Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations

We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schr?dinger equation in the energy space, after the infinite oscillation in time is removed. We also derive the optimal rate of convergence in . Received: 13 July 2000 / Published online: 1 February 2002     

Numerical reconstruction of an infrared wavefront utilizing an optical phase modulation device

We utilize nitroanisole, that absorbs infrared (IR) radiation as heat, as an optical modulation device based on a thermal process. The nitroanisole exhibits a thermal lens effect, i.e. a temperature dependent refractive index. Hence, the nitroanisole can induce phase modulation to visible light, in direct response to intensity of the incident IR radiation. The proposed method can be used to obtain the phase modulation distribution that corresponds to the IR intensity distribution, i.e. the IR hologram itself, on the nitroanisole by examining the phase map of visible light that is modulated upon passing through the nitroanisole. The IR wavefront can be reconstructed by calculating extracted IR holograms through the Fresnel transform. It is verified that both the amplitude and the phase of the IR wavefront can be reconstructed accurately by proposed method.     

Temperature and velocity measurement of a 3-D thermal flow field using thermo-sensitive liquid crystals

It is an important challenge to analyze a three-dimensional thermal flow field in engineering, science, and agriculture. For such an analysis, it is essential to measure physical quantities such as temperature and velocity over the entire thermal flow field. This paper presents a measurement system based on color image processing for temperature and velocity vector distributions in a three-dimensional thermal flow field. Flow visualization is accomplished by the use of thermo-sensitive liquid crystal tracers. An algorithm for the color-to-temperature transformation using a multi-layer feed-forward neural network is applied to three-dimensional natural convection in a rotating cylindrical cell. Two-dimensional temperature distributions in a slit plane are obtained by using the algorithm. A three-dimensional temperature distribution is consequently constructed by interpolating the two-dimensional distributions using the B-spline function. In addition, the Spatio-Temporal correlation method is applied to the natural convection to obtain a three-dimensional velocity vector distribution.     

A generalization of the weighted Strichartz estimates for wave equations and an application to self‐similar solutions

Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without a support restriction on the forcing term. The method of proof is based on expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self‐similar solutions to nonlinear wave equations are proved on up to five space dimensions. © 2006 Wiley Periodicals, Inc.     

Efficient [3,3]-sigmatropic rearrangement accelerated by a trifluoroacetyl group: synthesis of benzofurans under mild conditions

The [3,3]-sigmatropic rearrangement took place smoothly during the course of trifluoroacetylation of O-phenyloxime at below room temperature to give the dihydrobenzofuran or benzofuran as a result of concomitant cyclization.     

High-performance liquid chromatographic assay for glucuronidation activity of 7-ethyl-10-hydroxycamptothecin (SN-38), the active metabolite of irinotecan (CPT-11), in human liver microsomes

A simple and sensitive assay for glucuronidation activity of 7-ethyl-10-hydroxycamptothecin (SN-38), the active metabolite of irinotecan (CPT-11), in human liver microsomes by high-performance liquid chromatography (HPLC) with fluorescence detection is reported. The method was validated for the determination of SN-38 glucuronide (SN-38G) with respect to specificity, linearity, recovery, stability, precision, accuracy, and limits of detection and quantitation. There was no interference from matrix and non-enzymatic reactions. The calibration curve for SN-38G was linear from 5 to 500 nM. Average recoveries ranged from 98 to 100% in spiked human liver microsome samples, and the SN-38G was stable at 4 degrees C for at least 72 h. The newly developed method was found to be more sensitive and selective than previous methods using thin layer chromatography and HPLC. The limit of quantitation for SN-38G was 5 nM (2.5 pmol/assay). The intra- and inter-day precision and accuracy were less than 7 and 4%, respectively. The intra- and inter-day precision of enzyme assay for UDP-glucuronosyltransferase (UGT) activity toward SN-38 in human liver microsomes was less than 4%. With this improved sensitivity, the kinetics of SN-38 glucuronidation in human liver microsomes could be determined more precisely. Therefore, this method is applicable to in vitro study on the side effects and drug interactions of CPT-11 using small amounts of biological sample.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$