Recurrence relations with two indices and even trees

Recently, the author, Mansour, introduced a combinatorial problem, called Hobby's problem, to study different types of recurrence relations with two indices. Moreover, he presented several recurrence relations with two indices related to Dyck paths and Schröder paths. In this paper, we generalize Hobby's problem to study other types of recurrence relations with two indices for which a combinatorial method provides a complete solution. Combinatorially, we describe these recurrence relations as a set of lattice paths in the second octant of the plane integer lattice, and then we map bijectively these lattice paths to the set of even trees. Analytically, we use the kernel method technique to solve these recurrence relations.     

Recurrence relations in counting the pattern 13-2 in flattened permutations

We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method.     

Adaptive fuzzy control‐based projective synchronization of uncertain nonaffine chaotic systems

This article aims to introduce a projective synchronization approach based on adaptive fuzzy control for a class of perturbed uncertain multivariable nonaffine chaotic systems. The fuzzy‐logic systems are employed to approximate online the uncertain functions. A Lyapunov approach is used to design the parameter adaptation laws and to demonstrate the boundedness of all signals of the closed‐loop system as well as the convergence of the synchronization errors to bounded residual sets. Finally, numerical simulation results are presented to verify the feasibility and effectiveness of the proposed synchronization system based on fuzzy adaptive controller. © 2014 Wiley Periodicals, Inc. Complexity 21: 180–192, 2015     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.     

Ascents of size less than <Emphasis Type="Italic">d</Emphasis> in compositions

A composition of a positive integer n is a finite sequence π₁π₂...π _m of positive integers such that π₁+...+π _m = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if π _i+1 ≥ π _i +d (respectively, π _i < π _i+1 < π _i + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable generating function for the number of compositions of n according to the number of parts, ascents of size d or more, ascents of size less than d, descents and levels. Also, we extend the results of Brennan and Knopfmacher to the case of ascents of size less than d. More precisely, we determine the mean, variance and limiting distribution of the number of ascents of size less than d in the set of compositions of n.     

Pattern avoiding partitions and Motzkin left factors

Let L _n , n ≥ 1, denote the sequence which counts the number of paths from the origin to the line x = n ? 1 using (1, 1), (1, ?1), and (1, 0) steps that never dip below the x-axis (called Motzkin left factors). The numbers L _n count, among other things, certain restricted subsets of permutations and Catalan paths. In this paper, we provide new combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify four classes of the partitions of size n, all of which have cardinality L _n and each avoiding a set of two classical patterns of length four. We obtain a further generalization in one of the cases by considering a pair of statistics on the partition class. In a couple of cases, to show the result, we make use of the kernel method to solve a functional equation arising after a certain parameter has been introduced.     

Fourth-Order p-Laplacian Nonlinear Systems via the Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem

In this paper, existence results for a fourth-order nonlinear system are obtained. Both classical and vector versions of the Krasnosel’skiĭ’s fixed point theorem are used and a comparison of the obtained results to those from the literature is provided.     

Restricted 1-3-2 Permutations and Generalized Patterns

Recently, Babson and Steingrimsson (see [2]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 1-3-2 (or containing 1-3-2 exactly once) and an arbitrary generalized pattern t \tau on k letters, or containing t \tau exactly once. In several cases, the generating function depends only on k and can be expressed via Chebyshev polynomials of the second kind, and the generating function of Motzkin numbers.     

The behavior of poly(ε -caprolactone) and poly(ethylene oxide)-b-poly(ε -caprolactone) grafted to a poly(glycerol adipate) backbone at the air/water interface

The behavior of crystallizable poly(ε-caprolactone) (PCL) and poly(ε-caprolactone)-b-poly(ethylene oxide) (PCL-b-PEO) is studied at the air/water interface prior and after grafting to an amorphous poly(glycerol adipate) (PGA) backbone (PGA-g-PCL, PGA-g-(PCL-b-PEO)). Langmuir isotherms are measured and the structure formation in the monolayers on the water surface is followed by Brewster angle microscopy (BAM) and in Langmuir–Blodgett films after a transfer to silicon substrates by atomic force microscopy (AFM). It is observed that PGA-g-PCL forms significantly smaller crystals on the water surface and has smaller crystallization rate compared to PCL homopolymers of identical molar masses as the grafted chains. In contrast to crystals formed by linear PCL, the crystals formed by grafted PCL in PGA-g-PCL do not melt (readsorb at the water surface) in an expansion cycle on the Langmuir trough. Additionally, increasing the subphase temperature at constant surface area significantly above the melting point of linear PCL in bulk results in the formation of a mesophase, and it does lead to the disappearance of crystals. The isotherms of PGA-g-(PCL-b-PEO) show a transition at the surface pressure of ～10 mN/m. This is related to the fact that PEO chains leave the water surface and submerge into the subphase and/or the crystallization of PCL chains. The monolayer collapse appears in an extended plateau region starting at π values of ～30 mN/m. AFM images of Langmuir–Blodgett films reveal that PCL chains in PGA-g-PCL and PGA-g-(PCL-b-PEO) form lamellar crystals with a disk-shape and interconnected platelets, respectively.