scholarly journals Monomial dynamical systems of dimension one over finite fields

2011 ◽  
Vol 148 (4) ◽  
pp. 309-331 ◽  
Author(s):  
Min Sha ◽  
Su Hu
Keyword(s):  
Dynamical Systems ◽  
Finite Fields ◽  
Dimension One

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

scholarly journals PREDICTING PATTERN FORMATION IN PARTICLE INTERACTIONS

2012 ◽  
Vol 22 (supp01) ◽  
pp. 1140002 ◽  
Author(s):  
JAMES H. VON BRECHT ◽  
DAVID UMINSKY ◽  
THEODORE KOLOKOLNIKOV ◽  
ANDREA L. BERTOZZI
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Ground State ◽  
Self Assembly ◽  
Systems Approach ◽  
Particle Interactions ◽  
Space Filling ◽  
System Of Particles ◽  
Large Systems ◽  
Dimension One

Large systems of particles interacting pairwise in d dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere (in 3D) or a ring (in 2D). Localized, space-filling, co-dimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a nonlocal linear stability analysis for particles uniformly distributed on a d - 1 sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in self-assembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

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