On the extremal boundary of convex compact measures which represent a non-regular point in choquet simplex

1988 ◽  
pp. 211-213
Author(s):  
D. G. Keselman
Keyword(s):  
Regular Point ◽  
Convex Compact ◽  
Choquet Simplex

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

On minimal convex pairs of convex compact sets

1996 ◽  
Vol 67 (3) ◽  
pp. 226-238 ◽  
Author(s):  
Ryszard Urbański
Keyword(s):  
Convex Compact ◽  
Compact Sets ◽  
Minimal Convex ◽  
Convex Compact Sets

scholarly journals ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY

2009 ◽  
Vol 01 (04) ◽  
pp. 307-405 ◽  
Author(s):  
ALBERTO ABBONDANDOLO ◽  
MATTHIAS SCHWARZ
Keyword(s):  
Maximum Principle ◽  
Energy Level ◽  
Exact Sequence ◽  
Floer Homology ◽  
Energy Functional ◽  
Convex Compact ◽  
Closed Geodesics ◽  
Action Functional ◽  
Uniform Estimates ◽  
Rabinowitz Floer Homology

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

Čebyšev Sets in Hyperspaces over Rn

2009 ◽  
Vol 61 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Robert J. MacG. Dawson and ◽  
Maria Moszyńska
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Linear Structure ◽  
Convex Compact ◽  
Nearest Neighbour ◽  
Compact Sets ◽  
Strictly Convex ◽  
Convex Compact Sets

Abstract. A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over Rn, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

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