Numerical Verification Methods for Solutions of the Free Boundary Problem

We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.     

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

Existence of global decaying solutions to the exterior problem for the Klein–Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity

We prove the existence of global decaying solutions to the exterior problem for the Klein–Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity. To derive the required estimates of solutions we employ a ‘loan’ method.     

Phase reduction approach to synchronisation of nonlinear oscillators

Systems of dynamical elements exhibiting spontaneous rhythms are found in various fields of science and engineering, including physics, chemistry, biology, physiology, and mechanical and electrical engineering. Such dynamical elements are often modelled as nonlinear limit-cycle oscillators. In this article, we briefly review phase reduction theory, which is a simple and powerful method for analysing the synchronisation properties of limit-cycle oscillators exhibiting rhythmic dynamics. Through phase reduction theory, we can systematically simplify the nonlinear multi-dimensional differential equations describing a limit-cycle oscillator to a one-dimensional phase equation, which is much easier to analyse. Classical applications of this theory, i.e. the phase locking of an oscillator to a periodic external forcing and the mutual synchronisation of interacting oscillators, are explained. Further, more recent applications of this theory to the synchronisation of non-interacting oscillators induced by common noise and the dynamics of coupled oscillators on complex networks are discussed. We also comment on some recent advances in phase reduction theory for noise-driven oscillators and rhythmic spatiotemporal patterns.     

Coordination framework hosts consisting of 4-pyridyl-substituted carboxylic acid (PCA) dimers and 1D chains of Ni2+ and SCN-: a rational structural extension toward coordination framework hosts with large rectangular cavities

For the expansion of a rectangular cavity (RC) defined by two isonicotinic acid (isoH) dimers as bridging ligands and two SCN bridges, we conducted a structural extension based on the elongation of the bridging ligands by the replacement of isoH with longer 4-pyridyl-substituted carboxylic acid (PCA). For this purpose, the following three PCAs have been employed: trans-3-(4-pyridyl)propenoic acid (acrylH), 4-(4-pyridyl)benzoic acid (pybenH), and trans-3-(4-(4-pyridyl)phenyl)propenoic acid (pppeH). Self-assembly of Ni2+, SCN-, and each of four PCAs involving isoH, acrylH, pybenH, and pppeH in the presence of an aromatic guest gave four inclusion compounds formulated as [Ni(SCN)2(isoH)2].1/2(benz[a]anthracene) (1), [Ni(SCN)2(acrylH)2].1/2(benz[a]anthracene) (2), [Ni(SCN)2(pybenH)2].(pyrene) (3), and [Ni(SCN)2(pppeH)2](3/)(2).(benz[a]anthracene) (4). X-ray crystal structural determination of 1-4 revealed that the proposed structural extension was successful. Their crystal structures are layered structures of two-dimensional (2D) grid-type coordination frameworks (2D host layers) framed with bridging ligands of the corresponding PCA dimers and 1D chains consisting of Ni2+ ions and mu(1,3)-SCN- ions. The lengths of the PCA dimers are 12.269(5) A (isoH dimer), 16.890(4) A (acrylH dimer), 20.89(2) A (pybenH dimer), 25.387(3) A (pppeH dimer A), and 25.527(4) A (pppeH dimer B). Each 2D host layer has RCs defined by the two corresponding PCA dimers and the two SCN bridges. The dimensions of RCs are expanded in proportion to the increase in the lengths of the PCA dimers: 29.52 x 5.60-7.20 A2 (4) > 24.95 x 5.46-7.38 A2 (3) > 20.88 x 5.49-7.25 A2 (2) > 16.41 x 5.53-7.43 A2 (1). These expansions reflect the number of aromatic guests that can be included in RCs. RC of 1 include only one molecule of benz[a]anthracene, whereas RCs of 3 or 4 includes two molecules of pyrene or benz[a]anthracene, respectively. Comparison of the lengths between the PCA dimers and 4,4'-bipyridine-type ligands demonstrated that a design strategy-the preparation of a bridging ligand through self-assembly of two PCAs-is both efficient and particularly suitable for the preparation of very long bridging ligands.     

