scholarly journals Regular cyclic actions on complex projective space with codimension-two fixed points

1998 ◽  
Vol 65 (1) ◽  
pp. 51-67 ◽  
Author(s):  
Robert D. Little
Keyword(s):  
Fixed Point ◽  
Projective Space ◽  
Fixed Points ◽  
Cyclic Group ◽  
Complex Projective Space ◽  
Fixed Point Set ◽  
Codimension Two ◽  
Point Set ◽  
Isolated Point ◽  
Complex Projective

AbstractIfM2nis a cohomologyCPnandPis an odd prime, letGpbe the cyclic group of orderp. A TypeI I0Gpaction onM2nis an action with fixed point set a codimension-2 submanifold and an isolated point. A TypeI I0Gpaction is standard if it is regular and the degree of the fixed codimension-2 submanifold is one. If n is odd and M2nadmits a standardGpaction of TypeI I0, then every TypeI I0GpactionM2nis standard and so, if n is odd,CPnadmits aGpaction of TypeI I0if and only if the action is standard.

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

scholarly journals Finiteness of the Fixed Point Set for the Simple Genetic Algorithm

1995 ◽  
Vol 3 (3) ◽  
pp. 299-309 ◽  
Author(s):  
Alden H. Wright ◽  
Michael D. Vose
Keyword(s):  
Genetic Algorithm ◽  
Fixed Point ◽  
Fixed Points ◽  
Small Perturbation ◽  
Discrete Dynamical System ◽  
Fixed Point Set ◽  
Infinite Population ◽  
Simple Genetic Algorithm ◽  
Stable Manifold Theorem ◽  
Point Set

The infinite population simple genetic algorithm is a discrete dynamical system model of a genetic algorithm. It is conjectured that trajectories in the model always converge to fixed points. This paper shows that an arbitrarily small perturbation of the fitness will result in a model with a finite number of fixed points. Moreover, every sufficiently small perturbation of fimess preserves the finiteness of the fixed point set. These results allow proofs and constructions that require finiteness of the fixed point set. For example, applying the stable manifold theorem to a fixed point requires the hyperbolicity of the differential of the transition map of the genetic algorithm, which requires (among other things) that the fixed point be isolated.

Actions of finite p-groups on homology manifolds

2001 ◽  
Vol 131 (3) ◽  
pp. 473-486 ◽  
Author(s):  
BERNHARD HANKE
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Prime Order ◽  
Fixed Point Set ◽  
Poincare Duality ◽  
Cohomology Rings ◽  
Duality Space ◽  
Point Set ◽  
Homology Manifolds ◽  
Poincaré Duality Space

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

scholarly journals Properties of fixed point set of a multivalued map

2005 ◽  
Vol 2005 (3) ◽  
pp. 323-331 ◽  
Author(s):  
Abdul Rahim Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Best Approximation ◽  
Convex Banach Space ◽  
Multivalued Map ◽  
Multivalued Maps ◽  
Fixed Point Set ◽  
Strictly Convex Banach Space ◽  
Point Set

Properties of the set of fixed points of some discontinuous multivalued maps in a strictly convex Banach space are studied; in particular, affirmative answers are provided to the questions related to set of fixed points and posed by Ko in 1972 and Xu and Beg in 1998. A result regarding the existence of best approximation is derived.

σ4-Actions On Homotopy Spheres

1981 ◽  
Vol 33 (2) ◽  
pp. 275-281
Author(s):  
Chao-Chu Liang
Keyword(s):  
Fixed Point ◽  
Conjugacy Classes ◽  
Homotopy Sphere ◽  
Fixed Point Set ◽  
Codimension Two ◽  
Point Set

Let σ4 denote the group of all permutations of {a, b, c, d}. It has 24 elements, partitioned into five conjugacy classes: (1) the identity 1; (2) 6 transpositions: (ab), …, (cd); (3) 8 elements of order 3: (abc), …, (bcd); (4) 6 elements of order 4: (abcd), …, (adcb); (5) 3 elements of order 2: x = (ab)(cd), y = (ac)(bd), z = (ad)(bc).In this paper, we study the differentiate actions of σ4 on odd-dimensional homotopy spheres modelled on the linear actions, with the fixed point set of each transposition a codimension two homotopy sphere.A simple (2n – l)-knot is a differentiate embedding of a homotopy sphere K2n–l into a homotopy sphere Σ2n+1 such that πj(Σ – K) = πj(S1) for j < n.

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