Difference Problems and Differential Problems

We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a significant contribution to the understanding of the foundations of differential and integral calculus

Range descriptions for the spherical mean Radon transform

The transform considered in the paper averages a function supported in a ball in \RR^n over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for such transforms are important in all these areas, for instance when dealing with incomplete data, error correction, and other issues. Four different types of complete range descriptions are provided, some of which also suggest inversion procedures. Necessity of three of these (appropriately formulated) conditions holds also in general domains, while the complete discussion of the case of general domains would require another publication.Comment: LATEX file, 55 pages, two EPS figure

Krein formula and S-matrix for Euclidean Surfaces with Conical Singularities

We use Krein formula and the S-matrix formalism to give formulas for the zeta-regularized determinant of non-Friedrichs extensions of the Laplacian on Euclidean surfaces with Conical Singularities. This formula involves S(0) and we show that the latter can be expressed using the Bergman projective connection on the underlying Riemann surface.Comment: Accepted in Journal of Geometric Analysi

Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors

We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, "cubic" approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).Comment: V2: minor changes, and chapter 3: new results included; to appear in Forum Mathematicu

Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds

We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.Comment: 14 pages v2: minor change