Distribution Of Zeros Research Articles

On the distribution of zeros of general exponential polynomials

On the distribution of zeros of general exponential polynomials

Use of transmission and reflection complex time delays to reveal scattering matrix poles and zeros: Example of the ring graph.

We identify the poles and zeros of the scattering matrix of a simple quantum graph by means of systematic measurement and analysis of Wigner, transmission, and reflection complex time delays. We examine the ring graph because it displays both shape and Feshbach resonances, the latter of which arises from an embedded eigenstate on the real frequency axis. Our analysis provides a unified understanding of the so-called shape, Feshbach, electromagnetically induced transparency, and Fano resonances on the basis of the distribution of poles and zeros of the scattering matrix in the complex frequency plane. It also provides a first-principles understanding of sharp resonant scattering features and associated large time delay in a variety of practical devices, including photonic microring resonators, microwave ring resonators, and mesoscopic ring-shaped conductor devices. Our analysis involves use of the reflection time difference, as well as a comprehensive use of complex time delay, to analyze experimental scattering data.

Generalized Popoviciu Expansions for Bernstein Polynomials of a Rational Module

We study Bernstein polynomials for simple nonsmooth rational module functions. We show that these polynomials can be represented as special sums of regular structure. For historical reasons, the representations found are naturally called “generalized Popoviciu expansions.” To write generalized expansions, we develop a special formalism based on combinatorial calculations. Based on the formulas obtained, we propose a complete description of the character of convergence of the Bernstein polynomials studied in the complex plane. We also discuss the relationship of generalized Popoviciu expansions with the distribution of zeros of Bernstein polynomials. In the final part of the paper, we present additional new relations for Bernstein polynomials of the rational module function.

Low-lying zeros of L-functions for Quaternion Algebras

The density conjecture of Katz and Sarnak predicts that, for natural families of L-functions, the distribution of zeros lying near the real axis is governed by a group of symmetry. In the case of the universal family of automorphic forms on a totally definite quaternion algebra, we determine the associated distribution for a restricted class of test functions in the analytic conductor aspect. In particular it leads to non-trivial results on densities of non-vanishing at the central point.

Goss polynomials, q-adic expansions, and Sheats compositions

The zeroes of Goss polynomials Gk,Λ(X) for Λ=A:=Fq[T] and similar lattices Λ are studied. Generically, the zero distribution follows a simple pattern governed by the q-adic expansion of k−1. However, if q=pf with f≥2 is a proper power of the prime p, irregularities of the q-adic sum-of-digits function may lead to deviations from this pattern, to irregular zeroes, and an abundance of trivial zeroes of Gk,Λ compared to the generic formula. These phenomena are related to properties of the Sheats compositions of natural numbers n divisible by q−1. Among other things, we give a necessary and sufficient condition for the existence of irregular zeroes of Gk,A and a formula for the vanishing order of Gk,A at x=0.

Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

Let $$X$$ be a class of extended numerical functions on a domain $$D$$ of $$d$$ -dimensional Euclidean space $$\mathbb{R}^{d}$$ , $$H\subset X$$ . Given $$u\in X$$ and $$M\in X$$ , we write $$u\prec_{H}M$$ if there is a function $$h\in H$$ such that $$u+h\leq M$$ on $$D$$ . We consider this special preorder $$\prec_{H}$$ for a pair of subharmonic functions $$u$$ and $$M$$ on $$D$$ in cases when $$H$$ is the space of all harmonic functions on $$D$$ or $$H$$ is the convex cone of all subharmonic functions $$h\not\equiv-\infty$$ on $$D$$ . Main results are dual equivalent forms for this preorder $$\prec_{H}$$ in terms of balayage processes for Riesz measures of subharmonic functions $$u$$ and $$M$$ , for Jensen and Arens–Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements $$D\backslash S$$ of non-empty precompact subsets $$S\Subset D$$ . Applications to holomorphic functions $$f$$ on a domain $$D\subset\mathbb{C}^{n}$$ relate to the distribution of zero sets of functions $$f$$ under upper restrictions $$|f|\leq\exp M$$ on $$D$$ . If a domain $$D\subset\mathbb{C}$$ is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of $$D$$ , then our conditions for the distribution of zeros of $$f\neq 0$$ with $$|f|\leq\exp M$$ on $$D$$ are both necessary and sufficient.

