features will be presented and uncertainties estimated. Finally, the most important open questions will be listed and ways of addressing them outlined. Speaker: Dr Ralph Engel (Karlsruhe Institute of Technology (KIT)) • 3:30 PM 4:00 PM coffee break 30m Outside One West ### Outside One West #### Fermilab PO Box 500 Batavia, IL 60510 • 4:00 PM 4:30 PM Poster Highlight Talks One West ### One West #### Fermilab PO Box 500 Batavia, IL 60510 Short talks on featured posters Convener: Dr Henry Glass (Fermilab) • 4:30 PM 5:30 PM Poster Session I Atrium ### Atrium #### Fermilab PO Box 500 Batavia, IL 60510 All of the posters in this session will also be on display in Poster Session II. • 4:30 PM A Project of a Complex Setup at the Pamirs for Multi-Component Study of EAS and Parent PCRs in a Wide Energy Range Around the “Knee”. 1h A recommencement of CR researches with a unique X-Ray emulsion chamber (XREC) located at a high-altitude experimental site at the Pamirs (4360 m a.s.l.) in the framework of the Pamir-Chacaltaya International Scientific Research Center, recently established by the Governments of the Russian Federation and Tajikistan (2008), opens up a possibility for deep upgrading of the experimental setup and for deployment on its basis of a new complex one of 1 km2 in area for EAS multi-component study including electron, muon, optic and hadron components, as well as a fine structure of EAS cores. The main purpose of the project is a detailed and per elemental study of the PCR spectrum in a wide range of primary energies E0=30 x 106 TeV partially overlapping that of direct observations and containing the “knee” and other close intriguing irregularities of the spectrum. In addition, the designed setup will make it also possible to research a defuse γ-ray radiation with energy above 30 TeV in all northern hemisphere of the sky. The proposed project is based on a positive worldwide experience of creation of hybrid setups at mountain elevations which combines technique of EAS study by means of an array of spaced electronic detectors of charged particles with that of XRECs permitting to study a structure of EAS cores due to its high spatial resolution. A unique astronomical climate and high elevation of the Eastern Pamirs plateau provide excellent conditions for effective detecting of EAS Čerenkov light and particularly for detailed study of its space-angle characteristics, especially sensitive to the PCR composition. A spaced Čerenkov detector array of 245 x 245 m2 in area complemented with 4 wide field-of-view (≥20°) imaging atmospheric Čerenkov telescopes (IACT) of 3-4 m in diameter with angular resolution 0.5-1.0° will be employed for determining of space-angle distributions of individual EAS. The atmosphere quality control will be performed with lidar technique. One more Čerenkov light telescope with ring-like system of mirrors (R=80 m) and cylindric mosaic of PMT in the center of the ring, which is specially designed for detection of Čerenkov radiation of the PCR nuclei, is under simulation now. Speaker: Dr Alexander Borisov (P.N.Lebedev Physical Institute, RAS) • 4:30 PM Bistatic Radar: A New Method for Detecting Cosmic Rays 1h Progress in the study of high energy cosmic ray physics is limited by low flux. In order to collect substantial statistics above$10^{19}$~eV, the two largest ground arrays currently in operation cover 800~$\mbox{km}^2$(Telescope Array, Utah) and 3000~$\mbox{km}^2$(Auger Observatory, Argentina). The logistics and cost of an order-of-magnitude increase in ground array aperture is prohibitive. In the literature, radar detection experiments have been proposed but substantial results have not been reported. Here, we describe our plans to build and test a bistatic radar facility overlapping the Telescope Array (TA) in Delta, Utah. We have obtained an FCC license to broadcast a constant wave 54.1~MHz signal over the large TA ground array, with radar echoes to be received at our detection facility on the far side of the array. Systems monitoring and data logging systems are currently being developed. Our immediate goal is to detect cosmic rays in coincidence with TA by reflecting radar signals from the air shower ion core. Through subsequent detector advances we will seek to determine air shower geometry and energy. Speaker: Mr Isaac Myers (University of Utah Department of Physics and Astronomy) • 4:30 PM Constrains of Extragalactic Background Light expected from observation of distant metagalactic sources 1739+522 (z=1.375) and 3c454.3 (z=0.859) (by SHALON Cherenkov telescopes). 1h Extragalactic diffuse background radiation blocks the propagation of TeV γ-ray over large distances (z>0.1) by producing electron-positron pairs. As a result, primary spectrum of gamma-source is changed, depending on spectrum of background light. So, a hard spectra of Active Galactic Nuclei with high red shifts of 0.03 – 1.8 allow to determine an absorption by Extragalactic Background Light and thus spectrum of EBL. The redshifts of SHALON very high energy gamma-ray sources range from z=0.0183 to z=1.375. During the period 1992 – 2010, SHALON has been used for observations of the metagalactic sources NGC1275 (z=0.0183), SN2006gy (z=0.019), Mkn421 (z=0.031), Mkn501 (z=0.034), Mkn180 (z=0.046), OJ 287 (z=0.306), 3c454.3 (z=0.895), 1739+522 (z=1.375). Among them bright enough AGNs of BLLac type (Mkn421, Mkn 501) and FSRQ type (3c454.3, 1739+522) those spectra are resolved in the TeV energy band from 1 to ~20-30 TeV. Spectral energy distributions and images of distant Active Galactic Nuclei are presented. Spectral energy distribution of Extragalactic Background Light constrained from observations of Mkn421 (z=0.031), Mkn501 (z=0.034) 3c454.3 (z=0.859) and 1739+522(z=1.375) together with models and measurements are presented. Observations of distant metagalactic sources have shown that the Universe is more transparent to very high-energy gamma-rays than previously believed. Speaker: Dr Vera Yurievna Sinitsyna (P.N. Lebedev Physical Institute) • 4:30 PM Cosmic ray composition around the knee. 1h The Ne spectra for EAS and EAS with gamma-families are analyzed (Experiment "Hadron"-Tien-Shan).Presence thin structure (peaks) in EAS spectrum with gamma-families and necessity of simultaneous approximation of two spectra (EAS and EAS+γ) essentially the same mass composition limits possible models of nucleus individual spectra. The elementary variant of model when spectra of all five nuclear groups are similar is considered. Satisfactory approximation of both spectra Ne for EAS and EAS with gamma-families turns out in the assumption of magnetic rigidity of a break in spectra R=0.13 PV and presence of two peaks in the nuclear spectra at values of magnetic rigidity R=0.13 and 5.4 PV. This form of nuclear spectra permits to suggest two component CR composition. Presence of peaks in the nuclear spectra is explained by the contribution of radiation of single close source CR. Speaker: Prof. Sergey Shaulov (FIAN) • 4:30 PM Extensive air shower simulation for the Telescope Array surface detector 1h The history of ultra-high energy cosmic ray observation is now approaching 50 years. However, until quite recently, the full simulation of an extensive air shower was computationally impossible due to the vast quantity of daughter particles involved. However, with the advent of modern cluster computing, simulations that once would have taken years to complete can be done in a matter of hours or even minutes. Full shower simulations produced by a parallelization scheme employing the Karlsruhe Extensive Air Shower Simulation Code (CORSIKA) will be presented in conjunction with a dethinning'' technique that attempts to recover information lost by the CORSIKA statistical thinning algorithm. Detailed comparisons between simulated and real event sets will then be presented Speaker: Dr Benjamin Stokes (University of Utah) • 4:30 PM Fluctuation of TeV to EeV Energy Muons and the induced muon showers in Water 1h By using the integral methods in the muon propagation through water, we calculate the range fluctuation of high and ultra high energy muons. Many authors divide all radiative processes into two part, namely, the continuous part and stochastic part in their Monte Carlo simulation in order to consider the fluctuation in the both range and energies of the muons, while we treat all radiative processes as exactly as possible, without the introduction of the continuous parts in all radiative processes. The validity of our Monte Carlo method is checked by the corresponding analytical method which is methodologically independent on the Monte Carlo procedure. Accompanied electromagnetic showers are generated by the direct electron pair production, bremsstrahlung and photo-nuclear interaction. These showers are calculated by the exact Monte Carlo Method in one dimensional way. We report survival probabilities, their differential energy distributions, range distributions and examples of individual muon behavior. Speaker: Dr Nobusuke Takahashi (Hirosaki University) • 4:30 PM High-energy atmospheric neutrinos 1h High-energy neutrinos, arising from decays of mesons that were produced through the cosmic rays collisions with air nuclei, form unavoidable background noise in the astrophysical neutrino detection problem. The atmospheric neutrino flux above 1 PeV should be supposedly dominated by the contribution of charmed particle decays. These (prompt) neutrinos originated from decays of massive shortlived particles,$D^\pm$,$D^0$,$\overline{D}{}^0$,$D_s^\pm$,$\Lambda^+_c, compose the most uncertain fraction of the high-energy atmospheric neutrino flux because of poor explored processes of the charm production. Besides, an ambiguity in high-energy behavior of pion and especially kaon production cross sections for nucleon-nucleus collisions may affect essentially the calculated neutrino flux. There is the energy range where above flux uncertainties superimpose. A new calculation presented here reveals sizable differences, up to the factor of 1.8 above 1 TeV, in muon neutrino flux predictions obtained with usage of known hadronic models, SIBYLL 2.1 and QGSJET-II. This calculation of the atmospheric neutrino flux in the energy range 10 GeV-10 PeV is made within 1D approach to solve nuclear cascade equations in the atmosphere, which takes into account non-scaling behavior of the inclusive cross-sections for the particle production, the rise of total inelastic hadron-nucleus cross-sections and nonpower law of the primary cosmic ray spectrum. This approach was recently tested in the atmospheric muon flux calculations [Astropart. Phys. 30 (2008) 219]. The results of the neutrino flux calculations are compared with the Frejus, AMANDA-II and IceCube measurement data. Speaker: Prof. Sergei Sinegovsky (Institute of Applied Physics, Irkutsk State University) • 4:30 PM Impact of X-Ray Emulsion Chamber Response on Gamma-Family Observable Characteristics 1h Analysis of various data accumulated in X-ray emulsion chamber experiments, especially, data on gamma–hadron families with unusual characteristics (Centauros, aligned events etc.), requires a comprehensive computer code to simulate propagation of electromagnetic and various-type hadron particles through a sandwich-like medium of emulsion chambers as well as measuring procedures employed for emulsion chamber data processing. Such a new code, ECSim 2.1, has been recently elaborated on the basis of GEANT 3.21 package. As compared to the latter, the ECSim 2.1 takes into account the LPM effect for gamma-rays and electrons, uses new cross sections of muon interactions of different types allowing also for the LPM effect in pair generation, incorporates QGSJET or MC0/FANSY models for simulation of high-energy hadron interactions and accounts for production and interactions of charm particles. Besides, measuring and data treatment procedures employed in the Pamir experiment are simulated properly. An impact of X-Ray emulsion chamber response on gamma-family observable characteristics is discussed. Speaker: Dr Alexander Borisov (P.N.Lebedev Physical Institute, RAS) • 4:30 PM Integrated circuit of coordinate detector for detection of charged particles 1h New-type coordinate detector is considered which is based on special-purpose integrated circuit designed for detection of charged particles, local amplification and direct transmission of signal into computer. It is shown that such detectors make it possible to achieve a higher coordinate determination accuracy and processing speed as well as to bring down their cost as compared with modern detectors. It is possible to manufacture mosaic-structure large-sized detector panels with an active area-to-dead area ratio of not lower than ten. Detectors of this type could be applied in future space and balloon experiments. Speaker: Prof. Rauf Mukhamedshin (Institute for Nuclear Research of Russian Academy of Science) • 4:30 PM Modern status of high-mountain three-level ATHLET complex 1h Three-level (3340, 1750 and 850 m a.s.l) ATHLET (Almaty Three Level Experimental Technique) complex is built up for investigations in fields of cosmic ray (CR) physics, astrophysics and gamma-ray astronomy of superhigh energies. The ATHLET’s highest part has to include a) 1-km2-area ADRON-M facility with a “dense” location of detectors to detect electromagnetic, hadron, muon, neutron and radio EAS components with a high accuracy (~1 m) of determination of shower axes; b) specific shower array located at angle of ~45 degrees to detect showers in a wide range of zenith angles; c) GROZA complex for studying the nature of lightnings; d) “Muon beam” facility and classic seismic arrangements; e) a large instrumental complex to study low-energy components. Physical investigation goals are as follows. 1) Astrophysics of cosmic rays (energetic spectrum and mass composition of primary cosmic radiation at E0 = 10^14 – 2x10^18 eV). 2) Gamma-ray astronomy (at E>50 TeV) (by selecting muonless, hadronless and neutronless showers). 3) Study of high-energy hadron interactions with atmosphere nuclei and selection of models which could describe EAS observable features in the best way. 4) Search for new phenomena. 5) Analysis of relations between neutron physics and EAS. 6) Mechanisms of lightning discharge and their connection with EAS and other CR-induced phenomena, 7) Solar radiation and “cosmic weather”. 8) Seismology and EAS. Modern status of detectors of the ATHLET complex is considered. Speaker: Prof. Rauf Mukhamedshin for ATHLET Collaboration (Institute for Nuclear Research of Russian Academy of Science) • 4:30 PM Multiparticle production in nucleus-nucleus interactions at 14.6 A GeV 1h We present our observations on the various features from the 855 interactions of 14.6 A GeV 28Si in nuclear emulsion. Multiplicity distribution, mean multiplicities, multiplicity correlations of black, grey, shower and helium fragments are studied in this investigation. A comparative study of the results obtained from the interactions at 14.6 A GeV with other available data at the different energies per nucleon is also presented, which shows a good agreement with our experimental data. The study shows that production of grey particles has a linear dependence with shower particle multiplicity where as black particles exhibit a saturation effect, which describe the impact parameter dependence very well. Speaker: Mr Ashwini Kumar (Banaras Hindu University) • 4:30 PM Nucleon electromagnetic structure functions in extremely small x-region 1h We present results of caslculations of transverse and longitudinal cross sections of photoabsorption on the nucleon target, in a broad region of very small Bjorken x values and not very large photon virtualities, using the two-component model developed by authors in their previous works. The model is based on the generalized vector dominance concept and color dipole approaches. The detailed comparison of the theoretical predictions with the HERA data is given. Speaker: Prof. Edgar Bugaev (Institute for Nuclear Research) • 4:30 PM On the origins of the highest energy cosmic rays 1h Active galactic nuclei (AGNs) appear to be the most plausible source of ultra-high energy cosmic rays (UHECRs), yet there is currently no conclusive evidence for this hypothesis. Correlation between the arrival directions of some UHECRs and the positions of nearby AGNs has been reported for a sample of 27 UHECRs detected by the Pierre Auger Observatory (PAO 2007), although analyses of larger samples find a weaker signal (PAO 2010). Here we present a fully Bayesian analysis of the original PAO data, which makes use of more of the available information, and find, with 3 sigma confidence, that a subset of observed UHECRs originate from known AGNs listed in the Veron-Cetty and Veron (2006) AGN catalogue. We will extend our analysis to more homogeneous AGN catalogues such as the Swift BAT sample. Speaker: Ms Laura Watson (Imperial College London) • 4:30 PM On the Positron Fraction in Cosmic Rays and Models of Cosmic-Ray Propagation 1h The positron fraction observed by PAMELA and other experiments up to ~100 GeV is analyzed in terms of models of cosmic-ray propagation. It is shown that generically we expect the positron fraction to reach ~0.6 at energies of several TeV, and its energy dependence bears an intimate but subtle connection with that of the boron to carbon ratio in cosmic rays. The observed positron fraction can be fit in a model that assumes a significant fraction of the boron below ~10 GeV is generated through spallation of cosmic-ray nuclei in a cocoon-like region surrounding the sources, and the positrons of energy higher than a few GeV are almost exclusively generated through cosmic-ray interactions in the general interstellar medium. Such a model is consistent with the bounds on cosmic-ray anisotropies and other