Hurwitz Type Research Articles

Proofs of some conjectural Hurwitz type identities for three squares due to Cooper and Hirschhorn

Let rs(n) denote the number of representations of n as the sum of s squares of integers. Hurwitz provided eleven cases in which the generating function of r3(an+b) is a simple infinite product. In 2004, Cooper and Hirschhorn proved eleven identities of Hurwitz, another twelve of the same sort and eighty infinite families of similar results by utilizing some q-series techniques. Moreover, they further conjectured eighty-two infinite families of Hurwitz type identities for r3(n). In this paper, we confirm these conjectural infinite families of Hurwitz type identities by using the theory of modular forms.

Generating Function of q- and Elliptic Multiple Polylogarithms of Hurwitz Type

Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_2F_1$$\\end{document}. As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function r+1Fr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_{r+1}F_r$$\\end{document}. In this paper, we establish q- and elliptic analogues of this result. We introduce elliptic q-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain q- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.

Calibration Estimation of Population Proportion in Probability Proportional to Size Sampling in the Presence of Non-Response

Objectives: In this article, we addressed the problem of estimation of the finite population proportion under the Probability Proportional to Size (PPS) sampling technique, when the complete information is unavailable due to the presence of non-response. We developed calibrated estimators of the population proportion under PPS sampling in the presence of nonresponse based on the availability of auxiliary information. Methods: The expressions for the mean squared errors of the suggested estimators were developed to the first order of approximation. The developed estimators of the population proportion are compared with the design-based Horvitz-Thompson estimator and Horvitz-Thompson type calibration estimator which were obtained on the complete response units along with the design-based Hansen and Hurwitz type estimator in the presence of non-response. A Simulation study has also been conducted to support the performance of the developed estimators of population proportion with the help of two real datasets, by computing Percentage Absolute Relative Bias (%ARB) and Percentage Relative Root Mean Squared Error (%RRMSE) using R software. Findings: The simulation study supported the performance of the developed estimators of the finite population proportion based on %ARB and %RRMSE. The proposed calibration estimators of population proportion are more efficient than the other considered estimators in the presence of non-response. Novelty: The proposed new calibrated estimators have practical implications in the estimation of finite population proportions. Keywords: Auxiliary information, Calibration Approach, Non­response, Population proportion, PPS sampling

A new Schwarz-Pick lemma at the boundary and rigidity of holomorphic maps

In this paper we establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in CN in the spirit of the boundary Schwarz lemma of Burns–Krantz. Firstly, we focus on the case of the unit disk and prove a general boundary rigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases this result already includes new types of boundary versions of the lemmas of Schwarz–Pick, Ahlfors–Schwarz and Nehari–Schwarz. The proof is based on a new Harnack–type inequality as well as a boundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity results of Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theorems for sequences of conformal pseudometrics, which even in the interior case appear to be new. For instance, a first sequential version of the strong form of Ahlfors' lemma is obtained. As an auxiliary tool we establish a Hurwitz–type result about preservation of zeros of sequences of conformal pseudometrics. Thirdly, we apply the one–dimensional sequential boundary rigidity results together with a variety of techniques from several complex variables to prove a boundary version of the Schwarz–Pick lemma for holomorphic maps of strongly convex domains in CN for N>1.

ON THE DYNAMICS OF THE HODGKIN-HUXLEY NEURAL MODEL: A STABLE ANALYSIS USING HURWITZ POLYNOMIALS

In this paper, a study of the dynamics of the Hodgkin-Huxley neural model is conducted. This stability study is performed using the stability criteria for obtaining Hurwitz polynomials that provide necessary and/or sufficient conditions to analyze the dynamics of the model by studying the location of the roots of the characteristic polynomial associated with it. The main objective of the article is to analyze the stability of the Hodgkin-Huxley neural model using an analytical technique to establish Hurwitz type polynomials. This article describes and presents an analytical method to analyze the stability of the Hodgkin-Huxley neural model based on obtaining Hurwitz polynomials through stability criteria. This technique allows establishing the asymptotic stability of the neural model through the localization of the Eigen values of the matrix associated with the model. The results of this research allow establishing asymptotic stability conditions of the Hodgkin-Huxley neural model that allow analyzing the dynamics of the model. Through this article, it is possible to analyze the dynamics of the Hodgkin-Huxley neural model and allows drawing inferences regarding the initiation and transmission of action potentials in neurons and cardiac cells using the Hurwitz polynomial technique.

