Schreier rewriting beyond the classical setting

Using actions of free monoids and free associative algebras, we establish some Schreiertype formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras. This work was supported by Natural Sciences and Engineering Research Council of Canada (Grant No. 227060-04), Yuri Bahturin, National Science Foundation (Grant No. DMS-0700811) and Russian Fund for Basic Research (Grant No. 08-01-00573), Alexander Olshanskii     

Dedicated to H. Wielandt on the occasion of his 90th birthday

We determine some relevant structural properties of the group of lattice automorphisms of a nonperiodic modular group.     

Affinely Prime Dynamical Systems(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Singular Solutions to Conformal Hessian Equations(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

The authors show that for any ε∈]0, 1[, there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ? R~5 which belongs to C~(1,ε)(B) C~(1,ε+)(B).     

Clifford algebra approach to pointwise convergence of Fourier series on spheres Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.     

Invariant holomorphic extension in several complex variables Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Two fundamental problems on the invariant holomorphic extensions have been posed, which are naturally arose from our solution of the extended future tube conjecture and closely and deeply related to the general theory of Stein manifolds due to Cartan-Serre. In this paper, the relationship is presented between the two problems, the motivation of considering the problems, and the methods to approach the problems. We have also posed some questions and conjectures related to this two problems.     

FORMATION OF SINGULARITY FOR COMPRESSIBLE VISCOELASTICITY Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

The formation of singularity and breakdown of classical solutions to the threedimensional compressible viscoelasticity and inviscid elasticity are considered.For the compressible inviscid elastic fluids,the finite-time formation of singularity in classical solutions is proved for certain initial data.For the compressible viscoelastic fluids,a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.     

THE VACUUM IN NONISENTROPIC GAS DYNAMICS Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

We investigate the vacuum in nonisentropic gas dynamics in one space variable,with the most general equation of states allowed by thermodynamics.We recall physical constraints on the equations of state and give explicit and easily checkable conditions under which vacuums occur in the solution of the Riemann problem.We then present a class of models for which the Riemann problem admits unique global solutions without vacuums.     

LESSER KNOWN MIRACLES OF BURGERS EQUATION Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.     

The Mathematical Theory of Multifocal Lenses(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper presents the fundamental optical concepts of designing multifocal ophthalmic lenses and the mathematical methods associated with them. In particular, it is shown that the design methodology is heavily based on differential geometric ideas such as Willmore surfaces. A key role is played by Hamilton's eikonal functions. It is shown that these functions capture all the information on the local blur and distortion created by the lenses. Along the way, formulas for computing the eikonal functions are derived. Finally,the author lists a few intriguing mathematical problems and novel concepts in optics as future projects.     

LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

This paper is concerned with the low Mach number limit for the compressible Navier-Stokes equations in an exterior domain.We present here an approach based on Strichartz estimate defined on a non trapping exterior domain and we will be able to show the compactness and strong convergence of the velocity vector field.     

Isolated complex singularities and their CR links Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on C~n Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Given a principal value convolution on the Heisenberg group Hn = Cn×R, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on Cn. We also calculate the Dirichlet kernel for the Laguerre expansion on the group Hn.     

DECAY OF POSITIVE WAVES OF HYPERBOLIC BALANCE LAWS Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

Historically,decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws.Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes.Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time.This work presents two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism.The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics [7,17,24].