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...dache 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 have systematically carried out this study. This is ...

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...matics, has the capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or id... ... R. Padilla. Smarandache Algebraic Structures, Bulletin of Pure and Applied Sciences, Delhi, Vol. 17 E, No. 1, 119-121, (1998); http://www.gallup.un... ...is commutative we speak of a field. The characteristic of R is the smallest natural number k with kr = r + … + r (k times) equal to zero for all r ∈... ... Padilla, R. Smarandache Algebraic Structures. Bulletin of Pure and Applied Sciences, Delhi, Vol. 17 E, No. 1, 119-121, 1998. http://www.gallup.unm... ...santha Kandasamy, W.B. On Chain Semirings. Journal of Bangladesh Academy of Sciences, Vol. 16, 257 – 258, 1992. 5. Vasantha Kandasamy, W.B. Semiv... ...By the very definition of S-semidivision ring, the results are true. The natural question would be if a semiring S has S-subsemiring A, then can w... ... semigroup semiring. Z o S(n) is a Smarandache group semiring. Then the natural question would be what is the definition of Smarandache semigroup... ... Padilla, R. Smarandache Algebraic Structures. Bulletin of Pure and Applied Sciences, Delhi, Vol. 17 E, No. 1, 119-121, 1998. http://www.gallup.unm... ...iring and has no zero divisors and has k- semi algebras in them. The next natural question would be: Are all distributive lattices S-semifields? ...

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...ionary because group theory does not make one think of the unthinkable that can naturally occur but Smarandache semigroup explicitly and concretely e... ... 3. R. Padilla. Smarandache Algebraic Structures, Bulletin of Pure and Applied Sciences, Delhi, Vol. 17 E, No. 1, 119-121, (1998); http://www.gallu... ... 2. Padilla Raul, Smarandache algebraic structures, Bull of Pure and applied Sciences, Delhi, Vol. 17E, No. 1, 119-121, 1998. 13 CHAPTER ... ...every a, b ∈ G; a  b = b  a. A group, which is not abelian, is called naturally enough, non-abelian. Another natural characteristic of a g... ...en o(a) | o(G). Proof: With Lagrange's theorem already in hand, it seems most natural to prove the corollary by exhibiting a subgroup of G whose o... ... 1. Padilla Raul, Smarandache algebraic structures, Bull. of Pure and Applied Sciences, Delhi, Vol. 17E, No. 1, 119-121, 1998. 2. W.B.Vasantha K... ... 2. Padilla Raul, Smarandache algebraic structures, Bull of Pure and applied Sciences, Delhi, Vol. 17E, No. 1, 119-121, 1998. 3. W.B.Vasantha K... ...s AxB divides the S-semigroup S 2×2 into disjoint set of order 1, 2 and 4. The natural question would be "Is Smarandache double coset relation on a ... ... 2. Padilla Raul, Smarandache algebraic structures, Bull of Pure and applied Sciences, Delhi, Vol. 17E, No. 1, 119-121, 1998. 3. W.B.Vasantha K...

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...h more efficient. Genetic algorithms (GAs) are built upon the mechanism of the natural evolution of genetics. GAs emulate the biological evolutiona... ... with the binary operations of maximum and minimum, denoted by ⊕ and ⊗. For any natural n > 0, B (n) denotes the set of all n-dimensional column vec... ...is Godel algebra, in which the underlying set is the closed unit interval with natural ordering of real numbers. This max-min algebra will be denot... ...ome known patterns which are often called sample data. This process corresponds naturally to that of solving FRE and implicates a new approach. In t... ... (x) for all x ∈ ℵ. (1) Using some ideas from many-valued logic it is quite natural to extend that relation to a fuzzified, i.e., graded inclusi... ...zy Relation Equations in Chemical Engineering, In Recent Trends in Mathematical Sciences, Proc. of Int. Conf. on Recent Advances in Mathematical Sci...

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...ontroid of T. Counting the number of spanning trees in a graph occurs as a natural problem in many branches of science. Spanning trees were used by... ... Maps as a System Model for Failure Modes and Effects Analysis, Information Sciences, 88 (1996) 177-199. 51. Serre, J.P., Trees, Springer Vertage 19...

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...Number theory is an ancient subject, but we still cannot answer many simplest and most natural questions about the integers. Some old problems have been solved, but more arise. All the research for these ancient or new problems implicated and are still promoting the development of number theory and mathematics....

