DISTINGUISHING LINK-HOMOTOPY CLASSES BY PRE-PERIPHERAL STRUCTURES

1998 ◽  
Vol 07 (07) ◽  
pp. 925-944 ◽  
Author(s):  
JAMES R. HUGHES
Keyword(s):  
Open Problem ◽  
Group Homomorphism ◽  
Homotopy Classes ◽  
Minimal Weight ◽  
Ambient Isotopy ◽  
Link Homotopy

An open problem in link-homotopy of links in S3 is classification using peripheral invariants, analogous to that of Waldhausen for links up to ambient isotopy. An approach to such a classification was outlined by Levine, but shown not to be feasible by the author. Here, we develop an approach to finding classification counterexamples. The approach requires non-injectivity of a group homomorphism that is completely determined by minimal-weight commutator numbers (equivalent to the first non-vanishing [Formula: see text] invariants of Milnor). For non-injectivity, the minimal-weight commutator numbers must all be non-zero, and satisfy a certain system of polynomials, which we compute for 4- and 5-component links.

STRUCTURED GROUPS AND LINK-HOMOTOPY

1993 ◽  
Vol 02 (01) ◽  
pp. 37-63 ◽  
Author(s):  
JAMES R. HUGHES
Keyword(s):  
Automorphism Groups ◽  
Original Version ◽  
Theoretic Approach ◽  
Homotopy Classes ◽  
Structure Preserving ◽  
Three Sphere ◽  
Complex Program ◽  
Link Homotopy ◽  
Structured Groups

We study link-homotopy classes of links in the three sphere using reduced groups endowed with peripheral structures derived from meridian-longitude pairs. Two types of peripheral structures are considered — Milnor’s original version (called “pre-peripheral structures” in Levine’s terminology) and Levine’s refinement (called simply “peripheral structures”). We show here that pre-peripheral structures are not strong enough to classify links up to link-homotopy, and that Levine’s peripheral structures, although strong enough to distinguish those classes not distinguished by pre-peripheral structures, are also in all likelihood not strong enough to distinguish all link-homotopy classes. Following Levine’s classification program, we compare structure-preserving and realizable automorphisms, using an obstruction-theoretic approach suggested by work of Habegger and Lin. We find that these automorphism groups are in general different, so that a more complex program for classification by structured groups is required.

scholarly journals Detecting Whitney disks for link maps in the four-sphere

2017 ◽  
Vol 26 (12) ◽  
pp. 1750077
Author(s):  
Ash Lightfoot
Keyword(s):  
Open Problem ◽  
The Self ◽  
Complete Obstruction ◽  
Homotopy Invariant ◽  
Link Homotopy ◽  
Trivial Link

It is an open problem whether Kirk’s [Formula: see text]-invariant is the complete obstruction to a link map [Formula: see text] being link homotopic to the trivial link. The link homotopy invariant associates to such a link map [Formula: see text] a pair [Formula: see text], and we write [Formula: see text]. With the objective of constructing counterexamples, Li proposed a link homotopy invariant [Formula: see text] such that [Formula: see text] is defined on the kernel of [Formula: see text] and which also obstructs link null-homotopy. We show that, when defined, the invariant [Formula: see text] is determined by [Formula: see text], and is strictly weaker. In particular, this implies that if a link map [Formula: see text] has [Formula: see text], then after a link homotopy the self-intersections of [Formula: see text] may be equipped with framed, immersed Whitney disks in [Formula: see text] whose interiors are disjoint from [Formula: see text].

scholarly journals Multistring based matrices

2020 ◽  
Vol 29 (06) ◽  
pp. 2050038
Author(s):  
David R. Freund
Keyword(s):  
Homotopy Class ◽  
Open Problem ◽  
Chord Diagram ◽  
Homotopy Classes ◽  
Chord Diagrams ◽  
Text String ◽  
Reidemeister Moves

A virtual[Formula: see text]-string is a chord diagram with [Formula: see text] core circles and a collection of arrows between core circles. We consider virtual [Formula: see text]-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string [Formula: see text], Turaev associated a based matrix that encodes invariants of the virtual homotopy class of [Formula: see text]. We generalize Turaev’s method to associate a multistring based matrix to a virtual [Formula: see text]-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual [Formula: see text]-strings.

A Short Note on the Higher Level Version of the Krull–Baer Theorem

2011 ◽  
Vol 54 (2) ◽  
pp. 381-384
Author(s):  
Dejan Velušček
Keyword(s):  
Open Problem ◽  
Short Note ◽  
Group Homomorphism ◽  
Base Field ◽  
Skew Field ◽  
Level 1 ◽  
The Given ◽  
Crucial Ingredient

AbstractKlep and Velušček generalized the Krull–Baer theorem for higher level preorderings to the non-commutative setting. A n-real valuation v on a skew field D induces a group homomorphism . A section of is a crucial ingredient of the construction of a complete preordering on the base field D such that its projection on the residue skew field kv equals the given level 1 ordering on kv. In the article we give a proof of the existence of the section of , which was left as an open problem by Klep and Velušček, and thus complete the generalization of the Krull–Baer theorem for preorderings.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

scholarly journals Quantum Mean-Field Games with the Observations of Counting Type

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 7
Author(s):  
Vassili N. Kolokoltsov
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Open Problem ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Mean Field Games ◽  
Quantum Game ◽  
Quantum Games ◽  
Interesting Open Problem

Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.

A note on derived connections from semi-symmetric metric connections

2017 ◽  
Vol 67 (1) ◽  
pp. 221-226
Author(s):  
Adela Mihai
Keyword(s):  
Open Problem ◽  
Complex Structure ◽  
Kaehler Manifold ◽  
Different Types ◽  
Metric Connection

Abstract In this paper we construct examples of different types of connections starting from a semi-symmetric metric connection g, for example a connection which is a symmetric metric connection with respect to a conformally related metric, but symmetric non-metric with respect to the initial metric. We formulate an open problem: to find a parallel complex structure on a Kaehler manifold with respect to such a new connection.

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