Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.     

Analytic continuation and identities involving heat,Poisson, wave and Bessel kernels

In this article we use classical formulas involving the K–Bessel function in two variables to express the Poisson kernel on a Riemannian manifold in terms of the heat kernel. We then use the small time asymptotics of the heat kernel on certain Riemannian manifolds to obtain a meromorphic continuation of the associated Poisson kernel to all values of complex time with identifiable singularities. This result reproves in a different setting by different means a well–known theorem due to Duistermaat and Guillemin [DG 75]. Also, we develop analytic expressions for the heat kernel beyond asymptotic expansions. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evaluation of aortic stenosis severity using 4D flow jet shear layer detection for the measurement of valve effective orifice area

Aims

The objective of this study was to evaluate the potential of 4D flow MRI to assess valve effective orifice area (EOA) in patients with aortic stenosis as determined by the jet shear layer detection (JSLD) method.

Methods and Results

An in-vitro stenosis phantom was used for validation and in-vivo imaging was performed in 10 healthy controls and 40 patients with aortic stenosis. EOA was calculated by the JSLD method using standard 2D phase contrast MRI (PC-MRI) and 4D flow MRI measurements (EOA_JSLD-2D and EOA_JSLD-4D, respectively). As a reference standard, the continuity equation was used to calculate EOA (EOA_CE) with the 2D PC-MRI velocity field and compared to the EOA_JSLD measurements. The in-vitro results exhibited excellent agreement between flow theory (EOA = 0.78 cm²) and experimental measurement (EOA_JSLD-4D = 0.78 ± 0.01 cm²) for peak velocities ranging from 0.9 to 3.7 m/s. In-vivo results showed good correlation and agreement between EOA_JSLD-2D and EOA_CE (r = 0.91, p < 0.001; bias: − 0.01 ± 0.38 cm²; agreement limits: 0.75 to − 0.77 cm²), and between EOA_JSLD-4D and EOA_CE (r = 0.95, p < 0.001; bias: − 0.09 ± 0.26 cm²; limits: 0.43 to − 0.62 cm²).

Conclusion

This study demonstrates the feasibility of measuring EOA_JSLD using 4D flow MRI. The technique allows for optimization of the EOA measurement position by visualizing the 3D vena contracta, and avoids potential sources of EOA_CE measurement variability.     

ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems

In this note we present a new Rosenbrock solver which is third-order accurate for nonlinear parabolic problems. Since Rosenbrock methods suffer from order reduction when they are applied to partial differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. Steinebach modified the well-known solver RODAS of Hairer and Wanner to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third-order Rosenbrock solver for the nonlinear situation. Such a method exists with three stages and two function evaluations only. A comparison with other third-order methods shows the substantial potential of our new method.This revised version was published online in October 2005 with corrections to the Cover Date.     

关于椭圆曲线E_D:y~2=x~3-D~2x的BSD猜想

本文证明了形如ｙ~２＝ｘ~３－ｐ-１~２…ｐ-ｎ~２ｘ的一类椭圆曲线的Ⅲ群２-分支在一定条件下同构于 Ｚ／２Ｚ × Ｚ／２Ｚ,其阶为 ４,与Ｌ函数部分的相应结果和 Ｒｕｂｉｎ Ｋ．关于有复乘的椭圆曲线的重要结果一起,我们知道ＢＳＤ猜想对本文定理中的椭圆曲线成立．     

Convergence de mesures spectrales aléatoires et applications à des principes d'invariance

Nous présentons un cadre unificateur pour un certain nombre de résultats connus de convergence en loi pour des processus admettant une représentation linéaire par rapport à un bruit blanc faible pour lequel il existe un principe dinvariance faible. Nous montrons que ce principe dinvariance est équivalent à la convergence dune suite de mesures aléatoires spectrales. Ceci permet daffaiblir les hypothèses usuelles et donc dobtenir des résultats nouveaux de convergence en loi, notamment pour des processus à longue portée.     

On the Gelfand-Kirillov conjecture for quantum algebras

Let be a complex not a root of unity and be a semi-simple Lie -algebra. Let be the quantized enveloping algebra of Drinfeld and Jimbo, be its triangular decomposition, and the associated quantum group. We describe explicitly and as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.

     

Demonstration of an optical self-clocked time-division switch

A novel architecture for an optical time-division switch is investigated and demonstrated at a bit rate of 625 Mbit/s. The self-clocked configuration eliminates some of the limitations found in previous optical implementations of time-division switches: the switch operation is asynchronous, no read gate is required, and an optimal number of delay lines is used. Furthermore, this paper reports the first experimental demonstration of optical space-sorting demultiplexing.