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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 various articles in the theory of numbers (starting from 1975), updated many times. This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equations", published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with hi...

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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 its entries from modulo integers. However this sort of results are NP hard problems if we use reals or complex numbers....

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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. Secondly, the operation of × is defined and under × also these sets form only a semigroup. Several important properties about them are analysed. Several open conjectures are proposed....

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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 all this study is not only innovative and interesting but challenging. So this book which is only full of problems based on MOD structures will be a boon to researchers. Further the MOD series books of the authors will certainly be an appropriate guide to solve these problems....

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This work seeks to provide students and teachers with tools for assessing knowledge in all chapters of the current mathematics program. 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 subject matter included in the analytical mathematics syllabus of arithmetic in Romania, algebra and geometry from the lower secondary grades. These tests have been elaborated with the intention to offer proper support to those who use the workbook, assuring them the success and extra preparation for future exams. For all the suggested subjects options the authors present grading scales, solutions, problems solving suggestions and complete answers, depending on the difficulty and the specificity of each problem. Each test has been organized according to the models of the subjects given at recent high school entrance exams and are thus structured so that the entire examination syllabus is covered and the school workbooks are thus rhythmically covered. At the end of the workbook, the subjects giv...

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In this book authors for the first time construct a MOD Relational Maps model analogous to Fuzzy Relational Maps (FRMs) model or Neutrosophic Relational Maps (NRMs) model using the MOD rectangular or relational matrices. The advantage of using these models is that we are sure to get the MOD fixed point pair or a MOD limit cycle pair after a finite number of iterations. However we as in case of FRMs or NRMs need not at each stage threshold the resultant state vectors. We do updating when the on state which we started becomes zero. This is yet another advantage of using these newly constructed models. Finally the resultant state vector pair can be real MOD values or finite complex or neutrosophic or a dual number or a special quasi dual number or special dual like number....

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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 matrix special type of matrix topological spaces. Likewise using MOD interval natural neutrosophic subsets matrices semigroups we can build MOD interval natural neutrosophic matrix subset special type of topological spaces. The final chapter mainly proposes several open conjectures about the validity of the Kakutani’s fixed point theorem for all MOD special type of subset topological spaces....

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In this book authors for the first time introduce the notion of MOD subsets. The S-semigroups enjoy several interesting properties. The notion of MOD universal subset and MOD absorbing subsets are defined and developed. MOD natural neutrosophic subsets forms only a semigroup under ‘+’. In fact the main feature enjoyed by this structure is they have subset idempotents with respect to ‘+’. They are S-semigroups under ‘+’. These MOD natural neutrosophic subsets of all 6 types are only semigroups under ‘x’. These enjoy distinct properties depending on the subset that is used. Further in these cases two types of products can be used one zero dominated product and the other MOD natural neutrosophic zero dominated product. Both the product are different. Finally using these MOD subsets matrices and polynomials are defined and developed. The new notion of MOD subset matrices are defined and these collections also under ‘+’ (or ‘x’) are only semigroups. On similar lines MOD subset polynomials are defined and they also form only a semigroup under ‘+’ (or ‘x’). These new notions are interesting and researchers can find lots of applications whe...

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In this book authors for the first time introduce new mathematical models analogous to FCMs and NCMs. They have constructed 12 types of MOD Cognitive Maps models, such as: MOD Cognitive Maps model, MOD dual number Cognitive Maps model, MOD neutrosophic Cognitive Maps model, MOD finite complex number Cognitive Maps model, MOD special dual like number Cognitive Maps model, MOD special quasi dual number Cognitive Maps model. Apart from this they have defined MOD natural neutrosophic Cognitive Maps model, MOD natural neutrosophic-neutrosophic Cognitive Maps model, MOD natural neutrosophic dual number Cognitive Maps model, MOD natural neutrosophic special dual like number Cognitive Maps model, MOD natural neutrosophic special quasi dual number model and MOD natural neutrosophic finite complex number Cognitive Maps model. These model certainly are more appropriate than the existing ones. Finally they construct MOD interval Cognitive Maps model and describes it. The final chapter suggests a collection of problems for the interested reader....

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In this book authors for the first time introduce study and develop the notion of MOD graphs, MOD directed graphs, MOD finite complex number graphs, MOD neutrosophic graphs, MOD dual number graphs and so on. Likewise MOD directed natural neutrosophic graphs are defined. Further type I, type II and type III. MOD directed graphs and MOD natural neutrosophic graphs are defined and developed. This book has over 185 examples and over 250 figures. The notion of MOD bipartite graphs and MOD natural neutrosophic bipartite graphs are described. This book gives the probable applications of these concepts to MOD mathematical models like MOD Cognitive Maps model and MOD Relational Maps model which have been introduced by the authors. There are open conjectures which can help the researchers in graph theory. Several innovative results are obtained....

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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. Such study is new innovative and certainly lead to several special important applications in the field of science, engineering, technology medicine and social issues. This book gives lots of illustrative examples to make it easy for a non mathematician. Several problems are suggested about MOD special fixed point pairs and MOD limit cycle pairs for the interested reader....

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The authors 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....

