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...ral Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, ... ...n web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: s... ...e is infected, she is chased out of the home. In many cases she has no one to support her. Being in her early forties she has no parents or her rela... ...ffective prevention programs for these women, without putting in place social supports, such as means of independent living and health care. 10 ... ...s skilled training of both men and women for mutual negotiation. Unless these support systems are in place, women in relationship with HIV- infected... ...e deserted by their family members. Still we cannot say that there are strong supportive groups for these rural uneducated poor HIV/AIDS infected wo... ...hospital. None of them had any help from any of the women’s organizations or support groups. The advent of the disease 12 about which they were t... ... varies from women to women we are forced to analyze the problem using fuzzy mathematics. This chapter has two sections. In the first section we in... ...roduction of learning as encoding. We avoid all types of high and complicated mathematics. As our study is a psychological one, we give more import...

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...ator. Religions, however, are combinations of rituals designed by hu- mans to support them in their earthly existence. Specifically, it is the “opiu... ...a person actually acquires knowledge. You might say that it involves a lot of mathematics and dry rules. It has nothing to do with psychology or oth... ... the Creator, whether it’s the Psalms, Agadot, Tanya, the literal Torah, even mathematics and physics, our daily life routine or even Buddhism. Ever... ...r judgment. Why would we do this? Because the goal necessitates it. The facts support our opinion, and we accept our friend’s opinion against our co... ...f charms in his writings because they provide nothing more than psychological support. If I see that a certain object will save a person who comes to... ...! Man would like to discover something in the universe because he needs it as support. How- ever, besides us, there is only the Creator! Q: I am a UF... ...he full sense of the word: physically – to become a part of the body, to be a supporting organ – and spiritually – to try to think, experience, and ...

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...Path of Pain.” When we, the Jews, do what we must, the entire world will support and follow us. Everyone will see the benefit we bring to mankind, ... ...piritual work, and they are both important. Today, Kabbalists need public support in order to correct the world. The support must come, at least par... ...e. Kabbalists are in direct contact with spirituality, but they are also supported in their work by people. They raise 97 of 263 Interview... ... who wanted to attain the Creator from the collective and render certain support to the general public. A Kabbalist is a person who studies the sys... ...ly, Kabbalists developed a special methodology called Kabbalah. It is the mathematics of the human soul. It must be studied only at the last stage o...

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...ining of the sales personnel, special offers, dealerships, telemarketing and sales support). Attach a flow chart of the purchasing process from the... ... receipt of orders, invoicing, shipping. • Customer after-sales service (hotline, support, maintenance, complaints, upgrades, etc.). • Customer ... ...ers); • Raw materials: sources, cost and quality; • Relations with suppliers and support industries; • Import restrictions or licensing (where a... ...ntal information and may have some practical value." These hopeful inferences are supported by the work of other scholars, such as Paul Weller of ... ...rd. That volatility increases in the transition from bull to bear markets seems to support this pet theory. But how to account for surging volatili... ...LE - p16} 7. VARIANCE and STANDARD DEVIATION as measures of the difference between mathematics and reality. They are the measures of the frustrat...

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...rough 19 books to find the right one for a 1i\"8n incoming message; and look at the logistics to support such a ~"It.m: it tums out that the number of... ...ons center where there are guards and controls on access to prevent theft of equipment and their supporting materials. this KY-l cabinet was in fact a... ...e1, I think. Pint of aU, they. zeplelllDt a Wp invest­ ment; secondly, the usen have developed a supporting 10listic base for the syat8ID1, have train... ... that these protected wire· lines may~ needed. . Some"special rules govem communicationa used to support SlGINT operationa., IDd these rules bave been... ...ing about money. for example; and the best intentions are mNDinglees without budretary actiOD to support . ~"em. And it At no time frame Cor accomplis... ...ed by the real difficulty in differentiating various kinda of mathematical research from "crypto-mathematics" - notably in the burgeoning mathematical...

