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... another and can be read separately. Findings are illustrated with diagrams and tables. The latter have been kept to a minimum as it is often not the numbers but the general behaviour and pattern of numbers that matters. One of the facinations with number problems is that they are often easy to formulate but hard to solve – if ever, and if one finds a solution, new questio...

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...l Numbers - a few elementary studies on Smarandache sequences and other number problems Henry Ibstedt American Research Press Reho... ... The surprising behaviour of a sequence Mainly Natural Numbers - a few elementary studies on Smarandache sequences and other... ...randache/eBooks-otherformats.htm ISBN: 1-931233-73-X Standard Address Number: 297-5092 Printed in the United States of America 3 ... ...s and tables. The latter have been kept to a minimum as it is often not the numbers but the general behaviour and pattern of numbers that matters. O... ... behaviour and pattern of numbers that matters. One of the facinations with number problems is that they are often easy to formulate but hard to sol... ...m for n<1000 which is divisible by 333667 is S(999) from which we build S(108)=12…999_1000_…_1008_1008_…1000_999-…21 where we deconcatenate 100010...

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...he/eBooks-otherformats.htm ISBN: 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS ... ...kground in algebra and more specifically a strong foundation in loops and number theory. 6 This book introduces over 75 Smarandache concept... ...the introduction of a new class of loops of even order. We prove that the number of different representations of right alternative loop of even orde... ...n order (2n), in which square of each element is identity is equal to the number of distinct proper (2n – 1) edge colourings of the complete graph K... ... and PA(L). ii. Find conditions on the loop L so that PA(L) = SPA(L). 108 8. Can a Jordan loop be a Bruck loop? Moufang loop? Bol loop? 9. ... ..., 76 Absorption law, 12 ALBERT A.A., 29, 44, 113 A-loop, 5, 23, 29, 49, 108, 115-116 Algebraic lattice, 12 Alternating group, 14, 53 Alternating... ... R. H, 5, 15, 18-19, 22, 30, 49, 113 BURN R, 29, 113 C C-loop, 50, 108 CA-element, 25 CA-loop, 24-25, 28 Cauchy theorem for groups, 36 Cay... ..., 9-11, 14, 19-21, 24, 36-37, 39, 47-48, 51-53, 55-64, 67-72, 80-102, 108, 110-112 symmetric group, 9-10, 14 Group of permutations, 9 H ... ...8 J Join homomorphism, see Homomorphism, join Jordan loop, 27-28, 108 K KINYON, MICHAEL K., 5, 20, 23, 47, 49, 50, 91, 113, 115 L ...

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...Library (Romania), and Archives of State (Rm. Valcea, Romania). The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting. This book, following the ...

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...A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY Felice Russo AMERICAN RESEARCH PRESS Lupton USA 2000 Wrongnes... ...ache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this ... ...tics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and inte... ...ed problems, new Smarandache type sequences and new Smarandache Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Sma... ...ted to the analysis of Smarandache Double factorial function introduced in number theory by F. Smarandache ("The Florentin Smarandache papers" specia... ...34)|Zt(33), Zt(46)|Zt(45), Zt(75)|Zt(74), Zt(86)|Zt(85), Zt(90)|Zt(89), Zt(108)|Zt(107), Zt(172)|Zt(171), Zt(225)|Zt(224), Zt(242)|Zt(241), Zt(464)|Z... .... 1 ) ( 1 1 » ¥ = n n b where b(n) is the Smarandache reverse sequence. 108 References [1] Smarandache, Florentin, "Conjectures which Generalize A...

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...This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essen...

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... Hungary. ISBN: 1-931233-51-9 Standard Address Number 297-5092 Printed in the United States of America "...It is just... ...rt notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric the- or... ...s; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals esse... ...es or tetrahedrons. Other themes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophanti... ...sibility problems or questions related especially to the sequence of prime numbers. The author has solved various conjectures by Smarandache, Bencze, ... ... certain diophantine equations for particular arithmetic functions . . . . 108 21 On the diophantine equation a 2 +b 2 = 100a +b . . . . . . . . . . .... ... left side of (7): 6r 1 abc X ab(a +b) X AI: (9) D.F. Barrow (see [2], p.108) obtained another generalization of (3), by proving that X PA 2 X PA 0... ... then clearlyn = 4. Thereforen = 4 is the single solution of the equation. 108 The equation (n) =n +k, k = xed From (n) n + 1 (with equality on...

