Space Of Meromorphic Functions Research Articles

Zeta statistics and Hadamard functions

We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any “natural” sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

Subordination and superordination preserving properties for families of integral operators for meromorphic functions

In this paper we obtain subordination, superordination, and sandwich-type results related to certain family of integral operators defined on the space of meromorphic functions in the open unit disk. Also, an application of the subordination and superordination theorems to the Gauss hypergeometric function are considered, and the main new results generalize some previously well-known sandwich-type theorems.

Degrees of Cohomology Classes of Multisingularities in Hurwitz Spaces of Rational Functions

Hurwitz spaces are spaces of meromorphic functions with prescribed orders of poles on curves of a given genus. In this article, we derive new formulas for the degrees of the strata of Hurwitz spaces of genus 0 corresponding to functions that have two degenerate critical values with prescribed partitions of multiplicities of the preimages. More precisely, one of the critical values has an arbitrary multiplicity of the preimages, while the other is the simplest degenerate critical value.

A Class of Nonlinear Integral Operators Preserving Subordination and Superordination

In this paper we investigate subordination-preserving of a certain class of integral operators, which are defined on the space of meromorphic functions. Moreover, we consider an application of the subordination Theorem to the Gauss hypergeometric function.

Power bounded composition operators on spaces of meromorphic functions

We study composition operators with holomorphic symbols defined on spaces of meromorphic functions, when endowed with their natural locally convex topology. First, we show that such operators are well-defined, continuous and never compact. Then, we study the dynamics and prove that a composition operator is power bounded or mean ergodic if and only if the symbol is a nilpotent element in the group of automorphisms.

SANDWICH-TYPE THEOREMS FOR A CLASS OF INTEGRAL OPERATORS ASSOCIATED WITH MEROMORPHIC FUNCTIONS

The purpose of the present paper is to investigate some subordination and superordination preserving properties of certain integral operators de ned on the space of meromorphic functions in the puncture open unit disk. The sandwich-type theorems for these integral operators are also presented.

On the value distribution of L-functions of the Selberg class

In this paper, two limit theorems in the sense of the weak convergence of probability measures in the space of meromorphic functions and on the complex plane are proved for L-functions from a subclass of the Selberg class. The explicit form of the limit measures is given.

Some Properties of Certain Class of Integral Operators

The main object of this paper is to derive some inequality properties and convolution properties of certain class of integral operators defined on the space of meromorphic functions.

Fredholmness of Toeplitz operators and corona problems

A meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left-invertible in a space of meromorphic functions. The Fredholmness of Toeplitz operators with symbol G ∈ ( L ∞ ( R ) ) 2 × 2 is shown to be equivalent to that of a Toeplitz operator with scalar symbol γ : = det G , provided that the Riemann–Hilbert problem G ϕ + M = ϕ − M admits a solution such that the meromorphic corona problems with data ϕ ± M are solvable. The Fredholm properties are characterized in terms of ϕ ± M and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators.

Subordination and superordination results for the family of Jung-Kim-Srivastava integral operators

In this paper, we derive some subordination and superordination results associated with the family of Jung-Kim-Srivastava integral operators defined on the space of meromorphic functions. Several sandwich-type results are also obtained. 2010 Mathematics Subject Classifications. Primary 30C45; Secondary 30C80. .

Some Subclasses of Meromorphic Functions Associated with a Family of Integral Operators

Making use of the principle of subordination between analytic functions and a family of integral operators defined on the space of meromorphic functions, we introduce and investigate some new subclasses of meromorphic functions. Such results as inclusion relationships and integral-preserving properties associated with these subclasses are proved. Several subordination and superordination results involving this family of integral operators are also derived.

A Class of Nonlinear Integral Operators Preserving Double Subordinations

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain integral operators defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also considered.

A new hierarchy of integrable systems associated to Hurwitz spaces

In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of "times". Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux-Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.

A class of integral operators preserving subordination and superordination for meromorphic functions

The purpose of the present paper is to investigate several subordination- and superordination-preserving properties of a certain class of integral operators, which are defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also presented. Moreover, we consider an application of the subordination and superordination theorem to the Gauss hypergeometric function.

A Joint Discrete Limit Theorem in the Space of Meromorphic Functions for General Dirichlet Series

A joint discrete limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for general Dirichlet series is proved.

On 𝐿¹-norms of meromorphic functions with fixed poles

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces K B 1 {K_B^1} generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with L 1 L^1 -norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition B ′ ∈ L ∞ ( R ) B’\in L^\infty (\mathbb {R}) for the boundedness of the differentiation on K B 1 {K_B^1} . Our main tool is a construction of an unconditional basis of rational fractions in K B 1 {K_B^1} .

Limit theorems for the Estermann zeta-function. II

A limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function is obtained.

The Segal–Bargmann transform for noncompact symmetric spaces of the complex type

We consider the generalized Segal–Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal–Bargmann transform is a unitary map onto a certain L 2 space of meromorphic functions. For general functions, we give an inversion formula for the Segal–Bargmann transform, involving integration against an “unwrapped” version of the heat kernel for the dual compact symmetric space. Both results involve delicate cancellations of singularities.

Towards an intersection theory on Hurwitz spaces

Moduli spaces of algebraic curves are closely related to Hurwitz spaces, that is, spaces of meromorphic functions on curves. All of these spaces naturally arise in numerous problems of algebraic geometry and mathematical physics, especially in connection with string theory and Gromov-Witten invariants. In particular, the classical Hurwitz problem of enumerating the topologically distinct ramified coverings of the sphere with prescribed ramification type reduces to the study of the geometry and topology of these spaces. The cohomology rings of such spaces are complicated even in the simple case of rational curves and functions. However, the cohomology classes most important for applications (namely, the classes Poincaré dual to the strata of functions with given singularities) can be expressed in terms of relatively simple “basic” classes (which are, in a sense, tautological). The aim of the present paper is to identify these basic classes, to describe relations between them, and to find expressions for the strata in terms of them. Our approach is based on Thom's theory of universal polynomials of singularities, which has been extended to the case of multisingularities by the first author. Although the general Hurwitz problem still remains open, our approach enables one to achieve significant progress towards its solution and an understanding of the geometry and topology of Hurwitz spaces.

Discrete limit theorems for general Dirichlet series. III

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given.