Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

有限元插值误差的下界估计

基于一维区域上的拟一致剖分,证明了线性元插值误差的最优下界估计.基于此并利用超收敛理论,我们得到了有限元离散误差的上、下界.     

Identities for Anderson generating functions for Drinfeld modules

Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic $L$ -series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.     

Appropriate problems for learning and for performing—an issue for teacher training

Selecting, modifying or creating appropriate problems for mathematics class has become an activity of increaing importance in the professional development of German mathematics teachers. But rather than asking in general: “What is a good problem?” there should be a stronger emphasis on considering the specific goal of a problem, e.g.: “What are the ingredients that make a problem appropriate for initiating a learning process” or “What are the characteristics that make a problem appropriate for its use in a central test?” We propose a guiding scheme for teachers that turns out to be especially helpful, since the newly introduced orientation on outcome standards a) leads to a critical predominance of test items and b) expects teachers to design adequate problems for specific learning processes (e.g. problem solving, reasoning and modelling activities).     

离散时间Markov链几何非常返与代数非常返的判别准则

本文研究可数状态空间离散时间Markov链的几何非常返和代数非常返,利用某状态末离时的矩条件和某方程解的存在性,给出两种非常返性的判别准则.进一步,我们将所得结果应用于研究Geom/G/1排队模型的随机稳定性.     

Asymptotics for a resonance-counting function for potential scattering on cylinders

We study certain resonance-counting functions for potential scattering on infinite cylinders or half-cylinders. Under certain conditions on the potential, we obtain asymptotics of the counting functions, with an explicit formula for the constant appearing in the leading term.     

Jostling for position: local improvement for irregular cutting patterns

This paper introduces a new improvement heuristic for irregular cutting and packing problems. The method is based on a small number of repetitions of any leftmost placement policy and is particularly effective in situations where computation time is strictly limited but exceeds that required for a single pass approach. Both the algorithm and the geometry required for implementation are described in full and the results of computational experiments on a variety of data are presented. These results show that the algorithm is an effective technique for producing good packings.     

A Simulation Model for Planning Services for Renal Patients in Europe

A discrete-event simulation model is being developed to provide information for national and regional health planners about the effect of different treatment policies and practices on the increasing numbers of patients with irreversible kidney failure. The simulation program, written in Pascal using Pascal-SIM, describes the activities of individual patients whose progress through treatment depends on their characteristics and treatment history, as well as on treatments availability. Programming techniques have been developed to cope with the multiple scheduling of patient entities and the resultant queue-management problems. In order to provide simulation runs on request, for any population group, appropriate data is selected from the European Dialysis and Transplant Association Registry data base for use in the simulation program. Results are shown on a computer screen while the simulation is running, and are saved for further analysis to provide feedback to the requesting organization.     

An Interactive Algorithm for Vehicle Routeing for Winter — Gritting

In winter, when roads may become dangerously slippery due to frost, ice or snow, a de-icing agent (usually salt) is spread on them by a local authority for safety reasons. A gritter only needs to travel once down all those roads requiring treatment, as it can spread the salt onto both sides of the carriageway. The problem studied is how to design routes for gritters which will minimise costs. This problem is a type of Capacitated Arc Routeing Problem including consideration of multiple depot locations, limited vehicle capacities, time constraints on when roads must be gritted, roads with different priorities for gritting, the existence of one-way roads and salt-refilling locations. The objective function to be optimised depends on both the total distance travelled and the number and capacity of the gritters. A heuristic algorithm is devised with a computer program which allows user-interaction, and provides a practical tool for planning gritter routes. The model is linked to a GIS containing information on the road network for the County of Lancashire. Test results from the interactive algorithm are found to outperform another existing approach which solves the same problem.     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

An inequality for Steklov eigenvalues for planar domains

Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Graphical Diagnostics for Markov Models for Categorical Data

Markov models are widely used as a method for describing categorical data that exhibit stationary and nonstationary autocorrelation. However, diagnostic methods are a largely overlooked topic for Markov models. We introduce two types of residuals for this purpose: one for assessing the length of runs between state changes, and the other for assessing the frequency with which the process moves from any given state to the other states. Methods for calculating the sampling distribution of both types of residuals are presented, enabling objective interpretation through graphical summaries. The graphical summaries are formed using a modification of the probability integral transformation that is applicable for discrete data. Residuals from simulated datasets are presented to demonstrate when the model is, and is not, adequate for the data. The two types of residuals are used to highlight inadequacies of a model posed for real data on seabed fauna from the marine environment.

