Smooth Maps on a Manifold

2007 ◽  
pp. 57-62
Keyword(s):  
Smooth Maps

scholarly journals Invariants of Stable Maps between Closed Orientable Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 215
Author(s):  
Catarina Mendes de Jesus S. ◽  
Pantaleón D. Romero
Keyword(s):  
Point Of View ◽  
Stable Maps ◽  
Smooth Maps ◽  
Orientable Surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

Bifurcations in general piecewise-smooth maps

2007 ◽  
pp. 171-217 ◽  
Author(s):  
Mario di Bernardo ◽  
Alan R. Champneys ◽  
Christopher J. Budd ◽  
Piotr Kowalczyk
Keyword(s):  
Piecewise Smooth ◽  
Smooth Maps ◽  
Piecewise Smooth Maps

scholarly journals Efficiency of minimizing compositions of convex functions and smooth maps

2018 ◽  
Vol 178 (1-2) ◽  
pp. 503-558 ◽  
Author(s):  
D. Drusvyatskiy ◽  
C. Paquette
Keyword(s):  
Convex Functions ◽  
Smooth Maps

ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS

2009 ◽  
Vol 19 (05) ◽  
pp. 1709-1732 ◽  
Author(s):  
B. M. BAKER ◽  
M. E. KIDWELL ◽  
R. P. KLINE ◽  
I. POPOVICI
Keyword(s):  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Stimulus Period ◽  
Smooth Maps ◽  
Periodic Stimulation ◽  
Piecewise Smooth Maps ◽  
Dimensional Version ◽  
Bifurcation Law ◽  
The One

We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.

Smooth Maps

2021 ◽  
pp. 127-161
Author(s):  
Boris M. Makarov ◽  
Anatolii N. Podkorytov
Keyword(s):  
Smooth Maps

Influence of a square-root singularity on the behaviour of piecewise smooth maps

Nonlinearity ◽  
2010 ◽  
Vol 23 (2) ◽  
pp. 445-463 ◽  
Author(s):  
Viktor Avrutin ◽  
Partha Sharathi Dutta ◽  
Michael Schanz ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Smooth ◽  
Square Root ◽  
Smooth Maps ◽  
Piecewise Smooth Maps ◽  
Square Root Singularity

scholarly journals Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

2019 ◽  
Vol 12 (3) ◽  
pp. 303-332 ◽  
Author(s):  
Alexandra Convent ◽  
Jean Van Schaftingen
Keyword(s):  
Sobolev Space ◽  
Isometric Embedding ◽  
Riemannian Metrics ◽  
Higher Order ◽  
Necessary Condition ◽  
Smooth Maps ◽  
Homogeneous Sobolev Space ◽  
The One ◽  
Intrinsic Definition ◽  
Sobolev Spaces Between Manifolds

AbstractWe define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, {p\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.

Sign in / Sign up

Export Citation Format

Share Document