Maria Carla Tesi

Adsorption of a directed polymer subject to an elongational force

We consider several different directed walk models of a homopolymer adsorbing at a surface when the polymer is subject to an elongational force which hinders the adsorption. We use combinatorial methods for analyzing how the critical temperature for adsorption depends on the magnitude of the applied force and show that the crossover exponent φ changes when a force is applied. We discuss the characteristics of the model needed to obtain a re-entrant phase diagram.

Knotting in stretched polygons

The knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon. By examining the incidence of cut-planes in the polygon, we prove a pattern theorem in the stretching regime for large applied forces. This theorem can be used to examine the incidence of entanglements such as knotting and writhing. In particular, we prove that for arbitrarily large positive, but finite, values of the stretching force, the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases. In the case of writhing, we prove that for stretched polygons of length n, and for every function , the probability that the absolute value of the mean writhe is less than f(n) approaches 0 as n ? ?, for sufficiently large values of the applied stretching force.

Polymer entanglement in melts

We propose a new way of characterizing the entanglement complexity of concentrated polymer solutions and polymer melts. This involves considering a randomly chosen cube in the system, and investigating the entanglements between sub-chains in this cube. We present Monte Carlo calculations and scaling arguments for the density dependence of the entanglement complexity, and the way in which this behaviour scales with the size of the chosen cube.

div-curl Type Theorem, H-Convergence, and Stokes Formula in the Heisenberg Group

In this paper, we prove a div–curl type theorem in the Heisenberg group ℍ1, and then we develop a theory of H-convergence for second order differential operators in divergence form in ℍ1. The div–curl theorem requires an intrinsic notion of the curl operator in ℍ1 (that we denote by curlℍ), that turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of this definition, we prove an intrinsic Stokes formula for curlℍ. Eventually, we show that this notion is related to one of the exterior differentials in Rumins complex on contact manifolds.

Alzheimer's disease: a mathematical model for onset and progression

In this article we propose a mathematical model for the onset and progression of Alzheimers disease based on transport and diffusion equations. We regard brain neurons as a continuous medium and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons and ii) neuron-to-neuron prion-like transmission. We model these two processes by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The second equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. Our numerical simulations are in good qualitative agreement with clinical images of the disease distribution in the brain which vary from early to advanced stages.

A qualitative model for aggregation and diffusion of \(\beta \)-amyloid in Alzheimer’s disease

In this paper we present a mathematical model for the aggregation and diffusion of A

Self-averaging in the statistical mechanics of some lattice models

\beta

Entanglement complexity of semiflexible lattice polygons

amyloid in the brain affected by Alzheimer’s disease, at the early stage of the disease. The model is based on a classical discrete Smoluchowski aggregation equation modified to take diffusion into account. We also describe a numerical scheme and discuss the results of the simulations in the light of the recent biomedical literature.