Algebra Commons^™

Variance On Topics Of Plane Geometry, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

This book contains 21 papers of plane geometry. It deals with various topics, such as: quasi-isogonal cevians, nedians, polar of a point with respect to a circle, anti-bisector, aalsonti-symmedian, anti-height and their isogonal. A nedian is a line segment that has its origin in a triangle’s vertex and divides the opposite side in Q equal segments. The papers also study distances between remarkable points in the 2D-geometry, the circumscribed octagon and the inscribable octagon, the circles adjointly ex-inscribed associated to a triangle, and several classical results such as: Carnot circles, Euler’s line, Desargues theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s theorem, …

Algebraic Structures Using Subsets, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of subsets and giving algebraic structure to these subsets of a set started in the mid 18th century by George Boole. The first systematic presentation of Boolean algebra emerged in 1860s in papers written by William Jevons and Charles Sanders Peirce. Thus we see if P(X) denotes the collection of all subsets of the set X, then P(X) under the op erations of union and intersection is a Boolean algebra. Next the subsets of a set was used in the construction of topological spaces. We in this book consider subsets of a semigroup or a group or a …

Natural Product Xn On Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new product on matrices called the natural product. ...

Thus by introducing natural product we can find the product of column matrices and product of two rectangular matrices of same order. Further this product is more natural which is just identical with addition replaced by multiplication on these matrices. Another fact about natural product is this enables the product of any two super matrices of same order and with same type of partition. We see on supermatrices products cannot be defined easily which prevents from having any nice algebraic structure on the collection …

Neutrosophic Super Matrices And Quasi Super Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution. Here we introduce neutrosophic super matrices and show that the sum or product of two neutrosophic matrices is not in general a neutrosophic super matrix. Another interesting feature of this book is …

Innovative Uses Of Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, Indra Venkatbabu

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out the innovative applications of matrices defined, described and developed by them. Here they do not include the natural product on matrices newly described and defined by them in the book on ‘natural product ×n on matrices’.

This book is organized into seven chapters. The first one is introductory in nature. In the second chapter authors give the unique and new way of analyzing the data which is time dependent. We construct three types of matrices called Average Time Dependent data matrix (ATD matrix), Refined Time Dependent Data matrix (RTD matrix) and Combined Effective Time …

Special Dual Like Numbers And Lattices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b …

Dual Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Dual numbers was first introduced by W.K. Clifford in 1873. This nice concept has lots of applications; to screw systems, modeling plane joint, iterative methods for displacement analysis of spatial mechanisms, inertial force analysis of spatial mechanisms etc. In this book the authors study dual numbers in a special way. The main aim of this book is to find rich sources of new elements g such that g2 = 0. The main sources of such new elements are from Zn, n a composite number. We give algebraic structures on them. This book is organized into six chapters. The final chapter …

Algebraic Structures Using Super Interval Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The advantage of using super interval matrices is that one can build only one vector space using m × n interval matrices, but in case of super interval matrices we can have several such spaces depending on the partition on the interval matrix.

This book has seven chapters. Chapter one is introductory in nature, just introducing the super interval matrices or interval super matrices. In chapter two essential operations on super interval matrices are defined. Further in this chapter algebraic structures are defined on these super interval matrices using these operation. Using these super interval matrices semirings and semivector spaces …

Studies In Sampling Techniques And Time Series Analysis, Florentin Smarandache, Rajesh Singh

Branch Mathematics and Statistics Faculty and Staff Publications

This book has been designed for students and researchers who are working in the field of time series analysis and estimation in finite population. There are papers by Rajesh Singh, Florentin Smarandache, Shweta Maurya, Ashish K. Singh, Manoj Kr. Chaudhary, V. K. Singh, Mukesh Kumar and Sachin Malik. First chapter deals with the problem of time series analysis and the rest of four chapters deal with the problems of estimation in finite population. The book is divided in five chapters as follows: Chapter 1. Water pollution is a major global problem. In this chapter, time series analysis is carried out …

Algebraic Structures Using Natural Class Of Intervals, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Authors in this book introduce a new class of intervals called the natural class of intervals, also known as the special class of intervals or as natural intervals. These intervals are built using increasing intervals, decreasing intervals and degenerate intervals. We say an interval [a, b] is an increasing interval if a < b for any a, b in the field of reals R. An interval [a, b] is a decreasing interval if a > b and the interval [a, b] is a degenerate interval if a = b for a, b in the field of reals R. The natural class of intervals consists of the collection of increasing intervals, decreasing intervals and the degenerate intervals. Clearly R is contained in the natural …

Dsm Super Vector Space Of Refined Labels, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of supermatrices of refined labels. Authors prove super row matrix of refined labels form a group under addition. However super row matrix of refined labels do not form a group under product; it only forms a semigroup under multiplication. In this book super column matrix of refined labels and m × n matrix of refined labels are introduced and studied. We mainly study this to introduce to super vector space of refined labels using matrices. We in this book introduce the notion of semifield of refined labels using which …

Study Of Natural Class Of Intervals Using (–∞,∞) And (∞, –∞), Florentin Smarandache, W.B. Vasantha Kandasamy, D. Datta, H.S. Kushwaha, P.A. Jadhav

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors study the properties of natural class of intervals built using (–∞, ∞) and (∞, –∞). These natural class of intervals behave like the reals R, as far as the operations of addition, multiplication, subtraction and division are concerned. Using these natural class of intervals we build interval row matrices, interval column matrices and m × n interval matrices. Several properties about them are defined and studied. Also all arithmetic operations are performed on them in the usual way. The authors by defining so have made it easier for operations like multiplication, addition, finding determinant and …

Dsm Vector Spaces Of Refined Labels, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of DSm linear algebra of refined labels have been done by Florentin Smarandache, Jean Dezert, and Xinde Li.

