Stochastic domination: the contact process, Ising models and FKG measures

We prove for the contact process on $Z^d$, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate $\lambda$ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for ``downward'' FKG measures. As a consequence of this general result, we show that the plus and minus states for the Ising model on $Z^d$ dominate the same set of product measures. We show that this latter fact fails completely on the homogenous 3-ary tree. We also provide a different distinction between $Z^d$ and the homogenous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on $Z^d$ are never stochastically ordered, on the homogenous 3-ary tree, almost the complete opposite is the case. Next, we show that on $Z^d$, the set of product measures which the plus state for the Ising model dominates is strictly increasing in the temperature. Finally, we obtain a necessary and sufficient condition for a finite number of variables, which are both FKG and exchangeable, to dominate a given product measure.Comment: 27 page

Finitely dependent coloring

We prove that proper coloring distinguishes between block-factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block-factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. More precisely, Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and conjectured that no stationary k-dependent q-coloring exists for any k and q. We disprove this by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring, thus resolving the question for all k and q. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovasz local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block-factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving d dimensions and shifts of finite type; in fact, any non-degenerate shift of finite type also distinguishes between block-factors and finitely dependent processes

A Look at a Strict Construction of Section 2-207 of the Uniform Commercial Code From the Seller\u27s Point of View or What\u27s So Bad About Roto-Lith?

This is an examination of the workings of section 2-207 of the Uniform Commercial Code in the form contract between merchants. More specifically, the literal interpretation of the Section is to be investigated as to its effect on the practical formation of the sales contract A basic assumption of this comment is that the terms of the Code which may, under section 2-207 be read into a contract, are repugnant to the seller. This, I think, is obvious. It should, however, be kept in mind that, between merchants, both parties may be assumed to be big boys. Therefore, the problem of the mammoth corporation taking advantage of the helpless consumer, the overriding justification for the Code\u27s implied warranties and full spectrum of Buyer\u27s remedies, is not present

Integrals, Partitions, and Cellular Automata

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.Comment: Revised version. 28 pages, 2 figure

Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie--Weiss Ising model and includes as well all ferromagnetic Curie--Weiss Potts and Curie--Weiss Heisenberg models. By de Finetti's theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that ``ferromagnetism'' is not however in itself sufficient and also study in some detail the Curie--Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie--Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a ``formula'' for the extension which is valid in many cases.Comment: Published at http://dx.doi.org/10.1214/009117906000001033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

In vitro models for the study of the mechanisms of damage in age-related macular degeneration and Stargardt's disease

Includes bibliographical references (pages [167]-189).The retinal pigment epithelium (RPE) is interposed between the photoreceptor cells of the neural retina and the choriocapillaris, and lies on a bed of extracellular matrix called Bruch’s membrane. Dysfunction in one of these components can cause a cascade of events that results in RPE and photoreceptor cell death, which causes permanent vision loss. The most debilitating vision loss occurs in the macular region due to its high concentration of photoreceptor and RPE cells and its role in central vision. Mutations in the gene for the ATP-dependent binding cassette transport protein, ABCR, have been shown to result in a high accumulation of the autofluorescent pigment mixture, lipofuscin, within the RPE cells. The ABCR protein is located in the outer segments of the photoreceptor cells. Stargardt’s macular dystrophy (STDG), retinitis pigmentosa (RP) and cone/rod dystrophy (CRD) are associated with mutations in this gene. Additionally, mutations are implicated in some forms of age-related macular degeneration (AMD). Although AMD is a multi-factorial disease, the prolific accumulation of autofluorescent compounds in the RPE or in Bruch’s membrane is a positive indicator for the disease. Clearly, understanding the mechanisms of the ABCR protein function and lipofuscin-mediated damage to RPE cells are the foundation of vision loss by retinal maculopathies. The first part of the research presented in this dissertation describes the cloning, transfection and expression of the ABCR transporter protein that is often found defective in STGD patients and other retinal diseases. ABCR dysfunction leads to the prolific accumulation of lipofuscin in the RPE, which may in turn be related to the accumulation of age-related debris on Bruch’s membrane. Therefore, a RPE cell line that mimics the in vivo state was isolated and characterized, including melanin pigmentation, ECM and differentiation. These cells were then used to determine the phototoxicity of lipofuscin and the lipofuscin precursor, A2PE. Photooxidative damage to RPE-challenged A2PE was performed and the lipophilic extracts analyzed. From these studies, the mechanisms that damage RPE cells are addressed, which facilitate the progression of AMD, STGD and other retinal dystrophies.Ph.D. (Doctor of Philosophy

