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... 74 THE ARABIC/HINDU NUMBERS AND OUR DECIMAL SYSTEM 75 THE ORIGIN OF THE BRANCHES OF MATHEMATICS 76 THE ORIGIN OF ARITHMETIC 76 THE ORIGIN OF GEOMETRY 76 THE ORIGIN OF TRIGONOMETRY 77 THE ORIGIN OF ALGEBRA 78 THE ORIGIN OF ANALYTIC GEOMETRY 78 THE ORIGIN OF CALCULUS 79 THE PROPERTIES OF THE DIFFERENT MATHEMATICAL SCIENCES 80 WHAT IS ARITHMETIC ? 81 ...

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...ur solar system as a mechanical universe. RENE DESCARTES (1596-1650) French Philosopher and Mathematician 53 Created his Cartesian Coordinates and Analytic Geometry. ROBERT BOYLE (1627-1691) Irish Physicist and Chemist 53 Proposed his corpuscular theory, revived Democritus’ theory. Published The Sceptical Chemist and initiated modern chemistry.   WILLIAM GILBERT ...

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...o: Xiquan Pub lishingHouse,1993,pp54. [2] Tom M. Apostol. Introduction to Analytic Number Theory. New York: Springer Verlag,1976. ANEQUATIONINVOLV... ...s,XiquanPublishingHouse, Chicago,1993. [2] Tom M. Apostol. Introduction to Analytic Number Theory. New York: Springer Verlag,1976. ONTHEFUNCTIONϕ(N... ...’an,Shaanxi,P.R.China Abstract The main purpose of this paper is using the analytic methods to study the mean valuepropertiesoftheDirichlet’sdivisorfu... .... Chicago: Xiquan Pub lishingHouse,1993. [2] Apostol T M. Introduction to Analytic Number Theory. New York: Springer Verlag,1976. [3] Pan Chengdong a... ...: Springer Verlag,1976. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Sciencepress,1997. [4] Zhu Weiyi. On the k... ...ar number, because there are close relations between these numbers and the geometry. Now for any positive integern, we definearithmeticalfunctionc(n)as...

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...nd theories, especially if they run counter to the classical ones.” In a manner consistent with his unique point of view, he defined several types of geometry that are purposefully not Euclidean and that focus on structures that the rest of us can use to enhance our understanding of geometry in general....

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... a manner consistent with his unique point of view, he defined several types of geometry that are purposefully not Euclidean and that focus on struc... ... focus on structures that the rest of us can use to enhance our understanding of geometry in general. To most of us, Euclidean geometry seems self... ... and continued with Riemann, when they found counterexamples to the notion that geometry is precisely Euclidean geometry. This opened a whole univer... ...sely Euclidean geometry. This opened a whole universe of possibilities for what geometry could be, and many years later, Smarandache’s imagination h... ...rs later, Smarandache’s imagination has wandered off into this universe. The geometry associated with Gauss, Bolyai, and Lobachevski is now genera... ...w York: W.H. Freeman and Company. 13, Henle, M. (1997). Modern Geometries:The Analytic Approach. Upper Saddle River NJ: Prentice Hall. 14. Hilb...

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

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... contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geom... ...mes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophantine equations, some of which a... .... . . . . . . . . . . . . . . . . . . . . 27 9 On certain constants in the geometry of equilateral triangle . . . . . . . . . 35 10 The area of a Pyth... .... . . . . . . . . . . . . . . . . . . . . . 42 12 An arithmetic problem in geometry . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 2. Diophant... ...c PC 2 = constant (5) can be deduced. Usually, (4) (or (5)) is proved via Analytical Geometry. 3. Let nowP 1 ;P 2 ;P 3 be the projections of interior... ... Mat. 1/1979, p.13. 5. D.S. Mitrinovic (in cooperation with P.M. Vasi c), Analytic Inequalities, Springer Verlag, 1970. 6. J. S andor, Sur la foncti... ...8; 9g (30) See also [11]. Bibliography 1. T.M. Apostol, An introduction to analytic number theory, Springer Verlag, 1976. 2. R.K. Guy, Unsolved proble... ...near (or further) future. Bibliography 1. T.M. Apostol, An introduction to analytic number theory, Springer Verlag, 1976. 2. M. Deaconescu and J. S a... ...e nement of relation (6). Bibliography 1. T.M. Apostol, An introduction to analytic number theory, Springer Verlag, 1976. 187 2. J. S andor, Some di...

