The Pomraning-Eddington approximation to diffusion of light in turbid materials

The source-free diffusion problem of light in turbid media with generalized boundary conditions is considered. The intensity of light is considered as a sum of collimated and diffused radiance. In this way the problem is transformed to a source problem with a collimated source (problem 1). This problem is solved in terms of the corresponding source-free problem of simple boundary conditions (problem 2). The Pomraning-Eddington method is used to solve problem 2. Two coupled first-order differential equations are obtained involving the energy density and the radiation net flux. Weight functions are introduced in order to force the boundary conditions to be fulfilled. Numerical results are given and compared with previous calculations. The calculations show that the accuracy depends on the choice of the weight function.     

Random costs in combinatorial optimization

The random cost problem is the problem of indentifying the minimum in a list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. On the one hand this explains the bad performance of heuristic approaches to the number partitioning problem, but on the other hand it allows one to calculate the probability distributions of the optimum and suboptimum costs.     

Computation of discrete transparent boundary conditions for the 2D Helmholtz equation

We present a family of non-local transparent boundary conditions for the 2D Helmholtz equation. The whole domain, on which the Helmholtz equation is defined, is decomposed into an interior and an exterior domain. The corresponding interior Helmholtz problem is formulated as a variational problem in a standard manner, representing a boundary value problem, whereas the exterior problem is posed as an initial value problem in the radial variable. This problem is then solved approximately by means of the Laplace transformation. The derived boundary conditions are asymptotically correct, model inhomogeneous exterior domains and are simple to implement.     

A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

In this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.     

Estimation of accuracy of an asymptotic solution of the generalized Cauchy problem for the Boussinesq equation as applied to the potential model of tsunami with a “simple” source

Statement of the hydrodynamic problem in the framework of the potential tsunami model with “simple” source whose solution is chosen as the reference one. Generalized Cauchy problem for the Boussinesq equation and its reduction to the classical one. Analytical solution of the Cauchy problem for the Boussinesq equation. An explanation of the incorrectness of the formulation of the problem. Derivation of an approximate equation for the correct setting of the Cauchy problem. The known reference solution of the problem. An analytical solution of the correct problem and the derivation of its asymptotic representation in the “far zone.” Comparison of the graphs of the temporal history of wave height calculated by the formulas of the asymptotic and reference solutions. Estimation of the accuracy of the asymptotic solution by a three-level scale. Discussion. A remark concerning the referees.     

A method of linearization for Painlevé equations: Painlevé IV, V

A method to linearize the initial value problem of the Painlevé equations IV, V is given. The procedure involves formulating a Riemann-Hilbert boundary value problem on intersecting lines for the inverse monodromy problem. This boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space. Schlesinger transformations allow one to completely cover the parameter space. Special solutions are constructed from special cases of the Riemann problem as well.     

同调谐振子的变形

实现了在保持本征值谱结构不变情况下同调谐振子的几种变形.这些变形包括三维各向同性谐振子径向方程、开普勒径向方程、Morse势s态等问题.每一个变形体系对应着同调参量g的确定取值.同调谐振子有一定的普遍适应性,具有丰富的内容. 关键词： 同调谐振子 同调参量 产生算符 湮没算符     

An iterative solution of the three-colour problem on a random lattice

We study the generalisation of Baxter's three-colour problem to a random lattice. Rephrasing the problem as a matrix model problem we discuss the analyticity structure and critical behaviour of the resulting matrix model. Based on a set of loop equations we develop an algorithm which enables us to solve the three-colour problem recursively.     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

多级轴流压气机内部流场诊断问题的数值解法

多级轴流压气机内部流场诊断问题的数值解法张增产（清华大学力学系北京１０００８４）刘杰,叶大均（清华大学热能工程系北京１０００８４）关键词流场诊断;压气机１引言压气机是航空发动机中的核心部件。但是压气机的研制是一项既复杂又困难的工作,必须进行大量的实验...     

On an inverse spectral problem

We study an inverse spectral problem for the Sturm–Liouville problem. Such a problem arises in the study of micro- and nano-rods.     

