Analysis Commons^™

Interval Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy, Moon Kumar Chetry

Branch Mathematics and Statistics Faculty and Staff Publications

This book introduces several new classes of groupoid, like polynomial groupoids, matrix groupoids, interval groupoids, polynomial interval groupoids, matrix interval groupoids and their neutrosophic analogues.

Proposed Problems Of Mathematics (Vol. Ii), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The first book of “Problèmes avec et sans … problèmes!” was published in Morocco in 1983. I collected these problems that I published in various Romanian or foreign magazines (amongst which: “Gazeta Matematică”, magazine which formed me as problem solver, “American Mathematical Monthly”, “Crux Mathematicorum” (Canada), “Elemente der Mathematik” (Switzerland), “Gaceta Matematica” (Spain), “Nieuw voor Archief” (Holland), etc. while others are new proposed problems in this second volume.

These have been created in various periods: when I was working as mathematics professor in Romania (1984-1988), or co-operant professor in Morocco (1982-1984), or emigrant in the USA (1990-1997). I thank to …

Regular Functions On The Space Of Cayley Numbers, Graziano Gentili, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we present a new definition of regularity on the space Ç of Cayley numbers (often referred to as octonions), based on a Gateaux-like notion of derivative. We study the main properties of regular functions, and we develop the basic elements of a function theory on Ç. Particular attention is given to the structure of the zero sets of such functions.

Continuous Trace C*-Algebras, Gauge Groups And Rationalization, John R. Klein, Claude Schochet, Samuel B. Smith

Mathematics Faculty Research Publications

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A_ζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

Banach Algebras And Rational Homotopy Theory, Gregory Lupton, N. Christopher Phillips, Claude Schochet, Samuel B. Smith

Mathematics Faculty Research Publications

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GL_n(A), the group of invertible n x n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lc_n(A) denote the space of "last columns" of GL_n(A). We construct a natural isomorphism

Ȟ^s(Max(A);ℚ) ≅ π_2n-1-s(Lc_n(A)) ⊗ ℚ …

Superbimatrices And Their Generalizations, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The systematic study of supermatrices and super linear algebra has been carried out in 2008. These new algebraic structures find their applications in fuzzy models, Leontief economic models and data-storage in computers. In this book the authors introduce the new notion of superbimatrices and generalize it to super trimatrices and super n-matrices. Study of these structures is not only interesting and innovative but is also best suited for the computerized world. The main difference between simple bimatrices and super bimatrices is that in case of simple bimatrices we have only one type of product defined on them, whereas in case …

Groups As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Through this book, for the first time we represent every finite group in the form of a graph. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties like the subgroups of a group, normal subgroups of a group, p-sylow subgroups of a group and conjugate elements of a group …

Super Linear Algebra, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized …

N-Linear Algebra Of Type Ii, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of type II which is introduced in this book. In case of n-linear algebra of type II, we are in a position to define linear functionals which is one of the marked difference between the n-vector spaces of type I and II. However all the applications mentioned in n-linear algebras of type I can be appropriately extended to …

N- Linear Algebra Of Type I And Its Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

With the advent of computers one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure namely n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of nvector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several …

Special Classes Of Set Codes And Their Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce the notion of set codes, set bicodes and set n-codes. These are the most generalized notions of semigroup n-codes and group n-codes. Several types of set ncodes are defined. Several examples are given to enable the reader to understand the concept. These new classes of codes will find applications in cryptography, computer networking (where fragmenting of codes is to be carried out) and data storage (where confidentiality is to be maintained). We also describe the error detection and error correction of these codes. The authors feel that these codes would be appropriate to the …

The Barycenter Of The Numerical Range Of A Matrix, Sean A. Broughton, Roger G. Lautzenheiser, Thomas Werne

Mathematical Sciences Technical Reports (MSTR)

