For questions about functions which are symmetric in its arguments.

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38 views

Good Reason for Partitions Indexing Symmetric Functions?

I'm mostly unfamiliar with the study of symmetric functions. However, it's my understanding that: We are interested in, as a basic object, the vector spaces $\Lambda_n$ of symmetric polynomials in ...
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1answer
125 views

Proof of Newton Girard formula symmetric polynomials

Newton Girard formula states that for $k>2$: \begin{equation} p_k=p_{k-1}e_1-p_{k-2}e_2+\cdots +(-1)^{k}p_1e_{k-1}+(-1)^{k+1}ke_{k} \end{equation} where $e_i$ are elementary symmetric functions and ...
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45 views

Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$). Can we prove that the following problems are equivalent: $$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
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1answer
44 views

Solve the integral equation with symmetric kernel

I have the following integral equation with symmetric kernel $$g(x)=\cos \pi x +\lambda \int_{0}^{1} k(x,t)g(t)\,dt $$ where $k(x,t)$ is a symmetric kernel given by $$k(x,t)= \begin{cases} (x+1)t, ...
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136 views

A good equation system

Given $a,b,c$ positive numbers, solve the system $\sqrt{xy}+\sqrt{xz}-x=a$, $\sqrt{yz}+\sqrt{yx}-y=b$ and $\sqrt{zx}+\sqrt{zy}-z=c$, where $x,y,z\in \mathbb{R}$. This only a pretty question. I did ...
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1answer
52 views

Littlewoods First Principle problem

I am able to prove the first principle. Let $E$ be a measurable set of finite outer measure. Then for each $\epsilon > 0$, there is a finite disjoint collection of open intervals $I_k$ for which if ...
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32 views

Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural ...
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460 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
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137 views

Schur skew functions

Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If $s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$ ...
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21 views

A formula for the symmetric function $\sum_{i\neq j} \frac{z_i z_j}{(z_i-z_j)^2}$

In the course of an optimization problem, I encountered this expression $$S = \sum_{i\neq j} \frac{z_i z_j}{(z_i-z_j)^2},$$ where $z_1$, ... , $z_n$ are the roots of a polynomial $f(t)$ of degree $n$. ...
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47 views

A “nice” orthogonal basis for translation invariant symmetric polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
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37 views

Coefficients of Lagrange resolvent

I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$: \begin{alignat*}{2} p(X) &= ...
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69 views

Cauchy Identity for a specialized product of Schur polynomials

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from ...
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78 views

Roots of the derivative as symmetric (?) functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
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49 views

Symmetrizing a sequence of vectors

Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...
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141 views

Elementary symmetrical polynomial equations, whose solutions are known to be natural numbers.

Let $n_1,n_2,\dots,n_k$ be natural numbers (excluding 0), and for each $1\leq i\leq k$ let $\sigma_i(n_1,n_2,\dots,n_k)$ be the elementary symmetrical polynomial consisting of the sum of all products ...
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53 views

A Combinational identity using permutations

For a distribution {$p_1,p_2, …,p_m$}, with $p_i>0$ and$\sum_1^m{p_i}=1$ , let $J$ be a subset of size $j$, and $m>j\geq1$. It holds that: $$\int_0^1\prod_{i \in J} (x^{-p_i}-1) dx = ...
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97 views

Expansions of symmetric polynomials in terms of Jack symmetric polynomials

I was wondering if someone could help me with some Jack polynomial calculations. (I use the notation of I.G. Macdonald's book "Symmetric Functions and Hall Polynomials") Those of you familiar with ...
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90 views

Reference request on symmetric polynomials

Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,\ldots,x_n$ (and recall that $e_k=0$ if $k>n$). I know very little about these polynomials. I've just noticed this odd ...
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23 views

Is the following function symmetric?

I am reading the paper Hierarchical Clustering of a Mixture Model by Goldberger et al. On the page 2 of this paper they define the following function: $ d\big(\mathcal N(\mathbb\mu_1, ...
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45 views

System of equations problem

This two equations. How we can solve them? $$ \begin{cases} \frac{1}{\sqrt{x+1} +1} +\frac{1}{\sqrt{y+1} +1} =\frac{2}{3} \\ \sqrt{\frac{1}{x^{2}} +\frac{1}{y} } +\sqrt{\frac{1}{y^{2}} +\frac{1}{x} } ...
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50 views

finding the shortest distance of a hermitian matrix to a set of hermitian matricies with specific eigenvalues 2-norm

The title is more general, and all that I require is to show an inequality that I already have verified using random matrices in matlab. Let $\lambda_1 \leq ... \leq \lambda$ and $\mu_1 \leq ... \leq ...
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51 views

Empirical distribution generates exchangeable $\sigma$-algebra

I have a problem understanding the following statement from Klenke, p. 234: If we write $\Xi_n(\omega) := \xi_n \bigl(X(\omega)\bigr) = \frac{1}{n} \sum^n_{i=1} \delta_{X_i(\omega)}$ for the ...
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95 views

$\sigma$-algebra of events invariant under permutations

Let $X = (X_n)_{n\in\mathbb{N}}$ be a stochastic process with values in $E$. For $n \in \mathbb{N}$, define $$\mathcal{E}'_n := \sigma\bigl(F : F : E^\mathbb{N} \rightarrow \mathbb{R} \text{ ...
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95 views

Representation of symmetric functions

Let $n \in \mathbb{N}$. Show that every symmetric function $f\colon E^n \rightarrow \mathbb{R}$ can be written in the form $f(x) = g\Bigl(\frac{1}{n}\sum_{i=1}^n \delta_{x_i} \Bigr)$, where $g$ has ...
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74 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
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58 views

How do I solve these four simultaneous equations?

I have been trying hard to solve these equations. There are four equations in total: $$ \begin{align*} px^{p-1} + qx^{q-1} \lambda &= 0 \\ py^{p-1} + qy^{q-1} \lambda &= 0 \\ pz^{p-1} + ...
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56 views

Minimums of symmetric functions

In this problem minimize a function using AM-GM inequality we discussed about the minimum points of a simmetric function. Now I would like to ask to you this: If you have a rational function ...
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0answers
65 views

Dimension of the Image of Young Projectors corresponding to Tensor factors.

Suppose I define the action of the symmetric group on abstract tensors as shuffling indices. I know this is very naive. I apologise, I am a physicist and working on a problem that involves tensors ...
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50 views

Identity with symmetric rational functions

I am trying to prove this identity between rational functions involving symmetrization among variables. Let us consider a set of variables $\{p_1,\ldots,p_n\}$, which I indicate globally as ...
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0answers
210 views

Improper integral of odd function

I'm a student. In a recent assignment I was asked to find the mean of a Student's t multivariate distribution (which should be $\overline\mu$). I've divided the integral required to find the expected ...
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0answers
23 views

Power-sums analogue of Burgers equations

Let $D$ be a domain in $\mathbb C^2$. Let $f_1$, $\ldots$, $f_n$ be a family of holomorphic functions in $D$, satisfying Burgers-type equations: $$ \frac{\partial f_k}{\partial z_2} - f_k ...
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136 views

Image of the Sylvester matrix is the degree of the GCD

Let $P_k(F)$ denote the $F$-vector space of (univariate) polynomials of degree $\leq n$. Letting $F$ be a field lets everything be monic, but it seems sufficient to consider a ring $R$ such that the ...
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113 views

Probabilistic results on the elementary symmetric polynomials

The elementary symmetric polynomials of degree $k$ in $N$ variables are defined as $$e_k(x_1, \ldots, x_N) = \sum_{(i_1,\ldots,i_N) \in I_k^N}{x_1^{i_1}\ldots x_N^{i_N}}, \quad 0 \le k \le N$$ with ...
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0answers
29 views

When can $Tr(X^TAY)\geq 0$ when $A$ is positive semidefinite and $X,Y \neq 0$?

Under what conditions on $X,Y$ is $Tr(X^TAY)\geq 0$ when $A$ is square positive semidefinite but not the Identity matrix and $X,Y \neq \mathbb{0}$ (as in not equal to all matrix of all zeros)? $X,Y$ ...
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25 views

A simple PDE related to symmetric functions

This question was posed on mathoverflow but was put on hold. So far nobody has been able to give any hint either to the solution or to why it is trivial. Disclaimer: as far as I can tell this is not ...
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40 views

Integrals of a nonnegative, symmetric, periodic, continuous function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be such that $f$ is nonnegative, symmetric, $\ell$-periodic in both variables, zero exactly on the diagonal of its domain, and continuous. Specifically, $f$ has the ...