Locally flat PL submanifolds with codimension two

1967 ◽  
Vol 63 (1) ◽  
pp. 5-8 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Piecewise Linear ◽  
Normal Structure ◽  
Codimension Two ◽  
Basic Facts ◽  
Pl Bundle

We refer the reader to the IHES notes of Zeeman (14) for basic facts about PL (or piecewise-linear) manifolds. If Mm is a locally flat PL-submanifold of Qm+2, our object will be to study the normal structure of M in Q: one of our main results is:There exists a PL-bundle over M, with fibre a 2-simplex, which is PL-homeomorphic to a neighbourhood of M in Q; moreover, the bundle and homeomorphism are unique up to equivalence. We also make an application to smoothing theory.

Unknotting tori in codimension one and spheres in codimension two

1965 ◽  
Vol 61 (3) ◽  
pp. 659-664 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Piecewise Linear ◽  
Differential Topology ◽  
Standard Unit ◽  
Codimension Two ◽  
Codimension One

We shall present this paper in the framework and terminology of differential topology though all our arguments are valid in the piecewise linear ease also, under local un-knottedness hypotheses. In particular we use Rp for Euclidean space of dimension p, Sp−1 for the standard unit sphere in it, and Dp for the disc which it bounds.

TWO-PARAMETER BIFURCATION CURVES IN POWER ELECTRONIC CONVERTERS

2009 ◽  
Vol 19 (01) ◽  
pp. 349-357 ◽  
Author(s):  
E. FOSSAS ◽  
S. J. HOGAN ◽  
T. M. SEARA
Keyword(s):  
Duty Cycle ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Input Voltage ◽  
The State ◽  
System Response ◽  
State Variables ◽  
Power Electronic Converters ◽  
Codimension Two ◽  
Two Parameter

In this paper we prove the general result that, given a linear system [Formula: see text] where A is hyperbolic, u is piecewise linear and L-periodic, with [Formula: see text], then there exists a unique L-periodic solution x = xp(t) such that [Formula: see text]. We then consider a DC/DC buck (step-down) converter controlled by the ZAD (zero-average dynamics) strategy. The ZAD strategy sets the duty cycle, d (the length of time the input voltage is applied across an inductance), by ensuring that, on average, a function of the state variables is always zero. The two control parameters are v ref , a reference voltage that the circuit is required to follow, and ks, a time constant which controls the approach to the zero average. We show how to calculate d exactly for a periodic system response, without knowledge of the state space solutions. In particular, we show that for a T-periodic response d is independent of ks. We calculate period doubling and corner collision bifurcations, the latter occurring when the duty cycle saturates and is unable to switch. We also show the presence of a codimension two nonsmooth bifurcation in this system when a corner collision bifurcation and a saddle node bifurcation collide.

scholarly journals Unfolding Codimension-Two Subsumed Homoclinic Connections in Two-Dimensional Piecewise-Linear Maps

2020 ◽  
Vol 30 (03) ◽  
pp. 2030006 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Solution ◽  
Piecewise Linear ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Codimension Two ◽  
Codimension One ◽  
Border Collision Bifurcations ◽  
Homoclinic Connections ◽  
Linear Maps ◽  
Piecewise Linear Maps

For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such “subsumed” homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy [Formula: see text], in a two-parameter unfolding there exists an infinite sequence of roughly triangular regions within which the map has a stable single-round periodic solution. The result applies to both discontinuous and continuous maps, although these cases admit different characterizations for the border-collision bifurcations that correspond to boundaries of the regions. The result is illustrated with a discontinuous map of Mira and the two-dimensional border-collision normal form.

The microtubule associated protein (MAP) tau forms a new class of triple-stranded left-hand helical fibrous protein polymer

1992 ◽  
Vol 50 (1) ◽  
pp. 540-541
Author(s):  
G. C. Ruben ◽  
K. Iqbal ◽  
I. Grundke-Iqbal ◽  
H. Wisniewski ◽  
T. L. Ciardelli ◽  
...  
Keyword(s):  
Axial Ratio ◽  
Normal Structure ◽  
Paired Helical Filaments ◽  
Left Hand ◽  
New Class ◽  
Protein Polymer ◽  
Fibrous Protein ◽  
Protein Map ◽  
Neurotransmitter Transport ◽  
Alzheimer Neurofibrillary Tangles

In neurons, the microtubule associated protein, tau, is found in the axons. Tau stabilizes the microtubules required for neurotransmitter transport to the axonal terminal. Since tau has been found in both Alzheimer neurofibrillary tangles (NFT) and in paired helical filaments (PHF), the study of tau's normal structure had to preceed TEM studies of NFT and PHF. The structure of tau was first studied by ultracentrifugation. This work suggested that it was a rod shaped molecule with an axial ratio of 20:1. More recently, paraciystals of phosphorylated and nonphosphoiylated tau have been reported. Phosphorylated tau was 90-95 nm in length and 3-6 nm in diameter where as nonphosphorylated tau was 69-75 nm in length. A shorter length of 30 nm was reported for undamaged tau indicating that it is an extremely flexible molecule. Tau was also studied in relation to microtubules, and its length was found to be 56.1±14.1 nm.

Chaos in a manifold piecewise linear system

1981 ◽  
Vol 64 (10) ◽  
pp. 9-17 ◽  
Author(s):  
Toshimichi Saito ◽  
Hiroichi Fujita
Keyword(s):  
Linear System ◽  
Piecewise Linear ◽  
Piecewise Linear System
Sign in / Sign up

Export Citation Format

Share Document