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The dynamics of conjunctive and disjunctive Boolean networks
, 2008
"... The relationship between the properties of a dynamical system and the structure of its defining equations has long been studied in many contexts. Here we study this problem for the class of conjunctive (resp. disjunctive) Boolean networks, that is, Boolean networks in which all Boolean functions are ..."
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Cited by 10 (2 self)
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The relationship between the properties of a dynamical system and the structure of its defining equations has long been studied in many contexts. Here we study this problem for the class of conjunctive (resp. disjunctive) Boolean networks, that is, Boolean networks in which all Boolean functions are constructed with the AND (resp. OR) operator only. The main results of this paper describe network dynamics in terms of the structure of the network dependency graph (topology). For a given such network, all possible limit cycle lengths are computed and lower and upper bounds for the number of cycles of each length are given. In particular, the exact number of fixed points is obtained. The bounds are in terms of structural features of the dependency graph and its partially ordered set of strongly connected components. For networks with strongly connected dependency graph, the exact cycle structure is computed.
On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators
 MATH. COMP
, 2010
"... In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degree growth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates ..."
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Cited by 6 (6 self)
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In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degree growth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.
Fixed Points in Discrete Models for Regulatory Genetic Networks
, 2007
"... It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that ..."
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Cited by 3 (0 self)
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It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that enable the detection of steady states that are modeled by fixed points in discrete finite dynamical systems. We discuss two algebraic models, a univariate model and a multivariate model. We show that these two models are equivalent and that one can be converted to the other by means of a discrete Fourier transform. We give a new, more general definition of a linear finite dynamical system and we give a necessary and sufficient condition for such a system to be a fixed point system, that is, all cycles are of length one. We show how this result for generalized linear systems can be used to determine when certain nonlinear systems (monomial dynamical systems over finite fields) are fixed point systems. We also show how it is possible to determine in polynomial time when an ordinary linear system (defined over a finite field) is a fixed point system. We conclude with a necessary condition for a univariate finite dynamical system to be a fixed point system.
Monomial dynamical systems of dimension one over finite fields, Acta Arith.148
, 2011
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A mathematical formalism for agentbased modeling
 in Encyclopedia of Complexity and Systems Science
, 2009
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Acceleration of Finite Field Arithmetic with an Application to Reverse Engineering Genetic Networks
"... Finite field arithmetic plays an important role in a wide range of applications. This research is originally motivated by an application of computational biology where genetic networks are modeled by means of finite fields. Nonetheless, this work has application in various research fields including ..."
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Finite field arithmetic plays an important role in a wide range of applications. This research is originally motivated by an application of computational biology where genetic networks are modeled by means of finite fields. Nonetheless, this work has application in various research fields including digital signal processing, error correcting codes, ReedSolomon encoders/decoders, elliptic curve cryptosystems, or computational and algorithmic aspects of commutative algebra. We present a set of efficient algorithms for finite field arithmetic over GF(2 m), which are implemented on a High Performance Reconfigurable Computing platform. In this way, we deliver new and efficient designs on Field Programmable Gate Arrays (FPGA) for accelerating finite field arithmetic. Among the arithmetic operations, the most frequently used and time consuming operation is multiplication. We have designed a fast and spacesaving multiplier, which has been used for creating other efficient architectures for inversion and exponentiation which have in turn been used for developing a new and efficient architecture for finite field interpolation. Here, the bitlevel representation of the elements in GF(2 m) and some special structures in the formulation of multiplication and
ON THE ALGEBRAIC GEOMETRY OF POLYNOMIAL DYNAMICAL SYSTEMS
, 2008
"... This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks. It is shown that several problems relating to their struc ..."
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This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks. It is shown that several problems relating to their structure and dynamics, as well as control theory, can be formulated and solved in the language of algebraic geometry.
Examinateurs: Joan Daemen Principal Cryptographer
"... Département de formation doctorale en automatique École doctorale IAEM Lorraine Contrôle, synchronisation et chiffrement THÈSE présentée et soutenue publiquement le 03/10/2012 pour l’obtention du ..."
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Département de formation doctorale en automatique École doctorale IAEM Lorraine Contrôle, synchronisation et chiffrement THÈSE présentée et soutenue publiquement le 03/10/2012 pour l’obtention du
doi:10.1093/comjnl/bxm100 Computational Aspects of Monomial Dynamical Systems
, 2007
"... We consider the dynamics of x 7! x n, where n 2 is an integer, over the multiplicative group modulo p k, where k is a positive integer and p an odd prime. This paper is a review of earlier results by the author, but new results are also contained. Possible applications to pseudorandom number generat ..."
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We consider the dynamics of x 7! x n, where n 2 is an integer, over the multiplicative group modulo p k, where k is a positive integer and p an odd prime. This paper is a review of earlier results by the author, but new results are also contained. Possible applications to pseudorandom number generation will be discussed. The main results are a description of the preperiodic points and an algorithm to find the longest possible cycle. The preperiodic points form trees, all isomorphic as graphs to the preperiodic points of the fixed point 1. When n is a prime, different from p, we can describe the tree structure completely. A formula for the length of the longest cycle is presented. We can find one of the longest cycles of the monomial system using a primitive root modulo p k as an initial value.
LINEAR DYNAMICAL SYSTEMS OVER FINITE RINGS
, 810
"... Abstract. The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem ..."
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Abstract. The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system’s characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring R such that R  = q, where q is a positive integer, the algorithm determines whether a given linear system over R n is a fixed point system or not in time O(n 3 log(n log(q))). 1.