On μ-symmetric polynomials

2021 ◽  
pp. 2250233
Author(s):  
Jing Yang ◽  
Chee K. Yap
Keyword(s):  
Clustering Algorithm ◽  
Symmetric Functions ◽  
Complexity Analysis ◽  
Root Function ◽  
Linear Equations ◽  
Symmetric Polynomials ◽  
Root Functions ◽  
Multiplicity Formula ◽  
Polynomial Formula ◽  
Integer Polynomial

We study functions of the roots of an integer polynomial [Formula: see text] with [Formula: see text] distinct roots [Formula: see text] of multiplicity [Formula: see text], [Formula: see text]. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to [Formula: see text]-symmetric polynomials. We initiate the study of the vector space of [Formula: see text]-symmetric polynomials of a given degree [Formula: see text] via the concepts of [Formula: see text]-gist and [Formula: see text]-ideal. In particular, we are interested in the root function [Formula: see text]. The D-plus discriminant of [Formula: see text] is [Formula: see text]. This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that [Formula: see text] is [Formula: see text]-symmetric, which implies [Formula: see text] is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is [Formula: see text]-symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the [Formula: see text]-symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first.

scholarly journals A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

scholarly journals Towards the Calculation of Jack Polynomials

2021 ◽  
Author(s):  
◽  
Leigh Alan Roberts
Keyword(s):  
Recent Result ◽  
Symmetric Functions ◽  
Mathematical Statistics ◽  
Jack Polynomials ◽  
Symmetric Polynomials ◽  
Geometric Progressions ◽  
Elementary Symmetric Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

scholarly journals Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$

2020 ◽  
Vol 12 (1) ◽  
pp. 5-16
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Linear Combination ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Symmetric Polynomial ◽  
Complex Numbers ◽  
Symmetric Polynomials ◽  
Algebraic Basis

This work is devoted to study algebras of continuous symmetric, that is, invariant with respect to permutations of coordinates of its argument, polynomials and $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers resp., where $1\leq p < +\infty.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n).$ Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$ Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n),$ and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$

Equally Inclined Spheres

1962 ◽  
Vol 58 (2) ◽  
pp. 420-421 ◽  
Author(s):  
J. G. Mauldon
Keyword(s):  
Symmetric Functions ◽  
Linear Equations ◽  
Quadratic Equation ◽  
Cartesian Coordinates ◽  
Elementary Symmetric Functions ◽  
Equation Solving ◽  
Image Position

If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.

scholarly journals Analyze K-Value Selected Method of K-Means Clustering Algorithm to Clustering Province Based on Disease Case

2020 ◽  
Vol 9 (3S) ◽  
pp. 121-124
Keyword(s):  
Clustering Algorithm ◽  
Human Mobility ◽  
Linear Equations ◽  
K Value ◽  
Convergence Results ◽  
Disease Case ◽  
Strong Linear Correlation ◽  
The Government ◽  
The Relationship ◽  
Cluster Level

Disease cases throughout Indonesia has increased as seen from the Indeks Pembangunan Masyarakat (IPKM). Globalization has the effect of increasing human mobility across provinces, thus accelerating the process of spreading epidemics that could pose a threat for Indonesia. The speed of action from government is needed to reducing the level if outbreaks of the disease. For this reason, accuracy from the government is needed to solving this problem. The data were taken from data disease cases in 2015 which consisted of 34 provinces in Indonesia based on the Central Statistics Agency in Indonesian. In K-Means clustering, determining of K-value is needed because it affects in convergence results. To solve this problem, this research analyzes three methods of K-Value, there are Silhouette, Elbow, and Gap Statistics Methods.The result of testing three methods of determining K-value obtained execution times on Silhouette 13.09s, Elbow 14.76s, and Gap Statistics 20.28s. So, choosing Silhouette method produces 2 optimal clusters, there are low cluster level (C1) and high cluster level (C2). The correlation matrix to understand the relationship between each disease is performed and a value of 0.88 is obtained there is the strong linear correlation between Pneumonia and Pulmonary TB. Then, modeling the relationship between these two variables by fitting linear equations. The results of C1 cluster based on disease cases were obtained 32 provinces and for C2 cluster were 2 provinces there areWest Java and East Java. Based on the results of the clustering can be input to the Indonesian government to tackle disease cases in all provinces in Indonesia

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

2018 ◽  
Vol 27 (02) ◽  
pp. 1850015 ◽  
Author(s):  
Mikami Hirasawa ◽  
Kunio Murasugi
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Polynomial Formula ◽  
Integer Polynomial

Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.

scholarly journals On Determinantal Symmetric Functions

1934 ◽  
Vol 4 (1) ◽  
pp. 47-52
Author(s):  
Zia-ud-Din
Keyword(s):  
Symmetric Functions ◽  
Symmetric Polynomials ◽  
The Difference ◽  
Dual Identities ◽  
Image Position

§ 1. The theory of symmetric polynomials abounds in dual identities and symmetries of various kinds. It has been investigated from the determinantal standpoint largely by means of quotients of alternants, such as,the denominator being the difference product of a, b, c, …, a simple alternant.

scholarly journals Note on Dr Muir's Paper on a Peculiar Set of Linear Equations

1902 ◽  
Vol 23 ◽  
pp. 261-263
Author(s):  
Charles Tweedie
Keyword(s):  
Symmetric Functions ◽  
Elementary Theory ◽  
Linear Equations ◽  
Simple Method

In Dr Muir's Paper on a Peculiar Set of Linear Equations (communicated December 3, 1900) there occur two Determinants of the nth order, the expansions of which are given by Dr Muir, As the paper in question has so much to do with Symmetric Functions, the following simple method of obtaining their expansions may not prove uninteresting, based, as it is, upon the elementary theory of Symmetric Functions and the so-called Principle of Indeterminate Coefficients.

scholarly journals Towards the Calculation of Jack Polynomials

2021 ◽  
Author(s):  
◽  
Leigh Alan Roberts
Keyword(s):  
Recent Result ◽  
Symmetric Functions ◽  
Mathematical Statistics ◽  
Jack Polynomials ◽  
Symmetric Polynomials ◽  
Geometric Progressions ◽  
Elementary Symmetric Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

scholarly journals Lipschitz symmetric functions on Banach spaces with symmetric bases

2021 ◽  
Vol 13 (3) ◽  
pp. 727-733
Author(s):  
M.V. Martsinkiv ◽  
S.I. Vasylyshyn ◽  
T.V. Vasylyshyn ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Large Family ◽  
Symmetric Polynomials ◽  
Symmetric Basis ◽  
Several Variables ◽  
Symmetric Bases ◽  
Unbounded Subset

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

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