The class of infinite dimensional neat reducts of quasi‐polyadic algebras is not axiomatizable

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi‐polyadic algebras and quasi‐polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α ‐dimensional neat reducts of algebras in K_β is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cut‐elimination Theorems for Some Infinitary Modal Logics

In this article, a cut‐free system TLM_ω1 for infinitary propositional modal logic is proposed which is complete with respect to the class of all Kripke frames.The system TLM_ω1 is a kind of Gentzen style sequent calculus, but a sequent of TLM_ω1 is defined as a finite tree of sequents in a standard sense. We prove the cut‐elimination theorem for TLM_ω1 via its Kripke completeness.     

A Modeltheoretic Solution to a Problem of Tarski

Let 1 < n < ω and β > n. We show that the class Nr_nCA_β of n‐dimensional neat reducts of β‐dimensional cylindric algebras is not closed under forming elementary subalgebras. This solves a long‐standing open problem of Tarski and his co‐authors Andréka, Henkin, Monk and Németi. The proof uses genuine model‐theoretic arguments.     

On the boundedness of some potential‐type operators with oscillating kernels

We consider a class of multidimensional potential‐type operators with kernels that have singularities at the origin and on the unit sphere and that are oscillating at infinity. We describe some convex sets in the (1/p, 1/q)‐plane for which these operators are bounded from L_p into L_q and indicate domains where they are not bounded. We also reveal some effects which show that oscillation and singularities of the kernels may strongly influence on the picture of boundedness of the operators under consideration. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. 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)     

Spaces of Bessel‐potential type and embeddings: the super‐limiting case

We consider Bessel‐potential spaces modelled upon Lorentz‐Karamata spaces and establish embedding theorems in the super‐limiting case. In addition, we refine a result due to Triebel, in the context of Bessel‐potential spaces, itself an improvement of the Brézis‐Wainger result (super‐limiting case) about the “almost Lipschitz continuity” of elements of H^1+n/p_p (?ⁿ). These results improve and extend results due to Edmunds, Gurka and Opic in the context of logarithmic Bessel potential spaces. We also give examples of embeddings of Besselpotential type spaces which are not of logarithmic type. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

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)     

Ideal properties of the Dunford integration operator

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F‐valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach‐Saks property, separable Banach spaces not containing c₀, Banach spaces not containing c₀ or ?₁ and Asplund spaces not containing c₀.     

Pseudo asymptotically periodic mild solutions of semilinear functional integro‐differential equations in Banach spaces

In this paper, we introduce and investigate the functions of (μ,ν)‐pseudo S‐asymptotically ω‐periodic of class r(class infinity). We systematically explore the properties of these functions in Banach space including composition theorems. As applications, we establish some sufficient criteria for (μ,ν)‐pseudo S‐asymptotic ω‐periodicity of (nonautonomous) semilinear integro‐differential equations with finite or infinite delay. Finally, some interesting examples are presented to illustrate the main findings.     

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)     

On realization of the Kreĭn–Langer class Nκ of matrix‐valued functions in Pontryagin spaces

In this paper the realization problems for the Kre?n–Langer class N_κ of matrix‐valued functions are being considered. We found the criterion when a given matrix‐valued function from the class N_κ can be realized as linear‐fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii–Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space Π_κ with indefinite metric. We specify three subclasses of the class N_κ (R) of all realizable matrix‐valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in Π_κ . Alternatively we show that the class N_κ (R) can be realized as transfer matrix‐functions of some canonical impedance systems with self‐adjoint main operators in rigged spaces Π_κ . The case of scalar functions of the class N_κ (R) is considered in details and some examples are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the well‐posedness for Keller–Segel system with fractional diffusion

In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u₀,v₀) in critical Fourier‐Herz spaces with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd.     

A discrete representation of free MV‐algebras

We prove that the m ‐generated free MV‐algebra is isomorphic to a quotient of the disjoint union of all the m ‐generated free MV⁽ⁿ⁾‐algebras. Such a quotient can be seen as the direct limit of a system consisting of all free MV⁽ⁿ⁾‐algebras and special maps between them as morphisms (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal extensions of Π01 classes

A minimal extension of a Π⁰₁ class P is a Π⁰₁ class Q such that P ? Q, Q – P is infinite, and for any Π⁰₁ class R, if P ? R ? Q, then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q′ have the same Cantor‐Bendixson derivative. We show that for any class P which has a single limit point A, and that point of degree ≤ 0 , P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q. (© 2005 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.     

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)     

On the trace problem for Lizorkin–Triebel spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x₁ = 0 and x_n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x₁ = 0. For x₁ the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x_n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gδ‐pieces of canonical partitions of G‐spaces

Generalizing model companions from model theory we define companions of pieces of canonical partitions of Polish G‐spaces. This unifies several constructions from logic. The central problem of the paper is the existence of companions which form a G‐orbit which is a G_δ‐set. We describe companions of some typical G‐spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)