A Network Approach to a Geometric Packing Problem

We investigate a geometric packing problem (derived from an industrial setting) that involves fitting patterns of regularly spaced disks without overlap. We first derive conditions for achieving a feasible placement of a given set of patterns and then construct a network formulation that facilitates the calculation of such a placement. A heuristic utilizing this network representation is also outlined. Additionally, we show a connection to the well-known Periodic Scheduling Problem.     

A Differential Geometric Approach to the Helmholtz Equations and an Application to a Space with a Nontrivial Riemannian Metric

The necessary and sufficient conditions for variationality are obtained from the requirement that a differential two-form be closed. The classical Helmholtz equations are shown to follow from these equations. An application of these results to the case in which one of the functions in these equations is taken to be a Riemannian metric on a curved space is presented.     

Young Children's Composition of Geometric Figures: A Learning Trajectory

The purpose of this research is to chart the mathematical actions-on-objects young children use to compose geometric shapes. The ultimate goal is the creation of a hypothetical learning trajectory based on previous research, as well as instrumentation to assess levels of learning along the developmental progression underlying the trajectory. We tested both the developmental progression and the instrument through a series of studies, including formative studies (including action research by 8 teachers) and a summative study involving 72 children ages 3 to 7 years. Results provide strong support for the validity of the developmental progression's levels and suggest that children move through these levels of thinking in developing the ability to compose 2-dimensional figures. From lack of competence in composing geometric shapes, they gain abilities to combine shapes-initially through trial and error and gradually by attributes-into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes).     

A Purely Geometric Approach to the Problem of Computing the Projection of a Point on a Simplex

In this short note, we present a new proof of a classical algorithm for finding the projection of a point on the standard simplex. The new proof is based only on elementary geometric arguments.     

Lipschitz Continuity Results for a Class of Variational Inequalities and Applications: A Geometric Approach

We consider a class of parametric variational inequalities which, under suitable assumptions, admit an equivalent integral formulation. We study the Lipschitz continuity of the solution and treat in detail the case in which the parametric constraint set is a polytope. Finally, the results obtained are applied to the time-dependent traffic equilibrium problem.     

The Vanishing of the Theta Function in the KP Direction: A Geometric Approach

We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.     

A Constraint Satisfaction Approach to a Circuit Design Problem

A classical circuit-design problem from Ebers and Moll (1954) features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne (1993) in the box [0, 10]⁹. Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc-1 workstations) but they state that ‘at this time, we know no other method which has been applied to this circuit design problem and which has led to the same guaranteed result of locating exactly one solution in this huge domain, completed with a reliable error estimate’. The present paper gives a novel branch-and-prune algorithm that obtains a unique safe box for the above system within reasonable computation times. The algorithm combines traditional interval techniques with an adaptation of discrete constraint-satisfaction techniques to continuous problems. Of particular interest is the simplicity of the approach. This revised version was published online in July 2006 with corrections to the Cover Date.     

关联单形和一点的一类几何不等式

本文建立联系单形和一动点的几个新颖不等式,并提出几个有待进一步讨论的猜想.     

Geometric-Dynamic Approach to Billiard Systems: II. Geometric Features of Involutions

The following geometric features of billiard involutions are established and investigated: covariance, transformations related to reflections of billiard rays, types of symmetries, and the character of monotonicity and differentiability.     

Viability Constraints for Computing Solutions to the Lighthill-Whitham-Richards Model Involving Partial Autonomous Vehicle Flow

Set-Valued and Variational Analysis - This article proposes a new algorithm for computing solutions to mixed flow problems involving the classical Lighthill-Whitham-Richards (LWR) flow model. We...     

A Unified Approach to Optimal Feedback in the Infinite-Dimensional Linear-Quadratic Control Problem with an Inequality Constraint on the Trajectory or Terminal State

This paper develops a unified approach to optimal feedback controlin the infinite-dimensional linear-quadratic control problemwith an inequality constraint on the trajectory or terminalstate. A general abstract Hilbert-space framework for the problemis provided, and then the results are specified to concreteoptimal problems for a linear infinite-dimensional system whichallows unboundedness in the input and output operators. In allcases, the same technique is used in order to obtain the optimalcontrol in feedback form; in some cases, the differential Riccatiequation is derived for the optimal cost operator.     

A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data

In this article, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton–Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second-order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts dataset. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm. Supplementary materials including technical details, the R package fpca, and data analyzed by this article are available online.     

An Operational Research Approach to Planned Maintenance: Modelling P.M. for a Vehicle Fleet

A methodology is advanced for modelling the problem of planned preventive maintenance, thereby relating plant performance to maintenance activity. Ideas are discussed and developed within the context of a P.M. system for the maintenance of a vehicle fleet. Three specific questions are addressed: first, the omission of any desirable activities from the P.M. schedule, secondly, the presence of redundant items in the schedule, and thirdly, the consequences of different frequencies of application of the P.M. schedule. A fleet model is obtained relating the P.M. period to measures such as proportion of breakdown repairs and the cost per vehicle year.     

A Global Optimization Approach to a Water Distribution Network Design Problem

In this paper, we address a global optimization approach to a waterdistribution network design problem. Traditionally, a variety of localoptimization schemes have been developed for such problems, each new methoddiscovering improved solutions for some standard test problems, with noknown lower bound to test the quality of the solutions obtained. A notableexception is a recent paper by Eiger et al. (1994) who present a firstglobal optimization approach for a loop and path-based formulation of thisproblem, using a semi-infinite linear program to derive lower bounds. Incontrast, we employ an arc-based formulation that is linear except forcertain complicating head-loss constraints and develop a first globaloptimization scheme for this model. Our lower bounds are derived through thedesign of a suitable Reformulation-Linearization Technique (RLT) thatconstructs a tight linear programming relaxation for the given problem, andthis is embedded within a branch-and-bound algorithm. Convergence to anoptimal solution is induced by coordinating this process with an appropriatepartitioning scheme. Some preliminary computational experience is providedon two versions of a particular standard test problem for the literature forwhich an even further improved solution is discovered, but one that isverified for the first time to be an optimum, without any assumed boundson the flows. Two other variants of this problem are also solved exactly forillustrative purposes and to provide researchers with additional test caseshaving known optimal solutions. Suggestions on a more elaborate study involving several algorithmic enhancements are presented for futureresearch.     

带自由变量的广义几何规划全局求解的新算法

带自由变量的广义几何规划(FGGP)问题广泛出现在证券投资和工程设计等实际问题中.利用等价转换及对目标函数和约束函数的凸下界估计,提出一种求(FGGP)问题全局解的凸松弛方法.与已有方法相比,方法可处理符号项中含有更多变量的(FGGP)问题,且在最后形成的凸松弛问题中含有更少的变量和约束,从而在计算上更容易实现.最后数值实验表明文中方法是可行和有效的.