Questions tagged [symmetric-functions]

For questions about functions which are symmetric in its arguments.

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3
votes
1answer
127 views

prove that $\forall x \, \, \, h(a+x)=h(a-x) \Longleftrightarrow \mu=0.5$

The question is, prove that $$\forall x \, \, \, h(a+x)=h(a-x) \Longleftrightarrow \mu=0.5$$ where \begin{eqnarray} \label{eqpdf} h(x) &=&(x^2)\times \left\{ \begin{array}{cc} \int_{0.5}^{1}...
3
votes
5answers
123 views

Simplifying $\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}$

Simplify $$\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}\,.$$ I tried very hard but I am not being able to solve it easily I opened up everything and ...
2
votes
0answers
40 views

Jacobi–Trudi Determinants — how to use the Lindström–Gessel–Viennot Lemma to prove the second identity?

I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur ...
0
votes
0answers
32 views

Explicit expression for certain Schur polynomials

I am trying to find explicit expression for Schur polynomials of the form \begin{equation} s_\lambda(1,q,q^2,\dots, q^{d-1},q^{-c} ,q^{d+1}, \dots,q^{N-1})~, \end{equation} with $c\in \mathbb{Z}^+$ ...
1
vote
0answers
11 views

Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
1
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3answers
40 views

Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Consider the symmetric polynomial $$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$ How can we write $P$ as a polynomial in the variables $st,t+s$? The Fundamental theorem of symmetric polynomials implies this is ...
1
vote
1answer
30 views

What are the asymptotics of the $q$-binomial?

I have a rather basic question regarding the $q$-binomial $\begin{bmatrix}N \\ r \end{bmatrix}=\frac{(1-q^N)(1-q^{N-1} ) \dots (1-q^{N-r+1})}{ (1-q)(1-q^2)\dots(1-q^r) }$ as $N$ goes to infinity. ...
2
votes
1answer
83 views

Solving Functional Equation Involving Substitution

Suppose we seek a function of two variables $f(x,y)$ such that $f(x,y) = f(y,x)$ $f(x,y) = f\left(\frac{1+y}{x}, ~ y \right)$ Are there known techniques for approaching such questions? I already ...
0
votes
2answers
58 views

Relation between the roots and coefficient.

Let Let a, b and c be the roots of the equation $$x^3 +3x^2-1=0$$Then what is the value of expression $a^2b+b^2c+c^2a$. I got it done by evaluate the sum and difference of $a^2b+b^2c+c^2a$ and $ab^...
0
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0answers
10 views

How do you show a symmetric function can be factorized into a product of univariate functions?

I have a symmetric function given by $$f(x):=\frac{\det\left[{p+k\choose q+k}^{-1}{}_1F_1\left(\begin{matrix}q+k\\p+k\end{matrix};x_j\right)\right]_{j,k=1}^n}{\det[x_j^{k-1}]_{j,k=1}^n},$$ for some ...
0
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0answers
8 views

Are all odd harmonic functions are half-wave symmteric?

I know that the Fourier series expansion of a half-wave symmetric function contains only odd harmonics. Is the reverse true?
0
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2answers
34 views

Help to give/improve a proof on simple problem of Riemann integration

Let $K$ be a nonnegative, symmetric around zero, real function satisfying $$\int_{-1}^1K(u)du=1.$$ I want to show that $0<\int_0^1 K(u)u^2du$. This is intuitive but I'm struggling to show it. I ...
6
votes
1answer
171 views

Is the minimum of this optimization problem essentially unique?

Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Let $s&...
2
votes
1answer
69 views

Expansion of $(x+y)^n+(x+z)^n+(y+z)^n-x^n-y^n-z^n$ in terms of elementary symmetric polynomials

Consider the symmetric polynomial in $3$ variables $$ f_n(x,y,z)=(x+y)^n + (x+z)^n+(y+z)^n - x^n-y^n-z^n $$ where $n\geq 0$ is an integer. I'm inquiring if there is a closed formula for the ...
1
vote
1answer
27 views

Can Weierstrass's theorem be specialized to symmetric functions and symmetric polynomials?

