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...The theory of loops (groups without associativity), though researched by several mathematicians has not found a sound expression, for books, be it research level or otherwise, solely dealing with the properties of loops are absent. This...

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... B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, I... ... A 3 A 4 A 5 2 The picture on the cover represents the lattice of subgroups of the Smarandache loop L 15 (8). The lattice of subgroups of... ... But for the Smarandache concepts one wouldn't have studied the collection of subgroups of a loop. This book can be ordered in a paper bound rep... ... the loop into equivalence classes not using all elements but at the same time we cannot say the two sets partition in the same way. This is poss... ...a RIF loop? 7. Find a loop of even order which is Moufang and at the same time S-RIF loop. 93 Chapter four PROPERTIES ABOUT S-LOOPS In ... ....N, Topics in Algebra, Wiley Eastern Limited, (1993). 28. Higman.G, The units of group rings, Proc. London Math. Soc., Vol.46, No.2, 231-248 (1940...

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...Paradoxism is an avant-garde movement in literature, art, philosophy, science, based on excessive use of antitheses, antinomies, contradictions, parables, odds, paradoxes in creations. It was set up by the editor since 1980’s and promulgates a counter-time/ counter-sense creation. Paradoxism started as an anti-totalitarian pr...

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...rde movement in literature, art, philosophy, science, based on excessive use of antitheses, antinomies, contradictions, parables, odds, paradoxes in... ...eations. It was set up by the editor since 1980’s and promulgates a counter-time/ counter-sense creation. Paradoxism started as an anti-totalitaria... ...sm started as an anti-totalitarian protest against a closed society, Romania of 1980’s. Why was the movement based on contradictions? Because we li... ...losophical, or scientific terms, new procedures, methods, or even algorithms of creation. Paradoxism naturally arises and manifests. It is in folkl... ...allay the fears of those who might think this could be dangerous, after that time the kidneys shift up into overdrive and manufacture all that would... ...ion of the crisis of the subject (de-constructed, even philosophically in a time near to us, Deridda), a fact that brings a crisis of the communica... ...can also be considered as a future even unexpected of the traditional formal units, these ones appear presently, in a vehement opposition, as texts-...

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...introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is ...

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...to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036,... ...c.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: ... ... Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann A... ..., neutrosophic vector spaces and neutrosophic matrices. So we for the first time introduce and study these concepts. As our analysis in this book is... ... P I Z ) ( . If a,b ∈ Z(I ) \ P then two possibilities occur. a + b is odd times I or a + b is odd or a + b is even times I or even if a + b is eve... ...I or a + b is odd or a + b is even times I or even if a + b is even or even times I then a + b ∈ P, if a + b is odd or odd times I, a + b ∈ ) ( 2 )... ...e are analyzing some nodes / concepts which are divided into three disjoint units. Suppose we have 3 spaces say P, Q and R. We say some m nodes in ...

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...In this book for the first time we have ventured into the total analysis of migrant labourers in rural Tamil Nadu who are victims of HIV/AIDS using FCM, BAM and Neutrosophic Cognitive Maps. As in our study and analysis we felt several of the factors re...

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...Migration acquires great significance in the study of peoples and populations, for it not only involves the merely mathematical spatial redistribution of people, but also because it has enormous impact on livelihood, life-styles, employment, socioeconomic and political stabil...

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...W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE ANALYSIS OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING WITH HIV/AIDS USING FUZZY ... ...ce To Rural Tamilnadu In India XIQUAN Phoenix 2004 ANALYSIS OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING WITH HIV/AIDS USING FUZZY T... ...rence To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, I... ...y one of the authors Florentin Smarandache. In this book for the first time we have ventured into the total analysis of migrant labourers in rura... ...of migration and their vulnerability in catching HIV/AIDS. For the first time Neutrosophic Cognitive Maps are used in the study of migrant labourer... ... give us great insight into the minds of people with AIDS and at the same time some of the patients exhibit a reluctance to talk about the disease, ... ...e represent an object in a network as a pattern of activation of several units. If a unit or two responds incorrectly the overall pattern stays pre... ... time. Artificial neural networks consists of numerous simple processing units or neurons which can be trained to estimate sampled functions when w... ...te the correlations between causal associations among concurrently active units. But in FRMs we divide the very causal associations into two disjo...

