Mean value estimates in lattice point theory

LetQ(u) be a positive definite quadratic form inr2 variables with a real symmetric coefficient matrix of determinantD. Given a real vectorb with 0b _j <1, forx>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)x, letV(x) be the volume of this ellipsoid andP(x)=A(x)–V(x). Let . By introduction of a parameter we shall show how the treatment of estimates onP(x) and onM(x) can be unified.     

Lexicographic quasiconcave multiobjective programming

Summary Letf _i :A R ben real-valued objective functions on a convex setA -K ^m ,K:=R orC, n, mN. Letg: A R ⁿ be defined by , where for eachxA, (i ₁ (x), ..., i _n (x)) is a permutation of (1, ...,n) such that . In this paper we treat the problem of findingx ^*A such that , wherel-max denotes the lexicographic maximum. If the f_i's are strongly quasiconcave we can reduce the problem stepwise until finally it is in the form of a scalar programming problem. Further, we consider conditions for the existence and uniqueness of a solution and discuss the relationship of the problem to the vector maximum (i.e. Pareto) and maxmin (i.e. Chebychev) problems.

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Zusammenfassung f _i :AR seienn reellwertige Zielfunktionen über einer konvexen MengeA-K ^m ,K:=R oderC, n, mN. g:AR ⁿ sei definiert durch , wobei für jedesxA (i ₁ (x), ... i _n (x)) eine Permutation von (1, ...,n) derart ist, daß Wir betrachten das Problem, einx ^*A so zu finden, daß , wobeil-max das lexikographische Maximum bedeute. Falls dief _i stark quasikonkav sind, läßt sich das Problem stufenweise reduzieren, bis es schließlich die Gestalt eines skalaren Optimierungsproblems annimmt. Wir geben Existenz- und Eindeutigkeitsbedingungen an und besprechen Zusammenhänge mit dem Vektormaximumproblem (d.h. Pareto-Optimierung) und dem Maxmin-Problem (d.h. Tschebyscheff-Optimierung).

     

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.     

Studies of the electrochemical behavior of epinephrine at a homocysteine self-assembled electrode

The self-assembled electrode with the homocysteine monolayer (Hcy/Au) has been characterized by infrared spectroscopy and ac impedance spectroscopy in electrolyte. The Hcy/Au electrode is demonstrated to promote the electrochemical response of epinephrine (E) by cyclic voltammetry. A pair of well-defined redox waves was obtained and the calculated standard rate constant (k_s) is 2.1×10⁻² cm s⁻¹ at the self-assembled electrode. The reduction peak of E can be used to determine the concentration of E in presence of ascorbic acid (AA) owing to the Hcy/Au also promoting the electrochemical oxidation of AA.     

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.     

Reactivity of organolanthanide and organolithium complexes containing the guanidinate ligands toward isocyanate or carbodiimide: synthesis and crystal structures

The direct reactions of (C5H5)2LnCl with LiN=C(NMe2)2 proceeded at room temperature in THF under pure nitrogen to yield the lanthanocene guanidinate complexes [(C5H5)2Ln(mu-eta1:eta2-N=C(NMe2)2)]2 (Ln = Gd (1), Er (2)). Treatment of phenyl isocyanate with complexes 1 and 2 results in monoinsertion of phenyl isocyanate into the Ln-N(mu-Gua) bond to yield the corresponding insertion products [(C5H5)2Ln(mu-eta1:eta2-OC(N=C(NMe2)2)NPh)]2 (Ln = Gd (3), Er (4)), presenting the first example of unsaturated organic small molecule insertion into the metal-guanidinate ligand bond. Further investigations indicate that N,N'-diisopropylcarbodiimide does not react with complexes 1 and 2 under the same conditions; however, it readily inserts into the lithium-guanidinate ligand bond of LiN=C(NMe2)2. As a synthon of the insertion product Li[(iPrN)2C(N=C(NMe2)2)], its reaction with (C5H5)2LnCl gives the novel organolanthanide complexes containing the guanidinoacetamidinate ligand, (C5H5)2Ln[(iPrN)2C(N=C(NMe2)2)] (Ln = Yb (5), Er (6), Dy (7)). All complexes were characterized by elemental analysis and spectroscopic properties. The structures of complexes 1, 3, 5 and 7 were determined through X-ray single-crystal diffraction analysis.     

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