Locating a minisum annulus: a new partial coverage distance model

The problem is to find the best location in the plane of a minisum annulus with fixed width using a partial coverage distance model. Using the concept of partial coverage distance, those demand points within the area of the annulus are served at no cost, while for ‘uncovered’ demand points there will be additional costs proportional to their distances to the annulus. The objective of the problem is to locate the annulus such that the sum of distances from the uncovered demand points to the annulus (covering area) is minimized. The distance is measured by the Euclidean norm. We discuss the case where the radius of the inner circle of the annulus is variable, and prove that at least two demand points must be on the boundary of any optimal annulus. An algorithm to solve the problem is derived based on this result.     

Locating a minisum circle in the plane

We consider the problem of locating a circle with respect to existing facilities in the plane such that the sum of weighted distances between the circle and the facilities is minimized, i.e., we approximate a set of given points by a circle regarding the sum of weighted distances. If the radius of the circle is a variable we show that there always exists an optimal circle passing through two of the existing facilities. For the case of a fixed radius we provide characterizations of optimal circles in special cases. Solution procedures are suggested.     

Arbitrary-norm separating plane

A plane separating two point sets in n-dimensional real space is constructed such that it minimizes the sum of arbitrary-norm distances of misclassified points to the plane. In contrast to previous approaches that used surrogates for distance-minimization, the present work is based on a precise norm-dependent explicit closed form for the projection of a point on a plane. This projection is used to formulate the separating-plane problem as a minimization of a convex function on a unit sphere in a norm dual to that of the arbitrary norm used. For the 1-norm, the problem can be solved in polynomial time by solving 2n linear programs or by solving a bilinear program. For a general p-norm, the minimization problem can be transformed via an exact penalty formulation to minimizing the sum ofa convex function and a bilinear function on a convex set. For the one and infinity norms, a finite successive linearization algorithm can be used for solving the exact penalty formulation.     

Locating least-distant lines in the plane

In this paper we deal with locating a line in a plane. Given a set of existing facilities, represented by points in the plane, our objective is to find a straight line l minimizing the sum of weighted distances to the existing facilities, or minimizing the maximum weighted distance to the existing facilities, respectively. We show that for all distance measures derived from norms, one of the lines minimizing the sum objective contains at least two of the existing facilities. For the center objective we always get an optimal line which is at maximum distance from at least three of the existing facilities. If all weights are equal, there is an optimal line which is parallel to one facet of the convex hull of the existing facilities.     

<Emphasis Type="Italic">L</Emphasis>-optimal designs in the trigonometric regression model on the full circle

Singular L-optimal designs minimizing the sum of the variances of the estimates for different pairs of coefficients in the trigonometric regression models on the full circle are found.     

Locating a general minisum ��circle�� on the plane

We approximate a set of given points in the plane by the boundary of a convex and symmetric set which is the unit circle of some norm. This generalizes previous work on the subject which considers Euclidean circles only. More precisely, we examine the problem of locating and scaling the unit circle of some given norm k with respect to given points on the plane such that the sum of weighted distances (as measured by the same norm k) between the circumference of the circle and the points is minimized. We present general results and are able to identify a finite dominating set in the case that k is a polyhedral norm.     

k-Plane Clustering

A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the sum of the squares of the 2-norm distances to each of m₁ given points in the space. The plane is generated by an eigenvector corresponding to a smallest eigenvalue of an n × n simple matrix derived from the m₁ points. The algorithm was tested on the publicly available Wisconsin Breast Prognosis Cancer database to generate well separated patient survival curves. In contrast, the k-mean algorithm did not generate such well-separated survival curves.     

On the sum of distances along a circle

The arc distance between two points on a circle is their geodesic distance along the circle. We study the sum of the arc distances determined by n points on a circle, which is a useful measure of the evenness of scales and rhythms in music theory. We characterize the configurations with the maximum sum of arc distances by a balanced condition: for each line that goes through the circle center and touches no point, the numbers of points on each side of the line differ by at most one. When the points are restricted to lattice positions on a circle, we show that Toussaint's snap heuristic finds an optimal configuration. We derive closed-form formulas for the maximum sum of arc distances when the points are either allowed to move continuously on the circle or restricted to lattice positions. We also present a linear-time algorithm for computing the sum of arc distances when the points are presorted by the polar coordinates.     

