Rational maps of ℙn with prescribed fixed points and the smooth conic case

2014 ◽  
Vol 13 (08) ◽  
pp. 1450066 ◽  
Author(s):  
J. A. Vargas ◽  
A. S. Argáez
Keyword(s):  
Fixed Point ◽  
Complete Classification ◽  
Rational Maps ◽  
Plane Curves ◽  
Canonical Forms ◽  
Fixed Point Set ◽  
Constant Degree ◽  
Point Set ◽  
Quadratic Hypersurfaces

We construct rational maps of ℙn which have a prescribed variety as a component of their fixed point set. The resulting maps fix a pencil of lines for the case of hypersurfaces; thus including the cases of plane curves. We also determine the Cremona maps among the constructed ones for quadratic hypersurfaces. Our methods are based on associated matrices of forms of constant degree and the "triple action" of G = PGL n+1 on them. We include a complete classification of these maps and matrices for the case of the smooth conic curve in ℙ2. We obtain invariants and canonical forms for the orbits of our matrices under the triple action of G, modulo syzygies of a row vector. We obtain invariants and canonical forms for the orbits of the constructed rational maps under conjugation by G.

scholarly journals Infinitely Periodic Knots

1985 ◽  
Vol 37 (1) ◽  
pp. 17-28 ◽  
Author(s):  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Fixed Point Set ◽  
Finite Group Actions ◽  
Three Sphere ◽  
Knot Invariant ◽  
Periodic Transformation ◽  
Point Set ◽  
Fixed Point Sets

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S0, S1, or S2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.

Limiting cases of Boardman's five halves theorem

2013 ◽  
Vol 56 (3) ◽  
pp. 723-732 ◽  
Author(s):  
Michael C. Crabb ◽  
Pedro L. Q. Pergher
Keyword(s):  
Fixed Point ◽  
Dimensional Manifold ◽  
Fixed Point Set ◽  
Minimal Dimension ◽  
Point Set ◽  
Image Position

AbstractThe famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and ifis the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

scholarly journals On soft quasi-pseudometric spaces

2021 ◽  
Vol 22 (1) ◽  
pp. 17
Author(s):  
Hope Sabao ◽  
Olivier Olela Otafudu
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Soft Set ◽  
Fixed Point Set ◽  
Point Set ◽  
The Absolute ◽  
Pseudometric Space

<p>In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.</p>

scholarly journals Finite groups acting on hyperelliptic 3-manifolds

2020 ◽  
Vol 29 (04) ◽  
pp. 2050021
Author(s):  
Mattia Mecchia
Keyword(s):  
Fixed Point ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Fixed Point Set ◽  
Point Set

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to [Formula: see text] Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has [Formula: see text] components. In particular we prove that a simple group containing such an involution is isomorphic to [Formula: see text] for some odd prime power [Formula: see text], or to one of four other small simple groups.

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>

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