Counterexamples in optimal quadrature

It is widely believed that order of exactness is a good measure of the quality of an algorithm for numerical quadrature. We show that this is not the case, by exhibiting a situation in which the optimal algorithm does not even integrate constants exactly. We also show that there are situations in which the penalty for using equidistant nodes is unbounded. Finally, we show that the complexity of obtaining an -approximation can be an arbitrary function of , i.e., there is no hardest quadrature problem.     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

On cubature formulae with a minimal number of knots

Summary In this paper an approach is outlined to the two-dimensional analogon of the Gaussian quadrature problem. The main results are necessary and sufficient conditions for the existence of cubature formulae which are exact for all polynomials of degree m and which have a minimal number of 1/2k(k+1) knots,k=[m/2]+1. Ifm is odd, similar results are due to I.P. Mysovskikh ([5, 6]) which will be derived in a new way as a special case of the general characterization given here. Furthermore, it will be shown how this characterization can be used to construct minimal formulae of even degree.     

Some positive cotes numbers for the chebyshev weight function

We show that, for the Chebyshev weight function (1–x ²)^–1/2, the Cotes numbers for the quadrature rule with nodes at the zeros of the ultraspherical polynomialP _n ^/() are nonnegative if and only if –1/2<1.     

Natural spline functions,their associated eigenvalue problem

Summary We consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots. We show that they are wherem is the order of the spline andn, the number of knots.This result is of particular interest to prove optimality properties of the Generalized Cross-Validation Method and had been conjectured by Craven and Wahba in a recent paper.     

Solvent-induced electronic spectral shifts: Benzene·Arn revisited

Just what can be learned about cluster dynamics (and, more generally, about solvation dynamics) from spectral studies of small clusters that have been doped with a chromophore is still an open question. In the present work we suggest a novel procedure for calculating the shift in the electronic absorption spectrum of a chromophore deriving from the attachment to or the incorporation in a cluster. The particular system of interest here is benzene·Ar _n , for which experimental results are readily available although their interpretation has been a point of controversy. In addition, since the present formalism is equally applicable to a chromophore isolated in a bulk phase (either liquid or solid), we are able to venture an explanation for the apparent observation that the spectral shift of cluster-isolated benzene does not approach the asymptotic values characteristic of the bulk-isolated species.     

Some identities for products and degree raising of splines

In this paper it is shown how the algebraic product of two spline functions, each represented in terms of B-splines, can again be represented as a linear combination of suitable B-splines. As a corollary to this result we obtain an explicit representation of a given B-spline function in terms of B-splines of some arbitrary higher degree. This generalizes some known results for raising the degree by one. Recurrence relations for both products and degree raising are established that may be useful for computation.Communicated by Larry L. Schumaker.     

A class of nonmixing dynamical systems with monotonic semigroup property

In order to illustrate the class of conservative dynamical systems for which a Boltzmann entropy can be obtained under finite coarse-graining [2], we consider dynamical systems defined by the shift transformation on K , where K is any finite set of integers. We give a class of non-Markovian invariant measures that verify the Chapman-Kolmogorov equation (equivalent to a Boltzmann entropy) for any positive stochastic matrix and that are ergodic but not weakly mixing.     

High gain amplification of femtosecond pulses with low amplified spontaneous emission in a multipass dye cell

We present a multipass femtosecond amplifier pumped by a Nd-YAG laser where the gain medium is given by a variable length dye cell. This system allows amplification factors up to 5 · 10⁶ and output pulse energies of 150 J, while restricting the contribution of amplified spontaneous emission to 0.5% of the total output energy, without using any saturable absorber stage. A detailed study of the output pulsewidth as a function of the duration of the input chirped pulse is also presented.     

Laser-induced electron emission from an Au surface irradiated by single picosecond pulses at λ=2.94 μm. The intermediate region between multiphoton and tunneling effects

The photoinduced electron emission from the surface of a solid gold target irradiated by single picosecond pulses of an erbium laser is investigated. The applied laser intensity (5–120 GW/cm²) corresponds to the intermediate interaction region between the pure multiphoton and tunnel effects, where the decisive Keldysh-parameter, , is in the range 1<<12=n ₀. In the light intensity region which is free of surface heating (I _L<80 GW/cm²), the slope of the measured logarithmic intensity dependence of the photocurrent decreases from the n ₀=12 perturbative value down to n 5. Therefore the experiment shows that the Keldysh-type theories, which have recently been proved to describe correctly the ionization of atoms, are also valid to a certain extent in the case of the photoeffect in metals.