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Neutrosophic theory has representatives on all continent sand, therefore, it can be said to be a universal theory. On the other hand, according to the two volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018) about 150 researchers from 37 countries apply the idea and the neutrosophic method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics consists of the introduction of the degree of indeterminacy/neutrality (I) as independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus etc. and their applications in multiple fields have been extended and applied in various fields, such as communication, management and information technology. The present volume gathers the latest neutrosophic techniques, methodologies or mixed approaches, being thus a barometer of the neutrosophic research in 2020.

neutrosophic topology --- neutrosophic generalized topology --- neutrosophic generalized pre-closed sets --- neutrosophic generalized pre-open sets --- neutrosophic p T 1 2 space --- neutrosophic g p T 1 2 space --- generalized neutrosophic compact and generalized neutrosophic compact --- fuzzy operating characteristic curve --- fuzzy OC band --- Birnbaum-Sunders distribution --- single acceptance sampling plan --- aggregation operator --- decision making --- neutrosophic soft expert sets --- neutrosophic soft expert multiset --- neutrosophic sets --- neutrosophic multisets --- neutrosophic multigroups --- neutrosophic multisubgroups --- bipolar neutrosophic number (BNN) --- BNN improved generalized weighted HM (BNNIGWHM) operator --- BNN improved generalized weighted geometry HM (BNNIGWGHM) operator --- decision-making --- neutrosophic cubic set --- neutrosophic cubic hybrid geometric operator --- neutrosophic cubic Einstein hybrid geometric operator --- multiattributedecision-making (MADM) --- neutrosophic set --- Zhang-Zhang’s YinYang bipolar fuzzy set --- single-valued bipolar neutrosophic set --- bipolar fuzzy set --- YinYang bipolar fuzzy set --- multiple attribute group decision making (MAGDM) --- Linguistic neutrosophic --- LNN Einstein weighted-average operator --- LNN Einstein weighted-geometry (LNNEWG) operator --- semi-idempotent --- neutrosophic rings --- modulo neutrosophic rings --- neutrosophic semi-idempotent --- neutrosophic ring --- neutrosophic triplets --- idemponents --- special neutrosophic triplets --- acceptance number --- neutrosophic approach --- operating characteristics --- risks --- sample size --- probabilistic neutrosophic hesitant fuzzy set --- distance measure --- similarity measure --- entropy measure --- multi-criteria decision-making (MCDM) --- Neutrosophic Quadruple (NQ) --- Neutrosophic Quadruple set --- NQ vector spaces --- NQ linear algebras --- NQ basis --- orthogonal or dual NQ vector subspaces --- similarity index --- diagnosis --- process --- indeterminacy --- neutrosophic statistics --- time-truncated test --- Weibull distribution --- risk --- uncertainty --- neutrosophic --- neutrosophic logic --- fuzzy logic --- control chart --- neutrosophic numbers --- monitoring --- financial assets --- neutrosophicportfolio --- neutrosophic portfolio return --- neutrosophic portfolio risk --- neutrosophic covariance --- Abel-Grassmann’s neutrosophic extended triplet loop --- generalized Abel-Grassmann’s neutrosophic extended triplet loop --- strong inverse AG-groupoid --- quasi strong inverse AG-groupoid --- quasi Clifford AG-groupoid --- semigroup --- CA-groupoid --- regular CA-groupoid --- neutrosophic extended triplet (NET) --- Green relation --- multi-attribute group decision-making --- granular computing --- interval-valued neutrosophic information --- multigranulation probabilistic models --- merger and acquisition target selections --- dynamic neutrosophic environment --- dynamic interval-valued neutrosophic set --- unknown weight information --- single-valued neutrosophic linguistic set --- combined weighted --- logarithmic distance measure --- supplier selection --- fresh aquatic products --- MAGDM

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Neutrosophic theory has representatives on all continent sand, therefore, it can be said to be a universal theory. On the other hand, according to the two volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018) about 150 researchers from 37 countries apply the idea and the neutrosophic method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics consists of the introduction of the degree of indeterminacy/neutrality (I) as independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus etc. and their applications in multiple fields have been extended and applied in various fields, such as communication, management and information technology. The present volume gathers the latest neutrosophic techniques, methodologies or mixed approaches, being thus a barometer of the neutrosophic research in 2020.

