The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian α < 1/2, and global existence as well as analyticity of solution for α ≥ 1/2. We also prove the existence of solutions… (More)
Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur  showed that a certain class of weak solutions to the drift diffusion equation with initial data in L 2 gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies… (More)
We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs. A perfect matching in a graph is a set of its edges that… (More)
This work is in a stream (see e.g. , , , , ) initiated by a paper of Killip and Simon , an earlier paper  also should be mentioned here. Using methods of Functional Analysis and the classical Szegö Theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.
We show that the basins of zeroes under the gradient flow of the random potential U corresponding to a random Gaussian Entire Function f partition the complex plane into domains of equal area and that the probability that the diameter of a particular basin is greater than R is exponentially small in R.
Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions, and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering, and are often used to… (More)
Let N (f) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N (f)/n 2 tends to a positive constant a, and that N (f)/n 2 exponentially concentrates around a. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal… (More)
1. Introduction 7 2. Dyadic cubes and lattices 8 3. The Three Lattice Theorem 13 4. The forest structure on a subset of a dyadic lattice 17 5. Stopping times and augmentation 20 6. Sparse and Carleson families 21 7. From the theory to applications 29 8. The multilinear Calderón-Zygmund operators 30 9. Controlling values of T [f 1 ,. .. , f m ] on a cube 32… (More)
Denote by µ a the distribution of the random sum (1 − a) ∞ j=0 ω j a j , where P(ω j = 0) = P(ω j = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ a is supported on C a , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 − 2a), and iterating this process… (More)
We prove that the unit cube B n ∞ is a strict local minimizer for the Mahler volume product vol n (K)vol n (K *) in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.