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...P. Popescu, V. Seleacu, "About the behaviour of some new functions in the number theory" (to appear this year) . "Collection of Problems on Smarand... ...shbacher and Erhus University Press ISBN 1-879585-50-2 Standard Address Number 297-5092 Printed in the United States of America Preface The pre... ...published[ 1], the Smarandache function SCm) = n, where n is the smallest number such that m divides n'. and associated consequences has spawned man... ...s an extension ofan element of set (a). See Some Notions and Questions in Number Theory, edited by C Dumitrescu and V Seleacu, Erhus University Pres... ...ore specific points in the second. The first problem we will deal with is number (6) in [5]. Smarandache permutation sequence: 12,1342.135642, 1357... ...uous sense as this is just a restatement of the previous if For problem (108) it is clear that the set of non-Sma rand ache numbers must be empty, a...

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..., Padilla treated the Smarandache algebraic structures mainly with associative binary operation. Since then the subject has been pursued by a growing number of researchers and now it would be better if one gets a coherent account of the basic and main results in these algebraic structures. This book aims to give a systematic development of the basic non-associative algebra...

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...eBooks-otherformats.htm ISBN: 1-931233-61-6 Standard Address Number: 297-5092 Printed in the United States of America 3 ... ...ciative binary operation. Since then the subject has been pursued by a growing number of researchers and now it would be better if one gets a coheren... ...d of algebraic structures like semigroup, group etc. and a good foundation in number theory. In Chapter 1 we just recall the basic notations and s... ...ed to lessen the non abstractness of this structure. In this chapter, several number theoretic techniques are used. Chapter 4 starts with the def... ...(t, u) = Z n ∪ {e}) (t, u) where t ≠ u, (t, u) = 1; t, u ∈ Z n \ {0}}. 108 Hint: The operation ∗ on G = Z n ∪ {e} is a ∗ a = e for all a ∈... ... Alternative, 17, 20, 40-44, 56, 62, 65, 91, 92, 106-108 Associative Law, 11, 93 Automaton, 88, 93-96 Binary operation,... ...irect Product, groupoids finite order, 9, 11-12, 15, 29-30 G (n), 90-91, 107-108 G ∗ (n), 90-92, 108 G ∗∗ (n), 90-92, 108 110 G ∗∗∗ (n), 9... ...7-108 G ∗ (n), 90-92, 108 G ∗∗ (n), 90-92, 108 110 G ∗∗∗ (n), 90-92, 108 ideal, 22-23, 25-26, 33 idempotent element, 14, 90 idempotent gr... ... alphabet, 93-94, 97 IVAN NIVEN , 18 LIDL.R, 93, 104 Loops, 7, 16-18, 43, 108 MACLANE.S.S, 68 Monoid, 11-12, 96 Moufang groupoid, see under ...

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.../˜ smarandache/eBooks-otherformats.htm ISBN:1-931233-92-6 Standard Address Number: 297-5092 Printed in the United States of America Preface A combina... ...tablished in this chapter. The last section determines the relation of the number of embeddings and rooted maps of a graph on genus. A general equatio... ...ed total map and embedding polynomial of a graph................20 4.2 The number of rooted maps underlying a graph on genus...................24 Chap... ...ferences ..................................................................108 Index..................................................................... ...with empty boundary and analytic structures. If k(S),g(S) and χ(S) are the number of connected components of ∂S, the topological genus and the Euler c... ... six or seven distinct triangular disks. The vertices are classified by the number of the disks around them. A vertex around five, six or seven triangul... ...erlie regular maps on closed surfaces, J.London Math. Soc.(2)59(1999),100- 108. [21] G.Gromadzki, Abelian Groups of automorphisms of compact non-orien... ...bution for bouquets of circles, References 110 Discrete Math.248(2002),93-108. [32] L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries...

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...e set, i indeterminate in the set, and f false, where t, i, and f are real numbers taken from the sets T, I, and F with no restriction on T, I, F, no... ...the next papers. Keywords and Phrases: Non-standard analysis, hyper-real number, infinitesimal, monad, non-standard real unit interval, operations ... ...s the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all po... ... positive integers n one has xx x < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set,... ... an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-stan... ...false-neutrosophic connectors. Let's reconsider the previous notations. 108 2.12. Neutrosophic Topology. A) Definitions: - Let's construct... ...0 July 1999, http://forum.swarthmore.edu/epigone/sci.math/shelkelwhim. [108] Peirce, C. S., Essays in the Philosophy of Science, The Liberal Arts P...

