Questions tagged [symmetric-functions]
For questions about functions which are symmetric in their arguments.
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General form of symmetric function of 3 dimensional vectors
In the paper Deep Sets by Zaheer et al, they prove a theorem (eq 18 in appendix) that states that any general scalar symmetric function of $M$ variables can be written as
$$f(x_1, x_2, \dots, x_M) = \...
- 11
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Regarding the number of variables in Symmetric Functions
I'm studying Symmetric Functions and I came across a doubt that could be considered stupid but I need clarifications.
In the course I'm following we introduced symmetric functions as formal series of ...
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Exploiting symmetry of 4d integral
In my research, I have come across the following integral in four variables.
$$I=32\int_0^\infty f(x,y,w,z) dx dy dz dw$$
The function $f$ is invariant under permutation of any of the variables and ...
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Concluding a radial weak solution with radial test functions is a weak solution with all test functions
Let $B \subset \mathbb{R}^N$ be the unit ball and $E = \{u \in H^1_0(B) : u \text{ is radial}\}$. Define the functional $I : E \to \mathbb{R}$ by
$$
I(u) = \int_{B} |\nabla u(x)|^2 dx - \frac{1}{p}\...
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Largest normal distribution under a curve
I was wondering if I could have some insights on the following problem.
Suppose I have a probability distribution $F$, I want an algorithm that finds the largest normal distribution under $F$.
Say we ...
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3
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Building the tensor product of multiple algebras in sage?
I want to build $\Lambda\otimes\Lambda$ in Sage, where $\Lambda$ is the algebra of symmetric functions. You can build the algebra of symmetric functions in the Schur bases with SymmetricFunctions(QQ)....
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Defining Schwars rearrangement for other kind of symmetric domains
Given a Lebesgue measurable subset $E \subset \mathbb{R}^N$ we denote its $N-$dimensional Lebesgue measure
by $|E|$. Let $\Omega \subset \mathbb{R}^N$ a bounded measurable set and $u : \Omega \to \...
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Can $\csc(x)\csc(x + y)\sin(y)+\cot(y)$ be rewritten to make its symmetry in $x$ and $y$ more obvious?
I have a function $f(x, y) = \csc(x) \csc(x + y) \sin(y) + \cot(y)$, and I know from graphing it that the function is symmetric in $x$ and $y$ such that $f(x, y) = f(y, x)$. However, that fact isn't ...
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Formula for the chromatic symmetric function of a graph in terms of the graph's chromatic polynomial?
I know the chromatic symmetric function simplifies to the chromatic polynomial when 1's and 0's are subbed in for the x's. I was wondering if one could easily find the chromatic symmetric of a graph ...
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A proof of Pólya-Szegő inequality
Denote by $|A|$ the $N-$dimensional Lebesgue measure of a Borel set $A \subset \mathbb{R}^N$ and define
$$
A^\ast := B_{R}(0), \quad R = \left(\frac{N}{\omega_N}|A| \right)^{\frac{1}{N}},
$$
where $\...
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Prove that $n$ hypersurfaces with cyclic permutations of the last $n-1$ variables only intersect if all the variables have the same value.
Suppose that in $\mathbb{R}^n$, I have an equation $F(x,y,z,\dots,n)=0$ with the property that it is symmetric in the last $(n-1)$ variables $\{y,z,\dots,n\}$, but not in the first variable $x$. For ...
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Equality of two symmetric maps
Let $V$ and $W$ be two real vector spaces. I would like to show that two symmetric and multi-linear functions
$$\alpha_1,\alpha_2:V^n\to W$$
are equal if and only if
$$\forall v\in V:\alpha_1(v)=\...
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Why are the eigenfunctions of the Laplacian on a region with spherically symmetric boundary condition are not spherically symmetric?
For example the eigenfunctions of a spherically symmetric membrane can be found in https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane.
Then again I sometimes see people in Physics saying ...
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Algebraic integers: Explicit proof that $\prod_{i,j} (x^2+\alpha_i x+\beta_j)$ (with $\alpha_i$ and $\beta_j$ conjugates) has integer coefficients.
An algebraic integer is a complex number that is the root of a monic polynomial with integer coefficients. In Exercise 3 of Chapter 6 of Ireland and Rosen's book on number theory, we are asked to show ...
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How to symmetrize this vector expression?
In the context of a physics calculation, I have encountered an integral of the form:
$$\int d^3k_1d^3k_2F(\vec{k_1},\vec{k_2})$$
The notes that I'm reading tell me that I need to symmetrize the ...
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