Algebra Commons^™

Cohomology Of Frobenius Algebras And The Yang-Baxter Equation, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, Masahico Saito

Mathematics, Statistics and Data Science Faculty Works

A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.

Dynamic Sequent Calculus For The Logic Of Epistemic Actions And Knowledge, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

"Dynamic Logics (DLs) form a large family of nonclassical logics, and perhaps the one enjoying the widest range of applications. Indeed, they are designed to formalize change caused by actions of diverse nature: updates on the memory state of a computer, displacements of moving robots in an environment, measurements in models of quantum physics, belief revisions, knowledge updates, etc. In each of these areas, DL-formulas express properties of the model encoding the present state of affairs, as well as the pre- and post-conditions of a given action. Actions are semantically represented as transformations of one model into another, encoding the …

Full Newton Step Interior Point Method For Linear Complementarity Problem Over Symmetric Cones, Andrii Berdnikov

College of Graduate Studies: Theses & Dissertations

In this thesis, we present a new Feasible Interior-Point Method (IPM) for Linear Complementarity Problem (LPC) over Symmetric Cones. The advantage of this method lies in that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. By suitable choice of parameters we prove the global convergence of iterates which always stay in the the central path neighborhood. A global convergence of the method is proved and an upper bound for the number of iterations necessary to find ε-approximate solution of the problem is presented.

The Gordon Game, Anthony Frenk

EWU Masters Thesis Collection

In 1992, about 30 years after Gordon introduced group sequencings to construct row-complete Latin squares, John Isbell introduced the idea of competitive sequencing, the Gordon Game. Isbell investigated the Gordon Game and found solutions for groups of small order. The purpose of this thesis is to analyze the Gordon Game and develop a brute force method of determining solutions to the game for all groups of order 12 (up to isomorphism) as well as for abelian groups of order less than 20. The method used will be a depth first search program written in MATLAB. Consequently, group representation using matrices …

Relation Lifting, With An Application To The Many-Valued Cover Modality, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jirí Velebil

Engineering Faculty Articles and Research

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the “powerset monad” on categories, one is the preservation by T of “exactness” of certain squares. Both characterisations are generalisations of the “classical” results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks.

The results presented in this paper …

Nominal Coalgebraic Data Types With Applications To Lambda Calculus, Alexander Kurz, Daniela Petrişan, Paula Severi, Fer-Jan De Vries

Engineering Faculty Articles and Research

We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.

Nominal Computation Theory (Dagstuhl Seminar 13422), Mikołaj Bojanczyk, Bartek Klin, Alexander Kurz, Andrew M. Pitts

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 13422 “Nominal Computation Theory”. The underlying theme of the seminar was nominal sets (also known as sets with atoms or Fraenkel-Mostowski sets) and they role and applications in three distinct research areas: automata over infinite alphabets, program semantics using nominal sets and nominal calculi of concurrent processes.

Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. …

Topological Convolution Algebras, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce a new family of topological convolution algebras of the form ⋃p∈NL2(S,μp), where S is a Borel semi-group in a locally compact group G, which carries an inequality of the type ∥f∗g∥p≤Ap,q∥f∥q∥g∥p for p>q+d where d pre-assigned, and Ap,q is a constant. We give a sufficient condition on the measures μp for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.

Non-Commutative Stochastic Distributions And Applications To Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a non-commutative space of stochastic distributions, which contains the non-commutative white noise space, and forms, together with a natural multiplication, a topological algebra. Special inequalities which hold in this space allow to characterize its invertible elements and to develop an appropriate framework of non-commutative stochastic linear systems.

Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, …

Representation Formulas For Hardy Space Functions Through The Cuntz Relations And New Interpolation Problems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce connections between the Cuntz relations and the Hardy space H2 of the open unit disk D. We then use them to solve a new kind of multipoint interpolation problem in H2, where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves.

Relation Algebras As Expanded Fl-Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.

Subset Non Associative Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of non associative topological space is new and innovative. In general topological spaces are defined as union and intersection of subsets of a set X. In this book authors for the first time define non associative topological spaces using subsets of groupoids or subsets of loops or subsets of groupoid rings or subsets of loop rings. This study leads to several interesting results in this direction.

Over hundred problems on non associative topological spaces using of subsets of loops or groupoids is suggested at the end of chapter two. Also conditions for these non associative subset topological spaces …

Subset Polynomial Semirings And Subset Matrix Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the notion of subset polynomial semirings and subset matrix semirings. The study of algebraic structures using subsets were recently carried out by the authors. Here we define the notion of subset row matrices, subset column matrices and subset m × n matrices. Study of this kind is developed in chapter two of this book. If we use subsets of a set X; say P(X), the power set of the set X....

Hence if P(X) is replaced by a group or a semigroup we get the subset matrix to be only a subset matrix semigroup. If …

Subset Interval Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of groupoids is meager and we have recently introduced the new notion of subset groupoids and have studied them. It is interesting to keep on record that interval groupoids have been studied by us in 2010. Further when the subsets of a loop are taken they also form only a subset groupoid and not a subset loop. Thus we do not have the concept of subset interval loop they only form a subset interval groupoid. Special elements like subset interval zero divisors, subset interval idempotents and subset interval units are studied. Concept of subset interval groupoid homomorphism is …

Algebraic Structures Using Subsets, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of subsets and giving algebraic structure to these subsets of a set started in the mid 18th century by George Boole. The first systematic presentation of Boolean algebra emerged in 1860s in papers written by William Jevons and Charles Sanders Peirce. Thus we see if P(X) denotes the collection of all subsets of the set X, then P(X) under the op erations of union and intersection is a Boolean algebra. Next the subsets of a set was used in the construction of topological spaces. We in this book consider subsets of a semigroup or a group or a …

Algebraic Structures Using [0,N), Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce a new method of building algebraic structures on the interval [0, n). This study is interesting and innovative. However, [0, n) is a semigroup under product, × modulo n and a semigroup under min or max operation. Further [0, n) is a group under addition modulo n. We see [0, n) under both max and min operation is a semiring. [0, n) under + and × is not in general a ring. We define S = {[0, n), +, ×} to be a pseudo special ring as the distributive law is …

Set Theoretic Approach To Algebraic Structures In Mathematics - A Revelation, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out how sets in algebraic structure can be used to construct most generalized algebraic structures, like set linear algebra/vector space, set ideals in rings and semigroups. This sort of study is not only innovative but infact very helpful in cases instead of working with a large data we can work with a considerably small data. Thus instead of working with a vector space or a linear algebra V over a field F we can work with a subset in V and a needed subset in F, this can save both time and economy. The concept …

Fuzzy Neutrosophic Models For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, authors give the notion of different neutrosophic models like, neutrosophic cognitive maps (NCMs), neutrosophic relational maps (NEMs), neutrosophic relational equations (NREs), neutrosophic bidirectional associative memories (NBAMs) and neutrosophic associative memories (NAMs) for socio scientists. This book has six chapters. The first chapter introduces the basic concepts of neutrosophic numbers and notions about neutrosophic graphs which are essential to construct these neutrosophic models. In chapter two we describe the concept of neutrosophic matrices and the essential operations related with them which are used in the study and working of these neutrosophic models. However the reader must be familiar …