Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

Necessary and sufficient conditions for the existence of positive solutions to algebraic Riccati equations with indefinite quadratic term

Necessary and sufficient conditions are established in this paper for the existence of positive- and/or negative-definite solutions to the algebraic Riccati equation with indefinite coefficient. An iterative procedure is also given for computing such a solution.Project supported by the National Science Foundation of China and by the special program of the State Education Commission of China under grant 9033507.     

On the Interaction of Two Finite-Dimensional Quantum Systems

Interaction of quantum system S ^a described by the generalised × eigenvalue equation A| _s =E _s S ^a | _s (s=1,...,) with quantum system S ^b described by the generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) is considered. With the system S ^a is associated -dimensional space X ^a and with the system S ^b is associated an n-dimensional space X _n ^b that is orthogonal to X ^a . Combined system S is described by the generalised (+n)×(+n) eigenvalue equation [A+B+V]| _k = _k [S ^a +S ^b +P]| _k (k=1,...,n+) where operators V and P represent interaction between those two systems. All operators are Hermitian, while operators S ^a ,S ^b and S=S ^a +S ^b +P are, in addition, positive definite. It is shown that each eigenvalue _{k i} of the combined system is the eigenvalue of the × eigenvalue equation . Operator in this equation is expressed in terms of the eigenvalues _i of the system S ^b and in terms of matrix elements _s |V| _i and _s |P| _i where vectors | _s form a base in X ^a . Eigenstate | _k ^a of this equation is the projection of the eigenstate | _k of the combined system on the space X ^a . Projection | _k ^b of | _k on the space X _n ^b is given by | _k ^b =( _k S ^b –B)^–1(V– _k P})| _k ^a where ( _k S ^b –B)^–1 is inverse of ( _k S ^b –B) in X _n ^b . Hence, if the solution to the system S ^b is known, one can obtain all eigenvalues _{k i} } and all the corresponding eigenstates | _k of the combined system as a solution of the above × eigenvalue equation that refers to the system S ^a alone. Slightly more complicated expressions are obtained for the eigenvalues _{k i} } and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.     

Richtigkeit bei der Bestimmung von Spurenelementen in Lebensmitteln mit der Graphitrohrtechnik AAS

Ohne Zusammenfassung

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Accuracy of the graphite furnace AAS for trace determinations in foodstuffs

     

Potentiometrische Titration von Sulfat, Barium und Strontium mit Hilfe einer bleisensitiven Elektrode

Zusammenfassung Die bleisensitive Elektrode der Fa. Orion spricht neben Bleiionen auch auf Sulfationen an. Dies ermöglicht die potentiometrische Titration von Sulfat mit Bariumperchlorat sowie die Titration von Barium und Strontium mit Natriumsulfat. Der günstigste pH-Bereich liegt zwischen 5 und 7. Titriermedium sind Dioxan-Wasser-Gemische. Die Verfahren sind sehr empfindlich und erlauben in allen Fällen die Titration bis herab zu 5·10^–4 mMol.Anwendungsbeispiele für die Bestimmung von Schwefel in organischen Verbindungen nach Schöniger- oder Wickbold-Verbrennung sind angegeben.

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Potentiometric titration of sulphate, barium and strontium by means of a lead-sensitive electrode

The Orion lead electrode was found to be responsive not only to lead ions but to sulphate ions, too. This offers possibilities to titrate sulphate with barium perchlorate and reversely barium and strontium with sodium sulphate. The useful pH range is 5–7. Dioxan-water mixtures serve as solvents. The described methods are highly sensitive and allow determinations down to 5×10^–4 millimoles.Examples are given for the determination of sulphur in organic samples after combustion according to Schöniger or Wickbold.

     

Significance of Non-isothermal Kinetic Data. A statistical study

Arrhenius parameters values, in non-isothermal kinetic vaporisation processes for a series of compounds with related structures, have been calculated. This was made using a method of calculation that allows to find the most probable vaporisation mechanisms. According to this method DTG curves were compared with some theoretical ones reported in literature, whose shape results to be only a function of the mechanisms. In this way the choice of the mathematical functions which can be inserted in the kinetic equations, was influenced by the shape of the DTG plots and other thermal analysis signals thus allowing to choose the most probable mechanisms. The kinetic parameters derived from these mechanisms were compared, using statistical analysis, with those obtained from another method of calculation based on ‘a priori’ vaporisation mechanism chosen for the investigated liquid–gas transition. The standard deviations of the slope and of the intercept, together with the standard deviation and the square correlation coefficient (r ²) of the linear regression equations related to the mechanisms of the two methods were calculated. Student t-test, Fisher F-test, confidence intervals (c.i.) and residuals valueswere also given. Statistical analysis shows that the mechanisms obtained with the former method (diffusive and geometrical models) and the related Arrhenius parameters result to be more significant (in terms of probability) than the corresponding quantities of the latter for which a first-order model was chosen. This revised version was published online in August 2006 with corrections to the Cover Date.     