observations. Speaker: Mr Benjamin Burch (Washington University in St. Louis) • 4:30 PM Phenomenological approach to multiple particle production (2) 1h In our previous presentation we showed how well the rapidity density distributions and the transverse momentum (p_{T}) distributions at sqrt{s}=22.4, 546 and 1800 GeV are described by our phenomenological formulation. Based on the energy dependence of the values of the parameters, which are obtained by fitting the calculated distributions to those of the experiments, we examine how the present formulation describes the energy dependence of the p_{T} average, that of the multiplicity and the local p_{T} average along the rapidity y* in the forward region, obtained by UA7 Collaboration at sqrt{s}=630 GeV. Extrapolating the energy dependence of the parameters into higher energies, we discuss the multiplicity, inelasticity and the pseudo-rapidity density distribution at sqrt(s)=1.4 x 10^{3} GeV (LHC energy) and 4.5 x 10^{5} GeV (10^{20} eV in the laboratory energy), together with predictions by several models of multiple particle production. Speaker: Dr AKINORI OHSAWA (Institute for Cosmic Ray Research, University of Tokyo.) • 4:30 PM Pion Production Cross-section Measurements in p+C Collisions at the CERN SPS for Understanding Extensive Air Showers 1h An important approach to studying high-energy cosmic rays is the investigation of the properties of extensive air showers; however, the lateral distribution of particles in simulations of such showers strongly depends on the applied model of low-energy hadronic interactions. It has been shown that many constraints to be applied to these models can be obtained by studying identified-particle spectra from accelerator collisions, in the energy range of the CERN Super Proton Synchrotron. Here we present measurements of the pion production cross-section obtained by the NA61/SHINE experiment at the SPS, in proton-carbon collisions at the beam energy of 30 GeV from the years: 2007 and 2009. Further analyses of identified-particle yields in SHINE, in particular with a pion beam, are in preparation. Speaker: Dr Marek Szuba (Karlsruhe Institute of Technology) • 4:30 PM Search Sources of Cosmic Rays Ultrahigh Energy 1h The arrival directions of ultrahigh energy extensive air showers (EAS) by Yakutsk, AGASA and P. Auger data are considered. For the first time, the arrival directions of extensive air showers of ultrahigh energy, registered by Yakutsk EAS array more carefully are considered. It is found that the arrival directions of EAS Yakutsk data are correlated with pulsars from side Input of Local Arm Galaxy Orion. Also it is found that from this side the arrival directions of EAS by data AGASA are correlated with pulsars, the arrival directions of EAS by data P.Auger are correlated with pulsars from Outside of Local Arm Orion. It is shown the majority these pulsars have a short period of rotate around of their axes. The problem of cosmic ray origin is discussed. Speaker: Dr Aleksei A. Mikhailov (Yu.G. Shafer Institute of Cosmophysical Research and Aeronomy) • 4:30 PM Spectral Analysis, and Hardness-ratios Correlations of SGR 1900+14 Bursts 1h In the present study, we inspecte a refined sample of 117 bursts from SGR1900+14 observed with RXTE, PCA. We use 10 spectral-models, and the best fitting spectral-models has been found statistically to be the thermal bremsstrahlung and the power-law. Data are analyzed more by model-independent techniques. The global color-color diagrams are obtained with no distinguishable patterns as other objects like accretion disk neutron stars. Strong global correlations for burst timing and spectral properties with hardness-ratios has been found, and the most interesting ones are those between total hardness-ratios (soft/hard) and the bursts’ total counts. That is, the hardness-ratio decreases; in the mean; with the burst-total-counts (more photons = softer spectrum.) Also this result is confirmed by the strong correlations obtained between bursts’ total-counts and both hot-zone temperature (kT) and photon index (). Classification of bursts depending on the burst-duration and the total photons-contained will be taken into consideration in our future studies of bursts. Speaker: Mr Mohammed Hasan Soleiman Yussef (Cairo University, Faculty of Science, Physics department.) • 4:30 PM Studies of Emitted Particles in Nucleus-Nucleus Interactions at 4.5 A GeV/c 1h Analysis has been done for the emitted particles in (12C, 16O, 22Ne, 28Si) + Emulsion interactions at (4.1-4.5) A GeV/c. The multiplicity of the emitted particles; as a function of the mass-number of the interacting projectiles nuclei; has been calculated. The multiplicity distribution and the average-values of the emitted particles (the experimental-values) are compared with that calculated values from Monte-Carlo simulation (the code developed at high-energy lab; Cairo university : “modified cascade evaporated model” (MCEM). Strong correlation between the number of the recoiled nucleons has been observed. An agreement has been shown between the experimental values and the theoretical calculated ones. Speaker: Prof. Sayed Saleh (Cairo University) • 4:30 PM Study of primary cosmic rays at superhigh energies on the lunar surface and circumlunar orbit 1h Mathematical model of experimental conditions on research for primary cosmic radiation (PCR) on the lunar surface and circumlunar orbit is considered. The fundamental possibility of detection of PCR particles is shown by the use of simultaneous detection of three components produced by cascades in the lunar regolith (secondary neutrons, gamma-ray and radio emission) measured by detectors placed on the lunar surface as well detectors located aboard a circumlunar-orbit scientific satellite. The “Neutronium” project combining these approaches is considered. Results of simulations are given Speaker: Prof. Rauf Mukhamedshin (Institute for Nuclear Research of Russian Academy of Science) • 4:30 PM TeV emission from NGC1275 viewed by SHALON 15 year observations 1h Galaxy clusters have been consider as sources of TeV gamma-rays emitted by high-energy protons and electrons accelerated by large scale structure formation shocks, galactic winds, or active galactic nuclei. The Perseus cluster of galaxies is one of the best studied clusters due to its proximity and its brightness. Galaxy NGC 1275 is the central dominant galaxy of the Perseus Cluster of Galaxies and is of Seyfert galaxy class. NGC 1275 is known as powerful X-ray and radio source. Many studies explored correlations of X-ray radio optical and ultraviolet emission. In 1996 year a new metagalactic source was detected by SHALON at TeV energies. This object was identified with Seyfert galaxy NGC 1275 (with redshift z=0.0179); its image is presented. The maxima of the TeV gamma -ray, X-ray and radio emission coincide with the active nucleus of NGC 1275. In contrast, the X-ray and TeV emission disappears almost completely in the vicinity of the radio lobes. The correlation TeV with X-ray emitting regions was found whereas the integral gamma -ray flux for this source is found to be(0.78\pm0.13)\times10^{-12}cm^{-2}s^{-1}$at energies of$>0.8$TeV. The energy spectrum of NGC 1275 at 0.8 to 40 TeV can be approximated by the power law$F(> E_O) \propto E^k$, with$k=-2.25\pm0.10\$. The Seyfert galaxy NGC 1275 has been also observed with the Tibet Array (about 5 TeV) and then with Veritas telescope at energies about 300 GeV at 2009. The recent detection by the Fermi LAT of high-energy gamma-rays from the radio galaxy NGC 1275 makes the observation of the very high energy (E > 100 GeV) part of its broadband spectrum particularly interesting. The overall spectral energy distribution of NGC 1275 from the low energies to the TeV energies is presented. The spectrum of NGC 1275 from SHALON 15 year observations is also shown. The search for gamma-rays from radio galaxies is important for the understanding of the dynamics and structure of active galactic nuclei.
Speaker: Prof. Vera Georgievna Sinitsyna (P.N. Lebedev Physical Institute)
• 4:30 PM
The investigation of the hadronic interaction models using WILLI detector 1h
The WILLI detector, built in IFIN-HH Bucharest, in collaboration with KIT Karlsruhe, is a rotatable modular detector for measuring charge ratio for cosmic muons with energy < 1 GeV. It is under construction a mini-array for measuring the muon charge ratio in Extensive Air Showers. The EAS simulations have been performed with CORSIKA code. The values of the muon flux, calculated with semi-analytical formula, and simulated with CORSIKA code, based on DPMJET and QGSJET models for the hadronic interactions, are compared with the experimental data determined with WILLI detector. No significant differences between the two models and experimental data are observed. The measurements of the muon charge ratio for different angles-of-incidence, (performed with WILLI detector) shows an asymmetry due to the influence of magnetic field on muons trajectory; the values are in agreement with the simulations based on DPMJET hadronic interaction model. The simulations of muon charge ratio in EAS performed with CORSIKA code based on three hadronic interaction models (QGSJET2, EPOS and SYBILL) show relative small difference between models for H and for the Fe showers; the effect is more ronounced at higher inclination of WILLI detector. The future measurements should indicate which model is suitable.
Speaker: Dr Iliana Brancus (National Institute for Physics and Nuclear Engineering Horia Hulubei)
• 4:30 PM
The Measured Spectrum of the Telescope Array's Middle Drum Detector 1h
The Telescope Array's Middle Drum fluorescence detector was constructed using refurbished telescopes from the High Resolution Fly's Eye (HiRes) experiment. As such, there is a direct comparison between these two experiments' fluorescence energy spectra. A progress report will be presented based on over 2 years of collected data by the Middle Drum site of Telescope Array.
Speaker: Mr Douglas Rodriguez (University of Utah)
• 4:30 PM
Threshold Cerenkov detector with Radial Segmentation ( TCDRS ) 1h
I present the prototype Threshold Cerenkov Detector with Radial Segmentation; as a part of the detector development and implementation research. The detector has three concentric cylinders, each with a different dielectric medium, and four scintillators that triggers cosmic particles with a time of fly of 5 ns. The radiator is designed to produce more photons as the particles travels into the TCDRS and fewer photons as it leaves. The correlation between the number of photons produced in the cylinders and the particle momentum allows particles separation of one sigma, for e, μ, π, κ, and p up to 5 GeV/c. Details of the TCDRS Monte Carlo, construction, data collection and data analysis are presented.
Speaker: Dr Ely Leon (Chicago State University)
• 4:30 PM
Two source emission behavior of projectile fragments alpha in 84Kr interactions at around 1 GeV per nucleon 1h
The emission of projectile fragments alpha has been studied in 84Kr interactions with nuclei of the nuclear emulsion detector composition at relativistic energy below 2 GeV per nucleon. The angular distribution of projectile fragments alpha in terms of transverse momentum could not be explained by a straight and clean-cut collision geometry hypothesis of Participant – Spectator (PS) Model. Therefore, it is assumed that projectile fragments alpha were produced from two separate sources that belong to the projectile spectator region differing drastically in their temperatures. It has been clearly observed that the emission of projectile fragments alpha are from two different sources. The contribution of projectile fragments alpha from contact layer or hot source is a few percent of the total emission of projectile fragments alphas. Most of the projectile fragments alphas are emitted from the cold source.
Speaker: Dr Venktesh SINGH (Banaras Hindu University, Varanas 221 005, INDIA)
• 4:30 PM
Ultra-High Energy Muon Neutrino Propagation through the Earth and Induced Muon Energy Distribution near the One Cubic Kilometer Detector 1h
We calculate high and ultra-high energy upward-going muon neutrino propagation through the Earth and the induced muon energy distribution near the one cubic kilometer detector using the Monte Carlo simulation, according to neutral current interaction. The primary neutrino energies on the surface of the Earth are 1PeV, 1EeV, and 1ZeV. The mean free paths of ultra-high energy neutrino events generated by the deep inelastic scattering may be comparable with the diameter of the Earth or less than it. Therefore, the induced muon production distribution is influenced by the change of the densities interior to the Earth. Furthermore, in such situation, the contribution from the neutral current neutrino interaction to the induced muon production distribution cannot be neglected. We report several examples of the deep inelastic scattered depth of muon neutrino in the Earth and the induced muon energy distribution near the detector.
Speaker: Dr Nobusuke Takahashi (Hirosaki University)
• Wednesday, 30 June
• 8:30 AM 10:05 AM
Sensitivity of Monte Carlo models to data: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Dr Paolo Lipari (INFN Roma 1)
• 8:30 AM
New Development in EPOS 2 15m
Since 2006, EPOS hadronic interaction model is being used for very high energy cosmic ray analysis. Designed for minimum bias particle physics and used to have a precise description of SPS and RHIC heavy ion collisions, EPOS brought more detailed description of hadronic interactions in air shower development. Thanks to this model it was possible to understand why there was less muons in air shower simulations than observed in real data. With the start of the LHC era, a better description of hard processes and collective effects is needed to understand deeply the incoming data. I will describe the basic physics in EPOS and the new developments and constraints which are taken into account in EPOS 2, and their consequences on air shower development.
Speaker: Dr Tanguy Pierog (KIT, IK)
• 8:45 AM
KASCADE-Grande is a large detector array for the measurement of cosmic ray air showers in the primary energy range of 100 TeV to 1 EeV. Due to the multi-detector concept of the experimental set-up, various observables of the electromagnetic, the muonic and for lower primary energies also the hadronic particle component are measured for individual air showers. The experimental data are compared to predictions of CORSIKA simulations using high-energy hadronic interaction models (e.g. QGSJET or EPOS), as well as low-energy interaction models (e.g. FLUKA or GHEISHA). This contribution will summarize the results of such investigations. In particular, the validity of the new EPOS version 1.99 for EAS with energy around 100 PeV will be discussed.
Speaker: Dr Donghwa Kang (Karlsruhe Institute of Technology)
• 9:00 AM
Relation of Interaction Characteristics at Ultra-High Energies to Extensive Air Shower Observables 15m
Only by measurement of extensive air showers it is possible to explore the nature of cosmic ray particles at the highest energies. Most properties can only be obtained from the interpretation of air shower data and are thus depending on predictions of hadronic interaction models at ultra-high energies. We discuss different scenarios of model extrapolations from accelerator data to air shower energies and investigate their impact on the corresponding air shower predictions. For this purpose we developed an ad hoc model, which is based on the modification of the output of standard hadronic interaction event generators within the air shower simulation process. This model allows us to study the impact of changing interaction features on the air shower development. In a systematic study we demonstrate the resulting changes of important air shower observables and discuss them also in terms of the predictions of the Heitler model of air shower cascades.
Speaker: Dr Ralf Ulrich (PSU)
• 9:15 AM
Consequences of the LHC results in the interpretation of gama ray families and giant EAS data 15m
Present results of the LHC (up to 26 PeV in the Lab. system) are a very small lever arm for the extrapolation of models up to 100 EeV. However, the measurements of CMS exhibit a central pseudo rapidity density larger than the prediction of the different models. Introducing on this basis new guidelines, with larger multiplicities in the models inserted in thesimulation, we examine the consequences for gamma ray families and very large EAS. A special attention is given to the coplanar emission observed near 10 PeV : the case of large Pt's generated during the fragmentation of relativistic strings involving valence diquarks (partonic model+Schwinger mechanism)is explored as a possible source of alignments at this energy. At larger energies , the effects of those circumstances in the interaction fragmentation region are investigated, together with large multiplicities, as the possible origin of the small penetration power of proton initiated showers in the atmosphere. Associated statisticalbias generated by a sharp knee or ankle in the primary spectrum are also considered.
Speaker: Prof. Jean-Noël CAPDEVIELLE (APC, CNRS-University Paris Diderot)
• 9:30 AM
Sibyll with Charm 15m
The cosmic ray interaction event generator Sibyll is widely used in extensive air shower simulations for cosmic ray and neutrino experiments. Charm particle production has been added to the Monte Carlo with a phenomenological, non-perturbative model that properly accounts for charm production in the forward direction. As prompt decays of charm can become a significant background for neutrino detection, proper simulation of charm particles is very important. We compare charm meson and baryon production to accelerator data.
