Volume 24, Issue 6

Abstract. We define a generalized pseudohermitian structure on an almost CR manifold as a pair ( h , P ), where h is a positive definite fiber metric h on compatible with J , and is a smooth projector such that . We show that to each generalized pseudohermitian structure one can associate a canonical linear connection on the holomorphic bundle which is invariant under equivalence. This fact allows us to solve the equivalence problem in the case where is a kind 2 distribution. We study the curvature of the canonical connection, especially for the classes of standard homogeneous manifolds and 3-Sasakian manifolds. The basic formulas for isopseudohermitian immersions are also obtained in the attempt to enlarge the theory of pseudohermitian immersions between strongly pseudoconvex pseudohermitian manifolds of hypersurface type.

Abstract. The spaces and have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces, one has the classifying spaces of the finite groups , for an odd prime power q , and the exotic family of classifying spaces of 2-local finite groups . In this article we compute the loop space homology of and for all odd primes q , as Hopf algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.

Abstract. A class of interacting superprocesses arising from branching particle systems with continuous spatial motions, called superprocesses with dependent spatial motion (SDSMs), has been introduced and studied by Wang and by Dawson, Li and Wang. In this paper, we extend the model to allow discontinuous spatial motions. Under Lipschitz condition for coefficients, we show that under a proper rescaling, branching particle systems with jump-diffusion underlying motions in a random medium converge to a measure-valued process, called stable SDSM. We further characterize this stable SDSM as a unique solution of a well-posed martingale problem. To prove the uniqueness of the martingale problem, we establish the C 2+γ -regularity for the transition semigroup of a class of jump-diffusion processes, which may be of independent interest.

Abstract. Let p be a prime congruent to 1 or 3 modulo 8 so that the equation is solvable in integers. In this paper, we obtain closed-form expressions for a and b in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.

Abstract. We study theta lifts for . The theta-lift is realized via an integral transform with a Siegel theta series as kernel function. Since this Siegel theta series fails to be square integrable, it has to be regularized. The regularization is obtained by applying a suitable differential operator built from the Laplacian. For the regularized theta series we compute the theta lift for cusp forms. The regularized lift also gives a correspondence for non-cusp forms such as Eisenstein series. Also we obtain the spectral expansion of the theta series in either of its variables. As an application we prove a three dimensional analogue of Katok–Sarnak's correspondence using the Selberg transform.

Abstract. We study the holomorphic extension associated with power series, i.e., the analytic continuation from the unit disk to the cut-plane . Analogous results are obtained also in the study of trigonometric series: we establish conditions on the series coefficients which are sufficient to guarantee the series to have a KMS analytic structure. In the case of power series we show the connection between the unique (Carlsonian) interpolation of the coefficients of the series and the Laplace transform of a probability distribution. Finally, we outline a procedure which allows us to obtain a numerical approximation of the jump function across the cut starting from a finite number of power series coefficients. By using the same methodology, the thermal Green functions at real time can be numerically approximated from the knowledge of a finite number of noisy Fourier coefficients in the expansion of the thermal Green functions along the imaginary axis of the complex time plane.

Abstract. In joint work with E. Dubuc and D. Mundici, the first author extended Stone duality for boolean algebras to locally finite MV-algebras. On the topological side of the duality, one has to assign to each point of a boolean space a generalized natural number by way of a multiplicity, so as to make the assignment continuous with respect to the Scott topology of the lattice of generalized natural numbers. The continuous maps between such generalized multisets are to be multiplicity-decreasing with respect to the divisibility order of generalized natural numbers. In this paper we extend these results to the class of MV-algebras that are locally weakly finite , i.e., such that all their finitely generated subalgebras split into a finite direct product of simple MV-algebras. Using the Scott topology on the lattice of subalgebras of the real unit interval (regarded with its natural MV-algebraic structure), we construct a `real-valued multiset' over the (boolean) space of maximal ideals of a locally weakly finite MV-algebra. Building on this, we obtain a duality for locally weakly finite MV-algebras that includes as a special case the above-mentioned duality for locally finite MV-algebras. We give an example that shows that the duality established in this paper via the Scott topology cannot be extended, without non-trivial modifications, to larger classes of algebras.

Corrigendum to http://dx.doi.org/10.1515/forum.2010.045