Characteristics of (α,β)-Mean Li–Yorke Chaos of Linear Operators on Banach Spaces

2021 ◽  
Vol 31 (08) ◽  
pp. 2150120
Author(s):  
Shengnan He ◽  
Xiaoli Sun ◽  
Mingqing Xiao
Keyword(s):  
Linear Operators ◽  
Chaotic Operators ◽  
Finite Dimensional ◽  
Chaotic Operator ◽  
Special Cases ◽  
Generalized Kernel ◽  
The Mean ◽  
Different Characteristics ◽  
Finite Dimensional Banach Space ◽  
Unbounded Orbit

In this paper, we introduce a new concept, [Formula: see text]-mean Li–Yorke chaotic operator, which includes the standard mean Li–Yorke chaotic operators as special cases. We show that when [Formula: see text] or [Formula: see text], [Formula: see text]-mean Li–Yorke chaotic dynamics is strictly stronger than the ones that appeared in mean Li–Yorke chaos. When [Formula: see text] or [Formula: see text], it has completely different characteristics from the mean Li–Yorke chaos. We prove that no finite-dimensional Banach space can support [Formula: see text]-mean Li–Yorke chaotic operators. Moreover, we show that an operator is [Formula: see text]-mean Li–Yorke chaos if and only if there exists an [Formula: see text]-mean semi-irregular vector for the underlying operator, and if and only if there exists an [Formula: see text]-mean irregular vector when [Formula: see text], which generalizes the recent results by Bernardes et al. given in 2018. When [Formula: see text], we construct a counterexample in which it is an [Formula: see text]-mean Li–Yorke chaotic operator but does not admit an [Formula: see text]-mean irregular vector. In addition, we show that an operator with dense generalized kernel is [Formula: see text]-mean Li–Yorke chaotic if and only if there exists a residual set of [Formula: see text]-mean irregular vectors, and if and only if there exists an [Formula: see text]-mean unbounded orbit.

scholarly journals A block form of a singular pencil of operators and a method of obtaining it

2019 ◽  
Keyword(s):  
Differential Operator ◽  
Dimensional Space ◽  
Operator Pencil ◽  
Linear Operators ◽  
Singular Operator ◽  
Algebraic Equations ◽  
Finite Dimensional ◽  
Block Form ◽  
Special Cases ◽  
Kronecker Canonical Form

A block form of a singular operator pencil $\lambda A+B$, where $\lambda$ is a complex parameter, and the linear operators $A$, $B$ act in finite-dimensional spaces, is described. An operator pencil $\lambda A+B$ is called regular if $n = m = rk(\lambda A+B)$, where $rk(\lambda A+B)$ is the rank of the pencil and $m$, $n$ are the dimensions of spaces (the operators map an $n$-dimensional space into an $m$-dimensional one); otherwise, if $n \ne m$ or $n = m$ and $rk(\lambda A+B)<n$, the pencil is called singular (irregular). The block form (structure) consists of a singular block, which is a purely singular pencil, i.e., it is impossible to separate out a regular block in this pencil, and a regular block. In these blocks, zero blocks and blocks, which are invertible operators, are separated out. A method of obtaining the block form of a singular operator pencil is described in detail for two special cases, when $rk(\lambda A+B) = m < n$ and $rk(\lambda A+B) = n < m$, and for the general case, when $rk(\lambda A+B) < n, m$. Methods for the construction of projectors onto subspaces from the direct decompositions, relative to which the pencil has the required block form, are given. Using these projectors, we can find the form of the blocks and, accordingly, the block form of the pencil. Examples of finding the block form for the various types of singular pencils are presented. To obtain the block form, in particular, the results regarding the reduction of a singular pencil of matrices to the canonical quasidiagonal form, which is called the Weierstrass-Kronecker canonical form, are used. Also, methods of linear algebra are used. The obtained block form of the pencil and the corresponding projectors can be used to solve various problems. In particular, it can be used to reduce a singular semilinear differential-operator equation to the equivalent system of purely differential and purely algebraic equations. This greatly simplifies the analysis and solution of differential-operator equations.

scholarly journals On a conjecture of Lindenstrauss and Perles in at most 6 dimensions

1978 ◽  
Vol 19 (1) ◽  
pp. 87-97 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Banach Space ◽  
Extreme Points ◽  
Linear Operators ◽  
Dimensional Banach Space ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space

In [1] J. Lindenstrauss and M. A. Perles studied the extreme points of the set of all linear operators T of norm ≤ 1 from a finite dimensional Banach space X into itself. In particular they studied the question “When do these extreme points form a semigroup?”.

