Distribution Of Zeros Research Articles

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

A new approach to the problem of the zero distribution of type I Hermite—Pade polynomials for a pair of functions f1, f2 forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field which is posed on a two-sheeted Riemann surface.

Stable source reconstruction from a finite number of measurements in the multi-frequency inverse source problem

We consider the multi-frequency inverse source problem for the scalar Helmholtz equation in the plane. The goal is to reconstruct the source term in the equation from measurements of the solution on a surface outside the support of the source. We study the problem in a certain finite dimensional setting: from measurements made at a finite set of frequencies we uniquely determine and reconstruct sources in a subspace spanned by finitely many Fourier–Bessel functions. Further, we obtain a constructive criterion for identifying a minimal set of measurement frequencies sufficient for reconstruction, and under an additional, mild assumption, the reconstruction method is shown to be stable. Our analysis is based on a singular value decomposition of the source-to-measurement forward operators and the distribution of positive zeros of the Bessel functions of the first kind. The reconstruction method is implemented numerically and our theoretical findings are supported by numerical experiments.

Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre, and a related hierarchy by de Klerk, Hess, and Laurent. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.

On the existence of equivalent Dirichlet polynomials whose zeros preserve a topological property

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals.

Density of states engineering: normalized energy density of states band structure using the tridiagonal representation approach

We expand the quantum mechanical wavefunction in a complete set of square integrable orthonormal basis such that the matrix representation of the Hamiltonian operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction whose solution is a set of orthogonal polynomials in the energy. The polynomials weight function is the energy density of the system constructed using the Green's function, which is written in terms of the Hamiltonian matrix elements. We study the distribution of zeros of these polynomials on the real energy line based exclusively on their three-term recursion relations. We show that the zeros are generally grouped into sets belonging to separated bands on the orthogonality interval. The number of these bands is equal to the periodicity (multiplicity) of the asymptotic values of the recursion coefficients and the location of their boundaries depend only on these asymptotic values. Bound states (if they exist) are located at discrete zeros found in the gaps between the density bands that are stable against variation in the order of the polynomial for very large orders. We give examples of systems with a single, double and triple energy density bands.

On Singular Points of Meromorphic Functions Determined by Continued Fractions

It is shown that Leighton’s conjecture about singular points of meromorphic functions represented by C-fractions K∞n=1(a n z αn /1) with exponents α1, α2,... tending to infinity, which was proved by Gonchar for a nondecreasing sequence of exponents, holds also for meromorphic functions represented by continued fractions K∞n=1(a n A n (z)/1), where A1,A2,... is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.

Wideband multi-stage CROW filters with relaxed fabrication tolerances.

We present wideband and large free spectral range optical filters with steep passband edges for the selection of adjacent WDM communication channels that can be reliably fabricated with mainstream silicon photonics technology. The devices are based on three cascaded stages of coupled resonator optical waveguides loaded on a common bus waveguide. These stages differ in the number of resonators but are implemented with exactly identical unit cells, comprised of a matched racetrack resonator layout and a uniform spacing between cells. The different number of resonators in each stage allows a high rejection in the through port response enabled by the interleaved distribution of zeros. Furthermore, the exact replication of a unique cell avoids the passband ripple and high lobes in the stopband that typically arise in apodized coupled resonator optical waveguide based filters due to fabrication and coupling induced variations in the effective path length of each resonator. Silicon photonics filters designed for the selection of 9 adjacent optical carriers generated by a 100 GHz free spectral range comb laser have been successfully fabricated with 248 nm DUV lithography, achieving an out-of-band rejection above 11 dB and an insertion loss of less than 0.5 dB for the worst channels.

