... AMERICAN RESEARCH PRESS REHOBOTH 2002 A 1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops ... ... American Research Press Rehoboth, NM 2002 A 11 L A 7 A 8 A 6 A 10 A 1 A 2 A 3 A 4 A 5 2 T... ...epartment of Mathematics, Arlington, Texas 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box... ... An arithmetic function f(n) is said to be a multiplicative function if f(mn) = f(m) f(n) whenever (m, n) = 1. Notation: If x ∈ R (R the set of re... ...alogue of the very recently introduced loops by Michael K. Kinyon [40] in 2002, Smarandache ARIF loops, Smarandache RIF loops, Smarandache Steiner l... ...rate them with examples. Smarandache loops were introduced very recently (2002) [81]. Here we also give which of the well-known classes of loops are...

...eRe mari ÕI MICI / Smarandache Florentin / R >mnicu V >lcea: Offsercolor, 2002 129 p.; 20 cm. Bibliogr. ISBN: 973-85650-7-3 Tehnor... ... mari şi MICI Editura OFFSETCOLOR Râmnicu Vâlcea, 2002 4 5... ... Xico, International Writers and Artists Association, Moorhead University, MN, SUA, 1989. 3. Florentin Smarandache, NonPoems, Xiquan Publ. Hse., 199... ...n the World! (Aut-Arta, cea mai proast ă art ă din lume!), Râmnicu Vâlcea, 2002, edi ţie îngrijit ă de I. Soare). 62 Nu toate experimentele sunt... ... electronice şi ale revolu ţiei informa ţionale. 24-27 ianuarie 2002, Cluj-Napoca – New Mexico 107 ... ... Popescu and Paul Georgelin, Publishing House OFFSETCOLOR, Râmnicu Vâlcea, 2002, 164 pag. 6. Interviuri cu Florentin Smarandache, de Veronica Balaj ...

...eRe mari ÕI MICI / Smarandache Florentin / R >mnicu V >lcea: Offsercolor, 2002 129 p.; 20 cm. Bibliogr. ISBN: 973-85650-7-3 Tehnor... ... mari şi MICI Editura OFFSETCOLOR Râmnicu Vâlcea, 2002 4 5... ... Xico, International Writers and Artists Association, Moorhead University, MN, SUA, 1989. 3. Florentin Smarandache, NonPoems, Xiquan Publ. Hse., 199... ...n the World! (Aut-Arta, cea mai proast ă art ă din lume!), Râmnicu Vâlcea, 2002, edi ţie îngrijit ă de I. Soare). 62 Nu toate experimentele sunt... ... electronice şi ale revolu ţiei informa ţionale. 24-27 ianuarie 2002, Cluj-Napoca – New Mexico 107 ... ... Popescu and Paul Georgelin, Publishing House OFFSETCOLOR, Râmnicu Vâlcea, 2002, 164 pag. 6. Interviuri cu Florentin Smarandache, de Veronica Balaj ...

...f non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are ... ...udy of Smarandache near-rings is very recent; introduced only in the year 2002 [43, 118, 126]. DEFINITION 1.6.1: A near-ring is a set N together ... ...f rings. DEFINITION 1.6.5: A subnear-ring M of N is called invariant if MN ⊂ M and NM ⊂ M. Near-rings are called simple if it has no ideals. N p ... ...iii. ( Σα(m)m) ( Σµ(n)n) = Σγ(k)k; m, n, k ∈ L where γ (k) = Σ α (m) µ(n); mn = k ∈ L (whenever they are distributive other wise it is assumed to be ... ... () ; k ) k ( n ) n ( m ) m ( γ β α∑ = ∑ ∑ where γ (k) = Σ α (m) β (n) and mn = k; only when α (m)’s are distributive in N if α (m)’s are not distrib... ...tion we define bigroupoids. Study of groupoids and S-groupoids started in 2002 [114]; as books solely on groupoids is very meager. Here we define big... ...che semi- automaton and Smarandache automaton was defined only in the year 2002 [114]. DEFINITION 5.3.1: Y s = (Z, s s , A δ ) is said to be a S... ...several of its properties. Studies of S-semirings have been introduced in 2002 [122, 123]. Here we 1 b a 0 S 1 S 2 1 a b c 0 184 giv... ...8.1.8: A sub-binear-ring M of the binear-ring N is called bi-invariant if MN 1 ⊂ M 1 as M 1 N 1 ⊂ M 1 and M 2 N 2 ⊂ M 2 (where M = M 1 ∪ M 2 ...

