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... E-mail: steve_gd@hotmail.com Keywords: G-dimensional theory, mathematical physics, relativity, Smarandache hypothesis, gravity, quant... ...usion principle, we see a fourth spatial dimension in GDT has evolved from a mathematical convenience to a logical necessity. Let Universal Space be ... ...(7) mtanθ (17) mc 2 secθ (23) cr 2 /R 2 (25) 3.11 Three Logical Proofs of GDT Relativity Three recently developed statements define int... ...DT Relativity Three recently developed statements define intrinsic logical proofs in GD relativity. [50] 3.11.A Relativistic Volume Causality... ...is non-zero in another." [44] Again, GRT fails to meet the criteria of the mathematically non-trivial "strict VSE". In Section four the local regi... ...onditions for strict Very Strong Equivalence. In this context, GDT has clear mathematical superiority and raises a comparative paradox in SRT/GRT. 3... ...ollowing corollary theorems and hypotheses, some of which are logically and mathematically proven. From the intrusion principle, spatial curvature r...

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...andache loops in general do not form a modular lattice. Almost all the proofs of the theorem are given as exercise to the reader so that the read... ...rties, and state the famous classical theorems of Lagrange and Sylow. The proofs are left as exercises for the reader. DEFINITION 1.2.1: A non-em... ...of the from 1 + kp. Left for the reader to prove. For more about these proofs or definitions kindly refer I.N.Herstein [27] or S.Lang [34]. DE... ...perties are given for the deeper understanding of these concepts. Several proofs are left for the reader to prove. The concept of unique product gro... ... Conf., 33-37 (1995). 76. Vasantha.W.B, Loop algebras of Moufang loops, Mathematical Forum, Vol. XI, 21-25 (1997). 77. Vasantha.W.B, On Quasi l...

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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 published two further editions, in 1713 and 172...

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...te Johanne Fjamfle- die, A. R. in Obfervatorio Regio Grenovicenfi, 1725. Mathematical Elements of Natural Philofophy, confirm’d by Experiments, or a...

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...ole that logic plays in our modern world. One cannot understand the foundations of mathematics while lacking knowledge of the basics of logic and how proofs are constructed. Two of the first classes I took as a graduate student in mathematics were in the foundations of mathematics, and hardly a day goes by where I do not use some topic from those courses. Logic is also a f...

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...ons of mathematics while lacking knowledge of the basics of logic and how proofs are constructed. Two of the first classes I took as a graduate stud... ...fs in S, called the rules of inference. These rules are used to construct proofs, where the statements are: Given that this set of wffs true, w... ...ude that additional expressions are also true. Such inferences are called proofs in the theory. Example: The following is a proof in L. Hypo... ...om the notion of a proof in other areas of mathematics. In general formal proofs are more precise and sequential, with much less latitude in the tec... ...hat the expression holds for every value for the variable men. The formal mathematical term is “for all” and it is known as the universal quantifier... ...ion. Therefore, the expression 21 (Ex) x > z is mathematically equivalent. Once the values of the elements in the unive... ...three-valued logic is an extension of classical logic. In most cases, new mathematical structures are defined so that they are consistent with those... ...= min{ f 1 , f 2 , . . ., f k , f k+1 } Therefore, by the principle of mathematical induction, the formula is true for all n. Corollary: If ... ... A \/ B being an absolute contradiction. Therefore, by the principle of mathematical induction, we have reached the desired conclusion. The t...

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...This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in ...

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...f a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man w... ...ious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also l... ...f a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man w... ...ious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also l... ..., the proposition is also true for r+1. Therefore, by the principle of mathematical induction, the expression is true for all n. Note: If a 1 ,... ... matrix [S j,k ] ' nxn . Readers are encouraged to construct alternate proofs by using a combinatorial approach or other techniques. Section 9...

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...As an alternative to the existing logics we propose the Neutrosophic Logic to represent a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undefined, unknown, incompleteness, inconsistency, redundancy, contradiction. It is a non-classical logic. Eksioglu (1999) explains some of them: “Imprecis...