Speaker: Dr Eun-Joo Ahn (Fermilab)
• 9:45 AM
Phenomenological approach to multiple particle production (1) 20m
We describe the rapidity density distribution and the transverse momentum (p_{t}) distribution in multiple particle production, assuming a simple mechanism. It is an assumed mechanism that the newly produced particles are emitted isotropically from several emitting centers which are distributed on the rapidity axis in CMS. The energy distribution of the emitted particles is an exponential type in the rest frame of respective emitting centers. The distribution of the emitting centers is uniform between -y_{0} and y_{0} (y_{0}=ln(sqrt{s}/M)-lna_{2}, a_{2} an adjustable parameter). We can obtain the rapidity density distribution analytically, which can be transformed easily to the pseudo-rapidity density distribution and x-distribution. The rapidity density distribution and the p_{T} distribution by the present formulation describes well those of the experiments at various energies by adjusting values of the parameters (five in total). We show how well the experimental data at sqrt{s}=22.4, 546, and 1800 GeV are described by the present formulation.
Speaker: Dr AKINORI OHSAWA (Institute for Cosmic Ray Research, University of Tokyo.)
• 10:05 AM 10:35 AM
Extensive air shower experiments: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Gaurang Yodh (University of California Irvine)
• 10:05 AM
Cosmic ray data and their interpretation: the Tibet hybrid EAS experiment -- Primary energy spectra of Cosmic Rays at the knee and tests of hadronic interaction models -- 30m
The Tibet hybrid air shower experiment is composed by an air-shower core detector array and the air-shower array (and a large muon detector from October, 2010), that has been operated at Yangbajing (4300 m above sea level) in Tibet, China, since 1996. This multi-detector system is used for the search for high energy celestial gamma-ray and cosmic ray sources, and for the study of the chemical composition as well as the energy spectra of nuclear-components in the knee region. Both are aimed to investigate the origin of high energy cosmic rays through different approaches. In this talk, based on the chemical composition and the energy spectra of some individual nuclear components around the knee, we would like to discuss the sharp knee observed by our experiment and its relation with the contribution of possibly existing nearby source(s). We would also discuss the check of currently used hadronic interaction models by using new Tibet hybrid experimental data. We also plan to build a ground based large and complexγ/CR observatory at high altitude (4300m a.s.l.) within 10 years.
Speaker: Prof. Yuqian Ma (IHEP)
• 10:35 AM 11:05 AM
coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 11:05 AM 12:35 PM
Extensive air shower experiments: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Gaurang Yodh (University of California Irvine)
• 11:05 AM
The present status of the GRAPES-3 experiment 30m
The GRAPES-3 experiment is a high density array of 400 plastic scintillator detectors and a large (560 sq.m.) area muon detector located at Ooty at an altitude of 2200 m above sea level. The primary objective of this experiment is to study the high energy processes occurring in the universe through a systematic study of composition of primary cosmic rays below and above the knee', compact sources of multi-TeV gamma rays, diffuse flux of gamma rays and the solar accelerator through the impact of coronal mass ejections, solar flares etc. To achieve these objectives extensive in-house development of necessary instrumentation including plastic scintillator and high-speed signal processing electronics has been carried out. The development of high performance TDC and silicon photo-multiplier have the potential to complete change the nature of scientific problems that can now be addressed. During the talk some of these aspects would be highlighted.
Speaker: Prof. Sunil Gupta (Tata Institute of Fundamental Research)
• 11:35 AM
Results from the GAMMA experiment on Mt. Aragats - improved data 30m
Status of the GAMMA experiment is presented. The all-particle energy spectrum of the primary cosmic rays at energies 1 – 300 PeV has been obtained on the basis of the GAMMA experimental improved data. The irregularities of the energy spectrum above the knee are discussed in comparison with other experiments. An upper limit of Galactic diffuse gamma ray flux measured with the GAMMA experiment at energy about 175 TeV is also discussed.
Speaker: Dr Romen Martirosov (Yerevan Physics Institute, Yerevan, Armenia)
• 12:05 PM
Cosmic Ray Physics with IceTop and IceCube 30m
IceTop air shower array, as the surface component of the IceCube Neutrino Observatory at the South Pole, is now 92% complete and taking data with 73 stations. The detector will study the mass composition of primary cosmic rays from the knee up to about 1 EeV. In this talk the performance of IceTop, and the preliminary results in the energy range of 1 PeV to 80 PeV will be reported.
Speaker: Dr Serap Tilav (University of Delaware)
• 12:35 PM 1:30 PM
lunch 55m Cafeteria
### Cafeteria
#### Fermilab
PO Box 500 Batavia, IL 60510
• 1:30 PM 3:30 PM
Laboratory Tour Starting at Atrium / Main Entrance
### Starting at Atrium / Main Entrance
#### Fermilab
PO Box 500 Batavia, IL 60510
• 3:30 PM 4:00 PM
coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:00 PM 5:00 PM
Colloquium One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:00 PM
The Composition of Cosmic Rays: Questions, Surprises, and Recent Answers 1h
Even though cosmic rays have been observed for almost a century, they remain enigmatic messengers from distant regions in space, and many questions about their origin and acceleration are still open. Details of the composition and of the energy spectra of the individual components are required to find answers, but are increasingly difficult to obtain with increasing particle energies. We will review the present knowledge, emphasizing the energy region below the “knee” where direct observations are possible, and discuss current measurements, their implications, and future prospects. We also will discuss some of the challenges that are associated with recently reported data on rare components such as electrons, positrons, and anti-protons.
Speaker: Prof. Dietrich Müller (University of Chicago)
• Thursday, 1 July
• 8:30 AM 9:30 AM
Extensive air shower experiments: 3 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Suresh Tonwar (University of Maryland)
• 8:30 AM
The study of the cosmic ray energy spectrum in the interval 10^16 eV - 10^18 eV results of particular importance for several reasons, one of them is the possible existence of a second knee, other one is the possible presence of a galactic-extragalactic transition in the cosmic ray flux and another one is the prediction from some astrophysical models of a knee in the energy spectrum of the heavy component of galactic cosmic rays. To address these questions precise measurements of the arrival direction, energy and composition of cosmic rays in this energy regime need to be performed. For this purpose the KASCADE-Grande air-shower detector was built at the place of the Karlsruhe Institute of Technology. The detector covers a 0.5 km^2 surface with different arrays of detectors which allows to measure simultaneously the charged and muon components of the air-shower events. With this information a lot can be learned about the composition and energy of the primary cosmic ray particles. In this talk, the KASCADE-Grande detector is described and first results of the experiment are shown, mainly about the all-particle cosmic ray energy spectrum in the energy region from 10^16 eV to 10^18 eV.
Speaker: Dr Juan Carlos Arteaga-Velázquez (Instituto de Física y Matemáticas, Universidad Michoacana)
• 9:00 AM
Study of the longitudinal development of extensive air showers with the Muon Tracking Detector in KASCADE-Grande. 15m
The Muon Tracking Detector (MTD) in KASCADE-Grande experiment measures with high accuracy muon directions in EAS (Emu>800MeV). In addition, shower directions are determined by the surface detectors with high precision. These two conditions allow to study shower longitudinal development by means of quantities like muon production heights and muon pseudorapidities and lateral distributions of muon densities. Results of such investigations will be shown between 10^15 eV and 10^17 eV, for data and simulations based on CORSIKA with QGSJetII+Fluka2002.4 model combination and the new EPOS version 1.99. The muon pseudorapidity distributions will be studied in the predefined distance range to the shower core and compared to the simulations as well.The pseudorapidity distributions for muons which stem from above 15 km muon production height and which stem very likely from the first interactions are studied in more detail also in the context of geometric scaling in the near LHC energy range. This work was supported in part by the German-Polish bilateral collaboration grant (PPP-DAAD/MNiSW) for the years 2009-2010
Speaker: Dr Paul Doll (KIT-Karlsruhe)
• 9:15 AM
Behaviour of the EAS age parameter in the knee energy region 15m
We review the different definitions of the age parameter used in the lateral and longitudinal electron distributions. In order to remove ambiguities in the interpretation of the experimental data, we have compared simulations with CORSIKA carried simultaneously with the options NKG and EGS. The effect of the positron annihilation cross section missing in the NKG approach is pointed out for small and inclined EAS, near the axis ; the consequences of the electrons coming from muon decay at large distances from axis are also underlined. Distinguishing the longitudinal, lateral and local age parameters, correspondances and conversions between the 3 categories are inferred from the simulations. Finally, the age parameter derived by fitting the lateral profile of the electron distribution, is confirmed as a good indicator of the primary composition and the hadronicity of the cascade as far as some conditions are fullfilled concerning bands of istances to the axis and zenith angle, dependant slightly on the primary energy (examples in the interpretation from Kascade and Akeno data).
Speaker: Prof. Jean-Noël CAPDEVIELLE (APC, CNRS-University Paris Diderot)
• 9:30 AM 10:05 AM
Experiments above the Ankle: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Dietrich Muller (University of Chicago)
• 9:30 AM
Final Results from the High Resolution Fly's Eye (HiRes) Experiment 35m
Final results from the HiRes experiment on the spectrum, composition and anisotropy of ultra-high energy cosmic rays will be presented. Stereo and monocular data analysis will be described. The HiRes experiment has observed the Greisen-Zatsepin-Kuzmin cutoff. This analysis and evidence for a light composition of cosmic rays to the highest energies will be presented. Recent results on anisotropy relative to large scale structure of the universe will also be discussed.
Speaker: Prof. Pierre Sokolsky (University of Utah)
• 10:05 AM 10:35 AM
coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 10:35 AM 12:20 PM
Experiments above the Ankle: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Dietrich Muller (University of Chicago)
• 10:35 AM
Results from the Pierre Auger Observatory 35m
The Pierre Auger Observatory in the southern site of Mendoza, Argentina is the largest cosmic ray detector ever built. Since its completion in 2008, the Observatory is steadily taking data with 3000 km**2 of active detection area, accumulating an unprecedented statistics of high quality events. Results are presented on the energy spectrum of cosmic rays from 10**18 eV to the highest energy, on the anisotropy of the arrival direction of the highest energy cosmic rays, and on the nature and composition of cosmic rays.
Speaker: Prof. Paolo Privitera (University of Chicago)
• 11:10 AM
Measurement of UHECRs by the Telescope Array (TA) experiment 25m
The Telescope Array (TA) experiment, located in the west desert of Utah, USA, observes ultra-high energy cosmic rays (UHECRs) with energies above 10^18.5 eV. TA employs a surface detector (SD) array and 3 batteries of fluorescence detectors (FDs) to measure extensive air showers. The direction and the energy of incoming cosmic rays are measured by both detectors, and the results can be cross checked. The primary composition can be determined by the longitudinal shower development measured by the FD and the muon content inferred at the SD. A full detector is running since May, 2008. The design and the performance of TA, its operational status and the first year results will be presented in the meeting.
Speaker: Prof. Masaki Fukushima (ICRR, Univ. Tokyo)
• 11:35 AM
The Telescope Array Low Energy Extension (TALE) 15m
The Telescope Array (TA) experiment is the largest cosmic ray detector in the northern hemisphere. It also operates the largest scintillation counter array in the world. Together with the three fluorescence detectors (FDs), it is optimized to study cosmic rays as independent detectors and in hybrid mode at energies above the ankle structure. The TA low energy extension will add two additional fluorescence detectors along with an infill array. The first of these will operate in stereoscopic view with an existing TA FD to study in detail the 0.3-30 EeV range around the ankle, with more than a factor of five improvement in aperture at 1 EeV over HiRes. The Tower fluorescence detector, using larger mirrors, will operate in hybrid mode with the infill surface array to measure the spectrum, composition, and anisotropy of cosmic rays down to 30 PeV, well below the "second knee". Together, TA and TALE will be able to measure simultaneously all three known spectral features in the ultra high energy (UHE) regime. TALE will also study the transition from galactic to extragalactic cosmic ray flux, with fluorescence Xmax capabilities for the first time.
Speaker: Prof. Charles Jui (University of Utah)
• 11:50 AM
Analysis Techniques for the TA SD Detector 15m
Abstract: The Telescope Array experiment is the largest cosmic ray experiment in the northern hemisphere. It consists of a surface detector (SD) of 507 scintillation counters and three fluorescence stations overlooking the SD. We develop new techniques for estimating cosmic ray energies and calculating the aperture for TA SD which utilize an accurate CORSIKA Monte Carlo (MC) simulation of natural cosmic rays with appropriate energy spectrum, angular distribution, and composition so that the generated MC has all characteristics of the real data. The simulation is verified by detailed comparisons of MC distributions and fit results with those of the real data. Results of applying these analysis techniques to the actual TA SD data will be shown.
Speaker: Mr Dmitri Ivanov (Rutgers University)
• 12:05 PM
A Relation Between Charged Particles and Muons With Threshold Energy 1 GeV in Extensive Air Showers Registered at the Yakutsk EAS Array 15m
For a long time the three main components of extensive air showers have been measured at the Yakutsk array: the whole charged component, muons with e_{th} \ge 1 GeV and Cherenkov light. Using these data we reconstruct energy of primary cosmic particle (with quasi-colorimetric method), estimate the depth of shower maximum (by the shape of charged particles lateral distribution and a pulse shape of Cherenkov light response in differential detector, t_{1/2} ) and measure relative muon content at different core distances. In this work we consider a relation s_{mu} /s_{ch} between charged and muon components in showers and its fluctuations at fixed energies. The goal of this analysis is to make a comparison between experimental and computational data for different primaries and to obtain an estimation of cosmic rays mass composition in the ultra-high energy domain.
Speaker: Dr Stanislav Knurenko (Yu. G. Shafer Institute of cosmophysical research and aeronomy, SB RAS)
• 12:20 PM 1:20 PM
lunch 1h Cafeteria
### Cafeteria
#### Fermilab
PO Box 500 Batavia, IL 60510
• 1:20 PM 2:05 PM
Experiments above the Ankle: 3 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Jean-Noel Capdevielle (CNRS)
• 1:20 PM
The Depth of Maximum Shower Development and Its Fluctuations: Cosmic Ray Mass Composition at E<sub>0</sub> ≥ 10<sup>17</sup> eV 15m
We present a new data on Cherenkov light observations obtained during 1994-2009 period, after a modernization of the Yakutsk EAS array. A complex analysis of x_{max} and its fluctuations \sigma(x_{max}) was performed in a wide energy range. With the new data, accord- ing to QGSJet II model, an estimation was made of cosmic rays mass composition for E_0 \sim 10^{17} - 3 \times 10^{19} eV. The result points towards a mixed composition with a large portion of heavy nuclei at E_{0} \sim 10^{17} eV and the dominance of light nuclei at E_{0} \sim 1019 eV. The analysis of \sigma(x_max) energy dependence for the same energies qualitatively confirms this result. A shape of xmax distribution at fixed energy 1018 eV is analysed to make more precise conclusion on cosmic ray mass composition.
Speaker: Dr Stanislav Knurenko (Yu. G. Shafer Institute of cosmophysical research and aeronomy, SB RAS)
• 1:35 PM
The MIDAS Experiment: A New Technique for the Detection of Extensive Air Showers 15m
Recent measurements suggest free electrons created in ultra-high energy cosmic ray extensive air showers (EAS) can interact with neutral air molecules producing Bremsstrahlung radiation in the microwave regime. The microwave radiation produced is expected to scale with the number of free electrons in the shower, which itself is a function of the energy of the primary particle and atmospheric depth. Using these properties a calorimetric measurement of the EAS is possible. This technique is analogous to fluorescence detection with the added benefit of a nearly 100% duty cycle and practically no atmospheric attenuation. The Microwave Detection of Air Showers (MIDAS) prototype is currently being developed at the University of Chicago. MIDAS consists of a 53 feed receiver operating in the 3.6 to 4.2 GHz band. The camera is deployed on a 4.5 meter parabolic reflector and is instrumented with high speed power detectors and autonomous FPGA trigger electronics. We present the current status of the MIDAS instrument and an outlook for future development.
