An intractability result for multiple integration

Our aim is to show that in the worst case setting the integration problem is intractable. The implications of the intractability result for lattice methods are considered briefly in Section 3

CORRECTION TO "QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND”

ISSN:1446-1811ISSN:1446-873

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube $[0,1]^s$. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost $1/N$, where $N$ is the number of points, independently of dimension) to so-called “product and order dependent†(POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets. doi:10.1017/S144618111200007

Wavelet Methods in the Relativistic Three-Body Problem

In this paper we discuss the use of wavelet bases to solve the relativistic three-body problem. Wavelet bases can be used to transform momentum-space scattering integral equations into an approximate system of linear equations with a sparse matrix. This has the potential to reduce the size of realistic three-body calculations with minimal loss of accuracy. The wavelet method leads to a clean, interaction independent treatment of the scattering singularities which does not require any subtractions.Comment: 14 pages, 3 figures, corrected referenc

Testing the proposed link between cosmic rays and cloud cover

A decrease in the globally averaged low level cloud cover, deduced from the ISCCP infra red data, as the cosmic ray intensity decreased during the solar cycle 22 was observed by two groups. The groups went on to hypothesise that the decrease in ionization due to cosmic rays causes the decrease in cloud cover, thereby explaining a large part of the presently observed global warming. We have examined this hypothesis to look for evidence to corroborate it. None has been found and so our conclusions are to doubt it. From the absence of corroborative evidence, we estimate that less than 23%, at the 95% confidence level, of the 11-year cycle change in the globally averaged cloud cover observed in solar cycle 22 is due to the change in the rate of ionization from the solar modulation of cosmic rays

Growth and Structure of Stochastic Sequences

We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences leads to a wide variety of behaviors ranging from stretched exponential to log-normal to algebraic growth. Interestingly, the set of all possible sequence values has an intricate structure.Comment: 4 pages, 4 figure