scholarly journals Hilbert space embeddings of conditional distributions with applications to dynamical systems

2009 ◽  
Author(s):  
Le Song ◽  
Jonathan Huang ◽  
Alex Smola ◽  
Kenji Fukumizu
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Conditional Distributions

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Entropy of dynamical systems and perturbations of operators

1991 ◽  
Vol 11 (4) ◽  
pp. 779-786 ◽  
Author(s):  
Dan Voiculescu
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Lower Bound ◽  
Hausdorff Measure ◽  
Spectral Measure ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
Hilbert Space Operators ◽  
Neumann Class ◽  
Adjoint Operators

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.

Positive and Z-operators on Closed Convex Cones

2018 ◽  
Vol 34 ◽  
pp. 444-458
Author(s):  
Michael Orlitzky
Keyword(s):  
Game Theory ◽  
Hilbert Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Positive Operator ◽  
Convex Cone ◽  
Real Hilbert Space ◽  
Closed Convex Cone ◽  
Nonnegative Matrices ◽  
Finite Dimensional

Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

scholarly journals Skew products and minimal dynamical systems on separable Hilbert manifolds

1984 ◽  
Vol 4 (2) ◽  
pp. 213-224 ◽  
Author(s):  
A. Fathi
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Separable Hilbert Space ◽  
Countable Group ◽  
Locally Compact ◽  
Connected Manifold ◽  
Skew Products ◽  
Minimal Dynamical Systems

AbstractWe prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.

Entropy as an integral operator

2019 ◽  
Vol 69 (1) ◽  
pp. 139-146
Author(s):  
Mehdi Rahimi
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Integral Operator ◽  
Positive Operator ◽  
Kolmogorov Entropy ◽  
Kernel Operator

Abstract In this paper, we introduce the concept of entropy kernel operator for compact dynamical systems of finite Kolmogorov entropy. It is a compact positive operator on a Hilbert space. Then we state the Kolmogorov entropy in terms of the eigenvalues of the entropy kernel operator.

Hilbert space embeddings in dynamical systems

2003 ◽  
Vol 36 (16) ◽  
pp. 549-554 ◽  
Author(s):  
Alexander J. Smola ◽  
S.V.N. Vishwanathan
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems
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