Universal unfoldings in Banach spaces: reduction and stability

1979 ◽  
Vol 86 (1) ◽  
pp. 41-55 ◽  
Author(s):  
R. J. Magnus
Keyword(s):  
Banach Space ◽  
Dimensional Space ◽  
Local Theory ◽  
Finite Dimensional Space ◽  
Theoretical Discussion ◽  
Theoretical Interest ◽  
Infinite Dimensions ◽  
Reduction Procedure ◽  
Finite Dimensional ◽  
Point Set

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.

scholarly journals QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

2003 ◽  
Vol 46 (2) ◽  
pp. 421-433 ◽  
Author(s):  
David W. Kribs
Keyword(s):  
Structure Theorem ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Simple Analysis ◽  
Fixed Point Set ◽  
Open Problems ◽  
C Algebras ◽  
Finite Dimensional ◽  
Dilation Theory ◽  
Point Set

AbstractWe show that the representations of the Cuntz $C^*$-algebras $\mathcal{O}_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15

scholarly journals ON SOME PROJECTION METHODS FOR THE SOLUTION OF NONLINEAR EQUATIONS WITH NONDIFFERENTIABLE OPERATORS

1993 ◽  
Vol 24 (1) ◽  
pp. 1-7
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Operator Equation ◽  
Dimensional Space ◽  
Projection Methods ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Nondifferentiable Operator

We consider a nonlinear equation with a nondifferentiable operator in a Banach space. We approximate a solution of the nonlinear equation using an iteration, whose iterates can be obtained by solving a certain operator equation in a finite dimensional space.  

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

Estimation of the Hankel Singular Values of Fractional-Order Systems

2009 ◽  
Author(s):  
Jay L. Adams ◽  
Robert J. Veillette ◽  
Tom T. Hartley
Keyword(s):  
Fractional Order ◽  
Dimensional Space ◽  
Ritz Method ◽  
Singular Values ◽  
Finite Dimensional Space ◽  
Fractional Order Systems ◽  
Norm Estimates ◽  
Finite Dimensional ◽  
Hankel Singular Values ◽  
Hankel Norm

This paper applies the Rayleigh-Ritz method to approximating the Hankel singular values of fractional-order systems. The algorithm is presented, and estimates of the first ten Hankel singular values of G(s) = 1/(sq+1) for several values of q ∈ (0, 1] are given. The estimates are computed by restricting the operator domain to a finite-dimensional space. The Hankel-norm estimates are found to be within 15% of the actual values for all q ∈ (0, 1].

Series in a Finite-Dimensional Space

1997 ◽  
pp. 13-27
Author(s):  
Mikhail I. Kadets ◽  
Vladimir M. Kadets
Keyword(s):  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Finite Dimensional

Alternating projections and interpolation of stationary processes

1992 ◽  
Vol 29 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Mohsen Pourahmadi
Keyword(s):  
Missing Values ◽  
Stationary Process ◽  
Dimensional Space ◽  
Stationary Processes ◽  
Finite Dimensional Space ◽  
Alternating Projections ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Explicit Formulae ◽  
Linear Interpolator

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

Sign in / Sign up

Export Citation Format

Share Document