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...veloped the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Infor... ...native to the existing logics we propose the Neutrosophic Logic to represent a mathematical model of uncertainty, vagueness, ambiguity, imprecision, ... ... P., Upper and Lower Probabilities Induced by a Multivalued Mapping, Annals of Mathematical Statistics, 38, 325-339, 1967. [8] Dezert, J., Autonomo... ...alysis, Princeton University Press, Princeton, NJ, 1996. [37] Shafer, Glenn, A Mathematical Theory of Evidence, Princeton University Press, NJ, 1976... ...Western Section Meeting, Santa Barbara, CA, USA, Meeting # 951 of the American Mathematical Society, March 11-12, 2000, http://www.ams.org/amsmtgs/2... ...to be, A multidimensional logic approach” Carlos Gershenson [2] has generalized proofs: Everything is and isn't at a certain degree. (i.e., there is... ..., A multidimensional logic approach” Carlos Gershenson [2] has generalized the proofs: Everything is and isn't at a certain degree. (i.e., there is...

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.... Except for the introduction of smarandacheian notions by Prof. Florentin Smarandache such types of studies would have been completely absent in the mathematical world. Thus, Smarandache groupoids, which are groupoids with a proper subset, which is a semigroup, has several interesting properties, which are defined and studied in this book in a sequential way. This book as...

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...in Smarandache such types of studies would have been completely absent in the mathematical world. Thus, Smarandache groupoids, which are groupoids... ...arandache groupoid notions and give central importance only to them so several proofs in groupoids in chapter 2 and chapter 3 are left for reader to ... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 45 CHA... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 69 CHA... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 4. W. B. Vasantha Kan... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol. 1, 135-143, (2001). 4. W. B. Vasantha Ka...

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...c. Professors brought their handwritten manuscripts directly to the compositors, who did their best to decipher them. The compositors then passed the proofs to the “brainery”—the proofreaders who corrected typographical errors and edited for stylistic inconsistencies. To bring a common set of rules to the process, the staff of the composing room drew up a style sheet, whic...

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...s-3: 12. 254. In formulae, and elsewhere, put a thin space on each side of mathematical signs. Between letters forming products, and before superio... ...uscript. Copy for indices should be compiled from the special set of page- proofs furnished for this purpose, and promptly delivered to the printer... ... often caused by postponing these details till the last minute. READING OF PROOFS- Read and return your proofs promptly. In marking proof-sheets, ... ...o appear less so to the compositor. Be careful to answer all queries in the proofs. Delays and errors often result from not attending to them. Reme... ...ader may refer to it, should any question arise; and each successive set of proofs returned should be accom- panied by the previous marked set. This... ...s there. Contents of journals should be made up at the time the first page-proofs are read. Put your initial at the top of every galley you read o... ...be set in roman, 52; use of capitals in, 38. "Master," compounds of, 178. Mathematical signs, spacing of, 254. Measures, metric, how to designate,...

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...IONS 24 -- THE FOUNDATIONS OF OUR VALUES -- 24 SELF-CENTEREDASSUMPTIONS 32 GODBASEDASSUMPTIONS 41 CONCEPTS OF GOD 45 3 WHAT IS GOD LIKE? 46 THEISM 49 PROOFS FOR THE EXISTENCE OF GOD 53 IS AMERICA A CHRISTIAN NATION? 57 WHO GETS TO GO TO HEAVEN? 63 DEISM 65 PANTHEISM AND PANENTHEISM 67 AGNOSTICISM 68 ATHEISM 71 SATANISM 96 AN IDEA OF A BETTER SOCIETY AS A BASIC ASSUMPTION 9...

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... 9 —―But doesn‘t Russell‘s thinking refute his theory? Did his atheistic beliefs and his mathem... ...rom the input of his environment? It would seem that his materialism is quite counter to the religious beliefs of his society. And how did his mathem...

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...wrongly understood!), as well as - as much as I could understand- some of his mathematical, philosophical, enigmatical etc. creations. I recognize th... ... “our daily paradoxes” Smarandache has not certainly referred to the logical, mathematical or linguistic meaning of the word/notion “paradox”, but t... ...nsurgent from Balcesti-Gallup, his protest could not be expressed by means of mathematical formulae or notions and he had to appeal to literature. At... ...a rectitude that we can find only at the strong natures and brave people The mathematical qualities appear for both at secondary school already, man... ...nifesting themselves through contributions with problems and solutions in The mathematical newspaper. The literary debut of Ion Barbu was at 22 year... ... combinatory drama, paradoxist quatrain, paradoxist distich aso.) are as many proofs about Smarandache’s tendencies to break the (pretty vast) frame...

