In this paper, we consider different ways of generating dynamical systems on 3-manifolds. We first derive explicit differential equations for dynamical systems defined on generic hyperbolic 3-manifolds by using automorphic function theory to uniformize the upper half-space model. It is achieved via the modification of the standard Poincaré theta series to generate systems invariant within each ...

Problems of geometric probability for an arbitrary convex body of resolution in the euclidean space E3 has been investigated in [1]. In [9] Buffon’s problem is solved for a lattice of right-angled parallelepipeds in the 3-dimensional space. In this note we want to use the results in [5] for to solve problems of intersection for a particular lattice that we describe: the fundamental cell C0 of t...

which we call the fundamental domain. By previous results, this set also (essentially) represents equivalence classes of complex elliptic curves under isomorphism (τ ∈ D ↔ C/Λτ ). Lemma 1.1. The map π : D → Y (1) is a surjection, where π is the obvious projection π(τ) = SL2(Z)τ . Proof. This lemma states that D generates all of H under the action of SL2(Z); that is, it suffices to show that eve...

1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in Harish-Chandra’s investigations of the representations of semi-sim...

In this paper, we study a fundamental domain for the Siegel-Jacobi space Sp(g,Z)⋉H (g,h) Z \Hg × C .

Arithmetic Kleinian groups are arithmetic lattices in PSL2(C). We present an algorithm that, given such a group Γ, returns a fundamental domain and a finite presentation for Γ with a computable isomorphism.

Figure 1: Fundamental region R of the Modular Group, given by {z ∈ H : |z| > 1,− 2 ≤ Re(z) ≤ 1 2} The Klein Quartic curve is famous for its symmetries and many visualizations are created to capture these in lower dimensions. In fact, it has the largest possible automorphism group for its genus, the simple group of order 168. Yet, interest in this algebraic curve started with knowledge of the ex...

1 SL2(Z) and elliptic curves 2 1.1 SL2(Z) and the moduli of complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Fundamental region and a system of generators . . . . . . . . . . . . . . . . . . . . . 3 1.3 The Weierstrass ℘ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Nonsingular cubics and the invariant j . . . . . . . . . . . ...

Huntebrinker's research was supported by the Deutsche Forschungsgemeinschaft. Let H 3 be three-dimensional hyperbolic space and a group of isometries of H 3 that acts discontinuously on H 3 and that has a fundamental domain of finite hyperbolic volume. The Laplace operator of H 3 gives rise to a positive, essentially selfadjoint operator on L2( nH 3). The nature of its discrete spectrum dspec i...

Malaria kills more than 1 million people a year, mostly children in sub-Saharan Africa. Antimalarial drug resistance is one of the greatest challenges facing malaria control today. We present Ronald, a Domain-Specific Language to model the fundamental forces driving antimalarial drug resistance including drug pharmacokinetics and pharmacodynamics, drug regimens and parasite genotypes. Example o...