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...writing and was celebrated by critics. This book touches upon several themes: mathematics, science, computers, quantum physics, and artificial intel... ... through that period, and I wrote and published my first three books with his support in 1983. “After my teacher passed away, I began to develop and ... .... The first results left the scientists dumb- founded. Einstein, for instance, supported the view that the world was a lifeless machine. He thought t... ...e a vessel that leads us to knowing the point in the heart. The most abstruse mathematics in quantum mechanics can be a tool that enables us to reco... ...ou a short, easy introduction to quantum theory. I will not use sophisticated mathematics and I will only present terms you have probably heard befo... ...ence is a kind of extreme distillation of the me- chanical outlook down to the mathematics and the logic of me- chanical interactions. The scientific ... ...ce. The structure of this phenomenon is prearranged in perfect mechanism, its mathematics are known with utter preci- sion, and today this phenomenon... ...decided to discard the idea of the living entity. Professors that insisted on supporting the old concept were discharged from universities. Thus, mo...

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...amaica. In a sudden conversion, he published the Jamaica Letter (1815) in which he supported a model of government akin to the British parliamentar... ...States exports in 1860. Southerners, dependent on industrial imports as they were, supported free trade. Northerners were vehement trade protection... ... to blacks. For instance, a white mob lynched, in September 1862, forty four Union supporters in Gainesville, Texas. Similar events took place in S... .... The Elizabethan corset - as opposed to the Victorian one - was comfortable and supported the back. It evolved in Tudor times from the kirtle, s... ...f Violent Crime (NCAVC) Its mission is to combine investigative and operational support functions, research, and training in order to provide as... ...NEP.htm l Newton, Isaac Isaac Newton (1642-1727), the father of modern physics and mathematics, was an avowed and dedicated alchemist, mystic, theo... ...intellectuals, the lingua franca, akin to English today. Nor was he acquainted with mathematics until he was 30. Leonardo was born out of wedloc...

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...F. D. da Motta in a recent paper, [33] proposes the Smarandache Hypothesis, supported by recent theoretical and experimental findings, as a general ... ...vels? 26 The finding of cubic symmetry in the noble element progression supports the primary hypothesis of charge projection. Thus, a line drawn... ...GDT: B ≈ A → S (1.2) It is proposed that a sufficient body of supporting evidence has been presented in Sections 2-5 of this paper to ... ...presented at Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, December 21 - 24, 2000 University of Cr... ...presented at Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, December 21 - 24, 2000, University of C...

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...; - intelligible unintelligible language; - unsolved and open problems of mathematics like very nise poems of the spirit – we must scientificize th... ...] Weisstein, Eric W., Smarandache Paradox, în “CRC Concise Encyclopedia of Mathematics”, RC Press, Boca Raton, Florida, p. 1661, 1998. 20 iunie 20... ...ish, of curiosity, of incitement. What do you think about the paradoxes in mathematics and physics called <exact>. The paradoxist texts have a mean... ...e Antologia Paradoxist ă II, în revista (aten ţie la titlu!): “Humanistics Mathematics Network” (California). Dar şi invers: lirica în mate (nu stri... ...hbacher, redactor principal la revista american ă “Journal of Recreational Mathematics”), Italia (Dr. Felice Russo), Bulgaria (Krassimir Atanassov, ... ...s, Austin (1999) şi Los Alamos National Laboratory (Educational Networking Support Program, Gallup (1999). Activitate: A participat la olimpiade ş...

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...zzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field founded by Florentine Smarandache, is alike to that of Fuzzy Theory: both the fields imperatively questioned the dogmas of classical mathematics....

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...HE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology Madras Chennai – 600 036, Indi... ...tarted because of my perception at that time, that the world of classical mathematics – was a little too much of a black and white world, that the p... ...born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field founded by Floren... ...y Theory: both the fields imperatively questioned the dogmas of classical mathematics. Despite the fact that Fuzzy Algebra has been studied for ov... ... G generated by ξ in A is a fuzzy direct sum of fuzzy subgroup of G whose support is cycle i.e. for ξ = () { } I i ), x ( A t 0 x i i t i i ∈ ... ... finite valued if Im (X) is finite. We let X ∗ = {x ∈ R  X (x) > 0} the support of X. For all t ∈ [0, 1] we let X t = {x ∈ R | X(x) ≥ t} a level ... ...G is called the Smarandache fuzzy point (S-fuzzy point) with Smarandache support (S-support) x and value t is denoted by x t . It is very importa... ...bsemivector space, 342 S-subsystem, 434 S-sup property, 217, 229, 278 S-support, 229 S-syntactic near-ring, 414 S-tale, 224 S-T-fuzzy sub-bisemi... ...UTHOR Dr. W. B. Vasantha is an Associate Professor in the Department of Mathematics, Indian Institute of Technology Madras, Chennai, where she liv...

