scholarly journals Cuplength estimates in Morse cohomology

2016 ◽  
Vol 08 (02) ◽  
pp. 243-272 ◽  
Author(s):  
Peter Albers ◽  
Doris Hein
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Critical Points ◽  
Lagrangian Submanifolds ◽  
Type Equation ◽  
Base Case ◽  
Floer Theory ◽  
Hamiltonian Diffeomorphisms ◽  
Dirac Type ◽  
Bounded Below

The main goal of this paper is to give a unified treatment to many known cuplength estimates with a view towards Floer theory. As the base case, we prove that for [Formula: see text]-perturbations of a function which is Morse–Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we work with rather general assumptions the proof also applies in a variety of Floer settings. For example, this proves lower bounds (some of which were known) for the number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for Lagrangian submanifolds, translated points of contactomorphisms, and solutions to a Dirac-type equation.

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

scholarly journals A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS

2003 ◽  
Vol 05 (05) ◽  
pp. 803-811 ◽  
Author(s):  
YARON OSTROVER
Keyword(s):  
Symplectic Manifold ◽  
Isometric Embedding ◽  
Lagrangian Submanifolds ◽  
Canonical Embedding ◽  
Hamiltonian Diffeomorphisms

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham (M) into ℒ, f ↦ graph (f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

Growth rates of bending KdV solitons

1982 ◽  
Vol 28 (3) ◽  
pp. 469-484 ◽  
Author(s):  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Growth Rate ◽  
Lower Bounds ◽  
Acoustic Waves ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Type Equation ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Upper And Lower Bounds ◽  
Ion Acoustic Waves

Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.

Sharp upper bounds on perfect retrieval in the Hopfield model

1999 ◽  
Vol 36 (03) ◽  
pp. 941-950 ◽  
Author(s):  
Anton Bovier
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Upper Bound ◽  
Random Variables ◽  
Upper Bounds ◽  
Negative Association ◽  
Hopfield Model ◽  
Gradient Dynamics ◽  
One Step ◽  
The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

ONE-DIMENSIONAL CHAOS IN IMPULSED LINEAR OSCILLATING CIRCUITS

1993 ◽  
Vol 03 (04) ◽  
pp. 921-941 ◽  
Author(s):  
LAURA GARDINI ◽  
RENZO LUPINI
Keyword(s):  
Asymptotic Behavior ◽  
Fixed Points ◽  
Critical Point ◽  
Critical Points ◽  
Phase Plane ◽  
Global Bifurcation ◽  
Homoclinic Bifurcation ◽  
Combined Effect ◽  
Chaotic Oscillations ◽  
One Dimensional

The dynamics of a damped linear oscillating circuit subject to impulses is represented by a one-dimensional endomorphism (or noninvertible map) π: ℝ → ℝ. The asymptotic behavior of orbits in the phase-plane is characterized in terms of critical points and point singularities of π (fixed points or cycles). Their combined effect, that is, the merging of a critical point into a repelling cycle, causes a global bifurcation or a homoclinic bifurcation, with transition to chaotic oscillations.

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