Speaker: Mr Christopher Williams (University of Chicago)
• 1:50 PM
AIRFLY: Precise measurement of the absolute yield of fluorescence photons in atmospheric gases 15m
We present preliminary results from the most recent data on the absolute yield of fluorescence photons in atmospheric gases by the AIRFLY collaboration. Currently, the uncertainty in the yield forms the dominant contribution to the systematic uncertainty in the Pierre Auger Observatory's energy spectrum, and are at the level of 10%. Data were taken in 2009 and 2010 at the test beam facility, M-Test, at Fermilab using protons, electrons and pions, in nitrogen, air, and in non fluorescing gases like argon, and helium. The instrument is operated in two main modes. In the first, fluorescence photons are observed, whereas in the second, both Cherenkov as well as fluorescence are observed. Comparisons of the ratio of these measurements, combined with the known Cherenkov spectrum allows for the absolute yield to be determined with reduced systematic uncertainties. In addition, the absolute yield is found by comparing the fluorescence yield to the observed photon yield of a NIST calibrated laser source directed into the apparatus. The consistency of these independent calibrations indicates that a systematic uncertainty of 5% or better is within reach.
Speaker: Dr Frederick Kuehn (Fermilab)
• 2:05 PM 3:30 PM
Emulsion chambers: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 2:05 PM
On capability of high coordinate-resolution techniques to study superhigh-energy hadron-nuclear interactions 35m
Capability of high coordinate-resolution techniques to study features of hadron-nuclear interactions at superhigh-energies are considered by the example of X-ray emulsion chamber (XREC) techniques. Main results accumulated by this way are discussed. Sensitivity of this approach to hadron-nuclear interaction features is analyzed. Predictions for future LHC experiments are formulated. Some proposals on future experiments are given.
Speaker: Prof. Rauf Mukhamedshin (Institute for Nuclear Research of Russian Academy of Science)
• 2:40 PM
Hadronic- and electromagnetic-cores of air-showers observed by hybrid experiments at high mountains 35m
The Chacaltaya hybrid experiment together with emulsion chamber and EAS-array can detect air-showers by the air-shower array, the accompanied atmospheric families (a bundle of high energy electrons and gamma-rays) by emulsion chambers and hadrons by burst detectors just under the emulsion chambers. We study overall characteristics of the experimental data, gamma-families and hadron burst accompanied by air-showers, by studying various correlations between the three observable data, i.e, between families and air-showers, between bursts and air-showers, and between families and bursts, comparing with those of CORSIKA simulations using interaction models of QGSJET, SIBYLL and EPOS. The analysis shows that changes of chemical composition alone can not describe the global characteristics of the Chacaltaya hybrid data. That is, distributions of family energies are favorable to heavy-dominant composition of primary cosmic-rays but lateral distributions of families are favorable to proton-dominant composition. The Chacaltaya hybrid data are also compared with those of Tien-Shan and Tibet hybrid experiments. There are some discrepancies among the three experimental data though the details of experimental procedure is different. Discussions are given on the possible reason of the disagreement by comparing these experimental data with simulations.
Speaker: Dr Masanobu Tamada (Kinki University)
• 3:15 PM
Analysis of one hadron rich event 15m
Analysis on a especial event with a main characteristics of Centauro type events, i.e. mean transverse momentum of hadrons in an order of 1 GeV/c will be presented. In spite of this event (Centauro V) doesn’t show the aspect of pioneer event (Centauro I), that is the upper part of the detector has more particles than the lower part, the event Centauro V shows other common characteristics of Centauro I. Both two events has same value for the ratio height/radius of the spread area of particles, besides similar slope of the fractionally energy distribution of hadrons. As the discrimination and identification of hadronic showers is crucial, the analysis evolved construction of some kind of score tables, obtained with the use of parametric and non parametric statistics analysis, observing the photosensitive material (X-ray Films and Nuclear Emulsion Plates) and the comparison with computer simulated events behaviour inside the detector. Authors: S.L.C.Barroso1, A.O.deCarvalho2, J.A.Chinellato2, A.Mariano2, E.J.T.Manganote2,3, E.C.F.P.Vicente2 and E.H.Shibuya2 1Departamento de Ciências Exatas/UESB, 45083-900 Vitória da Conquista, BA 2Instituto de Física Gleb Wataghin'/UNICAMP, 13083-859 Campinas, SP 3Faculdade de Campinas/FACAMP, 13083-970 Campinas, SP
• 3:30 PM 4:00 PM
coffee break 30m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:00 PM 4:30 PM
Poster Highlight Talks II One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:30 PM 5:30 PM
Poster Session II Atrium
### Atrium
#### Fermilab
PO Box 500 Batavia, IL 60510
The posters for this session are the same as for Poster Session I.
• 6:00 PM 9:00 PM
Symposium Dinner 3h Chez Leon - Users' Center
### Chez Leon - Users' Center
#### Fermilab
PO Box 500 Batavia, IL 60510
Thursday, July 1 – Chez Leon at the Users Center
Cocktails at 6 pm – cash bar
Dinner at 7 pm
The dinner is scheduled to for two hours and will end at 9 pm
Buses will leave at 9 pm
• Friday, 2 July
• 8:30 AM 9:05 AM
Emulsion chambers: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
Convener: Prof. Akinori Ohsawa (University of Tokyo)
• 8:30 AM
“Some consequences of the results of cosmic ray investigations above the knee for LHC experiments ” 20m
During last tens years many unusual results which are very difficult to explain in frames of existing theories and models were obtained in cosmic ray investigations. But it is possible to explain all these results if to suppose that some new state of matter with effective mass about TeV and with large orbital momentum appears. This new state of matter can be, for example, quark-gluon plasma, some specific resonance state, principally new short-lived particle and even Higgs boson with very large mass (about TeV). In this talk, explanations of various unusual cosmic ray events in frame of this hypothesis are given and consequences for accelerator physics experiments (first of all, at LHC) are considered.
Speaker: Prof. Anatoly Petrukhin (National Research Nuclear University MEPhI)
• 8:50 AM
Proton Fraction in the PCR Flux at the Energy Range E_0=1-100 PeV According to the Pamir Experiment Data 15m
A detailed study of X-Ray emulsion chamber response with ECSim 2.1 computer package adopted from GEANT 3.21 code and suited for imitation of measuring procedures, employed in the Pamir experiment makes it possible to determine more accurately the proton fraction in the primary cosmic ray (PCR) flux at energies around the “knee” E_0=1-100 PeV. In particular, it is shown that the proton fraction in the PCR slowly decreases from 20% at E_0 ~ 1 PeV to 15% at E_0 ~ 10 PeV.
Speaker: Dr Alexander Borisov (P.N.Lebedev Physical Institute, RAS)
• 9:05 AM 10:05 AM
Anisotropy: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 9:05 AM
Cosmic magnetic fields, and implications for HE particle anisotropies 45m
Speaker: Prof. Philipp Kronberg (LANL/University of Toronto)
• 9:50 AM
How dark matter cares about topological superstrings 15m
Non-trivial toplogical properties of string world sheets with three boundaries can give rise to superpotentials which preserve supersymmetry but violate R-symmetry by two units. This results in four point functions which permit s-wave annihilation of two neutralinos into gauge bosons. If the topological partition function is such as to allow saturation of the WMAP dark matter density for low string scales (M_s \sim 2 TeV), the annihilation into monochromatic gamma rays is predicted to lie about a factor of 2 below the current H.E.S.S. measurement of gamma ray flux from the galactic center. Thus, it may be detectable in the next round of gamma ray observations.
Speaker: Prof. Luis Anchordoqui (University of Wisconsin Milwaukee)
• 10:05 AM 10:30 AM
coffee break 25m Outside One West
### Outside One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 10:30 AM 11:30 AM
Anisotropy: 2 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 10:30 AM
Tev Cosmic Ray Anisotropy in Milagro 45m
Using the Milagro data from 2000 to 2007 containing more than 95 billion events (the largest such data set in existence), we performed a harmonic analysis of the large-scale cosmic-ray anisotropy. We observe an anisotropy with a magnitude around 0.1% for cosmic rays with a median energy of 6 TeV. The dominant feature is a deficit region of depth 0.25% in the direction of the Galactic North Pole centered at 189 degrees right ascension. In addition, we made an unexpected discovery of a localized cosmic-ray anisotropy, showing up as two high significance regions of excess cosmic rays. Recently, both Tibet AS Gamma and ARGO have confirmed similar excesses co-located with the Milagro regions. These features appear on an angular scale of ~10 degrees and have a harder than the background cosmic ray distribution, and the spectrum appears to cut off around 10 TeV. In this talk these results will be discussed as well as possible explanations for this surprising result.
Speaker: Dr Jordan Goodman (University of Maryland)
• 11:15 AM
Gamma ray signatures of ultrahigh energy cosmic ray sources in magnetized environments 15m
The quest for sources of ultrahigh energy cosmic rays has long been associated with the search of their secondary gamma ray signatures. While propagating, the former indeed produce very high energy photons through the interactions with particles of the intergalactic medium, or by synchrotron emission in the presence of substantial magnetic fields. We examine the prospects for the detectability of gamma ray counterparts of ultrahigh energy cosmic ray sources in a general case, exploring a wide range of astrophysical parameters. We demonstrate the fair robustness of the gamma ray flux according to these parameters and that its normalization ultimately depends on the energy injected in the primary cosmic rays. We show that only very powerful and rare sources could be detectable with the current and upcoming instruments. We further demonstrate that if the extended emission of this signature is resolved (which should be the case with Fermi and CTA), such a detection should provide a distinctive proof of the propagation of ultrahigh energy cosmic rays. Finally, we also briefly discuss the detection of nearby sources, considering the radiogalaxy Cen A as a prototypical example.
Speaker: Dr Kumiko Kotera (University of Chicago)
• 11:30 AM 12:30 PM
Muons: 1 One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 11:30 AM
Measurement of cosmic muons - L3+C results 15m
The L3+C is a unique tool in detecting cosmic muons and measuring their momenta in the range of 15-3000 GeV/c. About 1.2 x 1010 cosmic muon events have been collected during its running period in 1999-2000. With these high quality data many results on cosmic rays and gamma rays have been obtained, for example, the measurement of the atmospheric muon spectrum and the muon charge ratio, the search for TeV anti-protons by the moon shadowing, the coincidence of muons with the solar flares, the search for transient flaring point sources by detecting the muon burst, the analysis of muon bundles and comparison to simulations, and so on. In this talk, above results as well as a few of remarks on the future muon experiment will be summarized and presented.
Speaker: Prof. Yuqian Ma (IHEP)
• 11:45 AM
Measurement of the charge ratio of atmospheric muons with the CMS detector 15m
A measurement is presented of the ratio of positive to negative muon fluxes from cosmic-ray interactions in the atmosphere, using data collected by the CMS detector at ground level and in the underground experimental cavern. Muons were detected in the momentum range from 3 GeV/c to 1 TeV/c. For muon momenta below 100 GeV/c the flux ratio is measured to be a constant 1.2766 ± 0.0032 (stat) ± 0.0032 (syst), the most precise measurement to date. At higher momenta an increase in the charge asymmetry is observed, in agreement with models of muon production in cosmic-ray showers and compatible with previous measurements by deep-underground experiments.
Speaker: Dr Gavin Hesketh (CERN)
• 12:00 PM
MINOS Cosmic Muon Results 15m
When high energy cosmic rays interact in the stratosphere, mesons are produced in the primary hadronic interactions. The MINOS experiment detects cosmic ray produced muons using two magnetized detectors at underground depths of 220 and 2080 mwe. The muon charge ratio and the variation of muon intensity with atmospheric temperature are used to obtain information on meson production by the primary cosmic rays in the atmosphere. The ratios of positive to negative pions, positive to negative kaons, and charged kaons to pions are obtained.
Speaker: Prof. Philip Schreiner (Benedictine University)
• 12:15 PM
Physics of high energy atmospheric muons 15m
In the first part of the talk the interesting new results of L3, MINOS and CMS collaborations are briefly discussed from theoretical point of view: an observational evidence of the rise in the muon charge ratio (L3 and MINOS data) at muon energies around 1 TeV and detailed studies of electromagnetic interactions of high energy muons (in a momentum range up to 1 TeV/c) in the medium of CMS detector. In the second part of the talk the recent calculations of atmospheric prompt lepton spectra are reviewed. The modern theoretical approaches to the problem of heavy quark production in high energy nucleon-nucleus interactions are briefly considered (color dipole formalism, saturation models). The recent new theoretical developments in the ancient problem of intrinsic charm are also discussed. The predictions for atmospheric muon spectrum in the region around 1 PeV (where the prompt muon contribution becomes to be dominant) are given.
Speaker: Prof. Edgar Bugaev (Institute for Nuclear Research)
• 12:30 PM 1:30 PM
lunch 1h Cafeteria
### Cafeteria
#### Fermilab
PO Box 500 Batavia, IL 60510
• 1:30 PM 3:30 PM
Summary lectures One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 1:30 PM
Experimental summary 40m
Speaker: Prof. Paul Sommers
• 2:10 PM
Theory summary 40m
Speaker: Prof. Angela Olinto
• 2:50 PM
Outlook 40m
Speaker: Prof. Francis Halzen
• 3:30 PM 4:00 PM
Director's Wine & Cheese 30m 2nd Floor Gallery
### 2nd Floor Gallery
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:00 PM 5:00 PM
Joint Experimental-Theoretical Physics Seminar One West
### One West
#### Fermilab
PO Box 500 Batavia, IL 60510
• 4:00 PM
Xmax from Auger and its interpretation 1h
Xmax, the depth of maximum number of charged particles in the atmosphere during the longitudinal development of an air shower, is a valuable parameter to understand the nature of cosmic rays. The behaviour of Xmax is closely related to the composition of the primary particle. Hadronic interaction models, which are tuned with accelerator data, are required to understand the composition. Hence past, present, and future accelerator data are crucial in shaping our understanding of cosmic rays. The southern Pierre Auger Observatory has observed nearly 4000 high quality events above 1 EeV with the fluorescence detector and at least one surface detector in coincidence. We describe the data collection criteria and the Xmax mean and fluctuations, and outline how cosmic rays can aid understanding of hadronic interactions beyond collider energy.
Speakers: Dr Eun-Joo Ahn (Fermilab), Dr Ralph Engel (KIT, Karlsruhe) | 2020-09-26 10:03:09 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5422939658164978, "perplexity": 3822.8479050043056}, "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-2020-40/segments/1600400238038.76/warc/CC-MAIN-20200926071311-20200926101311-00533.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-1-equations-and-inequalities-summary-exercises-on-solving-equations-exercises-page-149/5 | ## Precalculus (6th Edition)
Square both sides to obtain: $(\sqrt{x+2}+5)^2=(\sqrt{x+15})^2 \\(\sqrt{x+2}^2+2(\sqrt{x+2})(5)+5^2=x+15 \\x+2+10\sqrt{x+2}+25=x+15 \\x+10\sqrt{x+2}+27=x+15$ Isolate the terms with $x$ on the left side to obtain: $x+10\sqrt{x+2}-x=15-27 \\10\sqrt{x+2}=-12$ Divide 10 to both sides: $\dfrac{10\sqrt{x+2}}{10}=\dfrac{-12}{10} \\\sqrt{x+2}=-\dfrac{6}{5}$ RECALL: The square root of any real number is greater than or equal to zero. Thus, the value of $\sqrt{x+2}$ is greater than or equal to zero. This means that the value of $\sqrt{x+2}$ cannot be negative. Therefore, the given equation has no solution. | 2018-07-22 03:25:31 | {"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": 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": 0.6074363589286804, "perplexity": 62.77162056533562}, "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-2018-30/segments/1531676593004.92/warc/CC-MAIN-20180722022235-20180722042235-00248.warc.gz"} |
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Manager
Joined: 16 Jul 2011
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Does anyone know when the business week mba rankings for 2012 will be released? They release rankings every 2 years and the last one was in 2010 so it's time for the updated info....