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...y is an ancient subject, but we still cannot answer many simplest and most natural questions about the integers. Some old problems have been solved,... ...1], the de nition of Smarandache irrational root sieve is: from the set of natural numbers (except and ): -take off all powers of , ; -take off all p... ..., 1974. A NUMBER THEORETIC FUNCTION AND ITS MEAN V ALUE Gao Nan School of Sciences, Department of Mathematics, Xi’an Shiyou University, Xi’an, Shaanx... ... number; Identities; Riemann-zeta function. 1. Introduction For any given natural number and any positive integer , we call a -power complement numb... ... products; Proper divisor products; Asymptotic formula. 1. Introduction A natural number is called a divisor product of if it is the product of all p... ...as. 82 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY Lemma 1. For any natural number , we have the following identities where is the divisor fun... ... A HYBRID NUMBER THEORETIC FUNCTION AND ITS MEAN V ALUE Gao Nan School of Sciences, Department of Mathematics, Xi’an Shiyou University, Xi’an, Shaanx...

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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 the...

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...rom Sachs:[1] In a lecture that Einstein gave to the Prussian Academy of Sciences in 1921, he said the following: "Geometry predicates nothing ab... ... measured by a clock in the rocket. The units used in these equations are “natural coordinates” in which the velocity of light, c, is taken to be on... ... as two different areas of the same manifold. Such space of 0 ≤ g will be a natural generalization of the basic space-time of the General Theory of ... ...g this brief study, we conclude that instant displacements of particles are naturally permitted in the space-time of the General Theory of Relativit... ... "There Is No Speed Barrier In The Universe", <Bulletin of Pure and Applied Sciences>, Delhi, India, Vol. 17D (Physics), No. 1, p. 61, 1998; http:... ...Jason, “Superluminals and the Speed of Light”, Bulletin of Pure and Applied Sciences, Delhi, India, Vol. 20, Series D (Physics), No. 2, 107-110, 200... ...that are constrained in some way by geometric boundary conditions will have natural structures of energy levels. But the whole theory is very inadeq... ...d on geometric shapes, local curvature (the equivalent of a potential), and natural vibrations on the ("new') geometric shapes. These new ideas can ... ... had taken center stage. But then in an address to the Prussian Academy of Sciences in Berlin on 27 Jan 1921, Einstein gave in; and in 1922 these w...

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...acher 1 AMARNATH MURTHY S.E.(E&T) WELL LOGGING SERVICES OIL AND NATURAL GAS CORPORATION LTD CHANDKHEDA AHMEDABAD GUJARAT- 380005 INDIA ... ...licative Sequence ………………… 101 2.11 Decomposition of the Divisors of A Natural Number Into Pairwise Co- Prime Sets ………………… 103 2.12 On the Divi... ... fractions are fractions where the numerator is 1 and the denominator is a natural number. Our first point of interest is in determining all sets of ... ...eciprocal partition of unity for n (SRRPS(n)) is the set of all sets of n natural numbers such that the sum of the reciprocals is 1. More formally, ... ...two steps. Proposition A For every n, there exists a set of n distinct natural numbers, the sum of whose reciprocals is 1. Proof of propositio... ...andache Partition Type and Other Sequences”, Bulletin of Pure and Applied Sciences, Vol. 16 E(No. 2) 1997, page 240. 12. Amarnath Murthy, “On The ...

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...3. R. Padilla. Smarandache Algebraic Structures, Bulletin of Pure and Applied Sciences, Delhi, Vol. 17 E, No. 1, 119-121, (1998); http://www.gallup.... ... able to attribute to this is that it may be due to the fact that there is no natural way by which groupoids can be constructed. This book aim is two... ...ew classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 45 CHAPTER FOUR ... ...ew classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 69 CHAPTER FIVE ... ...ew classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 4. W. B. Vasantha Kandasamy, Smara... ...tate (z 1 ' , z 2 ') which is clear from the following circuit: Now a natural question would be do we have a direct product, which correspond... ...ew classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol. 1, 135-143, (2001). 4. W. B. Vasantha Kandasamy, Smar...

