scholarly journals Linear dynamics and recurrence properties defined via essential idempotents of

2016 ◽  
Vol 38 (1) ◽  
pp. 285-300 ◽  
Author(s):  
YUNIED PUIG DE DIOS
Keyword(s):  
Hypercyclic Operators ◽  
Linear Dynamics ◽  
Open Set ◽  
Banach Density ◽  
Backward Shifts ◽  
Satisfying Property

Consider $\mathscr{F}$, a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property ${\mathcal{P}}_{\mathscr{F}}$ if, for any non-empty open set $U$ in $X$, there exists $x\in X$ such that $\{n\geq 0:T^{n}x\in U\}\in \mathscr{F}$. Let $\overline{{\mathcal{B}}{\mathcal{D}}}$ be the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of a sequence of operators satisfying property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$, for which we have used a deep result of Bergelson and McCutcheon in the vein of Szemerédi’s theorem. It turns out that operators having property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\unicode[STIX]{x1D6FD}\mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.

scholarly journals Controllability of Continuous Bimodal Linear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Josep Ferrer ◽  
Juan R. Pacha ◽  
Marta Peña
Keyword(s):  
Linear Systems ◽  
Higher Dimensions ◽  
Linear Dynamics ◽  
Explicit Characterization ◽  
Separating Hyperplane

We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions.

scholarly journals CHARACTERIZATION OF POLYNOMIALS

2013 ◽  
Vol 11 (02) ◽  
pp. 1350015
Author(s):  
V. E. SÁNDOR SZABÓ
Keyword(s):  
Distributional Sense ◽  
Open Set ◽  
Derivatives Of

In 1954, it was proved that if f is infinitely differentiable in the interval I and some derivatives (of order depending on x) vanish at each x, then f is a polynomial. Later, it was generalized for the multivariable case. A further extension for distributions is possible. If Ω ⊆ Rn is a non-empty connected open set, [Formula: see text] and for every [Formula: see text], there exists m(φ) ∈ N such that (Dαu)(φ) = 0 for all multi-indices α satisfying ‖α‖ = m(φ), then u is a polynomial (in distributional sense).

scholarly journals Simultaneous dense and non-dense orbits for toral diffeomorphisms

2016 ◽  
Vol 37 (4) ◽  
pp. 1308-1322 ◽  
Author(s):  
JIMMY TSENG
Keyword(s):  
Baire Category ◽  
Geometric Characterization ◽  
The Other ◽  
Baire Category Theorem ◽  
Open Set ◽  
Category Theorem ◽  
Dense Orbits ◽  
Set Of Points ◽  
Toral Automorphisms

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.

A version of a theorem of Pliss for non-uniform and non-invertible dichotomies

2016 ◽  
Vol 147 (2) ◽  
pp. 225-243 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Continuous Time ◽  
Transversality Condition ◽  
Linear Operators ◽  
Bounded Solutions ◽  
Linear Dynamics ◽  
Exponential Dichotomies ◽  
The Past ◽  
Bounded Perturbation ◽  
Exponential Behaviour

For a dynamics on the whole line, for both discrete and continuous time, we extend a result of Pliss that gives a characterization of the notion of a trichotomy in various directions. More precisely, the result gives a characterization in terms of an admissibility property in the whole line (namely, the existence of bounded solutions of a linear dynamics under any nonlinear bounded perturbation) of the existence of a trichotomy, i.e. of exponential dichotomies in the future and in the past, together with a certain transversality condition at time zero. In particular, we consider arbitrary linear operators acting on a Banach space as well as sequences of norms instead of a single norm, which allows us to consider the general case of non-uniform exponential behaviour.

scholarly journals Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

2021 ◽  
Author(s):  
Adolfo Arroyo-Rabasa
Keyword(s):  
Arbitrary Order ◽  
Elliptic Systems ◽  
Young Measures ◽  
Constant Rank ◽  
State Problem ◽  
Functional Convergence ◽  
Open Set ◽  
Area Functional ◽  
Homogeneous Operators

