Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elementary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

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22 views

Hawking and Ellis's example of an extendible manifold

In S. Hawking's and G. Ellis's book "The Large-Scale Structure of Space Time", they discuss the notion of inextendible Lorentz manifolds $(M,g)$ (see chapter 3.1). A Lorentz manifold is ...
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36 views

Self-dual $126$ and $126^*$ in SO(10) or Spin(10) Irreps

We are looking at 126 or $126^*$ in SO(10) or Spin(10) Irreps. 126 is known as a complex total "anti-symmetric" and "self-dual" 5-index tensor irreps in Spin(10). This means the ...
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What's the difference between $v = a\cdot t$ and $\vec{v} = \int \vec{a} \, \mathrm dt$

In highschool, I learned $v = at$ and in university, I am learning $\vec{v} = \int \frac{\vec{F}}{m} \, \mathrm dt = \int \vec{a} \, \mathrm dt$. I understand one is for $v= at$ is for one-dimension ...
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Proof for First Null Point in Boresight

I was doing some of the exercises of a textbook about radars and encountered this exercise from the problem section (without answer); check question link or the Question: Two antennas are D distance ...
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1answer
37 views

Domain issues in transformation of the coordinate representation of a function

Start with a Manifold $M$ and define a function $f:M\rightarrow\mathbb{R}$. As usual, pick two charts $(U,x)$ and $(V,y)$ with $p \in U\cap V$ and $x:M \supset U \rightarrow x(U) \subset\mathbb{R}^n$. ...
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31 views

Showing the operator in the Maxwell eigenproblem is self-adjoint

Context I was reading article [1], where they write Maxwell's equations into the eigenproblem, using Dirac notation, $$ \hat A_k|H_k\rangle = (\omega/c)^2|H_k\rangle, $$ where the field $|H(x,t)\...
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60 views

Proof that poles on real axis count as “half” a residue when evaluating real integrals [duplicate]

There is a small formula for finding definite integral by complex analysis methods : If $f(x)$ contains cosine and sine functions along with polynomial functions then $f(x)$ can be treated as a real ...
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27 views

Question on the Schrödinger interpretation on one object being split into two

Suppose that there is an object (i.e. a wavefunction) containing $N+M$ particles, say, a person holding a mobile phone, which may or may not interact each other. Suppose also that the object needs to ...
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61 views

How to find test configurations to compute Futaki invariant?

I'm now reading 1204.2230 and 1512.07213 on K-stability as this is sort of related to superconformal field theory in physics. In these papers, the Hilbert series is used to compute the Futaki ...
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29 views

What is the correct choice of the contour in the case of undamped forced harmonic oscillator?

I am interested in finding the Green's function (GF) for the undamped forced harmonic oscillator equation: $$\Big(\frac{d^2}{dx^2}+\omega_0^2\Big)x(t)=f(t).$$ In order to find the GF, start by define ...
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85 views

Valeurs absolues ( absolute values ) [closed]

Exercice : soient $a, b$ deux réels et $c$ un réel positive montrer que si $|a|\le c$ et $|b|\le c$, Alors $|a+b| + |a-b|\le 2c$ montrer que : Si $|a+b| + |a-b|\le 2c$, alors $|a|\le c$ et $|b|\le c $...
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29 views

Tensor product of exponential operators

My quantum mechanics professor asked to show a demonstration of the following mathematical result: $$e^{X\otimes Y}=e^{X}\otimes e^{Y}$$ When $X$ and $Y$ are some normal operators. But I think that ...
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$c(t)$ geodesic $\Rightarrow$ $c(\alpha t)$ geodesic.

Let $S\subseteq \mathbb{R}^3$ be a regular surface with Riemannian metric $g$. Moreover let $p\in S$, $v\in T_pS$ and $c$ a geodesic with $c(0)=p$ and $c'(0)=v$. I want to show that for any constant $\...
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Finding the radius of curvature of the trajectory of a projectile.

The parabolic trajectory of a projectile has different radius of curvature at different points of time. Is there a way to find R of C for a simple projectile, thrown at an angle θ and initial ...
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42 views

Numerical quadrature for an improper multiple integral

In my numerical analysis course, my professor asked us to evaluate the integral $$2 \int_{0}^{1} \cdots \int_{0}^{1} \prod_{i<j}\left(\frac{u_{i}-u_{j}}{u_{i}+u_{j}}\right)^{2} \frac{d u_{1}}{u_{1}}...
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Dimensions of Physical objects doesn't make sense. [closed]

Can you show me a pure 1D or 2D object? A line and a plane have a thickness. A pure 1 D line is one with no width. But if its width is zero, it doesn't exist. Similarly, a 2D plane is perfectly ...
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39 views

Limit of integral containing $e^{i\omega L(\frac{\pi}{2}-\tau)}$ as $L\to \infty$

In one Physics paper several times one encounters a limit of the following form $$\lim_{L\to \infty}\int_0^\pi e^{\pm i\omega L(\frac{\pi}{2}-\tau)}\int_{S^2} \varepsilon(\hat{x})f(\tau,\hat{x})d\tau ...
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1answer
65 views