...tion of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using ...

...*************************** American Research Press Rehoboth 2002 József Sándor DEPARTMENT OF MATHEMATICS BABE Ş-BOLYAI UNIVER... ... American Research Press Rehoboth 2002 This book can be ordered in microfilm format from: Books on D... ...0-521-0600 (Customer Service) http://wwwlib.umi.com/bod/ Copyright 2002 by American Research Press Rehoboth, Box 141 NM 87322, USA E-mail: ... .... the so-called Pythagorean numbers) are given by a = (m 2 n 2 ); b = 2 mn;c = (m 2 +n 2 ) (if b is even) (2) where is arbitrary positive intege... ....e. (m;n) = 1 andm andn cannot be both odd or even). Clearly = ab 2 = 2 mn(m 2 n 2 ), integer. Letp be the semiperimeter of the triangle. From (2... ... ), integer. Letp be the semiperimeter of the triangle. From (2)p = (m 2 +mn); and denoting by r the inradius of a such triangle, it is well known th... ... are integers only if cjab, which, by (2) can be written as (m 2 + n 2 )j2 mn (m 2 n 2 ). Since (m;n) = 1, of di erent parity, it is immediate that ... ...n as one of the followings: (i) b = (m 2 +n 2 ), x = (m 2 n 2 ), u = 2 mn ; (ii) b = (m 2 +n 2 ), x = 2 mn , u = (m 2 n 2 ). We shall consider... ... (Gregory J. Chaitin, Conversations with a mathematician, Springer Verlag, 2002) 122 1 A note on S(n) Let S be the Smarandache function. We will prov...

...product of λ and A B written λ A B is defined to be λA B = 11 1n m1 mn aa aa λλ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ λλ ⎣ ⎦ ∪ 11 1n m1 mn b b bb λλ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ... ...ab bb i j + ∪+∀ . If we write in detail A B = 1111 11 1n 11 1n 1111 m1 mn m1 mn aabb aabb ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ C B = 2222 11 1n... ...bb aabb ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ C B = 2222 11 1n 11 1n 2222 m1 mn m1 mn aabb aabb ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ 13 A B + C B ... ... 13 A B + C B = 12 12 12 12 11 11 1n 1n 11 11 1n 1n 12 12 12 12 m1 m1 mn mn m1 m1 mn mn aa a a bb bb aa aa bb bb ⎡⎤ ⎡⎤ +++ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ... ...les, Introduction to Neutrosophic Logic, American Research Press, Rehoboth, 2002. 4. BIRKHOFF, G., and MACLANE, S., A Survey of Modern Algebra, Ma... ..., Stephen, Some topics pertaining to algebras of linear operators, November 2002. http://arxiv.org/pdf/math.CA/0211171 28. SHILOV, G.E., An Introdu... ...3, 2001 held at the University of New Mexico, published by Xiquan, Phoenix, 2002. 37. THRALL, R.M., and TORNKHEIM, L., Vector spaces and matrices,... ...NTHA KANDASAMY, W.B., Smarandache rings, American Research Press, Rehoboth, 2002. 45. VASANTHA KANDASAMY, W.B., Smarandache Semirings, Semifields a... ...rings, Semifields and Semivector spaces, American Research Press, Rehoboth, 2002. 46. VASANTHA KANDASAMY, W.B., Bialgebraic structures and Smaranda...