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... The sum of reciprocals of Zt(n) function is asymptotically equal to the natural logarithm of n: = » » + » N 1 n 0.739 b and 0.9743 a where b ln... ... 1 1 1 3 ) ( n n n n n n a n n Zt p and as known the sum of reciprocals of natural numbers diverges. Conjecture 1.1.2 K 8737 . 0 ) ( 1 » » = a ... ...tillo, Smarandache continued fractions, Bulletin of pure and applied Sciences, Vol. 17E, No.1, 1998, p. 149-151 [6] F. Russo, Some results about... ...tillo, Smarandache continued fractions, Bulletin of pure and applied Sciences, Vol. 17E, No.1, 1998, p. 149-151 [4] F. Russo, Some results about... ... sequences. The greatest discovery of all humankind may well have been the natural numbers. For many past aeons, philosophers and mathematicians have ... ...y past aeons, philosophers and mathematicians have studied the sequence of natural numbers, uncovering startling and mystifying truths. Mathematics, i... ...ering startling and mystifying truths. Mathematics, itself, began with the natural numbers and the study of their relationships. The demand for soluti... ...tillo, Smarandache continued fractions, Bulletin of pure and applied Sciences, Vol. 17E, No.1, 1998, p. 149-151 [3] F. Russo, Some results about... ....edu/~smarandache/russo3.htm [3] M. Bencze, Bulletin of Pure and Applied Sciences. Vol. 17 E(No. 1) 1998; p. 55-62 [4] http://www.gallup...

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...ly: object literature! 'The flying of a bird', for example, represents a "natural poem", that is not necessary to write down, being more palpable and... ... National Science Foundation grant proposal for Interdisciplinary Logical Sciences. 1.2. Neutrosophy, a New Branch of Philosophy A) Etymology: N... ... of it. Husserl's phenomenological epoch‚ is commuting not only from natural beliefs to an intellectual reflexion, but backwards as well. It i... ...ocre level of their times". Hence, any progress in arts, literature, and sciences lead to the society's decadence. "Man is born free; and anyw... ...omnium contra omnes (Lat.) (the war of everybody versus everybody), as a "natural status" (Hobbes from Plaut). And Spinoza oppositely with homo homi... ...n stay one near another, philosophers' not" (Schopenhauer). The absurd is natural, so the un-natural. 57 [See the lack of sense of the sense.] ... ...t you have you get bored of it. Man must live in accordance with the natural world around him (Pueblo Indian philosophy). While genius should ... ...TROSOPHIC ANTHROPOLOGY. And so on: neutrosology of the values, histories, sciences, arts. Philosophy reflects the existent from the non-existen... ...7 (which is 100% unstable) He gives Orientation Tables for Physical Sciences and Mechanics (Rugina 1989), for the Theory of Probability, for w...

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...knowledge representation, Preprint 232, Computer Center of the USSR Academy of Sciences, Novosibirsk, 1980 (in Russian). [32] Perez, Minh, Neutros... ...elief function in Dempster Shafer theory”, Foundations of Computing and Decison Sciences, Vol. 20, no. 4, pp. 287 306, 1995. [31] Klopotek M.A., “Iden... ...lief Structure in Dempster Shafer Theory”, Foundations of Computing and Decison Sciences, Vol. 21, no. 1, pp. 35 54, 1996. [32] Kyburg H.E., Jr., “Bay... ...c variable and its Application to Approximate Reasoning I,II,III” Informa tion Sciences, Vol. 8, Vol. 9, 1975. [73] Zadeh L.A.,“A Theory of Approxima... ...neng’s knowledge and outstanding views. When we see pennant and wind we will naturally believe we are right in our consciousness, however it is su... ...o Xiang) begins with: “Dao, daoable, but not the normal dao.” Referring to the natural law, we can say it is dao, but it doesn’t mean what we say. W... ...ustrated in Daodejing (Wang Bi, Guo Xiang) that whenever we capture dao as the natural law, universal method or logic, etc., what we capture can neve... ...y comprehensive way and made a thoroughgoing change in the concepts of today’s natural sciences. Part of the summary is shown bellow: Universe is ma... ...have the history of growth and death; on the other hand, the occurrence of the natural phenomenum of any kind is constrained by the internal inheren...

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...e of Systems Science Academy of Mathematics and Systems Chinese Academy of Sciences Beijing 100080, P.R.China maolinfan@163.com AUTOMORPHISM GROUPS OF... ...CHE GEOMETRIES Partially post-doctoral research for the Chinese Academy of Sciences American Research Press Rehoboth, NM 2005 This book can be ordere... ... F.Tian and X.D.Hu, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100080, P.R.China. Y.P.Liu and R.X.Hao, Department of App... ... this book is my post-doctor report in the Chinese Academy Preface iii of Sciences in 2005. Many colleagues and friends of mine have given me enthusi... ... for a Klein surface without boundary by using NEC groups. Since maps is a natural model for the Klein surfaces, an even more efficient approach is, per... ...pping by the definition, and vice via. ♮ A covering mapping π from M to M naturally induces a mapping π ∗ by the following condition: ∀x∈X α,β ,g∈Aut... ...has its own right,also its contribution to other branch of mathematics and sciences. Among those map ap- plication to other branch mathematics, one ty...

