In this book for the first time authors study MOD planes using modulo intervals. These planes unlike the real plane have only one quadrant so the study is carried aut in a compact space but infinite in dimension. The authors have given seven MOD planes viz real MOD planes (MOD real plane) finite complex MOD plane, neutrosophic MOD plane, fuzzy MOD plane, (or MOD fuzzy plane), MOD dual number plane, MOD special dual like number plane and MOD special quasi dual number plan...

In this book authors answer the question proposed by Florentin Smarandache “Does there exist neutrosophic numbers which are such that they take values differently and behave differently from I; the indeterminate?”. The authors have constructed a class of natural neutrosophic numbers. Further there are more natural neutrosophic numbers in the MOD interval [0, m). This concept is thoroughly analysed in the second chapter. Using all types of MOD planes and MOD intervals, th...

In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration. These drawbacks becomes a challenging issue. So study in this directi...

Understanding Federal Tyranny begins with a two-chapter overview, to provide a general framework to explain how government servants effectively became our political masters. Then, the last three chapters “follow the money” to prove true the outline in a specific case (the case of how our lawful money of gold and silver coin was effectively replaced with irredeemable paper currency). Understanding Federal Tyranny answers The Peculiar Conundrum— the odd phenomenon of how m...

In this book authors study the properties of finite real quaternion ring which was introduced in [2000]. Here a complete study of these finite quaternion rings are made. Also polynomial quaternion rings are defined, they happen to behave in a very different way. In the first place the fundamental theorem of algebra, “a nth degree polynomial has n and only n roots”, n is untrue in case of polynomial in polynomial quaternion rings in general. Further the very concept of de...

This research work will give a new direction for applications of fuzzy set theory particularly in algebraic logics, non-classical logics, fuzzy finite state machines, fuzzy automata, fuzzy languages, cognitive modeling, multiagent decision analysis and mathematical morphology. Introducing (∈,∈ ∨q_k)-fuzzy ideals, (∈_γ,∈_γ ∨q_δ)-fuzzy ideals and (∈_γ,∈_γ ∨q_δ)-fuzzy soft ideals in a new non-associative algebraic structure called Abel-Grassmann’s groupoid, discuss severa...

In this paper we examine the underlying concept of true World Missions and the conflict Media possess on it. Today we review the ‘process vs the Promises’ of God in his Word. Further was the camera’ cord-snake’ being the serpent in Billy’s Garden (Madison Square 1949) that would tempt him to succumb to its rather than to God’s injunction ‘have no gods before me?’ 1. Why did God admonished for assembling all of Israel (King David) 2. Israel in 29 AD had no mass communic...

King David and World Evangelism Southern Style King David was provoked by Satan to number Israel, “And again the anger of the Lord was kindled against Israel, and he moved David against them to say, Go, number Israel and Judah.” (2 Sam 24) Previously, “Samuel delivered an oration to the people of Israel that fairly sizzled with contempt for the idea of monarchy.” God needed to move swiftly and with reprehensibility. “The rule of kings…was strictly for the goyim.” Thi...

In this book the authors for the first time study special type of Euclid squares in the real plane, complex plane, neutrosophic plane, dual number plane and their specializations. This study can be visualized as a blend of algebra, geometry and analysis. 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. The authors define several types of algebraic structures on ...

In this book we introduce a new procedure called α-Discounting Method for Multi-Criteria Decision Making (α-D MCDM), which is as an alternative and extension of Saaty’s Analytical Hierarchy Process (AHP). It works for any number of preferences that can be transformed into a system of homogeneous linear equations. A degree of consistency (and implicitly a degree of inconsistency) of a decision-making problem are defined. α-D MCDM is afterwards generalized to a set of pref...

Study of MOD planes happens to a very recent one. Authors have studied several properties of MOD real planes. In fact unlike the real plane R × R which is unique MOD real planes are infinite in number. Further three other new types of MOD planes constructed using dual numbers, special dual number like numbers and special quasi dual numbers are introduced. Systematically algebraic structures on MOD planes like, MOD semigroups, MOD groups and MOD rings of different types a...

An innovative book from a different perspective to the run of the mill self-help books that basically goes intot the deep waters of the struggles and challenges that stands in the path of the young child aspiring to a dream and it cuts across the role of the pas expected by both the parents and their wards and provides a shining light for everyone frustrated with their inability to live their dreams

“You can do anything as long as you have the passion, the drive, and the support.” ~Sabrina Bryan Set in the plains of Nigeria in Sub-Saharan Africa, fiddling with some pieces of clothes, sat a little girl day dreaming about becoming the world`s foremost African fashion designer. She has pictures of clothes with amazing designs in her mind. But suddenly, a shadow bears upon her, she heard a voice saying to her, “Get up! Go and pick up your books, I have told you to sto...

In this book, you will find algebraic exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone work. The book can also be used as an extracurricular source, as the reader shall find enclosed important theorems and formulas, standard definitions and notions that ar...

In this book authors for the first time introduce the notion 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 conjectures are proposed. These new class of ...

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 pal...

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 spe...

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...

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. Likewi...

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 secon...

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 ...

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 a...