2,711 research outputs found
Mortality, Redundancy, and Diversity in Stochastic Search
We investigate a stochastic search process in one dimension under the
competing roles of mortality, redundancy, and diversity of the searchers. This
picture represents a toy model for the fertilization of an oocyte by sperm. A
population of independent and mortal diffusing searchers all start at
and attempt to reach the target at . When mortality is irrelevant, the
search time scales as for , where
is the diffusive time scale. Conversely, when the mortality rate of the
searchers is sufficiently large, the search time scales as ,
independent of . When searchers have distinct and high mortalities, a
subpopulation with a non-trivial optimal diffusivity are most likely to reach
the target. We also discuss the effect of chemotaxis on the search time and its
fluctuations.Comment: 5 pages, 1 figure, 2-column revtex format; updated version: error
corrected; general improvements; original figure deleted and 2 figures adde
First Invader Dynamics in Diffusion-Controlled Absorption
We investigate the average time for the earliest particle to hit a spherical
absorber when a homogeneous gas of freely diffusing particles with density
and diffusivity is prepared in a deterministic state and is
initially separated by a minimum distance from this absorber. In the
high-density limit, this first absorption time scales as
in one dimension; we also obtain the
first absorption time in three dimensions. In one dimension, we determine the
probability that the -closest particle is the first one to hit the
absorber. At large , this probability decays as ,
with analytically calculable. As a corollary, the
characteristic hitting time for the -closest particle scales
as ; this corresponds to superdiffusive but still subballistic motion.Comment: 13 pages, 5 figures, IOP format. Version 2: some minor typos fixed.
Version 3: some errors correcte
Extensions and Dilations for -dynamical Systems
Let be a unital -algebra and be an injective, unital
endomorphism of . A covariant representation of is a pair
consisting of a -representation of on a Hilbert space
and a contraction in satisfying .
It follows from more general results of ours that such a covariant
representation can be extended to a covariant representation (on a
larger space ) such that is a coisometry and it can be dilated to a
covariant representation (on a larger space ) with
unitary.
Our objective here is to give self-contained, elementary proofs of these
results which avoid the technology of -correspondences. We also discuss
the non uniqueness of the extension.Comment: 11 page
Morita Transforms of Tensor Algebras
We show that if and are -algebras and if (resp. ) is a
-correspondence over (resp. ), then a Morita equivalence between
and implements a isometric functor between the categories of
Hilbert modules over the tensor algebras of and
. We show that this functor maps absolutely continuous
Hilbert modules to absolutely continuous Hilbert modules and provides a new
interpretation of Popescu's reconstruction operator
Quantum Markov Processes (Correspondences and Dilations)
We study the structure of quantum Markov Processes from the point of view of
product systems and their representations.Comment: 44 pages, Late
Quantum Markov Semigroups (Product Systems and Subordination)
We show that if a product system comes from a quantum Markov semigroup, then
it carries a natural Borel structure with respect to which the semigroup may be
realized in terms of a measurable representation. We show, too, that the dual
product system of a Borel product system also carries a natural Borel
structure. We apply our analysis to study the order interval consisting of all
quantum Markov semigroups that are subordinate to a given one.Comment: Revised according to the referee's comments and suggestions; to
appear in International Journal of Mathematic
Representations of Hardy Algebras: Absolute Continuity, Intertwiners and Superharmonic Operators
Suppose is the tensor algebra of a
-correspondence and is the associated Hardy algebra.
We investigate the problem of extending completely contractive representations
of on a Hilbert space to ultra-weakly continuous
completely contractive representations of on the same Hilbert
space. Our work extends the classical Sz.-Nagy - Foia\c{s} functional calculus
and more recent work by Davidson, Li and Pitts on the representation theory of
Popescu's noncommutative disc algebra
Matricial Function Theory and Weighted Shifts
Let be the tensor algebra of a -correspondence
over a -algebra . In earlier work, we showed that the completely
contractive representations of , whose restrictions to
are normal, are parametrized by certain discs or balls
indexed by the normal -representations of . Each disc has
analytic structure, and each element gives rise to
an operator-valued function on
that is continuous and analytic on the interior. In this paper, we explore the
effect of adding operator-valued weights to the theory. While the statements of
many of the results in the weighted theory are anticipated by those in the
unweighted setting, substantially different proofs are required. Interesting
new connections with the theory of completely positive are developed. Our
perspective has been inspired by work of Vladimir M\"{u}ller in which he
studied operators that can be modeled by parts of weighted shifts. Our results
may be interpreted as providing a description of operator algebras that can be
modeled by weighted tensor algebras. Our results also extend work of Gelu
Popescu, who investigated similar questions
The Poisson Kernel for Hardy Algebras
This note contributes to a circle of ideas that we have been developing
recently in which we view certain abstract operator algebras ,
which we call Hardy algebras, and which are noncommutative generalizations of
classical , as spaces of functions defined on their spaces of
representations. We define a generalization of the Poisson kernel, which
``reproduces'' the values, on , of the
``functions'' coming from . We present results that are natural
generalizations of the Poisson integral formuala. They also are easily seen to
be generalizations of formulas that Popescu developed. We relate our Poisson
kernel to the idea of a characteristic operator function and show how the
Poisson kernel identifies the ``model space'' for the canonical model that can
be attached to a point in the disc . We also
connect our Poission kernel to various "point evaluations" and to the idea of
curvature
Canonical Models for Representations of Hardy Algebras
In this paper we study canonical models for representations of the Hardy
algebras that generalize the model theory of Sz.-Nagy and Foias for contraction
operators.
The Hardy algebras are non selfadjoint operator algebras associated with
-correspondences. This class includes the classical algebra,
free semigroup algebras, quiver algebras and other classes of algebras studied
in the literature
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