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 $N$ independent and mortal diffusing searchers all start at $x=L$ and attempt to reach the target at $x=0$. When mortality is irrelevant, the search time scales as $\tau_D/\ln N$ for $\ln N\gg 1$, where $\tau_D\sim L^2/D$ is the diffusive time scale. Conversely, when the mortality rate $\mu$ of the searchers is sufficiently large, the search time scales as $\sqrt{\tau_D/\mu}$, independent of $N$. 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 $\rho$ and diffusivity $D$ is prepared in a deterministic state and is initially separated by a minimum distance $\ell$ from this absorber. In the high-density limit, this first absorption time scales as $\frac{\ell^2}{D}\frac{1}{\ln\rho\ell}$ in one dimension; we also obtain the first absorption time in three dimensions. In one dimension, we determine the probability that the $k^{\rm th}$-closest particle is the first one to hit the absorber. At large $k$, this probability decays as $k^{1/3}\exp(-Ak^{2/3})$, with $A= 1.93299\ldots$ analytically calculable. As a corollary, the characteristic hitting time $T_k$ for the $k^{\rm th}$-closest particle scales as $k^{4/3}$; 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 C∗C^*-dynamical Systems

Let $A$ be a unital $C^*$-algebra and $\alpha$ be an injective, unital endomorphism of $A$. A covariant representation of $(A,\alpha)$ is a pair $(\pi,T)$ consisting of a $C^*$-representation $\pi$ of $A$ on a Hilbert space $H$ and a contraction $T$ in $B(H)$ satisfying $T\pi(\alpha(a))=\pi(a)T$. It follows from more general results of ours that such a covariant representation can be extended to a covariant representation $(\rho,V)$ (on a larger space $K$) such that $V$ is a coisometry and it can be dilated to a covariant representation $(\sigma,U)$ (on a larger space $K_1$) with $U$ unitary. Our objective here is to give self-contained, elementary proofs of these results which avoid the technology of $C^*$-correspondences. We also discuss the non uniqueness of the extension.Comment: 11 page

Morita Transforms of Tensor Algebras

We show that if $M$ and $N$ are $C^{*}$-algebras and if $E$ (resp. $F$) is a $C^{*}$-correspondence over $M$ (resp. $N$), then a Morita equivalence between $(E,M)$ and $(F,N)$ implements a isometric functor between the categories of Hilbert modules over the tensor algebras of $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$. 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 $\mathcal{T}_{+}(E)$ is the tensor algebra of a $W^{*}$-correspondence $E$ and $H^{\infty}(E)$ is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of $\mathcal{T}_{+}(E)$ on a Hilbert space to ultra-weakly continuous completely contractive representations of $H^{\infty}(E)$ 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 $\mathcal{T}_{+}(E)$ be the tensor algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. In earlier work, we showed that the completely contractive representations of $\mathcal{T}_{+}(E)$, whose restrictions to $M$ are normal, are parametrized by certain discs or balls $\overline{D(E,\sigma)}$ indexed by the normal $*$-representations $\sigma$ of $M$. Each disc has analytic structure, and each element $F\in \mathcal{T}_{+}(E)$ gives rise to an operator-valued function $\widehat{F}_{\sigma}$ on $\overline{D(E,\sigma)}$ 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 $H^{\infty}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\infty}$, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which ``reproduces'' the values, on $\mathbb{D}((E^{\sigma})^*)$, of the ``functions'' coming from $H^{\infty}(E)$. 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 $\mathbb{D}((E^{\sigma})^*)$. 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 $W^*$-correspondences. This class includes the classical $H^{\infty}$ algebra, free semigroup algebras, quiver algebras and other classes of algebras studied in the literature