Made available online by the Smarandache Notion Journal and the University of New Mexico - Gallup.

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. We structured this book using numbered Definitions, Theorems, Conjectures, Notes and Comments, in order to facilitate an easier reading but also to facilitate references to a specific paragraph. We divided the Bibliography in two parts, Writings by Florentin Smarandache (indexed by the name of books and articles) and Writings on Smarandache notions (indexed by the name of authors). We treated, in this book, about 130 Smarandache type sequences, about 50 Smarandache type functions and many solved or open problems of number theory. We also have, at the end of this book, a proposal for a new Smarandache type notion, id est the concept of “a set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors”, notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers. This encyclopedia is both for researchers that will have on hand a tool that will help them “navigate” in the universe of Smarandache type notions and for young math enthusiasts: many of them will be attached by this wonderful branch of mathematics, number theory, reading the works of Florentin Smarandache.

Articles, notes and problems on Smarandache Function, Pseudo-Smarandache function, Smarandache-simple functions, Inferior Smarandache Prime Part, Smarandache double factorial function, Generalized Smarandache Palindrome, Smarandache problems, Smarandache circular sequence etc.

The essential reference book on matrices—now fully updated and expanded, with new material on scalar and vector mathematics Since its initial publication, this book has become the essential reference for users of matrices in all branches of engineering, science, and applied mathematics. In this revised and expanded edition, Dennis Bernstein combines extensive material on scalar and vector mathematics with the latest results in matrix theory to make this the most comprehensive, current, and easy-to-use book on the subject. Each chapter describes relevant theoretical background followed by specialized results. Hundreds of identities, inequalities, and facts are stated clearly and rigorously, with cross-references, citations to the literature, and helpful comments. Beginning with preliminaries on sets, logic, relations, and functions, this unique compendium covers all the major topics in matrix theory, such as transformations and decompositions, polynomial matrices, generalized inverses, and norms. Additional topics include graphs, groups, convex functions, polynomials, and linear systems. The book also features a wealth of new material on scalar inequalities, geometry, combinatorics, series, integrals, and more. Now more comprehensive than ever, Scalar, Vector, and Matrix Mathematics includes a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. Fully updated and expanded with new material on scalar and vector mathematics Covers the latest results in matrix theory Provides a list of symbols and a summary of conventions for easy and precise use Includes an extensive bibliography with back-referencing plus an author index

This book covers only a part of the wide and diverse field of the Smarandache Notions, andcontains some of the materials that I gathered as I wandered in the world of Smarandache. Mostof the materials are already published in different journals, but some materials are new andappear for the first time in this book. All the results are provided with proofs._ Chapter 1 gives eleven recursive type Smarandache sequences, namely, the SmarandacheOdd, Even, Prime Product, Square Product (of two types), Higher Power Product (of twotypes), Permutation, Circular, Reverse, Symmetric and Pierced Chain sequences_ Chapter 2 deals with the Smarandache Cyclic Arithmetic Determinant and BisymmetricArithmetic Determinant sequences, and series involving the terms of the Smarandachebisymmetric determinant natural and bisymmetric arithmetic determinant sequences_ Chapter 3 treats the Smarandache function S(n)_ Chapter 4 considers, in rather more detail, the pseudo Smarandache function Z(n)_ And the Smarandache S-related and Z-related triangles are the subject matter of Chapter 5.To make the book self-contained, some well-known results of the classical Number Theory aregiven in Chapter 0. In order to make the book up-to-date, the major results of other researchersare also included in the book.At the end of each chapter, several open problems are given.

Papers on Smarandache function S(n), Erdos-Smarandache numbers, generalized intuitionistic fuzzy contra continuous functions and its applications, the asymptotic properties of triangular base sequence, linear operators preserving commuting pairs of matrices over semirings, two inequalities for the composition of arithmetic functions, compactness and proper maps in the category of generated spaces, and similar topics. Contributors: R. Ma, Y. Zhang, R. Dhavaseelan, M. Dragan, M. Bencze, Q. Yang, G. Mirhosseinkhani, C. Fu, S. S. Billing, B. Hazarika, and others.