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Intern
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It's usually released in late November.
Senior Manager
Status: schools I listed were for the evening programs, not FT
Joined: 16 Aug 2011
Posts: 389
Location: United States (VA)
GMAT 1: 640 Q47 V32
GMAT 2: 640 Q43 V34
GMAT 3: 660 Q43 V38
GPA: 3.1
WE: Research (Other)
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Kudos [?]: 46 [0], given: 50
The admissions stats are usually released every year though.
Current Student
Joined: 16 Sep 2010
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Concentration: Finance, Real Estate
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Businessweek came out with their new 2012 MBA rankings today. Here is the list:
30. Georgetown (McDonough)
29. Southern Methodist (Cox)
28. USC (Marshall)
27. Ohio State (Fisher)
26. Texas A&M (Mays)
25. Vanderbilt (Owen)
24. Maryland (Smith)
23. Georgia Tech (Scheller)
22. Emory (Gouizueta)
21. Yale
20. Notre Dame (Mendoza)
19. Texas-Austin (McCombs)
18. UCLA (Anderson)
17. North Carolina (Kenan-Flagler)
16. NYU (Stern)
15. Indiana (Kelley)
14.Columbia
13. UC Berkeley (Haas)
12. Dartmouth (Tuck)
11. Carnegie Mellon (Tepper)
10.Virginia (Darden)
9. MIT (Sloan)
8. Michigan (Ross)
7. Cornell (Johnson)
6. Duke (Fuqua)
5. Northwestern (Kellogg)
4.Stanford
3. Pennsylvania (Wharton)
2. Harvard
1. Chicago Booth
Here is the transcript discussing the rankings: http://staging.mzinga.com/n/pfx/forum.a ... ats&ptpw=y
They should post it to their website sometime in the not too distant future.
Current Student
Joined: 18 Nov 2011
Posts: 154
Schools: Johnson '15 (M)
GMAT 1: 730 Q47 V44
Followers: 4
Kudos [?]: 52 [0], given: 21
Looks like huge jumps for Carnegie Mellon (CMU) and Cornell! Congrats to those schools.
I see Yale is still sitting outside of the top 20.. hmm.
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OjilEye wrote:
I see Yale is still sitting outside of the top 20.. hmm.
BW has a ..lets call it unique way of looking at rankings. A vast majority of the overall score is for "student satisfaction" and "employer satisfaction", with a minor part of the score being made up by "intellectual capital".
While these are some of the factors necessary for ranking schools, in my opinion, its not all inclusive. I find their rankings interesting, but I offer them little weight in my own rankings.
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Expert's post
As an applicant, I didn’t take much stock in the BW ranking. A lot of its placements didn’t make intuitive sense as opposed to, say, US News.
After the b-school, having interacted and talked with many more students and graduates from other schools, I believe more and more in the BW ranking. BW ranking’s purpose is to gauge the school based on two important aspects of b-school: 1) your experience there, and 2) the perception of you by your future employers. At the end of the day, those are the two things that matters. Scores, acceptance rate, reputation, etc. may matter to an applicant. To a certain extent they may influence for recruiting and class discussion. I guarantee that as a student or recruiter those are not high on your priority list. Success in life isn’t about being the one with the highest score; it’s about being the one who can get things done. Being smart is only one of the factors.
I think an even better indicator of school’s quality is the recruiter rankings. Those ratings are harder to juice. If you look at the top schools by recruiter ranking, it start to make a lot of sense, especially at the top, where data points are plenty.
Where the rankings break down is the specifics. I bet if you look at the raw data, the difference between a #5 school and #9 is so minimal, that it may not be statistically significant. It’s like trying to place a bunch of fashion models in line according to their “beauty”. The ranking can only give you a general feel for the schools, it can never tell if it one school is absolutely better than another. There are too many variables at play here.
Intern
Joined: 13 Oct 2012
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GMAT 1: 740 Q48 V44
GPA: 3.91
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Kudos [?]: 8 [0], given: 10
Wow. CBS dropped big.
BW must of heard that I got in.
Current Student
Joined: 19 Apr 2012
Posts: 20
Concentration: Technology, Social Entrepreneurship
GMAT 1: 760 Q49 V45
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One of the interesting things, I think, is that the employer survey seems to lend weight to larger programs. It makes sense, when you think about it, that larger programs are sending more MBAs to more companies, and in larger numbers. If you break the schools into their typical perceived groupings, you notice that the smaller programs are almost always at the bottom of a given group (Stanford below H/W, for example).
As with any ranking, it means exactly what you allow it to mean. Either a program is right for you and your goals, or it's not. As someone interested in smaller programs, however, it was startling to see the trend.
One question I do have, though, is how the letter grades are assigned? It's strange to see A+ across the board for schools that are ranked well below others with a B or two.
Director
Affiliations: Columbia, Wharton, LBS
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From an admissions perspective, we never took much stock in these surveys. One thing that we noticed was that those schools with the most aggressive public relations initiatives always did the best. Also, we always gave our consumer - the business school applicants - credit in figuring out which schools were best for them -- as just because a school is ranked high in this survey does not mean that it is the right program for you. Candidates should choose the program that has the resources, focus, philosophy and energy that they are looking for; not choose a program based on some model that is not so transparent.
Kimberly Plaga
Manhattan Review
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Expert's post
DHokie2 wrote:
One of the interesting things, I think, is that the employer survey seems to lend weight to larger programs. It makes sense, when you think about it, that larger programs are sending more MBAs to more companies, and in larger numbers. If you break the schools into their typical perceived groupings, you notice that the smaller programs are almost always at the bottom of a given group (Stanford below H/W, for example).
You are definitely right; a smaller program does affect employer survey. Yield is important for employers, and there are only so many students available at the smaller programs. That aside, an important factor influencing the survey is the diversity of employers recruiting at the school. If a school is strongly aligned in a certain industry, and they may not get as many response from recruiters in other industries. This may be the case for NYU.
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GMAT 2: 680 Q45 V38
GMAT 3: 710 Q45 V42
GPA: 3.32
WE: Information Technology (Retail)
Followers: 21
Kudos [?]: 343 [0], given: 109
Has anyone spent time on the individual school pages, I love this new layout. Great free resource for information on schools IMO
Current Student
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Schools: MIT Sloan - Class of 2015
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highwyre237 wrote:
Has anyone spent time on the individual school pages, I love this new layout. Great free resource for information on schools IMO
Thanks for the heads up! I hadn't realized they had revamped their school profiles. I LOVE all the information BW provides in its school profiles.
Current Student
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Posts: 432
Concentration: Finance
GMAT 1: 700 Q44 V41
GMAT 2: 750 Q48 V46
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WE: Project Management (Military & Defense)
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I already posted my thoughts in the CBS thread but I'll reiterate them here. Rankings don't take location into account so I'm going to throw that out as well because individuals preference to location really doesn't say anything about the quality of the school. That being said, I don't believe for a minute that given the choice between Columbia and Ross, Tepper, Haas, Johnson or Fuqua anybody would turn down admission to Columbia. None of these are bad programs but nobody realistically believes Michigan is a better MBA than Columbia.
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mappleby wrote:
I already posted my thoughts in the CBS thread but I'll reiterate them here. Rankings don't take location into account so I'm going to throw that out as well because individuals preference to location really doesn't say anything about the quality of the school. That being said, I don't believe for a minute that given the choice between Columbia and Ross, Tepper, Haas, Johnson or Fuqua anybody would turn down admission to Columbia. None of these are bad programs but nobody realistically believes Michigan is a better MBA than Columbia.
Okay, this makes no sense.
1) US News, which many will consider the most legitimate, ranks Haas above Columbia.
2) I would argue there are relatively few cases where you would choose Columbia over the other schools on your list. If you look at a specialty ranking that isn't for finance, Columbia will rank lower than most of these schools for most specialties.
3) There are plenty of people that could get into Columbia that don't even apply there.
This is the problem. People have bias based on the schools that they are applying to. This bias gets even worse after you've attended a school.
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1
KUDOS
Columbia over Johnson or Tepper, probably so. Columbia over Haas, absolutely not.
Current Student
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Schools: MIT Sloan - Class of 2015
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1
KUDOS
Agreed. I could see someone choosing Fuqua/Ross/Haas over CBS. I would probably say the same about Tepper/Johnson, too, had I looked into them at all (they weren't on my radar). Everybody has their own motivations for applying to b school, after all.
Manager
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Schools: Stanford R1, Wharton R1 w/int, Chicago R1, HBS R2
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Yeah, I think Haas sits above Columbia for a lot of folks in their internal ranking. I know I never even considered applying to Columbia, but was very close to applying to Haas and probably would have had I not gotten into Booth first. The reason I like the BW rankings is because they do the best job of capturing how much attending a given program IMPROVES your position. It's true that people coming out of HBS and Stanford land some really fantastic gigs, but they also went in with some pretty amazing resumes. Therefore, it seems these top two do very little to actually improve one's opportunity set. On the flip side, lots of my friends at Chicago (including myself) came from sort of lesser known, or somewhat less prestigious firms pre-MBA, and then they were able to land at their absolute top choices for career and firm post business school. That to me signals that the school actually added a lot of value for us.
If your use of rankings is to see which school will produce alums that will earn the most \$ over the course of their career, yeah, USN is probably going to be more accurate. But I think for applicants who are a little more self aware of where they actually stand among the pool of applicants, BW is going to be more relevant.
Current Student
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Maybe it's just me but it seems like Columbia has been slipping. Big drops in applications, bad experiences with current students, etc. That was one school I never planned on applying to.
Posted from my mobile device
Intern
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Here's a very good article that I found on Poets and Quants: http://poetsandquants.com/2010/06/28/tu ... -rankings/
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Display posts from previous: Sort by | 2015-07-04 09:30:22 | {"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": 0.20127686858177185, "perplexity": 7524.627701680169}, "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-2015-27/segments/1435375096579.89/warc/CC-MAIN-20150627031816-00145-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://egeoscien.neigae.ac.cn/CN/Y2004/V14/I2/129 | • 论文 •
### RELATION BETWEEN PRECIPITATION AND SEDIMENT TRANSPORT IN THE DASHA RIVER WATERSHED
ZHANG Jian-chun1, ZHANG Wei1, LI Ji-hong1, SHI Zhi-gang2, PU Shen-yuan2
1. 1. College of Territorial Resources and Tourism, Anhui Normal University, Wuhu 241000, P. R. China;
2. Water Conservancy Department of Anhui Province, Hefei 230022, P. R. China
• 收稿日期:2003-12-15 出版日期:2004-06-20 发布日期:2011-12-15
• 作者简介:ZHANG Jian-chun(1964- ), male,a native of Nanjing, Jiangsu Province, professor, specialized in physical geography. E-mail: zhangjianchun@263.net
• 基金资助:
Under the auspices of the Natural Science Foundation of Anhui Education Office (No.2003KJ102) and Special Fund Project of Anhui Provincial Irrigation Office (No.2001-11)
### RELATION BETWEEN PRECIPITATION AND SEDIMENT TRANSPORT IN THE DASHA RIVER WATERSHED
ZHANG Jian-chun1, ZHANG Wei1, LI Ji-hong1, SHI Zhi-gang2, PU Shen-yuan2
1. 1. College of Territorial Resources and Tourism, Anhui Normal University, Wuhu 241000, P. R. China;
2. Water Conservancy Department of Anhui Province, Hefei 230022, P. R. China
• Received:2003-12-15 Online:2004-06-20 Published:2011-12-15
The study on sediment production and its relationship with climatic and hydrological factors in watershed is a major environment issue of concern in the international community. Based on the observational records covering the period from 1954 to 1999, the characteristics of precipitation changing over the Dasha River Watershed in Anhui Province and its relation to sediment yield were studied using tendency analysis and correlation analysis. Results showed that the precipitation of the Dasha River Watershed has high variability. In those 46 years, 34% of spring rainfall, 58% of summer rainfall and 30% of annual rainfall will be considered anomaly. The gray correlation analysis shows that sediment discharge correlates most closely with the frequency of the rainstorm with a daily precipitation above 100mm, secondly with the frequency of the rainstorm with a daily precipitation of 50-100mm, and thirdly with the number of rainy days. Their correlation coefficients are 0.98,0.90 and 0.85 respectively. In addition, the paper suggests the major countermeasures and methods for controlling of soil and water losses in this area.
Abstract:
The study on sediment production and its relationship with climatic and hydrological factors in watershed is a major environment issue of concern in the international community. Based on the observational records covering the period from 1954 to 1999, the characteristics of precipitation changing over the Dasha River Watershed in Anhui Province and its relation to sediment yield were studied using tendency analysis and correlation analysis. Results showed that the precipitation of the Dasha River Watershed has high variability. In those 46 years, 34% of spring rainfall, 58% of summer rainfall and 30% of annual rainfall will be considered anomaly. The gray correlation analysis shows that sediment discharge correlates most closely with the frequency of the rainstorm with a daily precipitation above 100mm, secondly with the frequency of the rainstorm with a daily precipitation of 50-100mm, and thirdly with the number of rainy days. Their correlation coefficients are 0.98,0.90 and 0.85 respectively. In addition, the paper suggests the major countermeasures and methods for controlling of soil and water losses in this area. | 2021-10-28 17:35:13 | {"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": 0.17976152896881104, "perplexity": 6541.88371125449}, "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/1634323588398.42/warc/CC-MAIN-20211028162638-20211028192638-00028.warc.gz"} |
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# Answer to Question #58860 in Macroeconomics for Frances Fitzgerald
Question #58860
In the international market for strawberries, Ireland is small and can be assumed to be unable to affect world prices. It imports strawberries at the price of 15 euros per box. The domestic supply and domestic demand curves for boxes of strawberries are given by QS = 60 + 20P and QD = 1225 – 15P respectively.
i. Assume Ireland is completely open to trade. What is the equilibrium price and quantity consumed? How much is produced domestically and how much is imported? Illustrate you answer on a diagram.
ii. Now consider the effect of an import quota of 400 boxes. What happens to the price of strawberries and the quantity consumed? How much is produced domestically and how much is imported? Illustrate you answer on a diagram.
iii. Who wins and who loses from the imposition of the quota? Discuss the effects on consumers, domestic producers and importers in terms of welfare changes. Illustrate you answer on a diagram.
Ireland imports strawberries at the price of 15 euros per box. QS = 60 + 20P, QD = 1225 – 15P.
i. If Ireland is completely open to trade, then the equilibrium price will be $15 and quantity consumed will be Qd = 1225 - 15*15 = 1,000 units. The amount of strawberries produced domestically is Qs = 60 + 20*15 = 360 and the amount imported is Qi = 1,000 - 360 = 640. ii. If the import quota of 400 boxes is imposed, then 400 boxes will be imported and produced domestically will be Qs = Qd - 400, 60 + 20P = 1225 - 15P - 400, 35P = 765 P =$21.86 is the price of strawberries.
The amount produced will be: Qs = 60 + 20*21.86 = 497 boxes.
iii. Producers will win and consumers will lose from the imposition of the quota.
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for any assignment or question with DETAILED EXPLANATIONS! | 2018-09-22 23:40:04 | {"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": 0.2803734540939331, "perplexity": 3351.1762348835937}, "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-2018-39/segments/1537267158766.53/warc/CC-MAIN-20180922221246-20180923001646-00011.warc.gz"} |
https://www.allanswered.com/post/awmjk/convolution-including-bose-function-yields-an-integral-over-a-bose-function-and-a-fermi-function-is-it-correct/ | ### Convolution including Bose function yields an integral over a Bose function and a Fermi function - is it correct?
