Resonant Leading Term Geometric Optics Expansions with Boundary Layers for Quasilinear Hyperbolic Boundary Problems

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2 Coulombel, J.-F., Gues, O., and Williams, M., 2011. Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems, Comm. Part. Diff. Eqs. 36 (2011), pp. 1797–1859.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] by Coulombel et al. studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such as approximately solving complex transport equations and result in the addition of boundary layers with exponential decay. This also prevents us from doing an error analysis based on almost periodic profiles as in [2 Coulombel, J.-F., Gues, O., and Williams, M., 2011. Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems, Comm. Part. Diff. Eqs. 36 (2011), pp. 1797–1859.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

单位球上小Bloch型空间之间的加权复合算子

对所有的0p、q∞,该文得到了Cn中单位球上小Bloch型空间β_0~p到β_0~q之间的加权复合算子T_ψ,φ为有界算子或紧算子的充要条件.     

A note on certain de Bruijn sequences with forbidden subsequences

We consider words over a finite alphabet with certain uniqueness properties (a subsequence of length k does not occur more than once) and distance properties (at least j other symbols separate the occurrence of the same symbol). The maximal length of these words is realised by linear de Bruijn sequences with certain forbidden subsequences. We prove the existence of these maximal sequences.     

Degree conditions for group connectivity

Let G be a 2-edge-connected simple graph on n≥13 vertices and A an (additive) abelian group with |A|≥4. In this paper, we prove that if for every uv∉E(G), max{d(u),d(v)}≥n/4, then either G is A-connected or G can be reduced to one of K_2,3,C₄ and C₅ by repeatedly contracting proper A-connected subgraphs, where C_k is a cycle of length k. We also show that the bound n≥13 is the best possible.     

Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.     

Numerical algorithms for solutions of Korteweg–de Vries equation

The nonlinear Korteweg–de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion‐based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

对Bloch定理中覆盖圆的估计

本文研究了Block定理覆盖圆问题.利用微分与积分估值法,对覆盖圆的半径做了更精确地估计,得到了两个相关定理,扩大了覆盖圆的范围.     

Products of B-patches

Products and tensor products of multivariate polynomials in B-patch form are viewed as linear combinations of higher degree B-patches. Univariate B-spline segments and certain regions of simplex splines are examples of B-patches. A recursive scheme for transforming tensor product B-patch representations into B-patch representations of more variables is presented. The scheme can also be applied for transforming ann-fold product of B-patch expansions into a B-patch expansion of higher degree. Degree raising formulas are obtained as special cases. The scheme calculates the blossom of the (tensor) product surface and generalizes the pyramidal recursive scheme for B-patches.