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.     

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.     

A Theorem for Numerical Verification on Local Uniqueness of Solutions to Fixed-Point Equations

We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods.

We already derived such a theorem in [11 N. Yamamoto ( 1998 ). A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem . SIAM J. Numer. Anal. 35 : 2004 – 2013 .[Crossref] , [Google Scholar]], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.     

Measurement of soil/dust arsenic by gas phase chemiluminescence

A gas phase chemiluminescence (GPCL)-based method for trace measurement of arsenic has been recently described for the measurement of arsenic in water. The principle is based on the reduction of inorganic As to AsH₃ at a controlled pH (the choice of pH governs whether only As(III) or all inorganic As is converted) and the reaction of AsH₃ with O₃ to produce chemiluminescence (Idowu et al., Anal. Chem. 78 (2006) 7088-7097). The same general principle has also been used in postcolumn reaction detection of As, where As species are separated chromatographically, then converted into inorganic As by passing through a UV photochemical reactor followed by AsH₃ generation and CL reaction with ozone (Idowu and Dasgupta, Anal. Chem. 79 (2007) 9197-9204). In the present paper we describe the measurement of As in different soil and dust samples by serial extraction with water, citric acid, sulfuric acid and nitric acid. We also compare parallel measurements for total As by induction coupled plasma mass spectrometry (ICP-MS). As(V) was the only species found in our samples. Because of chloride interference of isobaric ArCl⁺ ICP-MS analyses could only be carried out by standard addition; these results were highly correlated with direct GPCL and LC-GPCL results (r² = 0.9935 and 1.0000, respectively). The limit of detection (LOD) in the extracts was 0.36 μg/L by direct GPCL compared to 0.1 μg/L by ICP-MS. In sulfuric acid-based extracts, the LC-GPCL method provided LODs inferior to those previously observed for water-based standards and were 2.6, 1.3, 6.7, and 6.4 μg/L for As(III), As(V), dimethylarsinic acid (DMA) and monomethylarsonic acid (MMA), respectively.     

Koshikamide B, a cytotoxic peptide lactone from a marine sponge Theonella sp

Koshikamide B (1) has been isolated from two separate collections of the marine sponge Theonella sp. as the major cytotoxic constituent. Koshikamide B is a 17-residue peptide lactone composed of six proteinogenic amino acids, two D-isomers of proteinogenic amino acids, seven N-methylated amino acids, and two unusual amino acid residues. The unusual amino acids are N(delta)-carbamoylasparagine and 2-(3-amino-2-hydroxy-5-oxopyrrolidin-2-yl)propionic acid (AHPP); the former is first found as the constituent of peptides, whereas the latter is a new amino acid residue. The N-terminus of koshikamide B is blocked by a methoxyacetyl group. The structure of koshikamide B (1) has been determined by interpretation of spectral data and analysis of chemical degradation products. Koshikamide B (1) exhibits cytotoxicity against P388 murine leukemia cells and the human colon tumor (HCT-116) cell line with an IC50 value of 0.45 and 7.5 microg/mL, respectively.     

Syntheses, structures, and coordination chemistry of phosphole-containing hybrid calixphyrins: promising macrocyclic P,N2,X-mixed donor ligands for designing reactive transition-metal complexes

The syntheses, structures, and coordination chemistry of phosphole-containing hybrid calixphyrins (P,N2,X-hybrid calixphyrins) and the catalytic activities of their transition-metal complexes are reported. The 5,10-porphodimethene type 14pi-P,(NH)2,X- and 16pi-P,N2,X-hybrid calixphyrins (X = O, S, NH) are prepared via acid-promoted dehydrative condensation between a sigma4-phosphatripyrrane and the corresponding 2,5-bis[hydroxy(phenyl)methyl]heteroles followed by DDQ oxidation. Both spectroscopic and crystallographic data of the hybrid calixphyrins have revealed that the conformation and size of the macrocyclic platforms as well as the oxidation state of the -conjugated pyrrole-heterole-pyrrole (N-X-N) units vary considerably depending on the combination of heteroles. The sigma3-P,(NH)2,S- and sigma3-P,N2,S-hybrids react with Pd(OAc)2 and Pd(dba)2, respectively, to afford the same Pd(II)-P,N2,S-hybrid complex, in which the calixphyrin platform is regarded as a dianionic ligand. In the complexation with [RhCl(CO)2]2 in dichloromethane, the sigma3-P,N2,S-hybrid behaves as a neutral ligand to afford an ionic Rh(I)-P,N2,S-hybrid complex, whereas the sigma3-P,N2,NH-hybrid behaves as an anionic ligand to produce Rh(III)-P,N3-hybrid complexes. In the latter reaction, it is likely that a neutral Rh(I)-P,N3-hybrid complex, generated as a highly nucleophilic intermediate, undergoes C-Cl bond activation of the solvent. The complexation of AuCl(SMe2) with the sigma3-P,N2,X-hybrids (X = S, NH) leads to the formation of the corresponding Au(I)-monophosphine complexes. The spectral data and crystal structures of these metal complexes exhibit the hemilabile nature of the phosphole-containing hybrid calixphyrin platforms derived from the flexible phosphole unit and the redox active N-X-N units. The hybrid calixphyrin-palladium and -rhodium complexes catalyze the Heck reaction and hydrosilylations, respectively, implying that the metal center in the core is capable of activating the substrates under appropriate reaction conditions. The present results demonstrate the potential utility of the phosphole-containing hybrid calixphyrins as a new class of macrocyclic P,N2,X-mixed donor ligands for designing highly reactive transition-metal complexes.     

