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0
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1answer
27 views

Prove T is symmetric if and only if b=c

Assume T is a linear operator on R^3, that α={(1,1,1),(1,-1,0),(0,1,-1)} is a basis consisting of eigenvectors and that the corresponding eigenvalues of T are real numbers a,b,c. Prove that T is ...
0
votes
0answers
21 views

Identity with symmetric rational functions

I am trying to prove this identity between rational functions involving symmetrization among variables. Let us consider a set of variables $\{p_1,\ldots,p_n\}$, which I indicate globally as ...
2
votes
0answers
38 views

References for chromatic symmetric functions of hypergraphs

Define a hypergraph to be a pair $H = (V,E)$ where $V$ is a set of vertices and $E$ is any set of subsets of $V$ called edges. Thus if every edge $U \in E$ has only two elements, then the hypergraph ...
0
votes
1answer
39 views

Can convolution of two radially symmetric function be radially symmetric?

For example, take $x\in R^3$ and let $f(x)$ and $g(x)$ be radially symmetric. Can we prove that $f\ast g$ is also symmetric?
3
votes
1answer
37 views

What am I missing about Schur functions?

Let's say I only know the following about Schur functions: you give me a partition $\lambda$ of $d$ such that $\lambda$ has $n$ parts $\lambda_1,\ldots,\lambda_n$, and I can compute the Schur function ...
1
vote
0answers
119 views

Improper integral of odd function

I'm a student. In a recent assignment I was asked to find the mean of a Student's t multivariate distribution (which should be $\overline\mu$). I've divided the integral required to find the expected ...
3
votes
0answers
55 views

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 ...
1
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0answers
19 views

Power-sums analogue of Burgers equations

Let $D$ be a domain in $\mathbb C^2$. Let $f_1$, $\ldots$, $f_n$ be a family of holomorphic functions in $D$, satisfying Burgers-type equations: $$ \frac{\partial f_k}{\partial z_2} - f_k ...
1
vote
1answer
107 views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint ...
1
vote
2answers
110 views

find symmetric equation along line $y=-x$

generally i know that to find symmetric equation of function along line $y=x$,we should exchange $x$ and $y$ and solve back,but what about $y=-x$?should i repeat again the same procedure,but ...
2
votes
0answers
45 views

Expansions of symmetric polynomials in terms of Jack symmetric polynomials

I was wondering if someone could help me with some Jack polynomial calculations. (I use the notation of I.G. Macdonald's book "Symmetric Functions and Hall Polynomials") Those of you familiar with ...
1
vote
1answer
40 views

Alternative definition of complete homogeneous symmetric functions

I found this definition of symmetric functions: $g_n=\sum\limits_{i_1\leq i_2\leq ... \leq i_n} x_{i_1}x_{i_2}...x_{i_n}$ where for each integer $j$ at most $t$ of the numbers $i_1,i_2,...$ are equal ...
1
vote
0answers
70 views

Image of the Sylvester matrix is the degree of the GCD

Let $P_k(F)$ denote the $F$-vector space of (univariate) polynomials of degree $\leq n$. Letting $F$ be a field lets everything be monic, but it seems sufficient to consider a ring $R$ such that the ...
1
vote
2answers
77 views

A problem from <<Thinking in Problems>> by Roytvarf, Birkhauser

I got a problem, which turned to be from the book "Thinking in Problems How Mathematicians Find Creative Solutions" by Roytvarf, Chapter One, Jacobi Identities and Related Combinatorial Formulas : ...
-2
votes
2answers
117 views

Help me with this this system of equations

Help me with this system of equations $$a+b = 3 -c$$ $$\frac{1}{a}+\frac{1}{b}= \frac{5}{12}-\frac{1}{c}$$ $$ a^3+b^3 = 45 -c^3$$
1
vote
1answer
132 views

Decomposition of products of monomial symmetric polynomials into sums of them

I'm trying to make sense of the answer given in: this question I am stuck at the phrase 'where the partitions $\gamma$ result from adding, respectively, from $\alpha$ all distinct partitions obtained ...
0
votes
1answer
114 views

Anti-Symmetric Complex Polynomial

Let $f(x_1,...,x_n)$ be a complex polynomial. Show the following two conditions on $f$ are equivalent: i) for any transpositions $\tau$ we have $\tau.f=-f$ and ii) for any $\sigma \in S_n$ we have ...
2
votes
0answers
74 views