Weierstrass's theorem says that continuous functions can be uniformly approximated by polynomials. Can one have a similar theorem saying that symmetric functions can be uniformly approximated by ...
15
votes
2answers
242 views

Symmetric functions written in terms of the elementary symmetric polynomials.

[A recent post reminded me of this.] How can we fill in the blanks here: For any _____ function $f(x,y,z)$ of three variables that is symmetric in the three variables, there is a _____ function $...
0
votes
0answers
17 views

Logistic function symmetric (?) and $\tanh$ asymetric

From Modern Multivariate Statistical Techniques (Izenman), Exercise 10.2: "Show that the logistic function is symmetric, whereas the tanh function is asymetric." In the exercise immediately prior, ...
0
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0answers
10 views

Diagonal root of symmetric multilinear map

Let $F$ be a symmetric multilinear form from $(\mathbb{C}^n)^n$ to $\mathbb{C}$ with real coefficients (these are polynomials). By symmetric I mean that one may permute the arguments $$ \forall \sigma ...
0
votes
1answer
28 views

If $f(x,y)$ is symmetric, do the left and right derivatives need to be equal at $f(a,a)$ (in absolute value)

For $a$ in the domain, and the function being defined at $a$. And by $f_1$ I mean the derivative of the function w.r.t to its first argument (so $f_1(x,y)$ means the derivative w.r.t the first ...
1
vote
1answer
61 views

Suggestions on solving a system of equations

How do you suggest solving the following sytem? $$\begin{cases} \dfrac{12}{x}+\dfrac{12}{y}=1 \\ \dfrac{6}{x-2}+\dfrac{8}{y-6}=\dfrac{2}{3} \end{cases}$$ After simplifying the equations, I got $xy-...
0
votes
0answers
25 views

Finding kernel and image of integral operator over space of symmetric functions.

Let $\mathcal{F}_n$ be a function space over $\mathbb{R}^n$ containing suitably well-behaved real- (or maybe complex-) valued functions (e.g functions of rapid decrease or something weaker, maybe ...
1
vote
1answer
70 views

Irreducible Quartic equation.

Consider the set of equations $$\tag{A} x^{\frac 1 2}+y = 11 $$ $$\tag{B} x +y^{\frac 1 2} = 7 $$ With mere inspection and guessing, the Solution Set is ${4,9}$ . However I cannot not find a method ...
1
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0answers
22 views

Translating Operations on Solutions to that of Equations

Let $a, b$ be two algebraic numbers, $f(x), g(x)$ their minimal polynomials over $\mathbb{Q}$ respectively. Suppose the set of roots of $f$ is $$A = \{a=a_1, a_2, \cdots, a_n\}$$ and the set of roots ...
0
votes
1answer
77 views

non-singular Toeplitz submatrices

Let $p(x) = (1+x+x^2)^d$ for $d\ge 2$ and call its coefficients $$ p(x) = a_0 + a_1x+ a_2x^2 + \dots + a_{2d} x^{2d}. $$ Let $T(d)$ be the infinite upper triangular and Toeplitz matrix defined as $$ T(...
0
votes
1answer
73 views

what symmetry property is related to $f(x)+f(1/x)=1$?

enter image description herewe need to figure out the symmetry related to $$𝑓(𝑥)+𝑓(1/x)=1$$ function? What symmetry should $ f(x)+f(1/x)=1$ follow? My teacher gave me this problem and I'm quite ...
2
votes
2answers
43 views

A simple zeroes polynomial which constructs another one

Let $P(X) = (X-x_1)\ldots(X-x_N)$ be a complex polynomial with simple roots. I define $$Q(X) = P(X-a)-bP(X),$$ with $a\in\mathbb{C}$ and $b\neq 1$ so that $Q(X)$ is also a polynomial of degree $N$. ...
2
votes
1answer
60 views