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...AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036... ...ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail:... ...cademia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, University of Mysore, Karnataka, I... ...w, young, not-so-influential, revolutionary scientists and mathematicians who support the newer paradigm shift HI 7 Preface Th... ... (R, A) such that U (R, A) (x) > 0 and R (x, y) > 0 for all elements y in the support of U (R, A). Several other concepts of crisp orderings easil... ...uzzification is to select an element with the largest membership value from the support set. Since the defuzzification process is the last step in o... ... p p R M M x x x ∧ ∧ µ α µ = µ + (10) In this subsection we assume that the support set has finite elements and the number of pairs (the number ... ...possesses monotonicity. After a while we will find simulation results 101 support such a conclusion that if Eq. (3) has a solution at least the... ...al, and thus one defines the neutrosophic set. One gives 227 examples from mathematics, physics, philosophy, and applications of the neutrosoph...

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...dache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, Indi... ... Street, 2212 Sacele 3, Jud Brasov, Romania. Dr. Andrei Kelarev, Dept. of Mathematics, Univ. of Tasmania, Hobart, Tasmania 7001, Australia. Dr. A. R... ...asmania, Hobart, Tasmania 7001, Australia. Dr. A. R. T. Solarin, Dept. of Mathematics, Obafemi Awolowo University, (formerly University of Ife), Ile... ...tness to Dr. Minh Perez of American Research Press, whose constant mental support and encouragement made me write this book. I thank my husband D... ...op ring of a loop L over the ring R. Let ∑ = ∈ = m 1 i i RL m α α . Support of α denoted by supp α = {m i / α I ≠ 0}. |Supp α| denotes the nu... ...by supp α = {m i / α I ≠ 0}. |Supp α| denotes the number of elements in support of α. The augmentation ideal of RL denoted by W(RL) = { α = Σα i m... ...he study of complex semi simple Lie algebras refer Springer Monographs in mathematics 2001 in [47]. The reader has a lot of problem to study about S... ...arandache semigroup, International Conference on Combinatorial Methods in Mathematics, II meeting of the project 'Algebra, Geometria e Combinatoria'...

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...erican Research Press Rehoboth 2002 József Sándor DEPARTMENT OF MATHEMATICS BABE Ş-BOLYAI UNIVERSITY 3400 CLUJ-NAPOCA, ROMANIA Geome... ...n its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics..." (K.F. Gauss, Disquisitiones arithmeticae, G ottingen, 1801... ...ber of persons and organiza- tions from whom I received valuable advice or support in the preparation of this material. These are the Mathematics Depa... ...uable advice or support in the preparation of this material. These are the Mathematics Department of the Babe s-Bolyai University, the Domus Hun- gar... ...ntimate correlation between the Theory of numbers and other departments of Mathematics, not excluding geometry..." (Felix Klein, Evanston Colloquium L... ...bisector theorem. 11 Bibliography 1. F. Smarandache, Proposed problems of Mathematics, vol.II, Kishinev Univ. Press, Kishinev, Problem 61 (pp.41-42),...

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...are solved with the variables separated, the method- ology learned and preferred in mathematics. Only in conventional engineering is it standard prac... ...ems with the variables separated is the methodology learned and preferred in pure mathematics. Only in conventional engineering is it standard prac... ... as R is a variable if V is not proportional to I, and similarly for h and E. • Mathematics has no use for M because it generally complicates the ... ...nner. • The new engineering has no use for R, h, and E for the same reason that mathematics has no use for M. 1.3 The importance of separati... ...teration method. Or solve the problem graphically, or by trial-and-error. In mathematics, the separation of variables is so routine that the so... ... the resistance analyses of the same problems in Chapter 3. The answers strongly support the conclusion that electrical behavior V{I} should rep... ...the coefficient analyses of the same problems in Chapter 8. The answers strongly support the conclusion that convective heat transfer behavior sho... ...d to compare her/his solutions with those presented in Chapter 7. The comparison supports the following conclusions: • Proportional problems can... ...th the modulus analyses of the same problems in Chapter 12. The answers strongly support the conclusion that stress/strain behavior should repla...