AbstractWe give two characterizations, one for the class of generalized Young measures generated by $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A -free measures and one for the class generated by $${\mathcal {B}}$$ B -gradient measures $${\mathcal {B}}u$$ B u . Here, $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A and $${\mathcal {B}}$$ B are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized $${\mathcal {A}}$$ A -free Young measures in duality with the class of $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A -quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized $${\mathcal {B}}$$ B -gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of $$\mathrm {L}^1$$ L 1 -compensated compactness when concentration of mass is allowed. These include the failure of $$\mathrm {L}^1$$ L 1 -estimates for elliptic systems and the lack of rigidity for a version of the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $$\Omega $$ Ω , the inclusions $$\begin{aligned} \mathrm {L}^1(\Omega ) \cap \ker {\mathcal {A}}&\hookrightarrow {\mathcal {M}}(\Omega ) \cap \ker {{\,\mathrm{{\mathcal {A}}}\,}}\,,\\ \{{\mathcal {B}}u\in \mathrm {C}^\infty (\Omega )\}&\hookrightarrow \{{\mathcal {B}}u\in {\mathcal {M}}(\Omega )\} \end{aligned}$$ L 1 ( Ω ) ∩ ker A ↪ M ( Ω ) ∩ ker A , { B u ∈ C ∞ ( Ω ) } ↪ { B u ∈ M ( Ω ) } are dense with respect to the area-functional convergence of measures.

Time Changes of Symmetric Markov Processes

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Processes ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Energy Functional ◽  
Brownian Motions ◽  
Joint Distributions ◽  
Open Set ◽  
Symmetric Markov Processes ◽  
Time Changes

This chapter discusses the time change. It first relates the perturbation of the Dirichlet form to a Feynman-Kac transform of X and deals with characterization of the Dirichlet form (Ĕ,̆‎F) of a time-changed process. The chapter next introduces the concept of the energy functional of a general symmetric transient right process, as well Feller measures on F relative to the part process X⁰ of X on the quasi open set E₀ = E∖F. It derives the Beurling-Deny decomposition of the extended Dirichlet space (̆Fₑ,Ĕ) living on F in terms of the due restriction of E to F with additional contributions by Feller measures. Finally, Feller measures are described probabilistically as the joint distributions of starting and end points of the excursions of the process X away from the set F using an associated exit system. Examples related to Brownian motions and reflecting Brownian motions are also provided.

scholarly journals Entropy Bifurcation of Neural Networks on Cayley Trees

2020 ◽  
Vol 30 (01) ◽  
pp. 2050015 ◽  
Author(s):  
Jung-Chao Ban ◽  
Chih-Hung Chang ◽  
Nai-Zhu Huang
Keyword(s):  
Neural Networks ◽  
Complete Characterization ◽  
Excitable Media ◽  
Equilibrium Solutions ◽  
Shift Of Finite Type ◽  
Symbolic Space ◽  
Open Set ◽  
Cayley Trees ◽  
Network Topologies

It has been demonstrated that excitable media with a tree structure performed better than other network topologies, therefore it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characterization of entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change.

A look at soft neutrosophic spaces

2021 ◽  
pp. 1-22
Author(s):  
Arif Mehmood ◽  
Samer Al Ghour ◽  
Muhammad Ishfaq ◽  
Farkhanda Afzal
Keyword(s):  
Open Set ◽  
Separation Axioms ◽  
Definition Of

In this article, new definition of neutrosophic soft **  b -open set is introduced with the help of neutrosophic soft α-open set and neutrosophic soft β-open set. With the application of this new definition some neutrosophic soft separation axioms and neutrosophic soft other separation axioms are addressed with respect to soft points of the spaces. Suitable examples are provided for the clarification of different results. Soft countability results and its engagements with different other neutrosophic soft results are studied. In continuation, characterization of Bolzano Weirstrass Property with respect to neutrosophic soft results and neutrosophic soft compactness results are inaugurated.

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