Class Of Positive Definite Functions Research Articles

On a certain class of positive definite functions and measures on locally compact Abelian groups and inner-product spaces

Let L be a lattice in Rn and φ the L-periodic Gaussian function on Rn given by φ(x)=∑y∈Le−‖x−y‖2. The paper was motivated by the following observation: φ(x+y)φ(x−y) is a positive definite function of two variables x,y. We say that a positive definite function φ on an Abelian group G is cross positive definite (c.p.d.) if, for each z∈G, the function φ(x+y+z)φ(x−y) of two variables x,y is positive definite. We say that a finite Radon measure on a locally compact Abelian group is c.p.d. if its Fourier transform is a c.p.d. function on the dual group. We investigate properties of c.p.d. functions and measures and give some integral characterizations. It is proved that the classes of c.p.d. functions and measures are closed with respect to certain natural operations. In particular, products, convolutions and Fourier transforms of c.p.d. functions and measures are c.p.d., whenever defined.

Uniqueness of the Continuation of a Certain Function to a Positive Definite Function

In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| < a, can be extended to a positive definite one on R if and only if 0 < a ≤ 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| ≤ a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a ∈ [1/2,1] and Re ϕ(x) = 1 - |x|, |x| ≤ a, for some ϕ ∈ T, then ϕ(x) = (1 - |x|) +, x G R. In addition, for any a G (0,1/2), there exists a function ϕ ∈ T such that ϕ(x) = 1 - |x|, |x| ≤ a, but ϕ(x) ≠ (1 - |x|)+. Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.

Positive Definite Functions on Coxeter Groups with Applications to Operator Spaces and Noncommutative Probability

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.

Optimal stable control for nonlinear dynamical systems: an analytical dynamics based approach

This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov constraint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.

Sliding-Mode Enhanced Adaptive Motion Tracking Control of Piezoelectric Actuation Systems for Micro/Nano Manipulation

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper proposes a sliding-mode enhanced adaptive control methodology for piezoelectric actuation systems to track specified motion trajectories. This control methodology is proposed to overcome the problems of unknown or uncertain system parameters, nonlinearities including the hysteresis effect, and external disturbances in the piezoelectric actuation systems, without any form of feedforward compensation. In this paper, a special class of positive definite functions is employed to formulate the control methodology such that the closed-loop system stability can be guaranteed. The control formulation, stability analysis, and analytical closed-loop solution are presented. Furthermore, a precise tracking ability in following a specified motion trajectory is demonstrated in the experimental study. With the capability of motion tracking under the aforementioned conditions, the sliding-mode enhanced adaptive control methodology is very attractive in realising high-performance control applications in the field of micro/nano manipulation. </para>

A dichotomy for a class of positive definite functions

A dichotomy for a class of positive definite functions

Some positive definite functions on sets and their application to the Ising model

Consideration of correlation inequalities for Ising ferromagnets with arbitrary spins has led to the discovery of a class of positive definite functions on sets. These functions are linear combinations of the functions which enter into Muirhead's Theorem. We prove these functions to be positive definite and also show how they can be applied to the Ising problem to prove Griffiths second inequality for arbitrary spins.

An integral representation for a class of positive definite functions

is satisfied for all k, k' e K and x e G then ~b is said to be bi-invariant with respect to K. Now let B be the set of all elementary normalized positive definite functions on G, bi-invariant with respect to K, endowed with the weak-* topology induced by LI(G). (See [2], {} 30, for definitions and elementary properties of positive definite functions.) It is known that if G is unimodular, and K is a compact subgroup and if a certain subalgebra of LI(G ) is commutative, then corresponding to each bi-invariant positive definite function ~b on G there is a Radon measure/~ on B such that