"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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Complex analysis, cutoff integration

The diff-invariant distance between $z'$ and $z$ is (for short distances) $e^{w(z)}|z'-z|$, so a diff-invaraint cutoff would be at $|z'-z|=\epsilon e^{-w(z)}$. Then $ \int ...
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Formally evaluating integral to calculate electric or gravitational field.

I never understood how such integrals are calculated, formally. In a line is easy, just a line integral. In a surface, sometimes is easy, like in a disc. But, some surfaces, like sphere, it gets ...
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2answers
46 views

Initial value of Newton Raphson Method

I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it. The questions that I have are ...
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13 views

How the Jacobian is connected to the movement of particle from one domain to another? [on hold]

I am dealing with the proof of Reynold-Transport Theorem. There the Jacobian is used for the changing position of particles from one domain to another. Can anyone help me to understand what does ...
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2answers
41 views

Solving a differential equation numerically to plot particle path

I'm trying to plot the evolution of a particle in an accretion disk by solving the equation $$2X\frac{\partial X}{\partial\tau}=V_R(X,\tau)$$ where I have found $V_R$ numerically to be ...
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Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
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2answers
32 views

Calculate initial speed to launch the cat at specific spot

BACKGROUND: I’m trying to create a game where cat jumps from platform to platform, but as any other cat this furry devil won’t do the things I’m asking for. I want the cat to jump and land at the ...
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1answer
38 views

Angular momentum of an accretion disk

I need to plot the time evolution of the total angular momentum in an accretion disk. This confuses me because I thought this should be constant, since angular momentum has to be conserved? I'm given ...
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2answers
40 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
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83 views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
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115 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
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10 views

Examples of quasilinear wave equations

Consider a quasilinear wave equation equation of the form $\sum g^{ij}(u, Du)\partial_i\partial_j u = F(u, Du)$ on $R \times R^n$ subject to initial data $u(0,x)=g, \; \partial_t u(0,x)=h.$ Given ...
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56 views

What do the equations on this gate mean or relate to?

These equations are on gates to the John Dalton building in Manchester UK: $$X \geq Y \text{ iff } \not\exists \, x_r \leq Y \text{ & } X \leq Y_L $$ $$X = Y \text{ iff } X \leq Y \text{ & } Y ...
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15 views

Conformal transformation

The problem is following. This is an Exerciese of Polchinski $2.6$ (explanation about conformal field transformation) Consider the flat Euclidean metric $\delta_{ab}$ in $d$ dimensions. An ...
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27 views

Applications of matrices – CAT scans [closed]

Question 1 From the diagrams below, explain why one or two x-rays are insufficient in finding the location or number of tumours accurately, remembering that the x-ray source moves from left to right ...
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2answers
244 views

Calculating the linear output force a threaded rod and nut would produce based on the input torque

I would like to calculate the amount of linear thrust a threaded rod combined with a rotationally static nut would be able to produce based on the rotary force applied to the rod. The information ...
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1answer
62 views

Simple Harmonic Motion DE

The simple harmonic motion DE is $x''(t)=-x(t)$ I solved it as a homogenous linear equation and after inserting the Euler's formula into the equation, my solution becomes ...
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29 views

Lecture notes on holomorphic Yang-Mills theory

Some time ago I've found these lecture notes on the gauge theory. In particular, in these lecture notes the author introduces and studies the Yang-Mills equations in the case of real bundles and ...
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1answer
26 views

Trigonometry in projectile motion

I initially posted this question on Physics SE but got no responses probably because it's more related to maths than physics. A plane surface makes an angle $\bf X$ with the horizontal. From the ...
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61 views

My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why? [closed]

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it ...
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2answers
80 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
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15 views

Gauge covariant derivative on principal bundle over $\mathbb R^d$

I try to understand the physical definition of covariant derivative in gauge theories in terms of the exterior covariant derivative of vector-valued forms defined as the horizontal projection wrt a ...
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In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
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Compatibility between the Lax, Hamilton and St. Petersburg formalisms in terms of the conserved quantities.

In classical integrable models, in the discrete case we have the Lax equation discrete $\frac{dL_{n}}{dt} = M_{n+1}L_{n}-L_{n}M_{n}$ the Sklyanin algebra, $\lbrace T_{a}(u),T_{b}(v)\rbrace = ...
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1answer
37 views

When do isometries commute with the compatible derivative operator on a semi-Riemannian manifold?