Isolation and characterization of a spontaneously immortalized bovine retinal pigmented epithelial cell line

<p>Abstract</p> <p>Background</p> <p>The Retinal Pigmented Epithelium (RPE) is juxtaposed with the photoreceptor outer segments of the eye. The proximity of the photoreceptor cells is a prerequisite for their survival, as they depend on the RPE to remove the outer segments and are also influenced by RPE cell paracrine factors. RPE cell death can cause a progressive loss of photoreceptor function, which can diminish vision and, over time, blindness ensues. Degeneration of the retina has been shown to induce a variety of retinopathies, such as Stargardt's disease, Cone-Rod Dystrophy (CRD), Retinitis Pigmentosa (RP), Fundus Flavimaculatus (FFM), Best's disease and Age-related Macular Degeneration (AMD). We have cultured primary bovine RPE cells to gain a further understanding of the mechanisms of RPE cell death. One of the cultures, named tRPE, surpassed senescence and was further characterized to determine its viability as a model for retinal diseases.</p> <p>Results</p> <p>The tRPE cell line has been passaged up to 150 population doublings and was shown to be morphologically similar to primary cells. They have been characterized to be of RPE origin by reverse transcriptase PCR and immunocytochemistry using the RPE-specific genes <it>RPE65 </it>and <it>CRALBP </it>and RPE-specific proteins RPE65 and Bestrophin. The tRPE cells are also immunoreactive to vimentin, cytokeratin and zonula occludens-1 antibodies. Chromosome analysis indicates a normal diploid number. The tRPE cells do not grow in suspension or in soft agar. After ³H thymidine incorporation, the cells do not appear to divide appreciably after confluency.</p> <p>Conclusion</p> <p>The tRPE cells are immortal, but still exhibit contact inhibition, serum dependence, monolayer growth and secrete an extra-cellular matrix. They retain the <it>in-vivo </it>morphology, gene expression and cell polarity. Additionally, the cells endocytose exogenous melanin, A2E and purified lipofuscin granules. This cell line may be a useful <it>in-vitro </it>research model for retinal maculopathies.</p

Effects of Noise on Ecological Invasion Processes: Bacteriophage-mediated Competition in Bacteria

Pathogen-mediated competition, through which an invasive species carrying and transmitting a pathogen can be a superior competitor to a more vulnerable resident species, is one of the principle driving forces influencing biodiversity in nature. Using an experimental system of bacteriophage-mediated competition in bacterial populations and a deterministic model, we have shown in [Joo et al 2005] that the competitive advantage conferred by the phage depends only on the relative phage pathology and is independent of the initial phage concentration and other phage and host parameters such as the infection-causing contact rate, the spontaneous and infection-induced lysis rates, and the phage burst size. Here we investigate the effects of stochastic fluctuations on bacterial invasion facilitated by bacteriophage, and examine the validity of the deterministic approach. We use both numerical and analytical methods of stochastic processes to identify the source of noise and assess its magnitude. We show that the conclusions obtained from the deterministic model are robust against stochastic fluctuations, yet deviations become prominently large when the phage are more pathological to the invading bacterial strain.Comment: 39 pages, 7 figure

Nonlinear reactive systems viewed as Boolean dynamical systems

We present a stochastic, time-discrete boolean model which mimics the mesoscopic dynamics of the desorption reactions $A+A\to A+S$ and $A+A\to S+S$ in a 1D lattice. In the continuous-time limit, we derive a hierarchy of dynamical equations for the subset of moments involving contiguous lattice sites. The solution of the hierarchy allows to compute the exact dynamics of the mean coverage for both microscopic and coarse-grained initial conditions, which turn out to be different from the mean field predictions. The evolution equations for the mean coverage and the second order moments are shown to be equivalent to those provided by a time-continuous Master equation. The important role of higher order fluctuations is brought out by the failure of a truncation scheme retaining only two-particle fluctuation correlations.Comment: 30 pages, 9 figure