Mathematics Commons^™

Soft Neutrosophic Algebraic Structures And Their Generalization - Vol. 1, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and …

Pseudo Lattice Graphs And Their Applications To Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time authors introduce the concept of merged lattice, which gives a lattice or a graph. The resultant lattice or graph is defined as the pseudo lattice graph of type I. Here we also merge a graph with a lattice or two or more graphs which call as the pseudo lattice graph of type II. We merge either edges or vertices or both of a lattice and a graph or a lattice and a lattice or graph with itself. Such study is innovative and these mergings are adopted on all fuzzy and neutrosophic models which …

New Research On Neutrosophic Algebraic Structures, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures in a broader sense. The indeterminacy element “ I “ gives rise to a more bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three categories: such as neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical …

The Efficient Use Of Supplementary Information In Finite Population Sampling, Florentin Smarandache, Rajesh Singh

Branch Mathematics and Statistics Faculty and Staff Publications

The purpose of writing this book is to suggest some improved estimators using auxiliary information in sampling schemes like simple random sampling, systematic sampling and stratified random sampling. This volume is a collection of five papers, written by nine co-authors (listed in the order of the papers): Rajesh Singh, Mukesh Kumar, Manoj Kr. Chaudhary, Cem Kadilar, Prayas Sharma, Florentin Smarandache, Anil Prajapati, Hemant Verma, and Viplav Kr. Singh. In first paper dual to ratio-cum-product estimator is suggested and its properties are studied. In second paper an exponential ratio-product type estimator in stratified random sampling is proposed and its properties are …

Collected Papers, Vol. V, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Distance In Matrices And Their Applications To Fuzzy Models And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of distance between any two m  n matrices. If the distance is 0 or m  n there is nothing interesting. When the distance happens to be a value t; 0 < t < m  n the study is both innovating and interesting. The three cases of study which is carried out in this book are 1. If the difference between two square matrices is large, will it imply the eigen values and eigen vectors of those matrices are distinct? Several open conjectures in this direction are given. 2. The difference between parity check matrix and the generator matrix for the same C(n, k) code is studied. This will help in detecting errors in storage systems as well as in cryptography.

Solving Diophantine Equations, Florentin Smarandache, Octavian Cira

Branch Mathematics and Statistics Faculty and Staff Publications

In recent times, we witnessed an explosion of Number Theory problems that are solved using mathematical software and powerful computers. The observation that the number of transistors packed on integrated circuits doubles every two years made by Gordon E. Moore in 1965 is still accurate to this day. With ever increasing computing power more and more mathematical problems can be tacked using brute force. At the same time the advances in mathematical software made tools like Maple, Mathematica, Matlab or Mathcad widely available and easy to use for the vast majority of the mathematical research community. This tools don’t only …

Principal Angles And Approximation For Quaternionic Projections [Dataset], Terry A. Loring

Math and Statistics Datasets

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in the matrices over the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections.

Estimating Norms Of Commutators [Dataset], Terry A. Loring, Freddy Vides

Math and Statistics Datasets

We find estimates on the norm of a commutator of the form [f(x),y] in terms of the norm of [x,y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular we discuss |[x^2,y]| and |[x^{1/2},y]| for 0leq x leq 1. For larger values of delta = |[x,y]| we can rigorous calculate the best possible upper bound |[f(x),y]| leq eta_f(delta) for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the …

Estimating Norms Of Commutators, Terry A. Loring, Freddy Vides

Branch Mathematics and Statistics Faculty and Staff Publications

We find estimates on the norm of a commutator of the form $[f(x),y]$ in terms of the norm of $[x,y]$, assuming that $x$ and $y$ are bounded linear operators on Hilbert space, with $x$ normal and with spectrum within the domain of $f$. In particular we discuss $\|[x^2,y]\|$ and $\|[x^{1/2},y]\|$ for $0\leq x \leq 1$. For larger values of $\delta = \|[x,y]\|$ we can rigorous calculate the best possible upper bound $\|[f(x),y]\| \leq \eta_f(\delta)$ for many $f$. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the …

