Contractibility of the Solution Sets in Strictly Quasiconcave Vector Maximization on Noncompact Domains

We study the contractibility of the efficient solution set of strictly quasiconcave vector maximization problems on (possibly) noncompact feasible domains. It is proved that the efficient solution set is contractible if at least one of the objective functions is strongly quasiconcave and any intersection of level sets of the objective functions is a compact (possibly empty) set. This theorem generalizes the main result of Benoist (Ref.1), which was established for problems on compact feasible domains.The authors thank Dr. T. D. Phuong, Dr. T. X. D. Ha, and the referees for helpful comments and suggestions.     

Numerical solution of the heat equation with nonlinear boundary conditions in unbounded domains

The numerical solution of the heat equation in unbounded domains (for a 1D problem‐semi‐infinite line and for a 2D one semi‐infinite strip) is considered. The artificial boundaries are introduced and the exact artificial boundary conditions are derived. The original problems are transformed into problems on finite domains. The space semi‐discretization by finite element method and the full approximation by the implicit‐explicit Euler's method are presented. The solvability of the full discretization schemes is analyzed. Computational examples demonstrate the accuracy and the efficiency of the algorithms. Also, the behavior of blowing up solutions is examined numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 379–399, 2007     

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first‐ and second‐order splitting up method, of additive type, for the dependent variables is initially considered to solve the two‐ or three‐dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second‐order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

High‐resolution solid‐state NMR study of the domain structure and molecular motion in drawn films of a vinylidene fluoride and trifluoroethylene copolymer with 73 mol % vinylidene fluoride

The heterogeneous higher order structure and molecular motion in a single crystalline film of a vinylidene fluoride (VDF) and trifluoroethylene (TrFE) copolymer with 73 mol % VDF was investigated with the ¹H–¹³C cross‐polarization/magic‐angle spinning NMR technique. A transient oscillation was observed in plots of the ¹³C peak intensity versus the contact time for the CH₂, CHF, and CF₂ groups. On the basis of the extended cross‐relaxation theory of spin diffusion, we determined that the oscillation behavior was caused by the TrFE‐rich segments in the chain and that the crystal consisted of VDF‐rich and TrFE‐rich domains. The former had TrFE‐rich segments in VDF and TrFE fractions of 0.24 and 0.27, respectively, and the latter had VDF‐rich segments in a VDF fraction of 0.49. The spin–lattice relaxation time T_1ρH in the rotating frame for each group was minimal in the three temperature regions of β, α_b, and α_c (↑) on heating and in the two temperature regions of α_1D and α_c (↓) on cooling. The α_c (↑) and α_c (↓) processes depended on the first‐order ferroelectric phase‐transition regions on heating and cooling, respectively. The motional modes for the other processes were confirmed by the T_1ρH minimum behavior of the VDF and TrFE groups in the TrFE‐rich domain and the VDF‐rich segments in the VDF‐rich domain. The β and α_b processes were attributed to the flip–flop motion of the TrFE‐rich segments and the competitive motion of the TrFE‐ and VDF‐rich segments in the ferroelectric phase, respectively. The α_1D process was due to the one‐dimensional diffusion motion of the conformational defects along the chain in the paraelectric phase, accompanied by the trans and gauche transformation of the VDF conformers of ttg⁺tg^? and g⁺tg^?tt. The effect of the competitive motion of the TrFE‐rich segment on the thermal stability of the VDF‐rich segment in the chain near the Curie temperature was examined. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1026–1037, 2002     

Some remarks and problems on complex homogeneous domains

We discuss some problems on rigidity of canonical realizations of complex homogeneous domains at ℂⁿ Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Potential flow around obstacles using the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. The method works by weakening the governing differential equations in one co‐ordinate direction through the introduction of shape functions, then solving the weakened equations analytically in the other (radial) co‐ordinate direction. These co‐ordinate directions are defined by the geometry of the domain and a scaling centre. The method can be employed for both bounded and unbounded domains. This paper applies the method to problems of potential flow around streamlined and bluff obstacles in an infinite domain. The method is derived using a weighted residual approach and extended to include the necessary velocity boundary conditions at infinity. The ability of the method to model unbounded problems is demonstrated, together with its ability to model singular points in the near field in the case of bluff obstacles. Flow fields around circular and square cylinders are computed, graphically illustrating the accuracy of the technique, and two further practical examples are also presented. Comparisons are made with boundary element and finite difference solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

Markov-type Inequalities for Surface Gradients of Multivariate Polynomials

Let K ^d be a compact set with a smooth boundary and consider a polynomial p of total degree n such that p_C(K)1. Then we show that D_Tp(x)=o(n²) for any x Bd K and T a tangential direction at x. Moreover, the o(n²) term is given in terms of the modulus of smoothness of Bd K.     

非光滑区域上非自治Navier-Stokes 方程非齐边界问题的吸引子

本文研究在2维Lipschitz区域上Navier-Stokes方程的非齐边界问题的长时间行为，在外力是时间的拟周期下，通过引入双参过程的概念，证明一致吸引子A的存在性，并给出一致吸引子A的Hausdorff维数的上界估计。     

关于C.Fefferman定理与全纯映射的边界扩充

C．Fefferman定理证明了光滑有界强拟凸域之间的双全纯映射可以光滑延拓到边界，这个结果已经被推广到各种情形．其中Bell和Catlin以及Diederich和Fornaess独立地将其推广到拟凸域的逆紧全纯映射．本文较全面地综述了C．Fefferman定理的推广情况以及Bergman投射的边界正则性问题，同时对如何去掉Bell和Catlin以及Diederich和Fornaess定理条件中的拟凸性给出一个新观察，提出一个解决方向并且说明在具体情况下这个新观察确实是可以提供答案的．     

Optimal competence set expansion using deduction graphs

A competence set is a collection of skills used to solve a problem. Based on deduction graph concepts, this paper proposes a method of finding an optimal process so as to expand a decision maker's competence set to enable him to solve his problem confidently. Using the concept of minimum spanning tree, Yu and Zhang addressed the problem of the optimal expansion of competence sets. In contrast, the method proposed here enjoys the following advantages: it can deal with more general problems involving intermediate skills and compound skills; it can find the optimal solution by utilizing a 0–1 integer program; and it can be directly extended to treat multilevel competence set problems, and thus is more practically useful.This work was supported by the National Science Council, Taipei, Taiwan, Republic of China, Grant No. NSC-81-0301-H-009-501.