Some properties of horocycles on Damek-Ricci spaces

We prove that a Damek-Ricci space is symmetric if and only if the geodesic inversion preserves the set of horocycles.     

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number the Kronecker sequence m (m = 1,...,M) is equidistributed modulo one in the limit , and (ii) closed horocycles of length become equidistributed in the unit tangent bundle of a hyperbolic surface of finite area, as . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant the Kronecker sequence embedded in along a long closed horocycle becomes equidistributed in for almost all , provided that . This equidistribution result holds in fact under explicit diophantine conditions on (e.g. for = 2) provided that , with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for , our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence n² modulo one.     

Two algorithms for closed horocycles on the modular surface

We track the trajectories of individual directed horocycles on the modular surface. Our tracking is constructive, and we thus effectively establish topological transitivity and even line-transitivity for the horocyclic flow.     

Isometric Lagrangian immersion of horocycles of the hyperbolic plane in complex space

We prove that there exists an isometric Lagrangian immersion of a horocycle of the hyperbolic plane in the complex space ?², and there exists an isometric Lagrangian immersion of a horoball of hyperbolic (Lobachevski) space H ³ in the complex space ?³.     

Stable windings at the origin

In 1996, Bertoin and Werner demonstrated a functional limit theorem, characterising the windings of planar isotropic stable processes around the origin for large times, thereby complementing known results for planar Brownian motion. The question of windings at small times can be handled using scaling. Nonetheless we examine the case of windings at the origin using new techniques from the theory of self-similar Markov processes. This allows us to understand upcrossings of (not necessarily symmetric) stable processes over the origin for large and small times in the one-dimensional setting.     

Stable windings on hyperbolic surfaces

Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T ¹ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t ^{−1/(2δ−1)}, for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves. Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000     

Asymptotic oscillations

Given a function defined on the support of a ballean, we introduce the notion of slow oscillation in direction of a filter on X. We show that there exists a filter on X responsible for the rate of slow oscillation of f at infinity. We apply this result to the Stone-?ech compactifications of discrete groups.     

Asymptotic fractals

A new class of fractals which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure, is defined analytically and investigated.     

Asymptotic dimension

The asymptotic dimension theory was founded by Gromov [M. Gromov, Asymptotic invariants of infinite groups, in: Geometric Group Theory, vol. 2, Sussex, 1991, in: London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295] in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups.     

Asymptotic scheduling

We deal with the following scheduling problem: a finite set of jobs is given and each job consists in the execution of an infinite number of tasks. A task is a sequence of operations and each operation requires a specific machine. A machine can process only one operation at a time and preemption is not allowed. Performance measures of the processing system involve fixing a time horizon T, counting the number of tasks completed within T for each job and maximizing a specified function of these numbers to estimate the throughput of the schedule. Whilst computing the throughput for a given T is in general an extremely difficult problem, it is shown in this paper that the limit, as T tends to infinity, of the average throughput (i.e. the throughput divided by T) can be easily computed via Linear Programming under fairly mild conditions. This quantity, which may be called the asymptotic throughput, can be used to assess a bound on performance measures of real systems. Buffers play a crucial role and buffer sizes can be taken care of in assessing the system performance. Mathematics Subject Classification (2000):90B35, 90C05, 90C27, 90C90     

Asymptotic curves of submanifolds

Translated from Matematicheskie Zametki, Vol. 52, No. 5, pp. 3–12, November, 1992.     

Asymptotic behavior of waves

The asymptotic conjugation relation is established for all ƒL²(Rⁿ) under mild assumptions on and g, where denotes Fourier multiplication. The asymptotic estimate for finite energy solutions u of the wave equation is deduced from (*), along with generalizations to a class of first-order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient, and a weakened version for the Klein-Gordon equation. Also deduced from (*) is the fact that for a free Schrödinger particle the probability of being in the set tA at time t tends to the probability that the velocity is in A as t → ±∞.     

Asymptotic semismoothness probabilities

We call an integer semismooth with respect to and if each of its prime factors is , and all but one are . Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let be the asymptotic probability that a random integer is semismooth with respect to and . We present new recurrence relations for and related functions. We then give numerical methods for computing , tables of , and estimates for the error incurred by this asymptotic approximation.

     

Asymptotic existence of frame-GBTDs

Generalized balanced tournament designs(GBTDs) are an equivalent characterization of a class of equitable symbol weight codes. Motivated by the construction of GBTDs, we establish in this paper an asymptotic existence theorem for frame-GBTDs of type gnand block size k via decompositions of edge-colored complete digraphs into prescribed edge-colored subgraphs.