For questions about functions which are symmetric in its arguments.

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0
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1answer
13 views

Help with a Hypothesized Inequality with symmetric functions

I am trying prove the following inequality: Suppose $f(x)$ is a symmetric distribution about 0 (e.g. standard normal distribution), then: $\int f^2(x)dx \geq \int f(x-a)f(x+a)dx$ for any real $a$. ...
0
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0answers
19 views

How to solve such symmetric equations

I have the following symmetric system: $$f_i = \sum_j^n \tau_{ij}c_j^{-1}y_j$$ $$c_j = \sum_i^n \tau_{ij}f_i^{-1}y_i$$ $$\tau_{ij}=\tau_{ji}$$ $$F = \left[ \begin{matrix} f_1 \\ ... \\ ...
0
votes
1answer
35 views

Minima of symmetric polynomials subject to two symmetric constraints

The homogeneous symmetric polynomial of degree $k$ in $n$ variables is $$ f_k(x_1,x_2,\dots,x_n) = \sum_{i_1<i_2<\cdots<i_k}x_{i_1}x_{i_2}\cdots x_{i_k}. $$ Consider the following ...
2
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0answers
23 views

Symmetrized monomials under Weyl group?

Consider a given partition $\lambda=(\lambda_1,\lambda_2,...,\lambda_N)$ and start with the monomial $$z_1^{\lambda_1}z_2^{\lambda_2}...z_N^{\lambda_N}$$ in $N$ variables $z_1,z_2,...,z_N$. Now we ...
0
votes
1answer
58 views

How could I find an orthogonal basis of this bilinear form f?

Where $f : \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ corresponding to the quadratic form $q : \mathbb{R}^3 \rightarrow \mathbb{R}$, $q(x,y,z) = x^2 + 2xy + y^2 + 2yz + z^2$ I found ...
0
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0answers
11 views

Problem on Principal Component Analysis (P.C.A.)

Let $X \; = \; (X_1, X_2, \ldots, X_m)^T$ and $Y \; = \; (Y_1, Y_2, \ldots, Y_n)^T$. Let, $S$ = pooled variance-covariance matrix obtained from $X$ and $Y$. Let, $\alpha$ = principal component ...
0
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0answers
7 views

Proving that largest root (obtained via P.C.A.) is a symmetric function

Suppose, we are given $\textbf{X} = (X_1, X_2, \ldots,X_m)$ and $\textbf{Y} = (Y_1, Y_2, \ldots, Y_n)$. Also, we are given, S = pooled variance. If we implement Principal Component Analysis (P.C.A.) ...
0
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0answers
36 views

What can be said about $f$ in $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$ if $g$ is known?

Suppose you have the equailty $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$, $x,y\in\mathbb{R}$, and $g$ is known. What non-trivial facts about $f$ can be deduced from this? (Assume $f(x,y)\neq f(y,x)$...
2
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2answers
51 views

Calculate the value of the following integral: $ \int^{\frac{\pi}{2}}_0 \frac{\sin^n(x)}{\sin^n(x) + \cos^n(x)} $ [duplicate]

The question is show that $$ \int_0^{a} f(x) dx = \int_0^{a} f(a-x) dx $$ Hence or otherwise, calculate the value of the following integral $$ \int^{\frac{\pi}{2}}_0 \frac{\sin^n(x)...
0
votes
1answer
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, ...
2
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1answer
51 views

nth power symmetric polynomial in terms of Schurs polynomial

The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
1
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4answers
116 views

Factorize $(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$

I am unable to factorize this over $\mathbb{Z}:$ $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(y+z-x)(z+x-y)-8x^2y^2z^2$$ Since, this from an exercise of a book (E. J. Barbeau, polynomials) it must have a neat ...
6
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0answers
35 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 ...
1
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2answers
40 views

How to predict symmerty of parametric curve

Suppose we are given a curve as $$x^{2/3} + y^{2/3} = 1 $$ In parametric form it can be written as $$x=\cos^{3}(\theta)$$ and $$y=\sin^{3}(\theta)$$ now how can we predict if curve will be symmetric ...
0
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0answers
26 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 ...
1
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1answer
23 views

Maximizing symmetric functions on the unit cube

Let $f:[0,1]^n \to \mathbb{R}$. We say that $f$ is symmetric, if for every permutation $\sigma \in S_n$ and every $(x_1,..,x_n) \in [0,1]^n$ we have that $$f(x_1,..,x_n) = f(x_{\sigma(1)},...,x_{\...
0
votes
1answer
29 views

How to decide if $Q(\underline u,\underline v) $ on $\mathbb R^2$ is inner product or not?

