Über die Konvergenz des Relaxationsverfahrens bei nicht-negativen und diagonaldominanten Matrizen

Zusammenfassung In der folgenden Arbeit werden zunächst die Begriffe Gesamtschrittverfahren, Einzelschrittverfahren und Relaxationsverfahren allgemein formuliert und dann auf allgemeine lineare Gleichungssysteme angewandt. Im Spezialfall einer Matrix mit verschwindender Hauptdiagonale erhält man so die bekanntenJacobi-, Gauss-Seidel- und Relaxationsverfahren. Satz 1 macht eine Aussage über die Konvergenz des Einzelschrittverfahrens bei allgemeinen, nicht-negativen Matrizen. Der Beweis verläuft ähnlich wie in einem bereits 1948 vonStein undRosenberg [2] behandelten Spezialfall. Als Korollar ergibt sich eine Aussage über die Konvergenz des Relaxationsverfahrens bei nicht-negativen Matrizen. Es wird ferner der Satz 2 über die Konvergenz des Relaxationsverfahrens bei diagonaldominanten Matrizen beweisen.

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Summary In this paper we give a general definition what is meant by total-step-, single-step- and successive relaxation iterative method and we apply these concepts on systems of linear equations. In the special case of a matrix with zero diagonal entries we obtain the well knownJacobi-, Gauss-Seidel- and Relaxation iterative method. Theorem 1 gives conditions for the convergence of the singlestep-iterative method for general, non-negative matrices. The proof is similar to that given byStein andRosenberg in [2] (1948) for a special case. A corollary gives conditions for the convergence of the relaxation-iterative method for non-negative matrices. Further on we prove theorem 2 about the convergence of the relaxation-iterative method with diagonally dominant matrices.

     

Über die Nullstellen der ersten Ableitung von Besselfunktionen

Zusammenfassung Die Arbeit liefert einen Beitrag zur derLense'schen Vermutung über die Bewegung der Doppelnullstellen der AbleitungI′ _ν (z) der Besselfunktionen bei ver?nderlichem negativen ν.

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Summary This paper gives a contribution to the hypothesis ofLense regarding the movements of the second order zeros of the derivativesI′ _ν (z) of Bessel functions for every negativ variable ν.

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Mit 3 Textabbildungen

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HerrnJ. Lense zum 75. Geburtstag gewidmet.     

Ein Einschließungsverfahren für Nullstellen

Zusammenfassung Es wird ein Verfahren angegeben zur Bestimmung der Nullstellen einer Funktion bzw. eines Funktionensystems in einem beliebigen, vorgegebenen BereichB. Dabei sind keine Näherungswerte für die Nullstellen erforderlich. Es sind auch keine Vorkenntnisse über die Art und die Anzahl der inB gelegenen Nullstellen nötig. Das Verfahren beruht vielmehr auf einer geeigneten Abschätzung der Funktion bzw. des Funktionensystems. Es wird gezeigt, daß die einfache Intervallarithmetik für rationale Funktionen solche Abschätzungen liefert. Die Nullstellen werden durch obere und untere Schranken eingeschlossen. Dabei hängt die Genauigkeit von der Ziffernzahl der verwendeten Rechenanlage ab. Das Verfahren wurde an zahlreichen Beispielen erprobt. Einige davon sind angegeben.

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Summary In this paper we give a method for determining the zeros of a function or a system of functions in an arbitrary given regionB. Applying this method we need no approximations of the zeros. Further we need no preliminary knowledge of the order and the number of the zeros inB. Against this our method is based on a suitable estimation of the function or the system of functions. We show that the simple interval arithmetic for rational functions gives such estimations. As result the zeros are included by upper and lower bounds. The accuracy of the results depends only on the number of digits of the computer. The method was tested on many examples, some of which are given below.

     

Formalization and implementation of floating-point arithmetics

The paper is intended to show that floating-point arithmetic can be implemented in a way which leads to reasonable mathematical structures as described in chapters 5 and 6. It turns out for instance that all the rules of the minus-operator of the real numbers can be saved and that with respect to ≦ and ≧ inequalities can be manipulated as if they were real inequalities. These structures also occur in other fields of mathematics. They allow among others many theoretical considerations with floating-point arithmetics. Theorem 5.1 is the main result for the implementation. It reduces the structures to special properties of the rounding function. In chapter 3 these properties are derived as necessary conditions for an algebraic and order homomorphism between the real numbers and a floating-point system. Chapter 4 gives the algorithms for an implementation of floating-point arithmetics for all roudings of the set {?Δ□_μ,μ=0(1)b} (for definition see chapter 2) using short and long accumulators. It is an essential result that the whole implementation can be separated into fiveindependent steps as indicated in figure 2 and its context which means that an exchange of the rounding does not influence any other part of the algorithm. Although theorem 5.1 calls for monotone and antisymmetric roundings a computer is most flexibly equipped with the monotone downwardly directed rounding Δ since all roundings of the considered set can easily be generated by it and furthermore then also the directed roundings are available.     

