Weak saturation of ideals on Pκ(λ)

We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no (proper, fine, κ‐complete) ideal on P_κ(λ) is weakly λ⁺‐saturated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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)     

A note on parameter free Π1‐induction and restricted exponentiation

We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})$ (= Boolean combinations of Σ₁) theorems of IΠ^?₁ in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ^?_{n + 1} is conservative over IΣ^?_n with respect to $\mathcal {B}(\Sigma _{n+1})$ sentences cannot be extended to Π_{n + 2} sentences. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the Roots of σ‐Polynomials

Given a graph G of order n, the σ‐polynomial of G is the generating function where is the number of partitions of the vertex set of G into i nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729–756) proved that σ‐polynomials of graphs with chromatic number at least had all real roots, and conjectured the same held for chromatic number . We affirm this conjecture.     

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)     

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.     

Binary types in ℵ0‐categorical weakly o‐minimal theories

Orthogonality of all families of pairwise weakly orthogonal 1‐types for ?₀‐categorical weakly o‐minimal theories of finite convexity rank has been proved in 6 . Here we prove orthogonality of all such families for binary 1‐types in an arbitrary ?₀‐categorical weakly o‐minimal theory and give an extended criterion for binarity of ?₀‐categorical weakly o‐minimal theories (additionally in terms of binarity of 1‐types). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Index sets for ω‐languages

An ω‐language is a set of infinite sequences (words) on a countable language, and corresponds to a set of real numbers in a natural way. Languages may be described by logical formulas in the arithmetical hierarchy and also may be described as the set of words accepted by some type of automata or Turing machine. Certain families of languages, such as the languages, may enumerated as P₀, P₁, … and then an index set associated to a given property R (such as finiteness) of languages is just the set of e such that P_e has the property. The complexity of index sets for 7 types of languages is determined for various properties related to the size of the language.     

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.     

χ‐bounds,operations, and chords

A long unichord in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is long unichord free if it does not contain any long unichord. We prove a structure theorem for long unichord free graph. We give an time algorithm to recognize them. We show that any long unichord free graph G can be colored with at most colors, where ω is the maximum number of pairwise adjacent vertices in G.     

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)     

Perfect coloring and linearly χ‐bound P6‐free graphs

We derive decomposition theorems for P₆, K₁ + P₄‐free graphs, P₅, K₁ + P₄‐free graphs and P₅, K₁ + C₄‐free graphs, and deduce linear χ‐binding functions for these classes of graphs (here, P_n (C_n) denotes the path (cycle) on n vertices and K₁ + G denotes the graph obtained from G by adding a new vertex and joining it with every vertex of G). Using the same techniques, we also obtain an optimal χ‐binding function for P₅, C₄‐free graphs which is an improvement over that given in [J. L. Fouquet, V. Giakoumakis, F. Maire, and H. Thuillier, 11 , Discrete Math, 146, 33–44.]. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 293–306, 2007     

On quasi‐varieties of multiple valued logic models

We extend the concept of quasi‐variety of first‐order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi‐varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi‐varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the complexity of axiomatizations of the class of representable quasi‐polyadic equality algebras

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omegaUsing games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEA _α of representable quasi‐polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka’s methods of splitting, to adapt the quasi‐polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEA _n for $2< n <\omega$ and $l< n,$ $k < n$, k′ < ω are natural numbers, then Σ contains infinitely equations in which ? occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A (T,ψ)‐ψe decoupled scheme for a time‐dependent multiply‐connected eddy current problem

The aim of this paper is to develop a fully discrete ( T ,ψ)‐ψ_e finite element decoupled scheme to solve time‐dependent eddy current problems with multiply‐connected conductors. By making ‘cuts’ and setting jumps of ψ_e across the cuts in nonconductive domain, the uniqueness of ψ_e is guaranteed. Distinguished from the traditional T ‐ ψ method, our decoupled scheme solves the potentials T and ψ‐ ψ_e separately in two different simple equation systems, which avoids solving a saddle‐point equation system and leads to a remarkable reduction in computational efforts. The energy‐norm error estimate of the fully discrete decoupled scheme is provided. Finally, the scheme is applied to solve two benchmark problems—TEAM Workshop Problems 7 and IEEJ model. Copyright © 2013 John Wiley & Sons, Ltd.     

Order plus size of τ‐critical graphs

Let $G=\left(V,E\right)$ be a $\tau$‐critical graph with $\tau \left(G\right)=t$. Erd?s and Gallai proved that $\mid V\mid \le 2t$ and the bound $\mid E\mid \le \left(\genfrac{}{}{0ex}{}{t+1}{2}\right)$ was obtained by Erd?s, Hajnal, and Moon. We give here the sharp combined bound $\mid E\mid +\mid V\mid \le \left(\genfrac{}{}{0ex}{}{t+2}{2}\right)$ and find all graphs with equality.     

Categorical Abstract Algebraic Logic: Algebraic Semantics for (\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf{\pi }$\end{document})‐Institutions

Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π‐institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N‐rule based extension of an N‐rule based π‐institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok‐Rebagliato‐style characterization of those π‐institutions with a mono‐unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π‐institution to possess an algebraic semantics is provided.     

ω‐saturated quasi‐minimal models of Th(ℚω, +, σ, 0)

We show that (ℚ^ω, +, σ, 0) is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Amalgams and χ‐Boundedness

A class of graphs is hereditary if it is closed under isomorphism and induced subgraphs. A class of graphs is χ‐bounded if there exists a function such that for all graphs , and all induced subgraphs H of G, we have that . We prove that proper homogeneous sets, clique‐cutsets, and amalgams together preserve χ‐boundedness. More precisely, we show that if and are hereditary classes of graphs such that is χ‐bounded, and such that every graph in either belongs to or admits a proper homogeneous set, a clique‐cutset, or an amalgam, then the class is χ‐bounded. This generalizes a result of [J Combin Theory Ser B 103(5) (2013), 567–586], which states that proper homogeneous sets and clique‐cutsets together preserve χ‐boundedness, as well as a result of [European J Combin 33(4) (2012), 679–683], which states that 1‐joins preserve χ‐boundedness. The house is the complement of the four‐edge path. As an application of our result and of the decomposition theorem for “cap‐free” graphs from [J Graph Theory 30(4) (1999), 289–308], we obtain that if G is a graph that does not contain any subdivision of the house as an induced subgraph, then .     

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.