Maße für gerichtete Geraden und nicht-symmetrische Pseudometriken in der Ebene II

We continue our investigations started in a paper with the same title. The -length of a curve defined in this paper can be written as an integral along the curve. Furthermore, we determine necessary and sufficient conditions for the existence of a measure of orientated lines which generates a given non-symmetric pseudo-metric.

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Herrn Prof. Dr. H.-J. Kanold zum 65. Geburtstag gewidmet     

Ein asymptotischer Ausdruck für eine zahlentheoretische Funktion

Arithmetic formulae and an asymptotic expression for the order of magnitude of the arithmetic function ged (s, n) and of a generalisation of it are derived.     

Der quadratfreie Kern der Eulerschen φ-Funktion

It is shown by a simple induction that any square-free natural number is the square-free kernel of Euler's -function of infinitely many natural numbers. Furthermore every square-free natural number >1 is the square-free kernel of Euler's -function of infinitely many square-numbers.     

A variational proof of the Gauss-Bonnet formula

The classical Gauss-Bonnet formula has the form I(g_ij)=2, where I(g_ij) represents a sum of three terms each of which depends on the metric tensor g_ij. It is shown that the first variation I of I(g_ij) with respect to the metric g_ij vanishes and that for the Euclidean metric _ij we have I(_ij)=2. From this the formula I(g_ij)=2 follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(g_ij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor g_ij and its derivatives up to the second order is obtained with the aid of results of Rund [1].     

Spezifisches Verfahren zur quantitativen CO2-Bestimmung in natürlichen Wässern

The direct determination of CO₂ is often faulty because of certain contents of the natural water. Due to this reason a specific method was worked out. The water to be analysed and distilled water are placed side by side in a gas-tight vessel with an atmosphere free of CO₂, whereby mixing is avoided but gas-exchange is possible. After adjustment to the gas-equilibration the CO₂ content of the distilled water is determined e.g. by titration, measuring of conductivity or other methods, from where the CO₂ content in the natural water is calculated.     

Carbonyldinitrosyltris(fluorosulfato)tungstate(II) and ‐Molybdate(II) Anions: Synthesis,Spectroscopy, and Density Functional Theory Calculations

Carbonyldinitrosyltris(fluorosulfato)tungstate(II) and ‐molybdate‐(II) anions, [fac‐M(CO)(NO)₂(SO₃F)₃]^? (M=W, Mo), which are novel weakly coordinating anions that contain a metal carbonyl/nitrosyl moiety, have been generated in fluorosulfonic acid and completely characterized by multinuclear NMR, IR, and Raman spectroscopy as well as ESI mass spectrometry. ESI MS measurements performed for the first time on a superacidic solution system unambiguously reveal the formation of the monoanionic, mononuclear W and Mo complexes formulated as [M(CO)(NO)₂(SO₃F)₃]^? (M=W, Mo). Multinuclear NMR spectroscopic studies at natural abundance and ¹³C and ¹⁵N enrichment clearly indicate the presence of one CO ligand, two equivalent NO ligands, and two types of nonequivalent SO₃F^? groups in a 2:1 ratio. The IR and Raman spectra reveal that the two equivalent NO ligands have a cis conformation, thus indicating a fac structure. Density functional calculations at the B3LYP level of theory predict that these anions have a singlet ground state (¹A′) with a C_s symmetry along with C–O and N–O vibrational frequencies that are in agreement with the experimental observations. Mulliken population analysis shows that the monovalent negative charge is dispersed on the bulky sphere, the surface of which is covered by all the negatively charged O and F atoms with charge densities much lower than SO₃F^?, suggesting that [fac‐M(CO)(NO)₂(SO₃F)₃]^? (M=W, Mo) are weakly nucleophilic and poorly coordinating anions.     

The design of an irradiator for the continuous processing of liquid latex

This paper presents anew design concept for a gamma irradiation plant for the continuous processing of pumpable liquids. Typical applications of such a plant include

• * the irradiation vulcanisation of natural latex rubber

• * disinfection of municipal sewage sludge for agricultural use

• * sterilisation of liquids in the pharmaceutical and cosmetics industries

• * industrial processing of bulk liquids

The authors describe the design and operation of the latex irradiator now operating on a small production scale in Malaysia and proposed developments.

The design allows irradiation processing to be carried out under an inert or other gaseous environment.

State-of-the-art computer control system ensures the fully automatic processing operation needed by industrial computers.     

Phase separation of quaternary solubilized solutions or micro emulsion of hydrocarbons containing sodium oleate + phenol + benzene + water by freezing point measurements

Summary The depressions in freezing point measurements in quaternary solubilized solutions or micro-emulsion in sodium oleate + phenol + benzene + water are measured. Maxima and minima were observed, Gibbs' phase rule was applicable, eutectic points were detected, congruent melting points were obtained, loose combination of molecules or clusters, were formed, phases of the system transferred and finally unstable emulsion separated out. This is contrary to the views ofMcBain who called these solutions as thermodynamically stable. The results were compared with the conductivity, stability and heat of solubilization-concentration measurements.

