A new characterization of invariant subspaces of and applications to the optimal sensitivity problem

2005 ◽  
Vol 54 (6) ◽  
pp. 539-545 ◽  
Author(s):  
Kenji Kashima ◽  
Yutaka Yamamoto
Keyword(s):  
Invariant Subspaces ◽  
Sensitivity Problem

scholarly journals Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

2016 ◽  
Vol 38 (3) ◽  
pp. 1086-1117 ◽  
Author(s):  
GREGORY R. MALONEY ◽  
DAN RUST
Keyword(s):  
Invariant Subspaces ◽  
Finite Graph ◽  
One Dimension ◽  
One Dimensional ◽  
Substitution Tiling ◽  
Cohomological Invariants ◽  
Tiling Spaces ◽  
Tiling Space ◽  
Cech Cohomology

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

Frames and Stable Bases for Shift-Invariant Subspaces of L2(ℝd)

1995 ◽  
Vol 47 (5) ◽  
pp. 1051-1094 ◽  
Author(s):  
Amos Ron ◽  
Zuowei Shen
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Invariant Subspaces ◽  
Fourier Domain ◽  
Main Theme ◽  
Bessel Sequence ◽  
Analogous Property ◽  
Finite Set ◽  
Image Position

AbstractLet X be a countable fundamental set in a Hilbert space H, and let T be the operator Whenever T is well-defined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L2(ℝd), and for sets X of the form with Φ either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT* and T* T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators.

scholarly journals Closed sets of finitary functions between finite fields of coprime order

2020 ◽  
Vol 81 (4) ◽  
Author(s):  
Stefano Fioravanti
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Linear Transformation ◽  
Minimal Polynomial ◽  
Invariant Subspaces ◽  
Closed Sets ◽  
The Right ◽  
Coprime Order

AbstractWe investigate the finitary functions from a finite field $$\mathbb {F}_q$$ F q to the finite field $$\mathbb {F}_p$$ F p , where p and q are powers of different primes. An $$(\mathbb {F}_p,\mathbb {F}_q)$$ ( F p , F q ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the invariant subspaces of the vector space $$\mathbb {F}_p^{\mathbb {F}_q\backslash \{0\}}$$ F p F q \ { 0 } with respect to a certain linear transformation with minimal polynomial $$x^{q-1} - 1$$ x q - 1 - 1 . Furthermore we prove that each of these subsets of functions is generated by one unary function.

On the asymptotic output sensitivity problem for a discrete linear systems with an uncertain initial state

2020 ◽  
Vol 8 (1) ◽  
pp. 22-34
Author(s):  
S. Ben Rhila ◽  
◽  
M. Lhous ◽  
M. Rachik ◽  
◽  
...  
Keyword(s):  
Linear System ◽  
Closed Loop ◽  
The Other ◽  
Initial State ◽  
Control Laws ◽  
Finite Dimensional ◽  
Discrete Linear Systems ◽  
Sensitivity Problem ◽  
Theoretical Results

This paper studies a finite-dimensional discrete linear system whose initial state $x_0$ is unknown. We assume that the system is augmented by two output equations, the first one $z_i$ being representing measurements made on the unknown state of the system and the other $y_i$ being representing the corresponding output. The purpose of our work is to introduce two control laws, both in closed-loop of measurements $z_i$ and whose goal is to reduce asymptotically the effects of the unknown part of the initial state $x_0$. The approach that we present consists of both theoretical and algorithmic characterization of the set of such controls. To illustrate our theoretical results, we give a number of examples and numerical simulations.

scholarly journals Structural Characterization of Controllability in Timed Continuous Petri Nets using Invariant Subspaces

2020 ◽  
Vol 53 (2) ◽  
pp. 2087-2094
Author(s):  
César Arzola ◽  
C. Renato Vaázquez ◽  
Manuel Silva ◽  
Antonio Ramírez-Treviño
Keyword(s):  
Petri Nets ◽  
Structural Characterization ◽  
Invariant Subspaces ◽  
Continuous Petri Nets
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