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...In a lecture that Einstein gave to the Prussian Academy of Sciences in 1921, he said the following: "Geometry predicates nothing about relations of real things, but only geometry together with the purport of physical laws can do so... The idea of the measuring rod and the idea of ...

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... TODAY'S TAKE ON EINSTEIN’S RELATIVITY PROCEEDINGS OF THE CONFERENCE OF 18 FEB 2005 Cover Montage by Bob Wise Models of X-... ...: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann A... ...://wwwlib.umi.com/bod/ Peer Reviewers: Victor Pantcheliouga, Institute of Theoretical and Experimental Physics, Pushchino (Moscow region), Russ... ... foregoing comment, we should temporarily support the use of the length and time transformations as though they were physically real. Sachs then... ... A third non sequitur appears in connection with the twin paradox. Nahin, Time Machines, p.460:[2] Einstein himself, in his 1905 paper, specifi... ...instein himself, in his 1905 paper, specifically took the rate of a clock's timekeeping to be velocity-dependent only. However [there are] those who... ...ket and the parameter τ is the time measured by a clock in the rocket. The units used in these equations are “natural coordinates” in which the vel... ...to be one. In investigating the problem it is convenient to convert to cgs units and then to units in which time is expressed in years and distance... ...light years. This eliminates many powers of ten in the discussion. In cgs units the τ g terms must be divided by c to preserve the ratio to c natu...

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...In a world of chaotic alignments, traditional logic with its strict boundaries of truth and falsity has not imbued itself with the capability of reflecting the reality. Despite various attempts to reorient logic, there has remained an e...

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...ROSOPHIC COGNITIVE MAPS XIQUAN Phoenix 2003 CARE OF THE AIDS INFECTED PERSONS CREATION OF AWARENESS ABOUT AIDS PREVENT... ...E AIDS INFECTED PERSONS CREATION OF AWARENESS ABOUT AIDS PREVENTION OF SPREAD OF AIDS EPIDEMIC MEDICAL TREATMENT OF AIDS PATIENTS SOCIAL ... ...and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, I... ... book possibly bridges that gap. Second, we introduce here (for the first time) the concept of Neutrosophic Cognitive Maps (NCMs), which are a gener... ...t we are only able to give weightages for known concepts; and most of the time we exhibit an unconcern for indeterminate relationships between conce... ...and they are more beneficial owing to their capability of being economic, time-saving and sensitive. In the concluding sections of the first chapter... ...)]. c i = td(v i ) = od(v i ) + id(v i ). Transmitter variables are units whose od(v i ) is positive and their id(v i ) is 0. Receiver variabl... ...hose od(v i ) is positive and their id(v i ) is 0. Receiver variables are units whose od(v i ) is 0 and their id(v i ) is positive. Other variables w... ... with top down influences, a formal hierarchical system. Many transmitter units shows the flatness of a cognitive map where causal arguments are not...

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...Smarandache notions, which can be undoubtedly characterized as interesting mathematics, has the capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or identification as a substructure. Several researchers around the world working on Smarandache notions h...

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...emifields, and Semivector spaces W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036,... ...e power set P(X) where X = {a, b, c} which is a finite Smarandache semiring of order 16. This book can be ordered in a paper bound reprint from:... ...: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann A... ...characteristic of R is the smallest natural number k with kr = r + … + r (k times) equal to zero for all r ∈ R. We then write k = characteristic R. ... ... dimension three. Example 1.4.6: Let F be a field and F n = F × … × F (n times), F n is a vector space over F. A set of basis for F n over F is... ...a semiring. We say the semiring is of characteristic m if ms = s + … + s (m times) equal to zero for all s ∈ S. If no such m exists we say the chara... ...ties like subsemirings, ideals in semirings, zero divisors, idempotents and units in semirings. DEFINITION 2.2.1: Let S be a semiring. P a subset... ... a is b = (1/3, 4, 3/5, 2/7) ∈ S is such that ab = (1, 1, 1, 1). Thus S has units. DEFINITION 2.2.7: Let S and S' be two semirings. A mapping φ :... ...cation. M 2 ×2 is a non-commutative semiring with unit and has non-trivial units in it. PROBLEMS: 1. Find 3 subsemirings of the semiring S w...

