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Student ID: n5105269 Course: IT29 Honours in IT Primary Supervisor: Jim Hogan
"... I would first of all like to give special thanks to Dr James Michael Hogan and Dr Jin hai Cai for supervising my honours project. Thank you to Jim for introducing me to localisation of software research area, and for all academic and personal help. I would like to thank Jim for providing revision fo ..."
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I would first of all like to give special thanks to Dr James Michael Hogan and Dr Jin hai Cai for supervising my honours project. Thank you to Jim for introducing me to localisation of software research area, and for all academic and personal help. I would like to thank Jim for providing revision
In honour of Professor R. P. Bambah on his 80th birthday 1. Chowla’s problem
"... In 1969, Chowla raised the following question: Suppose f (n) is a rational valued, periodic function mod q where q(> 1) is a prime number. Then does the infinite series S = n=1 f (n) n (1.1) never vanish? Throughout the paper, we assume that f is a nonvanishing number theoretic function. Much e ..."
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In 1969, Chowla raised the following question: Suppose f (n) is a rational valued, periodic function mod q where q(> 1) is a prime number. Then does the infinite series S = n=1 f (n) n (1.1) never vanish? Throughout the paper, we assume that f is a nonvanishing number theoretic function. Much earlier, in 1949, Chowla [C] himself showed that S = 0 if f is an odd function and q prime with q−12 also prime. Around 1970, Siegel removed the restriction that q−12 is prime in the result of Chowla. In 1973, Baker, Birch and Wirsing [BBW] solved Chowla’s question completely. In fact, Chowla himself solved the question around the same time. The result of Baker, Birch and Wirsing is more general which we state below. Theorem A ([BBW]). Suppose f (n) is an algebraic valued, periodic function mod q. Then the infinite sum S defined in (1.1) does not vanish if f satisfies the following conditions. (i) f (r) = 0 if 1 < gcd(r, q) < q (ii) The cyclotomic polynomial q is irreducible over Q(f (1),..., f (q)). The affirmative answer to Chowla’s question is immediate since (i) is vacuous when q is prime and (ii) follows when f is rational valued as it is well known that q is 171 172 N. Saradha irreducible over Q. It was shown in [BBW] that conditions (i) and (ii) are necessary. For example, let q = p2 where p is prime and let f be defined by n=1 f (n) ns = (1 − p1−s)2ζ(s) where ζ(s) is the Riemann Zeta function. For p = 2, we get
Global Optimization Algorithms  Theory and Application
, 2011
"... This ebook is devoted to Global Optimization algorithms, which are methods for finding solutions of high quality for an incredible wide range of problems. We introduce the basic concepts of optimization and discuss features which make optimization problems difficult and thus, should be considered ..."
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Cited by 94 (26 self)
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This ebook is devoted to Global Optimization algorithms, which are methods for finding solutions of high quality for an incredible wide range of problems. We introduce the basic concepts of optimization and discuss features which make optimization problems difficult and thus, should be considered when trying to solve them. In this book, we focus on
Enhancing the Nuprl Proof Development System and Applying it to Computational Abstract Algebra
, 1995
"... This thesis describes substantial enhancements that were made to the software tools in the Nuprl system that are used to interactively guide the production of formal proofs. Over 20,000 lines of code were written for these tools. Also, a corpus of formal mathematics was created that consists of rou ..."
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Cited by 47 (4 self)
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This thesis describes substantial enhancements that were made to the software tools in the Nuprl system that are used to interactively guide the production of formal proofs. Over 20,000 lines of code were written for these tools. Also, a corpus of formal mathematics was created that consists of roughly 500 definitions and 1300 theorems. Much of this material is of a foundational nature and supports all current work in Nuprl. This thesis concentrates on describing the half of this corpus that is concerned with abstract algebra and that covers topics central to the mathematics of the co...
Distribution modulo 1 and the lexicographic world
, 2009
"... In honour of Paulo Ribenboim on the occasion of his 80th birthday ABSTRACT. We give a complete description of the minimal intervals containing all fractional parts {ξ2 n}, for some positive real number ξ, and for all n ≥ 0. ..."
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Cited by 4 (1 self)
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In honour of Paulo Ribenboim on the occasion of his 80th birthday ABSTRACT. We give a complete description of the minimal intervals containing all fractional parts {ξ2 n}, for some positive real number ξ, and for all n ≥ 0.
Lambda Calculus with Explicit Recursion
 Information and Computation
, 1996
"... This paper is concerned with the study of calculus with explicit recursion, namely of cyclic graphs. The starting point is to treat a graph as a system of recursion equations involving terms, and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible fo ..."
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Cited by 43 (5 self)
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This paper is concerned with the study of calculus with explicit recursion, namely of cyclic graphs. The starting point is to treat a graph as a system of recursion equations involving terms, and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible for firstorder term rewriting. Surprisingly, now the confluence property breaks down in an essential way. Confluence can be restored by introducing a restraining mechanism on the `substitution' operation. This leads to a family of graph calculi, which can be seen as an extension of the family of oecalculi (calculi with explicit substitution). While the oecalculi treat the letconstruct as a firstclass citizen, our calculi support the letrec, a feature that is essential to reason about time and space behavior of functional languages and also about compilation and optimizations of programs. CR Subject Classification (1991): D.1.1, D.3.1, F.1.1, F.4.1, F.4.2 Keywords & Phrases: lambda cal...
unknown title
"... Kanellakis Theory and Practice Award. This is one of the ACM’s top prizes and recognizes his role in developing Gröbner bases. We congratulate Bruno for this outstanding achievement and honour! ACM Official Press Release NEW YORK, May 13, 2008 – ACM (the Association for Computing Machinery) has reco ..."
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Kanellakis Theory and Practice Award. This is one of the ACM’s top prizes and recognizes his role in developing Gröbner bases. We congratulate Bruno for this outstanding achievement and honour! ACM Official Press Release NEW YORK, May 13, 2008 – ACM (the Association for Computing Machinery) has
Pi ˙ngala’s Fountain
"... Pi ˙ngala is a mathematician (c. 300 BCE) who authored Chandah.´sāstra (the science of Sanskrit metres). This is a brief note on a mathematical art project completed at the Chennai Mathematical Institute to honour Pi ˙ngala who was responsible for the discovery of binary expansion of numbers in the ..."
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Pi ˙ngala is a mathematician (c. 300 BCE) who authored Chandah.´sāstra (the science of Sanskrit metres). This is a brief note on a mathematical art project completed at the Chennai Mathematical Institute to honour Pi ˙ngala who was responsible for the discovery of binary expansion of numbers
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