Kupplungsreaktionen von Diazoniumbetainen mit 1,6-Methano[10]-annulenderivaten und ihre Cyclisierungsreaktionen zu Heterocyclen

Summary Coupling reactions of the diazoniumbetaines2, 4, 6a, 6b, 6c with derivatives of 1,6-methano[10]annulene1a–c and/or -naphthol7 yield the dyes3a–c, 5a, 5c, 8a–c, 9, 11. The syntheses of different triazine derivatives12a–c, 13a–c, 15, 16, and17a, 17c are described.

Herrn Kollegen Gerhard Schwenker mit den besten Wünschen zum 65. Geburtstag gewidmet     

Search

This paper reviews the theory of search. The main viewpoint is statistical but considerable space is spent on connections with other fields such as logic and optimisation. Three primitive ideas dominate: consistency, entropy, and Bayes' methods. The theory of screening provides one of the central examples and there is a section on nonsequential random search deriving from the work of Renyi. The game of Bulls and Cows provides a motivating example for both sequential and nonsequential algorithms.     

KT-nearfields of rank 2

Summary Nearfields of rank 2 over their kernel which are not Dickson nearfields have recently been constructed by H. Zassenhaus. Here it is shown that KT-nearfields of rank 2 over their kernels and of characteristic 2 are necessarily Dickson nearfields which are coupled to quadratic field extensions and their Dickson group is isomorphic to ₂. Using results of Gröger one obtains all KT-nearfields of rank 2 over the field of rational numbers.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

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.     

Problems with the ζ-function method

The -function method for evaluating effective actions is applied to an operator containing a potential localized on two parallel planes. This operator is characterized by a continuous spectrum and broken translation invariance. In this case, the -method leads to a divergent volume factor independent of the physical parameters. In a suitable regularization scheme, only the next to leading order term reproduces a physically interesting result.     

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.     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

On the monotonicity of sequences of quadrature formulae

Summary LetI(f)L(f)= _k=0 ^r ₌₀ ^vk–1 a _k f ⁽⁾(X _k ) be a quadrature formula, and let {S _n (f)} _n=1 be successive approximations of the definite integralI(f)= ₀ ¹ f(x)dx obtained by the composition ofL, i.e.,S _n(f)=L( _n ), where .We prove sufficient conditions for monotonicity of the sequence {S _n (f)} _n=1 . As particular cases the monotonicity of well-known Newton-Cotes and Gauss quadratures is shown. Finally, a recovery theorem based on the monotonicity results is presented