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The Tiling of the Hyperbolic 4D Space by the 120cell is Combinatoric
 Journal of Universal Computer Science
"... Abstract: The splitting method was defined by the author in [Margenstern 2002a, Margenstern 2002d]. It is at the basis of the notion of combinatoric tilings. As a consequence of this notion, there is a recurrence sequence which allows us to compute the number of tiles which are at a fixed distance f ..."
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Abstract: The splitting method was defined by the author in [Margenstern 2002a, Margenstern 2002d]. It is at the basis of the notion of combinatoric tilings. As a consequence of this notion, there is a recurrence sequence which allows us to compute the number of tiles which are at a fixed distance from a given tile. A polynomial is attached to the sequence as well as a language which can be used for implementing cellular automata on the tiling. The goal of this paper is to prove that the tiling of hyperbolic 4D space is combinatoric. We give here the corresponding polynomial and, as the first consequence, the language of the splitting is not regular, as it is the case in the tiling of hyperbolic 3D space by rectangular dodecahedra which is also combinatoric. 1 Key Words: cellular automata, hyperbolic plane
Arithmetic Dynamics
, 2002
"... This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi) conjugate a given continuous (or measurepres ..."
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This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi) conjugate a given continuous (or measurepreserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: • Betaexpansions, i.e., the radix expansions in noninteger bases; • “Rotational ” expansions which arise in the problem of encoding of irrational rotations of the circle; • Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodictheoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create “redundant” representations (those whose space of “digits ” is a priori larger than necessary)
The Subword Complexity of a TwoParameter Family of Sequences
"... We determine the subword complexity of the characteristic functions of a twoparameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a f ..."
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We determine the subword complexity of the characteristic functions of a twoparameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a fixed positive irrational number. The characteristic functions of such sequences have been shown to have subword complexity n + 1. We show that every sequence in the extended family has subword complexity O(n). 1 Introduction Denote by Z0 and Z?0 the set of nonnegative integers and positive integers respectively. Given two heaps of finitely many tokens, we define a 2player heap game as follows. There are two types of moves: 1. Remove any positive number of tokens from a single heap. 2. Remove k ? 0 tokens from one heap and l ? 0 from the other. Here k and l are constrained by the condition: 0 ! k l ! sk + t, where s and t are predetermined positive integers. The player who reaches a stat...
On the cost and complexity of the successor function
 In Proc. WORDS 2007
, 2009
"... Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the nth word of a genealogically ordered language L onto the (n+1)th word of L. We show that, if t ..."
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Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the nth word of a genealogically ordered language L onto the (n+1)th word of L. We show that, if the ratio of the number of elements of length n +1overthenumber of elements of length n of the language has a limit β>1, then the amortized cost of the successor function is equal to β/(β − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). 1
BetaIntegers As A Group
, 1999
"... this paper. The fiintegers are defined via a numeration system with base fi, see below. In ..."
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this paper. The fiintegers are defined via a numeration system with base fi, see below. In
Digital Blocks In Linear Numeration Systems
 Proceedings of the Number Theory Conference
, 1997
"... We establish quantitative refinements of recent results on the occurrence of blocks in digital expansions. Furthermore we extend these results to linear numeration systems. ..."
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We establish quantitative refinements of recent results on the occurrence of blocks in digital expansions. Furthermore we extend these results to linear numeration systems.
Combinatorial Representation of Generalized Fibonacci Numbers
, 1991
"... New formulae are presented which express various generalizations of Fibonacci numbers as simple sums of binomial and multinomial coefficients. The equalities are inferred from the special properties of the representations of the integers in certain numeration systems. ..."
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New formulae are presented which express various generalizations of Fibonacci numbers as simple sums of binomial and multinomial coefficients. The equalities are inferred from the special properties of the representations of the integers in certain numeration systems.
Algebraic synthesis of Fibonacci switched capacitor converters
 IEEE Conference on Microwaves, Communications, Antennas and Electronics Systems (COMCAS) 2011
"... Abstract — A simple algebraic approach to synthesis Fibonacci Switched Capacitor Converters (SCC) was developed. The proposed approach reduces the power losses by increasing the number of target voltages. The synthesized Fibonacci SCC is compatible with the binary SCC and uses the same switch networ ..."
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Abstract — A simple algebraic approach to synthesis Fibonacci Switched Capacitor Converters (SCC) was developed. The proposed approach reduces the power losses by increasing the number of target voltages. The synthesized Fibonacci SCC is compatible with the binary SCC and uses the same switch network. This feature is unique, since it provides the option to switch between the binary and Fibonacci target voltages, increasing thereby the resolution of attainable conversion ratios. The theoretical results were verified by experiments. Index terms — Charge pump, Fibonacci numbers, redundant number system, signeddigit representation, switched capacitor.
Odometers on regular languages
 Objective 8: Prevent the invasion of the zebra mussel into California. Goal 6: Water and Sediment Quality Improve
"... Abstract. Odometers or “adding machines ” are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A, <) is a total ..."
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Abstract. Odometers or “adding machines ” are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A, <) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a onetoone correspondence between N and L. In this general setting, the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer like continuity, injectivity, surjectivity, minimality,... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with βnumerations. 1.
OnLine Digit Set Conversion in Real Base
 Theoret. Comp. Sci
, 2000
"... Let fi be a real number ? 1. The digit set conversion between real numbers represented in fixed base fi is shown to be computable by an online algorithm, and thus is a continuous function. When fi is a Pisot number the digit set conversion is computable by an online finite automaton. 1 Introdu ..."
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Let fi be a real number ? 1. The digit set conversion between real numbers represented in fixed base fi is shown to be computable by an online algorithm, and thus is a continuous function. When fi is a Pisot number the digit set conversion is computable by an online finite automaton. 1 Introduction In computer arithmetic, online computation consists of performing arithmetic operations in Most Significant Digit First (MSDF) mode, digit serially after a certain latency delay [8]. This allows the pipelining of different operations such as addition, multiplication and division. It is also appropriate for the processing of real numbers having infinite expansions: it is well known that when multiplying two real numbers, only the left part of the result is significant. To be able to perform online addition, it is necessary to use a redundant number system (see [19], [8]). On the other hand, a function is computable by a finite automaton if it needs only a finite auxiliary storage me...