Galiliean group vs Poincare group and role of mass

In what follows I'm trying to generalize an approach from Landau & Lifschits Vol 1 -- unfortunately, the authors use it only in Vol 1 discussing Newtonian mechanics. I was trying to apply the same ...
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On Green function of $\nabla^2u(\vec{r})=\rho(\vec{r})= \delta\,(\vec{r}-\vec{r}_1)-\delta\,(\vec{r}-\vec{r}_2)$ in the semiplane $x > 0$ [closed]

Could someone please help me with this problem? I'm not good with math and that problem seems very challenging to me. Using the Green function, solve the following problem in the semiplane $x > 0$:...
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29 views

Constant invariant quantities

What is the intuition behid the constant invariant quantities and why do we need it? How is this related with probability theory? I was solving some maximization problem and when I saw the answer, the ...
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39 views

Who are the current leading researchers in the field of symmetry analysis of differential equations applied to physics? [closed]

I want to study symmetry analysis of differential equations to find exact solutions. I have heard of Ibragimov, Bluman, Hydon and Olver from textbooks. I would like to know which researchers are ...
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Coin tossing experiment : A geometric approach

A coin is tossed at random. We assume that the thickness of the coin is $0$ and in rotation,the vector of the normal applied to the heads side of the coin generates a cone.The axis of the cone makes ...
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the Holographic Principle

https://mathoverflow.net/questions/365765/this-is-topology-solved?noredirect=1#comment923604_365765 Assume extreme competence. The holographic principle, as described below in this question, is ...
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61 views

Ferris wheel Trig Question [duplicate]

Question: Suppose you wanted to model a Ferris wheel using a sine function that took 60 seconds to complete one revolution. The Ferris wheel must start 0.5 m above ground. Provide an equation of such ...
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56 views

Laplacian coupled with another equation over a two-dimensional rectangular region

I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation. The Laplacian is defined over $x\in[0,L], y\in[0,l]$. On manipulating the second equation (which I have ...
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1answer
95 views

A ferris wheel completes 2 revolutions in 30 seconds. Determine how far it has travelled in 15 seconds. The radius of the ferris wheel is 10 m. [closed]

If the Ferris wheel completes two revolutions in $30$ seconds, how many revolutions does the Ferris wheel complete in $15$ seconds? The radius of the Ferris wheel is 10 m. I'm stuck in this question, ...
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2answers
31 views

LHS where the argument of the function isn't explicit stated (vector equation)

The Lorentz force is given as $$ \mathbf F= q\left[\mathbf E(\mathbf r(t),t)+\mathbf v(t)\times \mathbf B(\mathbf r(t),t)\right] \tag 1 $$ where $\mathbf E, \mathbf B:\mathbb R^4\to\mathbb R^3$ are ...
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28 views

When finding the position function $s=f(t)$ from $v(t)$ at $f(0)=0$, why are the bounds of integration $0$ to $t$? [closed]

It comes from this question. A particle moves on a straight line with velocity function $v(t) = \sin \omega t \cos ^2\omega t$. Find its position function $s= f(t)$ if $f(0) = 0$. Thank you!
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26 views

Factorization method for solving differential equations

I want to solve the second order differential equation as follows $\gamma\xi\left(1-\xi x^2\right)W''+\left(-\xi x^2+\beta\gamma\xi^2x+1\right)W'+\beta x\xi W=0$. My suggestion is the factorized ...
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Finding a class $C$ of bipartite PPT states such that entanglement of $\rho \in C$ implies entanglement of $\rho + \rho^{\Gamma}$.

Consider an entangled bipartite quantum state $\rho \in \mathcal{M}_d(\mathbb{C}) \otimes \mathcal{M}_{d'}(\mathbb{C})$ which is positive under partial transposition, i.e., $\rho^\Gamma \geq 0$. As ...
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103 views

solution of $f'(x)+f(-x)=e^{-x^2}$

let $f$ be a function such that $f:\mathbb{R}\to \mathbb{R}$, I want to determine all functions of class $C^1$ such that $f'(x)+f(-x)=e^{-x^2}$ for all $x\in \mathbb{R}$, Now we have $f'(x)+f(-x)=e^{-...
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How I can solve this differential equation: $y'''(x)+ax\,y(x)=0$?

