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 Moore Bound for Irregular Graphs

 What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10). Received: June 27, 2000 Final version received: July 3, 2001     

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.     

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.     

Nuclear equation of state and compression modes

The nuclear matter (N = Z and no Coulomb interaction) incompressibility coefficient, K _nm , which is directly related to the curvature of the nuclear matter equation of state, is a very important physical quantity in the study of properties of nuclei, supernova collapse, neutron stars and heavy-ion collisions. We review the current status of K _nm and the experimental and theoretical methods used to determine the value of K _nm from the excitation crosssections σ(E) and the transition strength distributions S(E) of compression modes in nuclei. In particular, we will consider the isoscalar giant monopole resonance (ISGMR) and the isoscalar giant dipole resonance (ISGDR) and provide a simple explanation to the long standing problem of the conflicting results obtained for K _nm , deduced from experimental data on excitation cross sections for the ISGMR and data for the ISGDR.     

Elektrische Charakterisierung von Hochfrequenz-Niederdruck-Gasentladungen

In recent years there has emerged significant interest in low pressure radio frequency (rf) glow discharges which are used widely particulary in IC fabrication. Various parameters of the rf glow discharge have been found to be useful for its electrical characterization; however, there is no uniformity and agreement. Extensive experimental investigations on various discharge systems have shown, that the self-bias on the rf driven electrode, the complex conductivity and the breakdown characteristic are preferable parameters of rf discharges. Advantageously the self-bias and the complex conductivity should be presented in dependence on the pressure and the applied rf voltage. The discharge current cannot be measured quite accurately due to currents via leaky capacitors and the deviations from a sinusoidal form of the current due to nonlinearities.     

On Lattices of Uniformities

We study lattice theoretical properties of lattices of uniformities such as modularity, distributive laws and the existence of (relative) complements. For this the concepts of permutable uniformities (see Definition 3.1) and independent uniformities (see Definition 4.1) are important. Moreover, we show that e.g. the lattice of all lattice uniformities on a lattice L is a closed sublattice of the lattice of all uniformities on L.     

Romanovski polynomials in selected physics problems

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.     

Some examples of application of the metric entropy method

We show how the metric entropy method can be substituted for the dyadic chaining, to prove in a unified setting several classical results. Among them are Stechkin's theorem, Gál--Koksma theorems and quantitative Borel--Cantelli lemmas. We give simpler proofs and improve some of these results. Two classes of examples are given: firstly stationary Gaussian sequences with applications to upper and lower classes and the law of the iterated logarithm for subsequences, and secondly in Diophantine approximation relatively to Gál and Schmidt's theorems.