scholarly journals GENERALIZED SYNCHRONIZATION OF CHAOS VIA LINEAR TRANSFORMATIONS

1999 ◽  
Vol 09 (01) ◽  
pp. 215-219 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Synchronization ◽  
Linear Transformations ◽  
Driving System ◽  
Driven System ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Synchronization Manifold

Generalized synchronization (GS) of two chaotic systems is a generalization of identical synchronization. Usually, the manifold of GS is much more complex than the driven system and the driving system. In this paper, we study a special case of GS in which the synchronization manifold is linear (linear GS for short). In a theorem, we present the necessary and sufficient conditions under which a linear GS can be achieved between two chaotic systems. In particular, we study the linear GS of two Chua's circuits.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

ANTI-SYNCHRONIZATION OF A CLASS OF COUPLED CHAOTIC SYSTEMS VIA LINEAR FEEDBACK CONTROL

2006 ◽  
Vol 16 (04) ◽  
pp. 1041-1047 ◽  
Author(s):  
CHUANDONG LI ◽  
XIAOFENG LIAO
Keyword(s):  
Lyapunov Stability ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Stability Theorem ◽  
Linear Feedback ◽  
Generalized Synchronization ◽  
Lyapunov Stability Theorem ◽  
Linear Feedback Control ◽  
Relevant Variables ◽  
Special Case

As a special case of generalized synchronization, chaos anti-synchronization can be characterized by the vanishing of the sum of relevant variables. In this paper, based on Lyapunov stability theorem for ordinary differential equations, several sufficient conditions for guaranteeing the existence of anti-synchronization in a class of coupled identical chaotic systems via linear feedback or adaptive linear feedback methods are derived. Chua's circuit is presented as an example to demonstrate the effectiveness of the proposed approach by computer simulations.

scholarly journals Characterizations of the symmetrized polydisc via another family of domains

2021 ◽  
pp. 2150036
Author(s):  
Sourav Pal ◽  
Samriddho Roy
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Transformations ◽  
New Family ◽  
Spectral Interpolation ◽  
Polynomially Convex ◽  
Symmetrized Polydisc ◽  
Necessary And Sufficient

We find new characterizations for the points in the symmetrized polydisc [Formula: see text], a family of domains associated with the spectral interpolation, defined by [Formula: see text] We introduce a new family of domains which we call the extended symmetrized polydisc [Formula: see text], and define in the following way: [Formula: see text] [Formula: see text] We show that [Formula: see text] for [Formula: see text] and that [Formula: see text] for [Formula: see text]. We first obtain a variety of characterizations for the points in [Formula: see text] and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in [Formula: see text]. Also, we obtain similar characterizations for the points in [Formula: see text], where [Formula: see text]. A set of [Formula: see text] fractional linear transformations plays central role in the entire program. We also show that for [Formula: see text], [Formula: see text] is nonconvex but polynomially convex and is starlike about the origin but not circled.

scholarly journals On Finite Nilpotent Groups

1960 ◽  
Vol 12 ◽  
pp. 68-72 ◽  
Author(s):  
G. Bachman
Keyword(s):  
Cyclic Group ◽  
General Result ◽  
Sufficient Conditions ◽  
Nilpotent Groups ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Special Case

It is well known that if (n, ϕ(n)) = 1, where ϕ(n) denotes the Euler ϕ function, then the only group of order n is the cyclic group. This is a special case of a more general result due to Dickson (2, p. 201); namely, ifwhere the pi are distinct primes and each αi > 0, the necessary and sufficient conditions that the only groups of order n are abelian are (1) each αi ≤ 2 and (2) nois divisible by any p1 … , ps.We wish to establish a theorem which includes these two results. We let G(n) equal the number of groups of order n whereand we seek necessary and sufficient conditions on n so thatClearly, this problem is equivalent to finding necessary and sufficient conditions on n so that all existing groups of order n be nilpotent.

