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What are some of the methods used to test for the exactitude of a numerical solution, given that the analytical solution isn't available, and the numerical solution converges ?
You should also read about the Method of Manufactured Solutions (PDF) which will show you how to generate analytical solutions to your problem.... |
Many thanks to Bart Andrews for this contribution!
Question Show that a relation of the kind \(f(x, y, z) = 0\) between the three quantities x, y, and z implies the relation \[\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y = -1\] be... |
Consider the equations of motion of an object in an orbit around a central body. We use an inertial frame originating at centre of mass of the central body: $$ \frac{d\mathbf{r}}{dt} = \mathbf{V} \tag{1} $$ $$ \frac{d\mathbf{V}}{dt} = -\frac{\mu}{r^3}\mathbf{r} + \mathbf{A}_\text{ext} \tag{2} $$ where $\mathbf{r}$ is t... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphe... |
Hello, I've never ventured into char before but cfr suggested that I ask in here about a better name for the quiz package that I am getting ready to submit to ctan (tex.stackexchange.com/questions/393309/…). Is something like latex2quiz too audacious?
Also, is anyone able to answer my questions about submitting to ctan... |
In his classic paper
"Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in terms of $k$ (see the details here). Most of the series which Ramanujan provides are based on the following expressions... |
I am reading Pragmatic MPC (link) about the BGW protocol. I also cross-reference here (click on lecture link for slides) *The BGW Construction for the Information Theoretic Setting – Benny Pinkas I do not understand why the multiplication gate calculation can work.
First, let's say of $N$ parties, party $P_i$ has share... |
We know that a massless $\phi^4$ theory $$S=\int d^4x \left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{\lambda}{4!}\phi^4\right],$$ has conformal invariance at the classical level. But within the Coleman-Weinberg mechanism, at the one-loop level, quantum fluctuations will generate a vacuum expected value for $\p... |
In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a contravariant vector, and its covariant representation is written as $$x_{\mu} = \binom{ct}{-\textbf{x}}.$$ However, I know ... |
There is no general rule. However there is a class of bounded self-adjoint operators whose spectrum is made of a bounded set of isolated points (proper eigenvalues) -- except for $0$ at most -- and the eigenspaces associated to these eigenvalues are finite dimensional. They are the so-called
compact operators (this cla... |
The problem is:
Original Problem \[\begin{align}\min\>&\sum_i \color{darkred}x_i\\ & \mathbf{sd}(\color{darkred}x) \lt \color{darkblue}\alpha\\ & \color{darkred}x_i \in \{0,1\}\end{align}\]
Notes:
Here sdis the standard deviation. We assume \(x\) has \(n\) components. Of course, \(\lt\) is problematic in optimization. ... |
Loan balance after the ith payment principal or initial loan balance Interest rate Loan term Loan product, important parameter which fully specifies a loan Number of loan payments Time between loan payments Helpful collection of variables Fraction of loan term Repayment rate (dollars per time) Fraction of payment to in... |
I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on this question is to use the fundamental theorem of calculus or Jensen's inequality.
Assuming $f$ is integrable and twice... |
Normally I read the
Divergence Theorem written as (
\oiint doesn't exist here):
\begin{align} \oint_{\partial \Omega} \vec{F} \cdot \hat{n} \; dS = \iiint_{\Omega} \nabla \cdot \vec{F} \; dV \end{align}
Where $\partial \Omega$ is the boundary of region $\Omega$ and $dS$ are infinitesimal surface of $\partial \Omega$ an... |
This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in compactified Minkowski space time models.
It is also motivated deeply by Hamiltonian mechanics, symplectic geometry, and cont... |
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the conservation of this vector?
1)
Hamiltonian Problem. The Kepler problem has Hamiltonian
$$ H~=~T+V, \qquad T~:=~ \frac{p^2}{... |
I'm currently studying how to deduce Feynman rules for general theories, and I've managed to deduce them for $\phi^3$ and $\phi^4$ theories. Up to this point I've considered the same field for all cases, and deduced the Feynman rules by expanding the interacting term in the correlator and using Wick's theorem to make t... |
I understand the attractiveness and usefulness of infinite-series expansions such as Taylor expansions, but I wonder if they sometimes hide important aspects of the described system.
