In the study of photonic crystals, the question arises naturally: which crystals produce the largest band gaps? This question is investigated by means of an optimization-based evolution algorithm which, given two dielectric materials, seeks to produce a material distribution within the fundamental cell which produces a maximal band gap at a given point in the spectrum. The case of H-polarizatio...

Let α = −2+ √ −1 be a root of the polynomial p(x) = x+4x+5. It is well known that the pair (p(x), {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each x ∈ Z[α] admits a finite representation of the shape x = a0 + a1α + · · · + alα with ai ∈ {0, 1, 2, 3, 4}. The set T of points with integer part 0 in this number system T := ( ∞ X i=1 aiα −i , ai ∈ {0, 1, 2, 3, 4} ) is called the fun...

The Chazy equation y′′′ = 2yy′′ − 3y′2 is derived from the automorphic properties of Schwarz triangle functions S(α, β, γ; z). It is shown that solutions y which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of α, β, γ. The solutions are then systematically constructed. These analytic solutions provide all known and one new parametrization...

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group G. Such functions can always be restricted without loss of information to a fundamental region F̌ of the affine Weyl group. The members of each family satisfy basic orthogonality rel...

Overtone spectroscopy is used to observe the rovibrational spectra of the hydrogen isotopologues H2, HD, and D2 adsorbed in the metal–organic framework known as MOF-5. It is shown that the overtone spectrum facilitates the identification of hydrogen modes which are obscured in the fundamental region by the presence of MOF-5 features. Further, the overtone spectrum of H2 at the primary adsorptio...

Let OD be the ring of integers in the imaginary quadratic field Q( √ D) of discriminant D < 0. Then PGL2(OD) is a discrete subgroup of the isometry group PSL2(C) (= PGL2(C)) of hyperbolic 3-space H3 . The quotient space H/PGL2(OD) = XD is topologically a noncompact 3-manifold whose cusps (ends) are of the form S × [0,∞). The number of cusps of XD is known to be hD , the class number of OD . So ...

Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category Cm(A) of A is the m-left part Lm(A (m)) of the m-replicated algebra of A. Moreover, we obtain a one-toone correspondence between the tilting objects in Cm(A) (that is, the m-clusters) and those tilting modules in modA(m) for which all non projective-injective di...

A recent paper (Pitteri and Zanzotto, 1998) has proposed a framework for the study of the ‘arithmetic symmetry’ of multilattices (discrete triply periodic point sets in the affine space). The classical approach to multilattice symmetry considers the well known ‘space groups’, that is, the groups of affine isometries leaving a multilattice invariant. The ensuing classification counts 219 affine ...

Those groups r which act properly discontinuously and aillnely on II?’ with compact fundamental domain are classified. First it is shown that such a group f contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive alTine actions on IR’ are classified; this leads to a classification of atTine crystallographic ...

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc D whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in D, which is invariant under the action of a lattice subgroup Γ of U(1, 1), the group of isometries of D. In our case Γ generates a tiling of D with regular octagons. T...