On the construction of a surface from discrete derivative data and its extended surface using the least squares method

For given discrete derivative data in a rectangular region, we propose a method to generate an approximated surface which fits the given derivative data in the region and extends smoothly to a sufficiently large rectangular region. Such an extension is necessary in the generation of the surface in NC (numerical control) machine.     

On estimating the attraction regions of the equilibrium states of dynamic systems by the direct lyapunov method

The method of non-local reduction /1, 2/ is proposed for estimating the attraction regions of asymptotically stable equilibriums states of dynamic systems linked to the construction of Lyapunov functions in which the thoroughly investigated trajectories of two-dimensional systems are contained. The class of functions described below widens the class of functions obtained on the basis of Lur'e-Postnikov functions /3, 4/. That region entirely belongs to the attraction region of the system equilibrium state and is wider than the estimate of the attraction region estimated by the Lur'e-Postnikov function basis. The attraction region may exceed the limits of the interval in which the non-linearity satisfies the generalized Routh-Hurwitz conditions /5/. The algorithm proposed below contains the logarithms for specific conditions based on Lyapunov functions of the Lur'e type and the special condition of V.M. Popov.     

On the mushy region arising between two fluids in a porous medium

We study the mushy region arising between two fluids in a porous medium. We prove that the interior of the mushy region is an epigraph in the horizontal direction. Moreover, when the interior of the mushy region is empty, we give a necessary and sufficient condition to claim that the Lebesgue measure of this mushy region is zero.     

Extremal distributions for the prophet region in the independent case

In this paper, the prophet region in the independent case is investigated. A characterization of all processes is derived, leading to extreme points of this prophet region. Then we give a game-theoretical interpretation of the results and formulate a conjecture.     

湍流边界层外区扰动激发近壁区相干结构的一种机制

应用直接数值模拟的方法,研究了湍流边界层近壁区由于外区扰动的作用而导致相干结构产生的问题．结果表明:在湍流边界层近壁区的上边界存在的速度扰动,可以在近壁区内激发出相干结构,从而完善了单个相干结构的理论模型．     

关于求解随机用户均衡问题的截断拟牛顿型信赖域法研究

信赖域法是一种保证全局收敛性的优化算法,为避免Hessian矩阵的计算,基于拟牛顿校正公式构造了求解带线性等式约束的非线性规划问题的截断拟牛顿型信赖域法.首先给出了截断拟牛顿型信赖域法的构造过程及具体步骤;然后针对随机用户均衡模型中变量和约束的特点对算法进行了修正,并将多种拟牛顿校正公式下所得结果与牛顿型信赖域法的结果进行了比较,结果发现基于对称秩1校正公式的信赖域法更为合适.最后基于数值算例结果得到了一些在算法编程过程中的重要结论,对其它形式信赖域法的编程实现具有一定的参考意义.     

Approximate solution of the trust region problem by minimization over two-dimensional subspaces

The trust region problem, minimization of a quadratic function subject to a spherical trust region constraint, occurs in many optimization algorithms. In a previous paper, the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two-dimensional trust region problem. They showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem. This paper reports computational results showing that the two-dimensional minimization approach gives nearly optimal reductions in then-dimension quadratic model over a wide range of test cases. We also show that there is very little difference, in efficiency and reliability, between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization. These results may encourage the application of similar approximate trust region techniques in other contexts.Research supported by ARO contract DAAG 29-84-K-0140, NSF grant DCR-8403483, and NSF cooperative agreement DCR-8420944.     

Global asymptotical stability of the positive equilibrium of a logistic competitive model

Global asymptotical stability of the positive equilibrium (PE) of a dynamical system is one of the research focus in theoretical studies of both continuous and discrete bio-mathematical models. In this paper, we shall establish the global asymptotical stability of the PE of a discrete Logistic competitive model in certain planar region. Indeed, sufficient conditions, dependent only on the parameters of the model, are obtained to ensure the global asymptotical stability of the PE in this region. The parameter region that corresponds to these sufficient conditions can be illustrated graphically and several examples of such regions are presented. Our approach to establish the global asymptotical stability of the PE involves proving the global attractivity of the PE in the planar region concerned and a key process here is the derivation of the maxima of the related functions in the planar region.     

带形区域上SLE壳的性质

利用带形区域上SLE的性质与Schwarz反射原理,讨论了带形区域上SLE壳的性质.给出了R-对称共形映射与带形区域内壳的关系;得到了带形区域内由一对不相交的壳组成集合与Loewner共形映射之间的关系;导出了R-对称共形映射的提升在带形区域的壳空间内以及带形区域的壳对空间上的相关映射是连续.这就将上半平面上SLE壳的有关性质推广到了带形区域的情形.     