Dynamical Lee-Yang zeros for continuous-time and discrete-time stochastic processes.

We describe classical stochastic processes by using dynamical Lee-Yang zeros. The system is in contact with external leads and the time evolution is described by the two-state classical master equation. The cumulant generating function is written in a factorized form and the current distribution is characterized by the dynamical Lee-Yang zeros. We show that a continuous distribution of zeros is obtained by discretizing the time variable. When the transition probability is a periodically oscillating function of time, the distribution of zeros splits into many parts. We study the geometric property of the current by comparing the result with that of the adiabatic approximation. We also use the Floquet-Magnus expansion in the continuous-time case to study dynamical effects on the current at the fast-driving regime.

A Vector Equilibrium Problem for Symmetrically Located Point Charges on a Sphere

In this paper, we study a class of matrix-valued orthogonal polynomials (MVOPs) that are related to 2-periodic lozenge tilings of a hexagon. The general model depends on many parameters. In the cases of constant and $2$-periodic parameter values we show that the MVOP can be expressed in terms of scalar polynomials with non-Hermitian orthogonality on a closed contour in the complex plane. The 2-periodic hexagon tiling model with a constant parameter has a phase transition in the large size limit. This is reflected in the asymptotic behavior of the MVOP as the degree tends to infinity. The connection with the scalar orthogonal polynomials allows us to find the limiting behavior of the zeros of the determinant of the MVOP. The zeros tend to a curve $\widetilde{\Sigma}_0$ in the complex plane that has a self-intersection. The zeros of the individual entries of the MVOP show a different behavior and we find the limiting zero distribution of the upper right entry under a geometric condition on the curve $\widetilde{\Sigma}_0$ that we were unable to prove, but that is convincingly supported by numerical evidence.

Multiplication operator and average characteristic polynomial associated with exceptional Jacobi polynomials

Studying the multiplication operator associated with exceptional Jacobi polynomials, the zero distribution of the corresponding average characteristic polynomials is determined. Applying this result, the location of zeros of certain self-inversive polynomials is examined.

Polynomials of Least Deviation from Zero in Sobolev p-Norm

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<infty ) for the case p=1. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n-mathbf {d}^* zeros on the convex hull of the support of the measure, where mathbf {d}^* denotes the number of terms in the discrete part.

