On infinite products of functions in compact riemann surfaces

Letν′ be the complementary of a point ∞ in a compact Riemann surfaceν. The normal convergence in compact subsets ofν′ of an infinite product of meromorphic functions (with polynomic exponential singularities at ∞ of bounded degree) is shown in this paper to be equivalent to a certain type of convergence of the double series of Newton sums of the divisors of its factors. This applies, for instance, to products of Baker functions inν′ and to products of meromorphic functions inν. The result for this last case is also generalized to complementaries of arbitrary nonvoid finite subsets ofν. Research supported by SA30/00B.     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

Mixed weak continuity on generalized topological spaces

We introduce the notion of mixed weak (μ,ν₁ν₂)-continuity between a generalized topology μ and two generalized topologies ν₁, ν₂. We characterize such continuity in terms of mixed generalized open sets: (ν₁,ν₂)′-semiopen sets, (ν₁,ν₂)′-preopen sets, (ν₁,ν₂)-preopen sets [2], (ν₁,ν₂)′-β′-open sets and θ(ν₁,ν₂)-open sets [3]. In particular, we show that for a given mixed weakly (μ,ν₁ν₂)-continuous function, if the codomain of the given function is mixed regular (=(ν₁,ν₂)-regular), then the function is also (μ,ν₁)-continuous.     

Büchi’s problem for ultrametric meromorphic functions inside an open disk

Let K be a complete ultrametric algebraically closed field of characteristic zero such as ℂ _p . Büchi’s problem was solved for p-adic meromorphic functions in the whole field K. Here we show similar conclusions for meromorphic functions in an open disk that are not quotients of bounded analytic functions. The main method is the secondMain Theorem for p-adic meromorphic functions inside a disk, a specific p-adic theorem.     

A class of second order differential equations

We study the differential equationf″=N(f)f′ ² +M(f)f′+L(f), whereL, M, N are rational functions, and prove that if the differential equation has a transcendental meromorphic solutionf with order,p(f)>2, then the differential equation must be one of nine forms; and, moreover, we construct examples showing the existence of these nine forms with a transcendental meromorphic solution.     

A theorem on cyclic polytopes

LetC(ν, d) represent a cyclic polytope withν vertices ind dimensions. A criterion is given for deciding whether a given subset of the vertices ofC(ν, d) is the set of vertices of some face ofC(ν, d). This enables us to determine, in a simple manner, the number ofj-faces ofC(ν, d) for each value ofj (1≦j≦d−1).     

Normal families and fixed points

LetF be a family of meromorphic functions in a domainD and letk≥2 be a positive integer. If, for everyf ∈F, itsk-th iteratef ^k has no fixed point inD, thenF is normal inD. Supported by the NNSF of China (Grant No. 10471065), the SRF for ROCS, SEM., and the Presidential Foundation of South China Agricultural University.     

Normality and shared values

LetF be a family of meromorphic functions on the unit disc Δ and leta andb be distinct values. If for everyf∈F,f andf′ sharea andb on Δ, thenF is normal on Δ. The first author was supported by NNSF of China approved no. 19771038 and by the Research Institute for Mathematical Sciences, Bar-Ilan University.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

Dye’s theorem in the almost continuous category

Suppose X and Y are Polish spaces with non-atomic Borel probability measures μ and ν and suppose that T and S are ergodic measure-preserving homeomorphisms of (X, μ) and (Y, ν). Then there are invariant G _δ subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a homeomorphism ϕ: X′ → Y′ which maps μ|X′ to ν|Y′ and maps T-orbits onto S-orbits. We also deal with the case where T and S preserve infinite invariant measures.     

Escaping points of meromorphic functions with a finite number of poles

We establish several new properties of the escaping setI(f)={z∶f ⁿ (z)→∞ andf ⁿ(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)⊂J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.     

Partitioning series-parallel multigraphs into ν *-excluding edge covers

We prove that, for any given vertex ν* in a series-parallel graph G, its edge set can be partitioned into κ = min{κ′(G) + 1, δ(G)} subsets such that each subset covers all the vertices of G possibly except for ν*, where δ(G) is the minimum degree of G and κ′(G) is the edge-connectivity of G. In addition, we show that the results in this paper are best possible and a polynomial time algorithm can be obtained for actually finding such a partition by our proof.     

Two-sided bounds uniform in the real argument and the index for modified Bessel functions

Bounds uniform in the real argument and the index for the functionsa _ν (x)=xI′ _ν (x)/I′ _ν (x) andb _ν (x)=xK′ _ν (x)/K _ν (x), as well as for the modified Bessel functionsI _ν(x) andK _ν(x), are established in the quadrantx>0, ν≥0, except for some neighborhoods of the pointx=0, ν=0. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 681–692, May, 1999.     

On the growth of meromorphic functions with a radially distributed value

The purpose of this paper is to investigate the growth of meromorphic functions with a radially distributed value. The paper is closely related to a more recent result due to Fang and Zalcman [M.L. Fang and L. Zalcman, On the value distribution of f + a(f′) ⁿ , Sci. China Ser. A-Math. 38(2008), 279–285].     

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions inL ^p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2. This paper is a part of the author’s Ph.D. thesis supervised by V. Bergelson.     

A local extension theorem for proper holomorphic mappings in ℂ<Superscript>2</Superscript>

Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ².Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ².     

On the number of vertices and edges of the Buneman graph

In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph. In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is, that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible.     

Projections of 3-polytopes

Given any 3-dimensional convex polytopeP, and any simple circuitC in the 1-skeleton ofP, there is a convex polytopeP′ combinatorially equivalent toP, and a direction such that ifP′ is projected orthogonally in this direction, then the inverse image of the boundary of the projection is the circuit inP′ corresponding to the circuitC inP. Research supported by NSF Grant GP-8470.     

Fundamental solution of the cauchy problem corresponding to the one-speed linear Boltzman equation for anisotropic media

We consider the fundamental solution E (t,x,s;s ₀) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t+c(s,grad _x)+γ)E(t,x,s;s ₀)=γν∫ f((s, s′))E(t,x,s′; s₀)ds′+Ωδ(t)δ(x)δ (s−s ₀) that is assumed to be valid for any (t,x)∈Rⁿ⁺¹; morevoer, for t<0 the condition E(t,x,s; s₀)=0 holds. By using the Fourier-laplace transform in space-time arguments, the problem reduces to the study of an integral equation in the variables. For 0<ν≤1, the uniqueness and existence of the solution of the original problem are proved for any fixeds in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic type (f(.)=1), the solution of the integral equation is given in explicit form. In the approximation of “small mean-free paths,” various weak limits of the solution are obtained with the help of a Tauberian-type theorem, for distributions. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 319–332. Translated by Yu. B. Yanushanets.     

Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T _k :L ₂(ν)→L ₂(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T _k . An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions. It is well known that Mercer’s theorem has found important applications in various branches of mathematics, including probability theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of T _k in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L _∞(ν). For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L ₂(ν)→L ₂(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.