For example, take $F(x) = e^x$, whose Taylor expansion is $$F(x) = \sum_n \frac{x^n}{ n!}.$$ Sure, we can approximate $F(x)$ using the fi... |
Bernoulli Bernoulli Volume 19, Number 5A (2013), 2098-2119. Marked empirical processes for non-stationary time series Abstract
Consider a first-order autoregressive process $X_{i}=\beta X_{i-1}+\varepsilon_{i}$, where $\varepsilon_{i}=G(\eta_{i},\eta_{i-1},\ldots)$ and $\eta_{i}$, $i\in\mathbb{Z}$ are i.i.d. random var... |
Wave packet in 2D potential13 Oct 2018
Having some fun with Julia and simulating a Wave packet in 2D.
Main Idea
We will evolve the wavepacket in 2D by taking the tensor products between two 1D (x,y) spaces.
Wavepacket
The wavepacket is expressed as a Gaussian:
\[\Psi(x) = \frac{\sqrt{\Delta x}}{\pi^{1/4}\sqrt{\sigma}} ... |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this ps... |
Some tricks I've seen:
Tricks with notable products
$(a + b)^2 = a^2 + 2ab + b^2$
This formula can be used to compute squares. Say that we want to compute $46^2$. We use $46^2 = (40+6)^2 = 40^2+2\cdot40\cdot6 +6^2 = 1600 + 480 + 36 = 2116$. You can also use this method for negative $b$:$ 197^2 = (200 - 3)^2 = 200^2 - 2... |
This benchmark requires that the global optimization algorithm is run starting from the 100 randomly generated (minimized) Lennard-Jones structures. Runs that are greater than 0.001 energy units from the known global minimum are considered failures. If a run was successful, the number of force calls needed to reach the... |
Let $A$ denote a commutative ring. Then if $A$ is a field, we may deduce that every $A$-module is free. Does the converse hold? i.e. If every $A$-module is free, can we deduce that $A$ is a field?
I would think so.
Let $0\neq x\in A$ a non-invertible element. Then the ideal $Ax$ is proper. Now consider the quotient $A/... |
I have a friend who believes that 17% doesn't have to be equal to 0.17. Even though he says that 17% is equal to 0.17
on its own, he says that 17% at any other time is not equal to 0.17, referring to the argument that $17\%x \neq 0.17$. No matter how I try to explain it to him, he won't believe me when I say that 17% i... |
Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$.
Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract embeddings (ie, as $\mathbb{Q}$-algebras). Let $X_\mathbb{C},X_{\mathbb{C}}'$ be the base changes of $X_{\overline{K}}$ to $\math... |
Current browse context:
physics.atom-ph
Change to browse by: References & Citations Bookmark(what is this?) Physics > Atomic Physics Title: Photoionization of Xe and Rn from the relativistic random-phase theory
(Submitted on 30 Apr 2018 (v1), last revised 14 Feb 2019 (this version, v6))
Abstract: Photoionization cross ... |
EDIT (Aug 1):
I posted a small report with a more detailed proof on my blog; the reduction idea is the same, but the "gadget" used are better explained (you can also directly download the pdf from here)
The problem seems NP-complete and this is a possible reduction from SET COVER.
Suppose you have an universe $A$ of $n... |
So basically
$dS_t=\mu S_tdt+\sigma S_tdWt$
and
$\mu=r-\frac12\sigma^2$
I have just been thinking about this later equation. This is very interesting because it ties together risk-free rate, volatility and asset drift. I always like and try to look at equation from some simple perspective, for example assuming that som... |
10.8. RMSProp¶
In the experiment in Section 10.7, the learning rate of each element in the independent variable of the objective function declines (or remains unchanged) during iteration because the variable \(\mathbf{s}_t\) in the denominator is increased by the square by element operation of the mini-batch stochastic... |
I am confused about a statement made in the paper Linear Time Algorithm for Projective Clustering, section 5.1, second paragraph, second line.
Project clustering is a natural generalization of k-means clustering. Given a point set $P$, the goal to find $k$, $j$-dimensional affine spaces(flats) $F_1,\cdots,F_k$, such th... |
Algebraic topological model for a cell complex¶
This file contains two functions,
algebraic_topological_model()and
algebraic_topological_model_delta_complex(). The secondworks more generally: for all simplicial, cubical, and\(\Delta\)-complexes. The first only works for simplicial and cubicalcomplexes, but it is faster... |
10.6. Momentum¶
In Section 10.3, we mentioned that the gradient of the objective function’s independent variable represents the direction of the objective function’s fastest descend at the current position of the independent variable. Therefore, gradient descent is also called steepest descent. In each iteration, the g... |
If economic growth is indeed highly desirable (see this question), why must this growth be exponential? With finite resources, exponential growth might hit limits rapidly (or be impossible?). Why not express growth in linear rather than exponential terms?
Growth as is meant here "must" be nothing in particular. It is a... |
Scipy.optimize.linprog[1] recently added a sparse interior point solver [2]. In theory we should be able to solve some larger problems with this solver. However the input format is matrix based. This makes it difficult to express LP models without much tedious programming. Of course if the LP model is very structured t... |
Fortunately, and covering the points already made, there is a fairly complete answer to the determination of the excision map
$$\varepsilon_2 : \pi_2(B,C) \to \pi_2(X,A)$$when $X=A \cup B, C = A \cap B$ where the base point lies in $C$, and some other conditions, such as $A,B$ are open; we also need connectivity condit... |
I heard that mass and distance are the only deciding criteria for determining the gravitational pull. Keeping the distance constant, if mass would be the only deciding factor for gravitational pull, then every super-massive star capable of forming a black hole would absorb light in the first place, and thus it would al... |
Most of the question has already been answered in the comments. I'll summarise and expand on some of the points.
The OP asks why we have a requirement that the 2-norm of vectors be preserved under time evolution. The answer, already given in the comments, is as follows. Let $U(t)$ be the time evolution operator, and $\... |
Yes. The phase envelope (or equilibrium curve) of most liquids have a similar shape and can be approximated, to a reasonable degree, by using the Clausius-Clapeyron equation:
$$\Delta h_v=T(v^V-v^L)\frac{dP^s}{dT}$$
which requires finding ${dP^s}/{dT}$ from e.g. a vapor-pressure-temperature correlation like the Antoine... |
Show that L = $\{0^{2^n}| n\geq 0\}$ is not a context free language.
Let string $s = 0^{2^p}$. Then we know we can write $s$ as $s = uvxyz$. I know that |vy| > 0 and $|vxy| \leq p$.
So how do I show that $uv^2xy^2z$ is not in $L$.
Computer Science Stack Exchange is a question and answer site for students, researchers a... |
The integral is exactly the fractional part of $100!\,e$, or in other words $100!\ e-\lfloor100!\ e\rfloor\approx0.00999901019\ldots$
Apply integration by parts to the integral $I_n=\int_0^1e^{1-t}t^n\,dt$ (it's nicer
not to pull the $e$ out to the front) and we find for $n\geq1$, $$I_n=-1+nI_{n-1}$$
This gives us $$I_... |
@user193319 I believe the natural extension to multigraphs is just ensuring that $\#(u,v) = \#(\sigma(u),\sigma(v))$ where $\# : V \times V \rightarrow \mathbb{N}$ counts the number of edges between $u$ and $v$ (which would be zero).
I have this exercise: Consider the ring $R$ of polynomials in $n$ variables with integ... |
Hi,
I asked this question already on math.stackexchange but got no answer (link: https://math.stackexchange.com/questions/22155).
Our setting: An Euclidean vector bundle $(E, h, \nabla^E)$ over a Riemannian manifold (M,g) is said to have bounded geometry, if the norms of the curvature tensor $R^E$ and of all its covari... |
I've come across a question of which I can't fully understand the solution:
A space station is located in a gravity-free region of space. It consists of a large diameter, hollow thin-walled cylinder which is rotating freely about its axis. The cylinder is of radius $r$ and mass $M$.
Radial spokes, of negligible mass, c... |
The $\lambda$ value used in the original paper is arbitrary, but you can estimate that by assuming (in the simplest case) 2 assets and running the following model:
$\sigma^2_{12,t+1}$ $=$ $\lambda$$*$$\sigma^2_{12,t-1}$$+$$(1-\lambda)$$r_{1,t}$$*$$r_{2,t}$;
given $r_{1,t}$ and $r_{2,t}$ respectively as the returns for ... |
I'm looking for advice, or references, for a change of basis to my dependent variables that leads to a less computationally expensive scheme when solving a system of coupled polynomial equations. Below is an explanation of the problem, which will be somewhat involved.
The general idea is this. I have a system of equati... |
If I have a random too way overdeterminate systems of homogenous non-linear equations and few variables, What's the better way too find a solution, finding a the Groener Basis or just using linearization ?
Thanks
Cryptography Stack Exchange is a question and answer site for software developers, mathematicians and other... |
I need some help with a big O proof. I think I have a proof but I feel like some of the steps aren't logically compatible.
The Question: For all functions f,g with domain $\mathbb{N}$ that maps to $\mathbb{R}$, $log(f(n))\in \mathcal{O}(g(n)) \implies f(n) \in \mathcal{O}(3^{g(n)})$
I basically took the definiton of bi... |
Let $R(x)$ be a remainder upon dividing $x^{44}+x^{33}+x^{22}+x^{11} +1$ by the polynomial $x^4 +x^3 +x^2 +x +1$. Find: $R(1)+2R(2)+3R(3)$. Answer provided is $0$
closed as off-topic by user26857, Claude Leibovici, user91500, user1551, user223391 Jul 21 '16 at 4:45
This question appears to be off-topic. The users who v... |
On the history
See Butcher: A History of the Runge-Kutta method
In summary, people (Nystroem, Runge, Heun, Kutta,...) at the end of the 19th century experimented with success in generalizing the methods of numerical integration of functions in one variable $$\int_a^bf(x)dx,$$ like the Gauss, trapezoidal, midpoint and S... |
An equation such as Gauss' law $$ \nabla \cdot E(x) = \frac{\rho(x)}{\varepsilon_0} $$ is easy to interpret in physical terms. If we use the divergence theorem we have $$ \int_{\partial \Omega} E\cdot \hat{n} \ dx = \int_{\Omega} \frac{\rho(x)}{\varepsilon_0} dx $$ which says that the electric flux out of the domain $\... |
The Equation of the Trajectory of a particle can be obtained by eliminating the variable $t$ as we are doing in the equation of Trajectory of a parabolic Projectile. So, my question is if the equation of trajectory of a parabolic projectile is possible, then other simple ones like The Simple Pendulum should also be pos... |
I have two TeX files that I need to compare, and I thought texdiff could do this for me. Unfortunately, in the manual page, it reads
For texdiff to work, the following LaTeX code must be inserted in the preamble of the LaTeX document:
\usepackage{xcolor} \usepackage{ulem} \usepackage{changebar} \newcommand\TLSins[1]{\c... |
Two things to note here. First, subtracting inflation from the nominal interest rate
is an approximation to the real interest rate, but only in discrete time. Furthermore, the "true" relationship it's approximating isn't division of one rate by the other--you have to add 1 to all three of your quantities (inflation, re... |
Ok for real photons there is the formula when summing over the polarizations:
$$ \sum_{\lambda=\pm}\epsilon^{*\mu}_\lambda\epsilon^\nu_\lambda = -\eta^{\mu\nu}$$
But if I have a matrix element of the form: $$\epsilon^{*\mu}M_\mu\qquad \epsilon^\mu\bar{M}_\mu$$
So when I take the absoulte squared of that I have: $$|M|^2... |
I'd recommend "The Art of Molecular Dynamics Simulation" by D. C. Rapaport. The code samples are written in C. I'm not a huge fan of the programming style of the book, but at least it's not FORTRAN.Having said that, my advise would be to take any book where neighbour lists are explained (for instance the Frenkel & Smit... |
I am looking at two non-zero matrices, $A$ and $B$, both over $ \mathbb{R}$ and of dimension $n \times n$. I think I've proven that their product $AB$ can be equal to zero by the following:
In the first entry of $AB=0$, which is $ab_{11}$ = $\sum_{i=1}^{n} a_{1n}b_{n1}$, there exists some $a_{1k}b_{k1} = -\sum_{i=1}^{n... |
This task is more complex than the task to solve a quadratic equation, for example, and one must master a significant portion of a textbook – such as Georgi's textbook – and perhaps something beyond it to have everything he needs.