Buried points in complex dynamical systems and glassy transitions

<p indent=0mm>A new interdisciplinary research direction is addressed across the fields of statistical mechanics and complex dynamical systems together with specific research topics, such as the connection between buried points and glassy transitions. First, pertinent results on complex dynamical systems are delivered. Let<italic> f</italic> be a complex analytical mapping from the Riemann sphere (or the complex plane) to itself, i.e., <italic>f</italic> is rational or entire. The stable set <italic>F</italic>(<italic>f</italic>) of this dynamical system is called the Fatou set of <italic>f</italic>, and the unstable set <italic>J</italic>(<italic>f</italic>) the Julia set. Every component of <italic>F</italic>(<italic>f</italic>) is called a Fatou component. Obviously, the boundary of each Fatou component belongs to <italic>J</italic>(<italic>f</italic>). If a point in <italic>J</italic>(<italic>f</italic>) does not belong to the boundary of any Fatou component, then it is a buried point of <italic>f</italic>, which is often used to describe the topological complexity of the Julia set. In this quest, Makienko’s conjecture is undoubtedly one of the most important problems which is stated as follows: Let <italic>R</italic> be a rational mapping and <italic>F</italic>(<italic>R</italic>) contain infinitely many components. If every Fatou component is not completely invariant, then <italic>R</italic> has buried points. The known result is the following: If <italic>J</italic>(<italic>R</italic>) is disconnected, or connected and locally connected, then Makienko’s conjecture is true; if <italic>J</italic>(<italic>R</italic>) is disconnected and <italic>R</italic> has a buried point, then it must have buried components. Concerning entire functions, we have: If <italic>F</italic>(<italic>f</italic>) is not empty, then <italic>f</italic> has a buried point if and only if <italic>F</italic>(<italic>f</italic>) is disconnected. Following the above results, interesting topological properties on the sets of buried points are deduced. However, many open problems remain unsolved in regard to these special points. In the second part, we explained the topological complexity of the set of Yang-Lee zeros. In 1952, Yang C.N. and Lee T.D. established an analytic theory in statistical mechanics, summarized as the circle theorem: For statistical models like 2-dimensional lattice gas, the zeros of the partition function condense to the unit circle. Henceforth, the importance of the distribution of Yang-Lee zeros in the complex plane is emphasized for general models. In 1983, researchers including B. Derrida, L. De Seze and C. Itzykson found that the distribution of Yang-Lee zeros can be cast into a problem on Julia sets of renormalization transformations. In the third part, we describe the connection between buried points and glass transitions. Concerning the Potts model on a generalized diamond hierarchical lattice, we present a new type of distribution of Julia sets of renormalization transformations, which is called the real Feigenbaum-type distribution of buried points. Physically, this means that the limiting set of Yang-Lee zeros could contain an interval of the real axis, all of which are singularities of the free energy function. This important observation indicates that the exploration of renormalization transformation dynamics is essential to the understanding of the phenomenon of glassy transitions. Finally, we discussed the theoretical value and practical significance of the above research topics and directions. It is well known that <italic>Science</italic> published 125 most challenging scientific questions in 2005 to celebrate the Journal’s 125th anniversary, among which Question 47 is “What is the essence of the glassy substance?”. There are many theoretical models for glassy transitions, but up to now, a complete theory remains illusive. Perhaps, the connection between complex dynamics and glassy transition established above could shed new light on this old problem with the help of powerful tools from mathematics, as ever before.

Distributions of Zeros and Poles of N-Point Padé Approximants to Complex-Symmetric Functions Defined at Complex Points

Distributions of Zeros and Poles of N-Point Padé Approximants to Complex-Symmetric Functions Defined at Complex Points

On the distribution of zeros of all solutions of a first order nonlinear neutral differential equation

We investigate the distribution of zeros of all solutions of a non-autonomous nonlinear neutral differential equation that generalizes a lossless transmission network model. The neutral term is taken to be positive. We give several new estimations of the gap between adjacent zeros. The obtained results are supported by illustrative examples.

Padé Approximation and Hypergeometric Functions: A Missing Link with the Spectrum of Delay-Differential Equations

It is well known that rational approximation theory involves degenerate hyperge-ometric functions and, in particular, the Padé approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay systems known as multiplicity-induced-dominancy (MID), which opened a new direction in designing low-complexity controllers for time-delay systems by using a partial pole placement idea. Thanks to their relations to hypergeometric functions, we explore in this paper links between the spectrum of delay-differential equations and Padé approximations of the exponential function. This note exploits and further comments recent results from [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Comptes Rendus. Mathématique] and [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Bulletin des Sciences Mathématiques].

On distribution densities of algebraic points under different height functions

In the article we consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed degree. The distribution is introduced using a height function. We have obtained universal upper and lower bounds of the distribution density of such points using an arbitrary height function. We have shown how from a given joint density function of coefficients of a random polynomial of degree n, one can construct such a height function H that the polynomials q of degree n uniformly chosen under H[q] ≤1 have the same distribution of zeros as the former random polynomial.

