The end structure of a graph: recent results and open problems

This paper aims to give a brief introduction to a set of problems, old and new, concerned with one of the main and long-standing quests in infinite graph theory: how to represent the end structure of a given graph by that of a simpler subgraph, in particular a spanning tree. There has been a fair amount of activity in this field recently; we describe the latest results, as well as some of the new problems which these results suggest.     

On the Average Discrepancy of Successive Tuples of Pseudo-Random Numbers over Parts of the Period

 In the present paper we give an upper and a lower bound for the average value of the discrepancy of non-overlapping s-tuples of successive elements of a first order congruential pseudo-random-number generator (with prime modulus and maximal period). The estimates are – up to logarithmic factors – sharp also for short parts of the period. Received 30 January 1997; in revised form 2 May 1997     

Fluorinated poly(benzoxazole–imide)s

New poly(benzoxazole–imide)s have been prepared by polycondensation of bis(o-aminophenol)s, such as 3,3′ dihydroxybenzidine or 2,2-bis(3-amino-4-hydroxyphenyl)-hexafluoropropane, with diacid chlorides containing both imide and hexafluoroisopropylidene groups. The soluble polymer precursor resulting from the first step of polycondensation in N-methylpyrrolidinone (NMP) at low temperature was processed into thin flexible films, which were then thermally treated to reach the fully cyclized structure of poly(benzoxazole–imide)s, as observed by infrared spectra. An alternative way to produce the cyclization was by heating at reflux the solution of polymer precursor in NMP. Thermal stability, glass transition temperature, dielectric constant and its dependence on relative humidity have been studied and compared with those of related polybenzoxazoles and other heterocyclic Polymers.