Questions tagged [physics]
For questions about mathematical problems arising from physics, the natural science studying general properties of matter, radiation and energy.
50 questions with no upvoted or accepted answers
17
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Approximation of the effective resistance on Cayley graph
Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
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9
votes
0
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371
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Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?
First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion).
I am telling this since as it is, the title contains a meaningful question, but it ...
- 18.4k
8
votes
0
answers
1k
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Is there any physics theory which is similar to these analogies?
Since I am doing this little "research" project on my spare time and in my physical neighborhood there are not many people to discuss these ideas, I wanted to share with you a small point of ...
- 3,064
7
votes
0
answers
125
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Charge distribution of closed surfaces
Consider a closed surface $\Sigma$ which bounds a solid $\Omega$ in ${\mathbb R}^3$. Assume some electric charges, say totally $Q$, is distributed on $\Sigma$ and reaches an "equilibrium" state. In ...
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5
votes
0
answers
126
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Particles sent into the same direction with uniformly distributed speed
Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters ...
- 48.9k
5
votes
0
answers
240
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Does Dijkgraaf-Witten theory have a time-reversal symmetry?
By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...
- 3,001
5
votes
0
answers
274
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$S$-matrix in QED in 2d space-time
I am not completely sure that this question is appropriate for this site, but I have asked a similar question here https://physics.stackexchange.com/questions/271372/s-matrix-in-qed-in-2d-space-time ...
- 21.8k
4
votes
0
answers
116
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Superspace derivation of supersymmetric non-linear sigma model in Supersolutions by Deligne and Freed
I am having a little trouble understanding passage from the linear to the non-linear sigma model in Section 4.1 of Supersolutions by Deligne and Freed. Most of my confusion comes down to the ...
user482049
4
votes
0
answers
164
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List of Replica Symmetry results for different models?
Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have?
I am aware of some of the more famous results, e....
- 435
4
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0
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219
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Why do we care about simplicity of the spectrum in Oseledets' theorem?
Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana).
The simplicity of the spectrum has been studied ...
- 1,805
4
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0
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334
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Unusual generalization of the law of large numbers
I have seen in physical literature
an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
- 21.8k
4
votes
0
answers
373
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Bosonic String Theory
I would like clarification of 26 dimensional Bosonic String Theory. A definition would be, that this is free bosons compactified on a torus and orbifolded by a 2-point reflection group (or ...
- 515
3
votes
0
answers
214
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How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
From numerical experiments in Mathematica, I have found the following expression for the integral:
$$
\int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
- 31
3
votes
0
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108
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Are square configurations the only critical points of the energy on the circle?
$\newcommand{\S}{\mathbb{S}^1}$
$\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$
Define $E:M \to \mathbb{R}$ by
$$E(x_1,x_2,...
- 6,741
3
votes
0
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166
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geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold
It's my first post.
Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...
- 31
3
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0
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159
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Does there exist a compactly supported integrable function with infinite Coulomb energy?
The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that
$$
E[f] = \iint\limits_{\Omega\...
- 401
3
votes
0
answers
133
views
Partial Liouville equation
In my master's thesis, I worked on mathematical multi-scale models for muscle tissue. Now after finishing it, I would like to find out if one direction could be a research topic for my PhD.
At one ...
3
votes
0
answers
94
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Multiplicativity of $\zeta$-function regularized determinant
Let $A$ be a selfadjoint elliptic differential operator on a compact manifold. In mathematical physics and differential topology one often defines its determinant using the $\zeta$-function ...
- 21.8k
3
votes
0
answers
103
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Is there any connection between Lagrange points and the icosahedron?
Given the Newtonian two-body problem, one can ask if there are any orbits that allow a test particle to maintain a fixed configuration relative to the two bodies. In other words, in a frame that ...
- 1,444
3
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0
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145
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What is the relationship between complex time singularities and UV fixed points?
In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
- 418
3
votes
0
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804
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Children's drawings and Seiberg-Witten curves
This physics (bear with me for a while) paper seems to say something about Gal \bar Q/Q:
Children's Drawings From Seiberg-Witten Curves, hep-th/061108.
Let's ...
- 15.1k
2
votes
0
answers
74
views
Approximate solutions to $x''(t)=-cx + f(t)x$
I recently studied a problem which involved two particles joined by a harmonic spring moving in a potential and through some manipulation, I obtained the equation
$x''(t) = -\omega^2x + f(t)x$,
where $...
- 3,738
2
votes
0
answers
171
views
Is there an example Hamiltonian that is uncomputable?
In a paper from 2015 Toby S. Cubitt et al showed that the problem of determining the existence of a band gap in the excitation spectrum of a quantum many-body system, was undecidable. This result ...
- 123
2
votes
0
answers
263
views
What is the relationship between Riemannian and sympletic musical isomorphisms on the cotangent bundle?
Let $M$ be a smooth manifold. Its cotangent bundle naturally has a symplectic structure, and this gives rise to musical isomorphisms. These musical isomorphisms are the ones from physics that relate ...