I have done some attempts to solve the following ODE: $$y'''(x)+ax\,y(x)=0\,,$$ with $a >0$. I put $y(x)= e^{rx}$, with $r$ is an arbitrary real number, then $y'''=r^3 e^{rx}$, by substitution in ...
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56 views

How to write in a formal way the division by (dφ)²

Question: Consider a conic surface defined by: \begin{equation} z = h \left( 1-\frac{\rho}{a} \right) , \rho = \sqrt{x²+y²} \end{equation} generated by the revolution of the line \begin{equation} z = ...
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15 views

Approximation by finite subsets and strong resolvent convergence

Let $\mathbb{G}$ be an at most countable set (e.g., $\mathbb{G}=\mathbb{Z}^d$) and $H$ be a self-adjoint operator (not necessarily bounded) on $l^2(\mathbb{G})$. Let $\mathbb{G}_L$ be finite subsets ...
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Equivalence of eigenfunction correlatior

Consider a self-adjoint operator $H$ on $l^2 (\mathbb{Z}^d)$ and define the eigenfunction correlator on any Borel $I\subseteq \mathbb{R}$ as $$ Q(x,y;I) = \sup_\limits{F\in C(\mathbb{R}),||F||\le 1} |\...
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Dirac Notation: How $|f\rangle= \sum_{j=1}^{\infty}|\phi_j\rangle\langle\phi_j|f\rangle$?

In a text, it has been written that a function is $|f\rangle= \sum_{j=1}^{\infty}a_j|\phi_j\rangle$, how one can write that - $ |f\rangle= \sum_{j=1}^{\infty}|\phi_j\rangle\langle\phi_j|f\rangle$, ...
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1answer
75 views

How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality?

Let $\phi$ be a classical scalar field in $1+D$-dimensional spacetime with coordinates $(t,\vec x)$, and consder the equation of motion $$ \newcommand{\pl}{\partial} (\pl_t^2-\nabla^2)\phi+m^2\phi+ g\...
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Continuum limit of $ \sum_{r=1}^\infty \sum_{z=1}^\infty a_r |r z \rangle \langle r z | = \sum_{l=1} b_l |l \rangle \langle l | $?

Background Using bra-ket notation consider the identity (for integer $r$) : $$ \sum_{z=1}^\infty |r z \rangle \langle r z | = I \otimes \underbrace{\begin{bmatrix} 0 & 0 & 0 & 0 \\ ...
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1answer
29 views

Is there a math relation between blurriness and eye power?

In computers science, the main idea behind blurring a photo is to set the RGB value of every pixel to the average of all the values of its neighbours. For more details on what I mean by "blur&...
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51 views

To find the centre of gravity of the volume generated by revolution of the cardioid $r=a(1+\cos \phi)$ about the $x$-axis.

Answer to the above question is $\left(\frac {4a} 5,0\right)$. Also, please explain if there's something wrong with the question since I don't get whether to find CG of volume or surface generated by ...
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1answer
38 views

How can we solve this differential equation $d^nf(x)/dx^n=a$?

The differential equation $$\frac{d^nf(x)}{dx^n}=a.$$ My attempt I firstly solved the simpler version of it like $$\frac{df(x)}{dx}=a,$$$$\frac{d^2f(x)}{dx^2}=a,\ldots .$$ And got a generalistaion as $...
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23 views

Transformation of spatial coordinates

Please note that the transformed quantities will be indicated by $'$. Let me give some context first. The general approximate form of the potential energy $V$ is given to be $$V^{app} = q^T V q \tag 1$...
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116 views

How are spin network edges related to anti-symmetric projectors on the Hilbert space of the fundamental rep of SU(2)?

In the paper here https://arxiv.org/pdf/gr-qc/9905020.pdf we see an introduction to Spin-networks of the original Penrose type i.e an undirected open graph whose edges have labels that are irreducible ...
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1answer
48 views

If a circle and parabola touch each other and also have common root then what's the relationship between their coefficient?

Actually this question is from physics (projectile motion) but i believe its related more to maths here equation of parabola $$y=ax-5x^2-5(ax)^2$$ And that of circle $$x^2+y^2=(a/5(1+a^2))^2$$ where $...
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1answer
58 views

$SO(p,q)$ Fundamental Weights?

The weights in the $D^{n-1}$ and $D^{n}$ spinor representations of $SO(2n)$ are of the form $$\frac{1}{2}(\pm e_1 \pm e_2 \pm ... \pm e_{n-1} \pm e_n)$$ such that the products of all the $\pm 1$'s are ...
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79 views

Dirac Delta from cartesian to polar coordinates

An infinitely long wire carries a constant electric current $I$ along the $z$ axis. Thus, the current density $\mathbf{j}$ of the wire is given by, in cartesian coordinates: $$\mathbf{j}(\mathbf{r})=I\...
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1answer
26 views

Conservation of energy in three dimension

I'm trying to derive the conservation of energy in 3D from the equation $\vec{F}=m\vec{a}$. David Morin, in his book "Introduction to Classical Mechanics With Problems and Solutions" p. 138-...
3
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2answers
53 views

Explanation for behaviour of graph of $y=x^2e^{-x^2}$ (Maxwell-Boltzmann distribution)

Consider the function $$y=x^2e^{-x^2}$$ The graph initially behaves as a parabola then in later part exponential part of it dominates; i.e., the graph looks exponential after maximum of the curve. ...
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1answer
46 views

Real world applications of topological & symbolic dynamical systems and ergodic theory?

I have no background in any of these areas and was wondering whether these topics have any significant applications in applied mathematics and/or probability/statistics? From what I’ve read it seems ...

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