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.

Abstract.A new method for computing the discrete Fourier transform (DFT) of data endowed with linear symmetries is presented. The method minimizes operations and memory space requirements by eliminating redundant data and computations induced by the symmetry on the DFT equations. The arithmetic complexity of the new method is always lower, and in some cases significantly lower than that of its predecesor. A parallel version of the new method is also discussed. Symmetry-aware DFTs are crucial in the computer determination of the atomic structure of crystals from x-ray diffraction intensity patterns. (N d log N ) operations. Although this complexity bound cannot be improved for general DFT computations, some attempts to reduce the actual operation count and memory space requirements have been made for problems whose data is endowed with redundancies, such as x-ray crystal diffraction intensity data. In this article we review

A new backtracking algorithm is developed for generating classes of permutations, that are invariant under the group G 4 of rigid motions of the square generated by reflections about the horizontal and vertical axes. Special cases give a new algorithm for generating solutions of the classical n-queens problem, as well as a new algorithm for generating Costas sequences, which are used in encoding radar and sonar signals. Parallel implementations of this latter algorithm have yielded new Costas sequences for length n, 19 ≤ n ≤ 24.

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Recently criteria for determining when a certain type of nonlinear discrete dynamical system is a fixed point system have been developed. This theory can be used to determine if certain events modeled by those systems reach a steady state. In this work we formalize the idea of a "stabilizable" discrete dynamical system. We present necessary and sufficient conditions for a Boolean monomial dynamical control system to be stabilizable in terms of properties of the dependency graph associated with the system. We use the equivalence of periodicity of the dependency graph and loop numbers to develop a new O(n 2 log n) algorithm for determining the loop numbers of the strongly connected components of the dependency graph, and hence a new O(n 2 log n) algorithm for determining when a Boolean monomial dynamical system is a fixed point system. Finally, we show how this result can be used to determine if a Boolean monomial dynamical control system is stabilizable in time O(n 2 log n).