scholarly journals The Lie Group in Infinite Dimension

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
V. Tryhuk ◽  
V. Chrastinová ◽  
O. Dlouhý
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Vector Fields ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Symmetries Of Differential Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Infinitesimal Transformations

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

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.

scholarly journals Pro-Lie groups which are infinite-dimensional Lie groups

2009 ◽  
Vol 146 (2) ◽  
pp. 351-378 ◽  
Author(s):  
K. H. HOFMANN ◽  
K.-H. NEEB
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Exponential Function ◽  
Lie Group ◽  
Smooth Manifold ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Lie Groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

scholarly journals DIFFERENCE-DIFFERENTIAL GAME OF CONVERGENCE - EVASION IN HILBERT SPACE, II

2017 ◽  
Vol 20 (10) ◽  
pp. 74-83
Author(s):  
V.L. Pasikov
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Dynamic Game ◽  
Continuous Functions ◽  
Scientific School ◽  
Finite Dimensional Space ◽  
Space Of Continuous Functions ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Right

For conflict operated differential system with delay studying of dynamic game of convergence - evasion relatively functional goal set, now regarding evasion and solution of a problem of existence of alternative in the case under consideration is continued. In the work realization of condition of saddle point relatively to the right part of operated system is not supposed. Earlier similar tasks were set and solved for finite-dimensional space at scientific school of the academicianN.N. Krasovsky. For a case of infinite-dimensional space of continuous functions similar tasks were considered by the author. In the suggested work at theorem proving about convergence - evasion, the norm of Hilbert space is used.

ALMOST FRÉCHET DIFFERENTIABILITY OF LIPSCHITZ MAPPINGS BETWEEN INFINITE-DIMENSIONAL BANACH SPACES

2002 ◽  
Vol 84 (3) ◽  
pp. 711-746 ◽  
Author(s):  
WILLIAM B. JOHNSON ◽  
JORAM LINDENSTRAUSS ◽  
DAVID PREISS ◽  
GIDEON SCHECHTMAN
Keyword(s):  
Banach Spaces ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Lipschitz Mapping ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Lipschitz Mappings ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.2000 Mathematical Subject Classification: 46G05, 46T20.

scholarly journals The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes

2017 ◽  
Vol 60 ◽  
pp. 263-285 ◽  
Author(s):  
Nikolaos Kariotoglou ◽  
Maryam Kamgarpour ◽  
Tyler H. Summers ◽  
John Lygeros
Keyword(s):  
Markov Decision Processes ◽  
Dimensional Space ◽  
Decision Processes ◽  
Linear Program ◽  
Control Synthesis ◽  
Programming Approach ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Markov Decision

One of the most fundamental problems in Markov decision processes is analysis and control synthesis for safety and reachability specifications. We consider the stochastic reach-avoid problem, in which the objective is to synthesize a control policy to maximize the probability of reaching a target set at a given time, while staying in a safe set at all prior times. We characterize the solution to this problem through an infinite dimensional linear program. We then develop a tractable approximation to the infinite dimensional linear program through finite dimensional approximations of the decision space and constraints. For a large class of Markov decision processes modeled by Gaussian mixtures kernels we show that through a proper selection of the finite dimensional space, one can further reduce the computational complexity of the resulting linear program. We validate the proposed method and analyze its potential with numerical case studies.

scholarly journals Benign overfitting in linear regression

2020 ◽  
Vol 117 (48) ◽  
pp. 30063-30070 ◽  
Author(s):  
Peter L. Bartlett ◽  
Philip M. Long ◽  
Gábor Lugosi ◽  
Alexander Tsigler
Keyword(s):  
Linear Regression ◽  
Sample Size ◽  
Dimensional Space ◽  
Prediction Rule ◽  
Training Data ◽  
Finite Dimensional Space ◽  
Minimum Norm ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Regression Problems

The phenomenon of benign overfitting is one of the key mysteries uncovered by deep learning methodology: deep neural networks seem to predict well, even with a perfect fit to noisy training data. Motivated by this phenomenon, we consider when a perfect fit to training data in linear regression is compatible with accurate prediction. We give a characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy. The characterization is in terms of two notions of the effective rank of the data covariance. It shows that overparameterization is essential for benign overfitting in this setting: the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size. By studying examples of data covariance properties that this characterization shows are required for benign overfitting, we find an important role for finite-dimensional data: the accuracy of the minimum norm interpolating prediction rule approaches the best possible accuracy for a much narrower range of properties of the data distribution when the data lie in an infinite-dimensional space vs. when the data lie in a finite-dimensional space with dimension that grows faster than the sample size.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

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.

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