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The author introduces for the first time the concept of plithogeny in philosophy and, as a derivative, the concepts of plithogenic set / logic / probability / statistics in mathematics and engineering – and the degrees of contradiction (dissimilarity) between the attributes’ values that contribute to a more accurate construction of plithogenic aggregation operators and to the plithogenic relationship of inclusion (partial ordering). They resulted from practical needs in our everyday life, and several examples and applications of them are presented. Plithogeny is the genesis or origination, creation, formation, development, and evolution of new entities from dynamics and organic fusions of contradictory and/or neutrals and/or non-contradictory multiple old entities. Plithogeny pleads for the connections and unification of theories and ideas in any field. As “entities” in this study, we take the “knowledges” in various fields, such as soft sciences, hard sciences, arts and letters theories, etc....

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The topics discussed in this book are Int-soft semigroup, Int-soft left (right) ideal, Int-soft (generalized) bi-ideal, Int-soft quasi-ideal, Int-soft interior ideal, Int-soft left (right) duo semigroup, starshaped (∈, ∈∨ qk)-fuzzy set, quasi-starshaped (∈, ∈∨ qk)-fuzzy set, semidetached mapping, semidetached semigroup, (∈, ∈ ∨qk)-fuzzy subsemi-group, (qk, ∈ ∨qk)-fuzzy subsemigroup, (∈, ∈ ∨ qk)-fuzzy subsemigroup, (qk, ∈ ∨ qk)-fuzzy subsemigroup, (∈ ∨ qk, ∈ ∨ qk)-fuzzy subsemigroup, (∈, ∈∨ qkδ)-fuzzy subsemigroup, ∈∨ qkδ -level subsemigroup/bi-ideal, (∈, ∈∨ qkδ )-fuzzy (generalized) bi-ideal, δ-lower (δ-upper) approximation of fuzzy set, δ-lower (δ-upper) rough fuzzy subsemigroup, δ-rough fuzzy subsemigroup....

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In this book authors introduce the new notion of MOD rectangular planes. The functions on them behave very differently when compared to MOD planes (square). These are different from the usual MOD planes. Algebraic structures on these MOD rectangular planes are defined and developed. These semigroups have special and distinct features. Certainly these finite structures can find applications in several important problems related to automaton theory. The authors also mention about matrix and polynomial MOD rectangular natural neutrosophic semigroups. Several innovative results are obtained....

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In this book authors for the first time introduce the new notion of special subset vertex subgraph of subset vertex graphs. These subset vertex graphs takes the vertex set values from the power set P(X) of any set X. The main speciality of these subset vertex graphs is that once a set of subsets from P(X) is given, the edges of the graph are fixed in a unique way, so for a given collection of subset vertices the graph is always unique. The special subset vertex subgraphs of G are the ones, which have the same number of vertices as that of the subset vertex graph G. This special property enables one to use these subgraphs as fault tolerant graphs for fault tolerant networks. In this book authors define the notion of special subset vertex hyper subgraphs, which are better suited for fault tolerant graphs....

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In this book authors for the first time introduce the notion of subset vertex graph using the vertex set as the subset of the power set P(S), S is assumed in this book to be finite; however it can be finite or infinite. Theey have defined two types of subset vertex graphs, one is directed and the other one is not directed. The most important fact which must be kept in record is that for a given set of vertices there exists one and only one subset vertex graph be it of type I or type II. Several important and innovative features of these graphs are defined, described and developed in this book. The set S considered can be real or complex or indeterminate or dual numbers depending on the problem in hand. However by the definitions of the subset vertex graphs of type I and type II the edges or relation between two vertices are always real. Based on the property that in case of subset vertex graphs for a given set of vertices the graph is unique, these graphs are well suited for social networking. This is one of the special features enjoyed by these subset vertex graphs. The applications of these new types of subset vertex graphs are de...

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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 are not always guaranteed of getting the special subset vertex multisubgraphs. However, if we have subset vertex multigraphs which has special subset vertex multisubgraphs then we can have corresponding to them multinet works, these multi networks are fault tolerant networks, hence the study of these special type of multisubgraphs has become mandatory. Further in this book we introduce a new notion of super special subset vertex multigraphs which are built using the subset vertex multigraphs, especially when the subset ver...

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Neutrosophic theory and its applications have been expanding in all directions at an astonishing rate especially after of the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structures such as rough neutrosophic set, single valued neutrosophic rough set, bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been an important tool in the application of various areas such as data mining, decision making, e-learning, engineering, medicine, social science, and some more....

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The Neutrosophic Triplets were introduced by F. Smarandache & M. Ali in 2014 - 2016, and consequently the neutrosophic triplet group, ring, field – in general the neutrosophic triplet structures; while the Neutrosophic Extended Triplets were introduced by F. Smarandache in 2016 and consequently the neutrosophic extended triplet structures. A neutrosophic extended triplet is a neutrosophic triplet, where the neutral of x {denoted by e-neut(x) and called "extended neutral"} is allowed to also be equal to the classical algebraic unitary element (if any). Therefore, the restriction "different from the classical algebraic unitary element if any" is released. As a consequence, the "extended opposite" of x, denoted by e-anti(x), is also allowed to be equal to the classical inverse element from a classical group....