scholarly journals Detection of positive roots of a polynomial with five parameters

2001 ◽  
Vol 137 (1) ◽  
pp. 19-48
Author(s):  
Sui Sun Cheng ◽  
Yi-Zhong Lin
Keyword(s):  
Positive Roots ◽  
Roots Of A Polynomial

Computational Alternative to a Gain Setting Procedure of Brown and Campbell

1990 ◽  
Vol 112 (3) ◽  
pp. 516-517
Author(s):  
L. Kitis
Keyword(s):  
Control System ◽  
Frequency Response ◽  
Numerical Method ◽  
Multiple Solutions ◽  
Closed Loop ◽  
Linear Control ◽  
Function Evaluation ◽  
Resonant Frequencies ◽  
Positive Roots ◽  
Roots Of A Polynomial

A numerical method for setting the gain of a linear control system to obtain a specified peak closed-loop frequency response amplitude is presented. The essential computational step in the algorithm is the calculation of all positive roots of a polynomial. This step provides the set of all possible resonant frequencies. A gain constant for each of these frequencies is then found by a simple function evaluation. The possibility of multiple solutions is demonstrated by an example.

scholarly journals Symmetry of Narayana Numbers and Rowvacuation of Root Posets

2021 ◽  
Vol 9 ◽  
Author(s):  
Colin Defant ◽  
Sam Hopkins
Keyword(s):  
Weyl Group ◽  
Catalan Number ◽  
Recent Past ◽  
Positive Roots ◽  
Narayana Numbers ◽  
The Subject

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

2020 ◽  
pp. 1-10
Author(s):  
NGUYEN CONG MINH ◽  
LUU BA THANG ◽  
TRAN NAM TRUNG
Keyword(s):  
Polynomial Ring ◽  
Polynomial System ◽  
Combinatorial Nullstellensatz ◽  
Roots Of A Polynomial

Abstract Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

scholarly journals An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Rogelio Luck ◽  
Gregory J. Zdaniuk ◽  
Heejin Cho
Keyword(s):  
Null Space ◽  
Heat Conduction Problem ◽  
Hankel Matrix ◽  
Polynomial Equation ◽  
Transient Heat Conduction ◽  
Transcendental Equation ◽  
Integral Theorem ◽  
Transient Heat Conduction Problem ◽  
Cauchy’S Integral Theorem ◽  
Roots Of A Polynomial

This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals Multi-cluster complexes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Jean-Philippe Labbé ◽  
Christian Stump
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Simplicial Complexes ◽  
Cluster Complex ◽  
Geometric Properties ◽  
Cluster Complexes ◽  
Combinatorial Description ◽  
Compatibility Relation ◽  
Positive Roots ◽  
International Audience

International audience We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex. Nous présentons une famille de complexes simpliciaux appelés \emphcomplexes des multi-amas. Ces complexes généralisent le concept de complexes des amas et étendent la notion de multi-associaèdre de type ${A}$ et ${B}$ aux groupes de Coxeter finis. Nous étudions des propriétés combinatoires et géométriques de ces objets et, en particulier nous fournissons une description combinatoire simple de la relation de compatibilité sur l'ensemble des racines presque positives du complexe des amas.

scholarly journals Fractional Newton-Raphson Method

2021 ◽  
Vol 8 (1) ◽  
pp. 1-13
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz
Keyword(s):  
Fractional Calculus ◽  
Iterative Method ◽  
Complex Numbers ◽  
Initial Condition ◽  
The Real ◽  
Complex Roots ◽  
Newton Raphson ◽  
Roots Of A Polynomial ◽  
Raphson Method

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n, with n ∈ N. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

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