Least-squares fitting of circles and ellipses

Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g., computer graphics, coordinate meteorology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses insome least-squares sense without minimizing the geometric distance to the given points.In this paper we present several algorithms which compute the ellipse for which thesum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods.Circles and ellipses may be represented algebraically, i.e., by an equation of the formF(x)=0. If a point is on the curve, then its coordinates x are a zero of the functionF. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.Dedicated to Åke Björck on the occasion of his 60th birthday     

Orthogonal regression: a teaching perspective

A well-known approach to linear least squares regression is that which involves minimizing the sum of squared orthogonal projections of data points onto the best fit line. This form of regression is known as orthogonal regression, and the linear model that it yields is known as the major axis. A similar method, reduced major axis regression, is predicated on minimizing the total sum of triangular areas formed between data points and the best fit line. Either of these methods is appropriately applied when both x and y are measured, a typical case in the natural sciences. In comparison to classical linear regression, equation derivation for the slope of the major axis and reduced major axis lines is a nontrivial process. For this reason, derivations are presented herein drawing from previous literature with as few steps as possible to enable an easily accessible understanding. Application to eruption data for Old Faithful geyser, Yellowstone National Park, Wyoming and Montana, USA enables a teaching opportunity for choice of model.     

Linear separation and approximation by minimizing the sum of concave functions of distances

One recently proposed criterion to separate two data sets in Classification is to use a hyperplane that minimizes the sum of distances to it from all the misclassified data points, where misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study an extension of this problem: we seek the hyperplane minimizing the sum of concave nondecreasing functions of the distances of misclassified points to it. It is shown that an optimal hyperplane exists containing at least $d$ affinely independent points. This extends the result known for the minimization of the sum of distances, and enables to use combinatorial local-search heuristics for this problem. As a corollary, the same result is obtained for the approximation problem in which a hyperplane minimizing the sum of concave nondecreasing functions of the distances from a set of data points is sought.     

Extremal problems for the central projection

We consider area minimizing problems for the image of a closed subset in the unit sphere under a projection from the center of the sphere to a tangent plane, the central projection. We show, for any closed subset in the sphere, the uniqueness of a tangent plane that minimizes the area, and then the minimality of the spherical discs among closed subsets with the same spherical area.     

Odd and even hamming spheres also have minimum boundary

Combinatorial problems with a geometric flavor arise if the set of all binary sequences of a fixed length n, is provided with the Hamming distance. The Hamming distance of any two binary sequences is the number of positions in which they differ. The (outer) boundary of a set A of binary sequences is the set of all sequences outside A that are at distance 1 from some sequence in A. Harper [6] proved that among all the sets of a prescribed volume, the ‘sphere’ has minimum boundary.We show that among all the sets in which no pair of sequences have distance 1, the set of all the sequences with an even (odd) number of 1's in a Hamming ‘sphere’ has the same minimizing property. Some related results are obtained. Sets with more general extremal properties of this kind yield good error-correcting codes for multi-terminal channels.     

球面和射影平面上不可分地图的色和

给出了球面和射影平面上带根不可分地图的色和方程,从色和方程导出了球面和射影平面上带根一般不可分地图、二部地图的计数函数方程. 利用色和理论,研究不同类地图的计数问题,得到了一种研究计数问题的新方法. 此外,还得到了一些计数显示表达式.     

Efficient Location for a Semi-Obnoxious Facility

This paper deals with a location model for the placement of a semi-obnoxious facility in a continuous plane with the twin objectives of maximizing the distance to the nearest inhabitant and minimizing the sum of distances to all the users (or the distance to the farthest user) in a unified manner. For special cases, this formulation includes (1) elliptic maximin and rectangular minisum criteria problem, and (2) rectangular maximin and minimax criteria problem. Polynomial-time algorithms for finding the efficient set and the tradeoff curve are presented.     

Geometric fit of a point set by generalized circles

In our paper we approximate a set of given points by a general circle. More precisely, given two norms k ₁ and k ₂ and a set of points in the plane, we consider the problem of locating and scaling the unit circle of norm k ₁ such that the sum of weighted distances between the circumference of the circle and the given points is minimized, where the distance is measured by a norm k ₂. We present results for the general case. In the case that k ₁ and k ₂ are both polyhedral norms, we are able to solve the problem by investigating a finite candidate set.     

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

The classical problem of Apollonius is to construct circles that are tangent to three given circles in the plane. This problem was posed by Apollonius of Perga in his work “Tangencies.” The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls, find the smallest Euclidean ball that encloses all the balls in the first collection and intersects all the balls in the second collection. We also study a generalized version of the Fermat–Torricelli problem stated as follows: given two finite collections of Euclidean balls, find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.     

A logical characterization of the Euclidean plane,sphere and cylinder

Within a mathemtaical theory, a structured set is called homogeneous,if its ‘parts’ cannot be distinguished from each other by statements of that theory. In Euclidean space geometry, among all convex surfaces and among all complete connected orientable 2‐manifolds with at least one regular point, there are only three types of homogeneous objects: the Euclidean plane, sphere and cylinder.     

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C¹ smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.     

A noniterative multiproduct multiperiod production planning method

Production planning for multiple products on a single production facility over a time horizon requires minimizing the sum of production costs (regular time, overtime and subcontracting) and inventory carrying costs for meeting the known demands of the products. It is shown that the problem can be formulated and solved by a simple and noniterative method of ‘column minima’ even for multiple product situations.