Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

Determination of elastic moments in flat-plate and lift-slab structures by the moiré method

The paper presents details of an experimental investigation carried out to determine deflections and bending moments in continuous flat-plate and lift-slab structures under various loading combinations, using the moiré method. A suitable computational method was adopted to compute accurately deflections and moments from the moiré photographs, which thus replaced the usual graphical procedure.     

Resolution of RS-abscisic acid and the separation of abscisic acid metabolites from plant tissue by high-performance liquid chromatography

Attempts to resolve the enantiomers of racemic abscisic acid (ABA) by high-performance liquid chromatography on a chiral stationary-phase column were unsuccessful. However, reduction of RS-methyl ABA (RS-Me-ABA) with sodium borohydride generates a new chiral centre and one of the two isomeric products, the RS-Me-1',4'-cis-diol of ABA, was separated into its enantiomers by high-performance liquid chromatography on an optically active Pirkle column. High-performance liquid chromatography on a mu Bondapak C18 column separated the metabolites and conjugates of [2-14C]ABA fed to tomato shoots. The resolution method was used to measure the relative proportions of R and S enantiomers in the free acid liberated from conjugates of ABA.     

Applications of enzyme-catalysed reactions in trace analysis-IV Determination of beryllium and zinc by their inhibition of calf-intestinal alkaline phosphatase

Methods are described for the determination of beryllium (18-90 ng) and zinc (0.6-6 mug), based on their inhibition of calf-intestinal alkaline phosphatase.     

Inefficiency in packing squares with unit squares

It is shown that, in packing a square of side $\text{n +}\text{1}\text{2}$ with unit squares, the wasted space always has area $\text{? n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. This answers a question of Erdös and Graham.     

The preparation and cytotoxic properties of211at labelled concanavalin a bound to cell membranes

A method of labelling biologically active proteins with the alpha emmitting halogen²¹¹At is presented. The labelling procedure is discussed with reference to the chemistry of astatine. Proteins which have been labelled retain approximately 50% of their original biological activity. Using cell specific labelled proteins, dose response curves are given indicating that such reagents are extremely cytotoxic, D₃₇ human CML cells=5 atmos²¹¹At/cell. The research potential of²¹¹At labelled biologically active proteins is briefly discussed.     

Axisymmetric and asymmetric instabilities in elastic solid cylinders under finite axial strain

Zusammenfassung Das Vorkommen von rotationssymmetrischen und asymmetrischen Instabilitäten in elastischen Kreiszylindern unter grossen axialen Drucklasten ist bekannt. Es wird angenommen, dass das Material isotrop und inkompressibel ist und dass die dazugehörige Formänderungsenergiefunktion von der Mooneyschen Form ist. Der Übergang von asymmetrischem Knicken dünner Stäbe zu rotationssymmetrischem Beulen am Umfang von Scheiben ist klar ersichtlich. Die asymptotische Form der Lösung mit wachsendem Radius/Höhe-Verhältnis führt zu einem kritischen Ausdehnungsverhältnis, das mit Ergebnissen früherer Arbeiten übereinstimmt, die Beulen an Scheibenrändern untersuchen. Wo jedoch vorherige Arbeiten nur spezielle Arten von Mooney-Material in Betracht zogen, wird hier gezeigt, dass der kritische Wert für alle möglichen Mooney-Materialien gilt. Zum Schluss wird für den asymptotischen Fall nachgewiesen, dass die Instabilitäten auf Zonen nahe der freien Oberfläche beschränkt sind.