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...idged version of my article, and offered to send readers free copies of the galley proofs I had received from the AIChE Journal. Twenty-seven reade... ... ratios such as R, h, and E is that they combine the important variables. This is mathematically undesirable because non- linear problems are genera... ...I represents the numerical value of electric current in arbitrary dimensions. • Mathematical operations can be performed only on numbers pure num... ... operations can be performed only on numbers pure numbers, and numbers of things. Mathematical operations cannot be performed on things per se. Fo... ...e number of airplanes to determine the average number of people per airplane. • Mathematical operations can not be performed on dimensions becaus... ...to answer the question How many times does a second go into a foot? • Because mathematical operations can be performed only on num- bers, and be...

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...since f( )<f( 0 ) = l a m a . Bibliography 1. M. Bencze, OQ.591, Octogon Mathematical Magazine, vol.9, no.1, April 2001, p.670. 2. J. S andor, Geom... ... is true in any triangleABC, with b A 90 , (see [3], where ten distinct proofs are given, see also [4], pp.47-48). First we extend (2) in the follo... ...ding to the hypothenuse. Though, in their paper [1] they give two distinct proofs of (1), the given proofs are not illuminating, and the geometrical m... ...e has BK < BC, so we get a geometrical proof of 23 b +c<a p 2, many other proofs are given in [3]. But we can obtain a geometric proof of (4), too. I... ... c and D. Milo sevi c, Two inequalities for any right triangle, Octogon Mathematical Magazine, vol.9, no.1, 402-406. 2. J. S andor, On Emmerich’s ... .... Dacia, 1988. 3. M. Dinc a, About Erd os-Mordell’s inequality , Octogon Mathematical Magazine, vol.8, no.2, 2000, 481-482. 4. M. Dinc a, M. Bencze... ...r these three inequalities. The aim of this paper is to obtain certain new proofs, new applications and inequalities of type (1)-(3). 2. Inequality (1... ... 1 x 2 + 1 y 2 = 1 z 2 in integers, these Selected papers. 5. Ch.W. Trigg, Mathematical Quickies, New York, 1968. 58 2 On the equation 1 x 2 + 1 y 2 ... ...i surprize ^ n matematic a , Ed. Enc. Rom^ an a, 1975; English edition: Mathematical Quickies, New York, 1968. 4. B. Margit, Problem E:11248, Mat. ...

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...tary system, the atomic model or molecular models for substances, various mathematical models, etc. are examples of speculative models that could and... ...parated from its existential character. For this reason, according to the mathematical and philosophical principles, any existential model presuppose... ... the same length. (The demonstration can be performed according to strict mathematical rules). The natural conclusion is that in the case of intensi... ...ative modeling method is further confirmed by the validity offered by the mathematical modeling, such as the non-Euclidean geometries due to I.Bolyai... ...g. The accuracy of modeling (inner perfection) is justified logically and mathematically, the intrinsic value of a model being so much the more obvi... ...explain the lacking day detected electronically. There are numerous other proofs of the divine almightiness noted in the Bible, but such a scientifi... ...nevertheless, the credible reports, with witnesses and material (logical) proofs must be taken into consideration. A conclusive evidence for the ver... ...evertheless, denying the extraterrestrial Being due to lack of ″objective proofs″ is not an epistemological method for solving the problem, it can o... ...e problem, it can only postpone it. Science is bound to produce objective proofs, either in favor or against any advanced hypothesis, in this case, ...

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... or end and what, precisely, does it include? A fractal curve (boundary) is an apt mathematical treatment of this question. Indeterminacy can be de... ... are still widely deployed in many branches of the Humanities, the encroachment of mathematical modeling and statistics notwithstanding. Yet, mathe... ...quations account for a wealth of experiences and encapsulate numerous observations. Mathematical concepts and symbols do not correspond to material ... ...or do they alter reality or affect it in any way, shape, or form. One cannot map a mathematical structure or construct or number or concept into th... ...certainty. 5. Despite its aforementioned transigence, mathematics is invariant. A mathematical advance, regardless of how arcane or revolutionary,... ...urthermore, statistical significance and correlation should not be misconstrued as proofs of cause and effect. Consequently, there is no agreement...

