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... Florentin Smarandache. As a result, this book doesn't give a full-fledged analysis on loops and their properties. However, for the sake of readers w... ...f(mn) = f(m) f(n) whenever (m, n) = 1. Notation: If x ∈ R (R the set of reals), Then [x] denotes the largest integer that does not exceed x. Re... ... denotes the largest integer not exceeding x for any x ∈ R (R: the set of real numbers) and the remaining elements will decompose into cycles of len... ...dents / researchers to study as this Smarandache structures paves way for analysis of any structure is an exemplary way which cannot be done otherwi... ..., No.2, 371-375 (2000). 52. Riordn.J, An introduction to Combinatorial analysis, John Wiley and Sons Inc., (1964). 53. Robinson.D.A, Bol loops...

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...complete systematic introduction of all concepts together with a sequential analysis of these concepts. Examples are provided abundantly to make the... ...e in this book. Example 1.2.5: Let L be [a, b] any closed interval on the real line, [a, b] under the total order is a chain lattice. a b c ... ... is a division ring or a skew field. Example 1.3.5: Let R be the set of reals, R is a field of characteristic 0. Example 1.3.6: Let Z 28 = {0... ...aracteristic 0 are isomorphic to Q. Example 1.3.10: Let R be the field of reals. R has the subset Q ⊂ R and Q is a field; so R is not a prime fiel... ...'. Example 1.4.1: Let R[x] be the polynomial ring where R is the field of reals. R[x] is a vector space over R. Example 1.4.2: Let Q be the fie... ...over R. Example 1.4.2: Let Q be the field of rationals and R the field of reals. R is a vector space over Q. It is important and interesting to... ... researchers. Except for Smarandache notions such rich type of mathematical analysis would be completely absent in the realm of mathematics. 1. D...

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...tions for Smarandache concepts finds themselves accommodative in a better analysis by dissecting the whole structures into specified smaller structu... ...ationals is a field under usual addition and multiplication, R the set of reals is also a field; where as Z the set of integers is an integral domain... ...self under the operations of F is a field. For example in the field of reals R we have Q the field of rationals to be a subfield. If a field has ... ...s of R. As non-associative rings cannot be constructed using the set of reals or integers or rationals or complex or modulo integers without the a... ...x or modulo integers without the aid of another algebraic structure it is really difficult to give natural examples of non-associative rings. The ... ...ere Q + is the set of positive rationals and R + is the set of positive reals. DEFINITION 1.5.2: Let (S, +, ) be a semiring. We say the semiring... ...K 2n graphs. This question is an interesting one and is left for further analysis to the reader. In case of loops we have proved refer [109]. ... ...bson radical of RG is an ideal this ideal is also a quad ideal of RG. The analysis of bigroupoid birings to be a Moufang quad ring, Bol quad ring, Br... ...dy of how experiments can be organized systematically so that statistical analysis can be applied in an interesting problem which is carried out by s...

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...51;34512;45123;51234 | {z } 5 ; 123456;234561;345612;456123;561234;612345 | {z } 6 ;::: Let ]x[ be the largest natural number strongly smaller than th... ...5) In A5 the identity ¼(b) X k=1 ¼( n p k )=¼( n b ):¼(b)+ ¼( n 2 )¡¼( n b ) X k=1 ¼( n p ¼( n b )+k )( 6 ) is proved, under the condition b ¸2( b is ... ...richlet's identity X k·n f(k): X t=k g(t)= X k·n g(k): X s· n k f(ks); where g is also arbitrary arithmetic function. Putting there g(x)= lnx for ever... ...n f(k): X s· n k g(s)= X k·n g(k): X s· n k f(s); that is valid for arbitrary two arithmetic functions f and g,weput: g(x)´1; f(x)=lnx for any positiv... ...2 ) holds. In particular, it is possible to use µ(n) instead of ¸(n). Further, we will ¯nd an explicit formula for a (m) n when m¸2is ¯xed. Let for an... ...ov. On 25-th and 26-th Smarandache's problems. Notes on Number Theory and Discrete Mathematics, Vol. 9, 2003, No. 4, 99-104. [42] Yosida K., Functiona...

