A Note on the Two-Dimensional Rectangular Cutting-Stock Problem

The concept of orthogonality in the Two-Dimensional Rectangular Cutting-Stock Problem is introduced. None of the published approaches for solving this problem permit non-orthogonal cutting patterns. An example is given to show that this may prevent the optimal cutting pattern from being obtained. The application of orthogonality to the problem of obtaining the minimum area rectangular enclosure of a number of rectangles is also discussed.     

Two-Dimensional Shock Waves for a Model Problem

Mathematical Notes - The existence of nonclassical (two-dimensional) shock waves in Riemann’s problem is proved for a modification of the system of shallow water equations.     

A Stability Estimate for a Solution to a Two-Dimensional Inverse Problem of Electrodynamics

We consider the problem of finding the three coefficients c(x), (x), and q(x) in a hyperbolic equation. Here c(x) is the coefficient at the Laplace operator, (x) is the coefficient of the first time derivative, and q(x) is the coefficient of the lower-order term. The problem results from the inverse electrodynamic problem of finding the electrodynamic parameters of an isotropic medium under the assumption that the properties of the medium and the exterior current are independent of one coordinate. We suppose that the coefficients c(x)-1, (x), and q(x) are small in some norm and their supports are contained in some disk B. This is equivalent to the assumption that the electrodynamic parameters of the medium are close to constants. We suppose that the source initiating oscillations has the form of the impulse function (t)(x·) localized on the set t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The electromagnetic field excited by this source applied outside B is measured at points of the boundary of the domain B on some time interval of a fixed length T counted from the moment of arrival of the signal from the source for three different values of the parameter . It is proven that, for a sufficiently large T, these data determine the sought coefficients uniquely. We obtain a conditional stability estimate for a solution to the problem.     

曲面上的Steiner问题

本文将平面上的Steiner问题的一些结果推广到一般的正则曲面上,主要结果是 定理1 设A,B,C是正则曲面Σ上的三个点,若Σ上另一点P使Σ上的光滑弧长之和达到极小,则此三弧中每两弧在P之交角皆为120°。     

关于一类非线性双曲方程组的二维Cauchy问题

本文通过引进新的广义解定义，对一类非线性双曲方程组的二维Ｃａｕｃｈｙ问题，证明了解的存在唯一性．并且，解可能含δ波．     

Deformable Smooth Surface Design

A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin . It consists of one or more components, each tangent continuous and free of self-intersections and intersections with other components. The skin varies continuously with the weights and locations of the points, and the variation includes the possibility of a topology change facilitated by the violation of tangent continuity at a single point in space and time. Applications of the skin to molecular modeling and to geometric deformation are discussed. Received December 12, 1996, and in revised form December 4, 1997.     

约束非线性ι1问题的光滑近似算法

针对约束非线性ι1问题不可微的特点,提出了一种光滑近似算法.该方法利用“ “函数的光滑近似函数和罚函数技术将非线性ι1问题转化为无约束可微问题,并在适当的假设下,该算法是全局收敛的.初步的数值试验表明算法的有效性.     

约束非线性l_1问题的光滑近似算法

针对约束非线性l_1问题不可微的特点,提出了一种光滑近似算法.该方法利用" "函数的光滑近似函数和罚函数技术将非线性l_1问题转化为无约束可微问题,并在适当的假设下,该算法是全局收敛的.初步的数值试验表明算法的有效性.     

Analysis of a Two-Dimensional Thermal Cloaking Problem on the Basis of Optimization

For a two-dimensional model of thermal scattering, inverse problems arising in the development of tools for cloaking material bodies on the basis of a mixed thermal cloaking strategy are considered. By applying the optimization approach, these problems are reduced to optimization ones in which the role of controls is played by variable parameters of the medium occupying the cloaking shell and by the heat flux through a boundary segment of the basic domain. The solvability of the direct and optimization problems is proved, and an optimality system is derived. Based on its analysis, sufficient conditions on the input data are established that ensure the uniqueness and stability of optimal solutions.     

On a Problem of the Theory of Smooth Loops

Mathematical Notes -     

基于一类光滑函数法求解选址问题

选址问题是组合优化中一类有着重要理论意义和广泛实际背景的问题.在利用数学模型解决这类问题时经常会遇到非线性L_1问题,也就是不可微优化问题.为了解决这类问题,构造了适合于选址问题的一类新的光滑函数,并对这类光滑函数进行了性质描述,然后在此基础上提出了基于有效集法进行优化求解的计算步骤.最后,以实例证明了这类光滑函数应用在选址问题的优化求解上是有效的.     

Two-Dimensional (Two-Speed) Mode for a Nonstrictly Hyperbolic Problem

All types of solutions to the Riemann problem for a system of conservation laws that is nonstrictly hyperbolic in the sense of Petrovskii, but not in the sense of Friedrichs are described.     

A Weighted Version of the Jump Number Problem on Two-Dimensional Orders is NP-Complete

We prove the NP-completeness of a weighted version of the jump number problem on two-dimensional orders, by reducing the Maximum Independent Set on cubic planar graphs, using a geometrical construction.     

A Posteriori Error Analysis of a Fully-Mixed Finite Element Method for a Two-Dimensional Fluid-Solid Interaction Problem

In this paper we develop an a posteriori error analysis of a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements, and the fluid is supposed to occupy an annular region surrounding the solid, so that a Robin boundary condition imitating the behavior of the Sommerfeld condition is imposed on its exterior boundary. Dual-mixed approaches are applied in both domains, and the governing equations are employed to eliminate the displacement u of the solid and the pressure $p$ of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart-Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. As the main contribution of this work, we derive a reliable and efficient residual-based a posteriori error estimator for the aforedescribed coupled problem. Some numerical results confirming the properties of the estimator are also reported.     

Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface (Σ, g) with smooth boundary ?Σ. Explicitly, let λ₁(?Σ) = inf u∈W^1,2 (Σ,g),R ?Σ udsg=0,u6≡0 R Σ(|?gu|2 + u2 )dvg R ?Σ u2 dsg and H = n u ∈ W^1,2 (Σ, g) : Z Σ(|?gu|2 + u2 )dvg ? α Z ?Σ u2dsg ≤ 1 and Z ?Σ u dsg = 0o ,where W^1,2 (Σ, g) denotes the usual Sobolev space and ?g stands for the gradient operator.By the method of blow-up analysis, we obtain sup u∈H Z ?Σ e πu2 dsg ( < +∞, 0 ≤ α < λ1(?Σ),= +∞, α ≥ λ1(?Σ).Moreover, the author proves the above supremum is attained by a function uα ∈ H∩C∞(Σ)for any 0 ≤ α < λ1(?Σ). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363–386], [Yang,Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J. Math., 227, 2006, 177–200] and [Yang, Y., Extremal functions for TrudingerMoser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015,3161–3193]