"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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Intuition on the necessity of the Lipschitz condition and a physical example of an ODE

The Picard-Lindelöf theorem states that the initial value problem $$ y'(x) = F(x,y(x)), \ y(x_0) = y_0$$ will always have a unique solution on some closed interval containing $x_0$ assuming that the ...
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12 views

Integrate a multivariable funtion w.r. to one variable or make triple improper integral

I am using Matlab 2010Ra and I want to do triple integration on my function below: My function (first Fock state): $$ \psi(x) = \frac{1}{\pi^{\frac{1}{4}}} e^{-\frac{x^2}{2}} $$ In fact I have a ...
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1answer
14 views

inhomogeneous heat equation with mixed boundary conditons

Solve $$U_{t}=U_{xx}+u$$ with mixed boundary conditions $$U_x(0,t)=0, U(l,t)=0$$ and initial condition $$U(x,0)=\varphi(x)$$ I know that I have to use separation of variables and I have an idea of ...
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Levi civita symbol identity with n dimension

There is an identity $\displaystyle{\epsilon_{i_1...i_k i_{k+1}...i_n}\epsilon_{i_1...i_kj_{k+1}...j_n} =k!\epsilon_{i_{k+1}...i_n }}$ in wikipedia. https://en.wikipedia.org/wiki/Levi-Civita_symbol ...
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Solution for the inhomogeneous 3D heat equation with initial temperature distribution

Can anyone describe the general solution for the inhomogeneous 3-dimensional heat equation: $u_t = K\nabla^2u + \frac{1}{c\rho}f$, with initial condition $u(x, 0) = g(x)$, no boundary conditions. ...
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16 views

solve the initial value problem on the half line for the diffusion equation $U_x(t,0)=\sin t$ [on hold]

solve $U_t-U_{xx}=0$ for the half line with initial conditions: $$\quad Ux(t,0)=\sin t\\ U(0,x)=x$$
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22 views

Action variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian who doesn't depend on the new coordinates $w_k$, but only in the momenta,...
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1answer
39 views

Easier solution to first order non-linear differential equation?

Im am dealing with this differential equation: $$m\frac{dv}{dt}=mg-kv^2$$ where $m,g,k$ are constants. I am able to solve this by treating this as a separable differential equation, but that method ...
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81 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
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1answer
17 views

How to derive a logarithmic potential from Newtonian?

Suppose we believe that the formula for Newtonian potential in $R^3$ is correct: $\varphi(\bar{x}) = \frac{1}{|x|} = \frac{1}{r}$, disregarding the constant. What is the justification of the fact ...
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Trace identities for $\text{SO}(n)$

The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of $\text{SO}(n)$ of dimension $n$ and $n(n - 1)/2$ ...
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1answer
36 views

Solutions to Sturm-Liouville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Sturm-Liouville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
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0answers
11 views

Can Fluctuation-Dissipation Theorem Apply to Magnetic Forces in Multi-Spin Systems [closed]

Let's say I have multiple spin systems (atoms in a protein) in a solution of water and the spin systems are all producing a magnetic field $\mathrm{B_{loc}}$ that affects nearby spin systems. Will the ...
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5 views

Proof of Exponential Decay Pattern of Time Correlation Functions for $\mathrm{B_{loc}(t)}$ in NMR Spectroscopy

For a given protein, I know that the NMR Spectroscopy magnet generates a field $\mathrm{B_o}$ and that the interactions with the spins in the local environment generates a much smaller field $\mathrm{...
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3answers
339 views

A limit in a Feynman “proof” about Fermat's Theorem.

As perhaps some of you already know, Richard P. Feynman, the famous physicist tried a non-orthodox (in his usual way, I suppose) proof of the Fermat's Last Theorem. He tried a probabilistic "proof" ...
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17 views

Riesz measure in potential theory

I am studying Riesz measures associated to superharmonic funcions, following a book by Doob: Potential theory and its Probabilistic Counterpart. On page 51, the following theorem is introduced: If $u$...
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1answer
102 views

Complementary text for mathematical Quantum Mechanics lectures

I'm looking for a text to complement Frederic Schuller's lectures on QM. His approach is very mathematical -- in fact it looks like the first 12 of 21 lectures are just about the mathematical ...
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37 views

Mathematical Definition of Entropy and a Question about the Nature of Stat. Mechanics Approach

I have been studying for quite some time now about entropic functionals, including Boltzmann-Gibbs, Renyi, Kaniadakis and Tsallis, and I am familiar with the properties that a functional has to ...
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1answer
30 views

Infinite propagation speed for the Schrödinger equation

I've seen many articles making reference to the property of the infinite propagation speed for the solution of the linear Schrödinger equation; but i can't find a book giving a 'good' definition or a ...
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0answers
16 views

Maxwell–Boltzmann distribution average speed and second-order moment

Someone can give me a link where I can find the solution step by step of the following integrals. Otherwise,if there is someone so kind to solve them. $$dN(v)=4\pi N(\frac{m}{2\pi kT})^{\frac{3}{2}}...
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0answers
18 views

Number of states in microcanonical ensemble

for the non-physicists, all you need to know to answer my question is that I'm talking about a $6N$ dimensional space of the coordinates $\{\vec{q}_i,\vec{p}_i\}_{i=1} ^{N}$ which I call the phase ...
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1answer
26 views

Taylor expansion of Crystal Field potentials

I am trying to work through Michael Tinkham's "Group Theory and Quantum Mechanics". In discussing crystal field theory he uses the following example: We start with an atom at the origin. We want to ...
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2answers
508 views

Proof that the following function is a polynomial

I've been trying to get my head around this problem for a long time, yet I have not been able to make much progress. Let $\ell_0(j) = \left\lfloor \frac{1}{2}\left( \sqrt{8j^2 - 8j + 1} + 2j - 1 \...
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17 views

How to investigate the relationship between range and payload?

