Restricted partitions and q-Pell numbers

In this paper, we provide new combinatorial interpretations for the Pell numbers p _n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p _n . By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p _n . Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.     

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.     

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.     

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.     

Glyoxal sample preparation for high-performance liquid chromatographic detection of 2,4-dinitro-phenylhydrazone derivative: Suppression of polymerization and mono-derivative formation by using methanol medium

Some difficulties on a sampling of gaseous glyoxal using DNPH-silica cartridge were discussed, and an alternative sampling procedure was proposed. When glyoxal was sampled using the cartridge, it partially formed mono-hydrazone with various degrees, whereas it was quantitatively converted into its bis-hydrazone when sample gas was directly bubbled into a DNPH acidic solution. Additionally, glyoxal polymerized on the inner wall of the system during trapping to the silica cartridge. We found that the polymerization of glyoxal was effectively suppressed by dissolving glyoxal into methanol under reduced pressure; glyoxal in methanol readily reacted with DNPH in a hydrochloric acid solution and gave bis-hydrazone derivative, without any interference of methanol and any formation of mono-hydrazone derivative. When glyoxal was sampled by the proposed method, 102% of recovery was obtained, whereas it was reduced to 92.7% when the trapped glyoxal was dissolved into methanol under atmosphere.     

Approximation by polynomials with bounded coefficients

Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.     

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.     

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     

Restricted even permutations and Chebyshev polynomials

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.     

Crystallization and melting of poly(glycerol adipate)‐based graft copolymers with single and double crystallizable side chains

The crystallization and melting behavior of a series of poly(glycerol adipate) (PGA) based graft copolymers with either poly(ε‐caprolactone) (PCL), poly(ethylene oxide) (PEO), or PCL‐b‐PEO diblock copolymer side chains (i.e., PGA‐g‐PCL, PGA‐g‐PEO, and PGA‐g‐(PCL‐b‐PEO)) was studied using polarized light optical microscopy (POM), differential scanning calorimetry (DSC), small‐angle X‐ray scattering (SAXS), and wide‐angle X‐ray diffraction (WAXD). These results were compared with the behavior of the corresponding linear analogs (PEO, PCL, and PCL‐b‐PEO). POM revealed that spherulitic morphology was retained after grafting. However, spherulite radius as well as radial growth rate was significantly smaller in the graft copolymers. Evaluation of isothermal crystallization kinetics by means of the Avrami theory revealed that the nucleation density was much higher in the graft copolymers. The DSC results indicated that the degree of crystallinity decreased strongly upon grafting while the melting temperatures of PGA‐g‐PCL and PGA‐g‐PEO were found to be close to the values of neat PCL and PEO, respectively. This was attributed to the absence of specific thermodynamic interactions, and, additionally, to lamella thicknesses being similar to those of the homopolymers. The latter point was confirmed by SAXS measurements. In case of PCL‐b‐PEO diblock copolymers and PGA‐g‐(PCL‐b‐PEO) graft copolymers, the crystallization behavior and thus the resulting lamellar morphology is more complex, and a suitable model was developed based on a combination of DSC, WAXD, and SAXS data. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013 , 51, 1581–1591