Distribution Of Zeros Research Articles

Zero distribution of solutions of some linear differential equations

Assuming that A ( z ) = B 1 ( z ) e P ( z ) + B 2 ( z ) e ϱp ( z ) + Q ( z ) , where B 1 ( z ) , B 2 ( z ) , p ( z ) are nonzero polynomials, ϱ ∈ ( 0 , 1 ) , and Q ( z ) is an entire function of order less than deg ⁡ p , we study the oscillation of solutions of the linear differential equation f ( k ) + A ( z ) f = 0 . Some conditions on A ( z ) are given to guarantee that any non-trivial solution f satisfies λ ( f ) = ∞ or λ ( f ) ≥ σ ( A ) , where σ ( f ) and λ ( f ) denote respectively the order of f and the exponent of convergence of zeros of f. We also find some types of linear differential equations with zero-free solutions.

Maximum Likelihood Sequence Estimation Optimum Receiver Design with Channel Identification Based on Zero Distribution

In this paper, a novel maximum likelihood sequence estimation (MLSE) optimum receiver design is proposed to reduce the complexity of the conventional optimum receiver. We first propose a multipath channel classification method according to the zero distribution of the channel impulse response; that is, the multipath channel is divided into a minimum phase channel and a non-minimum phase channel. Then, an MLSE optimum receiver design with channel identification based on zero distribution is also proposed. In this design, the distribution of zero can be obtained directly by calculating the Z transform of the channel. Thus, the complexity is reduced. The number of the multiplication reduced is N(2N+1). In addition, 2N+1 memory units storing the autocorrelation sequence are reduced. The simulation results indicate that this channel classification method effectively represents the characteristics of the channel. Moreover, the MLSE optimum receiver proposed performs similarly and is less complex than the conventional one. The performance is greatly improved compared with the minimum mean square error (MMSE) equalizer.

Partition function zeros of zeta-urns

We discuss the distribution of partition function zeros for the grand-canonical ensemble of the zeta-urn model, where tuning a single parameter can give a first or any higher order condensation transition. We compute the locus of zeros for finite-size systems and test scaling relations describing the accumulation of zeros near the critical point against theoretical predictions for both the first and higher order transition regimes.

The Topology of Projective Codes and the Distribution of Zeros of Odd Maps

The Topology of Projective Codes and the Distribution of Zeros of Odd Maps

Analysing All-Optical Random Bit Sequences Using Gap-Based Approaches.

Quantum mechanical phenomena are revolutionizing classical engineering fields such as signal processing or cryptography. When randomness plays an important role, like in cryptography where random bit sequences guarantee certain levels of security, quantum mechanical phenomena allow new ways of generating random bit sequences. Such sequences have a lot of applications in the communication sector, e.g., regarding data transmission, simulation, sensors or radars, and beyond. They can be generated deterministically (e.g., by using polynomials, resulting in pseudo-random sequences) or in a non-deterministic way (e.g., by using physical noise sources like external devices or sensors, resulting in random sequences). Important characteristics of such binary sequences can be modelled by gap processes in conjunction with the probability theory. Recently, all-optical approaches have attracted a lot of research interest. In this work, an adaptation of the quantum key distribution setup is utilized for generating randomised bit sequences. The simulation results show that all-optically generated sequences very well resemble the theoretically ideal probability density characteristic. Additionally, an experimental optical setup is developed that confirms the simulation results. Furthermore, m-sequences show very promising results as well as Gold sequences. Additionally, the level of burstiness, i.e., the distribution of ones and zeros throughout the sequence, is studied for the different sequences. The results enable the finding that generator polynomials with concentrated non-zero coefficients lead to more bursty bit sequences.

ZEROS OF PARTITION FUNCTION FOR Z5-SYMMETRIC MODEL ON SQUARE LATTICE-PARTICULAR CASE

We study a statistical mechanics model of multiple-phase transitions called the ZQ-symmetric model on the square lattice for five possible spin directions (Q = 5). This study aims to investigate the existence of phase transition(s) and discuss the distribution of partition function zeros on the complex-temperature plane. Some cases of energy list χ are considered where they are written in an arbitrary arrangement of integer numbers (usually in decreasing value due to the angle of separation). The partition functions are computed using a transfer matrix approach, and their zeros are found numerically using the NewtonRaphson method. A Monte-Carlo Metropolis simulation is then performed to study the critical point of phase transition. The results are compared to the zeros distribution of the partition functions for the considered cases. Studying this generalised model is vital to mimic the system in the real world and to help experimental works use a more sustainable approach.

On the distribution of zeros of analytic functions in angles in C\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf{C} \\backslash \\{ {0}\\} $$\\end{document}

In this article some results on the value distribution theory of analyticfunctions defined in angles of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}$$\\end{document}, due mainly to B. Ja. Levin and A. Pfluger,will be extended to the more general situation where the functions are defined inangles of C\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\backslash\\{ 0\\}$$\\end{document}. More precisely, angles S(θ1,θ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S ( \ heta_{1},\ heta_{2}) $$\\end{document} with vertex at the origin will beconsidered and where a singularity at zero is allowed. An special class of thesefunctions are those of completely regular growth for which it is proved a basic resultwhich yields an expression of the density of its zeros in terms of the indicatorfunction.

