The density of a permutation pattern π in a permutation σ is the proportion of subsequences of σ of length |π| that are isomorphic to π. The maximal value of the density is found for several patterns π, and asymptotic upper and lower bounds for the maximal density are found in several other cases. The results are generalised to sets of patterns and the maximum density is found for all sets of l...

An abelian square is a nonempty string of length 2n where the last n symbols form a permutation of the first n symbols. Similarly, an abelian r’th power is a concatenation of r blocks, each of length n, where each block is a permutation of the first n symbols. In this note we point out that some familiar combinatorial identities can be interpreted in terms of abelian powers. We count the number...

We develop a multi-length scale (multi-fractal) theory for the effect of rock heterogeneity on the growth of the mixing layer of the flow of a passive tracer through porous media. The multi-fractal exponent of the size of the mixing layer is determined analytically from the statistical properties of a random velocity (permeability) field. The anomalous diffusion of the mixing layer can occur bo...

We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k d...

We present experimental data on the light scattering properties of linear chains of gold nanoparticles with up to six nanoparticles and an interparticle spacing of 1 nm. A red shift of the surface plasmon resonance with increasing chain length is observed. An exponential model applied to the experimental data allows determination of an asymptotic maximum resonance at a chain length of 10-12 par...

a chaos control method suggested by erjaee has been reviewed. it has been shown that thistechnique can be applied in various evolutionary systems of 2-dimensional types. the method has beenapplied for cases of the henon map, as well as burger’s map. the limitations of the control techniquehave also been discussed by considering the standard map and the gumowski-mira map. the resultsobtained thr...

We study the asymptotic regime for the relativistic Vlasov-Maxwell-FokkerPlanck system which corresponds to a mean free path small compared to the Debye length, chosen as an observation length scale, combined to a large thermal velocity assumption. We are led to a convection-diffusion equation, where the convection velocity is obtained by solving a Poisson equation. The analysis is performed in...

We show that every self-orthogonal code over $\mathbb F_q$ of length $n$ can be extended to a self-dual code, if there exists self-dual codes of length $n$. Using a family of Galois towers of algebraic function fields we show that over any nonprime field $\mathbb F_q$, with $q\geq 64$, except possibly $q=125$, there are self-dual codes better than the asymptotic Gilbert--Varshamov bound.

The time asymptotic decay of the variance of a passive scalar in a chaotic flow is studied. Two mechanisms for this decay, which involve processes at short and long length scales, respectively, are considered. The validity of the short length scale mechanism, which is based on Lagrangian stretching theory, is discussed. We also investigate the regimes of applicability and observable signatures ...