... 2 c 2 A ik F i F k + + 1 c 2 F m F m D+Δ m kn D k m F n − −h ik Δ m ik (D mn +A mn )F n =0, 4 D. Rabounski. A New Method to Measure the Speed of Grav... ... F n D+ 1 c 4 F k F k F i + 1 c 2 Δ i km F k F m − −h km ∂ ∂x k Δ i mn F n + Δ i kn Δ n mp −Δ n km Δ i np F p + +Δ i kn ∂F n ∂x m −Δ n ... ...utrosophic Probability, Set, and Logic. American Research Press, Rehoboth, 2002, 144 p. (see it in e-print: http://www.gallup.unm.edu/ smarandache). ... ...rms of the 6D angular momentum operators and a 6D pseudo-Euclidean metricη MN : ˆ M μν = Σ μν , ˆ M 56 = Σ 56 , (4.1) λΣ μ5 = ˆ x μ , R Σ μ6 = ˆ p... ...s and Fractals, v. 16, 2003, 539. 4. Castro C. Mod. Phys. Letters, v. A17, 2002, 2095–2103. 5. LaViolette P. Int. Jour. Gen. Sys., v. 11, 1985, 329. A... ...Fractals, v. 10 (2–3), 1999, 267. 17. Schuller F. Annals of Phys., v. 299, 2002, 174. 18. Lukierski J., Nowicki A., Ruegg H., Tolstoy V. Phys. Letters... ...-Camelia G. Phys. Letters, v. B510, 2001, 255. Int. J. Mod. Phys., v. D11, 2002, 35. Int. J. Mod. Phys., v. D11, 2002, 1643. 29. Greisen K. Phys. Rev.... ...ovo Cimento, v. B40, 1977, 27. 32. Low S. Jour. Phys. A Math. Gen., v. 35, 2002, 5711. J. Math. Phys., v. 38, 1997, 2197. 33. Vacaru S. Non-holonomic ... ...en the expression (1) which can be written more ex- plicitly as: |dX| 2 =G MN dX M dX N = dS 2 = = dσ 2 +L −2 dx μ dx μ +L −4 dx μν dx μν + ... + +L −...

...e product of λ and A B written λ A B is defined to be λA B = 11 1n m1 mn aa aa λλ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ λλ ⎣ ⎦ ∪ 11 1n m1 mn b b bb λλ ⎡ ⎤ ⎢ ⎥ ⎢ ... ...j ab bb i j + ∪+∀ . If we write in detail A B = 1111 11 1n 11 1n 1111 m1 mn m1 mn aabb aabb ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ C B = 2222 11 ... ...aabb aabb ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ C B = 2222 11 1n 11 1n 2222 m1 mn m1 mn aabb aabb ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ A B + C B = ... ...⎣⎦ A B + C B = 12 12 12 12 11 11 1n 1n 11 11 1n 1n 12 12 12 12 m1 m1 mn mn m1 m1 mn mn aa a a bb bb aa aa bb bb ⎡⎤ ⎡⎤ +++ ⎢⎥ ⎢⎥ ∪ ⎢⎥ ⎢⎥ ⎢⎥ ⎢... ...er, C. Introduction to Neutrosophic Logic, American Research Press, Rehoboth, 2002. 242 http://www.gallup.unm.edu/~smarandache/IntrodNeutLog ic.p... ...Specific Language Impairment, Artificial Intelligence in Medicine, 679, 1-18, 2002. 45. Goto, K. and T. Yamaguchi. Fuzzy Associative Memory Applic... ...ning Inference Amplification, Expert Systems with Applications, 22, 197- 211, 2002. 85. Lee, K.C., S. C. Chu and S.H. Kim. Fuzzy Cognitive Map-base... ... Florentin Smarandache editor, Xiquan, Phoenix, ISBN: 1-931233- 55-1, 147 p., 2002, also published in "Libertas Mathematica", University of Texas at... ...ed in "Libertas Mathematica", University of Texas at Arlington, 22, 175- 187, 2002. http://lanl.arxiv.org/ftp/math/papers/0211/0211465.pdf 91. Mad...

