In the present paper we obtain a Caratheodory-type selection theorem. This result is used in [9] to extend the theory of Nash equilibria, developed in [12, 4, 2, 7, 8, 10, and 13-171, to the setting of an arbitrary measure space of agents and an infinite dimensional strategy space. More specifically, we consider the following problem. Let T be a measure space, Z be a complete separable metric s...

Results concerning the global existence and uniqueness of mild solutions for a class of first-order abstract stochastic integro-differential equations with variable delay in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t are establish...

n be a continuous multifunction with compact, not necessarily convex values. If F is Lipschitz continuous, it was shown in [4] that there exists a measurable selection f of F such that, for every x 0 , the Cauchy problem ˙ x(t) = f (t, x(t)), x(0) = x 0 has a unique Caratheodory solution, depending continuously on x 0. In this paper, we prove that the above selection f can be chosen so that f (...

Let M be a close complex manifold and T M its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then M is a complex homogeneous manifold. It implies that every irreducible close Kähler manifold with ample tangent bundle is isomorphic to the projective space. This provides an alternative proof of Hartshore's conjecture in algebrai...

We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]-hard (and NP-hard) if the dimension is part of the input by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision vers...

The purpose of this paper is to extend and clarify the methods of the papers [G1] and [G2] by using the methods of [G3], incorporating ideas from Carnot-Caratheodory geometry such as those of the fundamental paper [NSW]. We will thereby prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe [DJ]. As in [G1] and [G2], we will associate a singular integral operat...

In 1914, the 1-dimensional case of the conjecture was proven in full generality by Caratheodory and Osgood-Taylor using elaborate methods from complex analysis. The 2-dimensional case was studied in the 1920s by Alexander who first circulated a manuscript claiming a proof but soon discovered a counterexample himself, the famous horned sphere. Later Alexander found an extra condition under which...

let  be a hilbert space of functions analytic on a plane domain  such that for every  in  the functional of evaluation at  is bounded. assume further that  contains the constants and admits multiplication by the independent variable , , as a bounded operator. we give sufficient conditions for  to be reflexive for all positive integers .

Suppose V is a finite set and C a collection of subsets of V that contains ∅ and V and is closed under taking intersections. Then the ordered pair (V, C) is called a convexity and the elements of C are referred to as convex sets. For a set S ⊆ V , the convex hull of S relative to C, denoted by CHC(S), is the smallest convex set containing S. The Caratheodory number, relative to a given convexit...