Property (quasi‐α) and the denseness of norm attaining mappings

We introduce property (quasi ‐α), which implies property (A) defined by Lindenstrauss [10] and whose dual property is property (quasi‐β) [2]. We consider relations between this property and other sufficient conditions for property (A), and study the denseness of norm attaining mappings under the conditions of these properties. In particular, if each of the Banach spaces X_k , 1 ≤ k ≤ n – 1, has property (quasi‐α) and X_n has property (A), then the projective tensor product X₁ ··· X_n has property (A). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A preservation theorem for theories without the tree property of the first kind

We prove the NTP₁ property of a geometric theory T is inherited by theories of lovely pairs and H‐structures associated to T . We also provide a class of examples of nonsimple geometric NTP₁ theories.     

A note on a lowest order divergence‐free Stokes element on quadrilaterals

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P₁ spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q₁‐P₀ element is also stable. Although this method is a subspace method of Qin‐Zhang's P₁‐P₀ element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.     

Closure and stable Hamiltonian properties in claw‐free graphs

In the class of k‐connected claw‐free graphs, we study the stability of some Hamiltonian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an infinite family of graphs G_k of arbitrarily high connectivity k such that the closure of G_k has the property while the graph G_k does not); (ii) traceability is a stable property even for k = 1; (iii) homogeneous traceability is not stable for k = 2 (although it is stable for k = 7). The article is concluded with several open questions concerning stability of homogeneous traceability and Hamiltonian connectedness. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 30–41, 2000     

A property of the lattice of equational theories

It had been conjectured that any algebraic lattice having a compact one could be represented as the lattice of equational theories extending some theory. However, we show that each lattice having such a representation satisfies a nontrivial quasidistributivity condition. In particular,M ₃ has no such representation.To the memory of András HuhnPresented by Walter Taylor.     

Functions that have read‐twice constant width branching programs are not necessarily testable

We construct a property on 0/1‐strings that has a representation by a collection of width‐3 read‐twice oblivious branching programs, but for which any two‐sided ?‐testing algorithm must query at least Ω(n^δ) many queries for some fixed ? and δ. This shows that Newman's result [Testing of functions that have small width branching programs, SIAM J Comput 31 (2002), 1557–1570] cannot be generalized to read‐k‐times functions for k > 1. In addition, we exhibit a property that has also a representation by a CNF formula of constant clause size. Hence, the nontestability results extend to properties that in addition have small (constant size) 0‐witnesses. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Envelopes,indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that B Σ_{n +1} (the fragment of Arithmetic given by the collection scheme restricted to Σ_{n +1}‐formulas) is a Π_{n +2}‐conservative extension of I Σ_n (the fragment given by the induction scheme restricted to Σ_n ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δ_{n +1}(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + B Σ_{n +1} is a Π_{n +2}‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Π_n ‐envelopes and Π_n ‐indicators for T . The analysis of Σ_{n +1}‐collection we develop here is also applied to Σ_{n +1}‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): B Σ_{n +1} and I Σ_{n +1} are Σ_{n +3}‐conservative extensions of their parameter free versions, B Σ^–_{n +1} and I Σ^–_{n +1}. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations

The coupled nonlinear Schrödinger–Boussinesq (SBq) equations describe the nonlinear development of modulational instabilities associated with Langmuir field amplitude coupled to intense electromagnetic wave in dispersive media such as plasmas. In this paper, we present a conservative compact difference scheme for the coupled SBq equations and analyze the conservative property and the existence of the scheme. Then we prove that the scheme is convergent with convergence order O(τ² + h⁴) in L_∞‐norm and is stable in L_∞‐norm. Numerical results verify the theoretical analysis.     

Generic Representation Theory of the Unipotent Upper Triangular Groups

It is generally believed (and for the most part it is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this article we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups U_n), and under a certain hypothesis relating the characteristic p to both n and the dimension d of a representation (specifically, p ≥ max(n, 2d)), Lie theory is completely sufficient to determine the representation theories of these groups. To finish, we mention some important analogies (both functorial and cohomological) between the characteristic zero theories of these groups and their “generic” representation theory in characteristic p.     

Model theory of a Hilbert space expanded with an unbounded closed selfadjoint operator

We study a closed unbounded self‐adjoint operator Q acting on a Hilbert space H in the framework of Metric Abstract Elementary Classes (MAECs). We build a suitable MAEC for such a structure, prove it is ?₀‐categorical and ?₀‐stable up to a system of perturbations. We give an explicit continuous axiomatization for the class. We also characterize non‐splitting and show it has the same properties as non‐forking in superstable first order theories. Finally, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.     

On definability of types and relative stability

In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.     

Increasing η ‐representable degrees

In this paper we prove that any Δ₃⁰ degree has an increasing η ‐representation. Therefore, there is an increasing η ‐representable set without a strong η ‐representation (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finitely generated submodels of an uncountably categorical homogeneous structure

We generalize the result of non‐finite axiomatizability of totally categorical first‐order theories from elementary model theory to homogeneous model theory. In particular, we lift the theory of envelopes to homogeneous model theory and develope theory of imaginaries in the case of ω‐stable homogeneous classes of finite U‐rank. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

Three-pencil lattices on triangulations

In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S ₃ on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.      

逆差商的一个紧凑行列式表示

本文给出了逆差商的一个紧凑行列式表示,从该表示式易知,一个函数在某ｎ＋１个点的ｎ阶逆差商与这些点的排序有关,但与前ｎ－１个点的局部换序无关．此外,还从另一角度定义倒差商,得出了倒差商与逆差商之间的关系以及倒差商的整体换序无关性．     

Non-existence of degree bounds for weighted sums of squares representations

Given a fixed family of polynomials , we study the problem of representing polynomials in the form

(*)

f=s₀+s₁h₁++s_rh_r