51
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1
27 days ago by
Micha had a nice problem concerning self energies in a slave-boson theory. There he had some nasty convolutions of Bose functions with spectral functions. When he asked me how to approach this problem e.g. numerically (problematic due to the divergence of the Bose function) we came up with the Idea of just a coordinate transformation. Although we had not much hope it would work, we pulled it through out of curiousity. I appended the results to this calculation and I now want to know: Is this a known phenomenon? Can we learn something from this transformation? Is it even allowed? Because I see some possible problems with diverging scales. Please note that the final integrals appear to be no convolution integrals anymore.
EDIT: there were sign mistakes in eq. 8, 10 and 11 - the final result is still correct, but now the intermediate solutions should be as well ;)
File attached: Integrals_over_Bose_function_convolutions_v1.1.pdf (46.85 KB)
2
26 days ago by
Upon a first reading and little bit of thought, I am not sure if we gain anything from this transformation, in theory or practice (numerics).
The major problem is that the transformation $\epsilon \mapsto \epsilon'$ with $\epsilon' = \frac{1}{\beta} \ln( 1 - e^{- \beta \epsilon})$, has a logarithmic singularity at $\epsilon = 0$. In particular, it cannot serve as a regularization scheme for integrations on the Bose-Einstein distribution. This singularity in the Bose-Einstein is really deep — perhaps, divine ;) I think one cannot remove it, because essentially, the very singularity leads to the Bose-Einstein condensation.
A secondary problem related to the final conclusions, Eqs. 14—15ff, is that due to the intricate nature of the transformation, it is hard to come up with a physical picture based on it: e.g., “that a convolution of a Bose function with some other function can be split up into an integral of the Bose function with another function over positive energies and a Fermi function with some other function over the whole energy range.” Is $\epsilon'$ really an “energy”? Hard to say.
Note that there is always very straight-forward relations between the Bose and Fermi distributions; e.g.,
$n_B(\varepsilon) = \frac{n_F(\varepsilon)}{2 n_F(\varepsilon) - 1} ~.$
So they can be always written in terms of each other. Does it resolve a problem? It depends.
1st
I agree, that the transformation has a logarithmic singularity and does not regularize the Bose-Einstein distribution function. This can very easily be seen in the final equations, where we have one integral yielding basically the same problem that we tried to solve in the first place. I find it however interesting, that the other integral (containing the Fermi-Dirac distribution function) does not show this particular problem of integrating over a divergence.
2nd
I also agree that the new coordinates are possibly not "physical" energies, although they are energies by dimension analysis. The idea is, that the original divergence (like $\frac{1}{\epsilon}$ close to zero, so logarithmic in the integral) should have been converted to a logarithmic divergence in the scale.
written 26 days ago by marv_the_great
1
Regarding the first part of your comment, “the other integral (containing the Fermi-Dirac distribution function) [is not] integrating over a divergence”, I'd say, it is quite common that one disentangles an originally divergent integral to a non-divergent and a divergent part. Yet, how one achieves this is extremely important and determines the applicability of the method.
written 26 days ago by AlQuemist
See the section on Sommerfeld expansion of Bose-Einstein distribution in the manuscript here.
written 26 days ago by AlQuemist
0
25 days ago by
A closely related topic is the Cauchy principal value of a Hilbert transform. Hilbert transform is one of the most divine transformation we have (see this), and we do it everyday in many-body calculations without naming it.
One of the stable methods to perform a finite Hilbert transform is the using the Cauchy principal value as described here: hilbert_cauchy.pdf (70.58 KB).
I believe one can find a similar method for the Bose-Einstein kernels; yet I have not tried it myself. | 2018-08-19 20:51:12 | {"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": 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": 0.8715429902076721, "perplexity": 518.8036476676086}, "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/1534221215393.63/warc/CC-MAIN-20180819204348-20180819224348-00384.warc.gz"} |
http://bkms.kms.or.kr/journal/list.html?pn=ahead | # Bulletin of theKorean Mathematical SocietyBKMS
ISSN(Print) 1015-8634 ISSN(Online) 2234-3016
HOME VIEW ARTICLES Ahead of Print Articles
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Final published versions of all papers are available under “All Issues” and also under “Current Issue” for papers included in the current issue.
* Paper information have been automatically generated from the information submitted by authors in the online submission system.
• ### Published online May 9, 2022
#### Rings with a right duo factor ring by an ideal contained in the center
Jeoung Soo Cheon, Tai Keun Kwak, Yang Lee, Zhelin Piao, and Sang Jo Yun
Abstract : This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We fi rst see that this property is seated between right duo and right pi-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W(R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W(R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W(R) is strongly regular, where W(R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D3(R) is right CIFD, where D3(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.
• ### Published online May 11, 2022
#### The characterisation of BMO via commutators in variable Lebesgue spaces on stratified groups
Dongli Liu, Jian Tan, and Jiman Zhao
Abstract : Let T be a bilinear Calder\'{o}n-Zygmund operator, b\in \cup_{q>1}L_{loc}^{q}(G). We firstly obtain a constructive proof of the weak factorisation of Hardy spaces, then we establish the characterization of BMO spaces by the boundedness of the commutator [b, T]_{j} in variable Lebesgue spaces.
• ### Published online May 11, 2022
#### Strong Classification of Extensions of Classifiable $C^*$-algebras
Søren Eilers, Gunnar Restorff, and Efren Ruiz
Abstract : We show that certain extensions of classifiable $C^*$-algebras are strongly classified by the associated six-term exact sequence in $K$-theory together with the positive cone of $K_0$-groups of the ideal and quotient. We use our results to completely classify all unital graph $C^*$-algebras with exactly one non-trivial ideal.
• ### Published online May 12, 2022
#### On the sizes of dual groups
Joungmin Song
Abstract : We give a formula for the sizes of the dual groups. It is obtained by generalizing a size estimation of certain algebraic structure that lies in the heart of the proof of the celebrated primality test by Agrawal, Kayal and Saxena. In turn, by using our formula, we are able to give a streamlined survey of the AKS test.
• ### Published online May 16, 2022
#### Constructions of regular sparse anti-magic squares
Guangzhou Chen, Wen Li, Bangying Xin, and Ming Zhong
Abstract : For positive integers $n$ and $d$ with $d Show More • ### Published online May 11, 2022 #### Characterizing S-flat modules and S-von Neumann regular rings by uniformity Xiaolei Zhang Abstract : Let$R$be a ring and$S$a multiplicative subset of$R$. An$R$-module$T$is called uniformly$S$-torsion provided that$sT=0$for some$s\in S$. The notion of$S$-exact sequences is also introduced from the viewpoint of uniformity. An$R$-module$F$is called$S$-flat provided that the induced sequence$0\rightarrow A\otimes_RF\rightarrow B\otimes_RF\rightarrow C\otimes_RF\rightarrow 0$is$S$-exact for any$S$-exact sequence$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. A ring$R$is called$S$-von Neumann regular provided there exists an element$s\in S$satisfies that for any$a\in R$there exists$r\in R$such that$sa=ra^2$. We obtain that a ring$R$an$S$-von Neumann regular ring if and only if any$R$-module is$S$-flat. Several properties of$S$-flat modules and$S$-von Neumann regular rings are obtained. Show More • ### Published online March 8, 2022 #### Free products of operator systems Florin Pop Abstract : In this paper we introduce the notion of universal free product for operator systems and operator spaces, and prove extension results for the Operator System Lifting Property (OSLP) and Operator System Local Lifting Property (OSLLP) to the universal free product. • ### Published online March 8, 2022 #### On toric Hamiltonian T-spaces with anti-symplectic involutions Jinhong Kim Abstract : The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let$(X, \omega, \mu)$be a toric Hamiltonian$T$-space, and let$\Delta=\mu(X)$denote the moment polytope. Let$\tau$be an anti-symplectic involution of$X$of$X$such that$\tau$maps the fibers of$\mu$to (possibly different) fibers of$\mu$, and let$p_0$be a point in the interior of$\Delta$. If the toric fiber$\mu^{-1}(p_0)$is real Lagrangian with respect to$\tau$, then we show that$p_0$should be the origin and, furthermore,$\Delta$should be centrally symmetric. In this paper, we also provide a simple example asserting that the condition of$\tau$preserving the fibration structure of$\mu$plays a crucial role in the proof of our main result, which thus disproves a general question stated without any restriction about the fibration structure of$\mu$. Show More • ### Published online May 11, 2022 #### On Eigensharpness and almost Eigensharpness of Lexicographic Products of Some Graphs Ahmad Abbasi and Mona Gholamnia Taleshani Abstract : The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) > max {p(G), q(G)}, where p(G) and q(G) are the numbers of positive and negative eigenvalues of the adjacency matrix of G, respectively. When equality is attained, G is said to be eigensharp and when b(G) = max {p(G), q(G)} + 1, G is called an almost eigensharp graph. In this paper, we investigate the eigensharpness and almost eigensharpness of lexicographic product of some graphs. • ### Published online May 12, 2022 #### Existence of the continued fractions of$\sqrt{d}$and its applications Jun Ho Lee Abstract : It is well known that the continued fraction expansion of$\sqrt{d}$has the form$[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]$and$a_1, \ldots, a_{l-1}$is a palindromic sequence of positive integers. For a given positive integer$l$and a palindromic sequence of positive integers$a_1, \ldots, a_{l-1}$, we define the set$S(l;a_1, \ldots, a_{l-1}) :=\{d\in \mathbb{Z} \,| \, d>0, \sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]\}$. In this paper, we completely determine when$S(l;a_1, \ldots, a_{l-1})$is not empty in the case that$l$is$4$,$5$,$6$, or$7$. We also give similar results for$(1+\sqrt{d})/2$. For the case that$l$is$4$,$5$, or$6$, we explicitly describe the fundamental units of the real quadratic field$\mathbb{Q}(\sqrt{d})$. Finally, we apply our results to the Mordell conjecture for the fundamental units of$\mathbb{Q}(\sqrt{d})$. Show More • ### Published online March 15, 2022 #### Finiteness and vanishing results on hypersurfaces with finite index in$\mathbb{R}^{n+1}$: a revision Nguyen Van Duc Abstract : In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in$\mathbb{R}^{n+1}$. When the hypersurface is stable minimal, we show that there is no nontrivial$L^{2p}$harmonic$1$-form for some$p$. The our range of$p$is better than those in \cite{DS}. With the same range of$p$, we also give finiteness results on minimal hypersurfaces with finite index. • ### Published online March 10, 2022 #### A generalization of w-linked extensions Xiaoying Wu Abstract : In this paper, the concepts of w-linked homomorphisms, the wφ-operation, and DWφrings are introduced. Also the relationships between wφ-ideals and w-ideals over a w-linked homomorphism φ : R → T are discussed.More precisely, it is shown that every wφ-ideal of T is a w-ideal of T. Besides,it is shown that if T is not a DWφring, then T must have an infinite number of maximal wφ-ideals. Finally we give an application of Cohen’s Theorem over w-factor rings, namely it is shown that an integral domain R is an SM-domain with w-dim(R) ≤ 1, if and only if for any nonzero w-ideal I of R, (R/I)w is an Artinian ring, if and only if for any nonzero element a ∈ R, (R/(a))w is an Artinian ring, if and only if for any nonzero element a ∈ R, R satisfies the descending chain condition on w-ideals of R containing a. Show More • ### Published online May 12, 2022 #### Application of Rothe's method to a nonlinear wave equation on graphs Yong Lin and Yuanyuan Xie Abstract : We study a nonlinear wave equation on finite connected weighted graphs. Using Rothe's and energy methods, we prove the existence and uniqueness of solution under certain assumption. For linear wave equation on graphs, Lin and Xie \cite{Lin-Xie} obtained the existence and uniqueness of solution. The main novelty of this paper is that the wave equation we considered has the nonlinear damping term$|u_t|^{p-1}\cdot u_t$($p>1$). • ### Published online March 10, 2022 #### Weighted integral inequalities for modified integral Hardy operators Duranta Chutia and Rajib Haloi Abstract : In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights$\omega, \rho, \phi$and$\psito hold the following weak type modular inequalities. \begin{align*} \mathcal{U}^{-1} \bigg ( \int \limits_{ \{ | \mathcal{I}f | > \gamma\}} \mathcal{U} \Big(\gamma \omega \Big ) \rho \bigg ) & \leq \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \Big ( C |f| \phi\Big) \psi \bigg ), \end{align*} where\mathcal{I}is the modified integral Hardy operators . We also obtain a necesary and sufficient condition for the following extra-weak type integral inequalities. \begin{align*} \omega \bigg ( \Big\{ |\mathcal{I}f| > \gamma \Big \} \bigg) &\leq \mathcal{U}\circ \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \bigg ( \dfrac{C |f| \phi}{\gamma} \bigg) \psi \bigg ). \end{align*} Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operators and its integral version. Show More • ### Published online March 10, 2022 #### Computation of Wedderburn decomposition of groups algebras from their subalgebra Gaurav Mittal and Rajendra Sharma Abstract : In this paper, we show that under certain conditions the Wedderburn decomposition of a finite semisimple group algebra\mathbb{F}_qG$can be deduced from a subalgebra$\mathbb{F}_q(G/H)$of factor group$G/H$of$G$, where$H$is a normal subgroup of$G$of prime order$P$. Here, we assume that$q=p^r$for some prime$p$and the center of each Wedderburn component of$\mathbb{F}_qG$is the coefficient field$\mathbb{F}_q$. • ### Published online May 12, 2022 #### A sharp integral inequality for compact linear Weingarten hypersurfaces Henrique de Lima, Fábio dos Santos, and Lucas Rocha Abstract : We establish a sharp integral inequality related to compact (without boundary) linear Weingarten hypersurfaces immersed in a locally symmetric Einstein manifold and we apply it to characterize totally umbilical hypersurfaces and isoparametric hypersurfaces with two distinct principal curvature, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Alías and Meléndez in reference [3] for the case of hypersurfaces with constant scalar curvature in the Euclidean round sphere. • ### Published online March 10, 2022 #### Uniqueness of meromorphic solutions of a certain type of difference equations Jun-Fan Chen and Shu-Qing Lin Abstract : In this paper, we study the uniqueness of two finite order transcendental meromorphic solutions$f(z)$and$g(z)$of the following complex difference equation $$A_{1}(z)f(z+1)+A_{0}(z)f(z)=F(z)e^{\alpha(z)}$$ when they share 0,$\infty$CM, where$A_{1}(z),A_{0}(z),F(z)$are non-zero polynomials,$\alpha(z)$is a polynomial. Our result generalizes and complements some known results given recently by Cui and Chen, Li and Chen. Examples for the precision of our result are also supplied. • ### Published online March 7, 2022 #### BEZOUT RINGS AND WEAKLY BEZOUT RINGS Haitham EL ALAOUI Abstract : In this paper, we study some properties of Bézout and weakly Bézout rings. Then, we investigate the transfer of these notions to trivial ring extensions and amalgamated algebras along an ideal. Also, in the context of domains we show that the amalgamated is an Bézout ring if and only if its a weakly Bézout ring. All along the paper, we put the new results to enrich the current literature with new families of examples of non-Bézout weakly Bézout rings. • ### Published online May 12, 2022 #### On right regularity of commutators Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung Ju Ryu, and Hyo Jin Sung Abstract : We study the structure of right regular commutators, and call a ring$R${\it strongly C-regular} if$ab-ba\in (ab-ba)^2R$for any$a, b\in R$. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring$R$, (i) if$R/W(R)$is commutative then$R$is commutative; and (ii) every prime factor ring of$R$is either a commutative domain or a noncommutative division ring, where$W(R)$is the Wedderburn radical of$R$. Show More • ### Published online March 16, 2022 #### Knots in homology lens spaces determined by their complements Kazuhiro Ichihara and Toshio Saito Abstract : In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let$M$be a homology lens space with$H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$and$K$a not null-homologous knot in$M$. We show that,$K$is determined by its complement if$M$is non-hyperbolic,$K$is hyperbolic, and$p$is a prime more than 7, or, if$M$is actually a lens space$L(p,q)$and$K$represents a generator of$H_1(L(p,q))$. • ### Published online March 16, 2022 #### ON COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF COORDINATEWISE NEGATIVELY ASSOCIATED RANDOM VECTORS IN HILBERT SPACES Vu Thi Ngoc Anh and Nguyen Thi Thanh Hien Abstract : This paper establishes the {Baum--Katz} type theorem and the {Marcinkiewicz--Zymund} type strong law of large numbers for sequences of coordinatewise negatively associated and identically distributed random vectors$\{X,X_n,n\ge1\}$taking values in a Hilbert space$H$with general normalizing constants$b_n=n^{\alpha}\widetilde L(n^{\alpha})$, where$\widetilde L(\cdot)$is the de Bruijn conjugate of a slowly varying function$L(\cdot).$The main result extends and unifies many results in the literature. The sharpness of the result is illustrated by two examples. • ### Published online March 16, 2022 #### The Kähler Different of a Set of Points in$\mathbb{P}^m\times\mathbb{P}^n$Nguyen T. Hoa, Tran N.K. Linh, Le N. Long, Phan T.T. Nhan, and Nguyen T.P. Nhi Abstract : Given an ACM set$\mathbb{X}$of points in a multiprojective space$\mathbb{P}^m\times\mathbb{P}^n$over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for$\mathbb{X}$. In$\mathbb{P}^1\times \mathbb{P}^1$, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kaehler different. However, this result fails to hold in$\mathbb{P}^m\times\mathbb{P}^n$for$n>1$or$m>1$. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that$\mathbb{X}$is a complete intersection of type$(d_1,...,d_m,d'_1,...,d'_n)$if and only if it has the Cayley-Bacharach property and the Kähler different is non-zero at a certain degree. When$\mathbb{X}$has the$(\star)$-property, we characterize the Cayley-Bacharach property of$\mathbb{X}$in terms of its components under the canonical projections. Show More • ### Published online May 4, 2022 #### Complete characterization of odd factors via the size, spectral radius or distance spectral radius of graphs Shuchao Li and Shujing Miao Abstract : Given a graph$G,$a$\{1,3,\ldots,2n-1\}$-factor of$G$is a spanning subgraph of$G$, in which each degree of vertices is one of$\{1,3,\ldots,2n-1\}$, where$n$is a positive integer. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of$G$to guarantee that$G$contains a$\{1,3,\ldots,2n-1\}$-factor. Then we determine an upper bound on the distance spectral radius of$G$to ensure that$G$has a$\{1,3,\ldots,2n-1\}$-factor. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible. • ### Published online May 4, 2022 #### Basic formulas for the double integral transform of functionals on abstract Wiener space Hyun Soo Chung Abstract : In this paper, we establish several basic formulas among the double-integral transforms, the double-convolution products, and the inverse double-integral transforms of cylinder functionals on abstract Wiener space. We then discuss possible relationships involving the double-integral transform. ## Current Issue ### March, 2022 Vol.59 No.2 ## Most Read • ### On the existence of Graham partitions with congruence conditions Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park Bull. Korean Math. Soc. 2022; 59(1): 15-25 https://doi.org/10.4134/BKMS.b200730 • ### On weighted Browder spectrum Preeti Dharmarha, Sarita Kumari Bull. Korean Math. Soc. 2022; 59(1): 1-13 https://doi.org/10.4134/BKMS.b200328 • ### Admissible balanced pairs over formal triangular matrix rings Lixin Mao Bull. Korean Math. Soc. 2021; 58(6): 1387-1400 https://doi.org/10.4134/BKMS.b200924 • ### One-sided fattening of the graph in the real projective plane Jaeyoo Choy, Hahng-Yun Chu Bull. Korean Math. Soc. 2022; 59(1): 27-43 https://doi.org/10.4134/BKMS.b201080 ## Most Downloaded • ### On weighted Browder spectrum Preeti Dharmarha, Sarita Kumari Bull. Korean Math. Soc. 2022; 59(1): 1-13 https://doi.org/10.4134/BKMS.b200328 • ### On the existence of Graham partitions with congruence conditions Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park Bull. Korean Math. Soc. 2022; 59(1): 15-25 https://doi.org/10.4134/BKMS.b200730 • ### Entire solutions of differential-difference equations of Fermat type Peichu Hu, Wenbo Wang, Linlin Wu Bull. Korean Math. Soc. 2022; 59(1): 83-99 https://doi.org/10.4134/BKMS.b210099 • ### Sasakian 3-Metric as a$\ast\$-Conformal Ricci Soliton Represents a Berger Sphere
Dibakar Dey
Bull. Korean Math. Soc. 2022; 59(1): 101-110
https://doi.org/10.4134/BKMS.b210125 | 2022-05-18 15:18:35 | {"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": 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": 0.8668157458305359, "perplexity": 2293.1389591206116}, "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-2022-21/segments/1652662522284.20/warc/CC-MAIN-20220518151003-20220518181003-00614.warc.gz"} |
https://brilliant.org/problems/all-about-that-base/ | # All about that Base!
Find the sum of the last 3 digits of $$7^{999}$$.
×
Problem Loading...
Note Loading...
Set Loading... | 2018-03-23 19:07:31 | {"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": 0.46217653155326843, "perplexity": 8553.833061020137}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-13/segments/1521257648431.63/warc/CC-MAIN-20180323180932-20180323200932-00205.warc.gz"} |
https://www.experts-exchange.com/questions/25991605/How-to-write-into-a-text-file-using-MS-Access-VBA-code-and-importing-that-text-file-into-Access.html | # How to write into a text file using MS Access VBA code and importing that text file into Access?
I am trying to write into an existing text file. The (text.txt) file contains some numbers such as...
123335
125455
145575
254875
I am trying to import that data into a table called "Invoice".
While importing this data from text file, I need to write "Field Name" on top of the text file and also change the data type to "text".
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Commented:
better to create another text file with the field name
Dim s
Open CurrentProject.Path & "\mytext.txt" For Input As #1
Open CurrentProject.Path & "\mytext2.txt" For Output As #2
Print #2, "FieldName"
Do Until EOF(1)
Line Input #1, s
Print #2, s
Loop
Close #1
Close #2
0
For example (you have to use Microsoft Scripting Runtime reference):
Sub Import()
Dim Fs As New Scripting.FileSystemObject
Dim Text As Scripting.TextStream
Dim S As String
Set Text = Fs.OpenTextFile("C:\text.txt")
S = "Field Name" + vbCrLf + S
Text.Close
Set Text = Fs.CreateTextFile("C:\text.txt", True)
Text.Write S
Text.Close
End Sub
0
Commented:
Assuming that you have all the values in a single string (CSV with the separator character being a vbCrLf), you can use code like this to save a string to a text file:
Public Sub SaveToTextFile(strString As String, strFilename As String)
Dim intChannel As Integer
'Close any open files
For intChannel = 1 To 511
Close #intChannel
Next intChannel
'Get a filenumber (aka: communication channel) to send information to the file
intChannel = FreeFile
'Open/Create/Overwrite the file, send the data to it, then close the file
Open strFilename For Output Access Write As #intChannel
'Write to the text file, then close it
Print #intChannel, strString
Close #intChannel
End Sub
Example:
SaveToTextFile "MyColumnHeader" & vbCrLf & "Hello" & vbCrLf & "There", "C:\Temp\MyFile.txt"
Will yeild a text file in C:\Temp named MyFile.txt (note: if it exists, it WILL be overwritten) and the contents of that file will be:
Hello
There
-----
To import the data, with the assumption that you will be CREATING new records into an existing table, you can do something like this:
Public Sub ImportFile(strPath As String, strName As String)
Dim strSQL As String
strSQL = "INSERT INTO tblDestinationTableName (FieldNameForData)" & _
" FROM [Text;FMT=Delimited;HDR=YES;CharacterSet=437" & _
";DATABASE=" & strPath & "].[" & strName & "] AS vMyData;"
CurrentDb.Execute strSQL, dbFailOnError
End Sub
Example:
ImportFile "C:\Temp", "MyFile.txt"
Will yeild two records in the destination table with "Hello" and "There" in the "FieldNameForData" field.
If my assumptions are incorrect, please let me know and I will help modify the code accordingly.
0
Commented:
... Oh ...
With the import method I have shown, you really DON'T need the field name as the first row of the text file. Also, I messed up the strSQL statement ...
Should be:
strSQL = "INSERT INTO tblDestinationTableName (FieldNameForData)" & _
" SELECT MyColumnHeader FROM [Text;FMT=Delimited;HDR=YES;CharacterSet=437" & _
";DATABASE=" & strPath & "].[" & strName & "] AS vMyData;"
Or... if you choose not to use a header ....
strSQL = "INSERT INTO tblDestinationTableName (FieldNameForData)" & _
" SELECT F1 FROM [Text;FMT=Delimited;HDR=NO;CharacterSet=437" & _
";DATABASE=" & strPath & "].[" & strName & "] AS vMyData;"
0
Author Commented:
I tried the first code, it's giving an error, "Bad file or Number", Run time error '52'.
Here's what I have exactly in my code:
Private Sub Command0_Click()
Dim s
Open CurrentProject.Path & "C:\test1.txt" For Input As #1
Open CurrentProject.Path & "C:\test2.txt" For Output As #2
Print #2, "Invoice"
Do Until EOF(1)
Line Input #1, s
Print #2, s
Loop
Close #1
Close #2
End Sub
0
Commented:
try this, the path to your file is not correct
Private Sub Command0_Click()
Dim s
Open "C:\test1.txt" For Input As #1
Open "C:\test2.txt" For Output As #2
Print #2, "Invoice"
Do Until EOF(1)
Line Input #1, s
Print #2, s
Loop
Close #1
Close #2
End Sub
0
Experts Exchange Solution brought to you by | 2018-08-19 19:18:32 | {"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": 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": 0.39662986993789673, "perplexity": 8569.688939830125}, "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-2018-34/segments/1534221215284.54/warc/CC-MAIN-20180819184710-20180819204710-00469.warc.gz"} |
https://www.cuemath.com/ncert-solutions/q-1-exercise-4-4-linear-equations-in-two-variables-class-9-maths/ | In the verge of coronavirus pandemic, we are providing FREE access to our entire Online Curriculum to ensure Learning Doesn't STOP!
# Ex.4.4 Q1 Linear Equations in Two Variables Solution - NCERT Maths Class 9
Go back to 'Ex.4.4'
## Question
Give the geometric representation of $$y = 3$$ as an equation
(i) in one variable
(ii) in two variables
Video Solution
Linear Equations In Two Variables
Ex 4.4 | Question 1
## Text Solution
#### Steps:
(i) Given,
Considering $$y = 3$$ is the equation in one variable
The representation of the solution on the number when $$y = 3$$ is treated as an equation in one variable
In one variable, $$y = 3$$ represents a point as shown in following figure.
(ii) Given:
Considering $$y = 3$$ is the equation in two variables
We know that $$y = 3$$ can be written as $$0.x + y = 0$$.
In two variables, $$y = 3$$ represents a straight line passing through point $$(0, 3)$$ and parallel to $$x$$-axis. It is a collection of all the points on the plane, having their $$y$$-coordinate as $$3.$$
Hence,
• When, $$x = 0$$, we get $$y = 3$$;
• When $$x = 2$$, we get $$y = 3$$;
• When $$x = -2$$, we get $$y = 3$$ are the solutions for the equations.
Plotting the points $$(0, 3) (2, 3)$$ and $$(–2, 3)$$ and on joining them we get the graph $$AB$$ as a line parallel to $$x$$-axis at a distance of $$3\,\rm units$$ above it
The graphical representation is shown below:
Video Solution
Linear Equations In Two Variables
Ex 4.4 | Question 1
Learn from the best math teachers and top your exams
• Live one on one classroom and doubt clearing
• Practice worksheets in and after class for conceptual clarity
• Personalized curriculum to keep up with school | 2020-04-07 13:24:05 | {"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": 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": 0.6321104168891907, "perplexity": 1349.081462357923}, "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-2020-16/segments/1585371799447.70/warc/CC-MAIN-20200407121105-20200407151605-00181.warc.gz"} |
http://mathhelpforum.com/calculus/188455-compute-limit-question-2-a.html | Math Help - Compute the Limit: Question 2
1. Compute the Limit: Question 2
I'm stuck on this one:
$\text{Compute } \lim_{u \rightarrow 1 } \frac{u^4-1}{u^3-1}$
$= \frac{1^4-1}{1^3-1}$
$= \frac{0}{0 } \text{Indeterminate}$
$\lim_{u \rightarrow 1 } \frac{u^4-1}{u^3-1}$
$= \lim_{u \rightarrow 1 } \frac{(u-1)(u+1)(u-1)(u-1)}{(u-1)(u+1)(u+1)}$
$= \lim_{u \rightarrow 1 } \frac{(u-1)(u-1)}{(u+1)}$
$= \frac{(1-1)(1-1)}{(1+1) }$
$= \frac{0}{2}$
2. Re: Compute the Limit: Question 2
Not good, Sparky. Please factor numerator and denominator again. Both are incorrect.
Not good, Sparky. Why did you substitute the limit value into a function you do NOT know is continuous? Never do that.
Do a little less manipulating little symbols and a lot more thinking.
3. Re: Compute the Limit: Question 2
Ok, here is another attempt at the following question:
$\text{Compute } \lim_{u \rightarrow 1 } \frac{u^4-1}{u^3-1}$
$u^4-1 = (u^2+1)(u^2-1)$
$u^3-1 = (u-1)(u^2+u+1)$
$\lim_{u \rightarrow 1 } \frac{(u^2+1)(u^2-1)}{(u-1)(u^2+u+1)}$
Where did I go wrong with my factorization?
4. Re: Compute the Limit: Question 2
so far, so good. you won't be able to factor $u^2+1$ further, but you CAN factor $u^2-1$. try that, and see if something cancels...
5. Re: Compute the Limit: Question 2
Originally Posted by Deveno
so far, so good. you won't be able to factor $u^2+1$ further, but you CAN factor $u^2-1$. try that, and see if something cancels...
$\text{Compute } \lim_{u \rightarrow 1 } \frac{u^4-1}{u^3-1}$
$u^4-1 = (u^2+1)(u^2-1)= (u^2+1)(u+1)(u-1)$
$u^3-1 = (u-1)(u^2+u+1)$
$= \lim_{u \rightarrow 1 } \frac{(u^2+1)(u+1)(u-1)}{(u-1)(u^2+u+1)}$
$= \lim_{u \rightarrow 1 } \frac{(u^2+1)(u+1)}{(u^2+u+1)}$
$= \frac{1+1+1+1}{1+1+1}$
$= \frac{4}{3}$
Is this correct?
6. Re: Compute the Limit: Question 2
It is a beautiful thing.
7. Re: Compute the Limit: Question 2
Originally Posted by sparky
$\text{Compute } \lim_{u \rightarrow 1 } \frac{u^4-1}{u^3-1}$
$u^4-1 = (u^2+1)(u^2-1)= (u^2+1)(u+1)(u-1)$
$u^3-1 = (u-1)(u^2+u+1)$
$= \lim_{u \rightarrow 1 } \frac{(u^2+1)(u+1)(u-1)}{(u-1)(u^2+u+1)}$
$= \lim_{u \rightarrow 1 } \frac{(u^2+1)(u+1)}{(u^2+u+1)}$
$= \frac{1+1+1+1}{1+1+1}$
$= \frac{4}{3}$
Is this correct?
your answer is correct, but the work is not. it is lucky for you that (1+1)(1+1) = 1+1+1+1.