Parameter spaces for the nucleation and the subsequent growth of cubic boron nitride films

The data existing in the literature about the deposition of cubic boron nitride thin films were reviewed critically in order to establish the parameter spaces of c-BN nucleation and growth. The ion energy E_i, the flux ratio F (=incoming ions/incoming boron atoms), the ion mass m_i, (or the ratio Ar/N₂, respectively), and the substrate temperature T_s, had already been identified as the decisive parameters which are, however, interdependent. Earlier data collections on c-BN deposition had shown that, irrespective of the deposition technique used, a well-defined c-BN region exists in the F/E_i parameter space, in which the deposition of c-BN is possible. Similar regions exist in the F/m_i and F/T_s parameter spaces. The present collection extends these older diagrams considerably, especially to the low energy region. From this extention it can be concluded that the momentum transfer concepts proposed in the literature fail to explain the data. Furthermore, the older collections were considered valid for nucleation and growth likewise. However, in recent years data have been published showing that the boundaries of the c-BN regions are different for nucleation and growth. After successful nucleation, subsequent growth can occur either at reduced ion bombardment (either energy or flux ratio or ion mass) and also at reduced temperatures. The existing data for this parameter reduction have been collected in this paper. It will be shown that the growth depends in a similar way as the nucleation on the (interdependent) ion bombardment parameters but no longer on temperature. This means that the nucleation and growth of c-BN are based on different, although in both cases ion-induced, mechanisms.     

Performance of mortars containing recycled fine aggregate from construction and demolition waste

Waste from construction and demolition accumulates in large quantities in the modern world. Recycled coarse aggregates derived from this waste can replace virgin aggregates used in the production of new concretes but the studies on the effect of using the fine fraction of this waste on the properties of new concrete have not yet led to clear conclusions. The present study evaluated the properties of recycled fine aggregates derived from two recycling plants using two different waste treatment procedures, as well as their effects on the properties of fresh and hardened mortars prepared using these aggregates at two water-to-cement ratios and three replacement ratios. It was found that the recycled aggregates were more porous than the natural aggregates and may have contained some organic matter. Setting times were longer when recycled aggregates replaced natural aggregates and strength and durability were reduced as well. Partial replacement of the fine aggregate is possible if an appropriate compensation of the water to cement ratio is applied.     

Grundlagen einer Maschinenintervallarithmetik

Zusammenfassung In der vorliegenden Arbeit wird eine in [1] angegebene Intervallarithmetik so erweitert, daß sie auch für Intervallfunktionen die exakten Ergebnisintervalle liefert. Es wird ferner die dieser Intervallarithmetik zugeordnete Maschinenintervallarithmetik angegeben, und es werden Eigenschaften dieser Arithmetik untersucht.

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Summary In the present paper we give an interval arithmetic, which is an extension of that given in [1]. It is one of the essential features of this new interval arithmetic, that in the case of interval functions we get the exact range of interval values. Further on we give a machine interval arithmetic corresponding to the new interval arithmetic and study properties of these arithmetics.

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Wir danken Herrn Prof.Nickel für zahlreiche Anregungen.     

Surface properties of differently prepared ultrananocrystalline diamond surfaces

Ultrananocrystalline diamond/amorphous carbon composite films have been deposited by microwave plasma chemical vapour deposition from 17% CH₄/N₂ mixtures at 600 °C. Thereafter the films were subjected to various treatments (plasma processes, UV/O₃ exposure) to obtain hydrogen, oxygen, and fluorine terminated surfaces, which then have been characterized with respect to their composition, roughness, wettability, and other properties. Among others, it will be shown that H- and F-terminated surfaces are very stable even if exposed for long time to air, while O-terminated ones are prone to contaminations. H- and O-termination can be patterned by applying the UV/O₃ treatment through a mask. Finally, it will be shown that a non-fouling poly(ethylene glycol) layer can be grafted directly on oxygen terminated surfaces by an atom transfer radical polymerization process using α-bromoisobutyryl bromide as an initiator.