Theoretical study of pyrazolate-bridged dinuclear platinum(II) complexes: interesting potential energy curve of the lowest energy triplet excited state and phosphorescence spectra

Four kinds of 3,5-dialkylpyrazolate(R2pz)-bridged dinuclear platinum(II) complexes [Pt2(mu-R2pz)2(dfppy)2] (dfppy=2-(2,4-difluorophenyl)pyridine; R2pz=pyrazolate in 1, 3,5-dimethylpyrazolate in 2, 3-methyl-5- tert-butylpyrazolate in 3, and 3,5-bis(tert-butyl)pyrazolate in 4) were theoretically investigated by the DFT(B3PW91) method. The Stokes shift of their phosphorescence spectra was discussed on the basis of the potential energy curve (PEC) of the lowest energy triplet excited state (T1). This PEC significantly depends on the bulkiness of substituents on pz. In 1 and 2, bearing small substituents on pz, one local minimum is present in the T1 state besides a global minimum. The local minimum geometry is similar to the S0-equilibrium one. The T1 state at this local minimum is characterized as the pi-pi* excited state in dfppy, where the dpi orbital of Pt participates in this excited state through an antibonding interaction with the pi orbital of dfppy; in other words, this triplet excited state is assigned as the mixture of the ligand-centered pi-pi* excited and metal-to-ligand charge transfer excited state ((3)LC/MLCT). The geometry of the T1-global minimum is considerably different from the S0-equilibrium one. The T1 state at the global minimum is characterized as the triplet metal-metal-to-ligand charge transfer ((3)MMLCT) excited state, which is formed by the one-electron excitation from the dsigma-dsigma antibonding orbital to the pi* orbital of dfppy. Because of the presence of the local minimum, the geometry change in the T1 state is suppressed in polystyrene at room temperature (RT) and frozen 2-methyltetrahydrofuran (2-MeTHF) at 77 K. As a result, the energy of phosphorescence is almost the same in these solvents. In fluid 2-MeTHF at RT, on the other hand, the geometry of the T1 state easily reaches the T1-global minimum. Because the T1-global minimum geometry is considerably different from the S0-equilibrium one, the phosphorescence occurs at considerably low energy. These are the reasons why the Stokes shift is very large in fluid 2-MeTHF but small in polystyrene and frozen 2-MeTHF. In 3 and 4, bearing bulky tert-butyl substituents on pz, only the T1-global minimum is present but the local minimum is not. The electronic structure of this T1-global minimum is assigned as the (3)MMLCT excited state like 1 and 2. Though frozen 2-MeTHF suppresses the geometry change of 3 and 4 in the T1 state, their geometries moderately change in polystyrene because of the absence of the T1-local minimum. As a result, the energy of phosphorescence is moderately lower in polystyrene than in frozen 2-MeTHF. The T1-global minimum geometry is much different from the S0-equilibrium one in 3 but moderately different in 4, which is interpreted in terms of the symmetries of these complexes and the steric repulsion between the tert-butyl group on pz and dfppy. Thus, the energy of phosphorescence of 3 is much lower in fluid 2-MeTHF than in frozen 2-MeTHF like 1 and 2 but that of 4 is moderately lower; in other words, the Stokes shift in fluid 2-MeTHF is small only in 4.     

Heteropolynuclear complexes of 3,5-dimethylpyrazolate [Pt2M4(Me2pz)8] (M = Ag, Cu). Highly luminescent character of the triplet excited state based on mixed-metal cores

The platinum dimer and heteropolynuclear platinum complexes of 3,5-dimethylpyrazolate, [Pt2M4(mu-Me2pz)8] [M = H (1), Ag (2), Cu (3)], were synthesized and structurally characterized. They exhibit yellow, sky-blue, and orange luminescence, respectively, in the solid state. The absorption bands of 2 and 3 are mainly assigned to the combination of the metal-metal-to-ligand charge-transfer and [Pt2 --> Pt2M4] transitions by the time-dependent density functional theory (DFT) method. DFT calculations also indicate that the emissive states of 2 and 3 are 3[Pt2 --> Pt2Ag4] and 3[Cu(d) --> Pt2Cu4], respectively.     

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.