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...Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analy...

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...SOME FUZZY AND NEUTROSOPHIC MODELS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 60003... ....ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail... ... ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road ... ...y at each stage but also the final result can easily be compared. 3. Saves time and economy. 6 This book is organized into four chapters. The ... ...aps (2003). Those who wish to simultaneously study several models that are at times time dependent or when a comparative analysis is needed can use ... ...of the bimatrix A B . 1.2. Neutrosophic Bimatrices Here for the first time we define the notion of neutrosophic bimatrix and illustrate them... ...romote the correlations between causal associations among concurrently active units. But in FRMs we divide the very causal associations into two dis... ...e units. But in FRMs we divide the very causal associations into two disjoint units, for example, the relation between a teacher and a student or re... ...tive Maps (NCMs) promote the causal relationships between concurrently active units or decides the absence of any relation between two units or the ...

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...ndache algebraists to develop and study several important and innovative properties about S-rings. Writing this book essentially involved a good deal of reference work. As a researcher, I felt that it will be a great deal better if we thrust importance on results given in research papers on ring theory rather than detail the basic properties or classical results that the s...

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... a 0 1 Smarandache Rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, I... ..., NM 2002 2 The picture on the cover is the lattice representation of the S-ideals of the Smarandache mixed direct product ring R = Z 3 × Z ... ...n a ring and a Smarandache ring. For, in a ring the lattice representation of ideals is always a modular lattice but we see in case of S-rings the la... ...s 3.1 Definition of Smarandache ring with examples 38 3.2 Smarandache units in rings 41 3.3 Smarandache zero divisors in rings 46 3.4 S... ...ldren Meena and Kama for single- handedly helping me by spending all their time in formatting and correcting this book. I dedicate this book to b... ... 3×3 . Does S 3×3 have idempotents? Does S 3×3 have zero divisors? Find units in S 3×3 . 11. For the semigroup Z 12 = {0, 1, 2, 3, … , 11} under... ...re not ideals. ii. Ideals. iii. Zero divisors. iv. Idempotents. v. Units. 12. Find in the semigroup S(21) right and left ideals. Does S(21) ... ...n infinite ring. DEFINITION 2.1.4: Let R be a ring if mx = x + …+ x (m-times) is zero for every x ∈ R, m a positive integer then we say character...

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...s are mainly built using a vector space over a field satisfying special identities called the Jacobi identity and Jordan identity respectively. Study of these algebras started as early as 1940s. Hence the study of non-associative algebras or even non-associative rings boils down to the study of properties of vector spaces or linear algebras over fields....

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...randache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, I... ... Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arb... ...lui Street, 2212 Sacele 3, Jud Brasov, Romania. Dr. Andrei Kelarev, Dept. of Mathematics, Univ. of Tasmania, Hobart, Tasmania 7001, Australia. Dr. A... ... school for the girls of untouchable and oppressed castes, at a period of time in history where women and the oppressed castes were denied all educa... ...ro divisors, Smarandache nilpotents, Smarandache idempotents, Smarandache units, Smarandache normal 31 elements, Smarandache semi-idempotents and... ...a unit which is not a SNA-unit. THEOREM 2.3.2: Let R be a NA-ring. All units in R in general need not be SNA-units of R. Proof: To prove this... ... ∈ R / aR is a S-quasi regular ideal of R}. PROBLEMS: 1. Find the units in Z 6 L 5 (2) which are SNA-units. 2. Find all quasi regular eleme... ... U = {0, 1 + a 2 , …, a 2n –1 + a 2n , …, sum of 2n elements taken at a time}. Now L ∗ = 1 + U, so |L ∗ | = 2 2n – 1 . Clearly L ∗ is a commutat... ... In our opinion such study for loop rings has not been carried out. Some times the collection of the substructures may be a distributive or a modul...

