Weak greedy algorithms[*]This research was supported by National Science Foundation Grant DMS 9970326 and by ONR Grant N00014‐96‐1‐1003.

Theoretical greedy type algorithms are studied: a Weak Greedy Algorithm, a Weak Orthogonal Greedy Algorithm, and a Weak Relaxed Greedy Algorithm. These algorithms are defined by weaker assumptions than their analogs the Pure Greedy Algorithm, an Orthogonal Greedy Algorithm, and a Relaxed Greedy Algorithm. The weaker assumptions make these new algorithms more ready for practical implementation. We prove the convergence theorems and also give estimates for the rate of approximation by means of these algorithms. The convergence and the estimates apply to approximation from an arbitrary dictionary in a Hilbert space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Detection of discontinuities in scattered data approximation

A Detection Algorithm for the localisation of unknown fault lines of a surface from scattered data is given. The method is based on a local approximation scheme using thin plate splines, and we show that this yields approximation of second order accuracy instead of first order as in the global case. Furthermore, the Detection Algorithm works with triangulation methods, and we show their utility for the approximation of the fault lines. The output of our method provides polygonal curves which can be used for the purpose of constrained surface approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Geometric demonstration of the fundamental theorems of the calculus

After the monumental discovery of the fundamental theorems of the calculus nearly 350 years ago, it became possible to answer extremely complex questions regarding the natural world. Here, a straightforward yet profound demonstration, employing geometrically symmetric functions, describes the validity of the general power rules for integration and differentiation. Differentiation and integration are readily seen to be reverse operations that compute slopes and under-areas of curves, without requiring tedious infinitesimal limits or infinite summation algebraic procedures. The areas under any two symmetric curves within a square combine to equal its square measure. Corresponding evaluated integrals of any symmetric pair were also found to add to that same area. The general power rules and the fundamental theorems are confirmed for an infinite number of functions containing exponents from the entire real number line, rational or irrational. Any particular equation represents the slope of its own under-area formula, as first discovered by Isaac Newton, where the rate that area accumulates at a point under a curve, traced at constant horizontal velocity, is the value of the curve at that point. Applications of the calculus in mathematics, physics and chemistry elucidated the orbital structure of the atom, vast scientific formula and secrets of the nature of light and gravity.     

Teaching proof in the context of physics

This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.     

共球有限点集的k号心及其性质

在n维欧氏空间中,建立了n维共球有限点集的k号心概念,并据此导出了一串有关的平行线、共线点、共点线及共球点定理.     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Kinematic Generation of Ruled Surfaces

This paper presents a method for kinematic generation of free-form ruled surfaces. The method is based on the kinematic displacement of lines. The ruled surfaces are represented as curves on a dual unit sphere. The curves are created by using the Lie Group structure of the dual space to generate dual displacement matrices for the lines. Free-form surfaces are created by repeated geodesic interpolation using the displacement matrices. An application for these surfaces is presented in five-axis cylindrical milling.     

Self-similar solutions to the hyperbolic mean curvature flow

This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.     

Menger概率度量空间中Caristi型混合不动点定理

本文在Menger概率度量空间中引进Caristi型混合不动点概念,得到两个混合不动点定理和两个集值映象序列的公共混合不动点定理,我们的定理改进和推广了Caristi不动点定理及相关的近期重要结果.     

On a high-dimensional objective genetic algorithm and its nonlinear dynamic properties

The revival of multi-objective optimization is mainly resulted from the recent development of multi-objective evolutionary optimization that allows the generation of the overall Pareto front. This paper presents an algorithm called HOGA (High-dimensional Objective Genetic Algorithm) for high-dimensional objective optimization on the basis of evolutionary computing. It adopts the principle of Shannon entropy to calculate the weight for each object since the well-known multi-objective evolutionary algorithms work poorly on the high-dimensional optimization problem. To further discuss the nonlinear dynamic property of HOGA, a martingale analysis approach is then employed; some mathematical derivations of the convergent theorems are obtained. The obtained results indicate that this new algorithm is indeed capable of achieving convergence and the suggested martingale analysis approach provides a new methodology for nonlinear dynamic analysis of evolutionary algorithms.     

有理B-样条曲线、曲面的包络性质

研究有理Bezier曲线和B-样条曲线、曲面的包络性质,愈来愈广泛,因为它从包络磨光的角度解释了曲线、曲面的一种几何构造特征,形象地说明了模型是由多边形或多面体逐步磨光的结果.     

乘积拓扑矢量空间中的不动点定理和广义矢量平衡问题组

利用零调映象的一个不动点定理,在乘积拓扑矢量空间内得到了某些新的不动点定理,作为应用,在乘积拓扑矢量空间内,对一类广义矢量平衡问题组证明了一些平衡存在性定理,这些定理推广了近期文献中的一些重要的已知结果.     

乘积空间中非线性算子的极大极小不动点定理及迭代法*

本文研究了乘积空间中非线性算子的极大极小不动点和迭代法.作为我们结果的推论,一些耦合不动点定理被获得、它们推广了由郭大钧和Lankshmikantham获得的耦合不动点定理.(见Nonlinear Anal、11(1986),623-632)和由兰在[4]、[6]中获得的结果.     

Applications of Michael's selection theorems to fixed point theory

Applying some of Ernest Michael's selection theorems, from recent fixed point theorems on u.s.c. multimaps, we deduce generalizations of the classical Bolzano theorem, several fixed point theorems on multimaps defined on almost convex sets, almost fixed point theorems, coincidence theorems, and collectively fixed point theorems. These results are related mainly to Michael maps, that is, l.s.c. multimaps having nonempty closed convex values.     

Fixed point theorems for better admissible multimaps on almost convex sets

We obtain new fixed point theorems on multimaps in the class Bp defined on almost convex subsets of topological vector spaces. Our main results are applied to deduce various fixed point theorems, coincidence theorems, almost fixed point theorems, intersection theorems, and minimax theorems. Consequently, our new results generalize well-known works of Kakutani, Fan, Browder, Himmelberg, Lassonde, and others.     

没有开纤维的广义拟均衡问题(英文)

In this paper, we first prove some new selection and fixed point theorems in generalized convex spaces. Then, we establish some existence theorems of quasi-equilibrium and generalized quasi-equilibrium without the conditions of open fibers, by applying our selection and fixed point theorems.     

概率度量空间与映象的不动点定理

概率度量空间的概念首先由Menger[7]提出,以后许多人对这一空间的理论和应用曾进行过某些讨论(见[1-9])。本文的目的是进一步研究这一空间中映象的不动点定理。在本文的§2中,我们得出了一些新型的不动点定理,这些结果改进和加强了引文[2,3,8]中某些主要结果。     

A primal–dual interior point method for nonlinear semidefinite programming

This paper is concerned with a primal–dual interior point method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a KKT point and the inner iteration (SDPLS) that calculates an approximate barrier KKT point. Algorithm SDPLS uses a commutative class of Newton-like directions for the generation of line search directions. By combining the primal barrier penalty function and the primal–dual barrier function, a new primal–dual merit function is proposed. We prove the global convergence property of our method. Finally some numerical experiments are given.     

Asymptotes and centers of affine algebraic curves

In the and century many beautiful theorems about the intersection of plane algebraic curves have been discovered. The book of Coolidge [C] contains a large collection of such results. One may ask which of these theorems can be generalized to curves in higher dimensional spaces. In this note we wish to discuss a generalization of a theorem of Waring and apply it to the question which affine algebraic curves have a unique "center". Received: 22 February 1995 / Accepted: 11 July 1995