The Polyakov path integral over bordered surfaces

Following the general scheme of the convariant path integral quantization of gauge systems, two alternative formulations of the first quantized closed bosonic string in a position representation are presented. In both approaches the covariant path integral representations of the propagator and of the higher order off-shell amplitudes are constructed. For a wide class of gauges the explicit formulae for off-shell amplitudes are obtained. This paper is the continuation of our previous work where the corresponding problems in the open string case were considered [20].     

Fibres of width 3 ordered sets

We show that in an ordered setP of width 3 there always exists a fibre (i.e., a subset intersecting each maximal nontrivial antichain) of size at most 11/18|P|. This improves previously known results.Research supported by Polish KBN Grant 2 2037 92 03.     

On mechano-chemical Calcium waves

The influence of mechano-chemical coupling on calcium concentration waves is considered. The propagation of calcium waves is described by a reaction–diffusion equation with the reaction term dependent on the mechanical stress responsible for the release of calcium. Similarly the balance of mechanical forces is influenced by the calcium concentration through the so-called traction force.     

Rulers and slaves in a random social group

LetD(n, d) be a digraph chosen at random from a family of alld-out-regular digraphs onn points. LetK(n, p) be a simple graph onn points in which each edge appears independently with probabilityp. A relationship between the properties of extreme in-degrees ofD(n,d) and extreme degrees ofK(n, p) in the case whend = (n – 1)p = o(n) is presented. A sociological interpretation is also provided.     

Intramolecular hydrogen bonds and hydrogen-bonded systems in di-Schiff bases of 4-methyl-isophthalaldehyde with 4-substituted anilines

Three di-Schiff bases of 2-hydroxy-5-methyl-isophthalaldehyde with 4-R-anilines (R=H, CH₃, OCH₃) and their 1:1 complexes with HClO₄ were studied by FT-IR, ¹H, and ¹³C NMR spectroscopy in acetonitrile and [²H₃]acetonitrile solutions, respectively. In di-Schiff bases intramolecular OH…N hydrogen bonds have been detected; however, they show no proton polarizability. Hydrogen-bonded systems with fast proton fluctuation and large proton polarizability have been found in the 1:1 complexes of di-Schiff bases with HClO₄.     

Bemerkungen zur colorimetrischen Bestimmung von Vanadium mit Phosphorwolframs?ure

Zusammenfassung Die colorimetrische Bestimmungsmethode von Vanadium mit Phosphorwolframsäure wurde in bezug auf den Einfluß der Reagenskonzentrationen untersucht. Außerdem wurde die Bestimmungsmöglichkeit in Gegenwart von Eisen geprüft. Es zeigte sich, daß die Konzentrationen an Phosphorsäure und Wolframat in weiten Grenzen geändert werden können, ohne die Bestimmungsgenauigkeit zu beeinflussen. Zu große Mengen Salpetersäure bewirken negative Fehler, indem die Extinktionen mit zunehmender HNO₃-Konzentration linear abnehmen. Nach der angegebenen Vorschrift kann Vanadium in Gegenwart von bis zu 150 mg Eisen(III)-ionen in einem Volumen von 25 ml bestimmt werden.     

Electron-transfer reduction of selected alcohols with alkalide K, K(15-crown-5)2 via organometallic intermediates

The course of the reaction of alkalide K⁻, K⁺(15-crown-5)₂1 with selected alcohols depends on the kind of alcohol and the mode of substrate delivery. In the case of methanol, potassium methoxide formed initially undergoes destruction at the excess of 1. It results in potassium oxide and methylpotassium. The latter opens the crown ether ring giving potassium tetraethylene glycoxide vinyl ether and methane. A similar course of the process is observed for propanol. Potassium glycidoxide is the main product formed in the reaction of 1 with glycidol. Its oxirane ring is opened at the excess of 1. Organopotassium alkoxides, i.e., potassium potassiomethoxide and dipotassium potassiopropane-1,2-dioxide are intermediate products of this reaction. They react then with the crown ether. Potassium methoxide, potassium enolate of acetaldehyde, dipotassium propane-1,2-dioxide and potassium tetraethylene glycoxide vinyl ether are the final products of this process.