... DATA SAMPLING (second edition) American Research Press Rehoboth 2002 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 0 . 0 512345... ...ent of Mathematics, UNM, USA. American Research Press Rehoboth 2002 3 This book can be ordered in microfilm format from: ... ...emand) Copyright 2002 by American Research Press & Authors Rehoboth, Box 141 NM 87322... ...istribution. Bain(1972) defined an unbiased estimator for b as b yy nK u im mn i m ∧ = − =− −           ∑ (,) 1 1 , (... ...− −           ∑ (,) 1 1 , (1.3) where () K n vv mn i m i m (,) =−       −       = − ∑ 1 1 1 E , ...

...s buried inside the body, i. e. outside the domain of the feld equations R mn =0. But if we accord to these equations an importance com- parable to th... ...per by J. Smoller and B. Temple. Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 (see also in www.geocities.com/theometria/Stavroulakis-1.pdf... ...ectric”. The spatial projection H ik =F ik =− 2 c A ik , H ik =h im h kn F mn =− 2 c A ik (17) of the feld tensor will be referred to as “magnetic”. S... ...ike any chr.inv.-pseudotensors [5, 6], by the formulae H i = 1 2 ε imn H mn , H i = 1 2 ε imn H mn , (23) ε ipq H i = 1 2 ε ipq ε imn H mn =H pq... ...ik = μc 4π 4A im A m∙ ∙k + 1 c 2 F i F k − − 1 3c 2 F m F m h ik + 4 3 A mn A mn h ik , (76) so viscous strengths of time density felds are anisot... ...temporal vacuum fuctuations of quantized felds. World Scientifc, Singapore 2002. 7. Coquereaux R., Jadcyk A. Reviews in Mathematical Physics, 1990, No... .... of Theor. Phys., 1986, v. 107, 561–575. 17. Castro C. Mod. Phys. Letts., 2002, v. A17, No. 32, 2095–2103. Class. Quan. Grav., 2003, v. 20, 3577–3592... ...kanen M. Topological Geometrodynamics I, II. Chaos, Solitons and Fractals, 2002, v. 13, No. 6, 1205, 1217. 20. Vladimorov V., Volovich I. and Zelenov ... ...per by J. Smoller and B. Temple. Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 (www.geocities.com/theometria/Stavroulakis-1.pdf). 13. Croth...

...n from V to W. If A is the associated matrix of T then. A = mm nm 11 mn nn 11 BO OC ×× ×× ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ where A is a (m 1 + n 1 ) × (m ... ... 1 C × is the associated matrix of T 2 : V 2 → W 2 and nm 1 O × and mn 1 O × are non zero matrices. Without loss of generality we can also... ... ∪−− −− ⎢⎥ ⎢⎥ −−−−− − ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ −−−−− −− −− ⎣⎦ ⎣⎦ 48. Does bimatrix 2002 3101 0120 1211 0110 0110 4002 0112 − ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ −− ⎜⎟ ⎜⎟ ∪ ⎜⎟ ⎜⎟ −− ... ...es, Introduction to Neutrosophic Logic, American Research Press, Rehoboth, 2002. 4. BIRKHOFF, G., and MACLANE, S., A Survey of Modern Algebra, Mac... ...rinceton, 1958. 12. HARVEY E. ROSE, Linear Algebra, Bir Khauser Verlag, 2002. 13. HERSTEIN I.N., Abstract Algebra, John Wiley,1990. 14. HERSTEI... ... Stephen, Some topics pertaining to algebras of linear operators, November 2002. http://arxiv.org/pdf/math.CA/0211171 32. SHILOV, G.E., An Introduc... ..., 2001 held at the University of New Mexico, published by Xiquan, Phoenix, 2002. 34. SMARANDACHE, Florentin, A Unifying field in Logics: Neutrosop...

...THETICS OF PARADOXISM (second edition) American Research Press Rehoboth 2002 2 Titu Popescu THE AESTHETICS OF PARADOXISM (second edition) Tran... ...orgelin, F. Smarandache, and L. Popescu American Research Press Rehoboth 2002 3 This book can be ordered in microfilm format from: Bell and H... ... 1-800-521-0600 http://www.umi.com/bod/ (Books on Demand) Copyright 2002 by American Research Press Rehoboth, Box 141 NM 87322, USA E-mail: ... ...International Writers and Artists Association, Moorhead State University, MN, USA, 1989; 27. LE PARDOXISME: un nouveau mouvement littéraire, Bergera...