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...CHABILITY: PERIYAR’S VIEW AND PRESENT DAY SITUATION A FUZZY AND NEUTROSOPHIC ANALYSIS 2.1 Analysis of untouchability due to Hindu religion usi... ...Analysis of untouchability due to Hindu religion using FCMs and NCMs 43 2.2 Analysis of discrimination faced by Dalits/ Sudras in the field of ed... ...Social inequality faced by Dalits and some of the most backward classes - an analysis using FCM and NCM 66 5 2.4 Problems faced by Dalits in th... ...ced by Dalits in the political arena due to discrimination – a FCM and NCM analysis 75 2.5 Study of Economic Status of Dalits due to untouchabi... ...the expert's opinion for this unsupervised data to obtain some idea about the real plight of the situation. This is just an illustration to show how... ...ut this section we assume the elements of the domain space are taken from the real vector space of dimension n and that of the range space are real ... ...al that the notion of neutrosophic logic play a vital role in several of the real world problems like law, medicine, industry, finance, IT, stocks ... ... the notion of neutrosophic concepts. In this book we assume all fields to be real fields of characteristic 0 all vector spaces are taken as real sp... ...l fields of characteristic 0 all vector spaces are taken as real spaces over reals and we denote the indeterminacy by ‘I ’ as i will make a confusi...

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...An associative ring is just realized or built using reals or complex; finite or infinite by defining two binary operations on it. But on the contrary when we want to define or study or even introduce a non-associative ring we need two separate algebraic...

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...Index 145 5 PREFACE An associative ring is just realized or built using reals or complex; finite or infinite by defining t... ... we recall them. DEFINITION 1.1.3: A bilinear form on a vector space V (real or complex) is a mapping [x, y] → (x, y) of V × V into the set of (re... ...n A itself is a S-K-vectorial space. The study of Smarandache basis is really a new concept. DEFINITION 1.2.2: Let V be a finite dimensional ve... ..., in fact an integral domain, Q the field of rationals and R the field of reals. L will denote a loop non-necessarily abelian but never a group so b... ...if p/n then Z p L is not regular. THEOREM 2.2.6: Let R be the field of reals and L a finite loop in which m i 2 = 1 for every m i ∈ L. Then the... ...o introduce Smarandache non- associative algebras we don't give a complete analysis of Jordan algebras. Further we are mainly interested in the algeb... ...Smarandache P-algebras using P- identities in groupoid rings. 120 4. Analysis of alternative algebras and Smarandache alternative algebras using... ... to these algebras. Further these algebras became little difficult for analysis as x 2 = 0 for all elements x in a Lie algebra. Yet we have intr...

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... Chapter three is completely devoted to the introduction, description and analysis of the Smarandache rings — element-wise, substructure-wise and als... ...e set of positive and negative rationals with zero R – denotes the set of reals, positive, negative with zero. Z n – set of integers modulo n. Z n ... ... i + α 2 j + α 3 k where all the numbers α 0 , α 1 ,α 2 and α 3 are real numbers. We declare two such symbols α 0 + α 1 i + α 2 j + α 3 k and ... ...f all 2 × 2 matrices         c 0 b a ; a is rational b and c are reals is right Artinian but not left Artinian. 5. Is the ring P = Z × Z ×... ...1 can never be S-units. We prove every unit in the field of rationals and reals are S-units. DEFINITIONS 3.2.1: Let R be a ring with unit. We say... ...contained in S. Hence the claim. Example 3.5.6: Let R be the field of reals. R[x] be the polynomial ring. Clearly R[x] is a S-ring. Now Q ⊂ R[x]... ...e ring of integers Z, 116 the field of rationals to be useless in the analysis of S-rings. So we have used S-ring level II to overcome this prob...