8. Re: Compute the Limit: Question 2
Originally Posted by Deveno
your answer is correct, but the work is not. it is lucky for you that (1+1)(1+1) = 1+1+1+1.
My work is not correct? Where did I go wrong?
9. Re: Compute the Limit: Question 2
In this step you're substituting wrong, you wrote:
$\lim_{u\to 1} \frac{(u^2+1)(u+1)}{u^2+u+1}=\frac{1+1+1+1}{1+1+1} =\frac{4}{3}$
It has to be:
$\lim_{u\to 1} \frac{(u^2+1)(u+1)}{u^2+u+1}=\frac{(1+1)(1+1)}{1+1 +1}=\frac{4}{3}$
Do you notice your mistake? Offcourse like Deveno said it's lucky that (1+1)(1+1)=1+1+1+1, but in other cases ... | 2016-07-28 02:22:40 | {"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": 0, "img_math": 0, "codecogs_latex": 32, "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": 0.5180104374885559, "perplexity": 2100.920553452952}, "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-2016-30/segments/1469257827781.76/warc/CC-MAIN-20160723071027-00037-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://hackage.haskell.org/package/repa-io-3.4.1.1/docs/Data-Array-Repa-IO-Timing.html | repa-io-3.4.1.1: Read and write Repa arrays in various formats.
Data.Array.Repa.IO.Timing
Description
Timing utilities used for benchmarks in the repa-examples package.
Synopsis
# Documentation
data Time Source #
Abstract representation of process time.
milliseconds :: TimeUnit Source #
microseconds :: TimeUnit Source #
cpuTime :: TimeUnit -> Time -> Integer Source #
wallTime :: TimeUnit -> Time -> Integer Source #
time :: IO a -> IO (a, Time) Source #
Time some IO action. Make sure to deepseq the result before returning it from the action. If you don't do this then there's a good chance that you'll just pass a suspension out of the action, and the computation time will be zero.
minus :: Time -> Time -> Time Source #
Subtract second time from the first.
plus :: Time -> Time -> Time Source # | 2017-07-21 16:08:21 | {"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": 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": 0.19659976661205292, "perplexity": 10011.45871712024}, "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-2017-30/segments/1500549423785.29/warc/CC-MAIN-20170721142410-20170721162410-00561.warc.gz"} |
https://math.stackexchange.com/questions/2071652/condition-for-two-line-to-be-coplanar | # Condition for two line to be coplanar
I want to know what is the condition for two lines to be coplanar . I searched it on internet.
I found that for coplanar the scalar product should be zero .
But I could not understand why it should be zero . And what are the three vectors whose scalar product is zero
• – user371838 Dec 25 '16 at 16:17
• @Rohan in that you have not mentioned about saclar product – Koolman Dec 25 '16 at 16:19
• In general we check for coplanarity like that. – user371838 Dec 25 '16 at 16:20
• @Rohan can you post an answer in detail explanation. – Koolman Dec 25 '16 at 16:25
• 3 vectors or 2 vectors? – Fawad Dec 25 '16 at 16:36
$$x=a+ut$$ $$y=b+vt$$ $$z=c+wt$$ where $(a,b,c)$ is a point of the line, $(u,v,w)$ the vector director and $t$ a parameter. | 2021-05-18 08:19:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5854029655456543, "perplexity": 439.41859025570386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989756.81/warc/CC-MAIN-20210518063944-20210518093944-00067.warc.gz"} |
https://codegolf.stackexchange.com/posts/171285/revisions | 2 added 22 characters in body
# CJam, 5(5 bytes) $294204.018... '$PB#
Try it online!
Explanation:
I derived it from Dennis' answer, but looked for combinations of numbers which would yield a higher result. I almost gave up, but I saw that P is the variable for $$\\pi\$$, and that $$\\pi^{11} \approx 294000\$$. The letter B has a value of 11 in CJam, giving the code above.
'$PB# Try it online! Explanation: I derived it from Dennis' answer, but looked for combinations of numbers which would yield a higher result. I almost gave up, but I saw that P is the variable for $$\\pi\$$, and that $$\\pi^{11} \approx 294000\$$. The letter B has a value of 11 in CJam, giving the code above. 1 # CJam, 5 bytes '$PB#
Try it online!
Explanation:
I derived it from Dennis' answer, but looked for combinations of numbers which would yield a higher result. I almost gave up, but I saw that P is the variable for $$\\pi\$$, and that $$\\pi^{11} \approx 294000\$$. The letter B has a value of 11 in CJam, giving the code above. | 2019-05-21 09:33:15 | {"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7770848274230957, "perplexity": 913.5517233626571}, "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-2019-22/segments/1558232256314.25/warc/CC-MAIN-20190521082340-20190521104340-00041.warc.gz"} |
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## Tikz chain labels for nodes with includegraphics
chainsdiagramsTikZAdd tags Information and discussion about graphics, figures & tables in LaTeX documents.
### Tikz chain labels for nodes with includegraphics
Hello everyone!
I'm trying to draw a network diagram using tikz and Cisco symbols.
I have the cisco symbols in the folder
Code: Select all • Open in writeLaTeX
figs/Cisco/3015_eps/
When I use the following code:
Code: Select all • Open in writeLaTeX
\usepackage{tikz}\usetikzlibrary{chains}\usetikzlibrary{positioning}\usepackage{epstopdf}\begin{tikzpicture}[start chain=going right,diagram item/.style={on chain,join}]\node[diagram item,label=Internet](Internet){\includegraphics[width=6pc]{figs/Cisco/3015_eps/cloud}};\node[diagram item,label=Backbone](Backbone){\includegraphics[width=6pc]{figs/Cisco/3015_eps/netranger}};\node[diagram item,label=BRAS](BRAS){\includegraphics[width=6pc]{figs/Cisco/3015_eps/router}};\node[continue chain=going right,diagram item,label=AAA](AAA){\includegraphics[width=5pc]{figs/Cisco/3015_eps/pad}};\node[continue chain=going below,diagram item,label=left:Switch](Switch){\includegraphics[width=6pc]{figs/Cisco/3015_eps/workgroup_switch}};\node[diagram item,label=left:DSLAM](DSLAM){\includegraphics[width=4.5pc]{figs/Cisco/3015_eps/dslam}};\node[diagram item,label=left:Filtre](Filtre){\includegraphics[width=1.5pc]{figs/Cisco/3015_eps/pad}};\node[start branch=1 going right,diagram item,label=right:phone](Phone){\includegraphics[width=6pc]{figs/Cisco/3015_eps/phone}};\node [diagram item,label=below:PC](PC){\includegraphics[width=6pc]{figs/Cisco/3015_eps/pc}};\end{tikzpicture}
I attached the result to this post.
What I want to do is to put the switch (and everything that's beneath it) under the BRAS not under the AAA I can't figure out how to do it.
And of course the labels are misplaced I don't understand why.
Can someone help please ? :)
Attachments
Untitled-1.jpg (53 KiB) Viewed 1452 times
megaflop
Posts: 2
Joined: Fri Dec 28th, 2012
### Re: Tikz chain labels for nodes with includegraphics
I forgot to mention that the Cisco icons could be found here http://www.cisco.com/web/about/ac50/ac47/2.html. Not that it really matters.
megaflop
Posts: 2
Joined: Fri Dec 28th, 2012
### Re: Tikz chain labels for nodes with includegraphics
Hi,
welcome to the board!
megaflop wrote:What I want to do is to put the switch (and everything that's beneath it) under the BRAS not under the AAA
I guess you need to make a branch before AAA instead of going below after AAA.
megaflop wrote:And of course the labels are misplaced I don't understand why.
Not when I compile your code. I got:
network.png (13.65 KiB) Viewed 1430 times
Some images are missing, perhaps an issue with the eps pictures, but that's not the question here. The labels look better than in your output, even if I did not change the code. Perhaps update your TikZ package if it's not a current version.
If there's still a problem, I suggest that you should post a minimal working example, i.e. a compilable short version. Your remaining packages and settings are unknown. And it would be better if you would add simple things such as class, \begin{document}, \end{document} etc. if you would like readers to test your code. Otherwise every reader has to fix and to complete the code again - not everybody does it. Also I postponed an answer until I had some time to deal with incomplete code.
Stefan
Stefan_K
Posts: 6517
Joined: Mon Mar 10th, 2008
Location: Hamburg, Germany
### Topic Tags
chainsdiagramsTikZ | 2014-12-22 05:38:05 | {"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": 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": 0.881087064743042, "perplexity": 5354.367087157106}, "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-2014-52/segments/1418802774464.128/warc/CC-MAIN-20141217075254-00147-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://openstudy.com/updates/50becf18e4b0de4262a05725 | ## anonymous 3 years ago Why does a planet follow an elliptical orbit and not circular?? I know keplers laws and i know the laws of gravitation.. but I wanna know if there is some reason why the planets follow an ellipse and not a perfect circle
1. anonymous
I think when you do the derivation using mathematics for a central force which varies 1 over r square... the result turns out to be an ellipse.. is there any intuitive way to come to this conclusion qualitatively?
2. anonymous
note: physics isn't intuitive. example: try to use intuition to understand special relativity. if you want the derivation for the ellipse form, you might want to try Exercise 11.9.32 from "Calculus" by James Stewart. nothing about forces though.
3. anonymous
physics isn't intuitive? :P.. what is wrong with you!!.. special relativity is not intuitive TO US cause.. we are not travelling at sub luminal speeds... it was intuitive to EINSTIEN.. if it wasn't.. we wouldn't have the theory in the first place.... Infact.. all the theories are intially laid down by making use of intuitions!!.. sometimes mathematics helps to derive stuff which are totally non intuitive... but at certain level intuition can always be made!!.. PHYSICS IS INTUITIVE mate!
4. anonymous
lol i'm meaning intuitive as in straightforward.
5. anonymous
the above question "is there any intuitive way to come to this conclusion qualitatively", if you count mathematics as intuition (like 1+1=2, duh, thing),then yes...
6. anonymous
lol obviously need not be straightforward.. but we can always generate some sort of intuition.. but i agree.. in this case. its not all that straight forward :P
7. anonymous
but i think.. its better to understand why the planets DO NOT REVOLVE in a circular orbit.. and thats because.. the initial speed does not satisfy the condition!
8. anonymous
if you want, i could give you some equations where you do a step-by-step derivation and develop your own intuition? technically....the initial speed doesn't really matter? lol how did you arrive to that conclusion?
9. anonymous
the initial speed matters.. if mv^2/r is not equal to the force of gravity.. then the object will not give a perfect circle..!!!
10. anonymous
"Uniform circular motion Every central force can produce uniform circular motion, provided that the initial radius r and speed v satisfy the equation for the centripetal force If this equation is satisfied at the initial moments, it will be satisfied at all later times; the particle will continue to move in a circle of radius r at speed v forever." form wiki :P http://en.wikipedia.org/wiki/Classical_central-force_problem
11. anonymous
but it's always equal to $$F_G$$. if initial speed just affect the altitude of orbit.
12. anonymous
NOOOOO!!!.. initial speed decides whether it ll be a circular orbit or not!
13. anonymous
otherwise the velocity will not be perpendicular to the force!!.. it ll have some tangential component
14. anonymous
i mean.. NON TANGENTIAL component :P
15. anonymous
ah. I saw it. though that might be why most planets are just almost circles.
16. anonymous
*orbits
17. anonymous
almost circles cause their eccentricities are too small!
18. anonymous
so the deviation from the required speed must be have been very small!!
19. anonymous
hmm,...which brings us to the problem....why? lol
20. anonymous
and how?!
21. anonymous
why they are not doing CIRCULAR ORBITTING :P..
22. anonymous
wait.. what why??? that depends upon the initial conditions.. when the planet was formed... why should the exact speed be matched!!?
23. anonymous
so basically those planets are just...lucky that they managed to do so? lol :P
24. anonymous
i dunno. .. you tell me :P..
25. anonymous
lol let's leave that to the NASA boys
26. anonymous
http://www.physicsforums.com/showthread.php?t=297095 this says.. drag forces must have reduced the velocity somehow.. !!
27. anonymous
i can't i am a teacher.. i need answers :P
28. UnkleRhaukus
|dw:1354694312598:dw|
29. UnkleRhaukus
|dw:1354694460654:dw|
30. anonymous
uncle what are you trying to suggest? :P
31. UnkleRhaukus
difference in energy
32. anonymous
i didn't get it :-/
33. anonymous
34. anonymous
Hmm - doesn't show the image for some reason. (I'm going to do a feature request, I guess ^_^) Anywhoo... as you can see in my ahm.. masterpiece ^^ The "effective potential Energy" is a sum of the (green) potential gravitational energy plus the centrifugal one. Now there are various cases - depending on the initial conditions. If the resulting energy is negative but there is radial-kinetic-energy, then that's an ellipse. The blue line I shows that case. r1 and r2 are the minimal and the maximal distance of the planet to the sun. If you do not have radial-kinetic-energy, you are going to have a circle. This is the minimal effetive energy you can possibly have - labeled II. The distance is a constant r0. If your energy is positive, then you are going to have a hyperbola. Meaning your celestial body will escape the gravitational field of the sun. (labelled III) Note here, that the point C is the minimal distance to the sun, the body is going to have. Also note that the transistion from elliptical to hyperbola (E = 0) is called a parabola. Please ask, what you don't understand yet, I'll elaborate. $E_{kin}^{\tan} = \frac{1}{2}\, m\, r^2\, \dot \phi ^2 = \frac{L^2}{2\, m\, r^2}$$E_{pot}^{eff} = E_{pot}(r) + \frac{L^2}{2\, m\, r^2} = -G \frac{m\, M}{r} + \frac{L^2}{2\, m\, r^2}$$E_{kin}^{rad} = \frac{1}{2}\, m \, \dot r ^2 = E - E_{pot}^{eff}$$\frac{d\, E_{pot}^{eff}}{dr}= 0 \rightarrow r_0 =\frac{L^2}{G \, m^2 \, M}$
35. anonymous
ok i need to study more i guess :D.. what is centrifugal energy?? isn't it just its inertia? I always hate the word centrifugal force... cause its not a real force!!
36. anonymous
|dw:1354712941270:dw|
37. anonymous
Well, why do you hate easy things? There is an acceleration acting on the body - I don't think you'll doubt that. Now just taking F = m*a and naming that the centrifugal force makes things easier. Yeah you can always choose a coordinate system, where you wouldn't need it, but why hate on easy principles? And about it being real or not.. that's pretty much a point-of-view-thingy.. Are you familiar with D'Alembert's priciple? [1] I meant the potential energy of the centrifugal force/movement. Does that name make it clearer? 1. http://en.wikipedia.org/wiki/D%27Alembert%27s_principle
38. anonymous
no.. i am not really familiar with that potential energy of centrifugal movement :-/.. its like potential to go straight???..
39. anonymous
Well hmm.. you split the effective potential in 2 parts. One part is the radial part $$\frac{m}{2} \dot r^2$$ giving you the kinetic energy of the radial movement. The other one is the 'angle-part' $$\frac{m}{2} r^2 \, \dot \phi^2$$that'll describe the energy of the tangential movement with a fixed distance r. This tangential part, we can also express with a constant angular momentum $$\frac{L^2}{2 m r^2}$$ NOW: because this part is only a function of r but not of the angle or the radial velocity, we simply say, it's part of the potential energy (which is also solely dependent on r). This sum, we now call the "effective potential energy" (the red line in my drawing).
40. anonymous
if u dont know then u sud read classical mechanics again :) | 2016-08-24 04:49:49 | {"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": 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": 0.7947335243225098, "perplexity": 1625.319930286138}, "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-2016-36/segments/1471982291015.10/warc/CC-MAIN-20160823195811-00013-ip-10-153-172-175.ec2.internal.warc.gz"} |