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..., Mohar and Thomassen [53] and White [70], especially the [53] for detail. The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics....

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...The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics. In fact, ma... ...zation or making combinatorial refinement for the Riemann surfaces, Riemann geometry and finally, the Smarandache geometries by maps is a very interesti... ...ntribution of combinatorics to classical mathematics, such as, the Riemann geometry and Smarandache geometries is realized. Those are the wish of mine... ...plex algebraic curves. For introducing this concept, it is need to enlarge analytic functions to those of dianalytic functions first. Now let f : A−→C ... ...f(z) = u(x,y)−iv(x,y)for certain functions u,v :A−→R. A mapping f :A−→C is analytic on A if ∂f ∂z = 0 (Cauchy-Riemann equation) and is antianlytic on ... ...to be dianalytic if its restriction to every connected com- ponent of A is analytic or antianalytic. Now we can formally define a Klein surface. A Klei... ...in surface viewed as an orientable surface with empty boundary and to be analytic functions. Whence, we have the following relation: {Riemann Suface... ...{(U 1 = d,φ 1 = 1 D )} in C are two Klein surfaces with empty boundary and analytic structures. If k(S),g(S) and χ(S) are the number of connected comp... ...able, 2−g(S)−k(S). if Sis non−orientable. 1.1.2 Definition of a Smarandache geometry By the history, we know that classical geometries include the Eucl...

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...____ 34 SELF-CENSORSHIP __ 34 WHERE ____________ 35 BIG LIE ____________ 35 APPEARANCES _____ 35 EXCESS ____________ 35 DESPAIR ___________ 35 ANALYTIC _________ 35 BACOVIAN _________ 35 UNINVITED GUEST _ 35 NAUGHTY _________ 36 ESSENCE __________ 36 IN HARD TIMES ____ 36 HARDLY ___________ 36 CONTRADICTORY __ 36 PHYLOSOPHY ______ 36 PARSIFAL __________ 36 ...

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...lved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry (edited by M. L. Perez), Xiquan Pub- lishing House, Phoenix, 2000.... ...itional potential energy which manifests the fractal and nondifferentiable geometry. 2 Non relativistic quantum mechanics 2.1 Quantum potential In the... ...established as a mere manifestation of the nondifferen- tiable and fractal geometry instead of being deduced from a postulated Schr¨ odinger equation.... ...e energy- momentum tensor, one solves the Einstein feld (i. e. space- time geometry) equations for the metric potentials, then one writes the geodesic... ...energy-momentum tensor, so that this is a looped system which has no exact analytical solution. In the case of a quantum mechanical particle considere... ... 2 )− − N(r)(xdx+ydy+zdz) 2 , r = p x 2 +y 2 +z 2 , (1) where,∀t,L,M,N are analytic functions such that, L,M,N >0. (2) In polar coordinates (1) become... ...(1) becomes, ds 2 =A(r)dt 2 −B(r)dr 2 −C(r)(dθ 2 +sin 2 θdϕ 2 ), (3) where analyticA,B,C >0 owing to (2). 68 S J. Crothers. On the General Solution to... ...istic units, ds 2 =A(r)dt 2 −B(r)dr 2 −C(r)(dθ 2 +sin 2 θdϕ 2 ), (1) where analytic A,B,C>0. The general solution to (1) for the simple point-mass is,... ...= L m , ρ 2 =r 2 +a 2 cos 2 θ, Δ=r 2 −rα+a 2 +q 2 , 0<r<∞. By applying the analytic technique of section 3, the general solution for the rotating poin...