Research & information: general --- Mathematics & science --- neutrosophic topology --- neutrosophic generalized topology --- neutrosophic generalized pre-closed sets --- neutrosophic generalized pre-open sets --- neutrosophic p T 1 2 space --- neutrosophic g p T 1 2 space --- generalized neutrosophic compact and generalized neutrosophic compact --- fuzzy operating characteristic curve --- fuzzy OC band --- Birnbaum-Sunders distribution --- single acceptance sampling plan --- aggregation operator --- decision making --- neutrosophic soft expert sets --- neutrosophic soft expert multiset --- neutrosophic sets --- neutrosophic multisets --- neutrosophic multigroups --- neutrosophic multisubgroups --- bipolar neutrosophic number (BNN) --- BNN improved generalized weighted HM (BNNIGWHM) operator --- BNN improved generalized weighted geometry HM (BNNIGWGHM) operator --- decision-making --- neutrosophic cubic set --- neutrosophic cubic hybrid geometric operator --- neutrosophic cubic Einstein hybrid geometric operator --- multiattributedecision-making (MADM) --- neutrosophic set --- Zhang-Zhang’s YinYang bipolar fuzzy set --- single-valued bipolar neutrosophic set --- bipolar fuzzy set --- YinYang bipolar fuzzy set --- multiple attribute group decision making (MAGDM) --- Linguistic neutrosophic --- LNN Einstein weighted-average operator --- LNN Einstein weighted-geometry (LNNEWG) operator --- semi-idempotent --- neutrosophic rings --- modulo neutrosophic rings --- neutrosophic semi-idempotent --- neutrosophic ring --- neutrosophic triplets --- idemponents --- special neutrosophic triplets --- acceptance number --- neutrosophic approach --- operating characteristics --- risks --- sample size --- probabilistic neutrosophic hesitant fuzzy set --- distance measure --- similarity measure --- entropy measure --- multi-criteria decision-making (MCDM) --- Neutrosophic Quadruple (NQ) --- Neutrosophic Quadruple set --- NQ vector spaces --- NQ linear algebras --- NQ basis --- orthogonal or dual NQ vector subspaces --- similarity index --- diagnosis --- process --- indeterminacy --- neutrosophic statistics --- time-truncated test --- Weibull distribution --- risk --- uncertainty --- neutrosophic --- neutrosophic logic --- fuzzy logic --- control chart --- neutrosophic numbers --- monitoring --- financial assets --- neutrosophicportfolio --- neutrosophic portfolio return --- neutrosophic portfolio risk --- neutrosophic covariance --- Abel-Grassmann’s neutrosophic extended triplet loop --- generalized Abel-Grassmann’s neutrosophic extended triplet loop --- strong inverse AG-groupoid --- quasi strong inverse AG-groupoid --- quasi Clifford AG-groupoid --- semigroup --- CA-groupoid --- regular CA-groupoid --- neutrosophic extended triplet (NET) --- Green relation --- multi-attribute group decision-making --- granular computing --- interval-valued neutrosophic information --- multigranulation probabilistic models --- merger and acquisition target selections --- dynamic neutrosophic environment --- dynamic interval-valued neutrosophic set --- unknown weight information --- single-valued neutrosophic linguistic set --- combined weighted --- logarithmic distance measure --- supplier selection --- fresh aquatic products --- MAGDM --- neutrosophic topology --- neutrosophic generalized topology --- neutrosophic generalized pre-closed sets --- neutrosophic generalized pre-open sets --- neutrosophic p T 1 2 space --- neutrosophic g p T 1 2 space --- generalized neutrosophic compact and generalized neutrosophic compact --- fuzzy operating characteristic curve --- fuzzy OC band --- Birnbaum-Sunders distribution --- single acceptance sampling plan --- aggregation operator --- decision making --- neutrosophic soft expert sets --- neutrosophic soft expert multiset --- neutrosophic sets --- neutrosophic multisets --- neutrosophic multigroups --- neutrosophic multisubgroups --- bipolar neutrosophic number (BNN) --- BNN improved generalized weighted HM (BNNIGWHM) operator --- BNN improved generalized weighted geometry HM (BNNIGWGHM) operator --- decision-making --- neutrosophic cubic set --- neutrosophic cubic hybrid geometric operator --- neutrosophic cubic Einstein hybrid geometric operator --- multiattributedecision-making (MADM) --- neutrosophic set --- Zhang-Zhang’s YinYang bipolar fuzzy set --- single-valued bipolar neutrosophic set --- bipolar fuzzy set --- YinYang bipolar fuzzy set --- multiple attribute group decision making (MAGDM) --- Linguistic neutrosophic --- LNN Einstein weighted-average operator --- LNN Einstein weighted-geometry (LNNEWG) operator --- semi-idempotent --- neutrosophic rings --- modulo neutrosophic rings --- neutrosophic semi-idempotent --- neutrosophic ring --- neutrosophic triplets --- idemponents --- special neutrosophic triplets --- acceptance number --- neutrosophic approach --- operating characteristics --- risks --- sample size --- probabilistic neutrosophic hesitant fuzzy set --- distance measure --- similarity measure --- entropy measure --- multi-criteria decision-making (MCDM) --- Neutrosophic Quadruple (NQ) --- Neutrosophic Quadruple set --- NQ vector spaces --- NQ linear algebras --- NQ basis --- orthogonal or dual NQ vector subspaces --- similarity index --- diagnosis --- process --- indeterminacy --- neutrosophic statistics --- time-truncated test --- Weibull distribution --- risk --- uncertainty --- neutrosophic --- neutrosophic logic --- fuzzy logic --- control chart --- neutrosophic numbers --- monitoring --- financial assets --- neutrosophicportfolio --- neutrosophic portfolio return --- neutrosophic portfolio risk --- neutrosophic covariance --- Abel-Grassmann’s neutrosophic extended triplet loop --- generalized Abel-Grassmann’s neutrosophic extended triplet loop --- strong inverse AG-groupoid --- quasi strong inverse AG-groupoid --- quasi Clifford AG-groupoid --- semigroup --- CA-groupoid --- regular CA-groupoid --- neutrosophic extended triplet (NET) --- Green relation --- multi-attribute group decision-making --- granular computing --- interval-valued neutrosophic information --- multigranulation probabilistic models --- merger and acquisition target selections --- dynamic neutrosophic environment --- dynamic interval-valued neutrosophic set --- unknown weight information --- single-valued neutrosophic linguistic set --- combined weighted --- logarithmic distance measure --- supplier selection --- fresh aquatic products --- MAGDM

