Aperiodic tiles

A set of tiles (closed topological disks) is calledaperiodic if there exist tilings of the plane by tiles congruent to those in the set, but no such tiling has any translational symmetry. Several aperiodic sets have been discussed in the literature. We consider a number of aperiodic sets which were briefly described in the recent bookTilings and Patterns, but for which no proofs of their aperiodic character were given. These proofs are presented here in detail, using a technique with goes back to R. M. Robinson and Roger Penrose. The research of Branko Grünbaum was supported in part by NSF Grants MCS8301971 and DMS-8620181.     

A microanalytical electrophoresis technique for the determination of polymorphic blood proteins for medical and forensic applications

Summary A Microanalytical Electrophoresis Technique for the Determination of Polymorphic Blood Proteins for Medical and Forensic Applications This paper describes the results obtained to date using an electro-phoretic technology (equipment and methodology) designed for simple, rapid, reliable, and economical determination of a number or polymorphic enzymes and proteins. The technology permits both the visualization and estimation of ng and pg quantities of enzymes and proteins. While methodology for the identification and quantification of specific blood proteins and enzymes is useful in many areas of biological research, it has practical application of particular value in medical diagnosis and in forensic science.

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Zusammenfassung Die Ergebnisse elektrophoretischer Untersuchungen und Bestimmungen einer Reihe polymorpher Enzyme und Proteine wurden mitgeteilt. Das angewandte Verfahren erlaubt die Sichtbarmachung und Bestimmung von ng- und pg-Mengen dieser Stoffe. Die Identifizierung und mengenmäßige Erfassung spezifischer Blutproteine und Enzyme ist für viele Gebiete biologischer Forschung nützlich. Die praktische Bedeutung für die medizinische Diagnostik und für forensische Zwecke wird hervorgehoben.

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Presented at the International Symposium on Microchemical Techniques, Davos, Switzerland, May 22–27, 1977.     

A noteworthy improvement of the 3-diazo-4-oxo-3,4-dihydroquinoline photosynthesis of indole-3-carboxamides

The use of benzene, in place of methylene chloride as solvent, allows the synthesis of amides of indole-3-carboxylic acid via the photolysis of 3-diazo-4-oxo-3,4-dihydroquinoline to occur in significantly higher yield and requiring shorter reaction times. This modification makes unnecessary the time consuming and costly chromatographic purification step, and allows the synthesis of novel amides which could not he isolated using the previous method.     

Syntheses of some azolopyridopyrimidines

Syntheses of isomeric azolopyridopyrimidines are described. As starting material the corresponding pyridopyrimidines were used. It could be established that in many cases a transformation of a functional group with the purpose to form an annelated five-membered ring proceeded with ring opening at the pyrimidine part. Subsequent ring closure with one carbon synthons gave then the desired tricyclic heterocycles.     

Shortness exponents of families of graphs

Known estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions.