# Tagged Questions

For questions about functions which are symmetric in its arguments.

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### Proof of Newton Girard formula symmetric polynomials

Newton Girard formula states that for $k>2$: $$p_k=p_{k-1}e_1-p_{k-2}e_2+\cdots +(-1)^{k}p_1e_{k-1}+(-1)^{k+1}ke_{k}$$ where $e_i$ are elementary symmetric functions and ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 \\ ... \\ ...
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### 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 ...
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### 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.) ...
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### 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 $f(x,y)\neq f(y,x)$...
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 ...