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...This book has five chapters. Chapter one is introductory in nature. Fuzzy linguistic spaces are introduced in chapter two. Fuzzy linguistic vector spaces are introduced in chapter three. Chapter four introduces fuzzy linguistic models. The final chapter suggests over 100 problems and some of them are at res...

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...Preface 5 Chapter One Introduction 7 Chapter Two Fuzzy Linguistic Spaces 9 Chapter Three Fuzzy Linguistic Set Vector Spaces 23 Chapter Four Fuzzy Linguistic Models 129 4.1 Operations on Fuzzy Linguistics Matrices 129 4.2 Fuzzy Linguistic Cognitive Models 139 4.3 Fuzzy Lingu...

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...In this book authors for the first time introduce the notion of special type of topological spaces using the interval [0, n). They are very different from the usual topological spaces. Algebraic structure using the interval [0, n) have been systemically dealt by the authors. Now using those algebraic str...

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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 ...

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...risp relations, Pauli Exclusion Principle and the Law of Included Multiple-Middle, weighted neutrosophic soft sets, interval valued neutrosophic soft topological spaces, generalization of soft neutrosophic rings and soft neutrosophic fields, neutrosophic refined similarity measure based on cosine function, and similar topics. Contributors: F. Yuhua, P. K. Maji, A. A. Sa...

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...ion 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 co...

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...Chapter one is introductory in nature and chapter two uses vector spaces to build quasi set topological vector subspaces. Not only we use vector spaces but we also use S-vector spaces, set vector spaces, semigroup vector spaces and group vector spaces to build set topological vector subspac...

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...To every quasi set topological vector subspace T relative to the set P  F, we have a lattice associated with it we call this lattice as the Representative Quasi Set Topological Vector subspace lattice (RQTV-lattice) of T relative to P. Whe...

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...Preface 5 Chapter One INTRODUCTION 7 Chapter Two QUASI SET TOPOLOGICAL VECTOR SUBSPACES 9 Chapter Three S-QUASI SET TOPOLOGICAL VECTOR SUBSPACES 59 Chapter Four NEW SET TOPOLOGICAL VECTOR SPACES 89 FURTHER READING 147 INDEX 149 ABOUT THE AUTHORS 152...

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... on mappings on neutrosophic soft classes, neutrosophic implications, hierarchical clustering algorithms, neutrosophic crisp sets, neutrosophic crisp topological spaces, neutrosophic logic for mental model elicitation and analysis, neutrosophic hypergroups and neutrosophic hyperrings, neutrosophic lattices, single valued neutrosophic similarity measures for multiple attrib...

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...troduce 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 possi...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...s BY T.G.JA´IY´E O. L´A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Smarandachely Precontinuous maps and Preopen Sets in Topological Vector Spaces BY SAYED ELAGAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Path Double Covering Number of Product Graphs BY A. NAGARAJAN, V. MAHESWARI, S. NAVANEETHAKRISHNAN. . . . . . 27...

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...A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combinatorial configuration with the classical mathematics, it fascinates more and more mathematician’s interesting. Its function and role in mathematics are...

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...n the United States of America Preface A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combina... ... formally define a Klein surface. A Klein surface is a Hausdorff, connected, topological space S together with a family = {(U i ,φ i ) |i ∈ I} such th... ... i U j )−→φ i (U i U j ). are dianalytic. The family is said to be a topological atlas on S. Chapter 1 Preliminary 2 The boundary of S is the s... ...s. If k(S),g(S) and χ(S) are the number of connected components of ∂S, the topological genus and the Euler characteristic of a surface S, then we have... ...a subspace) and with the Parallel Universes (because they combine separate spaces into one space) too([32]). Chapter 1 Preliminary 3 In [61], Smarand... ...e exist a triangulation{T i ,i≥ 1} on S. This theorem isfundamental forthe topologicalgraph theory, which enables us to discussion a surface combinato... ...ingerman in 1985 ([8]). The later is also related to the modular groups of spaces and need to investigate further itself. The questions are (i) how ca... ...t sectional curvaturec, then we knowthatM is isometric to one of the model spaces R n ,S R n or H R n. There is also a combinatorial map theory for hi...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...On the Bicoset of a Bivector Space Abstract: The study of bivector spaces was first intiated by Vasantha Kandasamy. The objective of this paper is to present the concept of bicoset of a bivector space and obtain some of its elementary properties. Key Words: bigroup, bivector space, bicoset,...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...uarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...Study of the Problems of Persons with Disability (PWD) Using FRMs BY W.B.VASANTHA KANDASAMY, A.PRAVEEN PRAKASH AND K.THIRUSANGU 01 Topological Multi-groups and Multi-fields BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 08 Shortest Co-cycle Bases of Graphs BY HAN REN AND JING HAN . . . . . . . . . . . . . . . ...

