Showing papers on "Extreme point published in 1997"

Automatic knowledge database generation for classifying objects and systems therefor

Abstract: A method for automatically generating a knowledge database in an object classification system having a digital image data source, and a computer, includes the steps of inputting digital image data corresponding to a plurality of training images, and characterizing the digital image data according to pre-defined variables, or descriptors, to thereby provide a plurality of descriptor vectors corresponding to the training images. Predetermined classification codes are inputted for the plurality of training images, to thereby define object class clusters comprising descriptor vector points having the same classification codes in N-dimensional Euclidean space. The descriptor vectors, or points, are reduced using a similarity matrix indicating proximity in N-dimensional Euclidean space, to select those descriptors vectors, called extreme points, which lie on the boundary surface of their respective class cluster. The non-selected points interior to the class cluster are not included in the knowledge database. The extreme points are balanced by eliminating functionally redundant extreme points from each class cluster to provide a preliminary knowledge database. Fine tuning of the preliminary knowledge database is performed by either deleting extreme points that tend to reduce the accuracy of the database, or adding new rules which enhance the accuracy of the data base. Alternately, the fine tuning step may be skipped.

Triangulating with high connectivity

Abstract: We consider the problem of triangulating a given point set, using straight-line edges, so that the resulting graph is “highly connected”. Since the resulting graph is planar, it can be at most 5-connected. Under the nondegeneracy assumption that no three points are collinear, we characterize the point sets with three vertices on the convex hull that admit 4-connected triangulations. More generally, we characterize the planar point sets that admit triangulations having neither chords nor complex (i.e., nonfacial) triangles. We also show that any planar point set can be augmented with at most two extra points to admit a 4-connected triangulation. All our proofs are constructive, and the resulting triangulations can be constructed in O(n log n) time. We conclude by stating several open problems. In particular, it is open whether a polynomial-time algorithm exists for determining whether a point set with no degeneracy restrictions and no restrictions on the number of extreme points admits a 4- or 5-connected triangulation.

Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball

Abstract: Introduction The Canonical Decomposition The Extremals $\partial^e$ Extensions to the Extremals Linear Extreme points in $\mathfrak C$ Numerical Ranges Unitary 2-Dilations Application to the inequality $|A|-\text {Re} (e^{i\theta}A)\ge 0$ Appendix References Index.

Network Methods for Head-dependent Hydro Power Scheduling

Abstract: We study short-term planning of hydro power with a nonlinear objective function. Given prices on the power one seeks to maximize the value of the production over a time-horizon. By assuming a bilinear dependency on head and discharged water we prove that the objective varies concavely when one sends flow along cycles. It follows that in each set of points of equal value containing a local optimum, there is an extreme point of the feasible set. This suggests computing stationary points by using a modified minimum cost network flow code. The model also allows us to derive explicit convex lower bounding functions of the objective. We present computational results for a real-sized hydro-power system.

Cartographic line simplication and polygon CSG formulae in O ( n log * n ) time

Abstract: The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. By using a data structure that supports splitting convex hulls and finding extreme points, both were known to have O(n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log*n) algorithm for maintaining convex hulls under splitting at extreme points. It opens the question of whether there is a practical, linear-time solution.

On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals

Abstract: A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space C(K, (X,w)) of continuous functions, under the supremum norm, from a compact Hausdorff topological space K into a Banach space X equipped with its weak topology w. Strongly extreme points of the unit ball of C(K, (X,w)) are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of K. It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of C(K, (X,w)). Lastly, some naturally arising and previously known extreme points of the unit ball of C(K, (X,w))∗ are shown to actually be strongly exposed points.

Extreme points in triangular uhf algebras

Abstract: We examine the strongly extreme point structure of the unit balls of triangular UHF algebras The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points Results concerning the geometry of the unit ball have a long history both in Banach space theory and in the theory of operator algebras This geometry can be affiliated with structural and algebraic properties Moreover, differences in geometric properties can prove useful in classification problems Two fundamental results are the Russo-Dye Theorem and Kadison’s Theorem on isometries More recently, there has been interest in the unit balls of nonselfadjoint operator algebras, especially nest algebras [1, 2, 3, 4, 15, 16] This paper concerns the unit balls of triangular UHF algebras These and the larger class of triangular AF algebras are nonselfadjoint analogues of the UHF and AF C*-algebras studied by Glimm and Bratteli Their theory has grown rapidly, cf [8, 9, 18, 19, 20] We focus on the extreme point structure, and our results have a different flavor than those for nest algebras Specifically, we study the strongly extreme points, those boundary points whose “stable character” with respect to approximations makes them behave well under direct limits Triangular UHF algebras are direct limits of full upper triangular matrix algebras Unit balls embed into unit balls in the direct limit scheme, and some types of embeddings respect the extreme point structure while others do not This leads to structural differences in the limit algebras The geometric structures of the unit balls of different triangular UHF algebras can be very dissimilar The convex hull of the strongly extreme points, even without closure, always contains the unit ball of the diagonal Theorem 7 shows that the two coincide if and only if the algebra is semisimple This is a characterization of a purely geometric property in terms of a purely algebraic one In contrast to Received by the editors January 11, 1996 and, in revised form, March 28, 1996 1991 Mathematics Subject Classification Primary 47D25, 46K50, 46B20

Extreme points of unit balls in Lipschitz function spaces

Abstract: We give a new characterization of the set ext (BX# ) of all extreme points of the unit ball BX# in the Banach space X # of all Lipschitz functions on a metric space X. This result is applied to get a total variation characterization of ext (BX# ) in the particular case when X is a convex subset of a Banach space. Let 0 ∈ X be an arbitrarily chosen point of a metric space X = (X, d) which consists of at least two distinct points. Following Lindenstrauss [3] denote by X the Banach space of all functions f : X →R such that f(0) = 0 and ‖f‖ = sup { |f(x)− f(y)| d(x, y) : x, y ∈ X, x 6= y } <∞. In other words, the Banach space X consists of all real-valued Lipschitz functions defined on X , which are equal zero at the distinguished point 0. In the following, we always assume that the distinguished point 0 is equal to the origin of the Banach space E, whenever X is a subset of E containing the origin of E. In the study of geometric Banach space theory and its various applications it is important to have a good characterization of the extreme points of unit balls. The investigation of the set of all extreme points ext(BX#) of the unit ball BX# of X # has been originated by Rolewicz [4] who has proved the following theorem. Theorem A. Let f be a function in [0, 1] # with ‖f‖ = 1. Then f ∈ ext(B[0,1]#) if and only if |f ′(x)| = 1 a.e. on [0, 1] . Moreover, he has shown in [5] that a similar result cannot hold for the space X = [0, 1] × [0, 1] with Euclidean metric. Next, Cobzas [1] has characterized the extreme points in X for a rather restricted class of metric spaces X. Recently, Farmer [2] has presented a new characterization of the set ext(BX#) without any additional restrictions on X. More precisely, he proved the following theorem. Theorem B. Let X be a metric space, and let f be a function in X with the norm ‖f‖ = 1. Then f ∈ ext(BX#) if and only if (i) x,y = 0 for all x, y ∈ X, Received by the editors November 13, 1995. 1991 Mathematics Subject Classification. Primary 46B20.

Deciphering Threshold Functions of k -Valued Logic

Abstract: The problem of deciphering threshold functions of k-valued logic of n arguments is considered. A polynomial deciphering algorithm is proposed which, given n, uses at most O (log n (k + 1)) appeals to the oracle.

Extreme points and retractions in Banach spaces

Abstract: ForT a completely regular topological space andX a strictly convex Banach space, we study the extremal structure of the unit ball of the spaceC(T,X) of continuous and bounded functions fromT intoX. We show that when dimX is an even integer then every point in the unit ball ofC(T, X) can be expressed as the average of three extreme points if, and only if, dimT< dimX, where dimT is the covering dimension ofT. We also prove that, ifX is infinite-dimensional, the aforementioned representation of the points in the unit ball ofC(T, X) is always possible without restrictions on the topological spaceT. Finally, we deduce from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a representation as the mean of three retractions of the unit ball onto the unit sphere.

