Related papers: Cycles in Repeated Exponentiation Modulo $p^n$

In these notes we investigate the rings of real polynomials in four variables, which are invariant under the action of the reflectiongroups [3,4,3] and [3,3,5]. It is well known that they are rationally generated in degree 2,6,8,12 and…

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we…

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number $n$ of elements, or a…

We consider systems characterized by the presence of a rapidly oscillating force. A general method is presented for the construction of the effective action governing the large-scale nonlinear dynamics of such systems order by order in…

Many sequences of $p$-adic integers project modulo $p^\alpha$ to $p$-automatic sequences for every $\alpha \geq 0$. Examples include algebraic sequences of integers, which satisfy this property for every prime $p$, and some cocycle…

We show that the Poincar\'e return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.

Periodic signals play an important role in daily lives. Although conventional sequential models have shown remarkable success in various fields, they still come short in modeling periodicity; they either collapse, diverge or ignore details.…

In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer…

The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…

Dynamical systems---by which we mean machines that take time-varying input, change their state, and produce output---can be wired together to form more complex systems. Previous work has shown how to allow collections of machines to…

Some polynomials $P$ with rational coefficients give rise to well defined maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto P(x)+r\Z$. More generally, there are polynomials in several variables with tuples of rational…

The orbits of the reversible differential system $\dot{x}=-y$, $\dot{y}=x$, $\dot{z}=0$, with $x,y \in R$ and $z\in R^d$, are periodic with the exception of the equilibrium points $(0,0, z)$. We compute the maximum number of limit cycles…

Answering a question of Bona, it is shown that for n>1 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,...,n} is 1/2 if n is odd and 1/2 - 2/(n-1){n+2) if n is even. Another result concerns…

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct…

We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree $k$ over $\mathbb{F}_q$ is induced by an action from…

We extend the notions of multipole and subsystem symmetries to more general {\it spatially modulated} symmetries. We uncover two instances with exponential and (quasi)-periodic modulations, and provide simple microscopic models in one, two…

A numerical technique used to solve boundary value problems is modified to find periodic steady-state solutions of nonautonomous dynamical systems. The technique uses a matrix representation of the time derivative obtained through…

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer. Let…

The Fubini numbers count the permutations of horse racing where ties are possible. The closely related $r$-horse numbers count the finishes of a horse race where some subset of $r$ horses agree to finish the race in a specific relative…

Self-regulatory models are common in nature, as described e.g. in (\cite{mur}), (\cite{ha}) and (\cite{Gb}).\\ Let us consider a system made up of a number of glands as a motivation. Each gland secretes a hormone that allows secretion in…