The real, nonsingular elliptic solutions of the Korteweg-deVries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is shown to be solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Specia...

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the cor...

Abstract. Let f1 = 1, f2 = 2, f3 = 3, f4 = 5, . . . be the sequence of Fibonacci numbers. It is well known that for any natural n there is a unique expression n = fi1 + fi2 + · · · + fiq , such that ia+1 − ia > 2 for a = 1, 2, . . . , q− 1 (Zeckendorf Theorem). By means of it we find an explicit formula for the quantity Fh(n) of partitions of n with h summands, all parts of them are the pairwis...

In a recent paper, Kaneko and Zagier studied a sequence of modular forms Fk(z) which are solutions of a certain second order differential equation. They studied the polynomials e Fk(j) = Y τ∈H/Γ−{i,ω} (j − j(τ))τ k, where ω = e2πi/3 and H/Γ is the usual fundamental domain of the action of SL2(Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e Fp−1(j) (mod p) is the...

For P ⊂ H3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/mij (mij ≥ 2 an integer), there is a naturally associated Coxeter group ΓP . Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O+(3, 1) = Isom(H3), with fundamental domain the original polyhedron P . In this paper, we provide a procedure...

We give a construction of a fundamental domain for the group PU(2, 1,Z[i]). That is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers Z[i]. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

From the regular hexagon of unit size remove circular holes of small radius ε > 0 centered at the vertices, obtaining a region Hε of area 3 √ 3 2 − πε2. For each pair (x, ω) ∈ Hε × [0, 2π] consider a point particle moving at unit speed on a linear trajectory with specular reflections when meeting the boundary. Denote by τhex ε (x, ω) the time it takes the particle to reach one of the holes. Thi...

Let F denote a totally real number field of degree d > 1 over Q, let p be a prime number and let χ be a totally odd Hecke character of finite order of F . Klingen and Siegel have shown that the values of the Hecke L-series L(χ, s) at integers n ≤ 0 lie in the algebraic closure Q ⊆ C of Q. In [14] Shintani gave another proof by constructing a nice fundamental domain (i.e. a finite disjoint union...