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...The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published in 1998. In the year 1999, bivector spaces was introduced. In 2001, concept of free De Morgan bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the const...

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...s, S-rings and S-NA-rings 19 1.5 Semirings, S-semirings and S-semivector spaces 35 1.6 Near-rings and S-near-rings 43 1.7 Vector spaces and S-vec... ...es 35 1.6 Near-rings and S-near-rings 43 1.7 Vector spaces and S-vector spaces 62 Chapter Two: BIGROUPS AND SMARANDACHE BIGROUPS 2.1 Bigroup... ...ties 166 4 Chapter Seven: BISEMIRINGS, S-BISEMIRINGS, BISEMIVECTOR SPACES AND S-BISEMIVECTOR SPACES 7.1 Bisemirings and its properties 17... ...operties 180 7.3 S-bisemirings and its properties 183 7.4 Bisemivector spaces and S-bisemivector spaces 190 Chapter Eight: BINEAR-RING SMARANDA... ...analogue and its applications 218 Chapter Nine: BISTRUCTURES, BIVECTOR SPACES AND THEIR SMARANDACHE ANALOGUE 9.1 Bistructure and S-bistructur... ... in this book purely in an algebraic vein so intermingling of category or topological concepts are totally absent in this book. DEFINITION 6.2.1: ...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...nd Cycles BY P. DEVADAS RAO, B. SOORYANARAYANA, M. JAYALAKSHMI . . . . . . . . . 48 On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces BY SAYED ELAGAN AND MOHAMAD RAFI SEGI RAHMAT . . . . . . . . . . . . 61 Achromatic Coloring on Double Star Graph Families BY VERNOLD VIVIN J., VENKATACHALAM M. AND AKBAR ALI M.M. . . . . . 71 Some Results on Super ...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...t Symmetric n-Marked Graphs BY P.SIVA KOTA REDDY, V. LOKESHA and GURUNATH RAO VAIDYA. . . . . . . . .41 Some Fixed Point Theorems in Fuzzy n-Normed Spaces BY SAYED KHALIL ELAGAN and MOHAMAD RAFI SEGI RAHMAT . . . . . . . . . . 45 A Result of Ramanujan and Brahmagupta Polynomials Described by a Matrix Identity BY R. RANGARAJAN . . . . . . . . . . . . . . . . 57 Biharm...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...arterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to othe...

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...hapter 3. Topology with Smarandache Geometry . . . . . . . . . . . . . . 91 3.1 Algebraic topology . . . . . . . . . . . . . . . . . . . . 92 3.1.1 Topological space. . . . . . . . . . . . . . . . . . . . . . . .92 3.1.2 Metric space . . . . . . . . . . . . . . . . . . . . . 95 3.1.3 Fundamental group. . . . . . . . . . . . . . . . . . . . . .96 3.1.4 Seifert and Van-...

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...randache Geometries 67 Chapter 4. Closed s-Manifolds 71 Closed s-Manifolds 71 Topological 2-Manifolds 81 Suggestions For Further Research 89 Refe... ..., it is now most likely to be associated with a class of differential geometric spaces called Riemannian manifolds. Here, geometry is studied throug... ...articular examples of one kind of space, and extending attention to non-uniform spaces brings much generality, applicability (e.g. general relativit... ...and Riemann sought to contradict. In fact, Riemann began the study of geometric spaces that are non-uniform with respect to the parallel postulate, ... ...nd we will see a wide variety of geometric structures in intuitively manageable spaces. Hilbert separated his set of axioms into groups, and we wi... ...e elliptic and a single hyperbolic vertex. In particular, an s-manifold that is topologically equivalent to the plane will be semi-paradoxist if all ... ...te Smarandache structure using topology. Here we will look at some of the basic topological structures obtainable by closed s-manifolds. Closed s-M... ...s, and has 12 vertices, all of them elliptic. It is a closed surface, and it is topologically equivalent to the unit sphere (or any sphere). Of all s... ...ally equivalent to the unit sphere (or any sphere). Of all s-manifolds that are topological spheres, this is the most regular, and the one most close...

