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...dies 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 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have ...

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...sef Sándor GEOMETRIC THEOREMS, DIOPHANTINE EQUATIONS, AND ARITHMETIC FUNCTIONS ************************************* AB/AC=(... ...A, ROMANIA Geometric Theorems, Diophantine Equations, and Arithmetic Functions Am... ...into ve chapters: Geometric the- orems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrat... ...le 20 may be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many papers study the famous Smarandache function, the sour... ...he author has discovered various generalizations, extensions, or analogues functions. Other topics are connected to the com- position of arithmetic fu... ...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, ... ... 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...

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...p://www.gallup.unm.edu/~smarandache/), the Smarandache Notions Journal always published by American Research Press, and several books on conjectures, functions, unsolved problems, notions and other proposed by Professor F. Smarandache in "The Florentin Smarandache papers" special collections at: the Arizona State University (Tempe, USA), Archives of American Mathematics (U...

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...A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY Felice Russo AMERICA... ...ys published by American Research Press, and several books on conjectures, functions, unsolved problems, notions and other proposed by Professor F. Sm... ...ing. This book, following the Smarandache spirit, presents new Smarandache functions, new conjectures, solved/unsolved problems, new Smarandache type ... ...erse. So let's start our observation. 4 Chapter I On some new Smarandache functions in Number Theory. A number-theoretic function is any function whi... ...t. Euler's function (n) j [3] is such, as are n!, º ß n , 2 n etc. The functions which are interesting from number theory point of view are, of co... ... = ? 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. ...

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...f unorganized Energy which then systematically organized itself into logical, harmonic organic shapes and dynamics of galaxies, stars and planets. ... ...e stated is: Everything is equally relative to everything because everything functions by the Principle of Dynamic Balance, and everything must obe... ...d everything must obey the Law of Relative Equality. The Universe works, and functions by the basis of seeking out, finding, and establishing relat... ...re Universe actively obeys only one single specific Law. Because of this, it functions very well. The Universe doesn’t balance-out vaguely, or gen... ...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... ...ality. Whenever any aspect of the Universe is examined: you will find that it functions in these Three Root Ways. What can be deduced from thes... ...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...

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...hile there are differences in the numbers, the general growth rate of both functions is the same. 5) Smarandache Concatenated Cubic Sequence 1, 18, 18... ...ly Natural Numbers”[2], Henry Ibstedt discusses at length the iteration of functions on the natural numbers. Most of these functions will either enter... ...owever, it does give us a place to start, in that there are many different functions that can be applied. For example, the iteration cycle could be a... ...ache, Pseudo-Smarandache, sum-of-divisors, Euler phi and number of factors functions. Therefore, we conclude this segment with some questions. Questio... ...clude this segment with some questions. Question: There are many different functions that define sequences that can be placed inside a matrix of zeros... ... 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... ... 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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... form of nutshell. The psi-curve just described is actually formed of harmonics, just as a musical sound is. Now, these harmonics actually comp... ... in Fig.26 is formed of a whole panoply of momentum helices (remember the harmonics?) as opposed to the single momentum state depicted in Fig.27, ... ... apart! Fig.31. Constructive Interference. 88 If the wave-functions are in phase, constructive interference occurs as in Fig.31. Wh...

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...4 8. On the 46-th Smarandache's problem 69 9. On the 78-th Smarandache's problem 72 10. On four Smarandache's problems 78 11. On four prime and coprim... ...a k;i = 8 < : 0; if "(k)<jij F("(k)¡jij); if "(k)¸jij ; (21) Let we put F(n)=G(H(n));n=0;1;2;::: (22) where H :N[f 0g!E and G : E!N[f 0g, are arbitrar... ...knownasMin¶ aÄ c's formula. It is given below ¼(n)= n X k=2 [ (k¡1)!+1 k ¡[ (k¡1)! k ]]: (2) Therefore, the problem for ¯nding of explicit formulae fo... ... 10 2 10 ]; 4 The results in this section are taken from [7] 47 a [log 10 n]+1 (n)´a k =n¡a 1 10 k¡1 ¡:::¡a k¡1 10: Obviously, k;a 1 ;a 2 ;:::;a k ar... ...= n Y k=1 k [ n k ] (14) holds. Proof. In the identity X k·n f(k): X s· n k g(s)= X k·n g(k): X s· n k f(s); that is valid for arbitrary two arithmeti... ...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...

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...ov Hydrodynamic Covariant Symp- lectic Structure from Bilinear Hamiltonian Functions. . . . . . . . . . . . . . . . . . . . 92 D. Rabounski, L. Boriss... ...nal feld are the proper radius and the curvature radius, both of which are functions of r. The variable r is actually a Euclidean parameter which is m... ...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 between the c... ...+dz 2 − −H |x−x 0 |dx+|y−y 0 |dy+|z−z 0 |dz 2 , (5) whereF,G,H>0 are functions of D= q |x−x 0 | 2 +|y−y 0 | 2 +|z−z 0 | 2 =|r−r 0 |, andP 0 (x 0... ... 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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... Table of Contents …………………. …………………… 4 Chapter 1 Smarandache Partition Functions ………………. 11 1.1 Smarandache Partition Sets, Sequences and Fun... ...Reciprocal Partition Theory ………………. 18 1.4 Generalization of Partition Functions, Introduction of the Smarandache Partition Function ………………. 1... ... A Few More Sequences ………………… 86 2.6 Some New Smarandache Sequences, Functions and Partitions ………………… 89 6 2.7 Smarandache Reverse Auto C... ...marandache Unary Sequence ………………… 106 2.13 Smarandache Dual Symmetric Functions and Corresponding Numbers of the Type of Stirling Numbers of ... ...s …………………. 147 3.1 Exploring Some New Ideas On Smarandache Type Sets, Functions and Sequences …………………. 147 9 3.2 Fabricating Perfect Square... ... 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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...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... ...sh left, center, flush right, or justified, and perform other typographic functions. Their hard-earned skills no longer needed, ―hot type‖ printer... ... At age eleven, having been taken to lectures and numerous other academic functions by his parents, Michael started to ―drop in‖ to any number of cl...

