Quasicrystals and the Wulff-Shape

Infinite sphere packings give information about the structure but not about the shape of large dense sphere packings. For periodic sphere packings a new method was introduced in [W2], [W3], [Sc], and [BB], which gave a direct relation between dense periodic sphere packings and the Wulff-shape, which describes the shape of ideal crystals. In this paper we show for the classical Penrose tiling that dense finite quasiperiodic circle packings also lead to a Wulff-shape. This indicates that the shape of quasicrystals might be explained in terms of a finite packing density. Here we prove an isoperimetric inequality for unions of Penrose rhombs, which shows that the regular decagon is, in a sense, optimal among these sets. Motivated by the analysis of linear densities in the Penrose plane we introduce a surface energy for a class of polygons, which is analogous to the Gibbs—Curie surface energy for periodic crystals. This energy is minimized by the Wulff-shape, which is always a polygon and in certain cases it is the regular decagon, in accordance with the fivefold symmetry of quasicrystals. Received June 1, 1997, and in revised form November 3, 1997.     

On some nonlinear evolution equations with strong dissipation,III

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with strong dissipation studied in previous papers [5, 6], where we proved the existence of global classical solutions with small data applying small energy techniques. This time, we prove the existence of a set of initial values which guarantees the solution to be global. We know the set is not bounded in the escalated energy spaces (Sobolev spaces). For the purpose, we establish approximate equations with another dissipative term which give a devised penalty to the solutions and lead the solutions to be bounded for all t > 0. Therefore we give an improvement to existence theories of equations describing a local statement of balance of momentum for materials for which the stress is related to strain and strain rate. These have been studied by many authors (cf. Greenberg et al. 19], Greenberg [10], Davis [3], Clements [2], Andrews [1], Yamada [12], Webb [13], etc.).     

Stabilisation faible interne locale de système élastique de Bresse

In [1], Alabau-Boussouira et al. (2011) studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. In this Note, our goal is to extend the results from Alabau-Boussouira et al. (2011) [1], by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we show that the system is exponentially stable. On the contrary, we establish a new polynomial energy decay rate.     

Exact Solutions of Compressible Flow Equations with Spherical Symmetry

In this paper, we construct spherically symmetric solutions of the equations of compressible flow, which are important in the theory of explosion waves in air, water, and other media. Following McVittie [ 1 ], we write a general solution form, in terms of velocity potential, as a product of a function of time and a function of a similarity variable. First, we find solutions to the equations of motion and continuity without reference to adiabatic or isentropic relation. These solutions are quite general and can be applied to nonadiabatic motions, such as the motions of interstellar gas clouds that lose energy by radiation. All the solutions found by McVittie [ 1 ] have linear velocity profile with respect to distance. We introduce a nonlinear form of the velocity function containing an arbitrary function of the similarity variable. Adiabatic conditions lead to a second-order ODE, which we discuss in some detail. We relate our work to the earlier investigations of Taylor [ 2 ], McVittie [ 1 ], and Keller [ 3 ].     

光测弹性理论中的耦联变分原理和广义耦联变分原理

在本文中,应用拉格朗日乘子法和高阶拉格朗日乘子法[1],我们系统地导出了光测弹性理论中的耦联势能原理,耦联余能原理和具有二类和三类变量的广义耦联势能原理和广义相联余能原理。     

求统计量极限分布的一个方法

一九四九年第一届Berkeley会议的论文集上刊印了一篇长达40多页的论文,在这篇论文(见[1])中,许宝騄先生用一个统一的方法,一举解决了一元、多元统计分析中近20个统计量的极限分布。这一方法的想法是非常明确的,它充分利用了数理统计的特点:样本是一组独立同分布的随机变量(或随机向量)。可惜的是后来的许多教科书和专著中,都没有把这一方法给以介绍和展开,其实,现在新提出的某些统计量,它们的极限分布是可以用这一方法求出的,因此,籍助几个重要的例子来说明这一方法还是值得的,本文就是为此目的而写的。     

影子价格与企业管理决策

本文利用线性规划与非线性规划模型，讨论了目标函数增量，影子价格及相应的常数项增量的特征区间之间的关系，从理论上对文[1]、[2]、[3]中的问题作出了解释。我们还给出了线性规划与非线性规划发生悖论的充要条件，对文[5]、[7]中的结果进行了推广。     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

Scaling Laws in Microphase Separation of Diblock Copolymers

Summary. In this note, we study a nonlocal variational problem modeling microphase separation of diblock copolymers ([22], [3], [21]). We apply certain new tools developed in [5] to determine the principal part of the asymptotic expansion of the minimum free energy. That is, we prove a scaling law for the minimum energy and confirm that it is attained by a simple periodic lamellar structure. A previous result of Ohnishi et al. [23] was for one space dimension. Here, we obtain a similar result for the full three-dimensional problem. Received February 20, 2001; accepted June 2, 2001 Online publication August 13, 2001     

A remark on the Lifshitz tail for Schrödinger operator with random magnetic field

In this note, we consider the Lifshitz singularity for Schrödinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrödinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.     

