Algebra Commons^™

Rational Functions Associated To The White Noise Space And Related Topics, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida’s white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product.

How Geometer’S Sketchpad Improves Student Learning, William George Awe

Mathematics Graduate Theses

This study investigated whether dynamic software improves learning and reviews different types of dynamic software. The study then narrows its focus to one type of dynamic software, Geometer’s Sketchpad and studies how Geometer’s Sketchpad, in itself, improves student learning. The researcher identified what dynamic software is, the importance of dynamic software and different types of dynamic software. The researcher also identifies how dynamic software is utilized in schools and researches how to improve the use of dynamic software in schools. Dynamic software, and Geometer’s Sketchpad, In particular, was found to improve learning across grade levels and with different levels of …

Carathéodory Functions In The Banach Space Setting, Daniel Alpay, Olga Timoshenko, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove representation theorems for Carathéodory functions in the setting of Banach spaces.

Special Fuzzy Matrices For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

This book is a continuation of the book, "Elementary fuzzy matrix and fuzzy models for socio-scientists" by the same authors. This book is a little advanced because we introduce a multi-expert fuzzy and neutrosophic models. It mainly tries to help social scientists to analyze any problem in which they need multi-expert systems with multi-models. To cater to this need, we have introduced new classes of fuzzy and neutrosophic special matrices. The first chapter is essentially spent on introducing the new notion of different types of special fuzzy and neutrosophic matrices, and the simple operations on them which are needed in …

Pi-Calculus In Logical Form, Marcello M. Bonsangue, Alexander Kurz

Engineering Faculty Articles and Research

Abramsky’s logical formulation of domain theory is extended to encompass the domain theoretic model for picalculus processes of Stark and of Fiore, Moggi and Sangiorgi. This is done by defining a logical counterpart of categorical constructions including dynamic name allocation and name exponentiation, and showing that they are dual to standard constructs in functor categories. We show that initial algebras of functors defined in terms of these constructs give rise to a logic that is sound, complete, and characterises bisimilarity. The approach is modular, and we apply it to derive a logical formulation of pi-calculus. The resulting logic is a …

François Viète, Between Analysis And Cryptanalysis, Marco Panza

MPP Published Research

François Viète is considered the father both of modern algebra and of modern cryptanalysis. The paper outlines Viète's major contributions in these two mathematical fields and argues that, despite an obvious parallel between them, there is an essential difference. Viète's 'new algebra' relies on his reform of the classical method of analysis and synthesis, in particular on a new conception of analysis and the introduction of a new formalism. The procedures he suggests to decrypt coded messages are particular forms of analysis based on the use of formal methods. However, Viète's algebraic analysis is not an analysis in the same …

Neutrosophic Rings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we define the new notion of neutrosophic rings. The motivation for this study is two-fold. Firstly, the classes of neutrosophic rings defined in this book are generalization of the two well-known classes of rings: group rings and semigroup rings. The study of these generalized neutrosophic rings will give more results for researchers interested in group rings and semigroup rings. Secondly, the notion of neutrosophic polynomial rings will cause a paradigm shift in the general polynomial rings. This study has to make several changes in case of neutrosophic polynomial rings. This would give solutions to polynomial equations for …

Fuzzy Interval Matrices, Neutrosophic Interval Matrices And Their Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The new concept of fuzzy interval matrices has been introduced in this book for the first time. The authors have not only introduced the notion of fuzzy interval matrices, interval neutrosophic matrices and fuzzy neutrosophic interval matrices but have also demonstrated some of its applications when the data under study is an unsupervised one and when several experts analyze the problem. Further, the authors have introduced in this book multiexpert models using these three new types of interval matrices. The new multi expert models dealt in this book are FCIMs, FRIMs, FCInMs, FRInMs, IBAMs, IBBAMs, nIBAMs, FAIMs, FAnIMS, etc. Illustrative …

Matrix-J-Unitary Non-Commutative Rational Formal Power Series, Daniel Alpay, D. S. Kalyuzhnyi-Verbovetzkii

Mathematics, Physics, and Computer Science Faculty Articles and Research

Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N-tuples of n × n matrices (with n = 1, 2, . . .) or operators on an infinite-dimensional separable Hilbert space. Moreover, such evaluations appear in control, optimization and stabilization problems of modern system engineering.