Indirect polarographic determination of phosmet O,O-Dimethyl-S-(phthalimidomethyl)dithiophosphate

Summary An indirect polarographic method for the determination of phosmet was elaborated. It is based on the fact that N-hydroxymethylphthalimide formed by the reaction of phosmet with sulphuric acid is polarographically active and exhibits a behaviour during electrolysis at a Hg dropping-electrode similar to that of phthalimide. The method has been applied to the determination of phosmet in apples. The sensitivity is 0.1 ppm.

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Indirekte polarographische Bestimmung von Phosmet O,O-Dimethyl-S-(phthalimidomethyl)dithiophosphat

Zusammenfassung Das Verfahren beruht darauf, daß N-Hydroxymethylphthalimid, das bei der Reaktion von Phosmet mit Schwefelsäure gebildet wird, polarographisch aktiv ist und sich bei der Elektrolyse mit der Quecksilber-Tropfelektrode ähnlich wie Phthalimid verhält. Die Methode wurde zur Bestimmung von Phosmet in Äpfeln angewandt. Die Empfindlichkeit beträgt 0,1 ppm.

     

Interferences in the determination of aluminium by means of atomic absorption spectrophotometry

Summary The following substances have been investigated with regard to their interfering effect in the determination of aluminium by atomic absorption spectrophotometry: HCl, H₂SO₄, HNO₃, H₃PO₄, alkali and alkaline earth metals, Cu, Fe, Zn, Ni, Mn, Co, Mo, Ti, Cr, V, Si. Ternary systems with La have been examined also. Si exerts a strong depressive effect, whereas other elements in certain concentration ranges exhibit positive effects.

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Störungen bei der atomabsorptionsspektralphotometrischen Bestimmung von Aluminium

Zusammenfassung Folgende Substanzen wurden hinsichtlich einer möglichen Beeinflussung der Aluminiumbestimmung untersucht: HCl, H₂SO₄, HNO₃, H₃PO₄, Alkalimetalle, Erdalkalimetalle, Cu, Fe, Zn, Ni, Mn, Co, Mo, Ti, Cr, V, Si. Ternäre Systeme mit La wurden ebenfalls geprüft. Si übt eine stark depressive Wirkung aus, während andere Elemente in bestimmten Konzentrationsbereichen positiv wirken.

Lecture given on the 2nd Czechoslovak Conference on Flame Speotroscopy, Zvíkov, June 5–8, 1973.     

Titrimetric determination of octacyanotungstate(IV) ions with N-bromosuccinimide

Summary N-Bromosuccinimide has been successfully applied in the determination of 1–50 mg of W(CN)₈ ^4–in approximately 6 M hydrochloric acid using Naphthol Blue Black indicator. The average error is ±0.42% and the average standard deviation is ±0.06 mg. The method is not applicable to the determination of Mo(CN) ₈ ^4– ions. Tetrakis-(N-hydro-2-carbocyclic acid-pyridinium)octacyanotungstate(IV) and -molybdate(IV) were used as primary standards for the preparation of standard Mo(IV) and W(IV) solutions. The composition of the latter compounds was verified by analysis and single crystal X-ray data were obtained.

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Titrimetrische Bestimmung von Octacyanowolframat(IV)-ionen mit N-Bromsuccinimid

Zusammenfassung N-Bromsuccinimid wurde mit Erfolg zur Bestimmung von 1–50 mg W(CN)₈ ^4–-Ionen in ungefähr 6 M Salzsäure mit Naphthol-Blau-Schwarz als Indicator verwendet. Der mittlere Fehler beträgt ±0,42 %, die mittlere Abweichung ±0,06 mg. Die Methode ist nicht brauchbar für die Bestimmung von Mo(CN)₈ ^4–-Ionen. Tetrakis(N-hydro-2-carbonylsäure-pyridinium) octacyanowolframat(IV) und -molybdat(IV) wurden als Primärstandards für die Herstellung von Standard-Mo(IV)- und -W(IV)-Lösungen benutzt. Die Zusammensetzung dieser Verbindungen wurde durch Analyse kontrolliert und die Einkristall-Röntgen-Daten wurden festgestellt.