Kakeya Theorem Research Articles

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

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.

On the Eneström–Kakeya theorem and its various forms in the quaternionic setting

On the Eneström–Kakeya theorem and its various forms in the quaternionic setting

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

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.

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

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.

On the Eneström–Kakeya theorem for quaternionic polynomials

On the Eneström–Kakeya theorem for quaternionic polynomials

Unions of lines in Rn$\mathbb {R}^n$

AbstractWe prove a conjecture of D. Oberlin on the dimension of unions of lines in . If is an integer, , and L is a set of lines in with Hausdorff dimension at least , then the union of the lines in L has Hausdorff dimension at least . Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear → linear argument of Bourgain and Guth.

On the zero bounds of polynomials and regular functions of a quaternionic variable

In this manuscript, we are concerned with the problem of locating the zeros of some special quaternionic polynomials and regular functions with restricted coefficients; namely quaternionic coefficients whose real components or their moduli satisfy suitable inequalities. The obtained results for this subclass of quaternionic polynomials and regular functions produce extensions of the classical Enestr?om-Kakeya theorem and its various variants from complex to the quaternionic setting.

Generalizations of the Eneström–Kakeya Theorem Involving Weakened Hypotheses

The well-known Eneström–Kakeya Theorem states that, for P(z)=∑ℓ=0naℓzℓ, a polynomial of degree n with real coefficients satisfying 0≤a0≤a1≤⋯≤an, then all the zeros of P lie in |z|≤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.

Zeros of one class of quaternionic polynomials

The goal of this paper is to study the properties of zeros of some special quaternionic polynomials with restricted coefficients, namely coefficients whose real and imaginary components satisfy suitable inequalities. We extend the well-known Enestr?m-Kakeya theorem and its various generalizations from complex to the quaternionic setting. The main tools used to derive the bounds for the zeros of these polynomials are the maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable.

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

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.

A unifying framework for generalizations of the Eneström–Kakeya theorem

The classical Enestrom-Kakeya theorem establishes upper and lower bounds on the zeros of a polynomial with positive coefficients that are explicit functions of those coefficients. We establish a unifying framework that incorporates this theorem and several similar ones as special cases, while generating new theorems of a similar type. These establish zero inclusion and exclusion regions consisting of a single disk or the union of several disks in the complex plane. Our framework is built on two basic tools, namely a generalization of an observation by Cauchy, and a family of polynomial multipliers. Its approach is transparent and reduces algebraic manipulations to a minimum.

The Eneström–Kakeya Theorem for polynomials of a quaternionic variable

The well-known Eneström–Kakeya Theorem states that a polynomial with real, nonnegative, monotone increasing coefficients has all its complex zeros in the closed unit disk in the complex plane. In this paper, we extend this result by showing that all quaternionic zeros of such a polynomial lie in the unit sphere in the quaternions. We also extend related results from the complex to quaternionic setting.

An Eneström–Kakeya theorem for new classes of polynomials

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Some generalizations of the Eneström–Kakeya theorem

Let p(z) = a0 + a1z + a2z2 + ... + anzn be a polynomial of degree n; where the coefficients aj, j = 0, 1, 2, ..., n, are real numbers. We impose some restriction on the coefficients and then prove some extensions and generalizations of the Eneström–Kakeya theorem.

Some extensions and generalizations of Eneström–Kakeya theorem

In this paper, we put restrictions on the coefficients of a polynomial in order to improve the bounds for their zeros in a specific region. Our results extend and generalise a number of previously well known theorems including Eneström–Kakeya theorem.

The Eneström–Kakeya theorem encounters the theory of orthogonal polynomials on the unit circle

A classical result due to Eneström and Kakeya gives some bounds for the moduli of the zeros of polynomials having a monotone sequence of non-negative (real) coefficients. The main subject of the present paper is a study of this fact with a view to the recurrence relations fulfilled by systems of orthogonal polynomials on the unit circle. In particular, we will be interested in the special case, where the zeros of the polynomials in question are not located on the boundary of the estimate which occurs in the Eneström–Kakeya theorem. Among other things, we will give characterizations of this case in terms of orthogonal polynomials and present an alternative approach to a well-known characterization due to Hurwitz. Furthermore, we will give some insight how one can apply the main results of this paper in the context of positive Hermitian Toeplitz matrices.

The endpoint multilinear Kakeya theorem via the Borsuk–Ulam theorem

We give an essentially self-contained proof of Guthʼs recent endpoint multilinear Kakeya theorem which avoids the use of somewhat sophisticated algebraic topology, and which instead appeals to the Borsuk–Ulam theorem.

On the zeros of polynomials: An extension of the Eneström–Kakeya theorem

This paper presents an extension of the Eneström–Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown ( K , L ) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994) [6].

Spectraloid operator polynomials, the approximate numerical range and an Eneström–Kakeya theorem in Hilbert space

Spectraloid operator polynomials, the approximate numerical range and an Eneström–Kakeya theorem in Hilbert space

A Note on Kakeya's Theorem on Algebraic Equations

A Note on Kakeya's Theorem on Algebraic Equations