Über substituierte 1,6-Dihydro-1,3,5-triazin-2,4-diamine, 1′,5′,6′,7′-Tetrahydrospiro[cyclopentan-1,4′-cyclopentapyrimidin]-2′(3′ H)-imine und das 6-Phenyl-2,4-pyrimidindiamin

Guanidine reacts with cyclohexanone, cycloheptanone, acetone and 3-pentanone, resp., in a molar ratio 2∶1 to give the 1,3,5-triazaspiro[5.5]undeca-and [5.6]dodeca-1,3-dien-2,4-diamines3 a and3 b resp. and the 6,6-dimethylresp. diethyl-1,6-dihydro-1,3,5-triazin-2,4-diamines3 d and3 e resp. On the contrary, action of guanidine on cyclopentanone yields not3 c, but the 1′,5′,7′-tetrahydrospiro[cyclopentane-1,4′-cyclopentapyrimidine]-2′(3′H)-imines2 c, 5 c and6 c resp., which are 1∶2- and 1∶3-condensates. Phenylacetone is transformed by guanidine (1∶2) to give 6-phenyl-2,4-pyrimidindiamine (8 f). The structure of the compounds cited is proved by NMR-, IR-, and (partially) mass spectra. The different courses of the formation of3 a, b, d, e, 2 c, 5 c and6 c resp. and8 f are also discussed. The structural formulae of some additional bases, which were synthesized from guanidine and cyclopentanone, 3-pentanone and phenylacetone resp. could not be established.     

A short note on Dedekind sums

We will give a new proof for the fact that the values of Dedekind sums are dense on the real line.     

A numerically stable algorithm for two server queue models

In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.     

On formal integration in composition rings of polynomials

This author wants to express his thanks to the School of Information Engineering at Teesside Polytechnic, Middlesbrough, England, for its support and hospitality during a visiting appointment of 3 months in 1989, when this paper was written.     

Minimum Weight Design for Structural Eigenvalue Problems by Optimal Control Theory*, †

The application of optimal control theory to minimum weight design of continuous one-dimensional structural elements subject to eigenvalue constraints is discussed. If not only the value of an eigenvalue is prescribed but also its position in the sequence of the ordered eigenvalues—for example, the critical buckling load of a column—the corresponding optimal control problem is shown to include necessarily all eigenvalues. Considering the unspecified eigenvalues as free parameters, necessary conditions for minimum weight design are derived. These conditions are compared with those obtained by use of variational methods. Attention is focused on the special case of multimodal solutions.     

Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H^s(T^d)$H^s\left({\mathbb{T}}^d\right)$ of fractional smoothness s>0$s>0$ for various equivalent norms including the classical one. The error is always measured in _L2(T^d)$L_2\left({\mathbb{T}}^d\right)$. Particular emphasis is given to the dependence of all constants on the dimension d$d$. We capture the exact decay rate in n$n$ and the exact decay order of the constants with respect to d$d$, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.     

Sign changes of Hecke eigenvalues of Siegel cusp forms of genus two

We shall show that the eigenvalues of a Hecke eigenform of integral weight and genus 2 not contained in the Maass space change signs infinitely often.

     

Traces of functions with a dominating mixed derivative in ℝ<Superscript>3</Superscript>

We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in ℝ³, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to p = 2. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.     

Über eine zum Kreisproblem verwandte Summe

An equivalent form of the classical circle problem is the unproved conjecture (3) . In this note the corresponding conjecture with _k (.) (the 1-periodic Bernoulli polynomials of orderk2) instead of ₁(.) is settled in an affirmative way (even with =0). Moreover, a generalized sum (6) is estimated by means of an analogue to a classical result due tovan der Corput.     

Interpolation on Sparse Grids and Tensor Products of Nikol'skij–Besov Spaces

We investigate the order of convergence of periodic interpolation on sparse grids (blending interpolation) in the framework of tensor products of Nikol'skij–Besov spaces. To this end, we make use of the uniformity of the considered tensor norms and provide a unified approach to error estimates for the interpolation of univariate periodic functions from Nikol'skij–Besov spaces.