In this book the authors introduce the notion of DSm vector spaces of refined labels. The reader is requested to refer the paper as the basic concepts are taken from that paper [35]. This book has six chapters. The first one is introductory in nature just giving only the needed concepts to make this book a self contained one. Chapter two introduces the notion of refined plane of labels, the three dimensional space of refined labels DSm vector spaces. Clearly …

Proposed Problems Of Mathematics (Vol. Ii), Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The first book of “Problèmes avec et sans … problèmes!” was published in Morocco in 1983. I collected these problems that I published in various Romanian or foreign magazines (amongst which: “Gazeta Matematică”, magazine which formed me as problem solver, “American Mathematical Monthly”, “Crux Mathematicorum” (Canada), “Elemente der Mathematik” (Switzerland), “Gaceta Matematica” (Spain), “Nieuw voor Archief” (Holland), etc. while others are new proposed problems in this second volume.

These have been created in various periods: when I was working as mathematics professor in Romania (1984-1988), or co-operant professor in Morocco (1982-1984), or emigrant in the USA (1990-1997). I thank to …

Neutrosophic Bilinear Algebras And Their Generalizations, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book introduces the concept of neutrosophic bilinear algebras and their generalizations to n-linear algebras, n>2. This book has five chapters. The reader should be well-versed with the notions of linear algebras as well as the concepts of bilinear algebras and n- linear algebras. Further the reader is expected to know about neutrosophic algebraic structures as we have not given any detailed literature about it. The first chapter is introductory in nature and gives a few essential definitions and references for the reader to make use of the literature in case the reader is not thorough with the basics. …

Groups As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Through this book, for the first time we represent every finite group in the form of a graph. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties like the subgroups of a group, normal subgroups of a group, p-sylow subgroups of a group and conjugate elements of a group …

Super Fuzzy Matrices And Super Fuzzy Models For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of supermatrix for social scientists was first introduced by Paul Horst. The main purpose of his book was to introduce this concept to social scientists, students, teachers and research workers who lacked mathematical training. He wanted them to be equipped in a branch of mathematics that was increasingly valuable for the analysis of scientific data. This book introduces the concept of fuzzy super matrices and operations on them. The author has provided only those operations on fuzzy supermatrices that are essential for developing super fuzzy multi expert models. We do not indulge in labourious use of suffixes or …

Special Fuzzy Matrices For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

This book is a continuation of the book, "Elementary fuzzy matrix and fuzzy models for socio-scientists" by the same authors. This book is a little advanced because we introduce a multi-expert fuzzy and neutrosophic models. It mainly tries to help social scientists to analyze any problem in which they need multi-expert systems with multi-models. To cater to this need, we have introduced new classes of fuzzy and neutrosophic special matrices. The first chapter is essentially spent on introducing the new notion of different types of special fuzzy and neutrosophic matrices, and the simple operations on them which are needed in …

A Unifying Field In Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability And Statistics - 6th Ed., Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

It was a surprise for me when in 1995 I received a manuscript from the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philosophy - revealing paradoxes - and logics. He had generalized the fuzzy logic, and introduced two new concepts: a) “neutrosophy” – study of neutralities as an extension of dialectics; b) and its derivative “neutrosophic”, such as “neutrosophic logic”, “neutrosophic set”, “neutrosophic probability”, and “neutrosophic statistics” and thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics. It was known to me his setting up in …

Auxiliary Information And A Priori Values In Construction Of Improved Estimators, Florentin Smarandache, Rajesh Singh, Pankaj Chauhan, Nirmala Sawan

Branch Mathematics and Statistics Faculty and Staff Publications

This volume is a collection of six papers on the use of auxiliary information and a priori values in construction of improved estimators. The work included here will be of immense application for researchers and students who employ auxiliary information in any form. Below we discuss each paper: 1. Ratio estimators in simple random sampling using information on auxiliary attribute. Prior knowledge about population mean along with coefficient of variation of the population of an auxiliary variable is known to be very useful particularly when the ratio, product and regression estimators are used for estimation of population mean of a …

Some Neutrosophic Algebraic Structures And Neutrosophic N-Algebraic Structures, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, for the first time we introduce the notion of neutrosophic algebraic structures for groups, loops, semigroups and groupoids and also their neutrosophic N-algebraic structures. One is fully aware of the fact that many classical theorems like Lagrange, Sylow and Cauchy have been studied only in the context of finite groups. Here we try to shift the paradigm by studying and introducing these theorems to neutrosophic semigroups, neutrosophic groupoids, and neutrosophic loops. We have intentionally not given several theorems for semigroups and groupoid but have given several results with proof mainly in the case of neutrosophic loops, biloops …

Sequences Of Numbers Involved In Unsolved Problems, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Here it is a long list of sequences, functions, unsolved problems, conjectures, theorems, relationships, operations, etc. Some of them are inter-connected. 1) Consecutive Sequence: 1,12,123,1234,12345,123456,1234567,12345678,123456789,12345678910, 1234567891011,123456789101112,12345678910111213,... How many primes are there among these numbers? In a general form, the Consecutive Sequence is considered in an arbitrary numeration base B.

References:

Student Conference, University of Craiova, Department of Mathematics, April 1979, "Some problems in number theory" by Florentin Smarandache.

Arizona State University, Hayden Library, "The Florentin Smarandache papers" special collection, Tempe, AZ 85287-1006, USA.

The Encyclopedia of Integer Sequences", by N. J. A. Sloane and S. Plouffe, Academic Press, San Diego, …

Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …

Metode De Calcul În Analiza Matematică, Florentin Smarandache, C. Dumitrescu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.