Volatility Estimation When the Zero-Process is Nonstationary

Financial returns are frequently nonstationary due to the nonstationary distribution of zeros. In daily stock returns, for example, the nonstationarity can be due to an upwards trend in liquidity over time, which may lead to a downwards trend in the zero-probability. In intraday returns, the zero-probability may be periodic: It is lower in periods where the opening hours of the main financial centers overlap, and higher otherwise. A nonstationary zero-process invalidates standard estimators of volatility models, since they rely on the assumption that returns are strictly stationary. We propose a GARCH model that accommodates a nonstationary zero-process, derive a zero-adjusted QMLE for the parameters of the model, and prove its consistency and asymptotic normality under mild assumptions. The volatility specification in our model can contain higher order ARCH and GARCH terms, and past zero-indicators as covariates. Simulations verify the asymptotic properties in finite samples, and show that the standard estimator is biased. An empirical study of daily and intradaily returns illustrate our results. They show how a nonstationary zero-process induces time-varying parameters in the conditional variance representation, and that the distribution of zero returns can have a strong impact on volatility predictions.

A Note on the Summation of the Incomplete Gamma Function

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

Phase Role in the Non-Uniformity of Main-Line Couplings in Asymmetric Extracted-Pole Inline Filters

The use of classical symmetrical polynomial definition to synthesize fully canonical inline filters with an asymmetrical distribution of the transmission zeros along the topology leads to the occurrence of uneven admittance inverter in the main-line. This form introduces some limitations to transform such topology into a ladder network. Despite circuital transformation can be used to accommodate both technology and topology, it is usual that extra reactive elements are necessary to implement phase shifts required to achieve the complete synthesis. This article introduces a novel method able to determine the required phase correction that has to be applied to the characteristic polynomials in order to equalize all the admittance inverters in the main path to the same value. It has been demonstrated that a suitable pair of phase values can be accurately estimated using a developed hyperbolic model which can be obtained from the transmission and reflection scattering parameters. To experimentally validate the proposed method, a Ladder-type filter with asymmetrical polynomial definition has been synthesized, fabricated, and measured, demonstrating the effectiveness of the developed solution.

Random number generation with arrays of coupled quantumdot micropillar lasers

The paper investigates the results of random number generation with arrays of coupled quantum-dot micropillar lasers. The micropillars array laser generation is modeled based on the rate equations for quantum dot lasers. The numerical simulation of the dynamics for the arrays of coupled quantum-dot micropillar lasers is carried out utilizing the semi-implicit Euler method, implemented in Julia programming language. The algorithm of random bit sequence generation consists of the following steps: sampling the values of the total field intensity for coupled micropillar lasers; normalizing and discretizing the obtained values per resolution of the 8-bit analog-to-digital converter; extracting the four least significant bits from the bit representation of the sampled values; concatenating the bit values in a single sequence. The possibility of the bit sequences generation having an equiprobable distribution of zeros and ones with a performance of up to 400 Gbit/s was shown utilizing a random number generator based on an array of coupled quantum-dot micropillar lasers for sequences with a length of 14285716 bits at a sampling rate of 100 gigasamples per second and four least significant bits extraction. The resulting bit sequences successfully passed 14 NIST 800-22 statistical tests for the p-value equal to 0.01. The proposed method can be applied to develop random number generators based on larger arrays of coupled quantum-dot micropillar lasers. The results can be utilized in the experimental implementation of random number generators based on arrays of coupled quantum-dot micropillar lasers.

On the Behavior of the Logarithm of the Riemann Zeta-Function

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $$\log {\zeta (s)}$$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the relation by some previous works. For example, Littlewood showed an upper bound of $$\log {\zeta (1/2 + it)}$$ by assuming the Riemann hypothesis in 1924. One of our results reveals that Littlewood’s upper bound can be proved without assuming a hypothesis as strong as the Riemann hypothesis.