636 research outputs found
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
The UP-Isomorphism Theorems for UP-algebras
In this paper, we construct the fundamental theorem of UP-homomorphisms in
UP-algebras. We also give an application of the theorem to the first, second,
third and fourth UP-isomorphism theorems in UP-algebras.Comment: 11 page
Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems
In this work there is established an optimal existence and regularity theory
for second order linear parabolic differential equations on a large class of
noncompact Riemannian manifolds. Then it is shown that it provides a general
unifying approach to problems with strong degeneracies in the interior or at
the boundary.Comment: To appear in "Recent Developments of Mathematical Fluid Mechanics",
Series: Advances in Mathematical Fluid Mechanics, Birkhaeuser-Verlag,
Editors: G. P. Galdi, J. G. Heywood and R. Rannacher. Some misprints of the
earlier version have been correcte
Spatial Mixing and Non-local Markov chains
We consider spin systems with nearest-neighbor interactions on an -vertex
-dimensional cube of the integer lattice graph . We study the
effects that exponential decay with distance of spin correlations, specifically
the strong spatial mixing condition (SSM), has on the rate of convergence to
equilibrium distribution of non-local Markov chains. We prove that SSM implies
mixing of a block dynamics whose steps can be implemented
efficiently. We then develop a methodology, consisting of several new
comparison inequalities concerning various block dynamics, that allow us to
extend this result to other non-local dynamics. As a first application of our
method we prove that, if SSM holds, then the relaxation time (i.e., the inverse
spectral gap) of general block dynamics is , where is the number of
blocks. A second application of our technology concerns the Swendsen-Wang
dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies
an bound for the relaxation time. As a by-product of this implication we
observe that the relaxation time of the Swendsen-Wang dynamics in square boxes
of is throughout the subcritical regime of the -state
Potts model, for all . We also prove that for monotone spin systems
SSM implies that the mixing time of systematic scan dynamics is . Systematic scan dynamics are widely employed in practice but have
proved hard to analyze. Our proofs use a variety of techniques for the analysis
of Markov chains including coupling, functional analysis and linear algebra
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