Some generalizations of the Borsuk–Ulam theorem and applications to realizing homotopy classes by embedded spheres

1973 ◽  
Vol 74 (2) ◽  
pp. 251-256 ◽  
Author(s):  
Roger Fenn
Keyword(s):  
Euclidean Space ◽  
Fixed Points ◽  
Piecewise Linear ◽  
Dimensional Sphere ◽  
Homotopy Classes ◽  
Ulam Theorem

In this paper, some theorems of the Borsuk-Ulam type (1) are given. One of these can be applied to show that certain homotopy classes in manifolds cannot be realized by embedded spheres. The n-dimensional sphere Sn is the subset of the euclidean spaceRn+l consisting of all points (x1, …,xn+1) satisfying . Let be a piecewise linear (PL) involution on Sn without fixed points.

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

scholarly journals Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems

2013 ◽  
Vol 22 (1) ◽  
pp. 41-46
Author(s):  
ANDREI BOZANTAN ◽  
◽  
VASILE BERINDE ◽  
Keyword(s):  
Fixed Points ◽  
Unconstrained Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Local Minima ◽  
Homotopy Algorithm ◽  
Unconstrained Optimization Problems ◽  
Piecewise Linear Homotopy ◽  
Computation Of Fixed Points ◽  
Modern Programming Language

This paper describes the main aspects of the ”piecewise-linear homotopy method” for fixed point approximation proposed by Eaves and Saigal [Eaves, C. B. and Saigal, R., Homotopies for computation of fixed points on unbounded regions, Mathematical Programming, 3 (1972), No. 1, 225–237]. The implementation of the method is developed using the modern programming language C# and then is used for solving some unconstrained optimization problems. The PL homotopy algorithm appears to be more reliable than the classical Newton method in the case of the problem of finding a local minima for Schwefel’s function and other optimization problems.

scholarly journals Regularity of mean-values

1988 ◽  
Vol 45 (1) ◽  
pp. 117-126
Author(s):  
Christopher Meaney
Keyword(s):  
Euclidean Space ◽  
Mean Value ◽  
Dimensional Sphere ◽  
Mean Values ◽  
Regular Elements ◽  
Integrable Functions ◽  
Spherical Mean ◽  
Group Of Isometries ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLetXbe either thed-dimensional sphere or a compact, simply connected, simple, connected Lie group. We define a mean-value operator analogous to the spherical mean-value operator acting on integrable functions on Euclidean space. The value of this operator will be written as ℳf(x, a), wherex∈Xandavaries over a torusAin the group of isometries ofX. For each of these cases there is an intervalpO<p≦ 2, where thep0depends on the geometry ofX, such that iffis inLp(X) then there is a set full measure inXand ifxlies in this set, the function a ↦ℳf(x, a) has some Hölder continuity on compact subsets of the regular elements ofA.

scholarly journals Typical representatives of free homotopy classes in multi-punctured plane

2019 ◽  
Vol 11 (03) ◽  
pp. 623-659
Author(s):  
Maxim Arnold ◽  
Yuliy Baryshnikov ◽  
Yuriy Mileyko
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Measure ◽  
Homotopy Class ◽  
Piecewise Linear ◽  
Homotopy Classes ◽  
Simple Method ◽  
Monte Carlo Techniques ◽  
Free Homotopy Class

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

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.

Finding Unstable Fixed Points From Time Series Data

2003 ◽  
Author(s):  
Takaaki Maehara ◽  
Mikio Nakai
Keyword(s):  
Time Series ◽  
Fixed Points ◽  
Time Series Data ◽  
Piecewise Linear ◽  
Duffing Oscillator ◽  
Conley Index ◽  
Series Data ◽  
Experimental Time ◽  
Index Pairs ◽  
Experimental Time Series

This study employs topological methods to extract unstable fixed points in phase space from both numerical and experimental time series data. Conley index of an isolated invariant subset and the R-B method can determine unstable fixed points contained in strange attractor from numerical time series data. For experimental time series data, the theorem for the relationship between index pairs and Conley index enables one to predict them with acceptable accuracy. As a corollary, some results for Duffing oscillator and piecewise linear system are shown.

Stability of fixed points placed on the border in the piecewise linear systems

2008 ◽  
Vol 38 (2) ◽  
pp. 391-399 ◽  
Author(s):  
Younghae Do ◽  
Sang Dong Kim ◽  
Phil Su Kim
Keyword(s):  
Fixed Points ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Piecewise Linear Systems

scholarly journals New Characterizations of the Clifford Torus and the Great Sphere

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1076 ◽  
Author(s):  
Sun Mi Jung ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Euclidean Space ◽  
Finite Type ◽  
Three Dimensional ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
Round Sphere ◽  
Clifford Torus ◽  
Great Sphere ◽  
Set Up

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

scholarly journals Homotopy Classification of Mappings of a 4-Dimensional Complex into a 2-Dimensional Sphere

1953 ◽  
Vol 5 ◽  
pp. 127-144 ◽  
Author(s):  
Nobuo Shimada
Keyword(s):  
Dimensional Sphere ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Cup Product

Steenrod [1] solved the problem of enumerating the homotopy classes of maps of an (n + 1)-complex K into an n-sphere Sn utilizing the cup-i-product, the far-reaching generalization of the Alexander-Čech-Whitney cup product [7] and the Pontrjagin *-product [5].

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