Mathematics Commons^™

New Trends In Neutrosophic Theory And Applications - Vol. 1, Florentin Smarandache, Surapati Pramanik

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, …

Special Type Of Fixed Points Of Mod Matrix Operators, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce a special type of fixed points using MOD square matrix operators. These special type of fixed points are different from the usual classical fixed points. A study of this is carried out in this book. Several interesting properties are developed in this regard. The notion of these fixed points find many applications in the mathematical models which are dealt systematically by the authors in the forth coming books. These special type of fixed points or special realized limit cycles are always guaranteed as we use only MOD matrices as operators with …

Semigroups On Mod Natural Neutrosophic Elements, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the notion of semigroups under + is constructed using the Mod natural neutrosophic integers or MOD natural neutrosophic-neutrosophic numbers or mod natural neutrosophic finite complex modulo integer or MOD natural neutrosophic dual number integers or MOD natural neutrosophic special dual like number or MOD natural neutrosophic special quasi dual numbers are analysed in a systematic way. All these semigroups under + have an idempotent subsemigroup under +. This is the first time we are able to give a class of idempotent subsemigroups under + by taking only those MOD natural neutrosophic elements

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 …

Mod Relational Maps Models And Mod Natural Neutrosophic Relational Maps Models, Florentin Smarandache, K. Ilanthenral, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Zadeh introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. Atanassov introduced the degree of non membership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t,i,f) = (truth, indeterminacy, falsehood). The words “neutrosophy” and “neutrosophic” were coined/invented by F. Smarandache in his 1998 book. Etymologically, “neutro-sophy” (noun) [French neutre 1), or complete information (sum of components = 1).

Mod Natural Neutrosophic Subset Topological Spaces And Kakutani’S Theorem, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time develop the notion of MOD natural neutrosophic subset special type of topological spaces using MOD natural neutrosophic dual numbers or MOD natural neutrosophic finite complex number or MOD natural neutrosophic-neutrosophic numbers and so on to build their respective MOD semigroups. Later they extend this concept to MOD interval subset semigroups and MOD interval neutrosophic subset semigroups. Using these MOD interval semigroups and MOD interval natural neutrosophic subset semigroups special type of subset topological spaces are built. Further using these MOD subsets we build MOD interval subset matrix semigroups and MOD interval subset …

Mod Cognitive Maps Models And Mod Natural Neutrosophic Cognitive Maps Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Zadeh introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. Atanassov introduced the degree of non membership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t,i,f) = (truth, indeterminacy, falsehood). The words “neutrosophy” and “neutrosophic” were coined/invented by F. Smarandache in his 1998 book. Etymologically, “neutro-sophy” (noun) [French neutre 1), or complete information (sum of components = 1).

Mod Natural Neutrosophic Subset Semigroups, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of MOD subsets using Zn ... On these MOD subsets the operation ‘+’ is defined, S(Zn) denotes the MOD subset and {S(Zn), +} happens to be only a Smarandache semigroup.

These S-semigroups enjoy several interesting properties. The notion of MOD universal subset and MOD absorbing subsets are defined and developed. {S(Zn), x } is also a semigroup and several properties associated with them are derived. MOD natural neutrosophic subsets forms only a semigroup under ‘+’. In fact the main feature enjoyed by this structure is they have subset idempotents …

Mod Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Zadeh introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. Atanassov introduced the degree of nonmembership/ falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/ neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood): http://fs.gallup.unm.edu/FlorentinSmarandache.htm Etymology. The words “neutrosophy” and “neutrosophic” were coined/ invented by F. Smarandache in his 1998 book. Neutrosophy: A branch of philosophy, introduced by F. Smarandache in 1980, which studies the origin, nature, and scope of neutralities, as well …

Special Type Of Fixed Point Pairs Using Mod Rectangular Matrix Operators, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling. Since all these Zn or I nZ or 〈Zn ∪ g〉 or 〈Zn ∪ g〉I or C(Zn) or CI(Zn) are all of finite order we are sure to arrive at a MOD fixed point pair or a MOD limit cycle pair …

Uses Of Sampling Techniques & Inventory Control With Capacity Constraints, Florentin Smarandache, Sachin Malik, Neeraj Kumar

Branch Mathematics and Statistics Faculty and Staff Publications

The main aim of the present book is to suggest some improved estimators using auxiliary and attribute information in case of simple random sampling and stratified random sampling and some inventory models related to capacity constraints. This volume is a collection of five papers, written by six co-authors (listed in the order of the papers): Dr. Rajesh Singh, Dr. Sachin Malik, Dr. Florentin Smarandache, Dr. Neeraj Kumar, Mr. Sanjey Kumar & Pallavi Agarwal. In the first chapter authors suggest an estimator using two auxiliary variables in stratified random sampling for estimating population mean. In second chapter they proposed a family …