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...United States of America JULY 2005 VOLUME 2 CONTENTS S. J. Crothers On the Geometry of the General Solution for the Vacuum Field of the Point-Mass. . ... ...ed from private donations. July, 2005 PROGRESS IN PHYSICS Volume 2 On the Geometry of the General Solution for the Vacuum Field of the Point-Mass Ste... ...wing unjustifed assumptions, ta- citly or otherwise; S J. Crothers. On the Geometry of the General Solution for the Vacuum Field of the Point-Mass 3 ... ...ired by Einstein’s theory. Mere assumption is not permissible. 2 The basic geometry of the simple point-mass The usual metric g s of the spacetime man... ...)dt 2 −B(r)dr 2 −C(r) dθ 2 +sin 2 θdϕ 2 , (2a) A,B,C>0, whereA,B,C are analytic functions. I emphatically remark that the geometric relations betw... ...J. Droste [9] and A. Eddington [10], that r 2 can be replaced by a general analytic function of r without destroying the spherical symmetry of (18). L... ... Schwarzschild’sR 2 and Hilbert’s r 2 can be replaced with any appropriate analytic functionC n (r) as given by (24), so the range and the bound- ary ... ...ntional conditions (see [19]) that must be met are inadequate, viz., 1. be analytic; 2. be Lorentz signature; 3. be a solution to Einstein’s free-spac... ...cle. Phys. Rev., 1958, v. 110, 965. 20. Levi-Civita T. Mechanics. — On the analytical expression that must be given to the gravitational tensor in Ein...

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...l and metaphorical comments – which are separated by blank rows. It is an analytical study, and it is related to multiple-valued logic because in alm... ...ugin | passed to that of "stable disequilibrium" (<Truth in the Abstract (Analytical) versus Truth in the Concrete (Empirical)>). A self-regulating ... ...trol. 27 We tried to de-formalize the Hilbert's for/mal/ization of geometry: by constructing an anti-model, which doesn't respect any of his... ...n in a hyperspace). Facts exist in isolation from other facts (= the analytic philosophy), and in connection as well with each other (= Wh... ...ent in the population. Philosophy is an ideational puzzle and, alike geometry, it circumscribes and inscribes an idea to and into a class of th... ... That's why I plead for a philosophical system without system. Not quite analytic philosophy. Congratulations for your failure! If you are defeate... ...comment and argue with them. Everything put in a form of short sections (analytic philosophy), systematically concatenated on themes, notions, cate... ...sal. From experimental psychology to experimental philosophy. "Geometry is exaggeration, philosophy is exaggeration, and so poetry. Every... ...e history of sciences: a) Felix Klein (1872), in his Erlangen program, in geometry, has proposed a common standpoint from which various branches of...

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...(H. Kohut. The Chicago Institute Lectures 1972-1976. Marian and Paul Tolpin (Eds.). Analytic Press, 1998) Kohut's contention is nothing less than re... ...ive dimension of the experience. If a psychological process lacks the evaluative or analytic elements, this lack does not question its existence or ... ...itain, girls have been outperforming boys in scholastic aptitude tests - including geometry and maths - since 1988. In an Op-Ed published by the ... ...er activists, says Meyerowitz, insist that gender and sexuality represent "distinct analytical categories". The New York Times wrote in its review o...

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... was immediately recognized as an important contribution and applied to problems in geometry, logic, mathematics, computation and physics. One of th... ...ehead concluded that mathematics was a branch of the logic of sets and that it is analytical. In other words: the language with which we analyse t... ...ure of things because they are, at bottom, self referential (equivalent to Kant's "analytic propositions"). IIC. Kant According to Kant, our sens... ...society of lack of self- awareness. But choice of words does not a coherence make. Analytic Summary of Kimball Lasch was a member, by conviction, o... ...sly critical sensibility exemplified by psychoanalysis and the degeneration of the 'analytic attitude' into an all out assault on ideals of every ki... ...ather than as a basis for enlightenment. The solution was to blend Marxism and the analytic method of Psychoanalysis (very much as Herbert Marcuse ... ...e sense organs that we do (eyes) and may not be acquainted with our mathematics and geometry. Reality can be successfully described and captured by ... ...ritain, girls have been outperforming boys in scholastic aptitude tests - including geometry and maths - since 1988. In an Op-Ed published by the ...

