scholarly journals On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

2019 ◽  
Vol 22 (6) ◽  
pp. 1089-1099
Author(s):  
Motoko Kato
Keyword(s):  
Fixed Points ◽  
Topological Dimension ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Thompson’S Group ◽  
Simple Action ◽  
Cube Complexes

Abstract We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

scholarly journals Hyperbolic and cubical rigidities of Thompson’s group V

2019 ◽  
Vol 22 (2) ◽  
pp. 313-345 ◽  
Author(s):  
Anthony Genevois
Keyword(s):  
General Criterion ◽  
Isometric Action ◽  
Group V ◽  
Cube Complex ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Fixed Point Properties ◽  
Thompson’S Group ◽  
Cube Complexes ◽  
Finitely Presented

Abstract In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson’s group V is hyperbolically elementary, and we deduce that it satisfies Property {({\rm FW}_{\infty})} , i.e., every isometric action of V on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals The Farkas lemma of Shimizu, Aiyoshi and Katayama

1985 ◽  
Vol 31 (3) ◽  
pp. 445-450 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Convex Programming ◽  
Infinite Dimensional ◽  
Farkas Lemma ◽  
Finite Dimensional

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

scholarly journals Free non-associative principal train algebras

1984 ◽  
Vol 27 (3) ◽  
pp. 313-319 ◽  
Author(s):  
P. Holgate
Keyword(s):  
Finite Number ◽  
Ground Field ◽  
Linear Forms ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Senses ◽  
Special Train ◽  
Complex Coefficients ◽  
Infinite Linear

The definitions of finite dimensional baric, train, and special train algebras, and of genetic algebras in the senses of Schafer and Gonshor (which coincide when the ground field is algebraically closed, and which I call special triangular) are given in Worz-Busekros's monograph [8]. In [6] I introduced applications requiring infinite dimensional generalisations. The elements of these algebras were infinite linear forms in basis elements a0, a1,… and complex coefficients such that In this paper I consider only algebras whose elements are forms which only a finite number of the xi are non zero.

A modified Krasnosellkii-Mann algorithms for computing fixed points of multivalued quasi-nonexpansive mappings in Banach spaces

2019 ◽  
Vol 28 (2) ◽  
pp. 191-198
Author(s):  
T. M. M. SOW
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Mann Iteration ◽  
Infinite Dimensional ◽  
Mann Algorithm

It is well known that Krasnoselskii-Mann iteration of nonexpansive mappings find application in many areas of mathematics and know to be weakly convergent in the infinite dimensional setting. In this paper, we introduce and study an explicit iterative scheme by a modified Krasnoselskii-Mann algorithm for approximating fixed points of multivalued quasi-nonexpansive mappings in Banach spaces. Strong convergence of the sequence generated by this algorithm is established. There is no compactness assumption. The results obtained in this paper are significant improvement on important recent results.

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