The Microstructure of (t, m, s)-Nets

In an article of A. B. Owen (1998, J. Complexity14, 466–489) the question about the distribution properties of digital (t, m, s)-nets in small intervals was raised. We give upper and lower bounds for the maximum number of points of a (t, m, s)-net in these intervals and also provide a way of improving the distribution properties in some cases.     

Halton-type sequences from global function fields

For any prime power q and any dimension s≥1, a new construction of (t, s)-sequences in base q using global function fields is presented. The construction yields an analog of Halton sequences for global function fields. It is the first general construction of (t, s)-sequences that is not directly based on the digital method. The construction can also be put into the framework of the theory of (u, e, s)-sequences that was recently introduced by Tezuka and leads in this way to better discrepancy bounds for the constructed sequences.     

On the weighted dyadic diaphony of digital (t,s)-sequences

In this note, we present results on the weighted dyadic diaphony of digital (t, s)-sequences over ℤ₂. These results are closely linked to the worst-case error for quasi-Monte Carlo integration in certain spaces of functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The number of lattice rules

A lattice rule is a quadrature rule for integration over ans-dimensional hypercube that employsN abscissas located on a lattice, chosen to conform to certain specifications. In this paper we determine the numberv _s(N) of distinctN-points-dimensional lattice rules. We show that, in general,

Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions f with positive exponent of smoothness s > 0 given on a domain . The norms on these spaces are defined via integral norms of the difference of the function f of order m > s treated as a function of the point of the domain and of the difference increment. For an arbitrary domain , we characterize these spaces in terms of the local approximations of the function by polynomials of degree m – 1.     

On the mixed problem for hyperbolic partial differential-functional equations of the first order

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order where is a function defined by z _(x,y)(t, s) = z(x + t, y + s), (t, s) [–, 0] × [0, h]. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.     

Monotonicity and supermodularity results for the Erlang loss system

For the Erlang loss system with s servers and offered load a, we show that: (i) the load carried by the last server is strictly increasing in a; (ii) the carried load of the whole system is strictly supermodular on {(s,a)|s=0,1,… and a>0}.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Some Bipartite Ramsey Numbers

Consider a complete bipartite graph K(s, s) with p = 2s points. Let each line of the graph have either red or blue colour. The smallest number p of points such that K(s, s) always contains red K(m, n) or blue K(m, n) is called bipartite Ramsey number denoted by rb(K(m, n), K(m, n)). In this paper, we show that

Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration

A detailed study of digital (t, m, s)-nets and digital (T,s)-sequences constructed over finite rings is carried out. We present general existence theorems for digital nets and sequences and also explicit constructions. Special attention is devoted to the case where the finite ring is a residue class ring of the integers. This study is motivated by the problem of numerical integration of multivariate Walsh series by quasi-Monte Carlo methods, for which we also provide a general error bound.The third author was supported by the CEI Project PACT, WP5.1.2.1.3.     

A Graham-Sloane Type Construction for s-Surjective Matrices

We give a construction of (n–s)-surjective matrices with n columns over using Abelian groups and additive s-bases. In particular we show that the minimum number of rows ms _q(n,n–s) in such a matrix is at most s ^s q ^n–s for all q, n and s.     

Logarithmic Estimates for the Density of Hypoelliptic Two-Parameter Diffusions

We consider a hypoelliptic two-parameter diffusion. We first prove a sharp upper bound in small time (s, t)[0, 1]² for the L^p-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of2² log p_s, t(x, y), as ↓0, where x is the initial condition of the diffusion, = , and p_s, t(x, y) is the density of the hypoelliptic two-parameter diffusion.     

The second and the third smallest distances on the sphere

Let s₁ (n) denote the largest possible minimal distance amongn distinct points on the unit sphere . In general, let s_k(n) denote the supremum of thek-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s₂(n) = s₁([n/3]). This equality holds forn > n₀ however s₂(12) > s₁(4).We set up a conjecture for s_k(n), that one can always reduce the problem of thek-th minimum distance to the function s₁. We prove this conjecture in the casek=3 as well, obtaining that s₃(n) = s₁([n/5]) for sufficiently largen.The optimal construction for the largest second distance is obtained from a point set of size [n/3] with the largest possible minimal distance by replacing each point by three vertices of an equilateral triangle of the same size . If 0, then s₂ tends to s₁([n/3]). In the case of the third minimal distance, we start with a point set of size [n/5] and replace each point by a regular pentagon.     

Anticipative Girsanov transformations

Summary The transformations of measures induced by with (K _s ) possibly anticipating the Wiener process (W _s ) is discussed and a Girsanovtype theorem under rather weak assumptions on (K _s ) is derived.     

Sylvester Waves in the Coxeter Groups

A new recursive procedure of the calculation of partition numbers function W(s, d ^m ) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s, d ^m ) and their periods (G) for the irreducible Coxeter groups and a list for the first twelve symmetric group _m are presented. A least common multiple (m) of the series of the natural numbers 1,2,...,m plays a role in the period ( _m ) of W(s, d ^m) in _m .     

Stochastic Partial Differential Equations on Manifolds~II. Nonlinear Filtering

The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold of . The existence of the conditional density with respect to a given measure on is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series.     

Spaces of Functions of Fractional Smoothness on an Irregular Domain

In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions with positive exponent of smoothness s > 0, defined on a domain . For a domain G with specific geometric properties, we establish the embedding B _pq ^s (G) = L _pq ^s (G) L _q (G), 1 < p < q < , with the relationship between the parameters defined by these geometric properties.     

Elliptic Systems with Nonlinearities of Arbitrary Growth

In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations: where is a bounded domain in We assume that and the function f is superlinear and with no growth restriction (for example f(s) = s e^s); then the system has a nontrivial (strong) solution.     

Linear programming bounds for codes in infinite projective spaces

We develop a technique for improving the universal linear programming bounds on the cardinality and the minimum distance of codes in projective spaces . We firstly investigate test functions P_j(m,n,s) having the property that P_j(m,n,s)<0 for somej if and only if the corresponding universal linear programming bound can be further improved by linear programming. Then we describe a method for improving the universal bounds. We also investigate the possibilities for attaining the first universal bounds.     

The Convergence Rate of Block Preconditioned Systems Arising from LMF-based ODE Codes

The solution of ordinary an partial differential equations using implicit linear multi-step formulas (LMF)is considered. More precisely, boundary value methods (BVMs), a class of methods based on implicit formulas will be taken into account in this paper. These methods require the solution of large and sparse linear systems x = b. Block-circulant preconditioners have been propose to solve these linear systems. By investigating the spectral condition number of , we show that the conjugate gradient method, when applied to solving the normalize preconditioned system, converges in at most O(log s) steps, where the integration step size is O(1/s). Numerical results are given to illustrate the effectiveness of the analysis.This revised version was published online in October 2005 with corrections to the Cover Date.