Uniform bounds under increment conditions

We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.

     

The complexity of cutting complexes

This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed.Bernard Chazelle wishes to acknowledge the National Science Foundation for supporting this research in part under Grant No. MCS83-03925. Herbert Edelsbrunner is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862.     

Pion correlations and resonance effects in \bar{p}p annihilation to 4\pi^0 at rest

We study correlations in the exclusive reaction at rest with complete reconstruction of the kinematics for each event. The inclusive distribution is fairly flat at small invariant mass of the pion pair while a small enhancement in the double differential distribution is observed for small invariant masses of both pion pairs. Dynamical models with resonances in the final state are shown to be consistent with the data while the stochastic HBT mechanism is not supported by the present findings. Received: 26 February 2002 / Revised version: 22 July 2002 / Published online: 30 August 2002     

Almost Sure Convergence and Square Functions of Averages Of Riemann Sums

Under a very moderate assumption on the Fourier coefficients of a periodic function, we show the convergence almost everywhere of the sequence of averages of its associated Riemann sums. The structure of the set of averages is analyzed by proving a spectral regularization type inequality, which allows to control the corresponding Littlewood-Paley square function.     

Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.     

A modular integer GCD algorithm

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC '93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U−bV)/p, where p is one of the prime moduli and b is the unique integer in the interval (−p/2,p/2) such that . When the algorithm is executed on a bit common CRCW PRAM with O(nlognlogloglogn) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/logn) time on the same number of processors.     

Neighborhood Complexes and Generating Functions for Affine Semigroups

Given a₁,a₂,...,a_n ∈ ℤ^d$, we examine the set, G, of all non-negative integer combinations of these a_i. In particular, we examine the generating function f(z) = ∑_{b ∈ G}z_b. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in ℤ_n. In the generic case, this follows from algebraic results of Bayer and Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.