Expected Number of Real Zeros of Gaussian Self-Reciprocal Random Algebraic Polynomials

Let $$Q_{n} (x) = \sum\nolimits_{i = 0}^{n} A_{i} x^{i}$$ be a random algebraic polynomial where the coefficients $$A_{0} ,A_{1} , \cdots$$ form a sequence of centered Gaussian random variables. Moreover, we assume that the increments $$\Delta_{j} = A_{j} - A_{j - 1}$$ , $$j = 0,1,2, \ldots ,n$$ are independent normal random variables with mean zero, finite variances $$\sigma_{j}^{2}$$ and the conventional notation of $$A_{ - 1} = 0$$ . The coefficients can be considered as $$n$$ consecutive observations of a Brownian motion. Assuming the symmetric property of $$A_{j}$$ ’s, we investigate some new and considerable results about the distribution of zeros. We prove that the expected number of real zeros of $$Q_{n} (x)$$ is asymptotically of order $$logn.$$

Zero distribution of random polynomials

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that under mild assumptions on the coefficients, their zeros are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane and quantify this convergence. In our results, random coefficients may be dependent and need not have identical distributions.

Mean growth and geometric zero distribution of solutions of linear differential equations

The aim of this paper is to consider certain conditions on the coefficient A of the differential equation f″ + Af = 0 in the unit disc which place all normal solutions f in the union of Hardy spaces or result in the zero-sequence of each non-trivial solution being uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.

On r‐gaps between zeros of the Riemann zeta‐function

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $\Theta$ and $\vartheta$, where $\gamma$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontrivial zeros of the Riemann zeta-function.

亚纯函数及其差分乘积的零点分布

亚纯函数及其差分乘积的零点分布

TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form $$ \sum_{\rho } R(\rho) x^{\rho}, $$ where the sum is over the non-trivial zeros $\rho$ of $\zeta(s)$, $R(x) \in \overline{\Q}(x) $ is a rational function over algebraic numbers and $x >0$ is a real algebraic number. In particular, we show that the function $$ f(x) = \sum_{\rho } \frac{x^{\rho}}{\rho} $$ has infinitely many zeros in $(1, \infty)$, at most one of which is algebraic. The transcendence tools required for studying $f(x)$ in the range $x<1$ seem to be different from those in the range $x>1$. For $x < 1$, we have the following non-vanishing theorem: If for an integer $d \ge 1$, $f(\pi \sqrt{d} x)$ has a rational zero in $(0,~1/\pi \sqrt{d})$, then $$ L'(1,\chi_{-d}) \neq 0, $$ where $\chi_{-d}$ is the quadratic character associated to the imaginary quadratic field $K:= \Q(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.

Generalization of value distribution and uniqueness of certain types of difference polynomials

In this paper, we investigate the distribution of zeros as well as the uniqueness problems of certain type of differential polynomials sharing a small function with finite weight. The result obtained improves and generalizes the recent results.

On Picard value problem of some difference polynomials

In this paper, we study the value distribution of zeros of certain nonlinear difference polynomials of entire functions of finite order.

On Global universality for zeros of random polynomials

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\Bbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $dd^cV_{Q}.$ In addition, in complex dimension one, we consider random linear combinations of orthonormal polynomials with respect to a regular measure in the sense of Stahl \& Totik and we prove similar results in this setting.

Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

In this paper, we obtain new sufficient conditions of boundedness of L-index in joint variables for entire function in Cn functions. They give an estimate of maximum modulus of an entire function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives and the distribution of zeros. In some sense, the obtained results are new for entire functions of bounded index and l-index in C too. They generalize known results of Fricke, Sheremeta, and Kuzyk.

Large Deviation Theorem for Zeros of Polynomials and Hermitian Random Matrices

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled.

Distribution of Zeros for Random Laurent Rational Functions

We study the asymptotic distribution of zeros for the random rational functions that can be viewed as partial sums of a random Laurent series. If this series defines a random analytic function in an annulus A, then the zeros accumulate on the boundary circles of A, being equidistributed in the angular sense, with probability 1. We also show that the equidistribution phenomenon holds if the annulus of convergence degenerates to a circle. Moreover, equidistribution of zeros still persists when the Laurent rational functions diverge everywhere, which is new even in the deterministic case. All results hold under two types of general conditions on random coefficients. The first condition is that the random coefficients are non-trivial i.i.d. random variables with finite \(\log ^+\) moments. The second condition allows random variables that need not be independent or identically distributed, but only requires certain uniform bounds on the tails of their distributions.

Rank metric codes and zeta functions

We define the rank metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.