四阶椭圆型方程奇异摄动问题的数值解

本文对一类四阶椭圆型方程奇异摄动问题建立了指数型拟合差分格式,并且证明了这种格式在能量范数意义下关于小参数ε的一致收敛性.最后,我们给出了数值结果.     

初生空化状态相似性问题

在前文中[1],讨论了热力学条件与初生空化的关系.在此基础上,引进空泡群的体积函数z₀(r),代替前文中的能量方程式,讨论相似流动体系中的空化状态相似性问题.分析结果表明,在两个满足Froude准则的相似流动体系中,空化状态是不相似的,其初生空化数随几何比尺的增加而增加.理论结果与实践是符合的.     

ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

Asymptotic equipartition of energy for equations of elasticity

In this paper we give a review of the equipartition of energy results of Goldstein and Sandefur [3], [4], [5] as well as proving a new result in the case of a particular fourth order equation. These results are then applied to the equations of elasticity to give a weak asymptotic orthogonality for the shear and pressure waves. In the case of boundary value problems in the interior of a bounded domain we get weak asymptotic orthogonality in the average.     

ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

Solving the Minimum Weighted Integer Coloring Problem

In this paper, we present, as we are aware of, the first combinatorialalgorithm specifically designed for the minimum weighted integercoloring problem (MWIP). We test the algorithm on randomly generated graphs with integer weights uniformly drawn from intervals [1, 1], [1, 2], [1, 5], [1, 10], [1, 15], and [1, 20]. We also use theproposed algorithm to test the quality of a simple, yet effectiveheuristic for the MWIP in the literature. We have observed from our test that: i( the algorithm is able to solve MWIP on graphs of up to 20 vertices when the average vertex weights is not too large; ii( The relative gap between the simple heuristic solutions and the optimal solution seems to decrease as the average vertex weight increases. A rough comparison with the state-of-the-art methods for the minimum unweighted coloring problem seems to suggest the advantage of solving MWIP directly.     

An energy consistent hybrid space-time Galerkin method for nonlinear thermomechanical problems

In this paper, we present an energy consistent hybrid space-time discretisation for thermomechanical coupled continuum dynamics, which is directly derived from the total energy itself. In comparison to a standard time approximation, we obtain the same advantages as for viscoelasticity [1], which are numerical stability and time step size independent balance laws as well as qualitative solutions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some nonlinear evolution equations with the strong dissipation,II

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with the strong dissipation studied in a previous paper [5]. In particular, we give sufficient conditions under which some of the equations have global solutions and at the same time we find steady state solutions of these equations which are exponentially stable as t → ∞. In the application, we improve the existence results to the equations which describe a local statement of balance of momentum for materials for which the stress is related to strain and strain rate through some constitutive equation (cf. Greenberg et al. [6], Greenberg [7], Davis [2], Clements [1], etc.).     

不可压缩理想势流绕两平行平板间圆柱流动的新的近似解析解

文献[1]得到了不可压缩理想势流绕两平行平板间圆柱流动的近似分析解.本文指出,用М.Е.Швец法[2]亦可得到与文献[1]完全相同的结果.如果用本文作者新提出的改进了的М.Е.Швец法[2]则可得到比文献[1]更加精确的结果.有算例.     

带非局部阻尼项的粘弹性方程解的衰减估计

In this paper,we consider the following viscoelastic equation u tt- △u +∫t 0 g(t-s)△u(s)ds + a(x)u t + u |u|r = 0 with initial condition and Dirichlet boundary condition.The decay property of the energy function closely depends on the properties of the relaxation function g(t) at infinity.In the previous works of [3,7,11],it was required that the relaxation function g(t) decay exponentially or polynomially as t → +∞.In the recent work of Messaoudi [12,13],it was shown that the energy decays at a similar rate of decay of the relaxation function,which is not necessarily dacaying in a polynomial or exponential fashion.Motivated by [12,13],under some assumptions on g(x),a(x) and r,and by introducing a new perturbed energy,we also prove the similar results for the above equation.