# Tagged Questions

For questions about functions which are symmetric in its arguments.

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 $a$. ...
19 views

45 views

29 views

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 ...
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 $n$. ...
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 ...
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 ...
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 $θ \in \Bbb R$. ...
46 views

67 views

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 ...
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 ...
22 views

79 views

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-... 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$$1answer 62 views ### Find relations on the real number: transitive and/or antisymmetric$$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 ...
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 \$\...