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.... . 20 C. Castro, M. Pavˇ siˇ c The Extended Relativity Theory in Clifford Spaces.. . . . .31 K. Dombrowski Rational Numbers Distribution and Resonanc... ... The last fact implies that signature conditions defning pseudo-Riemannian spaces are broken, so that photon teleportation manifests outside the basic... ...strictly non-degenerate metric feature g<0 by defnition. Pseudo-Riemannian spaces are a particular case of Riemannian spaces, where the metric is sign... ...and signatures) on the equal footing. Because the notion of dimension is a topological invariant, and the concept of a fxed dimension is lost due to t... ...le. For an extensive review of this Extended Relativity Theory in Clifford Spaces that encompasses the unifed dy- namics of all p-branes, for differen... ...ears earlier by Max Born on a Dual Relativity principle operating in Phase Spaces [49], [74] where there is an upper bound on the four-force (maximal ... ...994. Brekke L., Freund P. Phys. Reports, v. 231, 1993, 1. 41. Pitkannen M. Topological geometrodynamics I, II. Chaos, Solitons and Fractals, v. 13, No... ...imensions. C-space is a pandimensional continuum [14], [8]. Dimensions are topological invariants and, since the dimensions of the extended objects ch... ... 1994. Brekke L. and Freund P. Phys. Reports, v. 231, 1991, 1. Pitkanen M. Topological geometrodynamics, the website, http://www.physics.helsinki.f/ma...

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...ensors into pseudotensors in inhomoge- neous anisotropic pseudo-Riemannian spaces, are not phys- ically observable quantities. The completely antisymm... ... lines of the Riemannian space. In general, coordinate lines in Riemannian spaces are curved, inhomogeneous, and are not orthogonal to each other (if ... ...ion of the waves is given, according to Petrov’s classifcation of Einstein spaces and gravitational felds located therein. The world-lines deviation e... ...avitational waves is directly linked to the algebraical classi- fcation of spaces given by Petrov [37] ( Petrov classifcation ), according to which th... ...c spaces) complex domains,G/K [7, 8, 9] are not the same as the or- dinary topological boundaries (except in some special cases). The reason being tha... ...ical boundary. Shilov boundaries are the minimal subspaces of the ordinary topological boundaries which implement the Maldacena-’T Hooft-Susskind holo... ...mseddine-West formulation of Gravity, with a cosmolo- gical constant and a topological Gauss-Bonnet invariant term, can be obtained from an action ins... ... ≡Q 1 (very special case) since the circle is both the Shilov and ordinary topological boundary of the discD 1 . However,Q 2 ≡S 1 ×S 1 /Z 2 =S 1 ×RP 1... ... defnes the coset space B 6 . In this special case the Shilov and ordinary topological boundaries of B 6 coincide with S 5 [3]. Hence, following the s...

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...zations 84 1.8 Fuzzy near-rings and their properties 94 1.9 Fuzzy vector spaces and fuzzy bivector spaces 119 1.10 Fuzzy semigroups and their prope... ... fuzzy rings: definitions and properties 291 4.2 Smarandache fuzzy vector spaces and its properties 303 4.3 Smarandache fuzzy non-associative rings ... ...e fuzzy semirings and its properties 333 5.2 Smarandache fuzzy semivector spaces 341 5.3 Smarandache fuzzy non-associative semirings 352 5.4 Smaran... ...zy birings, fuzzy fields, fuzzy semirings, fuzzy near-rings, fuzzy vector spaces, fuzzy semigroups and fuzzy half- groupoids. The results used in the... ...tions, or to be more precise we are not interested in discussing concepts topologically or analytically like continuity, connected, increasing funct... ...A) = 1. iii. A is F-complete. Proof: Refer [72]. The study of fuzzy topological semigroups are not introduced or studied in this book. As the ... ...group G is called a g fuzzy topology. The pair (G, τ) is called a g-fuzzy topological space. Example 1.12.1: Let G = {1, –1} be the group with res... ... DEFINITION 1.12.3: Let (G 1 , τ 1 ) and (G 2 , τ 2 ) be any two g-fuzzy topological spaces. A function f: (G 1 , τ 1 ) → (G 2 , τ 2 ) is said to b... ...s fuzzy point by x 1 . DEFINITION 1.12.4: A fuzzy subset µ in a g-fuzzy topological space, (G, τ) is called a Q g -neighborhood of the fuzzy point...

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...ility, which supports incomplete (not completely known/determined) probability spaces (for 0 < n < 1 and i = 0, 0 [ t, f [ 1) or incomplete events w... ...letheist probability, which says that intersection of some disjoint probability spaces is not empty (for t = f = 1 and i = 0; some paradoxist probabi... ... "indeterminacy" - due to unexpected parameters hidden in some probability spaces, and let each component t, i, f be even boiling over 1 (overfl... ...such that (74) The interval , endowed with this topology, forms a neutrosophic topological space. Consider now two statements and , then one defines t... ...” and “absolute true event” the former being true in only some probability sub-spaces while the latter being true in all probability sub-spaces. Simi... ...lity sub-spaces. Similarly, events that are false in only some probability sub-spaces are classified as “relative false events” while events that ar... ...tems theory to form a science of complex systems in their own right, with many topologically isomorphic specialisms appearing (e.g. neural networks,... ...s, low resolution views that discard most of the connectivity effects (the two topological dimensions), i.e. how values interrelate to support or op...