For the 8-dimensional representation of $SU(3)$, things simplify because it's the "adjoin... |
One of the postulates of QM states that given a system in a state $|\psi\rangle$ and given an observable $A$ whose eigenstates are $|\phi_i\rangle$, then the state of the system can be expressed as a linear combination of them such that
$$|\psi\rangle=\sum_ic_i|\phi_i\rangle$$
and the probability of the eigenvalue $a_i... |
Added: a Stanford course on neural networks,cs231n,gives yet another form of the steps:
v = mu * v_prev - learning_rate * gradient(x) # GD + momentum
v_nesterov = v + mu * (v - v_prev) # keep going, extrapolate
x += v_nesterov
Here
v is velocity aka step aka state,and
mu is a momentum factor, typically 0.9 or so.(
v,
x... |
Given information:
The boiling point of benzene at atmospheric pressure is $353~\mathrm{K}$ the enthalpy of vaporization of benzene is $30.8~\mathrm{kJ~mol^{−1}}$ at this temperature. The molar heat capacities of the liquid and vapour are $136.1~\mathrm{J~K^{−1}~mol^{-1}}$ and $81.7~\mathrm{J~K^{−1}~mol^{-1}}$, respect... |
In this question, a
graph is a finite, undirected graph without loops or multiple edges, and a colouring of a graph is a proper vertex colouring. The product $G \times H$ of graphs $G$ and $H$ is the graph whose vertex-set is the product of the vertex-sets of $G$ and $H$ and whose edge-set is the product of the edge-se... |
Background on why I want this:
I'd like to check that suspension in a simplicial model category is the same thing as suspension in the quasicategory obtained by composing Rezk's assignment of a complete Segal space to a simplicial model category with Joyal and Tierney's "first row functor" from complete Segal spaces to... |
View Answer
question_answer1) If the quadratic equation \[p{{x}^{2}}-2\sqrt{5}\,px+15=0,\] has two equal roots then find the value of p. question_answer2)
In figure 1, a tower AB is 20 m high and BC, its shadow on the ground, is \[20\sqrt{3}\,m\] long. Find the sun's altitude.
View Answer
question_answer3) Two differen... |
Some tricks I've seen:
Tricks with notable products
$(a + b)^2 = a^2 + 2ab + b^2$
This formula can be used to compute squares. Say that we want to compute $46^2$. We use $46^2 = (40+6)^2 = 40^2+2\cdot40\cdot6 +6^2 = 1600 + 480 + 36 = 2116$. You can also use this method for negative $b$:$ 197^2 = (200 - 3)^2 = 200^2 - 2... |
Four Color Theorem (4CT) states that every planar graph is four colorable. There are two proofs given by [Appel,Haken 1976] and [Robertson,Sanders,Seymour,Thomas 1997]. Both these proofs are computer-assisted and quite intimidating.
There are several conjectures in graph theory that imply 4CT. Resolution of these conje... |
I have a table where I want more than one row in some cells. I am using parbox to achieve this but this makes some items in cells left aligned and some not. How can you fix this to make them all aligned the same way?
\documentclass{beamer}\usepackage{multirow}\begin{document} \begin{frame}\frametitle{Test page} \resize... |
And suppose that R has no periodic points of period n . Then (d, n) is one of the pairs \( (2,2) ,(2,3) ,(3,2) ,(4 ,2) \) , each such pair does arise from some R in this way .
The example of such pair is $$ 1. R(z) = z +\frac {(w-1)(z^2 -1)}{2z} ; it \ has \ no \ points \ of \ period \ 3 .$$
$$ 2. If R(z) = \frac {z^3 ... |
Briefly analyzed the characters of measuring three-phase electric power parameters,introduced a method of interface between the chip of S3C44B0X with 32-bit embedded processor of ARM7TDMI and the A/D application-specific chip of ADS7864.The structure can predigest hardware circuits,relieve the contradiction between the... |
A little background to Stirling’s Formula
Stirling’s approximation is vital to a manageable formulation of statistical physics and thermodynamics. It vastly simplifies calculations involving logarithms of factorials where the factorial is huge. In statistical physics, we are typically discussing systems of particles. W... |
I try to understand the actual intuition behind the logarithm properties and came across a post on this site that explains the multiplication and thereby also the division properties very nicely:
Suppose you have a table of powers of 2, which looks like this: (after revision)
$$\begin{array}{rrrrrrrrrr} 0&1&2&3&4&5&6&7... |
So, as far as I understand, Godel's theorem says that sufficiently strong theories such as Peano Arithmetic are negation-incomplete, which means that there exists a formula $\phi$ such that neither $\phi$ nor its negation $\lnot \phi$ are provable within the formal system.