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 …

Nidus Idearum. Scilogs, Ii: De Rerum Consectatione, 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 second book 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 Set Approach To Algebraic Structures, Florentin Smarandache, Madad Khan, Fazal Tahir

Branch Mathematics and Statistics Faculty and Staff Publications

Real world is featured with complex phenomenons. As uncertainty is inevitably involved in problems arise in various elds of life and classical methods failed to handle these type of problems. Dealing with imprecise, uncertain or imperfect information was a big task for many years. Many modelswerepresentedinordertoproperlyincorporateuncertaintyintosystem description, LotA.Zadeh in 1965 introduced the idea of a fuzzy set. Zadeh replaced conventional characteristic function of classical crisp sets which takes on its values in f0;1g by membership function which takes on its values in closed interval [0;1]. Fuzzy set theory is conceptually a very powerful technique to deal with another aspect or …

Mathematics. Possible Subjects For The High School Entrance Examination And The Capacity Examination In Romania, Florentin Smarandache, Constantin Coanda, Ionuț Ivanescu

Branch Mathematics and Statistics Faculty and Staff Publications

The present book tries to offer students and teachers knowledge evaluation tools for all the chapters from the current Romanian mathematics syllabus. In the evolution of teenagers, the phase of admission in high schools mobilizes particular efforts and emotions. The present workbook aims to be a permanent advisor in the agitated period starting with the capacity examination and leading to the admittance to high school. The tests included in this workbook have a complementary character as opposed to the many materials written with the purpose to support all those who prepare for such examinations and they refer to the entire …

Various Arithmetic Functions And Their Applications, Florentin Smarandache, Octavian Cira

Branch Mathematics and Statistics Faculty and Staff Publications

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, România). The book is based on …

Complements To Classic Topics Of Circles Geometry, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors’ aspiration and attraction, as a poet writes lyrics about spring according to his emotions.

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 …

Uncertainty Communication Solution In Neutrosophic Key, Florentin Smarandache, Bianca Tedorescu, Mirela Teodorescu

Branch Mathematics and Statistics Faculty and Staff Publications

This study is the first step of a research that points out the solving of uncertainties in process analysis. The research is based on Neutrosophy Theory (Smarandache, 1995), a new concept of states treatment with a generous applicability to sciences, like artificial intelligence (Vladareanu et al, 2014).

Quaestiones Neutrosophicae, Florentin Smarandache, Yale Landsberg

Branch Mathematics and Statistics Faculty and Staff Publications

The following dialogue contains cuts from different non-protocolar conversations, initially not intended for publication, held by the authors by email during the beginning of 2015 – on Neutrosophy and related topics.

Many thanks to all friends and dialogue partners who payed attention to Neutrosophy and connected areas, in emails, yahoo groups, social media, letters, private discussions.

Euclid Squares On Infinite Planes, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time the authors study the new type of Euclid squares in various planes like real plane, complex plane, dual number plane, special dual like number plane and special quasi dual number plane. There are six such planes and they behave distinctly. From the study it is revealed that each type of squares behave in a different way depending on the plane. We define several types of algebraic structures on them. Such study is new, innovative and interesting. However for some types of squares; one is not in a position to define product. Further under …

Techno-Art Of Selariu Supermathematics Functions, 2nd Volume, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

An ALBUM, according to the dictionary, is defined as "notebook for storing photos, postcards, stamps, lyrics, quotes etc.", which, in other words, means gatherings of "pieces" of the same "species". Or, in the new Techno-Art of Selariu SuperMathematics Functions ALBUM (the second book of Selariu SuperMathematics Functions), one contemplates a unique COMPOSITION, INTER-, INTRA- and TRANS-DISCIPLINARY. (Capitalizing here is not a futility, but a harmony with the TRUTH.) One caveat I am indebted to do, as a consequent "reader" – over time, I received the ALBUM, chapter by chapter, pace by pace, which gave me the time to analyze / …

A New Type Of Group Action Through The Applications Of Fuzzy Sets And Neutrosophic Sets, Florentin Smarandache, Mumtaz Ali

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy sets are the most significant tools to handle uncertain data while neutrosophic sets are the generalizations of fuzzy sets in the sense to handle uncertain, incomplete, inconsistent, indeterminate, false data. In this paper, we introduced fuzzy subspaces and neutrosophic subspaces (generalization of fuzzy subspaces) by applying group actions.Further, we define fuzzy transitivity and neutrosophic transitivty in this paper. Fuzzy orbits and neutrosophic orbits are introduced as well. We also studied some basic properties of fuzzy subspaces as well as neutrosophic subspaces.

