Equidistant Loci and the Minkowskian Geometries

1972 ◽  
Vol 24 (2) ◽  
pp. 312-327 ◽  
Author(s):  
B. B. Phadke
Keyword(s):  
Metric Spaces ◽  
Open Convex ◽  
Minkowskian Space ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Projective Metric ◽  
Affine Line ◽  
Finite Dimensional Banach Space ◽  
Homogeneous Norm ◽  
Symmetric Distance

The spaced of this paper is a metrization, with a not necessarily symmetric distance xy, of an open convex set D in the n-dimensional affine space An such that xy + yz = xz if and only if x, y, z lie on an affine line with y between x and z and such that all the balls px ≦ p are compact. These spaces are called straight desarguesian G-spaces or sometimes open projective metric spaces. The hyperbolic geometry is an example; a large variety of other examples is studied by contributors to Hilbert's problem IV. When D = An and all the affine translations are isometries for the metric xy, the space is called a Minkowskian space or sometimes a finite dimensional Banach space, the (not necessarily symmetric) distance of a Minkowskian space being a (positive homogeneous) norm. In this paper geometric conditions in terms of equidistant loci are given for the space R to be a Minkowskian space.

scholarly journals Conditions for a plane projective metric to be a norm

1973 ◽  
Vol 9 (1) ◽  
pp. 49-54 ◽  
Author(s):  
B.B. Phadke
Keyword(s):  
Affine Plane ◽  
Convex Set ◽  
The Novel ◽  
Open Convex ◽  
Real Affine ◽  
Mathematical Problems ◽  
Projective Metric ◽  
Affine Line ◽  
Unit Circles ◽  
Open Plane

Let R be a metrization with distance xy of an open convex set D in the 2-dimensional real affine plane such that xy + yz = xz whenever x, y, z lie on an affine line with y between x and z and such that all the balls px ≤ ρ are compact. The study of such metrics, called open plane projective metrics falls under the topic of Hilbert's Problem IV of his famous mathematical problems. In this paper it is proved that if in R the sets of points equidistant from lines lie again on lines then D must be the entire affine plane and the distance must in fact be a norm. The paper contributes to and gives extensions of similar results proved earlier. The novel features of the present result are that in the space collinearity of points x, y, z is taken only as a sufficient condition for the equality xy + yz = xz. Consequently the solution encompasses all normed linear planes, that is, norms whose unit circles are not necessarily strictly convex are also admitted.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

scholarly journals Approximations of Variational Problems in Terms of Variational Convergence

2017 ◽  
Vol 20 (K2) ◽  
pp. 107-116
Author(s):  
Diem Thi Hong Huynh
Keyword(s):  
Multiobjective Optimization ◽  
Metric Space ◽  
Metric Spaces ◽  
Equilibrium Problems ◽  
Variational Problems ◽  
Dimensional Case ◽  
Variational Convergence ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Definition Of

We show first the definition of variational convergence of unifunctions and their basic variational properties. In the next section, we extend this variational convergence definition in case the functions which are defined on product two sets (bifunctions or bicomponent functions). We present the definition of variational convergence of bifunctions, icluding epi/hypo convergence, minsuplop convergnece and maxinf-lop convergence, defined on metric spaces. Its variational properties are also considered. In this paper, we concern on the properties of epi/hypo convergence to apply these results on optimization proplems in two last sections. Next we move on to the main results that are approximations of typical and important optimization related problems on metric space in terms of the types of variational convergence are equilibrium problems, and multiobjective optimization. When we applied to the finite dimensional case, some of our results improve known one.

Dynamical systems: Approximate Lagrangians and Noether symmetries

2019 ◽  
Vol 16 (10) ◽  
pp. 1950160 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Kinetic Energy ◽  
Equations Of Motion ◽  
Noether Symmetry ◽  
Noether Symmetries ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Second Order Equations

We determine the approximate Noether point symmetries of the variational principle characterizing second-order equations of motion of a particle in a (finite-dimensional) Riemannian manifold. In particular, the Lagrangian comprises of kinetic energy and a potential [Formula: see text], perturbed to [Formula: see text]. We establish a convenient system of approximate geometric conditions that suffices for the computation of approximate Noether symmetry vectors and moreover, simplifies the problem of the effect of higher orders of the perturbation. The general results are applied to several practical problems of interest and we find extra Noether symmetries at [Formula: see text].

scholarly journals Affine line systems in finite-dimensional vector spaces

2003 ◽  
Vol 3 (s1) ◽  
Author(s):  
Rainer Löwen ◽  
Günter F. Steinke ◽  
Hendrik Van Maldeghem
Keyword(s):  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Affine Line

scholarly journals Actions of amenable equivalence relations on CAT(0) fields

2012 ◽  
Vol 34 (1) ◽  
pp. 21-54 ◽  
Author(s):  
MARTIN ANDEREGG ◽  
PHILIPPE HENRY
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Amenable Group ◽  
Equivalence Relations ◽  
General Notion ◽  
Rigidity Result ◽  
Finite Dimensional ◽  
Borel Field

AbstractWe present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

scholarly journals Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

2017 ◽  
Vol 38 (7) ◽  
pp. 2780-2800 ◽  
Author(s):  
RODOLPHE RICHARD ◽  
NIMISH A. SHAH
Keyword(s):  
Number Theory ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Reductive Groups ◽  
Natural Group ◽  
Homogeneous Dynamics ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Unipotent Flows ◽  
Linearization Techniques

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

scholarly journals Sums of finite-dimensional spaces

1969 ◽  
Vol 1 (3) ◽  
pp. 357-361 ◽  
Author(s):  
B.R. Wenner
Keyword(s):  
Metric Spaces ◽  
Dimension Theory ◽  
General Hypothesis ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Locally Countable

Analogues are developed to the sum theorems in the dimension theory of metric spaces. It is shown that, within the class of metric spaces, any locally countable, σ-locally finite, or closure-preserving sum of finite-dimensional sets is countable-dimensional. Similar results are obtained under the more general hypothesis of countable-dimensional rather than finite-dimensional sets.

scholarly journals Dynamical systems of finite-dimensional metric spaces and zero-dimensional covers

2013 ◽  
Vol 160 (3) ◽  
pp. 564-574 ◽  
Author(s):  
Yuki Ikegami ◽  
Hisao Kato ◽  
Akihide Ueda
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Finite Dimensional
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