In this paper, a theory of realization and minimal factorization of rational matrix-valued functions which are J-unitary on the imaginary …

Coalgebras And Their Logics, Alexander Kurz

Engineering Faculty Articles and Research

"Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics."

Introduction To Bimatrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in …

Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …

Introduction To Linear Bialgebra, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic …

Applications Of Bimatrices To Some Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. …

Rational Hyperholomorphic Functions In R4, Daniel Alpay, Michael Shapiro, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.

Point Evaluation And Hardy Space On A Homogeneous Tree, Daniel Alpay, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.

Weak Factorizations, Fractions And Homotopies, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

Coalgebras And Modal Expansions Of Logics, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

In this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise. Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of ‘spaces’ (X,A), ie sets X …

Basic Neutrosophic Algebraic Structures And Their Application To Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs or bipartite graphs. Thus the use of graphs has become inevitable in fuzzy models. The neutrosophic models are fuzzy models that permit …

Algebraic Semantics For Coalgebraic Logics, Clemens Kupke, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the …

Preface, Thomas Hildebrandt, Alexander Kurz

Engineering Faculty Articles and Research

No abstract provided.

A Unifying Field In Logics: Neutrosophic Logic Neutrosophy, Neutrosophic Set, Neutrosophic Probability (In Traditional Chinese), Florentin Smarandache, Feng Liu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Notes On Interpolation In The Generalized Schur Class. Ii. Nudelman's Problem, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

An indefinite generalization of Nudel′man’s problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides results on existence criteria for Pick-Nevanlinna and Carath´eodory-Fej´er interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel′man’s problem.

Stone Coalgebras, Clemens Kupke, Alexander Kurz, Yde Venema

Engineering Faculty Articles and Research

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. This yields a duality between the category of modal algebras and that of coalgebras over the Vietoris functor. Building on this idea, we introduce the notion of a Vietoris polynomial functor over the category of Stone spaces. …

Limits Highlights From Over 2000 Years Of Developments In Calculus Limits, Lauri A. Nevalainen

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.

Preface, Alexander Kurz

Engineering Faculty Articles and Research

No abstract provided.

Definability, Canonical Models, And Compactness For Finitary Coalgebraic Modal Logic, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

This paper studies coalgebras from the perspective of the finitary observations that can be made of their behaviours. Based on the terminal sequence, notions of finitary behaviours and finitary predicates are introduced. A category Behω(T) of coalgebras with morphisms preserving finitary behaviours is defined. We then investigate definability and compactness for finitary coalgebraic modal logic, show that the final object in Behω(T) generalises the notion of a canonical model in modal logic, and study the topology induced on a coalgebra by the finitary part of the terminal sequence.

A Note On Interpolation In The Generalized Schur Class. I. Applications Of Realization Theory, Daniel Alpay, T. Constantinescu, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

Realization theory for operator colligations on Pontryagin spaces is used to study interpolation and factorization in generalized Schur classes. Several criteria are derived which imply that a given function is almost the restriction of a generalized Schur function. The role of realization theory in coefficient problems is also discussed; a solution of an indefinite Carathéodory-Fejér problem is obtained, as well as a result that relates the number of negative (positive) squares of the reproducing kernels associated with the canonical coisometric, isometric, and unitary realizations of a generalized Schur function to the number of negative (positive) eigenvalues of matrices derived from …

Some Extensions Of Loewner's Theory Of Monotone Operator Functions, Daniel Alpay, Vladimir Bolotnikov, A. Dijksma, J. Rovnyak, A. Dijksma

Mathematics, Physics, and Computer Science Faculty Articles and Research

Several extensions of Loewner’s theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar- to matrix-valued functions of an operator argument. A notion of -monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.

Modal Predicates And Coequations, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show how coalgebras can be presented by operations and equations. This is a special case of Linton’s approach to algebras over a general base category X, namely where X is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic.