Searching for extensible Korobov rules

Extensible lattice sequences have been proposed and studied in [F.J. Hickernell, H.S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, in: G.H. Golub, S.H. Lui, F.T. Luk, R.J. Plemmons (Eds.), Proceedings of the Workshop on Scientific Computing (Hong Kong), Singapore, Springer, Berlin, 1997, pp. 209–215; F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138; F.J. Hickernell, H.Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286–300]. For the special case of extensible Korobov sequences, parameters can be found in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C.Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138]. The searches made to obtain these parameters were based on quality measures that look at several projections of the lattice. Because it is often the case in practice that low-dimensional projections are very important, it is of interest to find parameters for these sequences based on measures that look more closely at these projections. In this paper, we prove the existence of “good” extensible Korobov rules with respect to a quality measure that considers two-dimensional projections. We also report results of experiments made on different problems where the newly obtained parameters compare favorably with those given in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138].     

非平稳NA序列部分和的精确渐近性

本文探讨了非平稳NA序列部分和的精确渐近性．以前的文献在讨论NA序列此类极限性质时都附加有强平稳条件的限制,这必然会给一些问题的研究带来不便．周知,非平稳NA序列在许多实际问题中是大量存在的,所以解除强平稳条件的束缚具有较大的理论和实际意义,这正是本文的目的之所在,同时本文也将已有的一些结果包含成为特殊情形．     

Long zero-free sequences in finite cyclic groups

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2 in the additive group Z_n of integers modulo n. The main result states that for each zero-free sequence  of length ?>n/2 in Z_n there is an integer g coprime to n such that if denotes the least positive integer in the congruence class ga_i (modulo n), then . The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than n/2, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.     

Two classes of piecewise-linear difference equations with eventual periodicity three

In this paper, we will study two classes of difference equations which are piecewise-linear and of similar forms. We will show that all nontrivial solutions of one equation are eventually periodic with prime period three. We will show this result for one case of the second equation.     

Sigma function solution of the initial value problem for Somos 5 sequences

The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.

     

Tight Heffter Arrays Exist for all Possible Values

A tight Heffter array is an matrix with nonzero entries from such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from appears twice. We prove that exist if and only if both m and n are at least 3. If H has the property that all entries are integers of magnitude at most , every row and column sum is 0 over the integers, and H also satisfies ), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if . Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both are even.     

Tangent cones and Dini derivatives

We prove a property of the Bouligand tangent cone to the epigraph (or to the graph) of a locally Lipschitz function. It is also shown how this result can be used in determining Dini sequences. Finally, some relationships between such a cone and Dini derivatives are provided.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

混合序列的强大数律

讨论了混合序列部分和的强收敛性,所得结果推广和改进了文献[1]、[2],并推广了Marcinkiewicz强大数律．     

Compound Poisson Approximation of the Number of Exceedances in Gaussian Sequences

Consider a finite sequence of Gaussian random variables. Count the number of exceedances of some level a, i.e. the number of values exceeding the level. Let this level and the length of the sequence increase simultaneously so that the expected number of exceedances remains fixed. It is well-known that if the long-range dependence is not too strong, the number of exceeding points converges in distribution to a Poisson distribution. However, for sequences with some individual large correlations, the Poisson convergence is slow due to clumping. Using Steins method we show that, at least for m-dependent sequences, the rate of convergence is improved by using compound Poisson as approximating distribution. An explicit bound for the convergence rate is derived for the compound Poisson approximation, and also for a subclass of the compound Poisson distribution, where only clumps of size two are considered. Results from numerical calculations and simulations are also presented.