Begleitfiguren und Invariantensystem minimaler Differentiationsordnung von Kurven im reellenn-dimensionalen affinen Raum

For such curves we know a moving frame and a complete system of invariants (affine curvatures) with respect to the group of volume-conserving transformations (unimodular group) of the order 2n?1 respectively 2n (see [1], p. 171). In this paper we study for a curvec a moving frame and a system of invariants of the minimal ordern+1, respectivelyn+2, by means of a curve with vanishing affine curvatures that has contact of maximal order withc. Forn=3 this is a result ofA. Winternitz ([1], p. 171 or [3], p. 86).     

On the speed of convergence in the ergodic theorem

Given any ergodic invertible measure preserving transformation τ of [0,1] and any null-sequence (α _N ) of positive reals, there exists a continuousf such that $$\lim \sup \alpha _{\rm N}^{ - 1} \left| {N^{ - 1} \sum\limits_{k = 0}^{N - 1} {f \circ \tau ^k - \smallint f} } \right| = \infty a. e.,$$ i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence.     

Über geschlossene Regelflächen im elliptischen Raum

On a ruled closed surface in the elliptic 3-space two integral invariants are considered: the aperture distance of a curve orthogonal to the generating lines of , and the aperture angle of an orthogonally circumscribed tangent surfaces. By means of these integral invariants and by considering certain ruled surfaces associated to one finds the geometric meaning of further integral invariants. If is generated by the binormals of a curve one obtains some properties of closed curves in the elliptic 3-space.     

Heuristische Verfahren zur Tourenplanung bei der Hausmüllentsorgung in ländlichen Regionen

Zusammenfassung Tourenprobleme bei der Hausmüllsammlung lassen sich in die Reihe der Lieferplanprobleme einordnen. Zur Lösung praktischer Größenordnungen werden hier einfache heuristische Verfahren (Prioritätsregeln) vorgestellt, weniger um der Vielzahl heuristischer Verfahren weitere hinzuzufügen als vielmehr dem Praktiker einen vielversprechenden Weg beim Lösen seiner Probleme aufzuzeigen.

---

Summary The discussed solid waste collection problem shows the properties of the so called truck dispatching problem. To solve problems for typical problem structure heuristic rules are pointed out in order to show the planner an efficient way of problem solving.

---

---

Zusammenfassung dreier Vorträge im 4. planungstechnischen Seminar Tourenplanung bei der Abfallbeseitigung am Institut für Siedlungswasserwirtschaft der Universität Karlsruhe am 1.10.1976.     

Kriterien in der Theorie der Gleichverteilung

It is the aim of this paper to introduce two new notions of discrepancy. They are defined by the formulas $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z e^2 \pi i\omega \left( n \right)} \right)} - f\left( 0 \right)} \right|, and \hfill \\ \delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z \omega \left( n \right)} \right)} \cdot z - \int\limits_0^z {f\left( \zeta \right)d\zeta } } \right|, \hfill \\ \end{gathered} $$ wheref is a holomorphic function defined in the unit disc withf ^(k) (0)≠0 for allk∈?,r<1 is a positive number, and ω is a sequence in [0, 1]. The first of these discrepancies can be generalized for multidimensional sequences. ω is uniform distributed if and only if lim _N→∞ Δ _N ^r (ω;f)=0 resp. lim _N→∞δ _N ^r (ω;f)=0. These results are proved in a quantitative way by estimating the classical discrepancyD _N (ω) by means ofΔ _N ^r (ω;f) and δ _N ^r (ω;f): $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Phi \left( {\Delta _N^r \left( {\omega ;f} \right)} \right), \hfill \\ \delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Psi \left( {\delta _N^r \left( {\omega ;f} \right)} \right). \hfill \\ \end{gathered} $$ The functions Φ and Ψ only depend onf andr. These estimations are based on the inequalities ofKoksma-Hlawka andErdös-Turán.     

Gradient corrections in density functional theory calculations for surfaces: Co on Pd{110}

Ab initio total energy calculations have been performed for CO chemisorption on Pd{110}. Local density approximation (LDA) calculations yield chemisorption energies which are significantly higher than experimental values but inclusion of the generalised gradient approximation (GGA) gives better agreement. In general, sites with higher coordination of the adsorbate to surface atoms lead to a larger degree of overbinding with LDA, and give larger corrections with GGA. The reason is discussed using a first-order perturbation approximation. It is concluded that this may be a general failure of LDA for chemisorption energy calculations. This conclusion may be extended to many surface calculations, such as potential energy surfaces for diffusion.     

The Comparison of the Orientation of Sodium 3-Hydroxy-2-Naphthalenecarboxylate in the Cavity of β-Cyclodextrin and Heptakis-(2,3,6-Tri-O-Methyl)-β-Cyclodextrin

Conformational analysis of inclusion complexes of sodium 3-hydroxy-2-naphthalenecarboxylate with -cyclodextrin and heptakis-(2,3,6-tri-o-methyl)--cyclodextrin in D₂O was investigated by 1D and 2D ¹HNMR measurements. The results show that part of the naphthyl group of sodium 3-hydroxy-2-naphthalenecarboxylate is situated in the 2,3-OH side of the -cyclodextrin cavity asymmetrically while the whole naphthyl group is included in the heptakis-(2,3,6-tri-o-methyl)--cyclodextrin cavity with the caboxylate and hydroxy group close to the 6-OCH₃ group.     

Fe(II) and Co(II) pyridinebisimine complexes bearing different substituents on ortho- and para-position of imines: synthesis, characterization and behavior of ethylene polymerization

A series of 2,6-bis(imino)pyridyl iron(II) and cobalt(II) complexes [2,6-(ArNCMe)₂C₅H₃N]MCl₂ (Ar = 2,6-i-Pr₂C₆H₃, M = Fe: 3a, M = Co: 4a; Ar = 2,4,6-i-Pr₃C₆H₂, M = Fe: 3b, M = Co: 4b; Ar = 2,6-i-Pr₂-4-BrC₆H₂, M = Fe: 3c, M = Co: 4c; Ar = 2,4-i-Pr₂-6-BrC₆H₂, M = Fe: 3d, M = Co: 4d) has been synthesized, characterized, and investigated as precatalysts for the polymerization of ethylene in the presence of modified methylaluminoxane (MMAO). The substituents of pyridinebisimine ligands and their positions located significantly influence catalyst activity and polymer property. It is found that the catalytic activities of the iron complexes/MMAO systems are mainly dominated by electronical effect, while those of the cobalt complexes/MMAO systems are primarily controlled by hindering effect.