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21 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 ...
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0answers
28 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 ...
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0answers
15 views

Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
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2answers
44 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 ...
7
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1answer
356 views

The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
0
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1answer
11 views

Functions with symmetrical behaviour with respect to an axis or a plane

Suppose we have two functions with a symmetrical behaviour with respect to an axis. For the sake of simplicity, let $f(x)$ and $g(x)$ have a symmetrical behaviour with respect to the $y$ axis. A ...
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1answer
28 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 ...
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0answers
26 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 ...
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0answers
49 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
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1answer
89 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?
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1answer
41 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 ...
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0answers
132 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
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0answers
62 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 ...
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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
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1answer
163 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 ...
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2answers
143 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 ...
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0answers
52 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 ...
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1answer
41 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 ...
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0answers
74 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 ...
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2answers
79 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 : ...
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2answers
120 views

Help me with this this system of equations

Help me with this system of equations $$a+b = 3 -c$$ $$\frac{1}{a}+\frac{1}{b}= \frac{5}{12}-\frac{1}{c}$$ $$ a^3+b^3 = 45 -c^3$$
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1answer
165 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 ...
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1answer
142 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
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0answers
81 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 ...
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0answers
97 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 ...
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2answers
146 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 ...
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1answer
68 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 ...
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3answers
119 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.
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2answers
48 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 ...
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0answers
42 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
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1answer
171 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
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0answers
246 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 ...
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1answer
210 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
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1answer
291 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
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2answers
741 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
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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
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0answers
113 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 ...
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2answers
713 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 ...
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1answer
143 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 ...
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3answers
479 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
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0answers
117 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
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2answers
2k 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 ...