Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Summary A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal orderL ^p error estimates are derived for 2p. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.     

Waiakeamide, a Cyclic Hexapeptide from the Sponge Ircinia dendroides(1)

From the sponge, Ircinia dendroides, collected in Indonesia we isolated a new cyclic hexapeptide, waiakeamide (1). Its structure, consisting of three proline residues, two methionine sulfoxides, and one thiazolylphenylalanine, was elucidated by spectral analysis and chemical degradation. Isolation and structural elucidation of waiakeamide is described.     

A novel synthesis of (±)-descarboxyquadrone

The novel synthesis of descarboxyquadrone (3) and its model compound (4) by using the acid-catalyzed rearrangement of [4.3.2]propellanones is described.     

5-Amino-4-sulfanylphthalhydrazide as a chemiluminescence derivatization reagent for aromatic aldehydes in liquid chromatography

5-Amino-4-sulfanylphthalhydrazide (ASPH) was synthesized as a chemiluminescence derivatization reagent for aromatic aldehydes in liquid chromatography (LC). Benzaldehyde, 4-tolualdehyde, 4-chlorobenzaldehyde, 4-formylbenzoic acid, 4-hydroxybenzaldehyde and vanillin were used as model compounds to optimize the derivatization conditions. This reagent, ASPH, reacts selectively with aromatic aldehydes in the presence of sodium sulfite and disodium hydrogenphoshite in acidic medium at 100 degrees C to give the corresponding highly chemiluminescent 2-arylbenzothiazole derivatives. The resulting derivatives generated intense chemiluminescence by reaction with hydrogen peroxide and potassium hexacyanoferrate(III) in alkaline solution. The ASPH derivatives of aromatic aldehydes were separated by reversed-phase liquid chromatography with isocratic elution, and detected chemiluminometrically after mixing with oxidizing agents. The detection limits (signal-to-noise ratio = 3) for aromatic aldehydes are in the range 0.2-4.0 fmol for a 20-microl injection volume. Currently, the method is not effective for aliphatic aldehydes because of interfering LC peaks.     

Pyrimidine derivatives. IX . Synthesis of bromosubstituted 5-nitro-2,4(1H,3H)-pyrimidinedione derivatives

The following 5-nitro-2,4(1H,3H)-pyrimidinediones possessing bromo substituted side chains at the 1- and 6-positions were prepared by bromination of 3,6-dimethyl-1-(ω-hydroxyalkyl)-5-nitro-2,4(1H,3H)-pyrimidinediones 4a and 4b and its nitrates 2a and 2b . The three of mono-bromo derivatives are: 1-(ω-acetoxyalkyl and ω-hydroxyalkyl)-6-bromomethyl-3-methyl. 6a, 6b, 7a and 7b and 1-(ω-bromoalkyl)-3,6-dimethyl-2,4(1H,3H)- pyrimidinediones 8a and 8b . The one type dibromo derivatives are: 1-(ω-bromoalkyl)-6-bromomethyl-3-methyl-2,4(1H,3H)-pyrimidinediones 5a and 5b .     

In situ x-ray photoelectron spectroscopic and density-functional studies of Si atoms adsorbed on a C60 film

The interaction between C(60) and Si atoms has been investigated for Si atoms adsorbed on a C(60) film using in situ x-ray photoelectron spectroscopy (XPS) and density-functional (DFT) calculations. Analysis of the Si 2p core peak identified three kinds of Si atoms adsorbed on the film: silicon suboxides (SiO(x)), bulk Si crystal, and silicon atoms bound to C(60). Based on the atomic percent ratio of silicon to carbon, we estimated that there was approximately one Si atom bound to each C(60) molecule. The Si 2p peak due to the Si-C(60) interaction demonstrated that a charge transfer from the Si atom to the C(60) molecule takes place at room temperature, which is much lower than the temperature of 670 K at which the charge transfer was observed for C(60) adsorbed on Si(001) and (111) clean surfaces [Sakamoto et al., Phys. Rev. B 60, 2579 (1999)]. The number of electrons transferred between the C(60) molecule and Si atom was estimated to be 0.59 based on XPS results, which is in good agreement with the DFT result of 0.63 for a C(60)Si with C(2v) symmetry used as a model cluster. Furthermore, the shift in binding energy of both the Si 2p and C 1s core peaks before and after Si-atom deposition was experimentally obtained to be +2.0 and -0.4 eV, respectively. The C(60)Si model cluster provides the shift of +2.13 eV for the Si 2p core peak and of -0.28 eV for the C 1s core peak, which are well corresponding to those experimental results. The covalency of the Si-C(60) interaction was also discussed in terms of Mulliken overlap population between them.     

Synthesis and optical resolution of a new flavin with axial chirality and redox-dependent racemization

A new flavin (3-methyl-10-(2′-phenyl)phenylisoalloxazine) with restricted rotation about the C(1′)?N(10) single bond was synthesized and optically resoluted for the first time. (+)- Isomer (100% enantiomeric purity) did not racemize at ambient conditions, whereas it racemized invariably when it was reduced (by four different methods). The kinetic studies proved that the novel racemization mode is attributed to conversion of the “planar” oxidized form to the “bent” reduced form.     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.     

Transition metal-catalysed acylation of alpha,beta-unsaturated carbonyl compounds with acylstannanes

Acylstannanes were found to add to such alpha,beta-unsaturated carbonyl compounds as enones or ynoates in the presence of a nicel or palladium catalyst to give 2-stannyl-4-oxoalk-2-enoates or 1,4-diketones, whereas the three component coupling between acylstannanes, enones and aldehydes provided 2-hydroxymethyl 1,4-diketones.     

Einstein equations and Lagrangians on symmetry-restricted phase spaces

When Lagrangians are restricted to subspaces of the phase space satisfying certain symmetry conditions, they do not always yield the correct Einstein equations. An example was found by Hawking in the Lagrangian for spatially homogeneous universes. First we show that this problem is not limited to general relativity. Then we visualize certain terms which do not appear in Lagrangians on symmetry-restricted phase spaces but which nevertheless contribute to the Euler-Lagrange derivatives. We apply our general formula to the Schwarzschild metric, the Friedmann universes, and spatially homogeneous universes.