Decoration technique and surface physics

1978 ◽  
Vol 36 (1) ◽  
pp. 436-437 ◽  
Author(s):  
H. Bethge
Keyword(s):  
Diffusion Path ◽  
Special Importance ◽  
Surface Physics ◽  
Step Structure ◽  
Initial Stage ◽  
Special Cases ◽  
Atomic Surface ◽  
The Mean ◽  
Bulk Structure ◽  
Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

scholarly journals Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Bui Duc Tinh ◽  
Nguyen Quang Hoc ◽  
Dinh Quang Vinh ◽  
Tran Dinh Cuong ◽  
Nguyen Duc Hien
Keyword(s):  
Nearest Neighbor ◽  
Young Modulus ◽  
Thermodynamic Quantities ◽  
Atom Concentration ◽  
Main Metal ◽  
Special Cases ◽  
Analytic Expressions ◽  
Rigidity Modulus ◽  
Bcc Structure ◽  
The Mean

The analytic expressions for the thermodynamic and elastic quantities such as the mean nearest neighbor distance, the free energy, the isothermal compressibility, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure, the Young modulus, the bulk modulus, the rigidity modulus, and the elastic constants of binary interstitial alloy with body-centered cubic (BCC) structure, and the small concentration of interstitial atoms (below 5%) are derived by the statistical moment method. The theoretical results are applied to interstitial alloy FeC in the interval of temperature from 100 to 1000 K and in the interval of interstitial atom concentration from 0 to 5%. In special cases, we obtain the thermodynamic quantities of main metal Fe with BCC structure. Our calculated results for some thermodynamic and elastic quantities of main metal Fe and alloy FeC are compared with experiments.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

scholarly journals A New Approach for Analyzing the Reliability of the Repair Facility in a Series System with Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Renbin Liu ◽  
Yong Wu
Keyword(s):  
Renewal Process ◽  
Decomposition Method ◽  
Process Theory ◽  
Series System ◽  
New Approach ◽  
Numerical Examples ◽  
Special Cases ◽  
The Mean ◽  
Repair Facility

Based on the renewal process theory we develop a decomposition method to analyze the reliability of the repair facility in ann-unit series system with vacations. Using this approach, we study the unavailability and the mean replacement number during(0,t]of the repair facility. The method proposed in this work is novel and concise, which can make us see clearly the structures of the facility indices of a series system with an unreliable repair facility, two convolution relations. Special cases and numerical examples are given to show the validity of our method.

scholarly journals On Stable Perturbations of the Generalized Drazin Inverses of Closed Linear Operators in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Qianglian Huang ◽  
Lanping Zhu ◽  
Xiaoru Chen ◽  
Chang Zhang
Keyword(s):  
Banach Spaces ◽  
Null Space ◽  
Linear Operators ◽  
Special Cases ◽  
Stable Perturbation ◽  
Closed Linear Operators

We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space. As special cases of our results, we recover the perturbation theorems of Wei and Wang, Castro and Koliha, Rakocevic and Wei, Castro and Koliha and Wei.

Mathematical aspects of trapping modes in the theory of surface waves

1987 ◽  
Vol 183 ◽  
pp. 421-437 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Bound States ◽  
Constant Width ◽  
Minimum Energy ◽  
Horizontal Cylinder ◽  
Linear Operators ◽  
Inviscid Fluid ◽  
Infinite Length ◽  
Horizontal Canal ◽  
Special Cases ◽  
Characteristic Modes

A horizontal canal of infinite length and of constant width and depth contains inviscid fluid under gravity. The fluid is bounded internally by a submerged horizontal cylinder which extends right across the canal and has its generators normal to the sidewalls. Suppose that the fluid is set in motion by a surface pressure varying across the canal, then some of the energy is radiated to infinity while some of the energy is trapped in characteristic modes (bound states) near the cylinder. The existence of trapping modes in special cases was shown by Stokes (1846) and Ursell (1951); a general treatment, given by Jones (1953), is based on the theory of elliptic partial differential equations in unbounded domains. In the present paper a much simpler treatment is given which uses only the theory of bounded symmetric linear operators together with Kelvin's minimum-energy theorem of classical hydrodynamics.

scholarly journals A note on liftings of hermitian elements and unitaries

1980 ◽  
Vol 21 (1) ◽  
pp. 183-185
Author(s):  
C. K. Fong
Keyword(s):  
Banach Algebra ◽  
Present Note ◽  
Partial Answer ◽  
General Question ◽  
Quotient Mapping ◽  
Unitary Element ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
Special Cases

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u−1∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

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