Monotonicity of quotients of elementary symmetric polynomials

Let $n\in\mathbb{N}$. Let $\sigma_k$ ($0\leq k\leq n$) be the elementary symmetric polynomials: \begin{align} \sigma_0(x_1,\ldots,x_n):&=1 \\ \sigma_k(x_1,\ldots,x_n):&=\sum_{1\leq i_1<\...
1
vote
4answers
115 views

If $a+b+c = 4, a^2+b^2+c^2=7, a^3+b^3+c^3=28$ find $a^4+b^4+c^4$ and $a^5+b^5+c^5$ [closed]

I have tried to solve it but cannot find any approach which would lead me to the answer
2
votes
2answers
92 views

System of equation doesn't want to get solved

I've been trying to solve the following system of equations for hours, to no avail: $$(1)\;\frac{x}{y}+\frac{y}{x}=\frac{10}{3}\\\\$$ $$(2)\;x^2+y^2=8$$ Can anyone give me a hint?
1
vote
0answers
53 views

Continuously Differentiable Functions of Dodecahedron Symmetry

Which continuously differentiable functions $F(x,y,z)$ with closed form satisfy Dodecahedron Symmetry? Dodecahedron Symmetry xoy plane, $F(x,y,z) = F(x,y,-z)$ origin, $F(x,y,z) = F(-x,-y,-z)$ ...
-1
votes
1answer
114 views

Given numbers $a,b,c\geqq0$ and $-\frac{2}{11}\leqq k\leqq0$. Prove that $(k+1)^{6}(a+b+c)^{2}(\!ab+bc+ca\!)^{2}-81\prod\limits_{sym}(ka+b)\geqq0$ .

Problem. Given three numbers $a, b, c\geqq 0$ and $k= constant$ so that $- \dfrac{2}{11}\leqq k\leqq 0$. Prove that : $$(\!k+ 1\!)^{6}(\!a+ b+ c\!)^{2}(\!ab+ bc+ ca\!)^{2}\!- 81(ka+ b)(kb+ a)(kb+ c)(...
1
vote
1answer
112 views

Decomposing a symmetric function into elementary symmetric polynomials. [duplicate]

It is stated that any symmetric function can be expressed in terms of the elementary symmetric polynomials. I am trying to do that for the following generating function: \begin{equation} \prod_{1 \...
0
votes
1answer
46 views

Symmetric Functions on Infinite dimensions

Suppose $X = [0,1]^\infty$. Given $x\in X$ denote by $x(i,j)$ for $i<j$ by the element obtained by swapping the components $x_i$ with $x_j$ in $x$. A function $f:X\rightarrow\Bbb R$ is symmetric if ...
2
votes
1answer
98 views

Find all ordered pairs $(x,y)$ that satisfy both $\frac{3x-4y}{xy} = -8$ and $\frac{2x+7y}{xy} = 43$

The following is listed under the "multiple variable" category of my Algebra I homework. Find all ordered pairs $(x,y)$ that satisfy both $\frac{3x-4y}{xy} = -8$ and $\frac{2x+7y}{xy} = 43$ I can'...
3
votes
1answer
66 views

Solve a system of non-linear equations

How should I go around solving this system of non-linear equations? $$x+\frac{1}{y} =2 \frac{1}{3}$$ $$y+\frac{1}{z}=2\frac{3}{4}$$ $$z+\frac{1}{x}=-3\frac{1}{2}$$ I managed to solve it using ...
0
votes
2answers
48 views

Solve the following system of equations - (6).