I have read somewhere that it can be understoo... |
The point is the following:
Delta, $\Delta$, is defined as $\frac{\partial C}{\partial S}$, where $C$ is the value of the call option, and $S$ is the price of the underlying asset.
So, given that the value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is
$$C = N(d_... |
12.12. Neural Style Transfer¶
If you use social sharing apps or happen to be an amateur photographer, you are familiar with filters. Filters can alter the color styles of photos to make the background sharper or people’s faces whiter. However, a filter generally can only change one aspect of a photo. To create the idea... |
3.7. Concise Implementation of Softmax Regression¶
Just as Gluon made it much easier to implement linear regression in Section 3.3, we’ll find it similarly (or possibly more) convenient for implementing classification models. Again, we begin with our import ritual.
import d2lfrom mxnet import gluon, init, npxfrom mxnet... |
fskilnik wrote:
GMATH
practice exercise (Quant Class 15)
If the line segment AB is one of the sides of a regular polygon inscribed in the given circle with center O, how many sides does this polygon have?
(1) \(\,\alpha = {30^ \circ }\).
(2) The circumference of the circle is equal to\(\,4\pi \,\).
\({\rm{regular}}\,\,... |
Given a system with N identical Fermions, with spin $\frac{1}{2}$ and mass $m$, subject to the potential:$V(\vec{r}) = \frac{1}{2}m\omega^{2}(x^{2}+y^{2}+z^{2})$
and the single particle energy levels: $E_{\vec{n}} = \hbar\omega(n_{x}+n_{y}+n_{z}+\frac{3}{2})$,
Calculate the Density of states $g(\epsilon)$.
I know that ... |
I was testing some links within my produced PDF file and recognized that all of the links to an algorithm (from the
algorithm2e package) will jump to the beginning of the algorithm with skipping half of the algorithm's header and the caption of the algorithm. I use the
ruled option of the
algorithm2e package such that ... |
This is a Test of Mathematics Solution (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also see: Cheenta I.S.I. & C.M.I. Entrance Course
If the coefficients of a quadratic equat... |
So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$
For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ and get $2^1=2$, and so on, for $^{x} 2$.
$$\begin{align} & (0,1)\\ & (1,2) \\ & (2,4) \\ & (4,16) \\ & (16,65536) \\ \en... |
So I'm going through Niven's The Theory of Numbers, and it gives the definition that:
$$a \equiv b \pmod m \implies m \mid (a - b)$$
However, a few pages after this definition, it gives a theorem that states "if $\gcd(a, m) = 1$, then there is an $x$ such that $ax \equiv 1 \pmod m$. To prove this theorem, it states tha... |
Given a light pulse in vacuum containing a single photon with an energy $E=h\nu$, what is the peak value of the electric / magnetic field?
The electric and magnetic fields of a single photon in a box are in fact very important and interesting. If you fix the size of the box, then yes, you can define the peak magnetic o... |
Answer
The cost of sod is around $325.83 dollars.
Work Step by Step
1. Find the area of the semi circle and multiply it by 2 Let $S =$ area of the semi circle $S = \frac{1}{2}\pi r^{2}$ $S = \frac{1}{2} (3.14) (8^{2})$ $S = \frac{1}{2} (200.96)$ $S = 100.48$ sq ft There are 2 semi circles are on opposite ends and there... |
I've done some testing of different initial temperatures in my simulating annealing algorithm and noticed the starting temperature has an affect on the performance of the algorithm.
Is there any way of calculating a good initial temperature?
Computer Science Stack Exchange is a question and answer site for students, re... |
It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it follows that what we experience as a force is simply the difference between a geodesic on this curved surface and our perce... |
In our chemistry lab we performed the following experiment below.
Given that $K_\mathrm{sp}(\ce{AgCl}) = 8.2 \times 10^{-11}$ and $K_{\mathrm{sp}}(\ce{PbCl_{2}}) = 1.7 \times 10^{-5}$
Procedure: $0.2\,\mathrm{mL}$ each of $\mathrm{0.05\: M \: \ce{Ag+}}$ ions and $\mathrm{ 0.05\: M \: \ce{Pb^{2+}}}$ ions were added into... |
Let $f\in L^1(\mathbb R)$ s.t. $$\int_{\mathbb R}|f|=0\implies f=0\ a.e.$$
My attempt
Suppose $f$ continuous and that there is $y$ s.t. $|f(y)|\neq 0$. In particular, there is $\delta >0$ s.t. $f(x)\neq 0$ for all $x\in [y-\delta ,y+\delta ]$. By continuity, there is $m>0$ s.t. $|f(x)|\geq m$ for all $x\in [y-\delta ,y... |
This looks extremely easy, but then again it's late at night...
Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be
transcendental over $k$ if and only if every polynomial $P\in k\left[X\right]$ (with $X$ being an indeterminate) such that $P\left(a\right)=0$ must satisfy $P=0$.
L... |
A friend of mine asked me a question, which I considered trivial at first, but after a while gave rise to some doubts.
For instance, we have a potential well in 1 dimension defined by $$ V(x)= \begin{cases} +\infty &\text{if}& x<0 \text{ and } x>L\\ 0 &\text{if} &0\leq x\leq L \end{cases} $$ We know the wave function t... |
Under a scale transformation $$t\rightarrow \bar{t}=\mu t\hspace{0.3cm}\text{and}\hspace{0.3cm}\textbf{r}\to\bar{\textbf{r}}=\lambda\textbf{r},\tag{1}$$ Newton's law take the form $$m\frac{d^2\textbf{r}}{dt^2}=\textbf{F}\Rightarrow m\frac{d^2\bar{\textbf{r}}}{d\bar{t}^2}=\frac{\lambda}{\mu^2}\textbf{F}.\tag{2}$$ which ... |
I would like to ask a question about random walk. Campbell, Lo & Mackinlay defined the random walk, in the following way (RW3):
$$ cov[f(r_{t}),g(r_{t+k})]=0,\qquad k\neq0 $$
for all $f(\cdot)$ and $g(\cdot)$, where $f(\cdot)$ and $g(\cdot)$ are linear functions, and $\{r_{t}\}$ is a series of returns. So, the question... |
You can look at your problem in a different way. Look at the definition of norm , and then you can define your norm as:
$ \| x \|_{\mathcal{B}} = \sqrt{\sum_i \| x \|_{B_i}^2} = \sqrt{\sum_i \left( x^T,B_ix \right)^2} $
where $\mathcal{B} = \{B_i\}_i$ is the set of matrices $B_i$ that you have. You can do it if your ma... |
Getting all (complex) solutions of a non polynomial equation
Hi !
I was used to solve the following equation with Mathematica. \begin{equation} \alpha_1 + \alpha_2x + \alpha_3x^2 + x^4 + \frac{\alpha_4}{x^2-\alpha_0} + \frac{\alpha_5 x^2}{x^2-\alpha_0}=0 \end{equation} where $\alpha_i$ are constants.
The Mathematica fu... |
How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [commutative] ring and a simplicial [commutative] ring?
One could say many things about this, and I hope you get many replies! H... |
I am trying to vertical and horizontal align items in a table. I tried to follow what I found on the internet, including on this site, but it seems not to work for the last column. Here is what I have.
\begin{table}[!h]\begin{center}\begin{tabular}{|c||c|c|}\hlineDoping & $\alpha_b$ $\left(\textup{cm}^{-1}\right)$ & $\... |
CAD & 3D Printing
Category Archives: 3D
September 5, 2019 – 1:26 pm
On a recent trip I picked up a art-glass marble for my wife. She liked it so much she bought a nice lighted display stand for it. The stand wasn’t designed to display that particular marble though, so it didn’t work too well.
This was a problem easily ... |
I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve？Many thanks in advance!
McKean and Moll have written the nice book
Elliptic Curves: Function Theory, Geometry, Arithmetic that cleanly illustrates the connection between elliptic curves... |
The
Birch and Swinnerton-Dyer conjecture is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing and the first non-zero Taylor series coefficient of the L-function associated to an elliptic curve $E$ defined over $\Q$ at the central point $s=1$ to certain arit... |
The first one is expressible in terms of the modified Bessel function of the first kind:
$$1+\cfrac1{2+\cfrac1{3+\cfrac1{4+\cdots}}}=\frac{I_0(2)}{I_1(2)}=1.433127426722\dots$$
The second one, through an equivalence transformation, can be converted into the following form:
$$1+\cfrac1{\frac12+\cfrac1{\frac23+\cfrac1{\f... |
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