Narrow Results

In this paper, we significantly improve a result of the first author, published in an issue of Theoretical Computer Science in 2003. In this paper, the authors showed the existence of a weakly universal cellular automaton on the pentagrid with 22 states. The simulation used a railway circuit which simulates a register machine. In the present paper, using the same simulation tool, we lower the number of states for a weakly universal cellular automaton down to 9.

We discuss the isometry group structure of three-dimensional black holes and Chern–Simons invariants. Aspects of the holographic principle relevant to black hole geometry are analyzed.

We study gauge theories based on abelian p- forms on real compact hyperbolic manifolds. The tensor kernel trace formula and the spectral functions associated with free generalized gauge fields are analyzed.

We work with N-dimensional compact real hyperbolic space X_Γ with universal covering M and fundamental group Γ. Therefore, M is the symmetric space G/K, where G = SO₁(N, 1) and K = SO(N) is a maximal compact subgroup of G. We regard Γ as a discrete subgroup of G acting isometrically on M, and we take X_Γ to be the quotient space by that action: X_Γ = Γ∖M = Γ∖G/K. The natural Riemannian structure on M (therefore on X) induced by the Killing form of G gives rise to a connection p-form Laplacian 𝔏_p on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on Abelian p-forms on the real compact hyperbolic manifold X_Γ. The spectral zeta function related to the operator 𝔏_p, considering only the coexact part of the p-forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic functions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established.

A new and fast algorithm is presented in this paper for the automatic generation of aesthetic patterns with symmetries of the triangle groups T(p, q, r) by means of dynamical systems. Equivariant mappings with such symmetries are constructed. A modified convergence time scheme is described, which reflects the rate of convergence of various orbits and, at the same time, enhances the artistic appeal of generated images. Symmetrical chaotic attractors are generated by using a random iteration scheme. This method can be used to create a great variety of exotic symmetrical patterns.

We study Cantor staircases arising from different problems in Physics. Each staircase is endowed with the Farey structure in the vertical axis, the underlying cantordust set Ω having a fractal dimension strictly between 0 and 1. We find that length and distribution of stairsteps follow hyperbolic laws related to the Poincaré measure in the hyperbolic half-plane. The geometry of the underlying Ω is studied with the same tools.

In this paper we give necessary and sufficient conditions for discreteness of a subgroup of PSL(2,ℂ) generated by a hyperbolic element and an elliptic one of odd order with non-orthogonally intersecting axes. Thus we completely determine two-generator non-elementary Kleinian groups without invariant plane with real traces of the generators and their commutator. We also give a list of all parameters that correspond to such groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold ℍ³/Γ₃₅₃ has real parameters.

In this paper, we detail the geometrical approach of small cancellation theory used by Delzant and Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

For any g ≥ 2 we construct a graph Γ_g ⊂ S³ whose exterior M_g = S³\N(Γ_g) supports a complete finite-volume hyperbolic structure with one toric cusp and a connected geodesic boundary of genus g. We compute the canonical decomposition and the isometry group of M_g, showing in particular that any self-homeomorphism of M_g extends to a self-homeomorphism of the pair (S³,Γ_g), and that Γ_g is chiral. Building on a result of Lackenby we also show that any non-meridinal Dehn filling of M_g is hyperbolic, thus getting an infinite family of graphs in S² × S¹ whose exteriors support a hyperbolic structure with geodesic boundary.

Many three-dimensional manifolds are two-fold branched covers of the three-dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and includes many examples. It shows that there are three-dimensional homology spheres that do not two-fold branched cover any manifold, ones that only two-fold branched cover the three-dimensional sphere, ones that just two-fold branched cover a non-trivial manifold, and ones that two-fold branched cover both the sphere and non-trivial manifolds. When a manifold is surgery on a knot, the possible quotients via involutions generically correspond to quotients of the knot. There can, however, be a finite number of surgeries for which there are exceptional additional symmetries. The included proof of this result follows the proof of Thurston's Dehn surgery theorem. The paper also includes examples of such exceptional symmetries. Since the quotients follow the behavior of knots, a census of the behavior for knots with less than 11 crossings is included.

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

In this article, we construct some arithmetic hyperbolic links L such that S³\L is homeomorphic to ℍ³/Γ where Γ is not conjugate to a subgroup of any Bianchi group. We give two methods of construction illustrated by examples in and .

We present an efficient method of constructing hyperbolic patterns based on an asymmetric motif designed in the central hyperbolic polygon. Since there is no rotational symmetry in each hyperbolic polygon, a subset of the hyperbolic group elements has to be selected carefully so that the central hyperbolic polygon is transformed to the other polygons once and only once. An efficient labeling procedure is proved by considering the group presentation and can be easily implemented using the computer. Illustrative hyperbolic patterns are constructed from given asymmetric motifs for the symmetry group [$p$, $q$]${}^{+}$ which consists of all compositions of an even number of reflections.

Every surface in the Euclidean space ℝ³ admits a canonical Riemannian metric that has constant Gaussian curvature and is conformal to the original metric. Similarly, 3-manifolds can be decomposed into pieces that admit canonical metrics. Such metrics not only have theoretical significance in 3-manifold geometry and topology, but also have potential applications to practical problems in engineering fields such as shape classification.

In this paper we present an algorithm that is based on a discrete curvature flow to compute constant curvature metrics on 3-manifolds that are hyperbolic and have boundaries of a certain type. We also provide an approach to visualize such a metric by embedding the fundamental domain and universal covering in the hyperbolic space ℍ³. Some experimental results are given for both algorithms.

Furthermore, we propose an algorithm to automatically construct truncated tetrahedral meshes for 3-manifolds with boundaries. It can not only generate inputs to the curvature flow algorithm, but could also serve as an automatic tool for geometers and topologists to build simple models for complicated 3-manifolds, and therefore facilitate their research that requires such models.

We generalize the mass redistribution principle and apply it to prove the Bishop–Jones relation for limit sets of metrically proper isometric actions on real infinite-dimensional hyperbolic space. We also show that the Hausdorff and packing measures on the limit sets of convex-cobounded groups are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.

Regular tessellations of the hyperbolic plane play an important role in the design of signal constellations for digital communication systems. Self-dual tessellations of type $\{4g,4g\}$ with $g=2^n,3\cdot 2^n$, and $5\cdot 2^n$ have been considered where the corresponding arithmetic Fuchsian groups are derived from quaternion orders over quadratic extensions of the rational. The objectives of this work are to establish the maximal orders derived from $\{4g,4g\}$ tessellations for which the hyperbolic lattices are complete (the motivation for constructing complete hyperbolic lattices is their application to the design of hyperbolic lattice codes), and to identify the arithmetic Fuchsian group associated with a quaternion algebra and a quaternion order.

We consider a family of linearly elastic shells of the first kind (as defined in [2]), also known as non inhibited pure bending shells [7]. This family is indexed by the half-thickness ε. When ε approaches zero, the averages across the thickness of the shell of the covariant components of the displacement of the points of the shell converge strongly towards the solution of a "2D generalized membrane shell problem" provided the applied forces satisfy admissibility conditions [1,3]. The identification of the admissible applied forces usually requires delicate analysis.

In the first part of this paper, we simplify the general admissibility conditions when applied forces h are surface forces only, and obtain conditions that no longer depend on ε [5]: find h^αβ = h^αβ in L²(ω) such that for all η = (η_i) in V(ω), ∫_ω hⁱ η_i dω = ∫_ω h^αβγ_αβ(η)dω where ω is a domain of ℝ², θ is in and is the middle surface of the shells, where (γ_αβ (η)) is the linearized strain tensor of S and V(ω) = {η ∈ H¹(ω), η = 0on γ₀}, the shells being clamped along Γ₀ = θ(γ₀).

In the second part, since the simplified admissibility formulation does not allow to conclude directly to the existence of h^αβ, we seek sufficient conditions on h for h^αβ to exist in L²(ω). In order to get them, we impose more regularity to h^αβ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE's with h^αβ as unknowns. The existence of solutions depends both on the geometry of the shell and on the choice of h. We carry through the study of four representative geometries of shells and identify in each case a special admissibility functional space for h.

The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the extended Bianchi groups B_d, d = 5, 6, are used to reduce the problem of finding the discrete part of the Markov spectrum for the group B_d to the corresponding problem for one of its maximal Fuchsian subgroup. Hermitian points in the Markov spectrum of B_d are introduced for any d. Let H³ be the upper half-space model of the three-dimensional hyperbolic space. If ν is a hermitian point in the spectrum, then there is a set of extremal geodesics in H³ with diameter 1/ν, which depends on one continuous parameter. This phenomenon does not take place in the hyperbolic plane.

Applying the Klein model D² of the hyperbolic plane and identifying the geodesics in D² with their poles in the projective plane, the author has developed a method for finding the discrete part of the Markov spectrum for Fuchsian groups. It is applicable mostly to non-cocompact groups. In the present paper, this method is extended to cocompact Fuchsian groups. For a group with signature (0;2,2,2,3), the complete description of the discrete part of the Markov spectrum is obtained. The result obtained leads to the complete description of the Markov and Lagrange spectra for the imaginary quadratic field with discriminant -20.

Let H³ be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups Γ of an extended Bianchi group B_d, the extremality of the axis of hyperbolic F ∈ Γ in H₃ with respect to Γ implies its extremality with respect to B_d. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of B_d for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields with discriminants -120 and -132 are given. The second minima are also indicated for these fields.