Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

Preassociative aggregation functions

The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.Comment: arXiv admin note: text overlap with arXiv:1309.730

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page

Noncommutative Logic Systems with Applications in Management and Engineering

Zadeh's (min-max, standard) fuzzy logic and various other logics are commutative, but natural language has nuances suggesting the premises are not equal, with premises contributing to the conclusion according to their prominency. Therefore, we suggest variants of salience-based, noncommutative and non-associative fuzzy logic (prominence logic) that may better model natural language and reasoning when using linguistic variables. Noncommutative fuzzy logics have several theoretical and applicative motivations to be used as models for human inference and decision making processes. Among others, asymmetric relations in economy and management, such as buyer-seller, provider-user, and employer-employee are noncommutative relations and induce noncommutative logic operations between premises or conclusions. A class of noncommutative fuzzy logic operators is introduced and fuzzy logic systems based on the corresponding noncommutative logics are described and analyzed. The prominence of the operators in the noncommutative operations is conventionally assumed to be determined by their precedence. Specific versions of noncommutative logics in the class of the salience-based, noncommutative logics are discussed. We show how fuzzy logic systems may be built based on these types of logics. Compared with classic fuzzy systems, the noncommutative fuzzy logic systems have improved performances in modeling problems, including the modeling of economic and social processes, and offer more flexibility in approximation and control. Applications discussed include management and engineering problems and issues in the field of firms’ ethics or ethics of AI algorithms

The Polyhedral Geometry of Partially Ordered Sets

Pairs of polyhedra connected by a piecewise-linear bijection appear in different fields of mathematics. The model example of this situation are the order and chain polytopes introduced by Stanley in, whose defining inequalities are given by a finite partially ordered set. The two polytopes have different face lattices, but admit a volume and lattice point preserving piecewise-linear bijection called the transfer map. Other areas like representation theory and enumerative combinatorics provide more examples of pairs of polyhedra that are similar to order and chain polytopes. The goal of this thesis is to analyze this phenomenon and move towards a common theoretical framework describing these polyhedra and their piecewise-linear bijections. A first step in this direction was done by Ardila, Bliem and Salazar, where the authors generalize order and chain polytopes by replacing the defining data with a marked poset. These marked order and chain polytopes still admit a piecewise-linear transfer map and include the Gelfand-Tsetlin and Feigin-Fourier-Littelmann-Vinberg polytopes from representation theory among other examples. We consider more polyhedra associated to marked posets and obtain new results on their face structure and combinatorial interplay. Other examples found in the literature bear resemblance to these marked poset polyhedra but do not admit a description as such. This is our motivation to consider distributive polyhedra, which are characterized by describing networks by Felsner and Knauer analogous to the description of order polytopes by Hasse diagrams. For a subclass of distributive polyhedra we are able to construct a piecewise-linear bijection to another polyhedron related to chain polytopes. We give a description of this transfer map and the defining inequalities of the image in terms of the underlying network

On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1's and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of nxn matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.Comment: v2: published versio

Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested