Showing papers on "Piecewise linear function published in 1968"

Spectral density of piecewise linear first order systems excited by white noise

Abstract: The Fokker-Planck equation is used to develop a general method for finding the spectral density and other properties of first order systems governed by stochastic differential equations of the form dx dt + f(x) = n(t), where f(x) is piecewise linear and n(t) is stationary Gaussian white noise. For such systems, it is shown how the Laplace transformed Fokker-Planck equation can be solved to obtain the transformed transition probability density. By manipulation of this equation and its adjoint, an expression is derived for the spectral density and moments in terms of the transformed transition density and its derivatives. The results in several special cases are presented.

Free piecewise linear involutions on spheres

Abstract: If T is a piecewise linear fixed-point free involution on S, the orbit space Q = S/T is a PL-manifold homotopy equivalent to P»(JR) ~ P n [2J; the affirmative solution to the Poincaré conjecture implies that conversely for n&Z, 4 the double covering manifold of any such Q can be identified with S. Write In for the set of (oriented if n is even) PL-homeomorphism classes of manifolds Q homotopy equivalent to P \\ We will compute In for ti^Z, 4. Let Q be as above. We define a normal invariant rj(Q). Take a homotopy equivalence h: P-*Q (orientation-preserving if n is odd): this is unique up to homotopy. Approximate AX0 by a PL-embedding PXQ-*R{N>n); let v be the normal bundle of the embedding, which exists if N is large enough [5], and F: p—>e the fibre homotopy trivialisation induced by the homotopy equivalence [7], [10, 3.5]. Then (*>, F) induces a homotopy class TJ(Q) of maps P—»G/PL, which depends only on the PL-homeomorphism class of Q. We have thus defined 77: ƒ»—*[P, G/PL]: our description follows Sullivan [8], the main idea goes back to Novlkov [6]. We next compute [P, G/PL]. The homotopy groups of G/PL are known to be Z (in dimensions 4i), Z2 (in dimensions 4i+2), and 0 (in odd dimensions). Further, Sullivan [8] has shown that if finite groups of odd order are ignored, the only nonzero &-invariant is the first (which is 8Sq). We choose fundamental classes x2i G# (G/PLi Z2) (*V2), aE.H {P\\ Z2). Because of the fe-invariant, [P, G/PL]=~Z4: let y be an isomorphism. Further, denote by r the restriction [P»+\\ G/PL]-»[P, G/PL]. Then we have

Time-optimal control of a moving-coil linear actuator

Abstract: The time-optimal control problem for a moving-coil linear actuator has been worked out by means of functional analysis and a related graphical procedure requiring only data from an impulse response. On the basis of experience with a test model, there is good correlation between the theoretical and the experimental methods. While the latter is accurate over only very short distances, the usefulness of the technique can be extended as needed by operating the system in piecewise linear fashion. The nonlinearity of the coil inductance can be handled under computer control by including in the program the inductances for successive segments of travel.

On tubular neighborhoods of piecewise linear and topological manifolds.

Abstract: Analogues of the smooth tubular neighborhood theorem are developed for the topological and piecewise linear categories.

First order piecewise linear systems with random parametric excitation

Abstract: The Fokker-Planck equation is used to develop a general method for finding the spectral density and other properties of first order systems governed by stochastic differential equations of the form dx dt + f(x)[1 + m(t)] = n(t) , where f(x) is piecewise linear and m(t) and n(t) represent stationary Gaussian white noise. The method is similar to one used by the authors to deal with the case m(t) = 0, but is complicated by the possible existence of irregular (singular) points of the Fokker-Planck equation. Graphical results for some special cases are presented.

An Adaptive Procedure for Multiclass Pattern Classification

Abstract: —A more general adaptive procedure for determining linear or piecewise linear discriminant functions for multiclass pattern classification is proposed. The adaptive procedure is a many-pattern or group-pattern adaptation. The training sequence consists of groups of vectors in matrix form instead of single vectors. The convergence proof shows that this procedure terminates in a finite number of adaptions if the solution exists. A necessary and suffcient condition is developed for testing the linear separability of each subset of (d + 1) samples. Furthermore, the proposed procedure can be implemented with the addition of only a little complexity to existing systems. Computer simulations indecate satisfactory results.

Analysis of Elastic-Plastic Grid Systems

Abstract: An incremental load method for analyzing the piecewise linear behavior of elastic-plastic grid systems with various support conditions is presented. The assumption is made that the torsional rigidity of the members is negligible. The equilibrium and compatibility equations are written in terms of bending moments, plastic hinge rotations, load parameters and an elastic-plastic transition function. The use of this transition function allows the formulation to be applicable to every regime as well as facilitates the treatment of the reversal of plastic hinge rotations. A technique of linear scaling is employed to determine the points of transition including the state of incipient plastic collapse. The collapse mechanism is verified by means of the solution of a system of hinge rotation equations. The analysis provides the complete load-deflection history of the grid systems from the elastic regime, through all the intermediate regimes, up to and including the collapse regime. Numerical examples are given to illustrate the method of analysis.

Extension to continua of some minimum theorems in elastoplastic theory

Abstract: A study of the response of a continuum to given loads is carried out assuming a piecewise linear yield locus, normality, non-interacting yield planes and linear strain bardening. Minimum properties are determined for the plastic strain rates in the incremental case and for plastic strains satisfying holonomic and non-holonomic stress-strain laws, thus extending to continua the minimum properties determined elsewhere for discrete structural linear hardening systems.

Exponential approximation with piecewise linear error criteria

Abstract: The problem of approximating a continuous function on the interval [0, T] with a linear combination of real exponential functions is considered. Rather than restricting the analysis to mathematically well behaved error criteria, such as least-squares, the theory of distributions is used to analyze criteria based on piecewise linear weighting of the error. For such criteria, it is found that the first and second derivatives of the error measure E with respect to the parameters of the approximating function can be obtained from exact and easily evaluated formulae. A descent search to find the solution to the approximation problem by minimizing E is thus economically feasible, since no inaccurate and time-consuming numerical integrations are required to produce the needed derivatives. As an example of such a criterion, approximation in the L1 norm is treated.

The Numerical Solution of Singular Volterra Integral Equations

Abstract: It is possible to prove by Laplace transform analysis that (1.1) has a unique solution R(t) satisfying the foregoing conditions. However, this approach is not very useful for numerical purposes. We shall present a practical and efficient method of approximate solution. The successive approximations are piecewise linear. They converge to R(t) uniformly for 0 ? t < cc. The analysis is made both more difficult and more interesting by the facts that the equationi is homogeneous and is defined on an infinite interval, and the kernel is singular. The approximation method yields the existence of R (t) and some qualitative information as

A proof of Newman's theorem

Abstract: If Σn is a piecewise linear n-sphere and Qn is a piecewise linear n-ball which is a subpolyhedron of Σn thenis a piecewise linear n-ball.

Synthesis of piecewise linear characteristics

Abstract: An analytical method of synthesizing two or more non-linear characteristics put in parallel or in series, is described in the present study. The method is based on set theory and is restricted to n...

A sequential algorithm for piecewise linear classification functions

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