Algebraic Structures Using Natural Class Of Intervals

In this book the authors introduce and study the properties of natural class of intervals built using (-, ) and (, -). The operations on these matrices with entries from natural class of intervals behave like usual reals. So working with these interval matrices takes the same time as usual matrices. Hence, when applying them to fuzzy finite element methods or finite element methods the determination of solution is simple and time saving.

In this book we explore the possibility of extending the natural operations on reals to intervals and matrices. The extension to intervals makes us define a natural class of intervals in which we accept [a, b], a greater than b. Further, we introduce a complex modulo integer in Z_n (n, a positive integer) and denote it by iF with iF^2 = n-1.

This is the first volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation. The 78 authors are listed alphabetically. The introduction contains a short history of neutrosophics, together with links to the main papers and books. Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

In this fourteenth book of scilogs – one may find topics on examples where neutrosophics works and others don’t, law of included infinitely-many-middles, decision making in games and real life through neutrosophic lens, sociology by neutrosophic methods, Smarandache multispace, algebraic structures using natural class of intervals, continuous linguistic set, cyclic neutrosophic graph, graph of neutrosophic triplet group , how to convert the crisp data to neutrosophic data, n-refined neutrosophic set ranking, adjoint of a square neutrosophic matrix, neutrosophic optimization, de-neutrosophication, the n-ary soft set relationship, hypersoft set, extending the hypergroupoid to the superhypergroupoid, alternative ranking, Dezert-Smarandache Theory (DSmT), reconciliation between theoretical and market prices, extension of the MASS model by the incorporation of neutrosophic statistics and the DSmT combination rule, conditional probability of actually detecting a financial fraud, neutrosophic extension using DSmT combination rule, probabilistic information content, absolute and relative DSm conditioning rules, example of PCR5 with Zhang’s degree, PCR5 with degree of intersection, the most general form of SuperHyperAlgebra, on Crittenden and Vanden Eynden’s conjecture, use of special types of linear algebras and their generalizations, SuperMathematics, 3D-space in physics, neutrosophic physical laws, neutrosophy as a meta-philosophy, principle of interconvertibility matter-energy-information, neutrosophic philosophical interpretation, possible neutrosophic applications to Indian philosophy and religion, philosophical horizons in neutrosophy, clan capitalism, or artificial intelligence – email messages to research colleagues, or replies, notes, comments, remarks about authors, articles, or books, spontaneous ideas, and so on. Exchanging ideas with Mirela Teodorescu, Linfan Mao, Shondiin Silversmith, Mumtaz Ali, Vasantha W.B. Kandasamy, V. Lakshmana Gomathi Nayagam, Bharanidharan R., Michael Voskoglou, Said Broumi, Maissam Jdid, Sagvan Y. Musa, Mohammad Hamidi, Yaser Ahmad Alhasan, Nivetha Martin, Mohammad Khoshnevisan, Deqiang Han, Jean Dezert, Mircea Șelariu, Ștefan Vlăduțescu, Tudor Păroiu (in order of reference in the book).