Computing A Logarithm Of A Unitary Matrix With General Spectrum [Dataset], Terry A. Loring

Math and Statistics Datasets

We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix, and also skew-Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near -1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force an Hermitian output creates accuracy issues which are avoided by the considered algorithm. A modification is introduced to deal properly with the J-skew symmetric unitary matrices. Applications …

Examples Where The Conjunctive And Dempster’S Rules Are Insensitive, Florentin Smarandache, Jean Dezert, Valeri Kroumov

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we present several counter-examples to the Conjunctive rule and to Dempster rule of combinations in information fusion.

A Journey Into Quantization In Astrophysics, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

The present book consists of 17 select scientific papers from ten years of work around 2003-2013. The topic covered here is quantization in Astrophysics. We also discuss other topics for instance Pioneer spacecraft anomaly. We discuss a number of sub-topics, for instance the use of Schrödinger equation to describe celestial quantization. Our basic proposition here is that the quantization of planetary systems corresponds to quantization of circulation as observed in superfluidity. And then we extend it further to the use of (complex) Ginzburg-Landau equation to describe possible nonlinearity of planetary quantization. Some of these papers have been published in journal …

On Fuzzy Soft Matrix Based On Reference Function, Florentin Smarandache, Said Broumi, Mamoni Dhar

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we study fuzzy soft matrix based on reference function. Firstly, we define some new operations such as fuzzy soft complement matrix and trace of fuzzy soft matrix based on reference function. Then, we introduced some related properties, and some examples are given. Lastly, we define a new fuzzy soft matrix decision method based on reference function.

Relations Between Distorted And Original Angles In Str, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Using the Oblique-Length Contraction Factor, which is a generalization of Lorentz Contraction Factor, one shows several trigonometric relations between distorted and original angles of a moving object lengths in the Special Theory of Relativity.

On Improvement In Estimating Population Parameter(S) Using Auxiliary Information, Florentin Smarandache, Rajesh Singh

Branch Mathematics and Statistics Faculty and Staff Publications

The purpose of writing this book is to suggest some improved estimators using auxiliary information in sampling schemes like simple random sampling and systematic sampling. This volume is a collection of five papers, written by eight coauthors (listed in the order of the papers): Manoj K. Chaudhary, Sachin Malik, Rajesh Singh, Florentin Smarandache, Hemant Verma, Prayas Sharma, Olufadi Yunusa, and Viplav Kumar Singh, from India, Nigeria, and USA. The following problems have been discussed in the book: In chapter one an estimator in systematic sampling using auxiliary information is studied in the presence of non-response. In second chapter some improved …

Proceedings Of The First International Conference On Smarandache Multispace & Multistructures, Florentin Smarandache, Linfan Mao

Branch Mathematics and Statistics Faculty and Staff Publications

In recent decades, Smarandache’s notions of multispace and multistructure were widely spread and have shown much importance in sciences around the world. Organizedby Prof.Linfan Mao, a professional conference on multispaces and multistructures, named the First International Conference on Smarandache Multispace and Multistructure was held in Beijing University of Civil Engineering and Architecture of P. R. China on June 28-30, 2013, which was announced by American Mathematical Society in advance.

