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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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...s. The last chapter is completely devoted to research problems some of them are really very difficult for these problems may attract a research stud... ...inite group. 2.2 Some Examples of Groups Example 2.2.1: G consists of real numbers 1 and −1. G under multiplication is a group of order 2 ... ...G be the set of all 2 × 2 matrices         d c b a where a, b, c, d are real numbers such that ad − − − − bc ≠ 0. G is a group under matrix ... ...ache subsemigroup and obtain some interesting results about this study and this analysis of Smarandache subsemigroup has lead us to the definition o... ...bgroups in S none of whose order divides the order of S, these S-semigroups are really very innovative because they make the analog of the classical... ...e a ∈ G \ {e} with x = axa -1 that is ax = xa and a ~ x. Such things happen in reality also for consider S 4 the symmetric group of degree 4. ...

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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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...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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... of universal rationality>. Tackling the issue of rationality from such a realistic as well as idealistic position, should not be reason for catalog... ...theologian Marin Stefånescu stated in his ″Christian Philosophy″: < To be realistic does not mean to be materialistic or less idealistic; it means to... ...ritualize matter. We should not be mistaken by being idealistic without a reality or realistic without an ideal>. This cogitation, besides the explic... ...tion of the spatio-temporality into material (substantial), and spiritual reality, by two limits: the primary limit and the secondary limit. The mat... ...y, by two limits: the primary limit and the secondary limit. The material reality is comprised between the two limits and it has, on either of its s... ...spiritual, respectively, of the spatiotemporality, yet, upon an objective analysis, one can find that the hypotheses of the concept are of a scienti... ...conformist inability, have recourse to whimsical assumptions, in the last analysis of the same nature as the occultist ones. Any occultist ″speciali... ... the moral rules, especially those of the natural morals, and in the last analysis, the divine judgement takes effect, through monadic dissociation ... ... sacred predestination, and if these acts occur, they reflect in the last analysis some deficiencies of a moral nature. The manifestation of the rei...

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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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...s, solved/unsolved problems, new Smarandache type sequences and new Smarandache Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theory by F. Smarandache ("The Florentin Smarandache papers" special collection, University of Craiova Library, and Archivele Statului, Filiala Valcea) an...

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...che Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theo... ...euto-Smarandache-Totient harmonic series: ¥ = > 1 0 " " ) ( 1 n a number real a is a where n Zt Problem 47. Study the convergence of the series: ... ... can be made arbitrarily close to zero. Theorem 1.2.13 For any arbitrary real number 0 > e , there is some number 1 ‡ n such that e < n n Zw ) (... ...ct of distinct primes k p p p q K 2 1 = such that e < q 1 where e is any real number greater than zero. Now take a number n such that: 2 , , 2 1 2 1... ...ibuted uniformly in the interval ]0,1]. Conjecture 4.4 For any arbitrary real number 0 > e , there is some number 1 ‡ n such that e < n n Sdf ) ( L... ...W(n)=n-1 if n is repunit (that is all the digits are 1). In the following analysis we will exclude this two species of integers. With the Ubasic pro...