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...s, solved/unsolved problems, new Smarandache type sequences and new Smarandache Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theory by F. Smarandache ("The Florentin Smarandache papers" special collection, University of Craiova Library, and Archivele Statului, Filiala Valcea) an...

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...che Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theo... ... = ? Problem 46. Study the convergence of the Pseuto-Smarandache-Totient harmonic series: ¥ = > 1 0 " " ) ( 1 n a number real a is a where n Zt Pr... ...n. Problem 21. Study the convergence of the Pseudo-Smarandache-Squarefree harmonic series: ¥ = ˛ > 1 . 0 ) ( 1 n R a and a where n a Zw Problem 22.... ... ? Problem 22. Study the convergence of the Smarandache Double factorial harmonic series: R a and a where n Sdf n a ˛ > ¥ = 0 ) ( 1 1 Problem 23. ... ...W(n)=n-1 if n is repunit (that is all the digits are 1). In the following analysis we will exclude this two species of integers. With the Ubasic pro...

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...an communicate over space and time. Balance of power shifts from tribal chiefs to city-state warrior-kings and priests. -- 5. Symbols of Sound Demand Analysis-The alphabet makes the pen mightier than the sword, generating the powers of knowledge needed to create and govern empires. -- 6. China‘s InfoTech Siblings-For centuries, the Chinese keep to themselves ?the wasps’ se...

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...efs to city-state warrior-kings and priests. 5. Symbols of Sound Demand Analysis The alphabet makes the pen mightier than the sword, generating th... ...r than the powers of kings’ marching warriors. Symbols-of-Sound Demand Analysis To Abe Lincoln, the alphabet was ―the world‘s greatest inventio... ... Logan notes that every time a word is written, we repeat this sequential analysis or linear thinking. That, he claims, sublimely promotes the sk... ... or linear thinking. That, he claims, sublimely promotes the skills of analysis and matching that are so critical to developing logical, mathemat... ...metry—math applied to space  music—math applied to the time and sound of harmonics and tuning  astronomy—math applied to heavenly time and space.... ...until late in life that I became aware of the vital link between math and harmonics. I‘ve since noticed how common it is for scientists to enjoy musi...

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...f unorganized Energy which then systematically organized itself into logical, harmonic organic shapes and dynamics of galaxies, stars and planets. ... ...hematics based upon the base root of 10? It should be based upon the natural harmonic root of 12 but it isn’t. The answer is because the two symbol... ...o work properly and balance each other dynamically. The entire Universe is harmonically balanced, but…it is not balanced perfectly. If it were ba... ...the number 13 and the number 20 as sacred numbers. So… 13 x 20 = 360. Since harmonically, the number 13 is actually the number 1, it represented th... ...rong mathematical system based upon 10 instead of 12; you still end up with a harmonic balance because that is how the Universe is actually structur... ...amic of infinite, subtle, complexity and reflectivity that is beyond any true analysis. Your thought is not you. That part of you which initiates-... ...earn? What do we question and what do we take for granted? In the final analysis: the commitments and connections between people are forever. C... ...d of technological information… but no learning of the heart. In the final analysis: all heartless one-sided learning is evil, because it does not... ...oad of technological information… but no learning of the heart. In the final analysis: all heartless, one-sided learning is evil because it does no...

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... thenarmis@yahoo.com The black hole, which arises solely from an incorrect analysis of the Hilbert solution, is based upon a misunderstanding of the s... ...tion The variabler has given rise to much confusion. In the con- ventional analysis, based upon the Hilbert metric, which is almost invariably and inc... ...matter and charge requires the determination of the mappingψ. The orthodox analysis has completely failed to understand this and has consequently fail... ...nd this and has consequently failed to solve the problem. The conventional analysis simply looks at the Hilbert metric and makes the following unjusti... ... that the under action of gravitational waves the model will behave like a harmonic oscillator where the forcing power is in the Riemann- Christoffel ... ... x=2ω˙ y (8.23) and substitute ˙ y=¨ x/2ω into the second one. We obtain a harmonic oscillation equation ¨ x+4ω 2 x=0, (8.24) with respect to the rela... ...ns of motion for a free partic- le, equations (2.6), in a linear polarized harmonic wave a=Asin ω c (ct±x 1 ),b=0 are as follows U 0 +U 1 =ε= const, (... ...velocities U 2 (0) andU 3 (0) are added. Here we see that, besides regular harmonic oscillations, the termtcos ω c (ct±x 1 ) describes oscillations wi... ...in longitudinal oscillations — they are described by solution (9.10). Both harmonic oscillations and unbounded-rising oscillations exist there only if...