Let $M$ and $\tilde{M}$ be smooth manifolds, each with a metric $g_{ab}$ and $\tilde{g}_{ab}$, assumed here to be smooth symmetric invertible tensor fields, which are non-degenerate but not ...
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1answer
96 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
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2answers
52 views

Decomposition of a positive semidefinite self-adjoint operator?

If I have a positive semi-definite self-adjoint operator $H:D(H) \rightarrow L^2$, is it true that there is always a decomposition $H=A^* A$ available? If this is true, what can we say about the ...
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1answer
37 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
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1answer
813 views

Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first ...
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33 views

Subject to in equation

I have the formula below: $$\hat x_2=\arg\min\lVert x\rVert_2\quad\text{subject to}\quad A{x}=y.$$ But I didn't understand what was meant by "subject to" ? does $x$ is replaced by $x_2$? please can ...
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1answer
47 views

Solving Simplified Hamilton's Equation

I have a question on a project that I am working on. I have included a large amount of the background information so that all relevant information is included, however the question is as follows (it ...
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0answers
34 views

Equivalent descriptions of “flat space with non Euclidean metric” and “curved space with (local) Euclidean metric”: the case of Minkowski space.

FIRST: I start with the guiding idea: 1. we have the surface of a paraboloid (z = x2 + y2); its metric, in an infinitesimal neighbourhood of one of its points is (we can choose it) EUCLIDEAN; now, ...
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1answer
37 views

Math formulas on Clustering

I am currently studying Clustering in Machine Learning. I have found a document regarding guessing the right number of clusters. I am reading the first part of it, having difficulties in understanding ...
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1answer
34 views

Reference request: what is the relation between classical r-matrices and quantum R-matrices?

I learned from a professor that $$ R=Id+(q-1)r+ o(q-1), $$ where $R$ is a quantum $R$-matrix and $r$ is the corresponding classical $r$-matrix. Here $o(q-1)$ denotes a term of the form $A(q-1)^2$, ...
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Time of flight for a projectile with stated initial velocity and size at various distances [migrated]

I've been mooching round the internet looking for an answer to this one and can't find a ready resource, hence the question. In short, I'm trying to discover the flight time of a shotgun pellet at ...
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2answers
125 views

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
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1answer
56 views

Using metric to raise and lower indices

Everything I read on tensors makes it clear that using the metric matrix $g_{ab}$ and its inverse $g^{ab}$ to respectively lower and raise indices of a tensor is very important. As far as I know (and ...
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1answer
30 views

Rope question - integration

A 50-lb bucket is at the bottom of a 100-ft well. A 200 lb rope (also 100 ft long) is tied securely to the bucket. We will use rope to lift this bucket out of the wall, at a rate of 1 foot every ...
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1answer
310 views

cylinder-ray intersections equation

Can You Pleas Help with this one I found an article http://www.mrl.nyu.edu/~dzorin/rendering/lectures/lecture3/lecture3.pdf for Infinite cylinder-ray intersections And I don't know how they develop ...
3
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1answer
47 views

Is kinetic energy a positive definite quadratic form?

Recall (Arnold, Mathematical methods of classical mechanis, 4.19, B) Definition. Let $M$ be a riemannian manifold. The quadratic form on each tangent space $$ T = \frac{1}{2} \langle v, v ...
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2answers
172 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
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1answer
391 views

Lagrangian equations on Double Pendulum (Potential and kinetic Energy of double pendulum)

I have to prove the following equations using the Lagrangian equations, The figure shows the image, I know how to do lagrangian. I just don't know how to solve the kinetic and potential energies of ...
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2answers
96 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
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0answers
16 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
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1answer
45 views

The definition of scalar and vector concomitant of a metric

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: ...
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1answer
46 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
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Conformal group in two dimensions

In Conformal field theory, physicist says, the conformal group in two dimensions is infinite dimensional, so the associated with the infinity of generators and infinity conserved charges provided. Is ...
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1answer
27 views

Question about Logistic Regression - 2

How should I tell the difference between those two formulas in the circles below. I am studying logistic regression and I have faced two different formulas from two different documents. I don't know ...
4
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2answers
93 views

Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
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28 views

Properties of functional integration

this question comes from theoretical Physics, the issue being the so called Path Integral. The measure of this thing is something written as $[d\phi]=\prod_x d\phi(x)$ And this should be the limit ...