The purpose of this article is to establish a semi-group formula for the Riesz potentials of *L*^{p}-functions. As preparations, we study the Lizorkin space Φ(**R**^{n}) and investigate integral estimates of the Riesz potentials of functions in the spaces *L*^{p:r,s}(**R**^{n}).

In this paper we study the periodicity and the form of the solutions of the following systems of difference equations of order two

*x*_{n+1} = \frac{y_{n}x_{n-1}}{\pm x_{n-1}\pm y_{n}}, *y*_{n+1} = \frac{x_{n}y_{n-1}}{\pm x_{n}\pm y_{n-1}}, *n* ∈ ℕ_{0},

with nonzero real numbers initial conditions.

We describe explicitly the cohomology of the total complex of certain diagrams of invertible sheaves on normal toric varieties. These diagrams, called wheels, arise in the study of toric singularities associated to dimer models. Our main tool describes the generators in a family of syzygy modules associated to the wheel in terms of walks in a family of graphs.

We topologically characterize negatively curved Riemannian manifolds which are of cohomogeneity two under the action of a compact Lie group of isometries.

Let *R* be any ring. We prove that every right *R*-module is coretractable if and only if *R* is right perfect and every right *R*-module is small coretractable if and only if all torsion theories on *R* are cohereditary. We also study mono-coretractable modules. We show that coretractable modules are a proper generalization of mono-coretractable modules.