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Zusammenfassung Es wird die Gefrierpunktserniedrigung von quaternären solubilisierten Lösungen im System Natriumoleat-Phenol-Benzol-Wasser gemessen. Die verschiedenen Befunde (Maxima und Minima des Gefrierpunktes in Abhängigkeit von der Zusammensetzung, Anwendbarkeit der Gibbs'schen Phasenregel, Auftreten von Eutektika, kongruente Schmelzpunkte etc.) dienen zum Beweis, daß diese Lösungen thermodynamisch instabil sind.

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With 5 figures and 1 table     

Folded chain crystals as micro-phases

Summary A thermodynamic treatment of homo-polymer systems out of linear chains with folded chain crystals is developed outgoing from appropriate models for single component systems. An expansion of thermodynamics to multi-micro-phase systems the structure of which is partially or totaly frozen is indispensable. General properties of melt crystallized homopolymers with folded chain crystals can be recognized indeed when the thermodynamic formalisms developed are applied.

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Zusammenfassung Das Schmelzen in polymeren Einteilchensystemen mit Faltungskristallen einheitlicher Dicke kann thermodynamisch als Umwandlung 1. Ordnung in einer Richtung behandelt werden, wenn die Faltungslänge bis zur Umwandlungstemperatur konstant bleibt (Faltungslänge als innerer Zusatzparameter). Eine wesentliche begriffliche Erweiterung ist für eine phänomenologische Beschreibung mit den Mitteln der Thermodynamik unumgänglich, wenn eine Faltungskristallit-Dickenverteilung existiert, weil dann prinzipiell nur noch partielle Koexistenz bestimmter Fraktionen metastabiler autonomer Mikrophasen mit der Schmelze möglich ist. Partielles Aufschmelzen und Rektistallisation können so dann auch in Betracht genommen werden. Die entwickelten Konzeptionen bewähren sich in der Anwendung auf bekannte Experimente.

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Notation g ^c (y);g ^m (Y) molar Gibbs-free energy of a chain of a lengthy within an extended chain crystal and the melt rsp - g _o ^c ;g _o ^m molar free enthalpy of the unit in the crystal lattice and the melt rsp - g(y,y, f) molar Gibbs-function of an ideally folded chain crystal with the fold heighty _f - gco(y, y _ef,y _f) molar free enthalpy of the crystal corey _co - g ₀ ^ex ((y_ef) excess free enthalpy of the longitudinal layers of folded chain crystals - g ^f(y_ef,g _o ^ex ) molar free enthalpy of the longitudinal layers of the folded chain crystals - g ^tot molar free enthalpy of a chain of the lengthy within a folded chain crystal with longitudinal layers - h _o ^1c ,h _o ^m molar enthalpy of the chain unit within the crystal lattice and the melt rsp - h =h _o ^m -h _o ^c molar heat of fusion of the unit - C _p=C _p ^m -C _p ^c difference of the molar specific heat of a unit within the melt and within the chain crystal - h ^D molar defect enthalpy of local defects within the crystal lattice - h ^D molar defect enthalpy of the unit - s _o ^c ,s _o ^m molar entropy of the chain unit within the crystal lattice and the melt rsp - s _c ^m conformational entropy of a chain in the melt - s _gk conformational entropy of a chain of lengthy within a super-lattice as indicated in figure 5, - s molar entropy of fusion of the melt - s _n ^c nematic configurational entropy - T absolute temperature - T _M melting temperature of extended chain crystals of infinite size - T _M(y) melting temperature of extended chain crystals containing only chains of the lengthy - T _M (y, y _f) melting temperatureof folded chain crystals of the thicknessy _f composed of chains of the lengthy - T _M(y _f) melting temperature of folded chain crystals of the thicknessy _fy - _eh excess free enthalpy of the chain ends occupying crystallographic places - _ef excess free enthalpy of a single fold loop - z coordination number of the lattice - 7 Euler's constant - R Boltzmann's constant - y number of chain units - y _f height of lamelliform folded chain crystals - f=(y/y _f - 1) number of fold loops of a chain of a lengthy when being built into a folded chain crystal of the thicknessy _f - y _co thickness of the crystal core of the simplified twophase model - y _et average thickness of the surface layers of folded chain crystals - N _c number of crystallized units of a chain of the lengthy - x _c molar number of crystallized units of a chain of the lengthy - x _nc molar number of noncrystallized units - excess free enthalpy parameter - (y _f) thickness distribution of the fold heightsy _f With 15 figures and 2 tables