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... was immediately recognized as an important contribution and applied to problems in geometry, logic, mathematics, computation and physics. One of th... ...ehead concluded that mathematics was a branch of the logic of sets and that it is analytical. In other words: the language with which we analyse t... ...ure of things because they are, at bottom, self referential (equivalent to Kant's "analytic propositions"). IIC. Kant According to Kant, our sens... ...society of lack of self- awareness. But choice of words does not a coherence make. Analytic Summary of Kimball Lasch was a member, by conviction, o... ...sly critical sensibility exemplified by psychoanalysis and the degeneration of the 'analytic attitude' into an all out assault on ideals of every ki... ...ather than as a basis for enlightenment. The solution was to blend Marxism and the analytic method of Psychoanalysis (very much as Herbert Marcuse ... ...e sense organs that we do (eyes) and may not be acquainted with our mathematics and geometry. Reality can be successfully described and captured by ... ...ritain, girls have been outperforming boys in scholastic aptitude tests - including geometry and maths - since 1988. In an Op-Ed published by the ...

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...nds. Many results concerning the partition function were discovered using analytic functions by Euler, Jacobi, Hardy, Ramanujan and others. Other pr... ...ved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry, edited by M.L.Perez , Xiquan Publishing House, 2000. 11. Dr. ...

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...ity, the ability to see the bigger picture (synoptic view), interpersonal aptitude, analytic and synthetic skills, not to mention numeracy, literacy... ...images. Kekula dreamt the structure of the Benzen molecule, Einstein envisioned the geometry of space and so on. But, in the past, technology ended ...

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... unrecorded, that I still had loved The exercise and produce of a toil, Than analytic industry to me 385 More pleasing, and whose character I deem Is ... ...t having brought To land a single volume, saved by chance, 145 A treatise of Geometry, he wont, Although of food and clothing destitute, And beyond co...

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... unrecorded, that I still have lov’d The exercise and produce of a toil Than analytic industry to me More pleasing, and whose character I deem 405 Is ... ... deep, who having brought To land a single volume and no more— A treatise of geometry—was used, 165 Although of food and clothing destitute, And beyon...

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...en they thought of their friend Doctor Grim. One effect of his zealous and analytic instruction of the boy was very perceptible. Heretofore, though en... ...ellect. So that she had even some knowledge of the rudiments of Latin, and geometry, and algebra; inaccurate enough, but yet with such a briskness tha...

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...rew to be subjected to. For all these reasons then, and others perhaps too analytic to be verbally de veloped here, Ahab plainly saw that he must sti... ...ound me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, wi...

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...tter of it into synoptic tables. During the same years I learnt elementary geometry and al gebra thoroughly, the differential calculus, and other por... ...with Logic, in which I began at once with the Organon, and read it to the Analytics inclusive, but profited little by the Poste rior Analytics, whic... ...my father’s direction, my studies were carried into the higher branches of analytic psychology. I now read Locke’s Essay, and wrote out an ac count ... ...an the one I ultimately executed. Having done with Logic, we launched into Analytic Psy chology, and having chosen Hartley for our text book, we 71 ... ...oined together in Nature, to cohere more and more closely in our thoughts. Analytic habits may thus even strengthen the associations between causes an... ...he formation of my character anew, and create, in a mind now irretrievably analytic, fresh associations of plea sure with any of the objects of human... ...uc tive branches of natural philosophy, but the inappropriate one of pure geometry, which, not being a science of causa tion at all, does not requir... ...g so contained and implied, it could be new truth, and how the theorems of geometry, so different in appearance from the definitions and axioms, could...

