Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Microwave‐Promoted Ullmann Condensation of 2‐Aminopyridines with 2‐Chlorobenzoic Acids

A new efficient synthetic method of microwave‐promoted Ullmann condensation of 2‐aminopyridines with 2‐chlorobenzoic acids to yield various substituted 11H‐pyrido[2,1‐b]quinazolin‐11‐ones in dry media is presented. Results were compared with those obtained following the classical heating.     

A rapid and accurate determination of trace quantities of sulphur in tin

A method is described which permits the determination of 1–100 p.p.m. of sulphur in metallic tin. It is based on the complete combustion of the sample in a stream of oxygen and the determination of the change in electrical conductivity produced by the absorption of the gaseous combustion products (SO₂) in a suitable reaction liquid. Gas-borne tin (IV) oxide dust is removed from the oxidation products by electrostatic precipitation. The new technique is shown to be superior to the widely used methylene blue method.     

Recherches théoriques sur la structure électronique de molécules contenant des atomes hybridés en sp:

Résumé La théorie L. C. A. O. améliorée est appliquée à l'étude de la molécule C₃. Afin de satisfaire l'égalité nécessaire entre les éléments de matrice non diagonaux L _pq et L _qp, nous avons introduit un déplacement électronique le long des liaisons . Les charges nettes portées par les atomes terminaux sont: + 0,09. La distance C-C calculée (1,286 Å) est en bon accord avec l'expérience. Les énergies des deux premières transitions NV: ¹ _g ^{+ 1} _u ^– et¹ _g ^{+ 1}_u sont: 4,7 et 4,8 eV. Ces valeurs ne sont pas en contradiction avec l'expérience qui situe ces transitions au delà de 4,1 eV, limite à partir de laquelle l'absorption de l'atmosphère terrestre rend impossible l'étude des spectres cométaires.

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The improved LCAO theory is applied to the C₃ molecule. In order to satisfy the necessary equality between the non-diagonal matrix elements L _pq and L _qp, we have introduced an electron shift along the bonds. The net charges borne by the terminal atoms are: + 0.09. The computed C-C distance (1.286 Å) is in good agreement with experiment. The two first NV transition energies: ¹ _g ^{+ 1} _u ^– and ¹ _g ^{+ 1}_u are: 4.7 and 4.8 eV. These values are not in contradiction with experiment which places these transitions beyond 4.1 eV, limit from which the terrestrial atmospheric absorption makes impossible to study the comets' spectra.

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Zusammenfassung Die verbesserte LCAO-Theorie wird auf die C₃-Molekel angewandt. Um die nötige Gleichheit der nicht-diagonalen Matrixelemente L _pq und L _qp zu erhalten, haben wir eine Elektronenverschiebung längs der -Bindungen eingeführt. Die Ladungen der Endatome sind: + 0,09. Die berechnete C-C-Bindungslänge (1,286 Å) stimmt mit der Erfahrung überein. Die berechneten Energien der zwei ersten NV-Übergänge: ¹ _g ^{+ 1} _u ^– und ¹ _g ^{+ 1}_u sind: 4,7 und 4,8 eV. Diese Werte sind nicht in Widerspruch mit der Erfahrung, die diese Übergänge über 4,1 eV ansetzt; denn das Stadium der Kometenspektren ist über dieser Grenze wegen der Luftabsorption unmöglich.

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Nous remercions le Professeur P. Swings (Institut d'Astrophysique, Cointe-Sclessin, Belgique) ainsi que le Professeur Ch. Fehrenbach (Observatoire de Marseille), pour les échanges de vue que nous avons eu avec eux sur ce sujet.     

Die relativkonduktometrische Schwefelbestimmung im Dezimilligrammbereich in organischen Substanzen

Zusammenfassung Das Verfahren zur relativkonduktometrischen Schwefelbestimmung in organischen Substanzen im Milligrammbereich wird unter Verwendung einer Ultramikrowaage in den Dezimilligrammbereich übertragen. Die aus der Analyse einer Reihe von Testsubstanzen erhaltenen Resultate zeigen, daß die erzielte Genauigkeit in beiden Bereichen gleich gut ist.

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The relative conductometric sulfur determination in organic substances in the decimilligram range

Summary The procedure for the relative conductometric sulfur determination in organic materials in the milligram range has been carried over into the decimilligram region by using an ultramicro balance. The results obtained from the analyses of a number of test substances demonstrate that the accuracy attained in both regions is equal.

     

Recherches théoriques sur la structure électronique de molécules contenant des atomes hybridés en sp:

Résumé La méthode L. C. A. O. améliorée est utilisée pour étudier la structure électronique des molécules d'acétylène, de diacétylène et d'azote. Les énergies des transitions verticales calculées sont en bon accord avec l'expérience. Une relation entre l'indice de liaison mobile ( + ) et la distance interatomique, valable même lorsque les atomes sont de nature différente, est proposée.

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The improved LCAO theory is used to study the electronic structure of acetylene, diacetylene and nitrogen molecule. Calculated vertical transition energies are in good agreement with experiment. A relation between total mobile bond order ( + ) and bond distance is proposed, which applies also to atoms with different effective nuclear charges.

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Zusammenfassung Die elektronische Struktur der Moleküle Acetylen, Diacetylen und Stickstoff wird mit Hilfe der verbesserten LCAO-Methode untersucht. Die für die vertikalen Übergänge berechneten Energien stimmen mit dem Experiment befriedigend überein. Eine Beziehung zwischen Bindungsabstand und Gesamt--Bindungsindex, die auch für Atome verschiedener effektiver Kernladung gilt, wird vorgeschlagen.

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Que Monsieur G. Berthier (Laboratoire de Chimie Théorique de la Faculté des Sciences de Paris) qui a bien voulu relire notre manuscrit trouve ici l'expression de tous nos remerciements.     

Mathematics support--support for all?

Mathematics Support Centres are to be found in various formsin the majority of UK higher education institutions. They havebeen established in order to ease widespread and serious difficultiesthat a significant number of students have with mathematics,particularly at the school–university transition. Theyusually offer mathematics and/or statistics support to studentsacross the full range of disciplines studied. Anecdotal evidencesuggests that those students who make good use of such centresare not just those who struggle with mathematics. Many frequentusers are quite competent and simply want to do better. Thestudy reported here describes and analyses data from one cohortof engineering students. A novel aspect is the quantificationof the proportion of support centre visitors who fall into these,and other, categories. We conclude of the cohort in the study,mathematics support has improved the pass rate by 3%. Of thefailures, about half (4% of the sample total) could well havepassed had they attended the mathematics support centre regularly.Furthermore, the majority of those attending were not studentswho were in danger of failing. This has important implicationsnot only for the design of mathematics support provision, butalso for the performance of the high fliers. The methodologyoffers one way tackling the difficult task of evaluating theeffectiveness of mathematics support initiatives.