More than seven years ago, my first book on some of the Smarandache notions was published. The book consisted of five chapters, and the topics covered were as follows : (1) some recursive type Smarandache sequences, (2) Smarandache determinant sequences, (3) the Smarandache function, (4) the pseudo Smarandache function, and (5) the Smarandache function related and the pseudo Smarandache function related triangles. Since then, new and diversified results have been published by different researchers. The aim of this book to update some of the contents of my previous book, and add some new results.

This paper finds SS(n) for some particular cases of n.

Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities as well as their interactions with different ideational spectra. In all classical algebraic structures, the law of compositions on a given set are well-defined, but this is a restrictive case because there are situations in science where a law of composition defined on a set may be only partially defined and partially undefined, which we call NeutroDefined, or totally undefined, which we call AntiDefined. Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebra introduces NeutroAlgebra, an emerging field of research. This book provides a comprehensive collection of original work related to NeutroAlgebra and covers topics such as image retrieval, mathematical morphology, and NeutroAlgebraic structure. It is an essential resource for philosophers, mathematicians, researchers, educators and students of higher education, and academicians.

Multi attribute group decision making with VIKOR (VlseKriterijuska Optimizacija I Komoromisno Resenje) strategy has been widely applied to solving real-world problems. Recently, Pramianik et al. [S. Pramanik, S. Dalapati, S. Alam, and T. K. Roy. NCVIKOR based MAGDM strategy under neutrosophic cubic set environment, Neutrosophic Sets and Systems, 20 (2018), 95-108] proposed VIKOR strategy for solving MAGDM, where compromise solutions are not identified in neutrosophic cubic environment. To overcome the shortcomings of the paper, we further modify the VIKOR strategy by incorporating compromise solution in neutrosophic cubic set environment. Finally, we solve an MAGDM problem using the modified NC-VIKOR strategy to show the feasibility, applicability and effectiveness of the proposed strategy. Further, we present sensitivity analysis to show the impact of different values of the decision making mechanism coefficient on ranking order of the alternatives.

For software development companies, deciding which development methodology to use when starting a project is in many cases an important task to perform. The selection of development methodology can be modeled as a decision-making problem. Neutrosophy is a philosophical current that starts from Paradoxism. In this article, we describe a methodology selection proposal using neutrosophic numbers, from which the development methodology proposal is decided. The method allows taking into account the indeterminacy in decision-making, in addition to the use of linguistic terms that are more appropriate than numerical ones. The applicability of the proposal is confirmed through a demonstrative example. The paper ends with the conclusions and recommendations for future work.

Collection of papers from various scientists dealing with smarandache notions in science.

A collection of papers concerning Smarandache type functions, numbers, sequences, inteqer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc.

This volume is a collection of ten papers by contributors F. Smarandache, F. Yuhua, K. Mondal, S. Pramanik, S. Broumi, J. Ye, A. A. Salama,, N. Easa, S. A. Elhafez, M. M. Lotfy, L. Kong, Y. Wu, P. Biswas, B. C. Giri, A. Mukkerjee, and S. Sarkar, focusing on a new kind of algebraic structures called (T, I, F)- Neutrosophic Structures; Expanding Uncertainty Principle to Certainty-Uncertainty Principles with Neutrosophy and Quad-stage Methods; Rough Neutrosophic Multi-Attribute Decision-Making Based on Rough Accuracy Score Function; an Extended TOPSIS Method for Multiple Attribute Decision Making based on Interval Neutrosophic Uncertain Linguistic Variable; Review of Recommender Systems Algorithms Utilized in Social Networks based e-Learning Systems & Neutrosophic System; Fault Diagnosis Method of Gasoline Engines Using the Cosine Similarity Measure of Neutrosophic Numbers; Cosine Similarity Measure Based Multi-attribute Decision-making with Trapezoidal Fuzzy Neutrosophic Numbers; Thesis-Antithesis-Neutrothesis, and Neutrosynthesis; Negating Four Color Theorem with Neutrosophy and Quadstage Method; and A new method of measuring similarity between two neutrosophic soft sets and its application in pattern recognition problems.