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.

     

A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature

We consider the discretization in time of an inhomogeneous parabolicequation in a Banach space setting, using a representation ofthe solution as an integral along a smooth curve in the complexleft half-plane which, after transformation to a finite interval,is then evaluated to high accuracy by a quadrature rule. Thisreduces the problem to a finite set of elliptic equations withcomplex coefficients, which may be solved in parallel. The paperis a further development of earlier work by the authors, wherewe treated the homogeneous equation in a Hilbert space framework.Special attention is given here to the treatment of the forcingterm. The method is combined with finite-element discretizationin spatial variables.     

A periodic review production and maintenance model with random demand,deteriorating equipment,and binomial yield

In many environments, product yield is heavily influenced by equipment condition. Despite this fact, previous research has either focused on the issue of maintenance, ignoring the effect of equipment condition on yield, or has focused on the issue of production, omitting the possibility of actively changing the machine state. We formulate a Markov decision process model of a single-stage production system in which demand is random. The product yield has a binomial distribution that depends on the equipment condition, which deteriorates over time. The objective is to choose simultaneously the equipment maintenance schedule as well as the quantity to produce in a way that minimizes the sum of expected production, backorder, and holding costs. After proving some results about the structural properties of the optimal policy, numerical problems are used to compare this method to the typical approach of solving the maintenance and production problems sequentially. The results show that the simultaneous solution provides substantial gains over the sequential approach. In the cases studied, the proposed method resulted in an average cost savings of approximately 18%.     

The "overshoot" effect and sensory hearing impairment

The threshold for the detection of a brief tone masked by a longer-duration noise burst is higher when the tone is presented shortly after the onset of the noise than at longer delay times. This finding has been termed the "overshoot" effect [E. Zwicker, J. Acoust. Soc. Am. 37, 653-663 (1965)]. The present letter compared the size of the effect in the better and more impaired ear of six subjects with high-frequency unilateral or asymmetric hearing losses of sensory origin. Thresholds were measured for 5-ms 4-kHz tones presented 10, 200, and 390 ms after the onset of a 400-ms, 2- to 8-kHz noise burst. The better ear of each subject was tested using two noise levels, one equal in sound-pressure level and one equal in sensation level to that used for the impaired ear. Thresholds for all subjects and all ears decreased monotonically with increasing delay time, with the size of the effect typically 5 dB. Thus a small overshoot effect was observed regardless of hearing impairment.     

On Bateman's method for second kind integral equations

Summary Under suitable conditions, we prove the convergence of the Bateman method for integral equations defined over bounded domains inR ^d ,d1. The proof makes use of Hilbert space methods, and requires the integral operator to be non-negative definite. For one-dimensional integral equations over finite intervals, estimated rates of convergence are obtained which depend on the smoothness of the kernel, but are independent of the inhomogeneous term. In particular, for aC kernel andn reasonably spaced Bateman points, the convergence is shown to be faster than any power of 1/n. Numerical calculations support this result.     

Distribution of reaction products (theory). XI. H + F2

The availability for the first time of detailed rate constants k(V′, R′, T′) (where V′, R′ and T′ are product vibrational, rotational and translational excitation) for the highly exothermic reaction H + F₂ → HF(V′, R′) + F has prompted the 3D classical-trajectory study reported here. The potential-energy surface is found to be predominantly repulsive ($\text{A}$_⊥ ≈ 42%, R_⊥ ≈ 58%) corresponding to the rather low fractional conversion of reaction energy into vibration ((f′V) = 0.58 from experiment, and 0.56 from theory). In the homologous series of reactions H + X₂ (X  F, Cl, Br, I) the percentage of repulsive energy-release decreases for X  Cl, Br, I, but increases from X  F to Cl. It is shown that this cannot be due to charge in mass-combination, but can plausibly be explained by the anomolously short range of interaction between the separating X atoms in the case X  F. It is predicted that the more-forward scattered HF will be more rotationally excited. The form of the cross section function S_r(T) (where T is reagent translation) is analysed. In accordance with the expectation for a strongly exothermic reaction, it is found that S_r(T) rises more steeply than S_r(V) (where V is reagent vibrational energy). The effect on the product energy distribution conforms qualitatively to the “adiabatic” behaviour noted in previous work: ΔT → ΔT′ + ΔR′; ΔV → ΔV′. The explanation is to be found in reaction through more-compressed or more-extended intermediate configurations than are characteristic of room temperature reaction. We note the existence of an amplification effect: (ΔT′ + ΔR′)/ΔT ≈ 2, and ΔV′/ΔV ≈ 2.     

On the numerical evaluation of singular integrals

Product-integration rules of the form _–1 ¹ k(x)f(x)dx _i ⁼¹ⁿ w _ni f(x _ni ) are studied, with the points {w _ni } chosen to be the zeros of certain orthogonal polynomials, and the weights {w _ni } chosen to make the rule exact iff is any polynomial of degree less thann. If, in particular, the points are the Chebyshev points, and ifk L _p [–1, 1] for somep>1, then it is shown that the rule converges to the exact result for all continuous functionsf. With this choice of points, the practical application of the rule is shown to be straightforward in many cases, and to yield satisfactory rates of convergence. The casek(x)=|–x|, >–1, is studied in detail. Results of a similar, but weaker, kind are also obtained for other choices of the points {x _ni }.     

Recycling isoelectric focusing and isotachophoresis

A new apparatus for large scale preparative isoelectric focusing in free solution is described. Its main characteristic is rapid fluid recirculation with but short residues times in the electric field. For the first time the recycling principle was also applied to preparative isotachophoresis, by incorporating into the system a computer controlled leader counterflow. The rapid recirculation permits application of unusually high electrical power for rapid resolution, as it suppresses fluid flow instabilities arising from electrohydrodynamic effects.