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...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... ...sh left, center, flush right, or justified, and perform other typographic functions. Their hard-earned skills no longer needed, ―hot type‖ printers ... ... At age eleven, having been taken to lectures and numerous other academic functions by his parents, Michael started to ―drop in‖ to any number of cl...

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.../ (/) , (2.4) lies between 0 and 1, {i.e., 0 < w(p) ≤ 1} provided gamma functions exist, i.e., ) 2 / ( h p − > . Substituting (2.3) in (2.1) ... ... 11 The convex nature of the proposed statistic and the condition that gamma functions contained in w(p) exist, provides the criterion of choosing... ...table estimator. The class of such estimators are large. The following simple functions g(u,v) give even estimators of the class () () () () () ... ...only when the optimum values of constants involved in the estimator, which are functions of the unknown population parameters µ 0 , b and A, are k... ...ik, L.M. (1984): Alternative multivariate ratio estimators using geometric and harmonic means. Jour. Ind. Soc.Agri. Statist., 36, 110-118. Murthy, M...

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...by a simple use of L’Hˆ opital’s rule or by noticing that both coshξ;sinhξ functions approach infnity at the same rate ΔX =L P (coshξ−sinhξ)→0, (2.1... ...tten in terms of the Lorentz-boost rapidity parameterξ by using hyperbolic functions. The Lo- rentz dilation factor is cosh(ξ)=(1−β 2 frame ) −1/2 ; w... ...described as follows. A C-space worldline can be described in terms of fve functions x μ (τ), σ(τ) (all other C-space coordinates being kept constant)... ...size of the simplicial cells (p-loops), upon quantization of a generalized harmonic oscillator, for example, are given by multiples of the Planck scal... ...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 ... ...ral assumption due to the fact that complex valued quantum mechanical wave functions involve momentap μ and coordinatesx μ (e. g., a plane wave is giv... ... (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... ...y a simple use of L’Hˆ opital’s rule or by noticing that both coshξ; sinhξ functions approach infnity at the same rate ΔX =L P (coshξ−sinhξ)→0, (158...

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...job and sweep away the dirt of a capitalistic institution. How could she do functions that would keep it nice and operative looking? How could she e... ...fell through the curtain of their bedroom. The breathing of this friend was harmonic beauty and, at that moment, he halfway yearned for him. Evading ... ...she could only lay there in her tomb. Her sterile thoughts were filth. Her harmonic bliss in aloneness was, in illness, devestating and lonely as on... ...d of rain and so he went to a window but could not see the substance of this harmonic pattering in external darkness. He listened to its orphic sound... ...ted of the same cooking, shopping, and cleaning she always did plus maternal functions that varied with each woman like checking on him at strange hou... ... which she would watch for an hour each late afternoon on the Tijuana beach: harmonic and not missing a beat; the mesmerizing splashes cleansing the c...

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...o move his legs so why should I worry about his other bodily movements and functions; and I couldn't be prouder of a husband who explores the human so... ...e sweet bait of pleasure and so from the perspective of carrying out those functions that biological creatures were meant to, the physical and perhaps... ...d evoke little or no thought. Eating and eliminating wastes were necessary functions so the human mind refused to consider them abhorrent and animalis... ...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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...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... ... was to be sure to remember it.” Charley dried her eyes and entered on her functions, go ing in her matronly little way about and about the room and ...

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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... ... was to be sure to remember it.” Charley dried her eyes and entered on her functions, go ing in her matronly little way about and about the room and ... ...lean hands. Those interests are now paramount in this office. My digestive functions, as you may have heard me mention, are not in a good state, and r...

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...lean hands. Those interests are now paramount in this office. My digestive functions, as you may have heard me mention, are not in a good state, and r... ...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...

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... flying fingers, like Timotheus, from shift to shift —struck arpeggios and harmonic tones, but without exciting any of the astonishment which I had ex... ...ddes applied the tankard to his mouth, as if the discharge of these bodily functions by another had awakened his sympathies in an uncontrollable degre...

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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... ...rme jewellers, or “goldsmiths,” as they were styled, who performed all the functions of modern bankers from the period of the parliamentary war to the... ...st with Napoleon be was unequal to the fatigues of such a post. In all the functions of peace, however, he continued to be a useful servant to the las... ...e pretensions of all who stood in the way. At that time, and even yet, the functions of a literary man were very important in Germany; the popular min...

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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... ...erve the ideal in eating and drink- ing, in drawing the breath, and in the functions of life? Beauty must come back to the useful arts, and the dis- t... ...lly ideal and beautiful, draws us to the consideration of the na- ture and functions of the Poet, or the man of Beauty; to the means and materials he ...