Phosphatidylcholine vesicle-mediated decomposition of hydrogen peroxide

The decomposition of hydrogen peroxide (H2O2) was examined in aqueous solution (50 mM Tris-HCl buffer, pH 7.4, containing 100 mM NaCl) at 25 degrees C in pure buffer or in the presence of either vesicles or micelles formed from various phosphatidylcholines (PCs). In the absence of PCs, more than 90% of the initially added H2O2 (1.0 mM) remained intact after incubation for 120 h. The effect of the PCs on the decomposition of H2O2 was studied by using different PCs that varied in terms of number of carbon atoms in the two acyl chains n as well as in terms of the degree of unsaturation. PCs with short hydrocarbon chains (n = 4, 6-8) were dissolved in the buffer solution in the form of nonassociated monomers or as micelles in equilibrium with monomers at a fixed PC concentration of 10 mM. The presence of these short-chain PCs slightly enhanced the H2O2 decomposition rate. Micelles formed by non-lipid detergents (sodium cholate, Triton X-100, and sodium dodecylsulfate) had a similar effect. In marked contrast, PCs with long hydrocarbon chains (n > or = 10) dispersed in buffer solution as vesicles (liposomes) significantly enhanced the rate of H2O2 decomposition, with the most effective PC being 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) at 25 degrees C. This indicates that the packing density of the PC molecules influences the reactivity, presumably through the direct interaction of the PC assemblies with H2O2 molecules. Furthermore, in the case of vesicles formed from PCs with unsaturated acyl chains (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine, POPC; 1,2-dioleoyl-sn-glycero-3-phosphocholine, DOPC), carbon-carbon double bond oxidation did not occur extensively under the conditions used. This indicates that the observed effect of PCs on the decomposition of H2O2 is indeed related to the assembly structure (vesicle vs micelles vs monomers) and is clearly not related to the presence of unsaturated hydrocarbon chains. Fluorescence polarization measurements of two fluorescent probes embedded either in the acyl chain region of the vesicles (DPH, 1,6-diphenyl-1,3,5-hexatriene) or on the surface of the vesicles (TMA-DPH, 1-(4-trimethylammoniumphenyl)-6-phenyl-1,3,5-hexatriene iodide) show that the presence of H2O2 leads to a decrease in the fluidity of the lipid-water surface and not to a change in the fluidity of the hydrophobic region of the vesicle bilayer. This indicates that the decomposition of H2O2 is triggered through interactions between H2O2 and the polar head group area of PC vesicles.     

High-molecular-weight poly(p-phenyleneterephthalamide) by the direct polycondensation reaction with triphenyl phosphite

Poly(p-phenyleneterephthalamide) of high molecular weight was obtained when the polycondensation of terephthalic acid and p-phenylenediamine was carried out in N-methylpyrrolidone (NMP) that contained dissolved CaCl₂ and LiCl in the presence of pyridine. The molecular weight of the polymer obtained varied with the amount of pyridine relative to the metal salts and with the molar ratios of CaCl₂ to LiCl, the maximum η_inh value of 4.5 being obtained under the conditions Py/(CaCl₂ + LiCl) ≈ 2.5 (mol/mol), CaCl₂/LiCl ≈ 1.2 (mol/mol), and LiCl + CaCl₂ ≈ 4 g. Among the solvents tested, NMP was significantly effective for the reaction. Polycondensations of several combinations of other dicarboxylic acids and diamines were carried out with limited success.     

Manganese(II) o ctacyanotungstate(V)-based magnet containing a noncoordinated aromatic molecule

We have prepared a pillared layer magnetic material containing a noncoordinated aromatic molecule, [{MnII(pyrimidine)(H2O)}2{MnII(H2O)2}{WV(CN)8}2](pyrimidine)2.2H2O. This compound has one-dimensional channels (6.2 x 2.1 A) that are occupied by noncoordinated pyrimidine. The magnetization versus temperature plots showed the magnetic phased transition temperature (TC) was 47 K. The magnetization versus external magnetic field plots showed that the saturation magnetization (MS) value was 13.0 muB at 2 K. This MS value indicates that an antiferromagnetic interaction operates between the WV (S = 1/2) and MnII (S = 5/2) ions. The magnetic hysteresis loop showed that the coercive field (HC) was 17 G at 2 K.     

Establishment of de facto standard catalysts in the polypropylene industry

Polypropylene (PP) has become an indispensable material in our daily lives. Annual worldwide production of PP is now more than 30000000 tons and is predicted to grow at an annual rate of about 6% during the first decade of the 21st century. Commercial production of PP began in 1957 with the use of TiCl(3) catalysts established by Ziegler and Natta. However, the low activities and low stereospecificities of the catalysts resulted in large amounts of catalyst residue and atactic PP in the product, necessitating steps for their removal in commercial production. As a means of finding appropriate catalysts, we developed MgCl(2)-supported TiCl(4) catalysts, which basic concept was introduction of organic compounds onto the inorganic crystal catalyst surface. This addition led to remarkable enhancements in stereospecificity with extremely high activity. Use of the new catalysts enlarged and simplified the PP production process by eliminating the steps previously required for removal of catalyst residue and atactic PP. In addition, it greatly improved the properties of the PP, enabling a much wider range of PP applications by replacing metal and engineering plastics with the highly stereoregular PP. Therefore, these catalysts helped the rapid establishment of the current PP industry and now play a major role in production. The latest MgCl(2)-supported TiCl(4) catalyst is providing precise control of the isotactic PP structure. Future expectations for this type of catalyst are to acquire a single-site nature and to contribute to the creation of a new class of hybrid materials.     

Multiferroics by rational design: implementing ferroelectricity in molecule-based magnets

Multiferroic materials: A novel class of multiferroics based on organic-inorganic hybrid materials is synthesized. The ferromagnetic properties originate from the coordination network, whereas the ferroelectricity results from the polar organic ions (see picture, M=magnetization and P=polarization).