Degree theory for equivariant maps of finite-dimensional manifolds: Topological actions

1996 ◽  
pp. 31-42
Author(s):  
Alexander Kushkuley ◽  
Zalman Balanov
Keyword(s):  
Degree Theory ◽  
Equivariant Maps ◽  
Finite Dimensional

scholarly journals Degree theory for equivariant maps. I

1989 ◽  
Vol 315 (2) ◽  
pp. 433-433 ◽  
Author(s):  
J. Ize ◽  
I. Massab{ò ◽  
A. Vignoli
Keyword(s):  
Degree Theory ◽  
Equivariant Maps

Multi-clustered solutions for a singularly perturbed forced pendulum equation

2019 ◽  
Vol 150 (1) ◽  
pp. 387-417
Author(s):  
Salomé Martínez ◽  
Dora Salazar
Keyword(s):  
Degree Theory ◽  
Singularly Perturbed ◽  
Finite Dimensional Reduction ◽  
Unbounded Solutions ◽  
Asymptotic Profile ◽  
Pendulum Equation ◽  
Finite Dimensional ◽  
Highly Oscillatory ◽  
Forced Pendulum ◽  
Image Position

AbstractIn this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely $$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$Using a limiting energy function, we describe the behaviour of the solutions as the parameter ε approaches zero. We also prove the existence of a family of solutions having a prescribed asymptotic profile and exhibiting a highly rotatory behaviour alternated with a highly oscillatory behaviour in some open subsets of the domain. The proof relies on a combination of the Nehari finite dimensional reduction with the topological degree theory.

Degree theory for equivariant maps, the general 𝑆¹-action

1992 ◽  
Vol 100 (481) ◽  
pp. 0-0 ◽  
Author(s):  
Jorge Ize ◽  
Ivar Massabò ◽  
Alfonso Vignoli
Keyword(s):  
Degree Theory ◽  
Equivariant Maps

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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