Supplemental materials, including an R-package RMC with functions to perform the diagnostic measures on the class of models considered in this article, are at the journal’s website. The R-package is also available at CRAN.     

Error bounds for spline-based quadrature methods for strongly singular integrals

For the numerical evaluation of finite-part integrals with singularities of order p ⩾ 1, we give error bounds for quadrature methods based on spline approximation. These bounds behave in the same way as the optimal ones. The ideas of the proof are also useful for methods based on other approximation processes.     

Hypercontractivity for log-subharmonic functions

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on Rn and different classes of measures: Gaussian measures on Rn, symmetric Bernoulli and symmetric uniform probability measures on R, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for any symmetric measure on R. A log-Sobolev inequality (LSI) is deduced from the (SHC) for compactly supported measures on Rn, still for log-subharmonic functions. An analogous (LSI) is proved for Gaussian measures on Rn and for other measures for which we know the (SHC) holds. Our log-Sobolev inequality holds in the log-subharmonic category with a constant smaller than the one for Gaussian measure in the classical context.     

Method for solving the Boltzmann kinetic equation for polyatomic gases

A method is proposed for computing the collision operator of a generalized Boltzmann kinetic equation with allowance for energy transfer from translational to vibrational or rotational degrees of freedom. The collision operator is computed using a projection method on a uniform velocity grid. The operator satisfies the mass, momentum, and energy conservation laws and vanishes for an equilibrium velocity distribution function. Approximate models are suggested that provide savings on the computation of rotational-translational relaxation. Numerical examples are presented.     

Hyperplane method for reachable state estimation for linear time-invariant systems

A numerical algorithm is presented for generating inner and outer approximations for the set of reachable states for linear time-invariant systems. The algorithm is based on analytical results characterizing the solutions to a class of optimization problems which determine supporting hyperplanes for the reachable set. Explicit bounds on the truncation error for the finite-time case yield a set of so-called -supporting hyperplanes which can be generated to approximate the infinite-time reachable set within an arbitrary degree of accuracy. At the same time, an inner approximation is generated as the convex hull of points on the boundary of the finite-time reachable set. Numerical results are presented to illustrate the hyperplane method. The concluding section discusses directions for future work and applications of the method to problems in trajectory planning in servo systems.This research was supported in part by Digital Equipment Corporation through the American Electronics Association Fellowship Loan Program and by the National Science Foundation under Grant No. ECS-84-04607.     

Credibility premiums for the zero-inflated Poisson model and new hunger for bonus interpretation

The purpose of this paper is to explore and compare the credibility premiums in generalized zero-inflated count models for panel data. Predictive premiums based on quadratic loss and exponential loss are derived. It is shown that the credibility premiums of the zero-inflated model allow for more flexibility in the prediction. Indeed, the future premiums not only depend on the number of past claims, but also on the number of insured periods with at least one claim. The model also offers another way of analysing the hunger for bonus phenomenon. The accident distribution is obtained from the zero-inflated distribution used to model the claims distribution, which can in turn be used to evaluate the impact of various credibility premiums on the reported accident distribution. This way of analysing the claims data gives another point of view on the research conducted on the development of statistical models for predicting accidents. A numerical illustration supports this discussion.     

Coerciveness Estimate for Ventcel Boundary Value Problem for a Differential Equation

We give some simple algebraic conditions on the coefficients of a boundary value problem for a differential equations of Ventcel type, depending on a spectral parameter, which guarantee the existence, uniqueness of a solution and coerciveness estimate, for a spectral parameter lying in some sector.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.