673,277 research outputs found

    Continuity properties of integral kernels associated with Schroedinger operators on manifolds

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    For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation.Comment: 38 pages, major revision; to appear in Annales Henri Poincare (2007

    On the Continuity of Some Functions

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    We prove that basic arithmetic operations preserve continuity of functions.Institute of Informatics, University of BiaƂystok, Sosnowa 64, 15-887 BiaƂystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.CzesƂaw ByliƄski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesƂaw ByliƄski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesƂaw ByliƄski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesƂaw ByliƄski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesƂaw ByliƄski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesƂaw ByliƄski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.CzesƂaw ByliƄski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata DarmochwaƂ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata DarmochwaƂ and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Artur KorniƂowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.Artur KorniƂowicz and Yasunari Shidama. Intersections of intervals and balls in En/T. Formalized Mathematics, 12(3):301-306, 2004.Artur KorniƂowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117-124, 2005.Beata Padlewska and Agata DarmochwaƂ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Andrzej Trybulec and CzesƂaw ByliƄski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

    On the coarsest topology preserving continuity

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    We study a topology on a space of functions, called sticking topology, with the property to be the weakest among the topologies preserving continuity. In suitable frameworks, this topology preserves borelianity, local integrability, right continuity and other properties. It is coarser than the locally uniform convergence and it allows the presence of gliding humps as we show on examples. We prove relative compactness criteria for this topology and we consider some extensions.Comment: 14

    Refined Solutions of Time Inhomogeneous Optimal Stopping Games via Dirichlet Form

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    The properties of value functions of time inhomogeneous optimal stopping problem and zero-sum game (Dynkin game) are studied through time dependent Dirichlet form. Under the absolute continuity condition on the transition function of the underlying diffusion process and some other assumptions, the refined solutions without exceptional starting points are proved to exist, and the value functions of the optimal stopping and zero-sum game, which are finely and cofinely continuous, are characterized as the solutions of some variational inequalities, respectively

    Barrier functions for Pucci-Heisenberg operators and applications

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    The aim of this article is the explicit construction of some barrier functions ("fundamental solutions") for the Pucci-Heisenberg operators. Using these functions we obtain the continuity property, up to the boundary, for the viscosity solution of fully non-linear Dirichlet problems on the Heisenberg group, if the boundary of the domain satisfies some regularity geometrical assumptions (e.g. an exterior Heisenberg-ball condition at the characteristic points). We point out that the knowledge of the fundamental solutions allows also to obtain qualitative properties of Hadamard, Liouville and Harnack type
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