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...K.R.Narayanan was a lauded hero and a distinguished victim of his Dalit background. Even in an international platform when he was on an official visit to Paris, the media headlines blazed, ‘An Untouchable at Elysee’. He was visibly upset and it proved that a Dalit who rose up to such...

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... FUZZY AND NEUTROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e... ... K. Kandasamy e-mail: dr.k.kandasamy@gamil.com Translation of the speeches and writings of Periyar from Tamil by Meena Kand... ... ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road ... ...ence. Max Weber, perhaps the most outstanding comparative sociologist of all time, clearly defined the social identity of the Hindu in terms of cas... ...g is also provided. A study of this kind is being carried out for the first time. We have used mathematical tools to study this sociological probl... ...ons who take up to drugs, drinks etc. to forget their worries and unoccupied time. Under-employment forces them to do criminal acts E 1 E 3 E 2... ...romote the correlations between causal associations among concurrently active units. But in FRMs we divide the very causal associations into two dis... ...e units. But in FRMs we divide the very causal associations into two disjoint units, for example, the 19 relation between a teacher and a student ... ...tive Maps (NCMs) promote the causal relationships between concurrently active units or decides the absence of any relation between two units or the ...

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...This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smara...

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...nd Smarandache Sequences Editor’s Note This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with... ...ath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of... ... were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in transforming the papers into a coherent book was ... ... s 2 * s 3 * . . .* s t. . Multiplying the right side by one as many times as is required to make the number of terms in the product equal to L... ...bout F 1 and then the left terms omitted. If we depascalize one more time to get the sequence S d(-2) 1, -1, 2, -5, 13, -34, 89, -... ... . . . P r = sequence of primeth , primeth , . . . r times ,primes Open problem: If T n is the n th term of P r , then what... ...utes R = 12. We see that for the shortest routes, one has to travel 2m units of track length. There are routes of 2m+2 units in length up to the ... ... There are routes of 2m+2 units in length up to the longest that are 4m+4 units long. Definition: The Smarandache Route Sequence (SRS) is defined ...

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...The purpose of this book is to postulate some theories and test them numerically. Estimation is often a difficult task and it has wide application in social sciences and financial market. In order to obtain the optimum efficiency for som...

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...cond edition) Dr. Mohammad Khoshnevisan, Griffith University, School of Accounting and Finance, Qld., Australia. Dr. Housila P. Singh an... ...ing and Finance, Qld., Australia. Dr. Housila P. Singh and S. Saxena, School of Statistics, Vikram University, UJJAIN, 456010, India. Dr. Sarjind... ...ics, Vikram University, UJJAIN, 456010, India. Dr. Sarjinder Singh Department of Mathematics and statistics.University of Saskatchewan, Canada. Dr... ...to identical environmental conditions will fail at different and unpredictable times. The ‘time of failure’ or ‘life length’ of a component, measured... ...time of failure’ or ‘life length’ of a component, measured from some specified time until it fails, is represented by the continuous random variable ... ...(ii) the upper end point of the guessed interval, i.e., 2 β departs exactly two times from the true value β , i.e., 2 ∆ = 2. (iii) 1 − = ∆ q Thi... ... of size m. Suppose that y (1) , y (2) , …, y (m) are the y values of sample units in ascending order. Further, let t be an integer such that Y (... ...nalogues of the P ij (x,y) obtained from the population and m x the number of units in S m with X ≤ M X . Let ) ( ~ y F YA and ) ( ~ y F YB de... ... with X ≤ M X . Let ) ( ~ y F YA and ) ( ~ y F YB denote the proportion of units in the sample S m with X ≤ M X , and X>M X , respectively that...

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...The aim of this book is two fold. At the outset the book gives most of the available literature about Fuzzy Relational Equations (FREs) and its properties for there is no book that solely caters to FREs and its applications. Though w...