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... was written that will compute the first partial sums for this series, the harmonic series and the p-series with p = 1.1. The harmonic series is known... ...ed in figure 4. Figure 4 The plot with squares are the partial sums of the harmonic series, the plot of triangles the partial sums of the p-series and... ...milar to that for series number 10, although more slowly, so no additional analysis will be done. 12) Let f: N → R be a function defined by ... ... 67, 68, 69, 77, 83 Fractal Image 71 Generalized Fibonacci Sequence 68, 69 Harmonic Series 119, 120 Lucas Numbers 68, 69 Mandelbrot Set 71, 76, 83, 87...

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...efs to city-state warrior-kings and priests. 5. Symbols of Sound Demand Analysis The alphabet makes the pen mightier than the sword, generating th... ...r than the powers of kings’ marching warriors. Symbols-of-Sound Demand Analysis To Abe Lincoln, the alphabet was ―the world‘s greatest inventio... ... Logan notes that every time a word is written, we repeat this sequential analysis or linear thinking. That, he claims, sublimely promotes the skil... ...is or linear thinking. That, he claims, sublimely promotes the skills of analysis and matching that are so critical to developing logical, mathemat... ...metry—math applied to space  music—math applied to the time and sound of harmonics and tuning  astronomy—math applied to heavenly time and space.... ...until late in life that I became aware of the vital link between math and harmonics. I‘ve since noticed how common it is for scientists to enjoy musi...

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...size of the simplicial cells (p-loops), upon quantization of a generalized harmonic oscillator, for example, are given by multiples of the Planck scal... ... a ν =γ μ γ ν D μ a ν (57) where D μ is the covariant derivative of tensor analysis. Decomposing the Clifford product γ μ γ ν into its sym- metric and... ...e scalar polyparticle inC-space. A wave equation for a generalized C-space harmonic oscillator requires to introduce the potential of the form V =κX 2... ...ntum spacetime, resulting from of a Bose-Einstein condensate of theC-space harmonic oscillator. A C-space version of the Dirac Equation, representing ... ... (126) which reminds us of the creation/annihilation operators used in the harmonic oscillator. We shall now review the steps in [127] to reproduce th... ...nopoles in Maxwell’s theory was investigated by [89] and a recent critical analysis of “unifed” theories of gravity with electromagnetism has been pre... ...the notion of fractal “manifolds”. A scale- fractal calculus and a fractal-analysis construction that are esential in building the notion of a fractal... ...lgebras and physics. Edited by R. Ablam- owicz, B. Fauser. Vol 2: Clifford analysis. Edited by J. Ryan, W. Sprosig. Birkhauser, Boston, 2000. 7. Loune... ..., 2004, 19. 76. Choquet-Bruhat Y., DeWitt-Morette C. and Dillard-Bleick M. Analysis, manifolds and physics. Revised edition, North Holland Publ. Co., ...

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...en (24) yields the very interesting inequality ( 1 H n n X k=1 p d (k)) Hn ¸ n Y k=1 ( k p k) H [ n k ] ; where H m denotes the m-th partial sum of th... ...ov. On 25-th and 26-th Smarandache's problems. Notes on Number Theory and Discrete Mathematics, Vol. 9, 2003, No. 4, 99-104. [42] Yosida K., Functiona...

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... is very di cult to decide in this eld of Elementary Geometry - Algebra - Analysis) we note that (1) appears in [R. Sturm: Maxima und minima in der e... ...ce (e.g. "Heron trapeziums"). Bibliography 1. R.D. Carmichael, Diophantine analysis, John Wiley and Sons, New York, 1915. 2. R.K. Guy, Unsolved proble... ...ibid, 19(2001), 33-40. 7. J. S andor, On a function which generalized the harmonic series, C.R. Acad. Bulg. Sci. 41(1988), 19-21. 8. E.T. Whittaker, ... .... Sci. 41(1988), 19-21. 8. E.T. Whittaker, G.N. Watson, A course of modern analysis, Cambridge Univ. Press, 1969. 107 20 On certain diophantine equat... ...Discr. Math. 5(1999), No.2, 41-51. 2. W. Rudin, Principles of Mathematical Analysis, Second ed., Mc Graw-Hill Company, New York, 1964. 174 17 On the ... ... of divisors of n is A(d 1 ;:::;d s ) = d 1 +::: +d s s = (n) d(n) ; the harmonic mean is H(d 1 ;:::;d s ) = s 1 (n) = nd(n) (n) ; while the g... ...y 72(1965). 5. G. P olya and G. Szeg o, Aufgaben und Lehrs atze aus der Analysis , Berlin 1924, Band I. 6. P. Schweitzer, On an inequality related ...

