7,983 research outputs found
Towards a Convenient Category of Topological Domains
We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models
A Convenient Category of Domains
We motivate and define a category of "topological domains",
whose objects are certain topological spaces, generalising
the usual -continuous dcppos of domain theory.
Our category supports all the standard constructions of domain theory,
including the solution of recursive domain equations. It also
supports the construction of free algebras for (in)equational
theories, provides a model of parametric polymorphism,
and can be used as the basis for a theory of computability.
This answers a question of Gordon Plotkin, who asked
whether it was possible to construct a category of domains
combining such properties
Function Spaces on Singular Manifolds
It is shown that most of the well-known basic results for Sobolev-Slobodeckii
and Bessel potential spaces, known to hold on bounded smooth domains in
, continue to be valid on a wide class of Riemannian manifolds
with singularities and boundary, provided suitable weights, which reflect the
nature of the singularities, are introduced. These results are of importance
for the study of partial differential equations on piece-wise smooth domains.Comment: 37 pages, 1 figure, final version, augmented by additional
references; to appear in Math. Nachrichte
Representation of functionals of Ito processes and their first exit times
The representation theorem is obtained for functionals of non-Markov
processes and their first exit times from bounded domains. These functionals
are represented via solutions of backward parabolic Ito equations. As an
example of applications, analogs of forward Kolmogorov equations are derived
for conditional probability density functions of Ito processes being killed on
the boundary. In addition, a maximum principle and a contraction property are
established for SPDEs in bounded domains.Comment: 25 page
Degenerate backward SPDEs in domains: non-local boundary conditions and applications to finance
Backward stochastic partial differential equations of parabolic type in
bounded domains are studied in the setting where the coercivity condition is
not necessary satisfied and the equation can be degenerate. Some generalized
solutions based on the representation theorem are suggested. In addition to
problems with a standard Cauchy condition at the terminal time, problems with
special non-local boundary conditions are considered. These non-local
conditions connect the terminal value of the solution with a functional over
the entire past solution. Uniqueness, solvability and regularity results are
obtained. Some applications to portfolio selection problem are considered.Comment: arXiv admin note: substantial text overlap with arXiv:1211.1460,
arXiv:1208.553
Representation and uniqueness for boundary value elliptic problems via first order systems
Given any elliptic system with -independent coefficients in the upper-half
space, we obtain representation and trace for the conormal gradient of
solutions in the natural classes for the boundary value problems of Dirichlet
and Neumann types with area integral control or non-tangential maximal control.
The trace spaces are obtained in a natural range of boundary spaces which is
parametrized by properties of some Hardy spaces. This implies a complete
picture of uniqueness vs solvability and well-posedness.Comment: submitted, 70 pages. A number of maths typos have been eliminate
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