Showing papers on "Piecewise linear function published in 1969"

Lifting projections of convex polyhedra.

Abstract: If τ is a projection of a closed convex polyhedron P onto a convex polyhedron Q, then a lifting of Q into P is defined to be a single-valued inverse τ* of τ such that τ*(Q) is the union of closed faces of P The main result of this paper, designated the Lifting Theorem, asserts that there always exists a lifting r*, provided only that there exists at least one face of P on which τ acts one-to-one The lifting theorem represents a unifying generalization of a number of results in the theory of convex polyhedra and should prove useful as an investigative as well as a conceptual tool In the course of the proof, a special case of the Lifting Theorem is translated into linear programming terms and stated as the Basis Decomposition Theorem, which summarizes the behavior of a linear program as a function of its right-hand sides In particular, the fact that a lifting is necessarily a piecewise linear homeomorphism is reflected in the Basis Decomposition Theorem as the observation that the optimal solution of a linear program can always be chosen as a continuous function of the right-hand sides

Piecewise-quadratic and piecewise-linear Lyapunov functions for discontinuous systems†

Abstract: This paper develops necessary and sufficient conditions for the existence of Lyapunov functions of tho Lur'e type (‘ piecewise-quadratic ’) for relay-control systems and gives explicit formulae- for ‘optimum ’functions of this form for estimating regions of asymptotic stability in special cases, comparing these with results of numerical optimization. Certain similarities and advantages of ‘piecewise-linear ’Lyapunov functions are discussed. Among their benefits is algebraic simplification which allows one to compute ranges of time varying and non-linear parameters which preserve asymptotic stability in a given region in the state space; an example of this type is given.

Piecewise linear approximations of embeddings of cells and spheres in codimensions higher than two

Abstract: Recently the paper of Homma (Yokohama Math. J. 14 (1966), 47-54; MR 36 #892) which implies the possibility of piecewise linear approximation of piecewise linear manifolds in codimensions higher than two was found to contain an error, so that it is at present unclear whether the proof of this result can be completed using Homma's method. The present paper gives a proof of this result for the case of the elementary manifolds (cells and spheres), thus preserving the validity of two recently proved results whose proof were based on Homma's theorem. The method of proof used in this paper differs from Homma's method and is close to Connell's proof for approximation of stable homeomorphisms (Ann. of Math. (2) 78 (1963), 326-338; MR 27 #4238). Bibliography: 19 items

Tubular monopole of arbitrary dimensions: The radiation field

Abstract: The complete radiation field of a tubular monopole of arbitrary length and radius, which is base driven by a low-impedance coaxial transmission line, is derived. The integral equation for the current distribution, which includes a term for the feed-point correction is solved by the inversion of a matrix under the assumption that the current is a continuous, piecewise linear function. The far-zone electromagnetic field established by this current is expressed in closed form. It is found that the effect of thickness is to round off the nulls predicted by previous theory for vanishingly small radius and to enhance the high-angle radiation.

An equivalence theorem for embeddings of compact absolute neighborhood retracts

Abstract: In [2] the authors showed that if A is a k-dimensional polyhedron topologically embedded in En (2k +2 _ n, n _ 5) such that En -A is 1 =ULC, then for each E>0, there is an e-push h of (En, A) such that hi A: A--En is piecewise linear. Hence, a well-known theorem of Bing and Kister [1, Theorem 5.5 ] applies to prove Theorem 1 when Ao is a polyhedron. In fact, Theorem 5.5 of [1], together with the techniques of Homma [4] and Gluck [3] and the following engulfing theorem proved in [2], make our result possible.

Sequential Algorithm for the Design of Piecewise Linear Classifiers

Abstract: A sequential algorithm for designing piecewise linear classification functions without a priori knowledge of pattern class distributions is described. The algorithm combines adaptive error correcting linear classifier design procedures and clustering techniques under control of a performance criterion. The classification function structure is constrained to minimize design calculations and increase recognition through-put for many classification problems. Examples from the literature are used to evaluate this approach relative to other classification algorithms.

A branch-and-bound algorithm for allocation problems in which constraint coefficients depend on decision variables.