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...lem using FRE 196 2.6 Data compression with FRE 196 2.7 Applying FRE to Threat Analysis 196 2.8 FRE application to medical diagnosis 197 2.9 A fuz... ...4.10 Some properties of minimal solution for NRE 270 4.11 Application of NRE to Real-world problems 272 Chapter Five SUGGESTED PROBLEMS 279 Bi... ...sophic Relational Equations have a role to play for we see that in most of the real-world problems, the concept of indeterminacy certainly has its ... ...on 4.11 is completely devoted to suggest how one can apply NREs in the study of real world problems. We suggest many problems in chapter five for th... ... Neutrosophic Cognitive Maps (http://gallup.unm.edu/~smarandache/NCMs.pdf) and Analysis of Social Aspects of Migrant Labourers Living With HIV/AIDS... ...goodness of one value cannot compensate the badness of another value [117]. In reality there are situations that allow compensatability among the v... ... (=N 5 ). This clearly implies that all entries in the matrices P, Q, and R are real numbers from the unit interval [0, 1]. Assume now that the thre... ... we can only provide the decision maker with the Pareto optimal set for further analysis. For a problem with multiple linear objective functions, th... ...he transform function f and the threshold value θ, which is reasonable (see the analysis by [6]) in fact for our fuzzy neural network discussed late...

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...se from the need to define structures which were more compatible with the real world where the grey areas mattered. Lofti A Zadeh, the father of fuz... ...al sets were not natural, appropriate or useful notions in describing the real life problems, because every object encountered in this real physical... ...as o(µ ) = o(H). 13 THEOREM 1.2.6: Any subgroup H of a group G can be realised as a level subgroup of some fuzzy subgroup of G. The proof is... ... of its basic properties. In 1971 [112, 145] introduced fuzzy sets in the realm of group theory and formulated the concept of a fuzzy subgroup of a ... ... fuzzy polynomial near-rings. DEFINITION 1.8.13: Let R be the set of reals. The fuzzy polynomial near-ring R[x [0, 1] ] consists of elemen... ...dy of how experiments can be organized systematically so that statistical analysis can be applied in an interesting problem which is carried out by s... ... a number between 0 and 1 and it estimates the quality of any statistical analysis if E ≥ 0.75 the quality is good. Now, how does one construct a... ...ay help in even comparison of one BIBD with the another and also give the analysis of the common features. Thus the S-planar near-ring has more a... ... a number between 0 and 1 and it estimates the quality of any statistical analysis if E > 0.75 the quality is good. Here we give the constructio...

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... is very di cult to decide in this eld of Elementary Geometry - Algebra - Analysis) we note that (1) appears in [R. Sturm: Maxima und minima in der e... ...ce (e.g. "Heron trapeziums"). Bibliography 1. R.D. Carmichael, Diophantine analysis, John Wiley and Sons, New York, 1915. 2. R.K. Guy, Unsolved proble... .... Sci. 41(1988), 19-21. 8. E.T. Whittaker, G.N. Watson, A course of modern analysis, Cambridge Univ. Press, 1969. 107 20 On certain diophantine equat... ... for odd n, the theorem implies that C(n) =n 2 for n 3 and odd. Thus the real di culty in calculating C(n) is for n an even number. 169 2) The abov... ...ow de ne the following "additive analogue", which is de ned on a subset of real numbers. Let S(x) = minfm2N : x m!g; x2 (1;1) (1) as well as, its dua... ...Discr. Math. 5(1999), No.2, 41-51. 2. W. Rudin, Principles of Mathematical Analysis, Second ed., Mc Graw-Hill Company, New York, 1964. 174 17 On the ... ...Math. 34(1989), 7-14. 188 20 On certain inequalities for k 1. Letk be a real number andn 1 a positive integer. Letd 1 ;d 2 ;:::;d s be all distinc... ...1 n d i k ; so s X i=1 d k i =n k s X i=1 d k i (8) Let k;t > 0 be real numbers, and apply (6) to a i = d k=2 i , b i = d t=2 i (i = 1;s). T... ...alities [3] such that if a 1 a 2 ::: a s and b 1 b 2 ::: b s are real numbers, then s X i=1 a i ! s X i=1 b i ! s s X i=1 a i b i (12) ...