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... You can see it in how a living cell splits. You can even see it in abstract mathematics. All Mathematical equations are dynamic balanced splits. ... ...the theoretical abstract straight line of ancient Greek geometry and abstract mathematics which does not exist in the actual Universe; Science has m... ... by every verified fact that Science has been able to gather so far. Not by mathematics. Not by theories, Not by models. By actual facts that hav... ...of our Universe: not upon imaginary abstract concepts, or imaginary abstract mathematics, or imaginary abstract numbers, or imaginary abstract ideas... ...only answers the Intelligent Design theories of Christian Fundamentalists, it supports it. Except it supports it not as they would like it to be su... ...orts it. Except it supports it not as they would like it to be supported: it supports it organically, logically, scientifically: using simple commo... ...er half became freed from the earth; it was no longer used for mobility or to support the body’s weight. The complex muscle coordination required i... ...awareness: we have a god-given right to accumulate anything we wish. Science supports this unspoken assumption. It portrays animals in Nature film... ...edators in that plain hate and hound the most. The loner predator without any supporting social gang. The antisocial leopard. What have we learn...

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...The "diagnostic thresholds" between normal and abnormal are either absent or weakly supported. The polythetic form of the DSM's Diagnostic Crite... ...measurement and by organizing the results and presenting them using the language of mathematics. This does not atone for its primordial sin: that it... ...and practitioners with large axes to grind. Few, if any, theories in psychology are supported by modern brain research. All therapies and treatment ... ...d of consciousness (whatever that meant). None of these "nervous" speculations was supported by scientific, experimental evidence. Both sides of th... ...tain statements about relationships (mostly causal) between concepts - empirically-supported laws and/or propositions (statements derived from axio... ...d measurement and organizes the results, often presenting them in the language of mathematics. In some quarters, these practices lends it an air o... ...heir profession, by its literary and artistic qualities, by the dearth of empirical support for its assertions and fundaments, by the ambiguity of i... ...measurement and by organizing its results and presenting them using the language of mathematics. This does not atone for its primordial sin: that it...

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...e psyche, even taking W. Reich’s ideas as our reference point as a sort of mathematics of the spirit, what you propose as a therapeutic process beco... ...m loved back; it is disinterested in the sense that love for another means supporting their own good, and, to do so, I might sometimes have to renoun... ...mmunicates with the Psychological I and transmits the knowledge it needs to support the fecundated ovum’s development. The ovum can develop the villi...

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...slinn nodded. Her hand grabbed the back of the closest chair, holding on for support. A round wooden table, elegant despite its simplicity, stretch... ... sat down to eat, her soup almost scalded her face, her neck muscles tired of supporting her head on her shoulders. Geoffrey attempted to keep her a... ...th each blink. The prince had taken the seat beside her some time during her mathematics work, and she had defiantly ignored him when he said hello... ...eason to.” Aislinn embraced Adamina, taking the girl’s weight against her to support her. “Stop. You’ll… you’ll get… blood…,” Adamina struggled. ... .... The roll of the queen was to stand by her husband’s side, to provide silent support and produce heirs. But, she didn’t want to be a simple woman ... ...e pining for her husband while he went to war. She wanted to be a knight, to support her king and her kingdom. Kings commanded their armies from t...

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...are said to have cleared twenty five feet on level ground. Without factitious support, man is sure to come to earth again beyond that distance. The firs... ...mean that they should not play life, or study it merely, while the community supports them at this expensive game, but earnestly live it from beginnin... ...he experiment of living? Methinks this would exercise their minds as much as mathematics. If I wished a boy to know something about the arts and scien... ...ent of any, especially as it required only thirty or forty days in a year to support one. The laborer’s day ends with the going down of the sun, and h... ...There are those who have used all their arts to persuade me to undertake the support of some poor family in the town; and if I had nothing to do, — fo... ...d a sled might slide over the stumps; and instead of leaving a whole tree to support his corded wood, he would pare it away to a slender stake or spli...