Special Type Of Subset Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we construct special subset topological spaces using subsets from semigroups or groups or rings or semirings. Such study is carried out for the first time and it is both interesting and innovative. Suppose P is a semigroup and S is the collection of all subsets of P together with the empty set, then S can be given three types of topologies and all the three related topological spaces are distinct and results in more types of topological spaces. When the semigroup is finite, S gives more types of finite topological spaces. The same is true in case …

Subset Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of constructing non associative algebraic structures using subsets of a groupoid. Thus subset groupoids are constructed using groupoids or loops. Even if we use subsets of loops still the algebraic structure we get with it is only a groupoid. However we can get a proper subset of it to be a subset loop which will be isomorphic with the loop which was used in the construction of the subset groupoid. To the best of the authors’ knowledge this is the first time non associative algebraic structures are constructed using subsets. We get …

Indeterminate Masses, Elements And Models In Information Fusion, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper at the beginning, we make a short history of the logics, from the classical Boolean logic to the most general logic of today neutrosophic logic. We define the general logic space and give the definition of the neutrosophic logic. Then we introduce the indeterminate models in information fusion, which are due either to the existence of some indeterminate elements in the fusion space or to some indeterminate masses.

The best approach for dealing with such models is the neutrosophic logic, which is part of neutrosophy. Neutrosophic logic is connected with neutrosophic set and neutrosophic probability and statistics.

Clan Capitalism, Graph Distance, And Other Issues, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

This book consists of 6 papers focusing on social and economic issues. The topics covered include graph distance and optimal communication, migration in Jaipur, urbanization, clan capitalism, world population growth rate, and scientific inquiry. These papers were written in the period between 20092010. Hopefully the readers will find some new insights in this collection of papers.

Neutrosophic Emergencies And Incidences, Florentin Smarandache, Stefan Vladutescu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Oblique-Length Contraction Factor In The Special Theory Of Relativity, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper one generalizes the Lorentz Contraction Factor for the case when the lengths are moving at an oblique angle with respect to the motion direction. One shows that the angles of the moving relativistic objects are distorted.

The Hyperbolic Menelaus Theorem In The Poincare Disc Model Of Hyperbolic Geometry, Florentin Smarandache, Catalin Barbu

Branch Mathematics and Statistics Faculty and Staff Publications

In this note, we present the hyperbolic Menelaus theorem in the Poincar´e disc of hyperbolic geometry.

Filters Via Neutrosophic Crisp Sets, A. A. Salama, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

Correlation Coefficient Of Interval Neutrosophic Set, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we introduce for the first time the concept of correlation coefficients of interval valued neutrosophic set (INS for short). Respective numerical examples are presented.

Intuitionistic Neutrosophic Soft Set, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we study the concept of intuitionistic neutrosophic set of Bhowmik and Pal. We have introduced this concept in soft sets and defined intuitionistic neutrosophic soft set. Some definitions and operations have been introduced on intuitionistic neutrosophic soft set. Some properties of this concept have been established.

More On Intuitionistic Neutrosophic Soft Sets, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Intuitionistic Neutrosophic Soft Set theory proposed by S. Broumi and F. Samarandache [28], has been regarded as an effective mathematical tool to deal with uncertainties. In this paper new operations on intuitionistic neutrosophic soft sets have been introduced . Some results relating to the properties of these operations have been established. Moreover ,we illustrate their interconnections between each other.

N-Valued Refined Neutrosophic Logic And Its Applications To Physics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical valued logic to the general case of n-symbol or numerical valued logic. We show generalizations of 2-valued Boolean logic to fuzzy logic, also from the Kleene’s and Lukasiewicz’ 3-symbol valued logics or Belnap’s 4-symbol valued logic to the most general n-symbol or numerical valued refined neutrosophic logic. Two classes of neutrosophic norm (n-norm) and neutrosophic conorm (n-conorm) are defined. Examples of applications of neutrosophic logic to physics are listed in the last section. Similar generalizations can be done for n-Valued Refined Neutrosophic Set, …

Introduction To Neutrosophic Measure, Neutrosophic Integral, And Neutrosophic Probability, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, we introduce for the first time the notions of neutrosophic measure and neutrosophic integral, and we develop the 1995 notion of neutrosophic probability. We present many practical examples.

It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Neutrosophics study the indeterminacy. Indeterminacy is different from randomness. It can be caused by physical space materials and type of construction, by items involved in the space, etc.