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...S AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600... ...tm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-ma... ...m: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, A... ...r neutrosophic study. The second importance of this book is that for the first time we introduce the notion of Neutrosophic Relational Equations (... ...the social problems faced by bonded laborers etc. Chapter three for the first time defines several new neutrosophic concepts starting from the not... ...o minimize the total error ∑ υ υ = i i E E ) ( ) ( ( 7 ) at any time. 1.10 Unattainable Solution of FRE [35] have obtained a... ...e been used to study: (1). A typical chemical plant having several inter-linked units (2). Flow distribution in a pipe network and (3). A three stag... ...n a chemical plants A typical chemical plant consists of several interlinked units. These units act as nodes. The flowsheet is given in Figure 2.... ...ze waste gas in oil refinery. An oil refinery consists of several inter linked units. These units act as production units, refinery units and compr...

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...S AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600... ...tm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-ma... ...m: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, A... ...r neutrosophic study. The second importance of this book is that for the first time we introduce the notion of Neutrosophic Relational Equations (... ...the social problems faced by bonded laborers etc. Chapter three for the first time defines several new neutrosophic concepts starting from the not... ...o minimize the total error ∑ υ υ = i i E E ) ( ) ( ( 7 ) at any time. 1.10 Unattainable Solution of FRE [35] have obtained a... ...e been used to study: (1). A typical chemical plant having several inter-linked units (2). Flow distribution in a pipe network and (3). A three stag... ...n a chemical plants A typical chemical plant consists of several interlinked units. These units act as nodes. The flowsheet is given in Figure 2.... ...ze waste gas in oil refinery. An oil refinery consists of several inter linked units. These units act as production units, refinery units and compr...

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...The study of bialgebraic structures started very recently. Till date there are no books solely dealing with bistructures. The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published...

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...arandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, I... ... Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arb... ...er reviewed and recommended for publication by: Dr. Andrei Kelarev, Dept. of Mathematics, Univ. of Tasmania, Hobart, Tasmania 7001, Australia. Dr. M... ...an be made present in them -- thereby making the operation economical and time-saving. Bistructures are a very nice tool as this answers a major ... ...ctual delight and research productivity I have experienced in this span of time. I also thank my daughters Meena and Kama and my husband Dr. Kanda... ... which is a proper subset of R. Now we proceed on to define Smarandache units, Smarandache zero divisors and Smarandache idempotents in a ring. Th... ... semiring. We say the semiring is of characteristic m if mx = x + …+ x (m times) equal to zero for all x ∈ S. If no such m exists we say the charact... ...P is a non- commutative semiring. Now the notions of S-zero divisors, S-units and S-idempotents are defined in semirings as in case of rings. N... ...if S(P(I)) is a S-prime radical it has prime radical. The concept of S-units, S-idempotents and S-zero divisors in case of near-rings can be defi...

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...In this book, we study the subject of Smarandache Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field f...

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...DACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology Madras Chennai – 600 036, I... ...ations and their Smarandache analogues. The pictures on the left, composed of concentric circles, depicts the traditional conception of algebraic str... ...picts the traditional conception of algebraic structures, and the pictures of the right, with their liberal intersections, describe Smarandache algeb... ...arked that: "So, this whole thing started because of my perception at that time, that the world of classical mathematics – was a little too much of a... ...ith the Father of Fuzzy Sets, Lofti. A. Zadeh. Emphasizing about the long time it takes for a new subject to secure its place in the spotlight, he s... ...revious to that it was not accepted. There was an article in the New York Times about Bayesian things. It says this technology is 276 years old. Ano... ...at B ∗ is a field containg the identity of R and B(x) = B(x –1 ) for all units x in R, then B is called the fuzzy subfield in R. R is said to have a... ... It is interesting to note that the biring given in example 1.5.1 has no units but is a commutative biring of finite order. Example 1.5.2: Let R... ...one can define a µ so that Z 12 has several S-tips for 1, 3 and 4 are the units of the subgroup. DEFINITION 2.2.24: Let G be a S-semigroup. µ : G...