Measures of location in the plane

Summary There are many possible candidates for measures of location of asymmetric probability distributions. This difficulty is compounded for multivariate distributions. It is the purpose of this paper to characterize the set of all possible measures of location for a given bivariate probability distribution. A closed, convex region in the plane will be constructed, any point of which is a reasonable measure of location. Reasonable here refers to the invariance of the region under certain transformations and order relations. The size of this region can be used to characterize the degree of asymmetry that a distribution possesses.     

中国污染物排放强度及其收敛性的区域差异研究

本文运用回归和收敛模型对全国,东部,中部,西部污染物排放强度的差异和收敛性进行了实证分析.研究结果表明:我国各地区的污染物排放强度在不同程度上都有所下降,其中以东部下降幅度最大,西部次之,中部最小;经济增长对污染物排放强度的降低起到积极作用,第二产业占GDP比重对污染物排放强度起到正相关作用;全国,东部,中部,和西部地区都不存在绝对β收敛现象,但均存在条件收敛,且中部收敛速度最快,东部收敛速度最慢.     

On the Quasiconcave Bilevel Programming Problem

Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. In this paper, we consider the case in which both objective functions are quasiconcave and the constraint region common to both levels is a polyhedron. First, it is proved that this problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of connected faces of the polyhedron. Consequently, there is an extreme point of the polyhedron that solves the problem. Finally, it is shown that this model includes the most important case where the objective functions are ratios of concave and convex functions     

On the instability of the Riemann Hypothesis for varieties over finite fields

There exist perturbations of a rational function which remove zeroes and poles from a prescribed region as well as perturbations which add zeroes and poles to a prescribed region. We employ this to show the instability of the Riemann Hypothesis for zeta-functions of smooth projective varieties over finite fields.     

Note on the cubic decreasing region of the Chebyshev method

In this paper we reduce the two-dimensional cubic decreasing region considered in Hernandez and Salanova (2000) [1], [2] into one-dimensional region or interval for the Chebyshev method. It means that we find a simple sufficient condition for the semilocal convergence of the method.     

Cloaking via change of variables for the Helmholtz equation in the whole space

This paper is devoted to the study of a cloaking device that is composed of a standard near cloak based on a regularization of the transformation optics, i.e., a change of variables that blows up a small ball to the cloaked region, and a fixed lossy layer for the Helmholtz equation in the whole space of dimension 2 or 3 with the outgoing condition at infinity. We establish a degree of near invisibility, which is independent of the content inside the cloaked region, for this device. We also show that the lossy layer is necessary to ensure the validity of the degree of near invisibility when no constraint on physical properties inside the cloaked region is imposed. © 2010 Wiley Periodicals, Inc.     

Geometric existence theory for the control-affine nonlinear optimal regulator

For infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space.     

有界区间假设检验问题的研究

提出原假设为一个有界区间时的假设检验问题,给出了求这种假设检验的接受域、拒绝域、犯两类错误概率的方法,举例说明了这种假设检验在实际中的广泛应用.     

Computing the region of convergence for power series in many real variables: A ratio-like test

We give an elementary proof that the region of convergence for a power series in many real variables is a star-convex domain but not, in general, a convex domain. In doing so, we deduce a natural higher-dimensional analog of the so-called ratio test from univariate power series. From the constructive proof of this result, we arrive at a method to approximate the region of convergence up to a desired accuracy. While most results in the literature are for rather specialized classes of multivariate power series, the method devised here is general. As far as applications are concerned, note that while theorems such as the Cauchy-Kowalevski theorem (and its generalizations to many variables) grant the existence of a region of convergence for a multivariate Taylor series to certain PDEs under appropriate restrictions, they do not give the actual region of convergence. The determination of the maximal region of convergence for such a series solution to a PDE is one application of our result.     

On the complexity of extending the convergence region for Traub’s method

The convergence region of Traub’s method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results.     

Minimizing the area of a Pareto confidence region

A constrained optimization problem is formulated and solved in order to determine the smallest confidence region for the parameters of the Pareto distribution in a proposed family of sets. The objective function is the area of the region, whereas the constraints are related to the required confidence level. Explicit expressions for the area and confidence level of a given region are first deduced. An efficient procedure based on minimizing the corresponding Lagrangian function is then presented to solve the nonlinear programming problem. The process is valid when some of the smallest and largest observations have been discarded or censored, i.e., both single (right or left) and double censoring are allowed. The optimal Pareto confidence region is derived by simultaneously solving three (four) nonlinear equations in the right (double) censoring case. In most practical situations, Newton’s method with the balanced set as the starting point only needs a few iterations to find the global solution. In general, the reduction in area of the optimal Pareto region with respect to the balanced set is considerable if the sample size, n, is small or moderately large, which is usual in practice. This reduction is sometimes impressive when n is quite small and the censoring degree is fairly high. Two numerical examples regarding component lifetimes and fire claims are included for illustrative and comparative purposes.