Wednesdays at 4pm in Eckhart 202.
Prion desease are involved in a number of pathologies as scrappie and Kreutzfeld-Jacob, and are due to missfolding of a protein that becomes able to create polymers called fibrils. Models for the polymerization process involved in prion self-replication are well established and studied by a number of authors in the case where the dynamics coefficients do not depend on the size of polymers. However, several experimental studies indicate that the structure and size of the prion aggregates are determinant for their pathological effect. This motivated to take into account size dependent replicative properties of prion aggregates. We first show the history and biological issues, then develop the modeling of the process and finally prove a result concerning the dynamics of prion aggregates when a pathological state exists (high production of the normal protein). Then we study the strain phenomena and more specifically we wonder what specific replicative properties are determinant in strain propagation. We propose to interpret it also as a dynamical property of size repartitions.
The problem of understanding the parabolic hull of Brownian motion arises in two different fields. In mathematical physics this is the Burgers-Hopf caricature of turbulence (very interesting, even if not entirely turbulent). In statistics, the limit distribution we study was first considered by Chernoff, and forms the cornerstone of a large class of limit theorems that have now come to be called `cube-root-asymptotics'. It was in the statistical context that the problem was first solved completely in the mid-80s by Groeneboom in a tour de force of hard analysis. We consider another approach to his solution motivated by recent work on stochastic coalescence (especially work of Duchon, Bertoin, and my joint work with Bob Pego). The virtues of this approach are simplicity, generality, and the appearance of a completely unexpected Lax pair. If time permits, I will also indicate some tantalizing links of this approach with random matrices. This work forms part of my student Ravi Srinivasan's dissertation.
I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.
We investigate various properties of the Ablowitz-Ladik (AL) hierarchy of equation, whose starting point is an integrable discretization of the 1-dimensional cubic NLS. In particular, we will show how one can solve these equations by exploiting their algebraic structure and the connection to orthogonal polynomials. This leads to geometric considerations, which allow us to map the analytic and symplectic structures associate to the AL hierarchy onto the corresponding structures for the well-known Toda lattice.
The equations of elastodynamics and the Born-Infeld version of Maxwell's equations are examples of hyperbolic systems of conservation laws in which the principal entropy fails to be convex. It will be shown that this is compensated by the presence of "involutions" and supplementary "contingent entropies", so that the Cauchy problem is well-posed for classical solutions, which in turn are unique and stable within the broader class of admissible weak solutions.
We consider a quasi-static, volume preserving droplet model based on contact angle dynamics on a planar surface. Based on a gradient flow structure of the problem we derive a natural time-discretization scheme. The free boundary velocity of the time discrete solution is described in comparison with barrier functions. We will discuss the convergence in the continuum limit as well as the connection with standard viscosity solutions.
An important emerging scientific issue in many practical problems ranging from climate and weather prediction to biological science involves the real time filtering and prediction through partial observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom as well as the statistical accuracy of various strategies a new mathematical framework to address these issues has been developed by the speaker and collaborators at CIMS. One part of these ideas blends classical stability analysis for PDE's and their finite difference approximations, suitable versions of Kalman filtering, and stochastic models from turbulence theory to deal with the large model errors in realistic systems. Another aspect involves the development of test suites of statistically exactly solvable models and new NEKF algorithms for filtering and prediction for slow-fast system, moist convection; and turbulent tracers. Here a stringent suite of test models for filtering and stochastic parameter estimation is developed based on NEKF algorithms in order to systematically correct both multiplicative and additive bias in an imperfect model. There are both significantly increased filtering and predictive skill through the NEKF stochastic parameter estimation algorithms provided that these are guided by mathematical theory. The related research papers can be found at my NYU faculty website: http://www.math.nyu.edu/faculty/majda/
We study limiting behavior of rescaled size distributions in several models of clustering or coagulation dynamics, `solvable' in the sense that the Laplace transform converts them into nonlinear PDE. The scaling analysis that emerges has many connections with the classical limit theorems of probability theory, and an application to the study of shock clustering in the inviscid Burgers equation with random-walk initial data. I'll focus on recent progress regarding a `min-driven' clustering model related to domain coarsening dynamics in the Allen-Cahn equation.
To address important issues for climate change, ideas from rigorous mathematics, statistics, and statistical physics are blended with asymptotic/qualitative models and novel numerical algorithms/analysis in the contemporary applied math modus operandi. This lecture surveys efforts of the speaker and his collaborators in four important areas; (1) Clouds and their Climate Impact; (2) Fluctuation Dissipation Theorems and Climate Change Science; (3) Stochastic and Statistical Modelling of Low Frequency Variability; (4) Real Time Prediction from Partial Observations for Turbulent Systems. The related research papers can be found at my NYU faculty website: http://www.math.nyu.edu/faculty/majda/
We consider a new approach to a class of evolutionary PDEs where question of global existence or lack of it is tied to the asymptotics of solution to a non-linear integral equation in a dual variable whose solution has been shown to exist a priori. This integral equation approach is inspired by Borel summation of a formally divergent series for small time, but has general applicability and is not limited to analytic initial data. In this approach, there is no blow-up in the variable p, which is dual to 1/t or some power 1/t^n; solutions are known to be smooth in p and exist globally for p in R+. Exponential growth in p, for different choice of n, signifies finite time singularity. On the other hand, sub-exponential growth implies global existence. Further, unlike PDE problems where global existence is uncertain, a discretized Galerkin approximation to the associated integral equation has controlled errors. Further, known integral solution for p in [0, p_0], numerically or otherwise, gives sharper analytic bounds on the exponents in p and hence better estimate on the existence time for the associated PDE. We will also discuss particular results for 3-D Navier-Stokes and discuss ways in which this method may be relevant to numerical studies of finite time blow-up problems. (Joint work with O. Costin, G. Luo.)
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