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...ooks-otherformats.htm ISBN: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed i... ...∗ b = b ∗ a for all a, b ∈ S we say S is a commutative semigroup. If the number of elements in the semigroup S is finite we say S is a finite semi... ... G a • b = b • a. A group, which is not abelian, is called non-abelian. The number of distinct elements in G is called the order of G; denoted by ο ... ...a Smarandache commutative semigroup. 10 Let S be a S-semigroup, ο(S) = number of elements in S that is the order of S, if ο(S) is finite we say... ...s a subsemivector space of V and is not a S-subsemivector space of V. 108 Proof: Left for the reader to do the proofs as an exercise. THEO... ...ring II, 89, 90 S-linear operator, 109, 112 S-linear transformation, 106, 108-110 S-MC, 82, 90 S-mc, 85, 90, 112 S-mixed direct product, 86, 9... ... ideal, 73, 75, 111 S-pseudo ideal, 72, 75, 111 S-pseudo semivector space, 108 S-pseudo subsemiring, 71-73 S-pseudo subsemivector space, 108 S-... ...5, 80, 83-89, 111-112 S-subsemiring II, 87-89 S-subsemivector space, 104-108 S-unit, 79-80, 87 S-weak semifield, 97-98 S-weighted CSP, 112 S... ... W. B., 28, 52, 64, 93-94, 102, 110 Vector space, 7, 22-25, 58-62, 64, 106, 108-110, 112 W WERNER, KUICH, 50-52, 94 Z Zero divisor, 19-...

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...s.htm ISBN: 1-931233-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...) is a mapping [x, y] → (x, y) of V × V into the set of (real or complex) numbers such that (ax + bx', y) = a (x, y) + b (x', y) and (x, a... ...tor space V is a mapping [x, y] → (x, y) of V × V into the set of complex numbers such that (x, y) = ) x , y ( and (ax + bx', y) = a (x, y) + b (x',... ...(S-finite dimensional vector space) of K if every S-basis has only finite number of elements in it. It is interesting to note that if L is a finite ... ...ras of type B then SL B is a. S-Lie semi-simple. b. S-Lie simple. 108 Proof: Follows directly by the very definitions. Thus we see tha... ...G where G ∈ Z ∗∗∗ (n) which are a. not essential b. not S-essential 108. Characterize those groupoid rings Z p G which has a. non trivial ra... ...01 S-Bernstein algebra, 124 S-Bol groupoid, 61, 69 S-Cartan subalgebra, 108, 137 S-Cauchy element, 18 S-characteristic value, 10 S-characteristi... ...op, 18 S-left ideal, 60 S-left semi-ideal, 67 S-Lie algebra, 91, 96, 99-108 S-Lie algebra homomorphism, 99 S-Lie ideal, 100-102, 107 S-Lie isomo... ...107 S-Lie isomorphism, 99 S-Lie link related, 103 S-Lie semisimple, 107-108, 137 S-Lie simple, 107-108, 137 S-Lie subalgebra, 99 S-Lie superalge...

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...ington, Texas 76019, USA. ISBN: 1-931233-60-8 Standard Address Number: 297-5092 Printed in the United States of America 3 Forew... ...2 1 5 The order of evaluation of the connectives is marked by the numbers under the connectives. Therefore, the value of the proposition is... ...o free variables. Example: If the universe of discourse is the set of numbers U = { 1, 2, 3, 4, 5 }, then the expression (Ex) x > 0 is tru... ...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 ≤ x ... ... 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. 22 ... ... public static INLElement INL2or(INLElement nle1, INLElement nle2) 108 { float nle1true,nle1false; float nle2true,nle2false; fl...

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...hingOnline.com Peer reviewers: Ion Goian, Department of Algebra and Number Theory, University of Kishinev, Moldova. Sebastian Martin Ruiz, Av... ...che/eBooks-otherformats.htm ISBN: 1-931233-64-0 Standard Address Number: 297-5092 Printed in the United States of America 3 CONT... ...e properties locally makes the study easier and also gives way to greater number of newer concepts. Also localization of properties automatically co... ...uch that a -1 a = aa -1 = e. A group, which contains only a finite number of elements, is called a finite group, otherwise it is termed as an... ...te definitions in case of semigroup rings and study them. DEFINITION [108]: Let R be a ring not necessarily commutative. Let L denote the collec... ...ative to V, then P is a demi module relative to V, V a subring of R. 108 Proof: Left as an exercise for the reader to prove. DEFINITION 3... ...there exist a class of semigroup rings which are S-Noetherian? 191 108. Let the group ring RG be any S-ring. Depending on G and on R is it pos... ...n semi idempotents in group rings, Proc. of the Japan Acad., Vol. 61, 107-108, (1985). 80. Vasantha Kandasamy W.B., A note on the modular group ri... ...samy W.B., On subsemi ideal rings, Chinese Quat. J. of Math., Vol. 7, 107-108, (1992). 103. Vasantha Kandasamy W.B., On the ring Z 2 S 3 , The...