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... 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 essential need for an alternative system that could infuse into itself a representation of the real world. Out of this need arose the system of Neutro...

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...h and falsity has not imbued itself with the capability of reflecting the reality. Despite various attempts to reorient logic, there has remained an... ... alternative system that could infuse into itself a representation of the real world. Out of this need arose the system of Neutrosophy, and its conn... ...research papers which deal with FCMs, and the tool has been used to study real-world situations as varied as stock-investment analysis to supervisor... ...as been used to study real-world situations as varied as stock-investment analysis to supervisory system control, and child labor to community mobil... ...e concepts of FCMs, and also try to give an inclusive view of the various real-world problems to which FCMs have been applied. Though there are over... ...ter level. We have written this book as a maiden effort to inculcate into real-world problems the concept of indeterminacy, uncertainty and inconclu... ... systems; mobile robots and in intimate technology such as office plants; analysis of business performance assessment; formalism debate and legal ru... ...s to be better than that of the FCM model in several ways mainly when the analysis of the data can be treated as two disjoint entities. Thus its appl... ...12, 119]. This example illustrates the strengths and weaknesses of FCM analysis. FCM allows experts to represent factual and evaluative concepts ...

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...ersity of New Mexico Gallup, NM 87301, USA 0.1 Introduction to Non-Standard Analysis. In 1960s Abraham Robinson has developed the non-stan... ...s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines t... ...egers n one has xxx < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which include... ... unit interval. Actually, by “ - a” one signifies a monad, i.e. a set of hyper-real numbers in non-standard analysis: .( - a)= {a-x: x %‘ * ,... ...“ - a” one signifies a monad, i.e. a set of hyper-real numbers in non-standard analysis: .( - a)= {a-x: x %‘ * , x is infinitesimal}, and sim... ...– c + and c. Addition of non-standard finite numbers with themselves or with real numbers: - a + b = - (a + b) a + b + = (a + b) + - ... ...ion, roots, and powers of non-standard finite numbers with themselves or with real numbers. By extension let inf ] - 0, 1 + [ = - a and sup ] - 0,... ...Definition of Neutrosophic Components. Let T, I, F be standard or non-standard real subsets of ] - 0, 1 + [, with sup T = t_sup, inf T = t_inf... ...e appurtenances (t [ 1, and f [ 1 or I [ 1). 5 NEUTROSOPHIC STATISTICS: Analysis of events, described by the neutrosophic probability, means n...

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...root of two. Example: If the universe of discourse is the set of all real numbers x where 0 ≤ x ≤ 1.0, then (x) x 2 ≤ 1.0 is true (x) x 2 ... ...se The English equivalents of these predicates are The square of any real number between zero and one inclusive is less than or equal to one. ... ... and one inclusive is less than or equal to one. 22 The square of any real number between zero and one inclusive is less than or equal to the nu... ...and one inclusive is less than or equal to the number. The square of any real number between zero and one is less than zero. The connectives of ... ... predicates. Example: If the universe of discourse is the set of all real numbers x where 0 ≤ x ≤ 1.0, then ((x) x 2 ≤ 1.0) /\ ((x) x 2 ≤ x... ...n { A, B } = B, then max { A, B } = A. iii) This proof is done using case analysis. Case 1: A ≥ B ≥ C B /\ C = C and then A \/ C = A, so the left... ...¬( A /\ B ) = ¬A \/ ¬B DeMorgan’s Proof: i) The proof is by case analysis Case 1: If A or B is one, then the left side is zero. One of t... ...n the right are one, so the conjunction is 1. iii) The proof is by case analysis Case 1: If one of the values is zero, then the conjunction in ... ...( A /\ B ) = ¬A \/ ¬B DeMorgan’s Proof: i) The proof is by case analysis. Case 1: A ≥ B Then A \/ B = A and the negation on the left is...