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...sult is expected to have many applications, as well in physics as in other sciences, including biology [4]. It has been used, in particular, to sugges... ...ce (g=0) as two different areas of the same manifold. A space of g 0 is a natural general- ization of the basic space-time of the General Theory of R... ...nclude from this brief study that instant displace- ments of particles are naturally permitted in the space-time of the General Theory of Relativity. ... ... 3.1 The Machian Principle and Eddington-Dirac Large Numbers Coincidence A natural action associated with the invariant interval in Phase-Space given ... ...ing on a fundamental particle given by(mg) bound =m P (c 2 /L P )=m 2 P in natural units of =c=1. The Eddington-Dirac large numbers coincidence (and ... ...egrees of freedom than a relativistic top in ordinary spacetimes [46] and, naturally, to study the modifed Nambu- Poisson dynamics of p-branes [49] as... ... Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region, 142290, Russia E-mail: shnoll@iteb.ru †... ...cow, 119992, Russia ‡ Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, Russia As shown in our previous experiments fne structure... ...F. There is no speed barrier in the Universe. Bulletin of Pure and Applied Sciences, Delhi, India, v. 17D (Physics), No. 1, 1998, 61. 6. Rincon P. Tel...

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...t both neoclassical economics and classical thermal physics seek to model natural systems in terms of solutions to constrained optimization problems... ... which he or she would not want his or her portfolio value to fall then a natural choice for this floor would be the risk-free rate. While th... ...Bhattacharya and Smarandache, 2003). The theorem of consistent preference naturally implies that an protective put portfolio insurance strategy woul... ...from the previous generation will be reproduced based on the principal of natural selection. Following the reproduction function, 2(x – 1) number of... ... number of chromosomes included in the mating pool through the process of natural selection. Every g th chromosome in the mating pool is crossed wi... ...namical Market Prediction Utility, Journal of Ultra Scientist of Physical Sciences 15, 2003: 255-260. 129 Black, F. and R. Jones. Simplifying po...

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...eneral prescription of Eddington. Clearly, the correct form ofC(D(r)) must naturally yield the Droste/Weyl/(Hilbert) solution, as well as the true Sch... ...nt to his setting up of the meaning of the theory — all the postulates are naturally contained in the geometry of a four-dimensional pseudo- Riemannia... ...feld only in transitions of the electrons between their stationary orbits. Naturally, when an atom is in a stable state, all its elec- trons are locat... ...x for any V n have an ordinary geometric meaning as stationary curvatures. Naturally, the Riemannian curvatureV n in a two- dimensional direction is d... ...arallel lines a distance d apart. The problem was frst posed by the French naturalist Buffon in 1733. For l<d the probability is P = 1 2π Z 2π 0 dθ l|... ...., Bencze M. Smarandache paradoxist geometry. Bulletin of Pure and Applied Sciences, 1998, v. 17E, No. 1, 123–124. See also Smarandache Notions, book ... .... Cahill and Christopher M. Klinger School of Chemistry, Physics and Earth Sciences, Flinders University, GPO Box 2100, Adelaide 5001, Australia E-mai...

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...elihood. On the other hand, involuntary migration might take place due to natural disasters or war and the consequent persecution or in some of the ... ...they have concern or any form of binding with the family members under no natural circumstances they will stoop to betray them. For the hard money t... ...logical problems etc. Neural networks helps to solve these problems with natural mechanisms of generalizations. To over-simplify, suppose we repre... ...ond correctly to stimuli when neural networks operate similar 98 inputs naturally produce similar outputs. Most real world perceptual problems hav... ...h they are birds “Badness” shows up in a number of ways. Neural networks naturally develop this kind of category structure. The problems that neura... ...986) 65-75. 59. Kosko, B., “Fuzzy Entropy and Conditioning,” Information Sciences, 40 (1986) 165-174. 60. Kosko, B., “Hidden Patterns in Combined a... ... Spread of HIV-1 and HIV-2 Infections,” Proc. of the National Academy of Sciences USA, 91 (1994) 2407- 2414. 77. Raka, Sharon, Indian Women Workers...