Solve the following system of equations $$\large \left \{ \begin{aligned} x^2 + y^2 &= 8\\ \sqrt[2018]x - \sqrt[2018]y = (\sqrt[2019]y - \sqrt[2019]x)&(xy + x + y + 2020)\end{aligned} \right.$$...
0
votes
0answers
51 views

inner product in the algebra of shifted symmetric function, $\Lambda^*$

The ring of symmetric functions $\Lambda$ is defined as the ring of formal power series of bounded degree in coefficients $X_1,X_2...$ which are invariant under permutation of the coefficients. Given ...
0
votes
1answer
105 views

Why $\prod_i (1+x_i)$ is not considered a symmetric function, $\prod_i (1+x_i) \notin \Lambda$

The tittle of this question is the main question. Why it is necessary to not consider these kind of functions in $\Lambda$. My teacher told me that if one consider these functions to be in the ring ...
0
votes
1answer
44 views

non linear system of equations leading to quartic

The question asks to solve the system $$\begin{cases} \sqrt x + y = 16 \\ \sqrt y + x = 25 \end{cases}$$ Substitution leads to a fourth degree polynomial. Is there n easier way to solve it ? ...
3
votes
0answers
58 views

symmetry of partial differential equations (Heat equation)

Morning everyone, I am doing some problem sheets for my class in Partial differential equations where we dont have an actual textbook. we are given a pack on notes. I am having an issue discerning ...
3
votes
0answers
79 views

skew Schur identity

Let $\lambda$ be a partition of size at least 2, and let $n>0$ be an integer. Prove that $$s_{\lambda/(1)}.h_{n}=\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{...
1
vote
0answers
13 views

sign of a Symmetric bilinear form

given $\alpha_i > 0 $, $\beta_i \geq 1$ with $i=1$ or $2$ $R_i(t)=e^{-\alpha_i t^{\beta_i}}$ is Weibull survival function $\phi_i(t)=\int_{t}^{\infty}R_i(u)du$ $f(x,y)=R_1(x)R_1(y)\phi_2(x+y)$ ...
2
votes
0answers
23 views

If $E$ is a vector space, is there an established terminology for functions $f:E\times E\to\mathbb R$ whose values $f(x,y)$ only depend on $y-x$?

Let $E$ be a $\mathbb R$-vector space and $f:E\times E\to\mathbb R$ with $$f(x,y):=g(y-x)\;\;\;\text{for all }x,y\in E.$$ Is there an established terminology for such a function $f$? (I'm aware of ...
3
votes
1answer
137 views

Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law?

Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law on the interval $[-1,1]$ ? It is a lot of work to check on associativity imo. Maybe there is a shortcut around checking ...
0
votes
1answer
24 views

Where is a mistake equating $x$ derived from both equations of a system?

I know how to solve this, but why is the below reasoning wrong and leads to a mistake (I don't see any mistake!) Step 1: From first equation $x=\dfrac{8}{y}$ , and $y$ is not zero Step 2: From ...
2
votes
1answer
73 views

Properties of a step function

Consider the step function $\Delta: \mathbb{R}\rightarrow [0,1]$ $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $\lambda\equiv (\lambda_1,...,\lambda_J)$ is a ...
1
vote
1answer
111 views

shifted symmetric polynomials

BACKGROUND When defining shifted symmetric polynomials we do it in the following way: Let $\mu=(\mu_1,..., \mu_n)$ be a partition with length less or equal to $n$. Then $$s_{\mu}^*(x_1,...,x_n)=\...
1
vote
1answer
211 views

filtered algebra vs graded algebra

BACKGROUND When reading Okounkov-Olshanski's paper about shifted symmetric functions, they define $\Lambda^*$ to be the algebra of shifted symmetric functions. They say that $\Lambda_n^*$ is a ...
1
vote
2answers
34 views

skew Schur functions $C^{\lambda}_{\mu, \nu}$

When working with skew Schur functions, they can be defined as follows. Let $C^{\lambda}_{\mu, \nu}$ be the integers such that $$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$ ...
4
votes
3answers
163 views

$\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)

This question is related to this other question. When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...

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