Other Mathematics Commons^™

Special Subset Vertex Multisubgraphs For Multi Networks, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors study special type of subset vertex multi subgraphs; these multi subgraphs can be directed or otherwise. Another special feature of these subset vertex multigraphs is that we are aware of the elements in each vertex set and how it affects the structure of both subset vertex multisubgraphs and edge multisubgraphs. It is pertinent to record at this juncture that certain ego centric directed multistar graphs become empty on the removal of one edge, there by theorising the importance, and giving certain postulates how to safely form ego centric multi networks. Given any subset vertex multigraph we …

Neutrosophic Logic: The Revolutionary Logic In Science And Philosophy -- Proceedings Of The National Symposium, Florentin Smarandache, Huda E. Khalid, Ahmed K. Essa

Branch Mathematics and Statistics Faculty and Staff Publications

The first part of this book is an introduction to the activities of the National Symposium, as well as a presentation of Neutrosophic Scientific International Association (NSIA), based in New Mexico, USA, also explaining the role and scope of NSIA - Iraqi branch. The NSIA Iraqi branch presents a suggestion for the international instructions in attempting to organize NSIA's work. In the second chapter, the pivots of the Symposium are presented, including a history of neutrosophic theory and its applications, the most important books and papers in the advancement of neutrosophics, a biographical note of Prof. Florentin Smarandache in Arabic …

Strong Neutrosophic Graphs And Subgraph Topological Subspaces, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilantheral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of strong neutrosophic graphs. They are very different from the usual graphs and neutrosophic graphs. Using these new structures special subgraph topological spaces are defined. Further special lattice graph of subgraphs of these graphs are defined and described. Several interesting properties using subgraphs of a strong neutrosophic graph are obtained. Several open conjectures are proposed. These new class of strong neutrosophic graphs will certainly find applications in NCMs, NRMs and NREs with appropriate modifications.

Problems On Mod Structures, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time give several types of problems on MOD structures happens to be an interesting field of study as it makes the whole 4 quadrant plane into a single quadrant plane and the infinite line into a half closed open interval. So study in this direction will certainly yield several interesting results. The law of distributivity is not true. Further the MOD function in general do not obey all the laws of integration or differentiation. Likewise MOD polynomials in general do not satisfy the basic properties of polynomials like its roots etc. Thus over …

Nidus Idearum. Scilogs, I: De Neutrosophia, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Welcome into my scientific lab! My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). In this first books of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others …

Neutrosophic Graphs: A New Dimension To Graph Theory, Florentin Smarandache, Wb. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time have made a through study of neutrosophic graphs. This study reveals that these neutrosophic graphs give a new dimension to graph theory. The important feature of this book is it contains over 200 neutrosophic graphs to provide better understanding of this concepts. Further these graphs happen to behave in a unique way inmost cases, for even the edge colouring problem is different from the classical one. Several directions and dimensions in graph theory are obtained from this study. Finally certainly these new notions of neutrosophic graphs in general and in particular the …

Mod Functions: A New Approach To Function Theory, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration. Even finding roots of MOD polynomials happens to be very difficult as they do not follow the fundamental theorem of algebra, viz a nth degree polynomial …

Neutrosophic Crisp Set Theory, Florentin Smarandache, A.A. Salama

Branch Mathematics and Statistics Faculty and Staff Publications

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need …

Natural Neutrosophic Numbers And Mod Neutrosophic Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors answer the question proposed by Florentin Smarandache “Does there exist neutrosophic numbers which are such that they take values differently and behave differently from I; the indeterminate?”. We have constructed a class of natural neutrosophic numbers m 0I , m xI , m yI , m zI where m 0I × m 0I = m 0I , m xI × m xI = m xI and m yI × m yI = m yI and m yI × m xI = m 0I and m zI × m zI = m 0I . Here take m …

Fuzzy Abel Grassmann Groupoids, Florentin Smarandache, Madad Khan, Tariq Aziz

Branch Mathematics and Statistics Faculty and Staff Publications

Usually the models of real world problems in almost all disciplines like engineering, medical sciences, mathematics, physics, computer science, management sciences, operations research and articial intelligence are mostly full of complexities and consist of several types of uncertainties while dealing them in several occasion. To overcome these di¢ culties of uncertainties, many theories have been developed such as rough sets theory, probability theory, fuzzy sets theory, theory of vague sets, theory of soft ideals and the theory of intuitionistic fuzzy sets, theory of neutrosophic sets, Dezert-Smarandache Theory (DSmT), etc. Zadeh introduced the degree of membership/truth (t) in 1965 and dened …

