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Permuting Operations on Strings — Their Permutations and Their
 Twente University of Technology
"... Abstract — We study some lengthpreserving operations on strings that permute the symbol positions in strings. These operations include some wellknown examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on Archimedes spir ..."
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Abstract — We study some lengthpreserving operations on strings that permute the symbol positions in strings. These operations include some wellknown examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on Archimedes spiral. Such a permuting operation X gives rise to a family {p(X,n)}n≥2 of similar permutations. We investigate the structure and the order of the cyclic group generated by such a permutation p(X,n). We call an integer n Xprime if p(X,n) consists of a single cycle of length n (n ≥ 2). Then we show some properties of these Xprimes, particularly, how Xprimes are related to X ′primes as well as to ordinary prime numbers.
THE FELINE JOSEPHUS PROBLEM
"... Abstract. In the classic Josephus problem, elements 1, 2,..., n are placed in order around a circle and a skip value k is chosen. The problem proceeds in n rounds, where each round consists of traveling around the circle from the current position, and selecting the kth remaining element to be elimin ..."
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Abstract. In the classic Josephus problem, elements 1, 2,..., n are placed in order around a circle and a skip value k is chosen. The problem proceeds in n rounds, where each round consists of traveling around the circle from the current position, and selecting the kth remaining element to be eliminated from the circle. After n rounds, every element is eliminated. Special attention is given to the last surviving element, denote it by j. We generalize this popular problem by introducing a uniform number of lives ℓ, so that elements are not eliminated until they have been selected for the ℓth time. We prove two main results: 1) When n and k are fixed, then j is constant for all values of ℓ larger than the nth Fibonacci number. In other words, the last surviving element stabilizes with respect to increasing the number of lives. 2) When n and j are fixed, then there exists a value of k that allows j to be the last survivor simultaneously for all values of ℓ. In other words, certain skip values ensure that a given position is the last survivor, regardless of the number of lives. For the first result we give an algorithm for determining j (and the entire sequence of selections) that uses O(n2) arithmetic operations. “un gatto ha sette vite” 1.
The Feline Josephus Problem
"... 60 61 62 63 64 65 Theory of Computing Systems manuscript No. (will be inserted by the editor) ..."
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60 61 62 63 64 65 Theory of Computing Systems manuscript No. (will be inserted by the editor)
REPRESENTING REAL NUMBERS IN A GENERALIZED Numeration System
, 2009
"... ... numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show that our framework can be applied to the rational base numeration systems. ..."
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... numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show that our framework can be applied to the rational base numeration systems.