The asymptotic behaviour of semigroups of nonlinear contractions having large sets of fixed points

1978 ◽  
Vol 80 (3-4) ◽  
pp. 261-271 ◽  
Author(s):  
A. Pazy
Keyword(s):  
Asymptotic Behaviour ◽  
Fixed Points ◽  
Finite Codimension ◽  
Limit Sets ◽  
Large Sets ◽  
Finite Dimensional ◽  
Weil Theorem ◽  
Nonlinear Contractions

SynopsisThe asymptotic behaviour of semigroups of nonlinear contractions which have a set of fixed points containing a ball of finite codimension is studied. It is shown that the ω-limit sets of such semigroups are finite dimensional tori, and that an analogue of the classical Kronecker-Weil theorem holds for such semigroups.

Purity And Copurity in Systems of Linear Transformations

1976 ◽  
Vol 28 (4) ◽  
pp. 889-896
Author(s):  
Frank Zorzitto
Keyword(s):  
Vector Spaces ◽  
Finite Codimension ◽  
Linear Transformations ◽  
Complex Vector ◽  
Finite Dimensional ◽  
Quotient System

Consider a system of N linear transformations A1, … , AN: V → W, where F and IF are complex vector spaces. Denote it for short by (F, W). A pair of subspaces X ⊂ V, Y ⊂ W such that determines a subsystem (X, Y) and a quotient system (V/X, W/Y) (with the induced transformations). The subsystem (X, Y) is of finite codimension in (V, W) if and only if V/X and W / Y are finite-dimensional. It is a direct summand of (V, W) in case there exist supplementary subspaces P of X in F and Q of F in IF such that (P, Q) is a subsystem.

scholarly journals Residually finite dimensional algebras and polynomial almost identities

2020 ◽  
pp. 2250038
Author(s):  
Michael Larsen ◽  
Aner Shalev
Keyword(s):  
Lie Algebra ◽  
Finite Index ◽  
Open Group ◽  
Finite Codimension ◽  
Finite Dimensional Algebra ◽  
Nilpotent Ideal ◽  
Dimensional Algebra ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

scholarly journals On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

2019 ◽  
Vol 22 (6) ◽  
pp. 1089-1099
Author(s):  
Motoko Kato
Keyword(s):  
Fixed Points ◽  
Topological Dimension ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Thompson’S Group ◽  
Simple Action ◽  
Cube Complexes

Abstract We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

scholarly journals Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation

2003 ◽  
Vol 2003 (8) ◽  
pp. 409-427 ◽  
Author(s):  
Robert Willie
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Equation ◽  
Asymptotic Behaviour ◽  
Nonlinear Term ◽  
Damped Wave Equation ◽  
Order Ordinary Differential Equation ◽  
Finite Dimensional ◽  
Type Boundary ◽  
Large Diffusion

We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.

Finite-dimensional coagulation-fragmentation dynamics

2018 ◽  
Vol 28 (05) ◽  
pp. 851-868
Author(s):  
Jack Carr ◽  
Matab Alghamdi ◽  
Dugald B. Duncan
Keyword(s):  
Phase Transition ◽  
Asymptotic Behaviour ◽  
Numerical Approximation ◽  
Finite System ◽  
Numerical Results ◽  
Centre Manifold ◽  
Finite Dimensional ◽  
Time Period ◽  
Fragmentation Dynamics ◽  
Loss Of Mass

We examine a finite-dimensional truncation of the discrete coagulation-fragmentation equations that is designed to allow mass to escape from the system into clusters larger than those in the truncated problem. The aim is to model within a finite system the process of gelation, which is a type of phase transition observed in aerosols, colloids, etc. The main result is a centre manifold calculation that gives the asymptotic behaviour of the truncated model as time [Formula: see text]. Detailed numerical results show that truncated system solutions are often very close to this centre manifold, and the range of validity of the truncated system as a model of the full infinite problem is explored for systems with and without gelation. The latter cases are mass conserving, and we provide an estimate using quantities from the centre manifold calculations of the time period and the truncated system can be used for before loss of mass which is apparent. We also include some observations on how numerical approximation can be made more reliable and efficient.

Dissipative operators with finite dimensional damping

1982 ◽  
Vol 91 (3-4) ◽  
pp. 243-263 ◽  
Author(s):  
Harald Röh
Keyword(s):  
Separable Hilbert Space ◽  
Dissipative Operator ◽  
Finite Codimension ◽  
Spectral Mapping Theorem ◽  
Damped Wave Equation ◽  
Spectral Mapping ◽  
Dissipative Operators ◽  
Finite Dimensional ◽  
Maximal Dissipative Operator ◽  
Cayley Transformation

SynopsisLetG: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ andT(t) be theCosemigroup generated byG. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {xε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions forGare satisfied. Proofs are given in terms of the Cayley transformationT= (G+I)(G−I)−1ofG. The results are applied to the damped wave equationutt+ γutx+uxxxx+ ßuxx= 0, 0 ≦t< ∞ 0 <x< 1, β, γ ≧ 0, with boundary conditionsu(0,t) =ux(0,t) =uxx(1,t) =uxxx(1,t) = 0.

scholarly journals Fixed points of nonlinear contractions with applications

2021 ◽  
Vol 6 (9) ◽  
pp. 9378-9396
Author(s):  
Mohammed Shehu Shagari ◽  
◽  
Qiu-Hong Shi ◽  
Saima Rashid ◽  
Usamot Idayat Foluke ◽  
...  
Keyword(s):  
Fixed Points ◽  
Nonlinear Contractions
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