The location of critical points of polynomials

2019 ◽  
Vol 12 (07) ◽  
pp. 1950087
Author(s):  
Suhail Gulzar ◽  
N. A. Rather ◽  
F. A. Bhat
Keyword(s):  
Complex Plane ◽  
Simple Proof ◽  
Critical Points ◽  
Classical Result ◽  
Zeros Of Polynomials ◽  
Linear Combinations ◽  
Incomplete Polynomials ◽  
Location Of Zeros ◽  
Set Of Points

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

On the Location of Zeros of Polynomials

1967 ◽  
Vol 10 (1) ◽  
pp. 53-63 ◽  
Author(s):  
A. Joyal ◽  
G. Labelle ◽  
Q.I. Rahman
Keyword(s):  
Classical Result ◽  
Zeros Of Polynomials ◽  
Location Of Zeros

The different results proved in this paper do not have very much in common. Since they all deal with the location of the zeros of a polynomial, we have decided to put them in one place. Improving upon a classical result of Cauchy we obtain in § 2 a circle containing all the zeros of a polynomial. In § 3 we obtain an extension of the well known theorem of Enestrőm and Kakeya concerning the zeros of a polynomial whose coefficients are non-negative and monotonie.

A Short Proof of an Interpolation Theorem

1974 ◽  
Vol 17 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Simple Proof ◽  
Real Line ◽  
Short Proof ◽  
Interpolation Theorem ◽  
Operator Interpolation ◽  
Upper Half Plane ◽  
Positive Functions ◽  
Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

The universal multiplicity theory for analytic operator-valued functions

1995 ◽  
Vol 118 (2) ◽  
pp. 315-320 ◽  
Author(s):  
Jón Arason ◽  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Singular Set ◽  
Analytic Operator ◽  
Set Of Points

An analytic operator-valued function A is an analytic map A: D → L(E, E), where D = D(A) is an open subset of the complex plane C and E = E(A) is a complex Banach space. For such a function A the singular set σ(A) of A is defined as the set of points z ∈ D such that A(z) is not invertible. It is a relatively closed subset of D.

The shapes of a random sequence of triangles

1986 ◽  
Vol 18 (01) ◽  
pp. 156-169 ◽  
Author(s):  
G. S. Watson
Keyword(s):  
Limit Theorem ◽  
Ab Initio ◽  
Complex Plane ◽  
Random Sequence ◽  
Cubic Polynomial ◽  
Set Of Points ◽  
Independent And Identically Distributed

A triangle with vertices z 1, z 2 , z 3 in the complex plane may be denoted by a vector Z , Z = [z 1, z 2, z 3]t . From a sequence of independent and identically distributed 3×3 circulants { C j }∞ 1, we may generate from Z 1 the sequence of vectors or triangles { Z j }∞ 1, by the rule Z j = C j Z j–1 (j> 1), Z 1 = Z . The ‘shape’ of a set of points, the simplest case being three points in the plane has been defined by Kendall (1984). We give several alternative, ab initio discussions of the shape of a triangle, and proofs of a limit theorem for shape of the triangles in the sequence { Z j }∞ 1. In Appendix A, the shape concept is applied to the zeros of a cubic polynomial. Appendix B contains some further remarks about shape. Appendix C uses the methods of this paper to give proofs of generalizations of two old theorems on triangles.

SURVEY AND ADJUSTMENT OF CONTROLLER OF A GRAPH-ANALYTICAL METHOD TO A SECOND ORDER OBJECT

2020 ◽  
Vol 70 (2) ◽  
Author(s):  
Boris Grasiani
Keyword(s):  
Control System ◽  
Automatic Control ◽  
Control Systems ◽  
Quality Indicators ◽  
Automatic Control System ◽  
Complex Plane ◽  
Analytical Method ◽  
Second Order ◽  
Location Of Zeros ◽  
Zeros And Poles

This paper proposes to adjust the controller using a graph-analytical method in the complex plane. The various configurations regarding the location of zeros and poles to those of the object are also considered. Adjusted controllers are surveyed, such as they are integrated into control systems, and some of the quality indicators of an automatic control system are analyzed.

scholarly journals INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR 4-CORES

2012 ◽  
Vol 87 (2) ◽  
pp. 304-315 ◽  
Author(s):  
NAYANDEEP DEKA BARUAH ◽  
KALLOL NATH
Keyword(s):  
Theta Function ◽  
Simple Proof ◽  
Nonnegative Integer ◽  
Classical Result ◽  
Class Numbers ◽  
Acta Arith ◽  
Theta Function Identities

AbstractLetu(n) andv(n) be the number of representations of a nonnegative integernin the formsx2+4y2+4z2andx2+2y2+2z2, respectively, withx,y,z∈ℤ, and leta4(n) andr3(n) be the number of 4-cores ofnand the number of representations ofnas a sum of three squares, respectively. By employing simple theta-function identities of Ramanujan, we prove that$u(8n+5)=8a_4(n)=v(8n+5)=\frac {1}{3}r_3(8n+5)$. With the help of this and a classical result of Gauss, we find a simple proof of a result ona4(n) proved earlier by K. Ono and L. Sze [‘4-core partitions and class numbers’,Acta Arith.80(1997), 249–272]. We also find some new infinite families of arithmetic relations involvinga4(n) .

scholarly journals On the regions containing all the zeros of polynomials and related analytic functions

2021 ◽  
Vol 8 (2) ◽  
pp. 331-337
Author(s):  
Nisar Ahmad Rather ◽  
◽  
Ishfaq Dar ◽  
Aaqib Iqbal ◽  
◽  
...  
Keyword(s):  
Analytic Functions ◽  
Zeros Of Polynomials ◽  
Location Of Zeros ◽  
Standard Techniques

In this paper, by using standard techniques we shall obtain results with relaxed hypothesis which give zero bounds for the larger class of polynomials. Our results not only generalizes several well-known results but also provide better information about the location of zeros. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information on the zero bounds of polynomials than some known results.

scholarly journals A generalization of Lucas' theorem to vector spaces

1993 ◽  
Vol 16 (2) ◽  
pp. 267-276 ◽  
Author(s):  
Neyamat Zaheer
Keyword(s):  
Critical Points ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Linear Combinations ◽  
Inner Product Spaces ◽  
Complex Valued ◽  
Vector Valued

The classical Lucas' theorem on critical points of complex-valued polynomials has been generalized (cf. [1]) to vector-valued polynomials defined onK-inner product spaces. In the present paper, we obtain a generalization of Lucas' theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] inK-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims.

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