Potential scattering in the generator Coordinate method

The generator coordinate formulation of the quantum-mechanical scattering problem transforms the solution of a problem in Hilbert space into an equivalent problem in Banach space. The structure of this problem is, however, that of an ill-posed problem. In order to shed some light onto the numerical intricacies, we investigate a number of potential scattering systems. One finds that, notwithstanding the numerical uncertainties of the weight function, excellent results for phase shifts (and scattering solutions) can be obtained.     

Undressing the Kondo effect near the antiferromagnetic quantum critical point

The problem of a spin-1/2 magnetic impurity near an antiferromagnetic transition of the host lattice is shown to transform to a multichannel problem. A variety of fixed points is discovered asymptotically near the antiferromagnetic critical point. Among these is a new variety of stable fixed point of a multichannel Kondo problem which does not require channel isotropy. At this point Kondo screening disappears but coupling to spin fluctuations remains. In addition to its intrinsic interest, the problem is an essential ingredient in the problem of quantum critical points in heavy fermions.     

Modeling a waveguide with a nonlinear insert with a quadratic nonlinearity

The scalar problem of the scattering of a wave from a nonlinear insertion lying in the interior of a waveguide is reduced by the incomplete Galerkin method to the boundary value problem for a Hamiltonian system. The cases in which this problem admits a solution in finite terms are indicated. Examples are given to illustrate specific phenomena due to the nonlinearity of the problem.     

Interpretations of Quantum Theory in the Light of Modern Cosmology

The difficult issues related to the interpretation of quantum mechanics and, in particular, the “measurement problem” are revisited using as motivation the process of generation of structure from quantum fluctuations in inflationary cosmology. The unessential mathematical complexity of the particular problem is bypassed, facilitating the discussion of the conceptual issues, by considering, within the paradigm set up by the cosmological problem, another problem where symmetry serves as a focal point: a simplified version of Mott’s problem.     

Lieben Mei symmetry and conserved quantities of the Rosenberg problem

The Rosenberg problem is a typical but not too complicated problem of nonholonomic mechanical systems. The Lie—Mei symmetry and the conserved quantities of the Rosenberg problem are studied. For the Rosenberg problem, the Lie and the Mei symmetries for the equation are obtained, the conserved quantities are deduced from them and then the definition and the criterion for the Lie—Mei symmetry of the Rosenberg problem are derived. Finally, the Hojman conserved quantity and the Mei conserved quantity are deduced from the Lie—Mei symmetry.     

Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System

In this paper, we are concerned with the explicit construction of peakon solutions of the integrable twocomponent system with cubic non-linearity. We establish the spectral and inverse spectral problem associated to the Lax pairs of the system. The inverse problem is solved by the classical results of Stieltjes continued fractions, which also contributes a lot to the spectral problem. The explicit formulas are obtained from solutions of the inverse problem. The positivity of the spectral measures is implied by J. Moser's work on the Jacobi spectral problem.     

Retrospective on Quantization

Quantization is still a central problem of modern physics. One example of an unsolved problem is the quantization of Nambu mechanics. After a brief comment on the role of Harrison cohomology, this review concentrates on the central problem of quantization of QCD and, more generally, quark confinement seen as a problem of quantization. Several suggestions are made, some of them rather extravagant.     

二维金属光子晶体的带结构研究

采用一种新的平面波展开法研究金属光子晶体的带结构，即在传统平面波展开法的基础上，将“原问题”拓展，引入一个“新问题”，通过求解“新问题”得到“原问题”的带结构，并论证了它们之间的关系.为了准确求解“新问题”，引入辅助函数，将其色散关系等价为一个积分微分本征方程，求解这个本征方程得到“新问题”的带结构，从而由此导出“原问题”的带结构.最后，以正方晶格二维金属光子晶体为例，进行数值计算，得到了满意的结果. 关键词： 金属光子晶体 平面波展开法 带结构 正方晶格     

数字化机步旅抢修任务动态调度框架研究

针对一体化机动进攻作战中战场抢修任务动态分配缺乏定量化确定方法的问题,对数字化机步旅抢修任务动态调度的框架结构进行了研究。借鉴Hall三维结构方法论,从对象维、过程维和技术维3个维度建立了抢修任务动态调度的框架结构,并以此为基础,分析梳理出了抢修任务动态调度的7个核心子问题,总结了这些核心子问题的自身特性,为抢修任务动态调度的后续研究提供了宏观规划和微观指导。