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).     

The mixed joint universality for a class of zeta‐functions

A mixed joint limit theorem in the space of holomorphic functions, and a mixed joint universality theorem are proved for a rather general class of Euler products and periodic Hurwitz zeta‐functions.     

Hurwitz rational functions

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establish an analog of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.     

The joint value distribution of the Riemann zeta function and Hurwitz zeta functions II

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions. Received: 17 Apri1 2007     

Everywhere Differentiable Functions without Monotonicity Intervals and Transcendental Numbers

The class of everywhere differentiable functions without monotonicity intervals is considered in terms of number theory. A number-theoretic representation of the set of points of the unit interval is constructed using the classification of transcendental numbers proposed by K. Mahler, and a theorem on sufficient conditions for differentiable functions to belong to this class is stated. Results concerning the behavior of derivatives of functions from this class are presented. A mixed problem for the heat equation modeling heat transfer in a distributed system is considered. It is shown that the control function for this system can be everywhere differentiable but having no monotonicity intervals.     

Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions

A new class of generalized convex set-valued functions, termed nearly-subconvexlike functions, is introduced. This class is a generalization of cone-subconvexlike maps, nearly-convexlike set-valued functions, and preinvex set-valued functions. Properties for the nearly-subconvexlike functions are derived and a theorem of the alternative is proved. A Lagrangian multiplier theorem is established and two scalarization theorems are obtained for vector optimization.     

Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order

To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.     

Inequalities for Rational Functions

A new hyperbolic area estimate for the level sets of finite Blaschke products is presented. The following inversion of the usual Sobolev embedding theorem is proved:

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Hereris a rational function of degreenwith poles outside . This estimate implies a new inverse theorem for rational approximation of analytic functions with respect to areaL^pnorm.     

实轴上的双解析函数Riemann边值逆问题

提出了一类实轴上的双解析函数Riemann边值逆问题.先消去参变未知函数,再采用易于推广的矩阵形式记法,可把问题转化为两个实轴上的解析函数Riemann边值问题.利用经典的Riemann边值问题理论,讨论了该问题正则型情况的解法,得到了它的可解性定理.     

Superparabolic Functions Related to Second Order Hypoelliptic Operators

In this paper, we consider a wide class of second order hypoelliptic partial differential operators with nonnegative characteristic form. We prove a monotone smoothing theorem and a representation formula for superparabolic functions.     

On certain generating functions in positive characteristic

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums.     

Iteration of Analytic Normal Functions of Matrices

In this paper, the author proves that the classical theorem of Wolff in the theory of complex functions may be extended to the class of operator-valued functions f, where f is an analytic function from the open unit disc \Delta in the complex plane into a family of commutative normal operators on a certain n-dimensional complex Hilbert space, and ||f(z)||<1 holds for every z in \Delta.     

On the convergence of solutions of certain inhomogeneous fourth-order differential equations

The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function h lies in a closed subinterval of the Routh–Hurwitz interval.     

Continuity of Harmonic Functions for Non-local Markov Generators

We consider a class of pure jump Markov processes in ${\mathbb R}^d$ whose jump kernels are comparable to those of symmetric stable processes. We prove a support theorem, a lower bound on the occupation times of sets, and show that we can approximate resolvents using smooth functions.     

A limit theorem for the Hurwitz zeta function with an algebraic irrational parameter

A limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for the Hurwitz zeta function with an algebraic irrational parameter is obtained. Bibliography 6 titles. To the memory of Professor Yu. V. Linnik Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 125–134.     

On the Hurwitz Zeta Functions with Algebraic Irrational Parameter

Mathematical Notes - is well known that the Hurwitz zeta function ζ(s, α) with rational or transcendental parameter α is universal in the sense of Voronin, i.e., a wide class of...     

Omniscience Principles and Functions of Bounded Variation

A very weak omniscience principle is formulated, related omniscience principlesare considered, and the theorem that a function of bounded variation is the difference of two increasing functions is shown to be equivalent to the omniscience principle WLPO. It is a so shown that an arbitrary function (not necessarily strongly extensional) with located variation on an interval is the difference of two increasing functions.     

On an algebraic version of the Knizhnik–Zamolodchikov equation

A difference equation analogue of the Knizhnik?CZamolodchikov equation is exhibited by developing a theory of the generating function H(z) of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which H(z) is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples of non-positive integers. A generating function for these polynomials satisfies a similar difference equation to that of H(z), but in contrast to the fact that said polynomials have rational coefficients, the algebraic independence of the usual Hurwitz zeta functions is proven, and the Hurwitz polyzeta functions are shown to satisfy no algebraic relations other than those arising from the shuffle relations. The values of the NMBPs at z=1 provide a regularization of the multiple zeta values at tuples of negative integers, which is shown to agree with the regularization given in Akiyama et al. (Acta Arith. 98:107?C116, 2001). Various elementary properties of these values are proven.