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...ge does prevail, which means that any man may vote who lives in the State, supports himself, and assists to support the poor by means of poor rates. A... ...so. Immediately over the falls there was a suspension bridge, of which the supporting, or rather non-support- ing, pillars are still to be seen. But t... ...ye there appear to be twenty-five tubes; but each of the six side tubes is supported by a pier in the middle. A great part of the expense of the bridg... ...h statesmen, then I think that such an amalgamation should receive all the support which Downing Street can give it. The United States severed themsel... ...en conducted and animated by a different sentiment. National greatness and support of the law have been the idea of the North; national great- ness an... ...n intended that it should be given. But there are axioms in politics as in mathematics, which recommend themselves to the mind at once, and require no...

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... same of incredulity, and he that uses the word impossible outside of pure mathematics is lacking in pru dence. It should be remembered that Lactanti... ...his moment everything seems possible. My faith in all the truths that have supported me is tottering.’ For a while they remained silent. Arthur’s eyes... ...e full of tenderness. He put his arm affectionately round her shoulders to support her. ‘I’m afraid you’re quite exhausted, poor thing,’ he said. ‘I’m...

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...hopping. The unknown builders of the cabin had extended the side logs so as to support a cache at the rear. In this was stored the bulk of the party’s... ...d their replies seemed to relieve him. He hurried on. Next came two white men, support ing between them a woman. They also behaved as though drunken,... ...trapped to their shoul ders. Head straps, passing across the forehead, helped support these packs. Each man carried a rifle. They walked in a stooped ... ...d faint and dizzy and put out his free hand while he reeled, as though seeking support against the air. When he had steadied himself he stepped forwar... ... a passion for equity, and perhaps it was because of this that she excelled in mathematics. But she excelled in other things. She learned to read and ... ...m and out of doors, he signalled to the slaves, and the four tottering old men supported him on his feet as he met the returning revellers, upright, g...

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...ractitioners sought to disguise its nature as a social science by applying complex mathematics where common sense and direct experimentation would ... ...rd. That volatility increases in the transition from bull to bear markets seems to support this pet theory. But how to account for surging volatili... ...ome scholarly studies of researchers such as Stephen LeRoy and Richard Porter offer support - other, no less weighty, scholarship by the likes of Eu... ...al information and may have some practical value." These hopeful inferences are supported by the work of other scholars, such as Paul Weller of ... ... financial crises, there has been astonishingly little effort to provide empirical support for this belief." Yet, no one knows how to measure moral... ...ngdom to New Zealand - set up "innovation teams or units" to foster innovation and support it. Canada's is more than two decades old. The European ... ...ntists and mathematicians of the XX century. Apart from his great contributions to mathematics, computing and other fields, the minor fact that he ... ... three-dimensional extension of formal logic which is the set of RULES that govern mathematics. Then came the Macintosh and its emulation, the win... ...ependently (as is often the coincidence in science) the basics of a new branch in Mathematics (and logic): computability or recursive functions (l...

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...fference between killing (taking life) and letting die (not saving a life). This is supported by IE above. While there is a right not to be killed -... ... populations expand to no more than the number of individuals that the habitat can support (the habitat's carrying capacity). These are called K-se... ...t and present. Endless lists of prominent people of Jewish descent are produced in support of the above contention. Yet, governments are not the me... ...fference between killing (taking life) and letting die (not saving a life). This is supported by IE above. While there is a right not to be killed -... ...ures. People and firms lack the vision, the knowledge and the wherewithal needed to support competition. They fiercely oppose it and governments th... ...ractitioners sought to disguise its nature as a social science by applying complex mathematics where common sense and direct experimentation would ... ...ifferently to the same transformation? Arithmetic, formal logic, and, by extension, mathematics and physics deal with proving identities in equation... ...ecognized as an important contribution and applied to problems in geometry, logic, mathematics, computation and physics. One of the first questions... ...t infinite sets are organized in a hierarchy. Russel and Whitehead concluded that mathematics was a branch of the logic of sets and that it is ana...

