Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
Papers published on a yearly basis
Papers
More filters
Posted Content•
TL;DR: In this article, it was shown that a piecewise linear function on a convex domain in R d can be represented as a boolean polynomial in terms of its linear components.
Abstract: It is shown that a piecewise linear function on a convex domain in R d can be represented as a boolean polynomial in terms of its linear components.
91 citations
TL;DR: Both the machining error and the machined surface finish are explicitly considered and evaluated in the present work, and the primary direction of the generated iso-planar tool paths is derived from the projected boundary of the discrete points.
Abstract: This article presents a method of generating iso-planar piecewise linear NC tool paths for three-axis surface machining using ball-end milling directly from discrete measured data points. Unlike the existing tool path generation methods for discrete points, both the machining error and the machined surface finish are explicitly considered and evaluated in the present work. The primary direction of the generated iso-planar tool paths is derived from the projected boundary of the discrete points. A projected cutter location net (CL-net) is then created, which groups the data points according to the intended machining error and surface finish requirements. The machining error of an individual data point is evaluated within its bounding CL-net cell from the adjacent tool swept surfaces of the ball-end mill. The positions of the CL-net nodes can thus be optimized and established sequentially by minimizing the machining error of each CL-net cell. Since the linear edges of adjacent CL-net cells are in general not perfectly aligned, weighted averages of the associated CL-net nodes are employed as the CL points for machining. As a final step, the redundant segments on the CL paths are trimmed to reduce machining time. The validity of the tool path generation method has been examined by using both simulated and experimentally measured data points.
91 citations
05 Jun 2006
TL;DR: It is proved that for functions f on a 2-manifold such ε-simplification exists, and an algorithm to construct them in the piecewise linear case is given.
Abstract: We continue the study of topological persistence [5] by investigating the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function g an e-simplification of another function f if ¦¦f−g¦¦∞≤e, and the persistence diagrams of g are the same as those of f except all points within L1-distance at most e from the diagonal have been removed. We prove that for functions f on a 2-manifold such e-simplification exists, and we give an algorithm to construct them in the piecewise linear case.
91 citations
TL;DR: A versatile technique for designing computer algorithms for separating multiple-dimensional data (feature vectors) into two classes, referred to as classifiers, that achieve nearly Bayes-minimum error rates while requiring relatively small amounts of memory.
Abstract: We describe a versatile technique for designing computer algorithms for separating multiple-dimensional data (feature vectors) into two classes. We refer to these algorithms as classifiers. Our classifiers achieve nearly Bayes-minimum error rates while requiring relatively small amounts of memory. Our design procedure finds a set of close-opposed pairs of clusters of data. From these pairs the procedure generates a piecewise-linear approximation of the Bayes-optimum decision surface. A window training procedure on each linear segment of the approximation provides great flexibility of design over a wide range of class densities. The data consumed in the training of each segment are restricted to just those data lying near that segment, which makes possible the construction of efficient data bases for the training process. Interactive simplification of the classifier is facilitated by an adjacency matrix and an incidence matrix. The adjacency matrix describes the interrelationships of the linear segments {£i}. The incidence matrix describes the interrelationships among the polyhedrons formed by the hyperplanes containing {£i}. We exploit switching theory to minimize the decision logic.
91 citations
TL;DR: In this article, a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz is given.
Abstract: We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.
91 citations