Generating function of multiple polylog of Hurwitz type

AbstractWe introduce interpolated multiple Hurwitz polylogs and interpolated multiple Hurwitz zeta values. In addition, we discuss the generating functions for the sum of the polylogs/zeta values of fixed weight, depth, and all heights. The functions are expressed in terms of generalized hypergeometric functions. Compared with the pioneering results of Ohno and Zagier on the generating function, our setup generalizes the results in three directions, namely, at general heights, with a t-interpolation, and as a Hurwitz type. As an application, by fixing the Hurwitz parameter to rational numbers, the generating functions for multiple zeta values with level are given.

AN APPROACH TO THE STABILITY OF THE GAUSE-LOTKA-VOLTERRA COMPETITION ECOLOGICAL MODEL USING THE ROUTH-HURWITZ CRITERION

In the qualitative study of systems of ordinary differential equations, we can analyze the dynamics of the solutions by observing whether small variations or modifications in the initial conditions produce small changes in the future, this intuitive idea was formalized and studied by Lyapunov and reflects the concept of stability. For both linear and nonlinear systems of differential equations, we can analyze stability using criteria to obtain Hurwitz type polynomials. In this paper, we study the stability of a prey-predator model where three species compete considering the Gause effect generated by the Gause-Lotka-Volterra system, using an analytical stability technique to obtain Hurwitz-type polynomials called Routh-Hurwitz criterion. This criterion provides the necessary and/or sufficient conditions to analyze the dynamics of the system by studying the location of the roots of the characteristic polynomial associated with the system. This article describes and presents an analytical method to analyze the stability of the Gause-Lotka-Volterra model, based on the study of Hurwitz type polynomials. This technique allows establishing the asymptotic stability of the system by analyzing the location of the eigenvalues of the matrix associated with said system. The research results allow establishing a region of asymptotic stability of the Gause-Lotka-Volterra model that allows analyzing the interaction dynamics between the species.

A note on the stability of a modified Lotka-Volterra model using Hurwitz polynomials

In the analysis of the dynamics of the solutions of ordinary differential equations we can observe whether or not small variations or perturbations in the initial conditions produce small changes in the future; this intuitive idea of stability was formalized and studied by Lyapunov, who presented methods for the stable analysis of differential equations. For linear or nonlinear systems, we can also analyze the stability using criteria to obtain Hurwitz type polynomials, which provide conditions for the analysis of the dynamics of the system, studying the location of the roots of the associated characteristic polynomial. In this paper we present a stability study of a Lotka-Volterra type model which has been modified considering the carrying capacity or support in the prey and time delay in the predator, this stable analysis is performed using stability criteria to obtain Hurwitz-type polynomials.

A generalized regularization theorem and Kawashima's relation for multiple zeta values

Kawashima's relation is conjecturally one of the largest classes of relations among multiple zeta values. Gaku Kawashima introduced and studied a certain Newton series, which we call the Kawashima function, and deduced his relation by establishing several properties of this function. We present a new approach to the Kawashima function without using Newton series. We first establish a generalization of the theory of regularizations of divergent multiple zeta values to Hurwitz type multiple zeta values, and then relate it to the Kawashima function. Via this connection, we can prove a key property of the Kawashima function to obtain Kawashima's relation.

Adaptive Cluster Sampling-Based Design for Estimating COVID-19 Cases With Random Samples

During the COVID-19 pandemic, testing of all persons except those who are symptomatic, is not feasible due to shortage of facilities and staff This article focuses on estimating the number of COVID-19-positive persons over a geographical domain The Horvitz– Thompson and Hansen–Hurwitz type estimators under adaptive cluster sampling-based design have been suggested Two case studies are discussed to demonstrate the performance of the estimators under certain assumptions Advantages and limitations are also mentioned © 2021

Fine gradings on Kantor systems of Hurwitz type

We give a classification up to equivalence of the fine group gradings by abelian groups on the Kantor pairs and triple systems associated with Hurwitz algebras (i.e., unital composition algebras), under the assumption that the base field is algebraically closed of characteristic different from 2. The universal groups and associated Weyl groups are computed. We also determine, in the case of Kantor pairs, the induced (fine) gradings on the associated Lie algebras given by the Kantor construction.