Algebraic Structures On The Fuzzy Interval [0, 1), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce several algebraic structures on the special fuzzy interval [0, 1). This study is different from that of the algebraic structures using the interval [0, n) n ≠ 1, as these structures on [0, 1) has no idempotents or zero divisors under ×. Further [0, 1) under product × is only a semigroup. However by defining min(or max) operation in [0, 1); [0, 1) is made into a semigroup. The semigroup under × has no finite subsemigroups but under min or max we have subsemigroups of order one, two and so on. [0, 1) under + …

Special Pseudo Linear Algebras Using [0,N), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce some special type of linear algebras called pseudo special linear algebras using the interval [0, n). These new types of special pseudo interval linear algebras has several interesting properties. Special pseudo interval linear algebras are built over the subfields in Zn where Zn is a S-ring. We study the substructures of them. The notion of Smarandache special interval pseudo linear algebras and Smarandache strong special pseudo interval linear algebras are introduced. The former Sspecial interval pseudo linear algebras are built over the Sring itself. Study in this direction has yielded several interesting results. S-strong special …

Soft Neutrosophic Algebraic Structures And Their Generalization - Vol. 1, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and …

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 Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of constructing non associative algebraic structures using subsets of a groupoid. Thus subset groupoids are constructed using groupoids or loops. Even if we use subsets of loops still the algebraic structure we get with it is only a groupoid. However we can get a proper subset of it to be a subset loop which will be isomorphic with the loop which was used in the construction of the subset groupoid. To the best of the authors’ knowledge this is the first time non associative algebraic structures are constructed using subsets. We get …

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 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 [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 …

Innovative Uses Of Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, Indra Venkatbabu

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out the innovative applications of matrices defined, described and developed by them. Here they do not include the natural product on matrices newly described and defined by them in the book on ‘natural product ×n on matrices’.

This book is organized into seven chapters. The first one is introductory in nature. In the second chapter authors give the unique and new way of analyzing the data which is time dependent. We construct three types of matrices called Average Time Dependent data matrix (ATD matrix), Refined Time Dependent Data matrix (RTD matrix) and Combined Effective Time …

Special Quasi Dual Numbers And Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new notion called special quasi dual number, x = a + bg.

Among the reals – 1 behaves in this way, for (– 1)2 = 1 = – (– 1). Likewise –I behaves in such a way (– I)2 = – (– I). These special quasi dual numbers can be generated from matrices with entries from 1 or I using only the natural product ×n. Another rich source of these special quasi dual numbers or quasi special dual numbers is Zn, n a composite number. For instance 8 in Z12 is such that …

Non Associative Algebraic Structures Using Finite Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Authors in this book for the first time have constructed nonassociative structures like groupoids, quasi loops, non associative semirings and rings using finite complex modulo integers. The Smarandache analogue is also carried out. We see the nonassociative complex modulo integers groupoids satisfy several special identities like Moufang identity, Bol identity, right alternative and left alternative identities. P-complex modulo integer groupoids and idempotent complex modulo integer groupoids are introduced and characterized. This book has six chapters. The first one is introductory in nature. Second chapter introduces complex modulo integer groupoids and complex modulo integer loops.

Exploring The Extension Of Natural Operations On Intervals, Matrices And Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book extends the natural operations defined on intervals, finite complex numbers and matrices. The intervals [a, b] are such that a ≤ b. But the natural class of intervals [a, b] introduced by the authors are such that a ≥ b or a need not be comparable with b. This way of defining natural class of intervals enables the authors to extend all the natural operations defined on reals to these natural class of intervals without any difficulty. Thus with these natural class of intervals working with interval matrices like stiffness matrices finding eigenvalues takes the same time as …

Finite Neutrosophic Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time the authors introduce the notion of real neutrosophic complex numbers. Further the new notion of finite complex modulo integers is defined. For every C(Zn) the complex modulo integer iF is such that 2 Fi = n – 1. Several algebraic structures on C(Zn) are introduced and studied. Further the notion of complex neutrosophic modulo integers is introduced. Vector spaces and linear algebras are constructed using these neutrosophic complex modulo integers. This book is organized into 5 chapters. The first chapter introduces real neutrosophic complex numbers. Chapter two introduces the notion of finite complex …

Algebraic Structures Using Super Interval Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The advantage of using super interval matrices is that one can build only one vector space using m × n interval matrices, but in case of super interval matrices we can have several such spaces depending on the partition on the interval matrix.

This book has seven chapters. Chapter one is introductory in nature, just introducing the super interval matrices or interval super matrices. In chapter two essential operations on super interval matrices are defined. Further in this chapter algebraic structures are defined on these super interval matrices using these operation. Using these super interval matrices semirings and semivector spaces …

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 …

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 …

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 …

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. …

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 …