Sequences—Basic elements for discrete mathematics

Sequences are fundamental mathematical objects with a long history in mathematics. Sequences are also tools for the development of other concepts (e. g. the limit concept), as well as tools for the mathematization of real-life situations (e. g. growth processes). But, sequences are also interesting objects in themselves, with lots of surprising properties (e. g. Fibonacci sequence, sequence of prime numbers, sequences of polygonal numbers). Nowadays, new technologies provide the possibility to generate sequences, to create symbolic, numerical and graphical representations, to change between these different representations. Examples of some empirical investigation are given, how students worked with sequences in a computer-supported environment.     

Hooke numbers of polymers

Hooke numbers He ≡ σ_b/(E) are calculated from published ultimate tensile strengths σ_b, tensile moduli E, and ultimate elongations ?_b. Data for common thermoplastics and natural fibers each follow a function He = [1 + (?_b/?_crit)^ab]^?1/b with a Hookean region I (He = 1) at ?_b ? ?_crit, a non-Hookean region III at ?_b ? _crit, and a transition region II for ?_b ≈ ?_crit. Only non-Hookean regions III were found for semisimultaneous interpenetrating networks from polyisobutylene-polymethyl methacrylate, thermoplastic elastomers from segmented polyamide-polyethers, molecular composites from poly(p-phenylene benzobisthiazole) and poly[2,5(6)-benzimidazole], and three groups of various synthetic fibers. The Hooke numbers of lyotropic and thermotropic liquid-crystalline polymers vary with the heat treatment and depend on orientation angles for orientation angles greater than ca. 10°. Hooke numbers much greater than 1 are observed for highly stressed polymers. ©1995 John Wiley & Sons, Inc.     

Zur potentiometrischen Bestimmung des Kupfers neben gro?en Eisenmengen, insbesondere in Kiesabbr?nden, mit Chrom(II)-salz

Zusammenfassung Zur Bestimmung des Kupfers in sehr eisenreichen Proben, insbesondere zur serienmäßigen Bestimmung in Pyritabbränden wird das potentiometrische Verfahren nach Zintl und Schloffer vorgeschlagen. Der Abbrand wird mit Salzsäure und KClO₃ gelöst, das Kupfer vom größten Teil des Eisens durch Zementierung mit Elektronmetall abgetrennt und das zementierte Kupfer nach Filtration und Verglühen des Filters durch Titration mit Chrom(II)-sulfat bestimmt. Die Genauigkeit des Verfahrens beträgt 1–2% relativ.Diese Untersuchung bildet ein Beispiel dafür, daß in der analytischen Chemie durchaus kein Mangel an akademischen Methoden besteht, daß aber in den meisten Fällen deren Anwendung auf die praktische Analyse nicht ohne weiteres möglich ist. Wir widmen diese Arbeit Wilhelm Geilmann, der in seinen Arbeiten stets die Verbindung von Methode und Praxis pflegte.Der früheren Deutschen Wirtschaftskommission, der jetzigen Staatlichen Plankommission der Deutschen Demokratischen Republik, Zentralamt für Forschung und Technik, danken wir für die Unterstützung dieser Arbeit.     

On the discrete logarithm in the divisor class group of curves

Let be a curve which is defined over a finite field of characteristic . We show that one can evaluate the discrete logarithm in by operations in . This generalizes a result of Semaev for elliptic curves to curves of arbitrary genus.

     

Early solvent vapor exit in GC for coupled LC-GC involving concurrent eluent evaporation

Use of early solvent vapor exits for concurrent eluent evaporation with the loop-type interface has two purposes: protection of the GC detectors from large amounts of solvent vapors and more efficient discharge of the vapors to accelerate eluent evaporation and help avoiding broad solvent peaks. Use of a retaining pre-column after the uncoated pre-column can rule out losses of solute materials that form sharp peaks.     

Extraction of effective material parameters with application to sandwich structures

In this contribution we present a first-order numerical homogenization approach which allows for extracting effective linear elastic properties of heterogeneous materials. The approach is based on the window or self consistency method where a representative microscopic subdomain is embedded into a window of effective properties. Since these properties are not known in advance they have to be determined iteratively. For the discretization of the micro structures we use the Finite Cell Method, which is a fictitious domain method of higher-order. It is very well suited for efficiently discretizing complicated geometries stemming, for example, from tomography (CT-scans). In the numerical examples we will investigate a bending test of a sandwich plate which is composed of a polymeric core with thin faceplates made of Aluminum. Firstly, effective properties of the core are extracted and then applied to a macroscopic numerical model. The numerical results are validated by experiments. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)