Abstract: : A branch-and-bound algorithm is developed for a class of allocation problems in which some constraint coefficients depend on the values of certain of the decision variables Were it not for these dependencies, the problems could be solved by linear programming The algorithm is developed in terms of a strategic-deployment problem in which it is desired to find a least-cost transportation fleet, subject to constraints on men and/or materiel requirements in the event of certain hypothesized contingencies Among the transportation vehicles available for selection are aircraft with the characteristic that the amount of goods deliverable by an aircraft on a particular route in a given time period (called aircraft productivity and measured in kilotons per aircraft per month) depends on the ratio of type 1 to type 2 aircraft used on that particular route A model is formulated in which these relations are first approximated by piecewise linear functions A branch-and-bound algorithm for solving the resultant nonlinear problem is then presented; the algorithm solves a sequence of linear programming problems The algorithm is illustrated by a sample problem, and comments concerning its practicality are made (Author)

Piecewise linear groups and transformation groups

Abstract: The object of this paper is to show, via specific theorems, that the notions of topological group and transformation group become severely restricted when transposed to the piecewise linear category. Let us understand a piecewise linear (PL) group to be a topological group G together with a piecewise linear structure on G (i.e., triangulation of G as a locally finite simplicial complex), in terms of which the group multiplication G x G -> G and inversion G -> G are given by piecewise linear functions.

Adaptive classifiers for direction and regulation

Abstract: In the field of control, it is often necessary to classify a large set of different input vectors into a small number of different classes. The input vectors which for example represent the state of a controlled system, define points in the so-called “pattern space”, and the classification may be regarded as a separation of these points. The structures of two different trainable classifiers for linear and piecewise linear separation are described: the madaline and the learning-matrix. As a training method for the learning-matrix with minimum distance classification, two algorithms are introduced: the method of weighted shifting and the method of error projection.

A counter-example to the three balls problem

Abstract: We work throughout in the p.l. category (see (8)) consisting of polyhedra and p.l. (piecewise linear) maps. We are concerned with the following problem which is of importance in the theory of p.l. transversality.

Linear filtering and piecewise linear correlation functions

Abstract: For a class of piecewise linear correlation functions, it is shown that optimal linear mean-square filtering is achieved with a finite number of samples of the process for any finite observation interval. The class of correlation functions is defined by a particular property of the points at which they change slope. Conditions are discussed under which an arbitrary piecewise linear function is a correlation function. An example demonstrating various aspects of the theory is given, and applications of the theory are considered.

ELAS - A general-purpose computer program for the equilibrium problems of linear structures. Volume 2 - Documentation of the program

Abstract: A general purpose digital computer program for the in-core solution of linear equilibrium problems of structural mechanics is documented. The program requires minimum input for the description of the problem. The solution is obtained by means of the displacement method and the finite element technique. Almost any geometry and structure may be handled because of the availability of linear, triangular, quadrilateral, tetrahedral, hexahedral, conical, triangular torus, and quadrilateral torus elements. The assumption of piecewise linear deflection distribution insures monotonic convergence of the deflections from the stiffer side with decreasing mesh size. The stresses are provided by the best-fit strain tensors in the least squares at the mesh points where the deflections are given. The selection of local coordinate systems whenever necessary is automatic. The core memory is used by means of dynamic memory allocation, an optional mesh-point relabelling scheme and imposition of the boundary conditions during the assembly time.

The subset of piecewise-linear mappings is dense in the space of K-quasiconformal mappings of the plane

Abstract: For each index n from the set N of natural numbers, let ~n denote the regular net o/ equilateral triangles in the complex plane C, whose vertice set consists of the points [p+( 89 -~ with integers p and q. A mapping ~: C-~C is called linear, if there are constants a, b, cEC such that ~(z)= az +bz* +c; the superscript star denotes complex conjugation. A mapping ~: C-~C is said to be piecewise-linear with respect to the net ~ , if its restrictions to the triangles of ~n are linear mappings. We define the piecewise-linearized mapping ~<~>: C-->C for a mapping ~: C-~C with respect to the net }l~ as follows: ~ is piecewise-linear with respect to ~n, and it coincides with ~ on the vertice set of ~ . The set of continuous mappings ~: C-+C will be considered as a topological space with the compact-open topology; this induces convergence in the sense of uniform convergence on compact subsets. Approximation means convergence to a given mapping. Each continuous mapping ~: C-+C is approximated by its piecewise-linearized mappings ~<~>. In the subspace of quasiconformal mappings of the plane, there is the problem: can each ~ be approximated by ~ which are piecewise-linear with respect to Tl~?

Linear Filtering and Piecewise Correlation Functions Linear

Abstract: Absfracf-For a class of piecewise linear correlation functions, it is shown that optimal linear mean-square filtering is achieved with a finite number of samples of the process for any 6nite observation interval. The class of correlation functions is defined by a particular property of the points at which they change slope. Conditions are discussed under which an arbitrary piecewise linear function is a correlation function. An example demonstrating various aspects of the theory is given, and applications of the theory are considered.