Showing papers on "Piecewise published in 1979"

Invariant measures for Markov maps of the interval

Abstract: There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When the hypotheses fail in certain ways, this work shows that the same conclusion can still be gotten by applying the theorem mentioned to another transformation related to the original one by the method of inducing.

On the Dimension of Spaces Of Piecewise Polynomials in Two Variables

Abstract: The purpose of this paper is to discuss the dimension of linear spaces of piecewise polynomials defined on triangulations of the plane. Such spaces are of considerable interest in general approximation and data fitting, as well as in the numerical solution of boundary-value problems by the finite-element method. We begin by defining the notion of triangulation.

Pole assignment by piecewise constant output feedback

Abstract: The problem of polo assignment by piecewise constant output feedback is considered for linear time-invariant systems with infrequent observation. An algorithm for computing the output feedback gain is presented, together with necessary and sufficient conditions for pole assignment. The resultant output feedback gain is a periodic piecewise constant function of time.

On a numerically efficient method for computing multivariate B-splines

Abstract: In this paper an algorithm for the computation of smooth piecewise polynomials (multivariate B-spline) is given. The results of numerical calculation for twelve typical B-spline

Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements

Abstract: The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Local error estimates are given for the case when the finite element partitions are refined in a systematic fashion near corners. 0. Introduction. We assume that the reader is familiar with Part 1, [21, of this paper; some notation is briefly recollected in Section 1. General references to the literature were given in the Bibliography of Part 1. Of these references, the following are particularly relevant to our present situation: Babuska [1], Babuska and Aziz [21, Babuska and Rheinboldt [4], Babuska and Rosenzweig [5], Eisenstat and Schultz [1 1], Thatcher [36]. Let Q be a bounded simply connected plane polygonal domain with interior angles 0 2 denote the optimal order of the parameter h to which the spaces S" can approximate smooth functions in Lq norms. Furthermore, let Q2j, j = 1, . .. , M, be the intersection of Q with a disc of radius Rj centered at the jth vertex and such that Q2j contains no other vertex, and set Qo = 2\(UL, , 1). Also, put f3 = 7r/aj. In Part 1 we showed that with e > 0 arbitrarily small (see Part 1, Theorem 4.1 Received March 1, 1978. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. * This work was supported in part by the National Science Foundation. ? 1979 American Mathematical Society 0025-571 8/79/0000-0051 /$08.00 465 This content downloaded from 157.55.39.161 on Mon, 23 May 2016 06:01:22 UTC All use subject to http://about.jstor.org/terms 466 A. H. SCHATZ AND L. B. WAHLBIN for the precise hypotheses), IIU UhIIL (Q2i) r, we may take hM,k h (i.e., no refinement is necessary); whereas if OM r/2, no refinement is necessary at that vertex. If g3 2) the refinement process can be taken to start fairly close to the corner according to (0.10), and is less stringent than at the Mth vertex (even if f3 OM)The conditions (0.10)-(0.12) can also be motivated from simple approximation considerations, see Section 4. Let us remark that if an hr-E rate of convergence is desired only on the interior domain QO, then the weaker kind of refinement described in (0.10)-(0.12) suffices at each corner. To elucidate the above, let us give three examples. Example 0.1. A procedure for placing the nodes in the radial direction near VM. Consider the problem of how to place N + 1 nodes over [0, 1] so as to obtain an efficient approximation of the function xg (= gM) with piecewise polynomials of degree r 1. This problem was solved by Rice [1], who explicitly prescribed the location of the nodes so as to obtain a good approximation, asymptotically as N oo. Essentially, the N + 1 nodes xi, i =0, . . ., N, were taken as xi = (jIN)rl. In the two dimensional situation, one can, e.g., construct a triangular mesh near VM in the following fashion, Figure 1. Draw N + 1 radial lines (including the boundaries) from vM; along each of these mark down the N + 1 points xi. Then connect the ith points on the successive radial lines, thus obtaining a cobweb-like set of quadrilaterals. Now triangulate those by drawing one diagonal in each. The family of triangulations obtained in this simple way will, as N X oo, satisfy a maximum angle condition, but not a minimum angle one. In order to satisfy the latter, a more complicated construction would be necessary.

The Knapsack Sharing Problem

Abstract: The knapsack sharing problem has a utility or tradeoff function for each variable and seeks to maximize the value of the smallest tradeoff function (a maximin objective function). A single constraint places an upper bound on the sum of the non-negative variables. We develop efficient algorithms for piecewise linear, nonlinear, and piecewise nonlinear tradeoff functions and for any knapsack sharing problem with integer variables. These algorithms for the knapsack sharing problem extend the sharing problem algorithm in a companion paper to any piecewise linear, nonlinear, or piecewise nonlinear tradeoff functions.

On a class of transformations which have unique absolutely continuous invariant measures

Abstract: A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

On the finite time control of linear systems by piecewise constant output feedback

Abstract: The problem of finite time controllability of linear time invariant systems by piecewise constant output feedback is considered It is shown that complete controllability and complete observability are both necessary and sufficient conditions for output feedback controllability by piecewise constant gain Dead-beat output feedback controllers that drives the system to the origin in at most (μd × vd) steps, are also presented, where μd and vg are the indices of controllability and observability

Optimal feedback control via block-pulse functions

Abstract: Using block-pulse functions, a method is presented to determine the piecewise constant feedback controls for a finite linear Optimal control problem of a power system. The method is simple and computationally advantageous.

