For questions about functions which are symmetric in its arguments.

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0answers
33 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 ...
0
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0answers
33 views

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

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)? $X,Y$ ...
2
votes
2answers
49 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 ...
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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
votes
1answer
43 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 $\lambda \vdash n$ ...
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4answers
114 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
33 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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2answers
37 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
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 ...
1
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1answer
22 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) = ...
0
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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
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, ...
5
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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*} ...
0
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1answer
48 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
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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
67 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 ...
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1answer
15 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 $θ \in \Bbb R$. ...
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0answers
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} } ...
3
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1answer
93 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
51 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
130 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
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1answer
137 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
39 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
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 $n$ ...
2
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3answers
89 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
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0answers
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 = ...
0
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3answers
81 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
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1answer
131 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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0answers
52 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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0answers
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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0answers
96 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 ...
1
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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
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1answer
18 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
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1answer
84 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
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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 $$ ...
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2answers
275 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
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0answers
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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0answers
75 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
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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} + ...
3
votes
1answer
123 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
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1answer
61 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
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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 ...
1
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0answers
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 ...
4
votes
1answer
89 views

Properties of the 'forgotten' symmetric polynomials

In I.G. Mcdonald "Symmetric Functions and Hall Polynomials" pg.22, the forgotten symmetric functions $f$ are introduced very briefly as the result of applying an involution $\omega$ to the monomial ...
0
votes
1answer
55 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 ...
1
vote
1answer
89 views

On constructing symmetric positive definite kernels as sums of Gaussians

Let $\mathcal{X}$ be a non-empty set. We define $k\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ as a symmetric positive definite kernel. For instance, the Gaussian Radial Basis Function (RBF) is ...
1
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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 ...
1
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2answers
54 views

Designing a symmetric function

I need to design an analytical function that looks like this (See figure bellow). The idea is to control the angles "a" at the beginning and at the end. If the function depends on x (any kind of ...