A Comparison Of Combined Overlap Block Fuzzy Cognitive Maps (Cobfcm) And Combined Overlap Block Neutrosophic Cognitive Map (Cobncm) In Finding The Hidden Patterns And Indeterminacies In Psychological Causal Models: Case Study Of Adhd, Hojjatollah Farahani, Florentin Smarandache, Lihshing Leigh Wang

Branch Mathematics and Statistics Faculty and Staff Publications

In spite of researchers’ concerns to find causalities, reviewing the literature of psychological studies one may argue that the classical statistical methods applied in order to find causalities are unable to find uncertainty and indeterminacies of the relationships between concepts.

In this paper, we introduce two methods to find effective solutions by identifying “hidden” patterns in the patients’ cognitive maps. Combined Overlap Block Fuzzy Cognitive Map (COBFCM) and Combined Overlap Block Neutrosophic Map (COBNCM) are effective when the number of concepts can be grouped and are large in numbers. In the first section, we introduce COBFCM, COBNCM, their applications, and …

N-Valued Interval Neutrosophic Sets And Their Application In Medical Diagnosis, Said Broumi, Irfan Deli, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper a new concept is called n-valued interval neutrosophic sets is given. The basic operations are introduced on n-valued interval neutrosophic sets such as; union, intersection, addition, multiplication, scalar multiplication, scalar division, truthfavorite and false-favorite. Then, some distances between n-valued interval neutrosophic sets (NVINS) are proposed. Also, we propose an efficient approach for group multi-criteria decision making based on n-valued interval neutrosophic sets. An application of n-valued interval neutrosophic sets in medical diagnosis problem is given.

(T, I, F)-Neutrosophic Structures, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we introduce for the first time a new type of structures, called (T, I, F)-Neutrosophic Structures, presented from a neutrosophic logic perspective, and we show particular cases of such structures in geometry and in algebra. In any field of knowledge, each structure is composed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy, that structure is a (T, I, F)-Neutrosophic Structure.

Neutrosophic Axiomatic System, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we introduce for the first time the notions of Neutrosophic Axiom, Neutrosophic Axiomatic System, Neutrosophic Deducibility and Neutrosophic Inference, Neutrosophic Proof, Neutrosophic Tautologies, Neutrosophic Quantifiers, Neutrosophic Propositional Logic, Neutrosophic Axiomatic Space, Degree of Contradiction (Dissimilarity) of Two Neutrosophic Axioms, and Neutrosophic Model. A class of neutrosophic implications is also introduced. A comparison between these innovatory neutrosophic notions and their corresponding classical notions is made. Then, three concrete examples of neutrosophic axiomatic systems, describing the same neutrosophic geometrical model, are presented at the end of the paper.

Inteligencia Artificial Y Analítica De Datos Para Un Futuro Sostenible, Florentin Smarandache, Michael Y. Leyva Vazquez

Branch Mathematics and Statistics Faculty and Staff Publications

1-De los mitos a la Inteligencia Artificial moderna

2-Evolución e Impacto de la Inteligencia Artificial Generativa

3-Sostenibilidad en la Era de la Inteligencia Artificial: Impactos y Desafíos

3-Neutrosofía másalláde lo verdaderoy lo falso

Neutrosophic Precalculus And Neutrosophic Calculus, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic Analysis is a generalization of Set Analysis, which in its turn is a generalization of Interval Analysis.

Neutrosophic Precalculus is referred to indeterminate staticity, while Neutrosophic Calculus is the mathematics of indeterminate change.

The Neutrosophic Precalculus and Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the methods used to deal with such indeterminacy.

In this book, the author presents a few examples of indeterminacies and several methods to deal with these specific indeterminacies, but many other indeterminacies there exist in our everyday life, and they have to be studied …

Symbolic Neutrosophic Theory, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Symbolic (or Literal) Neutrosophic Theory is referring to the use of abstract symbols (i.e. the letters T, I, F, or their refined indexed letters T_j, I_k, F_l) in neutrosophics.

In the first chapter we extend the dialectical triad thesis-antithesis-synthesis (dynamics of A and antiA, to get a synthesis) to the neutrosophic tetrad thesis-antithesis-neutrothesis-neutrosynthesis (dynamics of A, antiA, and neutA, in order to get a neutrosynthesis).

In the second chapter we introduce the neutrosophic system and neutrosophic dynamic system. A neutrosophic system is a quasi- or –classical system, in the sense that the neutrosophic …