Let K be a totally real algebraic number field and O an order in a quaternion algebra A over K. Assume that the group O1 of units in O with reduced norm equal to 1 is embedded into PSL2(R) as an arithmetic Fuchsian group. It is shown how Ford’s algorithm can be effectively applied in order to determine a fundamental domain of O1 as well as a complete system of generators of O1.

We study symmetries of feedforward networks in terms of their corresponding groups and nd that these groups naturally act on and partition weight space. We specify an algorithm to generate representative weight vectors in a speciic fundamental domain. The analysis of the metric structure of the fundamental domain enables us to use the location information of weight vector estimates, e. g. for c...

The symmetries associated with the closed bosonic string partition function are examined so that the integration region in Teichmuller space can be determined. The conditions on the period matrix defining the fundamental region can be translated to relations on the parameters of the uniformizing Schottky group. The growth of the lower bound for the regularized partition function is derived thro...

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpin...

Let Y be a complex algebraic variety, G y Y an action of an algebraic group on Y , U ⊆ Y (C) a complex submanifold, Γ < G(C) a discrete, Zariski dense subgroup of G(C) which preserves U , and π : U → X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Γ\U . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalize...

An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. Properties of such functions will be described for compact simple Lie groups of all types. In particular, products of the functions decompose into their sums, the functions are periodic on copies of the fundamental region F of G, they are solutions o...

Previously we described an algorithm that can fill a region with an infinite sequence of randomly placed and progressively smaller shapes, producing a fractal pattern. If the algorithm is appropriately modified and the region is a fundamental region for one of the 17 “wallpaper” groups, one can obtain a fractal pattern with that symmetry group. This produces artistic patterns which have a pleas...

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpin...

For natural language understanding, it is essential to reveal semantic relations between words. To date, only the IS-A relation has been publicly available. Toward deeper natural language understanding, we semiautomatically constructed the domain dictionary that represents the domain relation between Japanese fundamental words. This is the first Japanese domain resource that is fully available....

1. H as homogeneous space for G = SL 2 (R) 2. Fundamental domain for Γ = SL 2 (Z) on H 3. Inversion and translation generate SL 2 (Z). 4. Re-enabling the action of SL 2 (R) 5. Fundamental domain for Γ θ and Γ(2) 6. Generators for Γ θ 7. Theta series are modular forms The real line R, being a group, is a homogeneous space in the sense that (of course) it acts transitively on itself. The upper ha...