I am interested in learning about the relationship between range and payload for an electric aircraft. How do I use math to investigate the relationship between range and payload for an electric ...
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10 views

Direction of arrival and distance to source

Suppose we have $3$ receivers $a,b,c$ and one wave source which are all located on the same $XY$ plane and we do not know the position of only $c$ on the plane. Assume a plane wave is transmitted $s(...
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48 views

Overview of Geometric analysis [closed]

Can anyone tell me what geometric analysis is about? After reading some articles I have a view that it uses PDE extensively for geometric problems. Am I right in this point? Also what kind of ...
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1answer
259 views

Can different choices of regulator assign different values to the same divergent series?

Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = ...
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505 views

A space more fundamental than Euclidean space

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which ...
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68 views

Understanding twisted differential forms

I'm trying to understand twisted differential forms. I do know that they are like regular differential forms but under coordinate transformations they pick up an extra factor of the sign of the ...
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0answers
50 views

Operator that kill the wave function

I have the following function in $x$. $\sum_{d=0}^{\infty} \frac{1}{\hbar^d}\frac{1}{d!}\left(\prod_{i=1}^{d-1}(1+i\hbar)^{m}\right)x^d$ I need a differential operator involving $(x,\frac{d}{dx},h,...
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1answer
77 views

Classifying continuous maps from closed 2-manifolds to various closed manifolds

I believe my question should be simple. The question is more physically oriented and originated from one of Witten's papers, "On Holomorphic Factorization of WZW and Coset Models", where he considered ...
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54 views

How can I understand the step by step calculations for the formula from the blog below?

I am studying clustering and found a useful article on the blog post here Finding the K in K-Means. But I am having difficulty in understanding the formulas below and how I can do step by step ...
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Cauchy horizon of a future Cauchy hypersurface

I'm studing on the book Semi-Riemannian geometry by O'Neil. I'm tryng to understand the proof of the Hawking's singularity theorem (theorem 55A in the book). What I don't understand is why if $S$ ...
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Biorthogonality of vectors

This question is equal parts math and physics, though I chose to ask it here because I am more concerned with the mathematics behind it, rather than physical implications. Let $\hat{K}$ be a non-...
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Convergence of a Sequence of Continuous Functions of Bounded Operators

Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, $\{A_n\}_n$ a sequence of self-adjoint operators in $\mathcal{B}\left(\mathcal{H}\right)$ (the bounded linear operators on $\mathcal{H}...
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0answers
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What is KL-Divergence? Why Do I need it? How do I use it?

I am currently studying KL Divergence. But It seems very confusing that I don't maybe understand why do I ever need it and what is that for? As I have been reading stuff about Mutual Information, it ...
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22 views

Why is this operator self-adjoint (or is it)?

I am reading literature on self-adjoint extensions of Hamiltonians (particle interaction) and I came across the following statement (in context of separating total momentum $P$): Operator $H$ ...
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Can somebody help me understand the formulas in the image below?

I got the image from this web site. The site talks about how to determine optimum number of clusters. I understand the first two but having a hard time to understand the last two. And what does "a ...
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1answer
81 views

Is the Entropy a Function or a Functional? [duplicate]

As in the title, I was wondering whether the entropy of a system (it can be any entropy, from Boltzmann to Renyi etc, it is of no importance) is a function or a functional and why? Since it is mostly ...
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2answers
21 views

When does $\frac{\partial}{\partial t}\int_0^x f(x') dx'=x \frac{\partial f}{\partial t}$?

Under what conditions do the following relation holds? $$\frac{\partial}{\partial t}\int_0^x f(x') dx'=x \frac{\partial f}{\partial t}$$ Should it be stated that $f(0)=0$? Let's say that I know the ...
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Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
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19 views

Factoring out a variable from an unknown multivariable function

I have a data set that follows the behavior of a function f that depends on a lot of different variables. Let's call two of those variables $a$ and $b$. The specific behavior I'm interested in is $f(a)...
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0answers
56 views

Future Space Opportunities for a Mathematician [closed]

I don't know if this question should be asked here or on "Mathematics Educators", however I'll post it here for the moment. I've just finished my first year of Mathematics and I do really like maths. ...
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1answer
41 views

Integral of Bessel Functions Multiplying “polynomials”

How can I compute the following integral: $$\int_{0}^{1} (1 - x^{2})^{\nu - \mu - 1} x^{\mu + 1} J_{\mu}(\alpha_{\nu}x) dx$$ where $\nu > \mu \geq 1$ and $J_{\nu }(\alpha_{\nu}) = 0$. The $J_{\nu}$ ...
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1answer
53 views

shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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40 views

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
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2answers
181 views

What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition? [closed]

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
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0answers
26 views

Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
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1answer
50 views

A Question Regarding Stone's Formula

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$ A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
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1answer
30 views

Can someone solve this in order to provide an example for solving central force problems?

Let $F(x)=x$ be a function that describes the magnitude and direction of a force that varies with distance from the origin. I understand that $m \frac {d^2} {dt^2} p(t) = F(p(t))$ is used to derive ...