Vanishing structure set of 3-manifolds

In this short note we update a result proved in Roushon (2007) [17]. This will complete our program of Roushon (2000) [13] showing that the structure set vanishes for compact aspherical 3-manifolds.     

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

Domains of infinitesimal generators of automorphism flows

We present a systematic characterization of the domain of a generator of a one parameter group on certain C∗-subalgebras of via finite-dimensional estimates. Our approach yields an example of a densely defined closed symmetric derivation on a C∗-subalgebras of whose domain is not closed with respect to the C¹-functional calculus. This completes and complements the earlier example of McIntosh (J. Funct. Anal. 30 (1977) 264). Our methods are partly based on the theory of adjoint C₀-semigroups.     

On the Monge–Ampère equation with boundary blow-up: existence,uniqueness and asymptotics

We consider the Monge–Ampère equation det D ² u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ?Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ?Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ?Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ?Ω and \(b\equiv 0\) on ?Ω.     

C*-Algebras of Singular Integral Operators with Shifts Having the Same Nonempty Set of Fixed Points

The C*-subalgebra of generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms with any nonempty set of common fixed points is studied. A symbol calculus for the C*-algebra and a Fredholm criterion for its elements are obtained. For the C*-algebra composed by all functional operators in , an invertibility criterion for its elements is also established. Both the C*-algebras and are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure. Submitted: April 30, 2007. Accepted: November 5, 2007.     

A Multidimensional Generalization of Shukla's 8ψ8 Summation

   Abstract. We give an r -dimensional generalization of H. S. Shukla's very-well-poised ₈ ψ ₈ summation formula. We work in the setting of multiple basic hypergeometric series very-well-poised over the root system A _r-1 or, equivalently, the unitary group U(r) . Our proof, which is already new in the one-dimensional case, utilizes an A _r-1 nonterminating very-well-poised ₆ φ ₅ summation by S. C. Milne, a partial fraction decomposition, and analytic continuation.     

A remark on Andrews-Askey integral

In this paper, we use the Andrews-Askey integral and the q-Chu-Vandermonde formula to derive a more general integral formula. Applications of the new integral formula are also given, which include to derive the q-Pfaff-Saalschütz formula and the terminating Sears's transformation formula.