Hyperbolic Fibrations and PDE

2010 Mathematics Subject Classification: 35L10, 35L90.

Interference in hyperbolic space

The interference in a phase space algorithm of Schleich and Wheeler [Nature 326, 574 (1987)] is extended to the hyperbolic space underlying the group SU(1,1). The extension involves introducing the notion of weighted areas. Analytic expressions for the asymptotic forms for overlaps between the eigenstates of the generators of su(1,1) thus obtained are found to be in excellent agreement with the numerical results.[S1050-2947(98)08602-8].

On the loss of regularity for a class of weakly hyperbolic operators

In this paper we determine bounds for the optimal loss of regularity in the Sobolev scale for a class of weakly hyperbolic operators. (C) 2009 Elsevier Inc. All rights reserved.

Category-theoretic fibration as an abstraction mechanism in information systems

This paper examines the problem of establishing a formal relationship of abstraction and refinement between abstract enterprise models and the concrete information systems which implement them. It introduces and justifies a number of reasonableness requirements, which turn out to justify the use of category theoretic concepts, particularly fibrations, to precisely specify a semantics for enterprise models which enables them to be considered as abstractions of the conceptual models from which the implementing information systems are built. The category-theoretic concepts are developed towards the problem of testing whether a system satisfies the fibration axioms, and are applied to case studies to demonstrate their practicability.

Solving MPCC problem with the hyperbolic penalty function

The main goal of this work is to solve mathematical program with complementarity constraints (MPCC) using nonlinear programming techniques (NLP). An hyperbolic penalty function is used to solve MPCC problems by including the complementarity constraints in the penalty term. This penalty function [1] is twice continuously differentiable and combines features of both exterior and interior penalty methods. A set of AMPL problems from MacMPEC [2] are tested and a comparative study is performed.

Solving a signalized traffic intersection problem with an hyperbolic penalty function

Mathematical Program with Complementarity Constraints (MPCC) finds many applications in fields such as engineering design, economic equilibrium and mathematical programming theory itself. A queueing system model resulting from a single signalized intersection regulated by pre-timed control in traffic network is considered. The model is formulated as an MPCC problem. A MATLAB implementation based on an hyperbolic penalty function is used to solve this practical problem, computing the total average waiting time of the vehicles in all queues and the green split allocation. The problem was codified in AMPL.

Numerical optimization experiments using the hyperbolic smoothing strategy to solve MPCC

In this work we solve Mathematical Programs with Complementarity Constraints using the hyperbolic smoothing strategy. Under this approach, the complementarity condition is relaxed through the use of the hyperbolic smoothing function, involving a positive parameter that can be decreased to zero. An iterative algorithm is implemented in MATLAB language and a set of AMPL problems from MacMPEC database were tested.

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

We prove that the stable holonomies of a proper codimension 1 attractor Λ, for a Cr diffeomorphism f of a surface, are not C1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.

Multidimensional schemes for hyperbolic conservation laws on triangular meshes

Genuinely multidimensional schemes, hyperbolic systems, wave equations, Euler equations, evolution Galerkin schemes, space-time conservative methods, high order accuracy, shock solutions

Exact Riemann solutions to two selected resonant hyperbolic systems

Magdeburg, Univ., Fak. für Mathematik, Diss., 2013

On Hyperbolic once-punctured-torus bundles II. Fractal tessellations of the plane

We describe fractal tessellations of the complex plane that arise naturally from Cannon-Thurston maps associated to complete, hyperbolic, once-punctured-torus bundles. We determine the symmetry groups of these tessellations.

Mixing invariants of hyperbolic 3-manifolds

Let M be a compact hyperbolic 3-manifold with incompressible boundary. Consider a complete hyperbolic metric on int(M). To each geometrically finite end of int(M) are traditionnaly associated 3 different invariants : the hyperbolic metric associated to the conformal structure at infinity, the hyperbolic metric on the boundary of the convex core and the bending measured lamination of the convex core. In this note we show how invariants of different types can be realised in the different ends.

Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a two-parameter framework and the nonlinear phenomena associated with such collision are considered. By dealing with planar discontinuous (Filippov) systems, some of such phenomena are pointed out through specific representative cases. A methodology for obtaining the corresponding bi-parametric bifurcation sets is developed.

Horospheres in hyperbolic geometry

In this paper we investigate the role of horospheres in Integral Geometry and Differential Geometry. In particular we study envelopes of families of horocycles by means of “support maps”. We define invariant “linear combinations” of support maps or curves. Finally we obtain Gauss-Bonnet type formulas and Chern-Lashof type inequalities.