Construction and separability of nonlinear soliton integrable couplings

A very natural construction of integrable extensions of soliton systems is presented. The extension is made on the level of evolution equations by a modification of the algebra of dynamical fields. The paper is motivated by recent works of Wen-Xiu Ma et al. (Comp. Math. Appl. 60 (2010) 2601, Appl. Math. Comp. 217 (2011) 7238), where new class of soliton systems, being nonlinear integrable couplings, was introduced. The general form of solutions of the considered class of coupled systems is described. Moreover, the decoupling procedure is derived, which is also applicable to several other coupling systems from the literature.Comment: letter, 10 page

Integrable discrete systems on R and related dispersionless systems

The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.Comment: 19 page

On Mathematics and Culture: Insights from an International School

We explore the factors that influence the relationship between mathematics and culture in the international school context. First, we share some thoughts about international schools in general and the international mathematics curriculum implemented at the middle grades level at our school in particular. Second, we present some interesting snapshots from our culturally-diverse mathematics classrooms

Bi-Hamiltonian structures for integrable systems on regular time scales

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of $\delta$-pseudo-differential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors is given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by $\Delta$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer hierarchies.Comment: 18 page

Composites of reactive silica nanoparticles and poly(glycidyl methacrylate) with linear and crosslinked chains by in situ bulk polymerization

Composites of poly(glycidyl methacrylate) (PGMA) and L-lysine-coated silica nanoparticles with varying contents were prepared by in situ bulk polymerization using benzoyl peroxide (BPO) as free radical initiator. Silica nanoparticles covered by L-lysine molecules were synthesized using emulsion method. Dynamic light scattering measurements confirmed that the particles are highly monodisperse with the diameter of 10 nm and free of aggregates in the monomer (glycidyl methacrylate, GMA). Upon polymerization of the homogeneous particle/monomer dispersion, aggregates of individual silica nanoparticles are observed by tapping mode atomic force microscope (AFM). Amine and/or carboxylic acid sites on particle surface covalently react with the oxirane groups of the polymer backbone. The aggregation was substantially suppressed by using a difunctional comonomer divinyl benzene (DVB) in polymerization. A three-dimensional polymer network, P(GMA-DVB), forms throughout the system. This structure leads to significant progress in particle dispersion, therefore in physical properties of the resulting composite. We demonstrated that the composites prepared by crosslinked chains are thermally more stable and mechanically stiffer than those prepared by linear ones.TÜBİTAK TBAG-109T905; TÜBİTAK TBAG-108T446; State Planning Organization (DPT-2003K120690-6

Phase Diagrams and Crossover in Spatially Anisotropic d=3 Ising, XY Magnetic and Percolation Systems: Exact Renormalization-Group Solutions of Hierarchical Models

Hierarchical lattices that constitute spatially anisotropic systems are introduced. These lattices provide exact solutions for hierarchical models and, simultaneously, approximate solutions for uniaxially or fully anisotropic d=3 physical models. The global phase diagrams, with d=2 and d=1 to d=3 crossovers, are obtained for Ising, XY magnetic models and percolation systems, including crossovers from algebraic order to true long-range order.Comment: 7 pages, 12 figures. Corrected typos, added publication informatio

R-matrix approach to integrable systems on time scales

A general unifying framework for integrable soliton-like systems on time scales is introduced. The $R$-matrix formalism is applied to the algebra of $\delta$-differential operators in terms of which one can construct infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are difference counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.Comment: 21 page

Preparation of diethyl malonate adducts from chalcone analogs containing a thienyl ring

Nine chalcone-diethyl malonate derivatives (4a-i) were prepared by the reaction of chalcone derivatives (3a-i) with diethyl malonate in the presence of a catalytic amount of KOt-Bu in CH2CI2 in good to excellent yields. The products were characterized by FTIR, 1H-NMR, 13C-NMR and elemental analyses. KEY WORDS: Michael addition, Chalcone, KOt-Bu, Diethyl malonate  Bull. Chem. Soc. Ethiop. 2010, 24(1), 85-91.