We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = C1, (ii) every N-gradable weak V module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including...

Let V be a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = C1, (ii) every N-gradable weak V -module is completely reducible and (iii) V is C2-cofinite. Then the q-traces of products of geometrically-modified intertwining operators are shown to satisfy modular invariant systems of differential equations which, for any fixed modular parameter, are re...

We prove that given a fixed radius r, the maximum density achieved by packings of the hyperbolic plane by radius r circles is the supremum of densities of “periodic packings” (those packings with cofinite symmetry). We also show that the maximal density function is continuous as a function of radius. Our investigations lead us to consider more general questions regarding the density of periodic...

in this paper we study the module contractibility ofbanach algebras and characterize them in terms the conceptssplitting and admissibility of short exact sequences. also weinvestigate module contractibility of banach algebras with theconcept of the module diagonal.

Let Γ be a nonelementary Kleinian group and H<Γ finitely generated, proper subgroup. We prove that if has finite covolume, then the profinite completions of H are not isomorphic. If index in Γ, there is onto which maps but does not. These results streamline existing proofs exist full-sized groups profinitely rigid absolute sense. They build on circle ideas can used to distinguish among subgroup...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

We elaborate a recursive-theoretic method to define real numbers which are random to any jump of the halting problem. This is obtained by generalizing a result of V. Becher and G. Chaitin dealing with the class of cofinite sets. As in the work of V. Becher and S. Grigorieff (but this time in the context of open sets in Cantor space) appeal to completeness phenomena replaces machine arguments. M...