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...ic science, as medical science whose object matter is things wholesome; or geometry whose ob- ject matter is magnitude: for it has not for its object ... ...n become geometri- 229 The Ethics of Aristotle cians who take Pleasure in geometry, and they apprehend particular points more completely: in like man... ... prove its truth if he should ask. But these terms are employed elsewhere (Analytica Post I cap. 11. sect. 10) to denote respectively particulars and ... ...nihil enim impedio. P 6, l. 1. Or “prove themselves good,” as in the Prior Analytics, ii 25, [Greek: apanta pisteuomen k.t l] but the other rendering ... ...st upon Compassion. P. 133, l. 24. It is the opening statement of the Post Analytics. P . 133, l. 27. Aristotle in his logical analysis of Induction, ... ...cs. P . 133, l. 27. Aristotle in his logical analysis of Induction, Prior. Analytics II. 25, defines it to be “the proving the in- herence of the majo... ...l to compare on the subject of Induction as the term is used by Aristotle, Analytica Prior. II 25 26 Analytica Post. I. 1, 3, and I. Topics VI I and X...

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...eld. Let a planet be wheeling with heavenly science, upon arches of divine geometry: suddenly, to us, it shall appear unaccountably ret- rograde; flyi... ...y of God’s attributes; suppose holiness or hap- piness, she feels, (though analytically she could not explain,) that God is not holy or is not happy b... ...es, atheistically given? We used to be told that there is no royal road to geometry. I don’t know whether there is or not; but I am sure there is no P... ... is sufficient (given a multitude of minds) to lead backwards or forwards, analytically or synthetically, into many of the rest. That is the principle... ...rts itself) as the richest and noblest; and for the same reason that makes geometry careless of language. The vilest jargon that ever was used by a sh... ... a fortiori applies, to religious truth, as one which lies far deeper than geometry in the spirit of man, one to which the inner attestation is profou... ...eding century, had been greatly improved by W allis, Savilian professor of geometry at Oxford, the improver of analytic mathematics, and the great his... ...oved by W allis, Savilian professor of geometry at Oxford, the improver of analytic mathematics, and the great historian of algebra. Algebra it was th...

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... Intro- duction, by O. F. Owen (Bohn’s Classical Library), 1848. Posterior Analytics, E. Poste, 1850; E. S. Bourchier, 1901; On Fallacies, E. Poste, 1... ...son in any science is the business of those who are skilful therein; as in geometry, of geometricians; in steering, of steersmen: but if some indi- vi...

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...valuable as illustrating the great truth, too gener- ally overlooked, that analytic power is a subordinate quality of the critic. On the whole, it may... ...ed unbounded. Indeed, it is very easily calcu- lated by means of spherical geometry, what a great extent of the earth’s area I beheld. The convex surf... ... conjecture. —Sir Thomas Browne. THE MENTAL FEATURES discoursed of as the analytical, are, in themselves, but little susceptible of analysis. We appr... ...y had turned outward the faces of their own. 118 Poe in Five V olumes The analytical power should not be confounded with ample ingenuity; for while t... ...ted general observation among writers on morals. Between ingenuity and the analytic ability there exists a difference far greater, indeed, than that b... ...enious are always fanciful, and the truly imaginative never otherwise than analytic. The narrative which follows will appear to the reader some- what ...

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...t this idea of the circle— this idea of it which, in regard to all earthly geometry, we con- sider as merely the mathematical, in contradistinction fr... ...ented to his inspection—he could have no difficulty in determining, by the analytic retrogradation, to what original impulse it was due. This power of...

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...have disqualified them, to my rigour; but they were stall right,” in the so analytic modern critical phrase, through their discreetly disavowing emulat... ...e sharp corners and hard edges, all the stony pointedness, the grand right geometry of his spreading Palladian church. Just so he was insensible to no... ...s, the first, had been the not inconsiderable length, to her later and more analytic consciousness, of this second wait for her husband’s reappearance....

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...s, just as mathematicians handle the same numeri- cal and spatial facts by geometry, by analytical geom- etry, by algebra, by the calculus, or by quat... ...thematicians handle the same numeri- cal and spatial facts by geometry, by analytical geom- etry, by algebra, by the calculus, or by quaternions, and ... ...religious constructions once more. For the moment, let me dryly pursue the analytic part of the task. Both thought and feeling are determinants of con...