Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

A new proof of the stable manifold theorem for hyperbolic fixed points on surfaces

We introduce a new technique for proving the classical stable manifold theorem for hyperbolic fixed points. This method is much more geometrical than the standard approaches which rely on abstract fixed point theorems. It is based on the convergence of a canonical sequence of “finite time local stable manifolds” which are related to the dynamics of a finite number of iterations.     

Nonlinear PDE Model for Image Restoration Using Second-Order Hyperbolic Equations

In this article we consider a novel nonlinear PDE-based image denoising technique. The proposed restoration model uses second-order hyperbolic diffusion equations. It represents an improved nonlinear version of a linear hyperbolic PDE model developed recently by the author, providing more effective noise removal results while preserving the edges and other image features. A rigorous mathematical investigation is performed on this new differential model and its well-posedness is treated. Next, a consistent finite-difference numerical approximation scheme is proposed for this nonlinear diffusion-based approach. Our successful image denoising experiments and method comparisons are also described.     

Lagrange and hermite interpolation by bivariate splines

We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.     

Hyperbolic Involutions and Quadratic Extensions

This is a variation on a theme of Bayer-Fluckiger, Shapiro, and Tignol related to hyperbolic involutions. More precisely, criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and involutions of the form σ ? τ and σ ? ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A, and ρ is an involution of a quaternion algebra are obtained.     

Dichotomy and trichotomy of difference equations

Dedicated to Professor Gerry Ladas on his sixtieth birthday in acknowledgement of his leading role of difference equations. We extend the notions of dichotomy and trichotomy to nonlinear ordinary difference equations. This is accomplished by using two completely different approaches. In the first approach we use a notion, independent of the nature of our difference equation, called tracking. In the second approach we introduce a discrete analogue of dichotomy and trichotomy in variation.     

LIMITATIONS ON PRESCRIBING THE HOPF DIFFERENTIAL FOR MINIMAL SURFACES IN H3

ABSTRACT

The Hopf map from minimal surfaces in H ³ to holomorphic quadratic differentials on H ² is considered. Whether or not a given quadratic differential is in the image of this map is determined by the global existence/non-existence of solutions to a certain quasi-linear PDE. It is shown that this map is not surjective.     

Hypercentral Unit Groups and the Hyperbolicity of a Modular Group Algebra

We classify groups G such that the unit group 𝒰 ₁(? G) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups 𝒰 ₁(KG).     

Free Subgroups of Groups with Nontrivial Floyd Boundary

Abstract

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205–218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.