How to decide if $Q(\underline u,\underline v) $ on $\mathbb R^2$ is inner product or not if $$ (\underline u, \underline v) = \underline u^T A\underline v$$ where $$A = \begin{pmatrix} -1 & -...
1
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0answers
25 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, \Sigma_1),\;\...
5
votes
1answer
168 views

A symmetric function

While working on a research problem on fuzzy metric spaces, I came across a special symmetric function $F_n:X^n\times (0,\infty)\to [0,1]$ i.e. \begin{equation*} F_n(x_1,x_2,\dots,x_n,t)=F_n(x_{\pi(1)...
0
votes
1answer
53 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, ...
2
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2answers
109 views

Mathematics Olympiad Question $a+b+c=7$, …

Given $a+b+c=7$ and $\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} = 0.7$, need to find $\frac{c}{a+b} + \frac{a}{b+c} + \frac{b}{a+c}$. I have noted that these two differ by a factor of $10$. So I ...
4
votes
0answers
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 $n$. ...
0
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0answers
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 ...
0
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2answers
75 views

A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
0
votes
1answer
16 views

Symmetries of a function imply certain properties of its Fourier coefficients

Exercise from Fourier Analysis: An Introduction by Stein and Shakarchi: Let $f$ be a $2\pi$-periodic Riemann integrable function defined on $\Bbb R$ with $f(θ + π) = f(θ)$ for all $θ \in \Bbb R$. ...
1
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0answers
46 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} } ...
3
votes
1answer
95 views

A symmetric system of nonlinear equations - how to solve?

So, I was adviced to ask a new question on my problem (as the first one wasn't very precise), that is to solve the system of equations: $$\begin{cases} x\cdot y=6 \\ x^y+y^x=17 \end{cases}$$ where: ...
1
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0answers
52 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 ...
2
votes
2answers
140 views

How to tell if a function has rotational symmetry? [closed]

How to spot rotational symmetry on(or in, of?) a function? If I have the function $f(x)={{5a^2+6ax+9x^2}\over {a+3x}}$, how can I know it has rotational symmetry about the point $(-{a\over 3},0)$? Is ...
0
votes
1answer
138 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 ...
1
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1answer
42 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 $...
3
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0answers
50 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 $n$ ...
2
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3answers
91 views

Solve the non-linear system of equations

For real $x,y,z>0$ solve the system of equation \begin{cases} \dfrac{1}{x}-3 y+4 z=5,\\ \dfrac{1}{y}-4 z+5 x=3,\\ \dfrac{1}{z}-5 x+3 y=4, \end{cases} It is easy to check out that $$ x =\...
2
votes
0answers
67 views

A Combinatorial 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 {$p_1,p_2, …,p_m$}, size $j$, and $m>j\geq1$. It holds that: $$\int_0^1\prod_{p_i \in J} (x^{-...
0
votes
3answers
83 views

Symmetric functions and roots of polynomials

$f = x^3-\frac{1}{2}x^2+1$ and their roots $a,b,c$. I want to find polynomial of degree 3 with roots $a^4,b^4,c^4$. I know that i need express $e_i(a^4,b^4,c^4)$ in terms of $e_i(a,b,c)$ those ...
1
vote
1answer
143 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 ...
1
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0answers
56 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 $n$-...
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0answers
97 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{ is ...
1
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0answers
97 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 to ...
1
vote
1answer
32 views

Conditions for global max of symmetric function to lie on diagonal

Assume $f:[0,1] \times [0,1]$ is symmetric, i.e. $f(x,y) = f(y,x) \;\;\forall x,y \in [0,1]$. Assume further that $f$ is smooth, and that for every $x \in [0,1]$ the map $\phi_{x}(y):=f(x,y)$ attains ...
0
votes
1answer
22 views

How do you define symmetry of a multidimensional function?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$. Then $f$ is symmetric if $f(x)=f(-x)$. How do you define symmetry for the function $f:\mathbb{R}^n \rightarrow \mathbb{R}$? for $f:\mathbb{R}^n \rightarrow \...
2
votes
1answer
85 views

(representation theoretic) meaning of sum over even rows of a Young tableau

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define $\...
4
votes
1answer
79 views

Can the natural proof of this algebraic identity be simplified?

Let $x^4+c_3x^3+c_2x^2+c_1x+c_0$ be a real polynomial with no real root. Then there are two pairs of conjugate complex roots, $a_1\pm b_1 i$ and $a_2\pm b_2 i$, and one has the identity $$ c_1^2-...
1
vote
2answers
292 views

Combinatorial definition of Hall–Littlewood polynomials (sum over SSYT?)

Hall–Littlewood polynomials $P_\lambda(x;t)$ is an important deformation of Schur polynomials forming a basis in the ring of symmetric polynomials over $\mathbb Z[t]$. There are various definitions, ...
3
votes
0answers
39 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) &= (...
1
vote
0answers
75 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ $$+(1\cdot2\cdot3+1\cdot2\cdot4+1\cdot3\...
1
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0answers
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} + qz^{q-...
3
votes
1answer
124 views

How find this real value $x+y+z $ if such this equation

let $x,y,z>0$ and such $$\begin{cases} \dfrac{x}{xy-z^2}=-\dfrac{1}{7}\\ \dfrac{y}{yz-x^2}=\dfrac{2}{5}\\ \dfrac{z}{zx-y^2}=-3 \end{cases}$$ show that: $$x+y+z=6$$
1
vote
1answer
62 views

Find relations on the real number: transitive and/or antisymmetric

$$I\ am\ searching\ for\ a\ relation\ on\ the\ real numbers\ (\mathbb R ),\ which\ sould\ be:$$ antisymmetric and transitive antisymmetric and NOT transitive NOT antisymmetric ,but ...
3
votes
0answers
71 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 $\...