The rank of the semigroup of transformations stabilising a partition of a finite set

Let $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.

Regular statistical convergence of multiple sequences

The concepts of statistical convergence of single and multiple sequences of complex numbers were introduced in [2] and [8], respectively. In this paper, we introduce the concept indicated in the title. We prove that if a d-multiple sequence is regularly statistically convergent, then its statistical limit can be computed as the iterated statistical limit of the statistical limits of its subsequences which correspond to an arbitrary partition of the index set {1,2,..., d}, d ≥ 2. As an application, we prove that if a function ƒ is in L

REVIEW OF CHAOS COMMUNICATION BY FEEDBACK CONTROL OF SYMBOLIC DYNAMICS

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also specifically to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the flow of a chaotic differential equation to carry a desired message.

COMPUTATION OF DIMENSION POLYNOMIALS

The consideration of differential versions of Hilbert dimension polynomials is due to A. Einstein [1] and E. Kolchin [2] (one can find the coverage of the theory of differential dimension polynomials in [6]). In this paper we introduce the notion of a dimension polynomial of a subset of ℕm associated with arbitrary partition of the set {1,…, m} into disjoint nonempty subsets (m∈ℕ, ℕ denoting the set of all nonnegative integers). The theory of such polynomials is developed. The importance of our considerations is connected with the fact that the computation of differential and difference dimen sion polynomials may be reduced to the computation of some dimension polynomials of subsets of ℕm where m∈ℕ (see [3, p. 115], [5]). We also give some methods and algorithms for computation of dimension polynomials.