...e best of my knowledge the concept on fuzzy bigroups appeared in the year 2002 [135]. 1.4 Fuzzy Rings and its properties In this section we r... ....8.30: A sub-binear-ring M of the binear-ring N is called bi-invariant if MN 1 ⊂ M 1 and M 2 N 2 ⊂ M 2 (where M = M 1 ∪ M 2 and N = N 1 ∪ N 2... ...bset of N which is a near-field) we have h (m + n) = h(m) + h(n), h (mn) = h(m) h(n) where h (m) and h (n) ∈ M 1 (M 1 is a proper subset o... ...ring homomorphism) if for all m, n ∈ M we have h (m + n) = h(m) + h(n), h(mn) = h(m)h(n) where h(m), h(n) ∈ M' (M' is a proper subset of N' which is... ...ton and Smarandache automaton was first defined by the author in the year 2002. The study of Smarandache semi automaton and uses the concept of grou... ... YIN, Pen Jia, Fuzzy complete sets, Fuzzy Sets and Systems, 131, 393-398 (2002). 73. LIU, W-J., Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Se... ... and languages: Theory and Applications, Chapman & Hall / CRC, New York, 2002. 96. MORDESON, John. N., and Malik, Davender S., Fuzzy commutative a... ..., Davender S., Fuzzy commutative algebra, World Sci. Pub. Co., Singapore, 2002. 97. MORSI, Nehad N., Note on "Normal fuzzy subgroups and fuzzy norma...

...A ⊗ x = b is solvable, or uniquely solvable, respectively, can be answered in O(mn) time. 84 THEOREM 1.13.6: Let A ∈ B (m, n), b ∈ B (m). The... ...mization Problems with Fuzzy Relation Constraints, Fuzzy Sets and Systems, 127 (2002) 147-164. [57, 60] 60. Lu, J., An Expert System Based on Fuzzy... ... Solving Fuzzy Relation Equations, Mathematics and Computers in Simulation, 59 (2002) 329-333. [85-86] 289 64. Michalewicz, Z. and Janikow, Z.,... ...y, Department of Mathematics, Indian Institute of Technology, Madras, November 2002. [9, 54] 84. Sanchez, E., Resolution of Composite Fuzzy Relatio... ...ral Network System to Study the Indian Politics, Varahimir J. of Math. Sci., 2 (2002) 44- 53. [167] 101. Vasantha Kandasamy, W.B., and Mary John, M.... ...the Disease Caused by Hazardous Waste From Textile Industries, Ultra Sci., 14 (2002) 248-251. [167] 102. Vasantha Kandasamy, W.B., Neelakantan, N.R...

...A ⊗ x = b is solvable, or uniquely solvable, respectively, can be answered in O(mn) time. 84 THEOREM 1.13.6: Let A ∈ B (m, n), b ∈ B (m). The... ...mization Problems with Fuzzy Relation Constraints, Fuzzy Sets and Systems, 127 (2002) 147-164. [57, 60] 60. Lu, J., An Expert System Based on Fuzzy... ... Solving Fuzzy Relation Equations, Mathematics and Computers in Simulation, 59 (2002) 329-333. [85-86] 289 64. Michalewicz, Z. and Janikow, Z.,... ...y, Department of Mathematics, Indian Institute of Technology, Madras, November 2002. [9, 54] 84. Sanchez, E., Resolution of Composite Fuzzy Relatio... ...ral Network System to Study the Indian Politics, Varahimir J. of Math. Sci., 2 (2002) 44- 53. [167] 101. Vasantha Kandasamy, W.B., and Mary John, M.... ...the Disease Caused by Hazardous Waste From Textile Industries, Ultra Sci., 14 (2002) 248-251. [167] 102. Vasantha Kandasamy, W.B., Neelakantan, N.R...