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...guaranteeing bank liabilities and recapitalization. In the UK alone, state support to the banking sector was £1.2 trillion, almost equivalent to count... ... Ray, Jake Johnston, Jose Antonio Cordero and Juan Antonio Montecino, “IMF-Supported Macroeconomic Policies and the World Recession: A Look at Forty-O... ...rnational commercial banks and private insurers became more risk averse to support cross-border trade and investments. The result was that demand for ... ...nels were full of stories about multi-billion private equity buyout deals. Supporters crowned private equity funds the “new kings of capitalism,” 2 w... ...els of forex reserves as an insurance policy against having to rely on IMF-supported bailout programs that come with strict conditionalities, such as ...

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...ir comments and recommendations on these papers: I. Balacenoiu, Department of Mathematics, University of Craiova, Romania; C. Corduneanu, Departmen... ...eutrosophic Probability and Statistics) Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA 0.1 In... ..., and start again. We may add rules that don’t preserve monotonicity. In bio-mathematics Heitkoetter and Beasley (1993-1999) present the evolutiona... ... Hence, the neutrosophic logic generalizes: - the intuitionistic logic, which supports incomplete theories (for 0 < n < 1 and i=0, 0 [ t, i, f [ 1)... ...= 1 and i = 0, and 0 [ t, i, f [ 1); from "CRC Concise Concise Encyclopedia of Mathematics", by Eric W. Weisstein, 1998, the fuzzy logic is "an exte... ...ely. Hence, the neutrosophic set generalizes: - the intuitionistic set, which supports incomplete set theories (for 0 < n < 1 and i = 0, 0 [ t, i, ... ... neutrosophic probability generalizes: - the intuitionistic probability, which supports incomplete (not completely known/determined) probability spa... ... Division Leclerc,92320 Chˆ atillon, France. This personal research work is not supported by Onera and does not reflect the position or policy of Onera... ...the measure of the partial belief that is committed exactly to (degree of truth supported exactly by ) by the body of evidence but not the total belie...

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...gy VI. Metaphors of the Net VII. The Solow Paradox VIII. Decision Support Systems Author: Sam Vaknin Contact Info: palma@unet.com.mk; s... ...together with Web advertising. Geocities - a community of free hosted, ad-supported, Web sites purchased by Yahoo! - is now selectively shutting dow... ...ff. Distributors, publishers and record companies also provide logistical support: warehousing, consolidated sales reporting and transaction auditin... ...f software or pirated copies of software without its manuals, updates and support - often lead to the purchase of a licence. Not so with plagiarism. ... ...ed from this arduous and convoluted tale? Are works of reference not self-supporting regardless of the revenue model (subscription, ad-based, print, ... ...sh, Russian, Romanian, and Farsi in progress. I wonder if we should count mathematics as a language? I was surprised at how many people were interes... ... mathematicians of the XX century. Apart from his great contributions to mathematics, computing and other fields, the minor fact that he was deeply ... ...mensional extension of formal logic which is the set of RULES that govern mathematics. Then came the Macintosh and its emulation, the windows GUI (... ...ly (as is often the coincidence in science) the basics of a new branch in Mathematics (and logic): computability or recursive functions (later to be ...

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... the body, the breath on my neck, special ways, his weight underneath me, supporting me, the sea coming through the windows. There is nothing better... ...t that night, as we mourned his death. Many of us. Corpses lodged against supports of the Puente Vecchio. Plagues. Madness. Work. We cherished him, ... ...eath the grey arches of the château bridge, I sat where I could study the supports, estimating their bulk and weight. No notebook. Too many unfinish... ...field of endeavor. Ambrogio cared nothing for geol- ogy. De Predis shunned mathematics. Boltraffio scorns cartography. Fra Luca shrugs off all but ch... ...plines. In this branch of science, the beam of light is best explained by mathematics and phys- ics. Since the axioms are long I will abridge them no... ...cartographers. Close friends, fellow artists, we looked to each other for support. “As I sketched Borgia I realized his animosity. Vitelli and I wer... ...edy was hounded off the boards. Walton had a couple of hecklers who were supported by the audience and broke up our play: we got eggs from many Wal...