Numerably Contractible Categories

We develop the notion of numerably contractible category and use it for describing conditions when a homotopy associative H-category has a homotopy inverse. We prove that complex categories are numerably contractible. The results play a role in Bak’s program for constructing delooping machines for global actions, small categories and related objects. (Received: February 2006)     

Kolmogorov＇s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov＇s ε-entropy for the bounded sets with small ＂tail＂ in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov＇s ε-entropy of the global attractor for lattice systems.     

A note on K-theory and triangulated categories

We provide an example of two closed model categories having equivalent homotopy categories but different Waldhausen K-theories. We also show that there cannot exist a functor from small triangulated categories to spaces which recovers Quillen’s K-theory for exact categories and which satisfies localization. Oblatum 28-V-2001 & 7-III-2002?Published online: 17 June 2002     

Fibrations of Simplicial Sets

There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen’s axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan’s Ex functor act on graphs and on nerves of small categories.     

Fourier-Mukai transform and index theory

Given a submersive morphism of complex manifoldsf: X→Y, and a complex vector bundleE onX, there is a relationship between the higher direct images of ε (the sheaf of holomorphic sections ofE) and the index of the relative Dolbeault complex twisted byE. This relationship allows one to yield a global and simple proof of the equivalence between the Mukai transform of stable vector bundles on a torusT of complex dimension 2 and the Nahm transform of instantons. We also offer a proof of Mukai’s inversion theorem which circumvents the use of derived categories by introducing spectral sequences of sheaves onT (this is related to Donaldson and Kronheimer’s proof, but is automatically global and somehow simpler). The general framework developed in the first part of this paper may be applied to the study of the Mukai transform for more general varieties. Research partly supported by the Italian Ministry for University and Research through the research projects ‘Metodi geometrici e probabilistici in fisica matematica’ and ‘Geometria reale e complessa,’ and by the Spanish DGICYT through the research projects PB91-0188 and PB92-0308. This paper has been typeset by using theAMS-TEX macro package     

Generalized McCormick relaxations

Convex and concave relaxations are used extensively in global optimization algorithms. Among the various techniques available for generating relaxations of a given function, McCormick’s relaxations are attractive due to the recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures. This article leverages the recursive nature of McCormick’s relaxations to define a generalized form which both affords a new framework within which to analyze the properties of McCormick’s relaxations, and extends the applicability of McCormick’s technique to challenging open problems in global optimization. Specifically, relaxations of the parametric solutions of ordinary differential equations are considered in detail, and prospects for relaxations of the parametric solutions of nonlinear algebraic equations are discussed. For the case of ODEs, a complete computational procedure for evaluating convex and concave relaxations of the parametric solutions is described. Through McCormick’s composition rule, these relaxations may be used to construct relaxations for very general optimal control problems.     

Cohomological stratification of diagram algebras

The class of cellularly stratified algebras is defined and shown to include large classes of diagram algebras. While the definition is in combinatorial terms, by adding extra structure to Graham and Lehrer’s definition of cellular algebras, various structural properties are established in terms of exact functors and stratifications of derived categories. The stratifications relate ‘large’ algebras such as Brauer algebras to ‘smaller’ ones such as group algebras of symmetric groups. Among the applications are relative equivalences of categories extending those found by Hemmer and Nakano and by Hartmann and Paget, as well as identities between decomposition numbers and cohomology groups of ‘large’ and ‘small’ algebras.     

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field.     

Global well-posedness for a coupled modified KdV system

We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modified Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura’s Transform that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura’s Transform that takes a KdV system to the system we are considering. To overcome this difficulty we developed a new proof of the sharp global well-posedness result for the single mKdV equation without using Miura’s Transform. We could successfully apply this technique in the case of the mKdV system to obtain the desired result.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field. The author was partly supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

Word problem solving and pictorial representations: insights from an exploratory study in kindergarten

The aim of this study was to investigate how pictorial representations with different semiotic characteristics affect additive word problem solving by kindergartners. The focus of the study is on three categories of additive problems (change problems, combine problems and equalize problems) and on representational pictures with different semiotic characteristics: (a) pictures in which the problem quantities are represented in pictorial form, that is, as groups of illustrated objects (PP pictures), (b) pictures in which the quantities are represented partly in pictorial form and in symbolic form (PS pictures), and (c) pictures in which the quantities are represented in symbolic form (SS pictures). Data were collected from 63 kindergartners using a paper-and-pencil test. Results showed that the semiotic characteristics of representational pictures had a strong and significant effect on performance. Children’s performance was higher in the problems with PP pictures but declined in the problems with PS and SS pictures. However, the differences in children’s performance across the problems with different representational format varied between the problem categories and their mathematical structures. The semiotic characteristics of representational pictures had an important role in the establishment of close relations between children’s solutions in problems in different categories. Detailed analysis of children’s answers to the problems revealed a number of picture-related difficulties. Findings are discussed and directions for future research are drawn considering the methodological limitations of the study.     

S-types of Global Towers of Spaces and Exterior Spaces

The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak ℕ-S-equivalences is called the category of exterior ℕ-S-types. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories) given by the inversion of classes of week equivalences. The family of neighbourhoods ‘at infinity’ of an exterior space can be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of S-types of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models localized up to weak equivalences can be used to give some algebraic models of S-types. The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431.     

The Cauchy Problem of the Nonlinear Schrodinger Equations in R^1＋1

In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.     

On the Beauville form of the known irreducible symplectic varieties

We study the global geometry of O’Grady’s ten-dimensional irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.     

Perturbation analysis of global error bounds for systems of linear inequalities

This paper studies the existence of a uniform global error bound when a system of linear inequalities is under local arbitrary perturbations. Specifically, given a possibly infinite system of linear inequalities satisfying the Slater’s condition and a certain compactness condition, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if the original system is bounded or its homogeneous system has a strict solution. Received: April 12, 1998 / Accepted: February 11, 2000?Published online July 20, 2000     

Reflective Luck and Belief Ownership

A belief is reflectively lucky if it is a matter of luck that the belief is true, given what a subject is aware of on reflection alone. Various epistemologists have argued that any adequate theory of knowledge should eliminate reflective luck, but doing so has proven difficult. This article distinguishes between two kinds of reflective luck arguments in the literature: local arguments and global arguments. It argues that local arguments are best interpreted as demanding, not that one be reflectively aware of the reliability of the sources of one’s beliefs, but that one’s beliefs be attributable to one as one’s own. The article then argues that global arguments make illegitimate demands, because they require that we be ultimately answerable for our beliefs. In the end, the article argues that epistemologists should shift their focus away from reflective luck and toward the conditions under which beliefs are attributable to cognitive agents.     

Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case

We consider the Cauchy problem for the Hopf equation corresponding to the two-dimensional Navier–Stokes equations. The global uniqueness of a solution is proved under the condition that the mean energy of the initial measure is finite, without any additional conditions, i.e., under the same assumptions as in the existence theorem. Thereby we receive a sharp improvement of Foias’s result obtained in the 1970’s about the uniqueness of a spatial statistical solution. Bibliography: 12 titles.     

Complex Lie algebras corresponding to weighted projective lines

The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.     

Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the ‘∨-premorphisms’ part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (J. Algebra 141:422–462, 1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for (ordered) ∧-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.