Periodic boundary conditions on the pseudosphere

We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the needed mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and we illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.

Errors in chromosome segregation during oogenesis and early embryogenesis

Errors in chromosome segregation occurring during human oogenesis and early embryogenesis are very common. Meiotic chromosome development during oogenesis is subdivided into three distinct phases. The crucial events, including meiotic chromosome pairing and recombination, take place from around 11 weeks until birth. Oogenesis is then arrested until ovulation, when the first meiotic division takes place, with the second meiotic division not completed until after fertilization. It is generally accepted that most aneuploid fetal conditions, such as trisomy 21 Down syndrome, are due to maternal chromosome segregation errors. The underlying reasons are not yet fully understood. It is also clear that superimposed on the maternal meiotic chromosome segregation errors, there are a large number of mitotic errors taking place post-zygotically during the first few cell divisions in the embryo. In this chapter, we summarise current knowledge of errors in chromosome segregation during oogenesis and early embryogenesis, with special reference to the clinical implications for successful assisted reproduction

The Convolution Algebra of Schwartz Kernels Along a Singular Foliation

Motivated by the study of Hörmander’s sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal. © Copyright by THETA, 202