- 557
2
votes
0
answers
37
views
Difference Between Total Least Squares Plane and Plane Orthogonal to Principal Axis of Inertia Tensor
Given a finite set $P$ of points in $\mathbb{E}^3$ , one can calculate an approximating plane either as the solution of a Total Least Squares problem or by interpreting the problem physically, ...
- 13.2k
2
votes
0
answers
131
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Questions about using mathematical methods to prove the Caratheodory's Concept of Temperature
Caratheodory's Concept of Temperature is not Carathéodory's theorem.
I have tried,but I found nothing about this question by searching online.
This is what I have seen in a thermodynamics textbook; ...
- 21
2
votes
0
answers
99
views
1D Schrödinger Equation with Measure-Valued Coefficients
I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
- 21
2
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0
answers
103
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Is this correct: Inflection points of Euler number graph in Island-Mainland transition correspond to spanning cluster site percolation threshold?
I'm writing with respect to the paper Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice.
Here's a link to a PDF ...
user119567
2
votes
0
answers
192
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Diffusion equation on mixing of diffusing particles
I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diffusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...
2
votes
0
answers
115
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The condition of maximality in branching rules of $SO$ group representations
Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
- 1,893
1
vote
0
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65
views
Some details about relationship between central charges and second cohomology group of the Lie algebra
S. Weinberg in his book "The quantum theory of fields" talks about central charge that appear in Lie algebra of a given Lie group. To be more precise, on page 83 in the book, he computes the ...
- 287
1
vote
0
answers
204
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Are causally isomorphic spacetimes Wick-related?
Take the time-orientable spacetimes $(M_1,g_1)$ and $(M_2,g_2)$ that are locally(to be clarified below) Wick-related and both are globally Wick-rotatable(to be clarified below) to the same Riemannian ...
- 612
1
vote
0
answers
228
views
Is the topological dimension of spacetime fixed for causally isomorphic spacetimes?
Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are not homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
The Lorentzian metrics $g_1$ and $...
- 612
1
vote
0
answers
170
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Order isomorphism + manifold homeomorphism => path topology homeomorphism?
Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
Let's call this map $\phi: (M_1, \...
- 612
1
vote
0
answers
80
views
Biot-Savart-like integral for a toroidal helix
The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...
- 255
1
vote
0
answers
93
views
How to smoothly interpolate gravitational field between trajectories in high dimension?
I'm looking for the adequate numerical interpolation technique to solve the following problem. This is probably trivial for physicists who study gravitational fields, but I didn't find clear answers ...
- 19
1
vote
0
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142
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A question about Roger Penrose's spin networks and mathematical formalization?
Let $a,b,c$ be "units" in the spin network.
Then there are there are the following three requirements to fulfill (according to the relevant Wikipedia entry):
$a,b,c \in \mathbb{N}$
Triangle ...
- 3,064
1
vote
0
answers
183
views
polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:
$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
- 11
1
vote
0
answers
75
views
Is there an analytic formula (or even a name...) for a plane curve with curvature inversely proportional to x?
I'm interested in plane curves with curvature inversely proportional to distance from the axis:
$$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x(...
1
vote
0
answers
84
views
Diffraction across an absorbing wall
I need help finding the procedure for the solution of the following differential equation.
This is equation is: Find $u:\mathbb{R}^2 \to \mathbb{C}$ such that for $C>0$
$ \begin{cases} u_{xx}+ ...
1
vote
0
answers
501
views
Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143
0
votes
0
answers
100
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I'm looking for the NLab page on particle species
This is just a reference request.
I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently.
If someone can point ...
- 2,369
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votes
0
answers
97
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The specific connection between the Hecke operator and the t'Hooft Operator
As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...
0
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0
answers
117
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Cardinal Invariants and Physics
There are many applications of topology to physics, but I wonder if there is a known application of cardinal invariants to physics.
0
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0
answers
77
views
Energy-minimizing set of discrete points in a bounded domain
Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain.
Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize
$$
\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}
$$
over all ...
- 401
0
votes
0
answers
92
views
Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
- 141
0
votes
0
answers
3k
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What is a self-consistent equation in percolation theory
I was reading papers about percolation theory in which I was confused by the expression "self-consistent equation", for example in Temporal percolation in activity-driven networks. I read some ...
- 211
0
votes
0
answers
127
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Geometric interpretation of table with permutations and inversions
Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg.
$n=1,...,6$, $k=1,....
0
votes
0
answers
238
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Simple question on the foundations of spin foam formalism
To make it simple, take the spin foam formalism of ($SU(2)$) 3D gravity. My question is about the choice of the data that will replace the (smoothly defined) fields $e$ (the triad) and $\omega$ (the ...
- 733
-1
votes
1
answer
437
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Harmonic function in infinite domain in $\mathbb{R}^3$, constant on the boundary and decaying as $1/r^2$
EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\...
- 21.8k