Search results for "Hilbert"

On differences and similarities in the analysis of Lorenz, Chen, and Lu systems

2015

Currently it is being actively discussed the question of the equivalence of various Lorenz-like systems and the possibility of universal consideration of their behavior (Algaba et al., 2013a,b, 2014b,c; Chen, 2013; Chen and Yang, 2013; Leonov, 2013a), in view of the possibility of reduction of such systems to the same form with the help of various transformations. In the present paper the differences and similarities in the analysis of the Lorenz, the Chen and the Lu systems are discussed. It is shown that the Chen and the Lu systems stimulate the development of new methods for the analysis of chaotic systems. Open problems are discussed.

UN METODO INNOVATIVO DI ANALISI MODALE OPERATIVA PER L’IDENTIFICAZIONE STRUTTURALE

2022

Floquet spectrum for two-level systems in quasiperiodic time-dependent fields

1992

We study the time evolution ofN-level quantum systems under quasiperiodic time-dependent perturbations. The problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged Hilbert space, or equivalently of a generalized Floquet operator. We discuss criteria for the appearance of pure point as well as continuous spectrum, corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior. We discuss two types of mechanisms that lead to instability. The first one is due to near resonances, while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation. We tre…

Generalized Riesz systems and quasi bases in Hilbert space

2019

The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ such that $\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.

The Study of Dynamic Objects Identification Algorithms Based on Anisotropic Properties of Generalized Amplitude-Phase Images

2018

The article presents some results of dynamical objects identification technology based on coincidence matrixes of templates and tested objects’ amplitude-phase images (APIm) calculated with discrete Hilbert transforms (DHT). DHT algorithms are modeled on basis of isotropic (HTI), anisotropic (HTA), generalized transforms – AP-analysis (APA) and the difference (residual) relative shifted phase (DRSP-) images to calculate the APIm. The identified objects are recognized as members of classes modeled with 3D templates – images of different types airplanes rotated in space. The dynamic anisotropic properties of APIm causes the increasing of sensitivity to circular angle rotation and make possibl…

An Innovative Structural Dynamic Identification Procedure Combining Time Domain OMA Technique and GA

2022

In this paper an innovative and simple Operational Modal Analysis (OMA) method for structural dynamic identification is proposed. It combines the recently introduced Time Domain–Analytical Signal Method (TD–ASM) with the Genetic Algorithm (GA). Specifically, TD–ASM is firstly employed to estimate a subspace of candidate modal parameters, and then the GA is used to identify the structural parameters minimizing the fitness value returned by an appropriately introduced objective function. Notably, this method can be used to estimate structural parameters even for high damping ratios, and it also allows one to identify the Power Spectral Density (PSD) of the structural excitat…

Weighted Sobolev spaces and exterior problems for the Helmholtz equation

1987

Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ R n , it is shown here that radiating solutions satisfy a condition of the form ( 1 + r ) − 1 2 ( ln ( e + r ) ) − 1 2 δ u ∈ L 2 ( Ω ) , 0 < δ < 1 2 …

Integration of functions ranging in complex Riesz space and some applications in harmonic analysis

2015

The theory of Henstock—Kurzweil integral is generalized to the case of functions ranging in complex Riesz space R and defined on any zero-dimensional compact Abelian group. The constructed integral is used to solve the problem of recovering the R-valued coefficients of series in systems of characters of these groups by using generalized Fourier formulas.

A remarkable equality referring to spline functions in Hilbert spaces

2010

In the introduction of this paper is presented the definition of the generalized spline functions as solutions of a variational problem and are shown some theorems regarding to the existence and uniqueness. The main result of this article consists in a remarkable equality verified by the generalized spline elements, based on the properties of the spaces, operator and interpolatory set involved, which can be used as a characterization theorem of the generalized spline functions in Hilbert spaces.

U(N) tools for loop quantum gravity: the return of the spinor

2011

We explore the classical setting for the U(N) framework for SU(2) intertwiners for loop quantum gravity (LQG) and describe the corresponding phase space in terms of spinors with appropriate constraints. We show how its quantization leads back to the standard Hilbert space of intertwiner states defined as holomorphic functionals. We then explain how to glue these intertwiners states in order to construct spin network states as wave-functions on the spinor phase space. In particular, we translate the usual loop gravity holonomy observables to our classical framework. Finally, we propose how to derive our phase space structure from an action principle which induces non-trivial dynamics for the…