Sets of Generators of a Commutative and Associative Algebra(1)

1971 ◽  
Vol 14 (3) ◽  
pp. 315-319
Author(s):  
D. Ž. Djoković
Keyword(s):  
Recent Result ◽  
Associative Algebra ◽  
Polynomial Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quotient Algebra ◽  
Algebra 1 ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Special Case

AbstractLet A be a finite dimensional commutative and associative algebra with identity, over a field K. We assume also that A is generated by one element and consequently, isomorphic to a quotient algebra of the polynomial algebra K[X]. If A=K[a] and bi=fi(A), fi(X) ∊ K[X], 1≤i≤r we find necessary and sufficient conditions which should be satisfied by fi(X) in order that A = K[b1, …, br].The result can be stated as a theorem about matrices. As a special case we obtain a recent result of Thompson [4].In fact this last result was established earlier by Mirsky and Rado [3]. I am grateful to the referee for supplying this reference.

scholarly journals Re-nnd SOLUTIONS OF THE MATRIX EQUATION AXB=C

2008 ◽  
Vol 84 (1) ◽  
pp. 63-72 ◽  
Author(s):  
DRAGANA S. CVETKOVIĆ-ILIĆ
Keyword(s):  
Inverse Problem ◽  
Linear Algebra ◽  
Matrix Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Explicit Solutions ◽  
Matrix Inverse ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Special Case

AbstractIn this article we consider Re-nnd solutions of the equation AXB=C with respect to X, where A,B,C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A=I we obtain the results from a paper of Groß (‘Explicit solutions to the matrix inverse problem AX=B’, Linear Algebra Appl.289 (1999), 131–134).

scholarly journals Simultaneity of Synchronization and Antisynchronization in a Class of Chaotic Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhi Liu ◽  
Rongwei Guo ◽  
Yi Qi ◽  
Cuimei Jiang
Keyword(s):  
Numerical Simulations ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Synchronization Phenomenon ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Theoretical Results

In this paper, a new synchronization phenomenon, that is, the simultaneity of synchronization and antisynchronization, is investigated for a class of chaotic systems. First, for a given chaotic system, necessary and sufficient conditions for the simultaneity of synchronization and antisynchronization are proved. Then, based on these conditions, all solutions of such synchronization phenomenon for a given chaotic system are derived. After that, physical controllers that are not only simple but also implementable are designed to realize the simultaneity of synchronization and antisynchronization in the above system. Finally, illustrative examples based on numerical simulations are used to verify the validity and effectiveness of the above theoretical results.

Stationarity and Persistence in the GARCH(1,1) Model

1990 ◽  
Vol 6 (3) ◽  
pp. 318-334 ◽  
Author(s):  
Daniel B. Nelson
Keyword(s):  
Fractional Order ◽  
Sufficient Conditions ◽  
Conditional Variance ◽  
Necessary And Sufficient Conditions ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Special Case

This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift converges almost surely to zero, while IGARCH(1,1) with a positive drift is strictly stationary and ergodic. We examine the persistence of shocks to conditional variance in the GARCH(l.l) model, and show that whether these shocks "persist" or not depends crucially on the definition of persistence. We also develop necessary and sufficient conditions for the finiteness of absolute moments of any (including fractional) order.

Norm Convergence of Tn

1977 ◽  
Vol 29 (6) ◽  
pp. 1340-1344 ◽  
Author(s):  
Glenn R. Luecke
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Necessary And Sufficient Conditions ◽  
Norm Convergence ◽  
Continuous Linear ◽  
Linear Transformations ◽  
Necessary And Sufficient

Throughout this paper X will denote a complex Banach space and all operators T will be assumed to be continuous linear transformations from X into X. If T is an operator then ┘(T), γ(T), and R(T) will denote the spectrum of T, the spectral radius of T, and range of T, respectively. This paper contains necessary and sufficient conditions for the (norm) convergence of {Tn} when T is an operator on X.

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