Superconvergence of collocation methods for Volterra integral equations of the first kind

Abstract: Collocation methods for solving first-kind Volterra equations in the space of piecewise polynomials possessing finite (jump) discontinuities at their knots and having degreem≧0 are known to have global order of convergencep=m+1. It is shown that a careful choice of the collocation points (characterized by the Lobatto points in (0, 1]) yields convergence of order (m+2) at the corresponding Legendre points.

An Approximation Procedure for Nonsymmetric, Nonlinear Hyperbolic Systems with Integral Boundary Conditions

Abstract: Optimal order error estimates for two linearizations of a collocation process using tensor products of continuous, piecewise linear functions in space and time are derived for a class of nonsymmetric, nonlinear hyperbolic systems with integral boundary conditions. This procedure is a variant of the so-called box scheme.Error estimates are also derived for a generalization of these procedures to collocation based on continuous, piecewise polynomials of degree r in space tensored with continuous, piecewise linear functions in time.

The Approximate Solution of Singular Integral Equations

Abstract: We present a survey of numerical methods for solving Cauchy singular integral equations on both open and closed arcs in the plane. For completeness, necessary theory is reviewed, particularly the method of regularization. For closed arcs we discuss collocation methods based on piecewise polynomial or rational representations of the solution. Emphasis here, as for the open arc case, is on regularizable equations. For open arcs a detailed discussion is given of a degenerate kernel method developed recently by Dow and Elliott. In addition to this, a generalization of a Galerkin method due to Karpenko is presented. Attention is drawn to the relation of Cauchy singular equations and solving rectangular systems of linear equations. The possibility of exploiting this for the direct solution of such equations is discussed, and some direction for future research is given.

A Study of ${\text{PC}}^1 $ Homeomorphisms on Subdivided Polyhedrons

Abstract: In this paper we consider the problem of establishing conditions when a given piecewise continuously differentiable mapping is a homeomorphism of a closed convex polyhedral set. These conditions are a generalization of the ones used by Gale–Nikaido and are similar in spirit to those of Mas-Colell. For the special case when the mapping is piecewise linear, we give an apparently new sufficiency condition for the mapping to be a homeomorphism of $R^n $. The results are further extended to include the case when the Jacobians may be singular.

Improved orders of approximation derived from interpolatory cubic splines

Abstract: Lets be a cubic spline, with equally spaced knots on [a, b] interpolating a given functiony at the knots. The parameters which determines are used to construct a piecewise defined polynomialP of degree four. It is shown thatP can be used to give better orders of approximation toy and its derivatives than those obtained froms. It is also shown that the known superconvergence properties of the derivatives ofs, at specific points of [a, b], are all special cases of the main result contained in the present paper.

Invariant *‐product quantization of the one‐dimensional Kepler problem

Abstract: It is shown that, with proper interpretation, phase space trajectories for the one‐dimensional Kepler problem retain their significance after quantization. The classical and quantum theories are both entirely conventional; only the Wigner correspondence (Weyl quantization rule) is not. The usual quantum theory may be obtained by so(2,1) ‐invariant *‐product quantization (generalized Moyal mechanics). [The analog in three dimensions is so(4,2) invariant *‐quantization; it is believed that this should encounter no new difficulties of principle.] New results concerning invariant *‐products on polynomials, in the case of so(2,1), are presented. The Kepler problem is the first known example of a nonanalytic *‐representation. Piecewise analytic *‐representations are defined and are shown to provide a general framework for invariant quantization on singular orbits of semisimple groups. When piecewise analytic *‐representations are allowed, then the specification of an invariant *‐product on polynomials is no longer sufficient to determine a unique quantum theory.

Global analysis on PL-manifolds

Abstract: The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with "constant volume density" implies smoothing. A geometric realization of PL(m)/O(m) is given in terms of Riemannian metrics. A graded differential complex 0*(M) is constructed: it appears as a subcomplex of Sullivan's complex of piecewise differentiable forms. In the complex 0*(M) the operators d, * , 8, A are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator. INTRODUCTION This paper is a slightly modified version of my Ph.D. thesis at the Massachusetts Institute of Technology. The departure point of this study was a problem proposed by I. M. Singer [S], that is the problem of the geometrical realization of Sullivan's orientation class [Su2]. This study has to be considered as a first approach towards the solution of this problem. In this paper we extend, by analogy, some techniques of partial differential equations involved in the solution of the signature theorem, from the category of smooth manifolds to that of combinatorial manifolds. To our knowledge, this paper constitutes the first tentative step of this nature. We show that some phenomena in the smooth case persist even in the combinatorial case, while others do not. This paper is divided into three chapters. After the first two preparatory sections of Chapter I, we define in ?3 a graded differential complex Q*(M) which is a subcomplex of Sullivan's complex A *(M) of piecewise differentiable forms. In the complex S*(M) the Received by the editors June 21, 1977 and, in revised form, June 22, 1978. AMS (MOS) subject classifications (1970). Primary 57D10, 58-xx, 58G05; Secondary 57C99, 58G10.

Die Projektivitätengruppe der Moulton-Ebenen

Abstract: The group of projectivities is calculated for the Moulton-Planes Mk, k>1: it does not depend on k and consists of all “piecewise projective” mappings of the circle onto itself.

Conform and mixed finite element schemes for the Dirichlet problem for the Bilaplacian in plane domains with corners

Abstract: In plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh-Clough-Tocher elements on regular triangulations as conform C1-finite elements a new error analysis is performed without higher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2nd order and its Laplacian are approximated simultaneously by C0-finite elements being piecewise polynomials.