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...raphs, neutrosophic fields, neutrosophic matrices and neutrosophic vector spaces. We provide many illustrations and applications relating NCMs. In t... ..., and Sculptural Presence. In our practice we simultaneously explore both spaces; artistic and technical constraints and opportunities mutually info... ...rmediate relation exists within the domain elements or node and the range spaces elements. The number of elements in the range space need not in gen... ...as nodes and causalities as edges. It represents causal relations between spaces D and R . Let D i and R j denote that the two nodes of an FRM. ... ...sociated with employees or employer. We prefer to choose 8 × 6 matrix for spaces X and Y respectively. Using the fuzzy matrix we find a best relatio... ...anics, Physical Review, 113 (1959) 1165-1177. 67. Lewin, K. Principles of Topological Psychology, McGraw Hill, New York, 1936. 68. Liu, F., and F. ...

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...ntermediate relation exists within the domain elements or node and the range spaces elements. The number of elements in the range space need not in... ...c, as nodes and causalities as edges. It represents causal relations between spaces D and R . 122 Let D i and R j denote that the two nodes o... ...125 concepts and causalities as edges. It represents causal relations between spaces D i and R i i = 1, 2. The directed bigraph may be disjoint or... ... called a limit bicycle. Now we illustrate the model. FRBMs has two domain spaces and two range spaces which may or may not have common nodes. ... ...ces which may or may not have common nodes. Examples 3.5.1: When both the spaces are identical, we illustrate by example in the case of employee... ...echanics, Physical Review, 113, 1165-1177, 1959. 88. Lewin, K. Principles of Topological Psychology, McGraw Hill, New York, 1936. 89. Liu, F., and...

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...orks; - to create contradictory literary and artistic works in scientific spaces (using scientific: symbols, meta-language, matrices, theorems, lemm... ...there are more (concentric or not) universes; in a space: more spaces in a time: more times in a move: more moves We meet... ...; - create contradictory literary and artistic works in scientific spaces (using scientific: symbols, meta-language, matrices, theorems,... ...f “)”.} The interval NT, endowed with this topology, forms a neutrosophic topological space. 2.13. Neutrosophic Sigma-Algebra. - The collection of ... ..., which supports incomplete (not completely known/determined) probability spaces (for 0 < n < 1, 0 [ t, f [ 1) or incomplete events whose probabilit... ...ist probability, which says that intersection of some disjoint probability spaces is not empty (for t = f = 1 and i = 0; some paradoxist probabilitie...

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...erintuitively, they are wrapped around space rather than in it - very much like the topological chimeras the Mobius strip, or the Klein bottle). Ma... ...abi-Yau space wrapped on every 4-dimensional point in our universe. But Calabi-Yau spaces are not fixed entities. New ones can be created every tim...

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...ol. 50, 71-84, (1991). 76. Ursul N. I, Connectivity in weakly Boolean Topological rings, IZV Acad. Nauk Moldor. Vol. 79, 17-21, (1989). 206 ... ...Vasantha Kandasamy W.B., Smarandache Semirings, Semifields and Semivector spaces, American Research Press, Rehoboth, NM, (2002). 158. Vasantha Ka...

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...rate intangibles such as the beauty of a sunset or the liberating sensation of open spaces. "Green" accounting - adjusting the national accounts to ... ...erintuitively, they are wrapped around space rather than in it - very much like the topological chimeras the Mobius strip, or the Klein bottle). Ma... ...labi-Yau space wrapped on every 4-dimensional point in our universe. But Calabi-Yau spaces are not fixed entities. New ones can be created every tim... ...niches (moons and planets and other galaxies and inter-planetary and inter-galactic spaces). Long before Man evolves into another animal through ge...

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...rate intangibles such as the beauty of a sunset or the liberating sensation of open spaces. "Green" accounting - adjusting the national accounts to ... ...erintuitively, they are wrapped around space rather than in it - very much like the topological chimeras the Mobius strip, or the Klein bottle). Ma... ...labi-Yau space wrapped on every 4-dimensional point in our universe. But Calabi-Yau spaces are not fixed entities. New ones can be created every tim... ...niches (moons and planets and other galaxies and inter-planetary and inter-galactic spaces). Long before Man evolves into another animal through ge...