Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions.     

Quantaloids,enriched categories and automata theory

This article is intended to be an survey article outlining how the theory of quantaloids and categories enriched in them provides an effective means of analyzing both automata and tree automata. The emphasis is on the unification of concepts and how categorical methods provide insight into various calculations and theorems, both illuminating the original presentation as well as yielding conceptually simpler proofs. Proofs will be omitted and the emphasis is on providing the reader (even a relatively inexperienced one) with an understanding of the basic constructions and results.     

Involutive and relational quantaloids

We present a discussion of involutive and relational quantaloids in a fashion similar to the one for quantales developed by C. J. Mulvey and J. W. Pelletier, C. Brown and D. Gurr, and S. Valentini. We introduce two classes of involutive quantaloids and prove that every quantaloid is isomorphic to a relational quantaloid. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 475–492, October–December, 1999.     

\boldsymbol{\mathcal{Q}}-*-Categories

We consider the theory of categories enriched in an involutive quantaloid : the -*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in to define -*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.      

Sheaves on quantaloids

We generalize the well-known notions of a singleton, completeQ-set, presheaf, and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary quantaloids. We show that every completeQ-set can be viewed as a sheaf and vice versa. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 133–171, April–June, 2000.     

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.     

Multi-start methods for combinatorial optimization

Multi-start methods strategically sample the solution space of an optimization problem. The most successful of these methods have two phases that are alternated for a certain number of global iterations. The first phase generates a solution and the second seeks to improve the outcome. Each global iteration produces a solution that is typically a local optimum, and the best overall solution is the output of the algorithm. The interaction between the two phases creates a balance between search diversification (structural variation) and search intensification (improvement), to yield an effective means for generating high-quality solutions. This survey briefly sketches historical developments that have motivated the field, and then focuses on modern contributions that define the current state-of-the-art. We consider two categories of multi-start methods: memory-based and memoryless procedures. The former are based on identifying and recording specific types of information (attributes) to exploit in future constructions. The latter are based on order statistics of sampling and generate unconnected solutions. An interplay between the features of these two categories provides an inviting area for future exploration.     

Accessible aspects of 2-category theory

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the structure of monoidal, but not strict monoidal, categories) then the 2-category in question is accessible. Furthermore, we explore the flexible limits that such 2-categories possess and their interaction with filtered colimits.     

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.     

Bayesian neural network learning for repeat purchase modelling in direct marketing

We focus on purchase incidence modelling for a European direct mail company. Response models based on statistical and neural network techniques are contrasted. The evidence framework of MacKay is used as an example implementation of Bayesian neural network learning, a method that is fairly robust with respect to problems typically encountered when implementing neural networks. The automatic relevance determination (ARD) method, an integrated feature of this framework, allows us to assess the relative importance of the inputs. The basic response models use operationalisations of the traditionally discussed Recency, Frequency and Monetary (RFM) predictor categories. In a second experiment, the RFM response framework is enriched by the inclusion of other (non-RFM) customer profiling predictors. We contribute to the literature by providing experimental evidence that: (1) Bayesian neural networks offer a viable alternative for purchase incidence modelling; (2) a combined use of all three RFM predictor categories is advocated by the ARD method; (3) the inclusion of non-RFM variables allows to significantly augment the predictive power of the constructed RFM classifiers; (4) this rise is mainly attributed to the inclusion of customer/company interaction variables and a variable measuring whether a customer uses the credit facilities of the direct mailing company.     

函子范畴的Recollement

函子范畴是—类重要的范畴,因为许多常见的范畴都是函子范畴,并且任意给定的范畴都可以通过Yoneda引理嵌入到一个函子范畴,而函子范畴具有比原范畴更好的性质。本文证明了Abel范畴的recollement可以自然诱导两类函子范畴的recollment．应用到k-线性范畴,得到k．线性Abel范畴的recollement可以自然诱导其模范畴的recollement．     

Almost split sequences in tri-exact categories

We study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend a number of existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories by numerous researchers. As applications, we obtain some new results on the existence of almost split triangles in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations.     

The Leibniz formula in algebraic K-theory

The paper may be viewed as an addendum to a paper of Thomason and Throbaugh, where the K-theory of algebraic varieties is equipped with relative K-groups. It is proved that this enriched K-theory satisfies the Panin—Smirnov axioms for ring cohomology theories of algebraic varieties. In particular, it is proved that the Leibniz formula, describing the interaction between multiplication and differential, holds in this case. The language of symmetric spectra and of monoidal model categories is used. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 264–292.     

Bayesian networks with a logistic regression model for the conditional probabilities

Logistic regression techniques can be used to restrict the conditional probabilities of a Bayesian network for discrete variables. More specifically, each variable of the network can be modeled through a logistic regression model, in which the parents of the variable define the covariates. When all main effects and interactions between the parent variables are incorporated as covariates, the conditional probabilities are estimated without restrictions, as in a traditional Bayesian network. By incorporating interaction terms up to a specific order only, the number of parameters can be drastically reduced. Furthermore, ordered logistic regression can be used when the categories of a variable are ordered, resulting in even more parsimonious models. Parameters are estimated by a modified junction tree algorithm. The approach is illustrated with the Alarm network.     

The Popescu-Gabriel theorem for triangulated categories

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.     

Adjoint Functors and Triangulated Categories

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.     

On the modelling of bubble plumes in a liquid pool

Turbulent, bubble plumes are investigated numerically using the commercial, Computational Fluid Dynamics (CFD) code CFX-F3D. A six-equation, two-fluid model approach is adopted, in which interphase momentum exchange models include buoyancy, drag, added mass, lift and turbulent dispersion effects. Particular attention is paid to turbulence modelling, in which generation and dissipation resulting from interaction between bubbles and liquid are specifically taken into account within the context of an extended k − ϵ turbulence model. Results from a number of calculations are presented and compared against published, experimental bubble plume data. It is suggested that existing bubble/liquid interaction models for plumes may be grouped into three categories: those which produce lateral bubble spreading, those which diffuse the ambient liquid velocity field, and those which couple the plume to the surrounding liquid and thereby ultimately govern the pool mixing behaviour.     

Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.     

On one-sided torsion pair

Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.