More on regular and decomposable ultrafilters in ZFC

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: (a) If m ≥ 1 and the ultrafilter D is (_m(λ⁺ⁿ), _m(λ⁺ⁿ))‐regular, then D is κ ‐decomposable for some κ with λ ≤ κ ≤ 2^λ (Theorem 4.3(a^')). (b) If λ is a strong limit cardinal and D is (_m(λ⁺ⁿ), _m(λ⁺ⁿ))‐regular, then either D is (cf λ, cf λ)‐regular or there are arbitrarily large κ < λ for which D is κ ‐decomposable (Theorem 4.3(b)). (c) Suppose that λ is singular, λ < κ, cf κ ≠ cf λ and D is (λ⁺, κ)‐regular. Then: (i) D is either (cf λ, cf λ)‐regular, or (λ^', κ)‐regular for some λ^' < λ (Theorem 2.2). (ii) If κ is regular, then D is either (λ, κ)‐regular, or (ω, κ^')‐regular for every κ^' < κ (Corollary 6.4). (iii) If either (1) λ is a strong limit cardinal and λ^<λ < 2^κ, or (2) λ^<λ < κ, then D is either λ ‐decomposable, or (λ^', κ)‐regular for some λ^' < λ (Theorem 6.5). (d) If λ is singular, D is (μ, cf λ)‐regular and there are arbitrarily large ν < λ for which D is ν ‐decomposable, then D is κ ‐decomposable for some κ with λ ≤ κ ≤ λ^<μ (Theorem 5.1; actually, our result is stronger and involves a covering number). (e) D × D^' is (λ, μ)‐regular if and only if there is a ν such that D is (ν, μ)‐regular and D^' is (λ, ν^')‐regular for all ν^～ < ν (Proposition 7.1). We also list some problems, and furnish applications to topological spaces and to extended logics (Corollar‐ies 4.6 and 4.8) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sperner spaces and first‐order logic

We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of pointline incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo‐elementary class, it is not elementary nor even ??_∞ω‐axiomatizable. We also axiomatize the first‐order theory of this class.     

Pseudo asymptotically periodic solutions for Volterra integro‐differential equations

In this paper, we propose a new class of functions called pseudo ‐asymptotically ω‐periodic function in the Stepanov sense and explore its properties in Banach spaces including composition results. Furthermore, the existence and uniqueness of the pseudo ‐asymptotically ω‐periodic mild solutions to Volterra integro‐differential equations is investigated. Applications to integral equations arising in the study of heat conduction in materials with memory are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

h‐monotonically computable real numbers

Let h : ? → ? be a computable function. A real number x is called h‐monotonically computable (h‐mc, for short) if there is a computable sequence (x_s) of rational numbers which converges to x h‐monotonically in the sense that h(n)|x – x_n| ≥ |x – x_m| for all n andm > n. In this paper we investigate classes h‐ MC of h‐mc real numbers for different computable functions h. Especially, for computable functions h : ? → (0, 1)_?, we show that the class h‐ MC coincides with the classes of computable and semi‐computable real numbers if and only if Σ_i∈?(1 – h(i)) = ∞and the sum Σ_i∈?(1 – h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then h‐ MC = SC (the class of semi‐computable reals) no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h‐ MC = c‐ MC . Finally, if h is monotone and unbounded, then h‐ MC contains all ω‐mc real numbers which are g‐mc for some computable function g. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of μ−pseudo almost periodic solutions to some evolution equations

In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace‐Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of μ‐pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd.     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

μ‐Pseudo almost automorphic solutions of abstract fractional integro‐differential neutral equations with an infinite delay

The aim of this work is to study μ‐pseudo almost automorphic solutions of abstract fractional integro‐differential neutral equations with an infinite delay. Thanks to some restricted hypothesis on the delayed data in the phase space, we ensure the existence of the ergodic component of the desired solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the kernel space for an L0‐linear function

A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L⁰‐linear function f from a random normed module S to the algebra is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.     

On the Range of Two Convolution Operators

Let _(ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′_(ω)(ℝ) we give necessary conditions for the inclusion T_ν( _(ω)(ℝ)) ⊂ T_μ( _(ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis .     

(ω,c)-asymptotically periodic functions,first-order Cauchy problem,and Lasota-Wazewska model with unbounded oscillating production of red cells

In this paper, we study a new class of functions, which we call (ω,c)-asymptotically periodic functions. This collection includes asymptotically periodic, asymptotically antiperiodic, asymptotically Bloch-periodic, and unbounded functions. We prove that the set conformed by these functions is a Banach space with a suitable norm. Furthermore, we show several properties of this class of functions as the convolution invariance. We present some examples and a composition result. As an application, we prove the existence and uniqueness of (ω,c)-asymptotically periodic mild solutions to the first-order abstract Cauchy problem on the real line. Also, we establish some sufficient conditions for the existence of positive (ω,c)-asymptotically periodic solutions to the Lasota-Wazewska equation with unbounded oscillating production of red cells.     

Fixed points of (α,ψ)‐contractions in Hausdorff partial metric spaces

The concept of (α,ψ)‐contractions was introduced by Samet et al. in this paper, we introduce (α,ψ)‐generalized contractions in a Hausdorff partial metric space. We discuss its significance and obtain some common fixed point theorems for a pair of self‐mappings. Some examples are given to support the theory.     

Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ω‐categorical weakly o‐minimal expansions of Boolean lattices

We study ω‐categorical weakly o‐minimal expansions of Boolean lattices. We show that a structure ?? = (A,≤, ?) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o‐minimal if and only if it is ω‐categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ?. We propose other related examples of weakly o‐minimal ω‐categorical models in this framework, and we examine the internal structure of these models.     

Graphs and ranks of monoids

For a finite monoid S, let ν(S) (ν_d(S)) denote the least number n such that there exists a graph (directed graph) Γ of order n with End(Γ)?S. Also let rank(S) be the smallest number of elements required to generate S. In this paper, we use Cayley digraphs of monoids, to connect lower bounds of ν(S) (ν_d(S)) to the lower bounds of rank(S). On the other hand, we connect upper bounds of rank(S) to upper bounds of ν(S) (ν_d(S)).     

A partition property of a mixed type for Pκ(λ)

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on P_κ(λ) that satisfy the square brackets partition relation , where μ is a cardinal ≥2. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical calculation of the discrete spectra of one‐dimensional Schrödinger operators with point interactions

In this paper, we consider one‐dimensional Schrödinger operators S_q on with a bounded potential q supported on the segment and a singular potential supported at the ends h₀, h₁. We consider an extension of the operator S_q in defined by the Schrödinger operator and matrix point conditions at the ends h₀, h₁. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator . Moreover, we provide closed‐form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self‐adjoint and nonself‐adjoint problems involving general point interactions described in terms of δ‐ and δ′‐distributions.     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

χ‐bounded families of oriented graphs

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets of orientations of P₄ (the path on four vertices) similar statements hold. We establish some positive and negative results.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)