Incomplete Multi-Poly-Bernoulli Numbers and Multiple Zeta Values

Using the restricted and associated Stirling numbers of the second kind, we define the incomplete multi-poly-Bernoulli numbers which generalize poly-Bernoulli numbers. We study their analytic and combinatorial properties. As an application, we present a new infinite series representation of the multiple zeta values via the Lambert W-function. We also give similar results for incomplete Bernoulli numbers of Hurwitz type and incomplete multi-poly-Bernoulli-star numbers.

A note on certain maximal curves

ABSTRACTWe characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely . We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of type through class field theory.

On complex zeros off the critical line for non-monomial polynomial of zeta-functions

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelof hypothesis for the Riemann zeta-function is equivalent to the Lindelof hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

Total nonnegativity of matrices related to polynomial roots and poles of rational functions

In this paper totally nonnegative (positive) matrices are considered which are matrices having all their minors nonnegative (positive); the almost totally positive matrices form a class between the totally nonnegative matrices and the totally positive ones. An efficient determinantal test based on the Cauchon algorithm for checking a given matrix for falling in one of these three classes of matrices is applied to matrices which are related to roots of polynomials and poles of rational functions, specifically the Hankel matrix associated with the Laurent series at infinity of a rational function and matrices of Hurwitz type associated with polynomials. In both cases it is concluded from properties of one or two finite sections of the infinite matrix that the infinite matrix itself has these or related properties. Then the results are applied to derive a sufficient condition for the Hurwitz stability of an interval family of polynomials. Finally, interval problems for a subclass of the rational functions, viz. R-functions, are investigated. These problems include invariance of exclusively positive poles and exclusively negative roots in the presence of variation of the coefficients of the polynomials within given intervals.

The Lazard formal group, universal congruences and special values of zeta functions

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist–Meurman–type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group (based on earlier works of the author). Their role in the theory of L L –genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann–Hurwitz–type zeta functions.

Hurwitz Type Results for Sum of Two Triangular Numbers

Let t_2(n) denote the number of representations of n as a sum of two triangular numbers and t_(a,b)(n) denote number of representations of n as a sum of a times triangular number and b times triangular number. In this paper, we prove number of results in which generating functions of t_2(n) and t_(1,3) are in�finite product. We also establish relations between t_(1,3)(n); t_(1,12)(n); t_(3,4)(n); t_2(n) and t_(1,4)(n).

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

This paper is a direct continuation of the author's recent investigations [4] on the non-degenerate truncated matricial Stieltjes problem. Inspired by a characterization of scalar Hurwitz polynomials via continued fractions (see e.g. Gantmacher [15]) a class of matrix Hurwitz type polynomials is introduced. It is shown that this class is intimately related with Stieltjes positive definite sequences of matrices and can be expressed in terms of the associated Stieltjes quadruple of sequences of left orthogonal matrix polynomials. It is shown that a monic matrix polynomial is a matrix Hurwitz type polynomial if and only if the sequence of its Markov parameters is a Stieltjes positive definite sequence.

Hidden structures of knot invariants

We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion and discuss its properties. We also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Then we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.

Total Nonnegativity of Infinite Hurwitz Matrices of Entire and Meromorphic Functions

In this paper we completely describe functions generating the infinite totally nonnegative Hurwitz matrices. In particular, we generalize the well-known result by Asner and Kemperman on the total nonnegativity of the Hurwitz matrices of real stable polynomials. An alternative criterion for entire functions to generate a Pólya frequency sequence is also obtained. The results are based on a connection between a factorization of totally nonnegative matrices of the Hurwitz type and the expansion of Stieltjes meromorphic functions into Stieltjes continued fractions (regular $$C$$ -fractions with positive coefficients).