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...om 9 2 = 81, and the sum of the digits of 81 = 9. For purposes of this analysis, we will consider one to be a trivial solution and ignore it. We ... ... r=1 The sequence of numbers is 1, 3, 10 , … Note: The harmonic series ∑ 1/n satisfies the inequality log(n+1) < ∑ 1/n < log n... ...zed Smarandache Star Function 40-46 Geometric Mean 172-173 Hardy 19 Harmonic Series 178 213 Index of Beauty 169-171, 174-175 Jacobi ...

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...e in his mouth. Experience (he smiled again, for as much as this ceaseless analysis of the intricacies of his life might seem morbid to others were th... ...r all, his aim had initially been to drive to the sea, to see and hear the harmonic continuum of wave upon wave slamming its froth against the shore, ...

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... us away with sentimentalism. Senti- mental people, in her phrase, ‘fiddle harmonics on the strings of sensualism,’ to the delight of a world gaping f... ...d, and goaded by an appetite, sustained by sheer gratifications. Fiddle in harmonics as it may, it will have these gratifications at all costs. Should... ...person of T om Redworth was left on British land; but for some reason past analysis, intermixed, that is, among a swarm of sensations, Diana named her... ...dy. They sparkled in review and preserved their integ- rity under critical analysis. The reading of them hurried him in pursuit of her from house to h... ...he cried. And is our public likely to muster the slightest taste for comic analysis that does not tumble to farce? The doubt re- duced her whole MS. t... ...an was justly mated with a devious filmy sentimentalist, likely to ‘fiddle harmonics on the sensual strings’ for him at a mad rate in the years to com... ...idea of her loving. The woman who talked of the sentimentalist’s ‘fiddling harmonics,’ herself stressed the material chords, in her attempt to escape ...

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...ly exists as yet among you—a nation given up to philosophical theories, to analysis and discus- sion, and always torn by civil disturbances. Modern mu... ...will hear the ominous melody the maestro has engrafted on to this profound harmonic composition, wor- thy to compare with the most elaborate structure...

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...The coroner is to sit in the first floor room at the Sol’s Arms, where the Harmonic Meetings take place twice a week and where the chair is filled by ... ...most awful shapes. He is conducted by the beadle and the land lord to the Harmonic Meeting Room, where he puts his hat on the piano and takes a Winds... ...mmodation he has provided a spe cial little table near the coroner in the Harmonic Meeting Room) should see all that is to be seen. For they are the ... ... melts into the shadowy night and then flares out of it strong in gas. The Harmonic Meeting hour arriving, the gentleman of professional celeb rity t... ...oll, tippy tol li doll, Dee! The jingling piano at last is silent, and the Harmonic friends rally round their pillows. Then there is rest around the l... ...It shall pollute, this very night, the choice stream (in which chemists on analysis would find the genuine nobility) of a Norman house, and his Grace ...

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...d. The Sol skilfully carries a vein of the prevailing interest through the Harmonic nights. Little Swills, in what are profession ally known as “patt... ...It shall pollute, this very night, the choice stream (in which chemists on analysis would find the genuine nobility) of a Norman house, and his Grace ...

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... like ourselves, to see the effect of passionate mu- sic and “deep-inwoven harmonics” upon the feeling of an idiot, we may conceive what we mean. Such... ...racter, for instance, that could employ the fiftieth part of that profound analysis which has been ap- plied to Hamlet, to Falstaff, to Lear, to Othel...

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...uman embryo predict the light? the ear of Handel predict the witchcraft of harmonic sound? Do not the constructive fingers of Watt, Fulton, Whittemore... ... teachings are tu- itions. In that deep force, the last fact behind which analysis cannot go, all things find their common origin. For the sense of b... ...ure is not that which may easily be read, but is inex- haustible. The last analysis can no wise be made. We judge of a man’s wisdom by his hope, knowi... ...es only upon those of tender age, and al- though a beauty overpowering all analysis or comparison and putting us quite beside ourselves we can seldom ... ...et them then, we shall not want them, for we are already they. In the last analysis, love is only the reflec- tion of a man’s own worthiness from othe... ...iation of these. So in this great society wide lying around us, a critical analysis would find very few spontaneous actions. It is almost all custom a...