The numerical range W(A) of an nxn matrix A is the totality of the scalar products <Ax,x> as x varies over all unit vectors in Cⁿ The barycenter (center of mass) of the numerical range is defined geometrically as the center of mass of W(A) considered as a planar lamina with variable density and also as a limit of sample averages (<Ax₁,x₁>+...+<Ax_N,x_N>)/N. Under a wide range the sampling schemes it is shown that the barycenter is the average of the spectrum …

Auxiliary Information And A Priori Values In Construction Of Improved Estimators, Florentin Smarandache, Rajesh Singh, Pankaj Chauhan, Nirmala Sawan

Branch Mathematics and Statistics Faculty and Staff Publications

This volume is a collection of six papers on the use of auxiliary information and a priori values in construction of improved estimators. The work included here will be of immense application for researchers and students who employ auxiliary information in any form. Below we discuss each paper: 1. Ratio estimators in simple random sampling using information on auxiliary attribute. Prior knowledge about population mean along with coefficient of variation of the population of an auxiliary variable is known to be very useful particularly when the ratio, product and regression estimators are used for estimation of population mean of 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 …

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 …

Some Neutrosophic Algebraic Structures And Neutrosophic N-Algebraic Structures, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, for the first time we introduce the notion of neutrosophic algebraic structures for groups, loops, semigroups and groupoids and also their neutrosophic N-algebraic structures. One is fully aware of the fact that many classical theorems like Lagrange, Sylow and Cauchy have been studied only in the context of finite groups. Here we try to shift the paradigm by studying and introducing these theorems to neutrosophic semigroups, neutrosophic groupoids, and neutrosophic loops. We have intentionally not given several theorems for semigroups and groupoid but have given several results with proof mainly in the case of neutrosophic loops, biloops …

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 …

N-Algebraic Structures And S-N-Algebraic Structures, Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, for the first time we introduce the notions of Ngroups, N-semigroups, N-loops and N-groupoids. We also define a mixed N-algebraic structure. We expect the reader to be well versed in group theory and have at least basic knowledge about Smarandache groupoids, Smarandache loops, Smarandache semigroups and bialgebraic structures and Smarandache bialgebraic structures. The book is organized into six chapters. The first chapter gives the basic notions of S-semigroups, S-groupoids and S-loops thereby making the book self-contained. Chapter two introduces N-groups and their Smarandache analogues. In chapter three, Nloops and Smarandache N-loops are introduced and analyzed. Chapter four …

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 …

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.

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.

Algebra În Exercijii Şi Probleme Pentru Liceu, Florentin Smarandache, Ion Goian, Raisa Grigor, Vasila Marin

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Asupra Unor Noi Functii În Teoria Numerelor, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Performantele matematicii actuale,ca si descoperirile din viitor isi au,desigur, inceputul in cea mai veche si mai aproape de filozofie ramura a matematicii, in teoria numerelor. Matematicienii din toate timpurile au fost, sunt si vor fi atrasi de frumusetea si varietatea problemelor specifice acestei ramuri a matematicii. Regina a matematicii, care la randul ei este regina a stiintelor, dupa cum spunea Gauss, teoria numerelor straluceste cu lumina si atractiile ei, fascinandu-ne si usurandu-ne drumul cunoasterii legitatilor ce guverneaza macrocosmosul si microcosmosul. De la etapa antichitatii, cand teoria numerelor era cuprinsa in aritmetica, la etapa aritmeticii superioare din perioada Renasterii, cand teoria …

Reproducing Kernel Krein Spaces Of Analytic Functions And Inverse Scattering, Daniel Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

The purpose of this thesis is to study certain reproducing kernel Krein spaces of analytic functions, the relationships between these spaces and an inverse scattering problem associated with matrix valued functions of bounded type, and an operator model.

Roughly speaking, these results correspond to a generalization of earlier investigations on the applications of de Branges' theory of reproducing kernel Hilbert spaces of analytic functions to the inverse scattering problem for a matrix valued function of the Schur class.

The present work considers first a generalization of a portion of de Branges' theory to Krein spaces. We then formulate a general …