Sheaves on Involutive Quantaloids

We extend the well-known notions of a singleton, complete -set, presheaf and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary involutive quantaloids. We show that sheaves on and complete -sets come to the same thing. This paper can be considered as a symmetric version of an earlier work of the author.     

Sheaves and sheafification on Q-sites

We introduce noncommutative sites over a quantale, the so-called Q-sites, and define appropriate presheaves and sheaves over these. We show how most of the technical machinery which allows to construct sheaves associated to arbitrary presheaves in the commutative cases transposes to this setting. This allows us to define and study sheafification in this new, noncommutative context.     

Generalized Burnside rings and group cohomology

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.     

Similarity mass and approximate reasoning

It can reflect the nature of approximate reasoning and meet more application expectations to design the approximate reasoning matching schemes and the corresponding algorithms with similarity relation Q instead of equivalence relation R. In this paper, based on similarity relation Q, we introduce type V matching scheme and corresponding approximate reasoning type V Q-algorithm with the given input A* and knowledge A → B. Besides, we present completeness of type V and its perfection on knowledge base K in Q-logic ℂ _Q in this paper.     

Solvability of Rado systems in D-sets

Rado's Theorem characterizes the systems of homogeneous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of D-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.     

Some Sets of Type (m,n) in Cubic Order Planes

A (4,9)-set of size 829 in (2,5³) is constructed, as is a (4,11)-set of size 3189 in (2,7³).     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.     

Transition function for the Toda chain

We use the method of Λ-operators developed by Derkachov, Korchemsky, and Manashov to derive eigenfunctions for the open Toda chain. Using the diagram technique developed for these Λ-operators, we reproduce the Sklyanin measure and study the properties of the Λ-operators. This approach to the open Toda chain eigenfunctions reproduces the Gauss-Givental representation for these eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 371–390, March, 2007.     

A finite characterization ofK-matrices in dimensions less than four

The class of realn × n matricesM, known asK-matrices, for which the linear complementarity problemw – Mz = q, w 0, z 0, w ^T z =0 has a solution wheneverw – Mz =q, w 0, z 0 has a solution is characterized for dimensionsn <4. The characterization is finite and practical. Several necessary conditions, sufficient conditions, and counterexamples pertaining toK-matrices are also given. A finite characterization of completelyK-matrices (K-matrices all of whose principal submatrices are alsoK-matrices) is proved for dimensions <4.Partially supported by NSF Grant MCS-8207217.Research partially supported by NSF Grant No. ECS-8401081.     

Sheaves on abelian surfaces and strange duality

We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering a question raised by Göttsche et al. (K-theoretic Donaldson invariants via instanton counting. arXiv:math/0611945).     

On Algebraic Anosov Diffeomorphisms on Nilmanifolds

This article is devoted to the algebraic approaches to Anosov diffeomorphisms. All examples of Anosov diffeomorphisms known so far are connected directly or indirectly with compact nilmanifolds. We consider some new necessary conditions for existence of these diffeomorphisms on nilmanifolds. We demonstrated the absence of Anosov diffeomorphisms on some classes of nilmanifolds. We also prove some results on Lie groups and lattices in them.     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

A Decomposition of a Weaker Form of Continuity

The aim of this paper is to give a decomposition of a weaker form of continuity, namely gα^**-continuity by providing the concepts of K-sets and K-continuity. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Sobolev and Capacitary Inequalities for Contractive Besov Spaces over d-sets

We derive Sobolev inequalities for Besov spaces B ^p,p (F), 0<<1, 1p< on d-sets F in R ⁿ , dn, from a metric property of the Bessel capacity on R ⁿ . We first extend Kaimanovitch's result on the equivalence of Sobolev and capacitary inequalites for contractive p-norms in a general setting allowing unbounded Lévy kernels. A simple part of the Jonsson–Wallin trace theorem for Besov spaces and some basic properties of Bessel and Besov capacities on R ⁿ are then utilized in getting the desired inequalities. When p=2, the Besov space being considered is a non-local regular Dirichlet space and gives rise to a jump type symmetric Markov process M over the d-set. The upper bound of the transition function of M and metric properties of M -polar sets are then exhibited.     

Optimum experimental design for a regression on a hypercube-generalization of Hoel's result

In the note Hoel's result (1965, Ann. Math. Statist., 36, 1097–1106) is generalized to a large family of experimental design optimality criterions. Sufficient conditions for optimality criterion are given, which ensure existence of the optimum experimental design measure which is a product of design measures on lower dimensional domains.     

Cozero-accessible points

We define a point x to be cozero-accessible if for each dense open set U, there is a cozero-set CU such that . It is shown to be independent of Martin's Axiom that there are cozero-accessible points in .