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...oks-otherformats.htm ISBN: 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...f p ) x ( 3 2 1 Clearly µ is a fuzzy subgroup of a group G and o(µ ) = number of elements of the set {x ∈ G | µ (x) = µ (e)} = number of elements... ...by an example. 25 Example 1.2.10: Let X = N be the set of all natural numbers and µ be defined by µ (x) = x 1 for every x ∈ X. Now consider th... ...n = R be any chain of ideal of a ring R. Let t 0 , t 1 , …, t n be some numbers lying in the interval [0, 1] such that t 0 > t 1 > … > t n . Then... ...id to be a fuzzy ideal of M if it satisfies the following conditions: 108 i. µ (x + y) ≥ min {µ (x), µ (y)}. ii. µ (-x) = µ (x). iii. µ (x) =... ...poid of T. ii. f –1 (µ) is a S-fuzzy sub-bigroupoid of X. Problem 3.4.108: Prove a fuzzy subset λ of a S-bigroupoid X = X 1 ∪ X 2 is said to be... ...rove µ s where s ∈ Im µ is a semi-primary bi-ideal of R. Problem 4.5.108: Prove if µ is any S-fuzzy semiprimary bi-ideal of a S-biring R then t... ...OSH, Shamik, Fuzzy k-ideals of semirings, Fuzzy Sets and Systems, 95, 103-108 (1998). 43. GILL, A., Introduction to the Theory of Finite State Machi... ...ongruences on a groupoid under semibalanced maps, Fuzzy Sets and Systems, 108, 111-116 (1999). 436 46. HALL, Marshall, The theory of groups, Macmi...

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...ormats.htm ISBN: 1-931233-71-3 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...y G is a abelian or a commutative group. If the group G has only a finite number of elements we call G a group of finite order otherwise G is said t... ...s a subgroup of G then o(H) is a divisor of o(G). Note: o(G) means the number of elements in G it will also be denoted by |G|. CAUCHY THEOREM (... ...ROUPS): Suppose G is a finite abelian group and p/o(G) where p is a prime number, then there is an element a ≠ e ∈ G such that a p = e. ... ... a 1 a 4 a 4 a 2 a 1 a 5 e a 3 a 5 a 5 a 4 a 3 a 2 a 1 e 108 Clearly |L| = 15. Now p = 5 is prime and p/15 but L has no element o... ... 107. Characterize those S-bigroupoids which are S-Cauchy bigroupoids. 108. Construct using Smarandache notions a S-bicascade. 109. Obtain some... ...MY, W. B., On semi idempotents in group rings, Proc. Japan Acad., 61, 107-108 (1985). 54. VASANTHA KANDASAMY, W. B., A note on the modular group r... ...s of an operator loop over Z, Acta Ciencia Indica, 26, 129-131 (2000). 108. VASANTHA KANDASAMY, W. B., On a new class of semivector spaces, Varah... ... loop, definition of, 9 262 Bruck near-ring, 49 C Cauchy biloop, 108 Cauchy element, 85, 115, 121, 248 Cauchy theorem, 8, 70, 72, 74, 78, ...

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... ISBN: 1-931233-48-9 Standard Address Number 297-5092 Printed in the United States of America 4 De... ...his way if the trajectories of the movements did no bend due to the great number of interacting bodies. Furthermore, due to the movement over curved... ... transportation means for rational extraterrestrial beings. The increased number of reports on the presence of and ″contact″ with extraterrestrial b... ...e as an explanation for the presence, around the Inca temples, of a great number of human skeletons mutilated in the same manner). By admitting the ... ... refer to my own experience. My personal oneiric activity was marked by a number of dreams that repeated themselves almost identically for a great n... ...igious unification 10. Date of birth of Jesus V. The moral criterion 108 1. A hermeneutics of some natural moral norms 2. Nietzsche and ...