Suppose you know the locations of post offices or cell phone satellites, and you want to know what regions they serve. Or maybe you know the locations of atoms in a crystal, and you want to know what a fundamental region looks like. There are lots of reasons you might want to make a tiling around a given discrete set of points. A natural way to do it is with “Voronöı tessellations”—so natural, ...

We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves.

Let F be a field and F[x, y] the ring of polynomials in two variables over F. Let f ∈ F[x, y] and consider the residue class ring R := F[x, y]/fF[x, y]. Our first aim is to study digit representations in R, i.e., we ask for which f each element r ∈ R admits a digit representation of the form d0 + d1x + · · · + d`x with digits di ∈ F[y] satisfying degy di < degy f . These digit systems are motiv...

Let Γ be a discrete subgroup of PSL(2,R) of infinite covolume with infinite conjugacy classes. Let Ht be the Hilbert space consisting of analytic functions in L(D, (Im z)dzdz) and let, for t > 1, πt denote the corresponding projective unitary representation of PSL(2,R) on this Hilbert space. We denote by At the II∞ factor given by the commutant of πt(Γ) in B(Ht). Let F denote a fundamental doma...

Waves of fashionable ideas shape the practice and research of information systems (IS). What forces drive idea waves in IS? This research takes the first step to empirically study IS idea waves in inter-organizational communities through the lens of organizing visions. Introduced by Swanson and Ramiller (1997), an organizing vision is a focal community idea for applying information technologies...

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for Q( √ 5), Q( √ 2), and Q( √ 3) have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the...

Rankin and Swinnerton-Dyer [R, S-D] prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A := {eiθ : π 2 ≤ θ ≤ 2π 3 }. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique “gap function” in...

We study lattices with a non-compact fundamental domain of small volume in hyperbolic space Hn. First, we identify the arithmetic lattices in Isom+Hn of minimal covolume for even n up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for n = 11 and n = 13 in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results d...

We study the relationship between stable sampling sequences for bandlimited functions in Lp(Rn) and the Fourier multipliers in Lp. In the case that the sequence is a lattice and the spectrum is a fundamental domain for the lattice the connection is complete. In the case of irregular sequences there is still a partial relationship. 2010 Mathematics Subject Classification: 42A45, 42B15, 42C15, 62...

Let X = Γ\G/K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has Q-rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D0. We prove that D0 is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ-con...