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Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Ordered algebraic structures --- 512.74 --- Abelian varieties --- Modular functions --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Varieties, Abelian --- Geometry, Algebraic --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Abelian varieties. --- Modular functions. --- Abelian extension. --- Abelian group. --- Abelian variety. --- Absolute value. --- Adele ring. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic equation. --- Algebraic extension. --- Algebraic number field. --- Algebraic structure. --- Algebraic variety. --- Analytic manifold. --- Automorphic function. --- Automorphism. --- Big O notation. --- CM-field. --- Characteristic polynomial. --- Class field theory. --- Coefficient. --- Complete variety. --- Complex conjugate. --- Complex multiplication. --- Complex number. --- Complex torus. --- Corollary. --- Degenerate bilinear form. --- Differential form. --- Direct product. --- Direct proof. --- Discrete valuation ring. --- Divisor. --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Existential quantification. --- Field of fractions. --- Finite field. --- Fractional ideal. --- Function (mathematics). --- Fundamental theorem. --- Galois extension. --- Galois group. --- Galois theory. --- Generic point. --- Ground field. --- Group theory. --- Groupoid. --- Hecke character. --- Homology (mathematics). --- Homomorphism. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Lie group. --- Linear combination. --- Linear subspace. --- Local ring. --- Modular form. --- Natural number. --- Number theory. --- Polynomial. --- Prime factor. --- Prime ideal. --- Projective space. --- Projective variety. --- Rational function. --- Rational mapping. --- Rational number. --- Real number. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scientific notation. --- Semisimple algebra. --- Simple algebra. --- Singular value. --- Special case. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Theorem. --- Vector space. --- Zero element.

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The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology.In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems.Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.

Braid theory --- Braids, Theory of --- Theory of braids --- Braid theory. --- Algebraic topology --- Knot theory --- Representations of groups --- 512.54 --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Knots (Topology) --- Low-dimensional topology --- 512.54 Groups. Group theory --- Groups. Group theory --- Knot theory. --- Representations of groups. --- Addition. --- Alexander polynomial. --- Algebraic structure. --- Automorphism. --- Ball (mathematics). --- Bijection. --- Braid group. --- Branched covering. --- Burau representation. --- Calculation. --- Cartesian coordinate system. --- Characterization (mathematics). --- Coefficient. --- Combinatorial group theory. --- Commutative property. --- Commutator subgroup. --- Configuration space. --- Conjugacy class. --- Corollary. --- Covering space. --- Dehn twist. --- Determinant. --- Diagram (category theory). --- Dimension. --- Disjoint union. --- Double coset. --- Eigenvalues and eigenvectors. --- Enumeration. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Faithful representation. --- Finite set. --- Free abelian group. --- Free group. --- Fundamental group. --- Geometry. --- Group (mathematics). --- Group ring. --- Groupoid. --- Handlebody. --- Heegaard splitting. --- Homeomorphism. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Identity element. --- Identity matrix. --- Inclusion map. --- Initial point. --- Integer matrix. --- Integer. --- Knot polynomial. --- Lens space. --- Line segment. --- Line–line intersection. --- Link group. --- Low-dimensional topology. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix group. --- Matrix representation. --- Monograph. --- Morphism. --- Natural transformation. --- Normal matrix. --- Notation. --- Orientability. --- Parity (mathematics). --- Permutation. --- Piecewise linear. --- Pointwise. --- Polynomial. --- Prime knot. --- Projection (mathematics). --- Proportionality (mathematics). --- Quotient group. --- Requirement. --- Rewriting. --- Riemann surface. --- Semigroup. --- Sequence. --- Special case. --- Subgroup. --- Submanifold. --- Subset. --- Symmetric group. --- Theorem. --- Theory. --- Topology. --- Trefoil knot. --- Two-dimensional space. --- Unimodular matrix. --- Unit vector. --- Variable (mathematics). --- Word problem (mathematics). --- Topologie algébrique

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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.

Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété