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Modulo One Uniform Distribution of the Sequence of Logarithms of Certain Recursive Sequences
 Fibonacci Quarterly
"... Let {x.}° ° be a sequence of real numbers with corresponding fractional parts {/3.}°°, where 0. = x. [x.] and the bracket denotes the greatest integer function. For each n> 1, we define the function F on [ 0,1] so that F (x) is the number of those terms among /31 $ • • • , /3R whichlie in the ..."
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Let {x.}° ° be a sequence of real numbers with corresponding fractional parts {/3.}°°, where 0. = x. [x.] and the bracket denotes the greatest integer function. For each n> 1, we define the function F on [ 0,1] so that F (x) is the number of those terms among /31 $ • • • , /3R whichlie in the interval [0,x), divided by n. Then {x.} is said to be uniformly distributed modulo one iff n lim oo—*n F (x) = x for all x € TO,
What is a Random Sequence
 The Mathematical Association of America, Monthly
, 2002
"... there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a ..."
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there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a
Resource relocation on . . .
, 2010
"... The necessary information to optimally serve sequential requests at the vertices of an undirected, unweighted graph with a single mobile resource is a known result of Chung, Graham, and Saks; however, generalizations of this concept to directed and weighted graphs present unforeseen and surprising c ..."
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The necessary information to optimally serve sequential requests at the vertices of an undirected, unweighted graph with a single mobile resource is a known result of Chung, Graham, and Saks; however, generalizations of this concept to directed and weighted graphs present unforeseen and surprising changes in the necessary lookahead for strategic optimization. A pair of edges of unequal weights and opposite orientation can serve to simulate a communication or transportation connection with asymmetric costs, as may arise in a transportation network from prevailing winds or elevation changes, or in a communication network from aDSL or a similar technology. This research explores the complications introduced by asymmetric connections within even very small networks. We consider the dynamic relocation problem on a twovertex system and find that, even in this simplest possible asymmetric graph, the necessary lookahead for optimal relocation may be arbitrarily large. This investigation also gives rise to a lineartime algorithm to determine the optimizing realtime response to any request sequence which uniquely determines an optimal response.
A TRANSFORMATION TO SOLVE INDEFINITE QUADRATIC EQUATIONS IN INTEGERS
"... Abstract. The paper proposes a new method, called the Fast Quadratic Transform (FQT), to solve the general indefinite twovariable quadratic equation in integers. The paper presents the new approach, discusses its properties, and provides a comparative evaluation with the classical technique. The FQ ..."
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Abstract. The paper proposes a new method, called the Fast Quadratic Transform (FQT), to solve the general indefinite twovariable quadratic equation in integers. The paper presents the new approach, discusses its properties, and provides a comparative evaluation with the classical technique. The FQT is demonstrated to be markedly superior for all cases in which it applies, including examples for more than sixty percent of the discriminants through two hundred. Consider the equation Prologue
Strong Normality of Numbers
"... “... the problem of knowing whether or not the digits of a number like √ 2 satisfy all the laws one could state for randomly chosen digits, still seems... to be one of the most outstanding questions facing mathematicians.” Émile Borel [Borel 1950] Champernowne’s number is the bestknown example of a ..."
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“... the problem of knowing whether or not the digits of a number like √ 2 satisfy all the laws one could state for randomly chosen digits, still seems... to be one of the most outstanding questions facing mathematicians.” Émile Borel [Borel 1950] Champernowne’s number is the bestknown example of a normal number, but its digits are highly patterned. We present graphic evidence of the patterning and review some relevant results in normality. We propose a strong normality criterion based on the variance of the normal approximation to a binomial distribution. Allmost all numbers pass the new test but Champernowne’s number fails to be strongly normal. 1
PERIODIC SOLUTIONS OF ARBITRARY LENGTH IN A SIMPLE INTEGER ITERATION
, 2005
"... We prove that all solutions to the nonlinear secondorder difference equation in integers yn+1 = ayn − yn−1, {a∈R: a  < 2, a = 0,±1}, y0, y1 ∈ Z, are periodic. The firstorder system representation of this equation is shown to have selfsimilar and chaotic solutions in the integer plane. Copyr ..."
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We prove that all solutions to the nonlinear secondorder difference equation in integers yn+1 = ayn − yn−1, {a∈R: a  < 2, a = 0,±1}, y0, y1 ∈ Z, are periodic. The firstorder system representation of this equation is shown to have selfsimilar and chaotic solutions in the integer plane. Copyright © 2006 Dean Clark. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Searching for ApéryStyle Miracles [Using, InterAlia, the Amazing AlmkvistZeilberger Algorithm]
, 1934
"... Like in all the joint articles of the authors, this article is not the main point. It may be viewed as a user’s manual for the much more important Maple package, NesApery (BTW, nes means ‘miracle ’ in Hebrew). It may be obtained directly from the following url: ..."
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Like in all the joint articles of the authors, this article is not the main point. It may be viewed as a user’s manual for the much more important Maple package, NesApery (BTW, nes means ‘miracle ’ in Hebrew). It may be obtained directly from the following url:
RAMANUJAN’S EISENSTEIN SERIES AND POWERS OF
"... Abstract. In this article, we use the theory of elliptic functions to construct theta function identities which are equivalent to Macdonald’s identities for A2,B2 and G2. Using these identities, we express, for d = 8, 10 or 14, certain theta functions in the form ηd(τ)F (P,Q,R), where η(τ) is Dedeki ..."
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Abstract. In this article, we use the theory of elliptic functions to construct theta function identities which are equivalent to Macdonald’s identities for A2,B2 and G2. Using these identities, we express, for d = 8, 10 or 14, certain theta functions in the form ηd(τ)F (P,Q,R), where η(τ) is Dedekind’s etafuncion, and F (P,Q,R) is a polynomial in Ramanujan’s Eisenstein series P,Q, and R. We also derive identities in the case when d = 2. These lead to a new expression for η26(τ). This work generalizes the results for d = 1 and d = 3 which were given by Ramanujan on page 369 of the “Lost Notebook”. 1.