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...Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also pub...

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...triplo minoris eCt yuadruplo vel. f&up10 minus, POlW 6 BE-~FEOSOP~-~I& NATURALTS DEFINITX- QNEb. Pcxrs attrafbnes 4% impul~us mdein fenCr1 acce... ..., & nimis lonpm e&t omnia hut fpeltantia demonitrare. 20 N-HLOSOPHIE NATURALIs A XIDM .4~.4, fe, Proinde centrum illud a motuum zqualibus mutat... ...altem perfefle efaitica, cujufmodi nulla reperiuntur in com- pC&tkJllibus naturalibus ; addo quod experimenta jam defcripta fuccedunt in corporibus ... ...orem ut R T 4 x T Tctib, sld STq x PVmb. II . . .jd JO PHI:LOSOPHIrE NATURALIS id efi, ut SYx RTq ad ST cub. x T V cub. l P I- C?Lb. 9 five ... ...initam p e ipfi S T T pamlklam, & in ea INI,2 capiendo 81 p~jLoSOPMI& NATURALIS capiendo fernper p e szqualem 5” r ; & agenda re&as B e, Cr con:... ...k of the MarquiTs da I’ HoJjita[, Honorary F’eilow of the Academy Royal d Sciences. 4O. IX. Hiiloria CceleItis Britannica, tribus Voluminibus conten... ...dbiflet of the ldte I;i~ncb liing, atId Mcmlw of the Acadcn~y of Arts and Sciences ; neccfi:lry for young ~.~‘Jeople who woul~l apply to that noble ...

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...College, EC, USA Florentin Smarandache Department of Mathematics and Sciences The University of New Mexico at Gallup, USA ... ...er "primordial forces" (C&W) (meaning primarily gravitation) normally and naturally propagate through empty space but it clearly does not set the sp... ..., There is no Speed Barrier in the Universe, Bulletin of Pure and Applied Sciences, Vol.17D (Physics), No. 1, p. 61, Jan-Jun, 1998. [4] Read all ... ...aw of nature and on p.76: The systems to which the physicist must refer natural phenomena...owe their effectiveness to... the objects present in t... ...ndition. But finally the ramjet bites again, and SS Alpha is back in her natural environment - sailing along at a constant ¼G acceleration through ... ...u space on a beeline to the center of the galaxy, will be shown to spiral naturally in accordance with the galactic Coriolis effect when mapped onto... ...There Is No Speed Barrier In The Universe", <Bulletin of Pure and Applied Sciences>, Vol. 17D (Physics), No. 1, p. 61, 1998; http://www.gallup.unm.e... ...ason, “Superluminals and the Speed of Light”, Bulletin of Pure and Applied Sciences, Delhi, India, Vol. 20, Series D (Physics), No. 2, 107-110, 2001,...

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................................53 5. A Triple Inequality with Series and Improper Integrals, by Florentin Smarandache, in Bulletin of Pure and Applied Sciences, Vol. 25E, No. 1, 215-217, 2006.........54 6. Immediate Calculation of Some Poisson Type Integrals Using SuperMathematics Circular Ex-Centric Functions, by Florentin Smarandache & Mircea Eugen..........................

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................................53 5. A Triple Inequality with Series and Improper Integrals, by Florentin Smarandache, in Bulletin of Pure and Applied Sciences, Vol. 25E, No. 1, 215-217, 2006.........54 6. Immediate Calculation of Some Poisson Type Integrals Using SuperMathematics Circular Ex-Centric Functions, by Florentin Smarandache & Mircea Eugen..........................

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...most significant discoveries in chemistry and physics are presented chronologically to illustrate their contributions to the creation of the chemical sciences during the last 21 centuries....

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...ine pleasure and challenge for me to try to express the full extent of my emotions and reasons for writing this book on the STORY OF THE ATOM AND THE SCIENCES, with special reference to the CHEMICAL SCIENCES. In one sentence, I can distill the essence of the purpose for this study by simply stating that it has been a labor of love that transcended the written word becaus...

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...r 50 Proponent of the heliocentric theory of our solar system with the Sun and not the Earth at its center. CHARLES R. DARWIN (1809-1882) English Naturalist 51 Proponent of the theory of evolution. The Origin of Species by Means of Natural Selection. THE SECOND GENERATION IN SCIENTIFIC DEVELOPMENT 53 IN THE PHYSICAL, CHEMICAL AND MATHEMATICAL SCIENCES 53 GALILE...