A global two-dimensional version of Smale’s cancellation theorem via spectral sequences

2015 ◽  
Vol 36 (6) ◽  
pp. 1795-1838 ◽  
Author(s):  
M. A. BERTOLIM ◽  
D. V. S. LIMA ◽  
M. P. MELLO ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Sequence Analysis ◽  
Spectral Sequence ◽  
Matrix Theory ◽  
Chain Complex ◽  
Connection Matrix ◽  
Local Version ◽  
Two Dimensional ◽  
Dimensional Version ◽  
Cancellation Theorem ◽  
Flows On Surfaces

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

scholarly journals Spectral sequences in Conley’s theory

2009 ◽  
Vol 30 (4) ◽  
pp. 1009-1054 ◽  
Author(s):  
O. CORNEA ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Spectral Sequence ◽  
Flow Line ◽  
Reverse Flow ◽  
Chain Complex ◽  
Connection Matrix ◽  
Spectral Sequences ◽  
Direct Connection ◽  
Zero Differential ◽  
Action Type ◽  
A Chain

AbstractIn this paper, we analyse the dynamics encoded in the spectral sequence (Er,dr) associated with certain Conley theory connection maps in the presence of an ‘action’ type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix Δ to provide a system that spans Er in terms of the original basis of C and to identify all of the differentials drp:Erp→Erp−r. In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating E0p and E0p−r in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications.

Continuation and bifurcation associated to the dynamical spectral sequence

2013 ◽  
Vol 34 (6) ◽  
pp. 1849-1887 ◽  
Author(s):  
R. FRANZOSA ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Spectral Sequence ◽  
Directed Graphs ◽  
Chain Complex ◽  
Morse Decomposition ◽  
Connection Matrix ◽  
Transition Matrices ◽  
Dynamic Interpretation ◽  
Connection Matrices ◽  
Parameterized Family ◽  
Bifurcation Behavior

AbstractIn this paper we consider a filtered chain complex $C$ and its differential given by a connection matrix $\Delta $ which determines an associated spectral sequence $({E}^{r} , {d}^{r} )$. We present an algorithm which sweeps the connection matrix in order to span the modules ${E}^{r} $ in terms of bases of $C$ and gives the differentials ${d}^{r} $. In this process a sequence of similar connection matrices and associated transition matrices are produced. This algebraic procedure can be viewed as a continuation, where the transition matrices give information about the bifurcation behavior. We introduce directed graphs, called flow and bifurcation schematics, that depict bifurcations that could occur if the sequence of connection matrices and transition matrices were realized in a continuation of a Morse decomposition, and we present a dynamic interpretation theorem that provides conditions on a parameterized family of flows under which such a continuation could occur.

scholarly journals Geometry of the 3D Pythagoras' Theorem

2016 ◽  
Vol 8 (6) ◽  
pp. 78 ◽  
Author(s):  
Luis Teia
Keyword(s):  
Three Dimensional ◽  
Transformation Process ◽  
Two Dimensional ◽  
Natural Evolution ◽  
Dimensional Version ◽  
Pythagoras Theorem

This paper explains step-by-step how to construct the 3D Pythagoras' theorem by geometric manipulation of the two dimensional version. In it is shown how $x+y=z$ (1D Pythagoras' theorem) transforms into $x^2+y^2=z^2$ (2D Pythagoras' theorem) via two steps: a 90-degree rotation, and a perpendicular extrusion. Similarly, the 2D Pythagoras' theorem transforms into 3D using the same steps. Octahedrons emerge naturally during this transformation process. Hence, each of the two dimensional elements has a direct three dimensional equivalent. Just like squares govern the 2D, octahedrons are the basic elements that govern the geometry of the 3D Pythagoras' theorem. As a conclusion, the geometry of the 3D Pythagoras' theorem is a natural evolution of the 1D and 2D. This interdimensional evolution begs the question -- Is there a bigger theorem at play that encompasses all three?

Solutions to the Tipping Point Problem

2012 ◽  
Vol 106 (1) ◽  
pp. 60-63
Keyword(s):  
Ice Cream ◽  
Tipping Point ◽  
Rectangular Prism ◽  
Point Problem ◽  
Two Dimensional ◽  
Dimensional Version ◽  
Fixed Area

The problem posed in MT August 2011 (vol. 105, no. 1, pp. 62-66) asked readers to consider the two-dimensional version of tipping a bowl (assumed to be a rectangular prism) to spoon out the last little bit of melted ice cream. Here is the essence of the problem: Given a fluid region of fixed area A contained in a rectangle whose width is W, find a formula for the fluid depth D when the container is tilted through a known angle T that is measured from horizontal.

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.

Microstrip Gas Counters

2014 ◽  
pp. 31-51
Keyword(s):  
Glass Surface ◽  
Two Dimensional ◽  
Large Area ◽  
Proportional Chamber ◽  
Microelectronic Technology ◽  
Multiwire Proportional Chamber ◽  
Dimensional Version ◽  
A Chain ◽  
New Generation ◽  
First Time

In this chapter, the first micropattern gaseous detector, the microstrip gas counter, invented in 1988 by A. Oed, is presented. It consists of alternating anode and cathode strips with a pitch of less than 1 mm created on a glass surface. It can be considered a two-dimensional version of a multiwire proportional chamber. This was the first time microelectronic technology was applied to manufacturing of gaseous detectors. This pioneering work offers new possibilities for large area planar detectors with small gaps between the anode and the cathode electrodes (less than 0.1 mm). Initially, this detector suffered from several serious problems, such as charging up of the substrate, discharges which destroyed the thin anode strips, etc. However, by efforts of the international RD28 collaboration hosted by CERN, most of them were solved. Although nowadays this detector has very limited applications, its importance was that it triggered a chain of similar developments made by various groups, and these collective efforts finally led to the creation of a new generation of gaseous detectors-micropattern detectors.

S-matrix Theory

2020 ◽  
pp. 622-675
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Matrix Theory ◽  
Point Of View ◽  
Integrable Models ◽  
Geometrical Interpretation ◽  
Coupling Constants ◽  
Two Dimensional ◽  
Recursive Structure ◽  
Mathematical Point ◽  
S Matrix ◽  
Dynamical Principle

Chapter 17 discusses the S-matrix theory of two-dimensional integrable models. From a mathematical point of view, the two-dimensional nature of the systems and their integrability are the crucial features that lead to important simplifications of the formalism and its successful application. This chapter deals with the analytic theory of the S-matrix of the integrable models. A particular emphasis is put on the dynamical principle of bootstrap, which gives rise to a recursive structure of the amplitudes. It also covers several dynamical quantities, such as mass ratios or three-coupling constants, which have an elegant mathematic formulation that is also of easy geometrical interpretation.

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