608 research outputs found
Recovery of the absorption coefficient in radiative transport from a single measurement
In this paper, we investigate the recovery of the absorption coefficient from
boundary data assuming that the region of interest is illuminated at an initial
time. We consider a sufficiently strong and isotropic, but otherwise unknown
initial state of radiation. This work is part of an effort to reconstruct
optical properties using unknown illumination embedded in the unknown medium.
We break the problem into two steps. First, in a linear framework, we seek
the simultaneous recovery of a forcing term of the form (with known) and an isotropic initial condition using
the single measurement induced by these data. Based on exact boundary
controllability, we derive a system of equations for the unknown terms and
. The system is shown to be Fredholm if satisfies a certain
positivity condition. We show that for generic term and weakly
absorbing media, this linear inverse problem is uniquely solvable with a
stability estimate. In the second step, we use the stability results from the
linear problem to address the nonlinearity in the recovery of a weak absorbing
coefficient. We obtain a locally Lipschitz stability estimate
Feller semigroups with boundary conditions
This expository paper is devoted to the problem of construction of Feller semigroups with Ventcelâ(Wentzell) boundary conditions for elliptic Waldenfels operators.Intuitively our result may be stated as follows: We can construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until itâdiesâat which time it reaches the set where the absorption phenomenon occurs.In the Proceedings of the Third International Conference on Functional Analysis and Approximation Theory(F. Altomare et al., eds.
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by omega-automata, so
called omega-automatic structures, and prove that relations defined over such
structures in first-order logic expanded by the first-order quantifiers `there
exist at most many', 'there exist finitely many' and 'there exist
modulo many' are omega-regular. The proof identifies certain algebraic
properties of omega-semigroups. As a consequence an omega-regular equivalence
relation of countable index has an omega-regular set of representatives. This
implies Blumensath's conjecture that a countable structure with an
-automatic presentation can be represented using automata on finite
words. This also complements a very recent result of Hj\"orth, Khoussainov,
Montalban and Nies showing that there is an omega-automatic structure which has
no injective presentation
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by
omega-automata, so called omega-automatic structures, and prove
that relations defined over such structures in first-order logic
expanded by the first-order quantifiers `there exist at most
many\u27, \u27there exist finitely many\u27 and \u27there exist
modulo many\u27 are omega-regular. The proof identifies certain
algebraic properties of omega-semigroups.
As a consequence an omega-regular equivalence relation of countable
index has an omega-regular set of representatives. This implies
Blumensath\u27s conjecture that a countable structure with an
-automatic presentation can be represented using automata
on finite words. This also complements a very recent result of
Hj"orth, Khoussainov, Montalban and Nies showing that there is an
omega-automatic structure which has no injective presentation
Brownian Motions on Metric Graphs
Brownian motions on a metric graph are defined. Their generators are
characterized as Laplace operators subject to Wentzell boundary at every
vertex. Conversely, given a set of Wentzell boundary conditions at the vertices
of a metric graph, a Brownian motion is constructed pathwise on this graph so
that its generator satisfies the given boundary conditions.Comment: 43 pages, 7 figures. 2nd revision of our article 1102.4937: The
introduction has been modified, several references were added. This article
will appear in the special issue of Journal of Mathematical Physics
celebrating Elliott Lieb's 80th birthda
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