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...or the results for this would make our book unwieldy and enormous in size. For proofs, one can refer the papers that have been cited in the bibliog... ...their properties It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from X to Y, binary re... ...ominated criterion vectors is called the non-dominated set. In the absence of a mathematical specification of the decision maker’s utility function,... ... is a critical element, then p ij ≥ 0. It is left for the reader to supply proofs for the following lemmas: Lemmas 1.8.3: For column, j if t... ... S (A, b): = { x ∈ B (n); A ⊗x = b}, i.e. S(A, b) : = S* (A, b) ∩ B (n). The proofs of the following theorems are left as an exercise for the rea... ...ere only the basic notation and formulation of theorems, without repeating any proofs. In the notation M = {1, 2, …, m}, N = {1, 2, …, n}, P = {1... ...ity to determine outputs for a given set of inputs without using conventional, mathematical models. FLC follows the general strategy of control wor... ... with sup-t composition nor adjoint equation. Therefore, simple control is not “mathematically correct”. However, solutions of fuzzy relational equa... ...of the fluid under study their by changing the flow rates while analyzing as a mathematical model. So in the study of flow rates in chemical plants...

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...or the results for this would make our book unwieldy and enormous in size. For proofs, one can refer the papers that have been cited in the bibliog... ...their properties It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from X to Y, binary re... ...ominated criterion vectors is called the non-dominated set. In the absence of a mathematical specification of the decision maker’s utility function,... ... is a critical element, then p ij ≥ 0. It is left for the reader to supply proofs for the following lemmas: Lemmas 1.8.3: For column, j if t... ... S (A, b): = { x ∈ B (n); A ⊗x = b}, i.e. S(A, b) : = S* (A, b) ∩ B (n). The proofs of the following theorems are left as an exercise for the rea... ...ere only the basic notation and formulation of theorems, without repeating any proofs. In the notation M = {1, 2, …, m}, N = {1, 2, …, n}, P = {1... ...ity to determine outputs for a given set of inputs without using conventional, mathematical models. FLC follows the general strategy of control wor... ... with sup-t composition nor adjoint equation. Therefore, simple control is not “mathematically correct”. However, solutions of fuzzy relational equa... ...of the fluid under study their by changing the flow rates while analyzing as a mathematical model. So in the study of flow rates in chemical plants...

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...roups. It is pertinent to mention that Smarandache notions on all algebraic and mathematical structures are interesting to the world of mathematicia... ...of us, for we have been considering mappings from the very earliest days of our mathematical training. DEFINITION: If S and T are nonempty sets, ... ...groups are the only algebraic structures with a single binary operation that is mathematically so perfect that an introduction of a richer structure... ... In this chapter, we just recall some theorems in group theory with proofs. The main purpose for giving the proof is that when we try to... ...ylow subgroup. Thus, the classical three parts of theorems due to Sylow with proofs will be given in this chapter. It is interesting to note that... ...oofs will be given in this chapter. It is interesting to note that out of three proofs were given to the first Sylow's theorem, which clearly enable...

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... We give only the basis and in several places we leave the reader to supply proofs for our main motivation is introduce several new concepts and the... ...16. JACOBSON, N., Structure of Rings, Colloquium Publications, 37, American Mathematical Society, 1956. 17. JOHNSON, T., New spectral theorem for v... ...bility). He contributed with proposed problems and solutions to the Student Mathematical Competitions. His latest interest is in information fusion ...

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...ult). Supplementary Reading 1. Birkhoff, G. Lattice Theory. American Mathematical Society, Providence, R.I., 1967. 2. Birkhoff, G. and Ba... ...swer? Supplementary Reading 1. Birkhoff, G. Lattice Theory. American Mathematical Society, Providence, R.I., 1967. 2. Birkhoff, G. and Ba... ... S-subsemivector space of V. 108 Proof: Left for the reader to do the proofs as an exercise. THEOREM 6.3.4: Let V = S 1 × … × S n where S... ...ve one among researchers. Except for Smarandache notions such rich type of mathematical analysis would be completely absent in the realm of mathemat...

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...ed two other well-known theorems concerning the Smarandache function. For proofs of these theorems, see[2]. Theorem 19: Ifp is a prime, then S(p) = ... ...yor may not have a solution, depending on how the terms are defined. In a mathematical sense, the word assertion is used to refer to a statement that... ...t consequence of Godel's Theorem that there are questions for which it is mathematically impossible to form an answer Therefore, the answer to the f...