The extreme points of a class of functions with positive real part

Abstract: In spite of its elegance, extreme point theory plays a modest role in complex function theory. In a series of papers Brickman, Hallenbeck, Mac Gregor and Wilken determined the extreme points of some classical families of analytic functions. An excellent overview of their results is contained in [4]. Of fundamental importance is the availability of the extreme points of the set P of functions f analytic on the unit disc, with positive real part, normalized by f(0) = 1. These extreme points can be obtained from an integral representation formula given by Herglotz in 1911 [5]. A truly beautiful derivation of ExtP was given by Holland [6]. In this note we present yet another method, based on elementary functional analysis. As an application we determine the extreme points of the set F of functions f analytic on the unit disc, with imaginary part bounded by π 2 and normalized by f(0) = 0. They were originally determined by Milcetich [7] but our derivation is simpler. Finally we determine the extreme points of the set Pα of functions f ∈ P for which | arg f | ≤ απ2 for some constant α < 1. These were earlier described by Abu-Muhanna and Mac Gregor [1].

Extreme point solutions to the diagonal dominance problem and stability analysis of uncertain systems

Abstract: This paper provides sufficient conditions for the robust stability of interval and affine linear multi-input/multi-output (MIMO) uncertain systems, based on Rosenbrock's direct Nyquist array. The robust Gershgorin row/column, as well as the generalised diagonal dominance measures of transfer function matrices are re-defined for such systems, and the corresponding problems are solved based on using either vertices or edge points of the parameter space. An illustrative example is given.

The AR-property for Roberts’ example of a compact convex set with no extreme points Part 2: Application to the example

Abstract: In this second part of our paper, we apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR.

Robust control via convex approximation of the stability region

Abstract: A Schur invariant transformation on the coefficient space of polynomials is used to obtain a sufficient stability condition in terms of simplexes. The order of polynomials could be increased step by step starting from some low order stable simplex. A stability measure is introduced as a minimal distance between the extreme points of interval system and the surface of the stable simplex. A straightforward robust controller design procedure is proposed for interval plants. Some low order examples are given.

Optimal Control Problems with Constraints

Abstract: This chapter presents the general approach of constrained problems. Almost all of the statements are all purpose and apply to any model: Markovian and non-Markovian, with the finite and infinite horizon, with the discount factor and with the average loss. The investigation is based on the general theory of convex programming [151, 155, 191]: the Lagrange function is introduced, the corresponding version of the Kuhn-Tucker theorem is formulated, and the essence of constraints is discussed. An individual section is devoted to the most important properties of the strategic measures space. The algorithm constructing a saddle point for the Lagrange function is detailed for problems with integral functionals. The elementary example illustrating all of the theoretical reasonings is presented at the end of the chapter.

The AR-property for Roberts’ example of a compact convex set with no extreme points Part 1: General result

Abstract: We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.

Leading Coefficients and Extreme Points of Homogeneous Polynomials

Abstract: Recently, H. Hakopian proved that the squares of a bivariate homogeneous polynomial and of its gradient have, in general, the same set of maximum points on the sphere. A generalization of this result mentioned can be gained by a method O. D. Kellogg used years ago in the estimate of some coefficient functionals.

Extreme point result for robuststability of discrete systems with complex coefficients in a diamond

Abstract: Robust stability of a continuous-time system with coefficients of the characteristic polynomial varying in a diamond can be considered as a 'dual' problem to Kharitonov's theorem on interval polynomials. This paper aims at developing similar results for discrete-time systems. Specifically, it shows that stability of a family of polynomials, whose complex coefficients lie in a diamond of a transformed parameter space, can be determined by simply checking four (eight) extreme polynomials. This result can be viewed as a counterpart of Kharitonov's result (strong version) for discrete-time systems.

A simple extreme point sufficient condition for checking instability of interval matrices

Abstract: This paper provides a simple, analytical sufficient condition for checking the presence of instability of interval matrices with asymptotically stable vertex matrices.

The Lancaster’s Probabilities on R2 and Their Extreme Points

Abstract: Given two probabilities μ and ν on R, the Lancaster’s probabilities on R 2 are the probabilities $$\sigma (dx, dy)= \mu (dx) K (x, dy) = u (dy) L(y, dx)$$ with margins μ and ν such that for all \(n \epsilon N, \int_R y^n(x, dy)\) and \( \int_R y^n L(x, dy)\) are polynomials with degree less or equal to n. This lecture reviews the properties of the convex set of these measures.