# Tagged Questions

"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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\mathcal{H}... 0answers 22 views ### Biorthogonality of vectors This question is equal parts math and physics, though I chose to ask it here because I am more concerned with the mathematics behind it, rather than physical implications. Let$\hat{K}$be a non-... 0answers 15 views ### What is KL-Divergence? Why Do I need it? How do I use it? I am currently studying KL Divergence. But It seems very confusing that I don't maybe understand why do I ever need it and what is that for? As I have been reading stuff about Mutual Information, it ... 0answers 22 views ### Why is this operator self-adjoint (or is it)? I am reading literature on self-adjoint extensions of Hamiltonians (particle interaction) and I came across the following statement (in context of separating total momentum$P$): Operator$H$... 1answer 53 views ### shifting integration variable and taking derivative seemingly giving problem I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ... 1answer 81 views ### Is the Entropy a Function or a Functional? [duplicate] As in the title, I was wondering whether the entropy of a system (it can be any entropy, from Boltzmann to Renyi etc, it is of no importance) is a function or a functional and why? Since it is mostly ... 0answers 25 views ### Can somebody help me understand the formulas in the image below? I got the image from this web site. The site talks about how to determine optimum number of clusters. I understand the first two but having a hard time to understand the last two. And what does "a ... 2answers 21 views ### When does$\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 ... 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 ... 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$\...
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 ...