A design method for filter to achieve arbitrary distribution of multiple transmission zeros without cross-coupling

This paper proposes a method for introducing transmission zeros (TZs) in filters using dual-mode resonators. By loading a dual-mode resonator as a second-order extracted-pole unit (EPU) at the input or output of the filter, a pair of finite TZs can be introduced without cross-coupling. This approach reduces the size of the filter while generating two finite transmission zeros compared to conventional methods. Since the dual-mode resonator is characterized by its separable odd and even modes, and by adjusting the resonant frequency of the EPU, it can achieve an arbitrary distribution of TZs. It is worth noting that the EPU does not affect the insertion loss of the filter because it operates mainly outside the passband. Furthermore, this work delves into the EPU’s mechanism for generating TZs and the method of adjusting TZ position by designing filters of second, fourth, sixth, and eighth orders, providing a solid foundation for the creation of higher-order filter circuits.

Zero Distribution of Finite Order Bank–Laine Functions

Zero Distribution of Finite Order Bank–Laine Functions

Applications of multiple orthogonal polynomials with hypergeometric moment generating functions

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their Q-linear independence.

Probabilistic zero bounds of certain random polynomials

Abstract This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy’s bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

Zeros of meromorphic functions of the form [formula omitted]

We study zeros distribution for meromorphic functions of the form ∑ncn(z−tn)2, where ∑n|cn||tn|2<∞. We prove an analog of a well-known theorem of Keldysh and discuss a relation between zero-free functions of this form and second order differential equations with polynomial coefficients.

Winding Number, Density of States, and Acceleration

Abstract Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schrödinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy, respectively. We will prove that the winding number is in fact Avila’s acceleration, it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schrödinger operators and Avila’s global theory. Communicating Editor: Prof Svetlana Jitomirskaya

Distributions of zeros and masses of entire and subharmonic functions with restrictions on their growth along the strip

Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complex plane $\mathbb C$. The following problem dates back to F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$, for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, there exists an entire function $f\neq 0$ of exponential type that vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $\mathrm Z$ and $\mathrm W$ (which lie only on the positive semiaxis). Several generalizations of this criterion were established by the author of the present paper in the late 1980s for "complex" $\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, we solve more involved problems in a more general subharmonic framework for distributions of masses on $\mathbb C$. All the previously mentioned results can be obtained from the main results of this paper in a much stronger form (even for the initial formulation for distributions of points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$ of exponential type). Some results of the present paper are closely related to the famous Beurling-Malliavin theorems on the radius of completeness and a multiplier.

Adjustable Jacobi-Fourier Moment for Image Representation.

The widely adopted Jacobi-Fourier moment (JFM) is limited by its inability to effectively capture spatial information. Although fractional-order JFM ( FOJFM) introduces spatial information through a fractional-order parameter, the control of spatial information remains inadequate. This limitation stems from the insufficient control over zeros distribution associated with the used moment's radial kernel. To address this issue, we generalize both JFM and FOJFM into a transformed JFM. A transformed function with four parameters is designed, and adjustable JFM (AJFM) is proposed. Two parameters correlate to increasing velocities on the left and right parts of the transformed functions, enabling zeros quantities of radial kernel fall in the left and right parts of the interval. The other two parameters segment the transformed function, adjusting regions where different quantities of zeros fall in. This refined control over the radial kernel's zero distribution enhances the versatility of feature extraction by the AJFM, governed by the introduced parameters. Experimental results demonstrate that AJFM, with properly chosen parameters, can emphasize specific regions within an image more effectively.

On the zero bounds of quaternionic polynomials with restricted coefficients

The primary objective of this paper is to explore extensions of the classical Enestorm-Kakeya theorem and its diverse generalizations regarding the distribution of polynomial zeros, transitioning from the complex domain to the quaternionic context.

Distribution of zeros of entire functions of exponential growth

Для систем уравнений с бесконечным числом корней иногда удается получить теоремы типа Кушниренко-Бернштейна-Хованского, заменяя вычисление числа корней на вычисление их асимптотической плотности. Мы рассматриваем системы целых функций экспоненциального роста в пространстве $\mathbb C^n$ и вычисляем асимптотику усредненного распределения корней в терминах геометрии выпуклых тел, расположенных в комплексном векторном пространстве. Библиография: 11 названий.

Adaptive fixed-time proximal gradient method for non-smooth optimization: the fractional approach

Non-smooth optimization has now played an important role in many fields, such as l1-regulation problem and sparse recovery problem. In this paper, a novel fractional fixed-time proximal gradient method is proposed for solving the composite optimization problems with non-smooth terms. At first, according to the distribution of zeroes of Mittag-Leffler functions, a novel fixed-time convergence theory with a fractional adaptive gain is proposed. Based on the optimal condition analysis of the transformed problem using the augmented Lagrangian function, a novel fixed-time proximal gradient method is then proposed and the fixed-time convergence property is rigorously proven. Furthermore, the proposed algorithm is applied to solve the sparse recovery problem in compressive sensing, and the restricted isometry condition required for the application of the algorithm is analyzed to achieve fixed-time recovery of the signal. Finally, simulation examples are provided to validate all the results.

Distribution of zeros of functions with exponential growth

For systems of equations with an infinite number of roots one can sometimes establish results of the type of the Kushnirenko-Bernstein-Khovanskii theorem by replacing the calculation of the number of the roots by the calculation of the asymptotic density of these roots. We consider systems of entire functions with exponential growth in $\mathbb C^n$ and calculate the asymptotic behaviour of the averaged distribution of their zeros in terms of the geometry of convex bodies in a complex vector space. Bibliography: 11 titles.

Fractional calculus of the Lerch zeta function – part II

This paper concerns the fractional derivative of the Lerch zeta function. The author already dealt with its functional equation. He reduced its computational cost and proved an approximate functional equation for this fractional derivative. Here, we study the mean square of this fractional derivative. Moreover, we deal with the distribution of zeros, showing some zero‐free regions.