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29 pages, 452 KiB

Open AccessEditor’s ChoiceArticle

Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems

by Manuel De la Sen

Mathematics 2024, 12(19), 3023; https://doi.org/10.3390/math12193023 - 27 Sep 2024

Viewed by 809

Abstract

This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays. Full article

8 pages, 250 KiB

Open AccessArticle

The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions

by Robert Gardner and Matthew Gladin

AppliedMath 2023, 3(4), 722-729; https://doi.org/10.3390/appliedmath3040038 - 17 Oct 2023

Viewed by 1153

Abstract

Motivated by results on the location of the zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström–Kakeya theorem and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain regions. We do so by introducing a reversal in monotonicity conditions on the real and imaginary parts of the coefficients and also on their moduli. The conditions imposed are less restrictive than many of those in the current literature and hence apply to polynomials not covered by previous results. The results presented naturally apply to certain classes of lacunary polynomials. In particular, the results apply to certain polynomials with two gaps in their coefficients. Full article

11 pages, 786 KiB

Open AccessArticle

A Probabilistic Version of Eneström–Kakeya Theorem for Certain Random Polynomials

by Sajad A. Sheikh, Mohammad Ibrahim Mir, Javid Gani Dar, Ibrahim M. Almanjahie and Fatimah Alshahrani

Mathematics 2023, 11(19), 4061; https://doi.org/10.3390/math11194061 - 25 Sep 2023

Cited by 1 | Viewed by 1551

Abstract

This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results. Full article

(This article belongs to the Section D1: Probability and Statistics)

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13 pages, 264 KiB

Open AccessArticle

Generalizations of the Eneström–Kakeya Theorem Involving Weakened Hypotheses

by Robert Gardner and Matthew Gladin

AppliedMath 2022, 2(4), 687-699; https://doi.org/10.3390/appliedmath2040040 - 7 Dec 2022

Cited by 3 | Viewed by 2726

Abstract

The well-known Eneström–Kakeya Theorem states that, for

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

, a polynomial of degree n with real coefficients satisfying

0 \leq a_{0} \leq a_{1} \leq \dots \leq a_{n}

[...] Read more.

The well-known Eneström–Kakeya Theorem states that, for

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

, a polynomial of degree n with real coefficients satisfying

0 \leq a_{0} \leq a_{1} \leq \dots \leq a_{n}

, then all the zeros of P lie in

| z | \leq 1

in the complex plane. Motivated by recent results concerning an Eneström–Kakeya “type” condition on real coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients and concerning the moduli of the coefficients. In this way, our results generalize the other recent results. Full article

10 pages, 261 KiB

Open AccessArticle

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

by Azmeer Nordin and Mohd Salmi Md Noorani

Mathematics 2021, 9(11), 1268; https://doi.org/10.3390/math9111268 - 1 Jun 2021

Cited by 2 | Viewed by 2128

Abstract

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results. Full article

(This article belongs to the Section C2: Dynamical Systems)

8 pages, 436 KiB

Open AccessArticle

On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

by Pavel Trojovský

Mathematics 2020, 8(8), 1387; https://doi.org/10.3390/math8081387 - 18 Aug 2020

Cited by 3 | Viewed by 3017

Abstract

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence

{(T_{n}^{(k)})}_{n \geq 0}

. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

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