Reference request on symmetric polynomials

Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,\ldots,x_n$ (and recall that $e_k=0$ if $k>n$). I know very little about these polynomials. I've just noticed this odd ...
1
vote
0answers
94 views

Probabilistic results on the elementary symmetric polynomials

The elementary symmetric polynomials of degree $k$ in $N$ variables are defined as $$e_k(x_1, \ldots, x_N) = \sum_{(i_1,\ldots,i_N) \in I_k^N}{x_1^{i_1}\ldots x_N^{i_N}}, \quad 0 \le k \le N$$ with ...
5
votes
2answers
144 views

Symmetry of a Plücker function

Let $d \in \mathbb{N}$ and let $I$ be a set. Let $\omega : I^d \times I^d \to \mathbb{R}$ be a function, denoted by $(a_1,\dotsc,a_d,b_1,\dotsc,b_d) \mapsto a_1 \cdots a_d | b_1 \cdots b_d$, with the ...
0
votes
1answer
65 views

Convergence of series of elementary symmetric functions

Let $x_1,x_2,x_3,\ldots$ be an infinite sequence of real numbers (or assume they're complex numbers if you find that convenient). Let $e_0,e_1,e_2,e_3,\ldots$ be the elementary symmetric functions of ...
-2
votes
3answers
113 views

If $f(x,y) = f(y,x)$, does it follow that $x=y$? [closed]

If $f(x,y) = f(y,x)$, does it follow that $x=y$? If yes, please show a proof. If no, please demonstrate a counter-example. Thank you.
1
vote
2answers
43 views

Finding correct symmetry axis

before I ask for anything I must admit I'm working hard to understand this beautiful subject. Thanks in advance. $$ f(x)= 2(x)^2+8x+5 $$ Acoording to the graph of this function, there is a x-axis ...
3
votes
0answers
41 views

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 ...
2
votes
1answer
162 views

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
5
votes
0answers
210 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
9
votes
1answer
193 views

What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?

The product of monomial symmetric polynomials can be expressed as $m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$ for some constants $c_{\lambda\mu}^{\nu}$. In the case of Schur ...
1
vote
1answer
272 views

Doubt with parametric and symmetric equations

In the line through $P(0, 0, 0)$ and is perpendicular to $x=y-5$, $z=2y-3$, when we solve the equations and get the symmetric equations in order to find the vectors $V_1$ and $V_2$, why the normal ...
3
votes
2answers
686 views

Scalar Product for Vector Space of Monomial Symmetric Functions

Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$: $$ P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} . $$ Since ...
2
votes
1answer
147 views

Symmetric polynomials have unique expressions as polynomials in symmetric elementary functions

Let $s_i$ be the symmetric elementary functions. For example, $s_1=x_1+\cdots+x_n$. Suppose a polynomial $p(z_1,\ldots,z_n)\in R[z_1,\ldots,z_n]$ satisfies $p(s_1,\ldots,s_n)=0$ in ...
2
votes
0answers
110 views

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 ...
7
votes
2answers
690 views

Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?

Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via $$ \prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N. $$ How can one get the monomial symmetric functions ...
0
votes
1answer
139 views

Roots of elementary monomials

Let $m_\lambda(X_1,X_2,...X_N)$ be a monomial symmetric function with partition $\lambda$. For example: $$ m_{(3,1,1)}(X_1,X_2,X_3) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $$ Is there a general ...
-1
votes
3answers
463 views

How do I find out the symmetry of a function?

For example, how do I know that with: $$f(x_1,x_2,x_3,x_4)=\frac{x_1 x_2+x_3 x_4-x_2 x_3-x_1 x_4}{x_1 x_2+x_3 x_4-x_1 x_3-x_2 x_4}$$ $f$ has the property: ...
5
votes
0answers
112 views

Schur skew functions

Let $\lambda,\mu,\nu$ be some partitions. Let's denote with $s_\lambda,s_\mu,s_\nu$ the Schur functions associated to these partitions. If $s_\mu s_\nu=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda$ ...
4
votes
2answers
1k views

Sum of cubed roots

I need to calculate the sums $$x_1^3 + x_2^3 + x_3^3$$ and $$x_1^4 + x_2^4 + x_3^4$$ where $x_1, x_2, x_3$ are the roots of $$x^3+2x^2+3x+4=0$$ using Viete's formulas. I know that ...