"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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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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1answer
27 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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9 views

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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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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2answers
506 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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496 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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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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3answers
337 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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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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2answers
75 views

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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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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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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0answers
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
100 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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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 ...
12
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1answer
258 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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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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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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2answers
428 views

Centre of Mass and Moment of Inertia of a sphere - spherical cap

I have been given a sphere of radius a, from this sphere a cap of hight h is cut off. 1) What is the centre of mass of the rest of the sphere? 2) What is the moment of inertia regarding the axis of ...
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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 ...
3
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1answer
553 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
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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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0answers
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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0answers
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 ...
3
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0answers
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 ...
2
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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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1answer
641 views

Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
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15 views

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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33 views

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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22 views

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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0answers
15 views

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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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 ...
5
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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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25 views

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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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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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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3answers
464 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\...
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22 views

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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1answer
118 views

proof of$\frac{\partial^2 f(x,y)}{\partial x\partial y}$=$\frac{\partial^2 f(x,y)}{\partial y\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\...
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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. ...
2
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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
56 views

Proving an integral relation (isotropic function)

In Hansen-McDonald's book Theory of Simple Liquids the following relation is often used: We want to evaluate the integral $$\int_V f(\vec r_1, \vec r_2) d \vec r_1$$ We observe that if the function ...