Rings of normal and nonnormal numbers

We construct an uncountable ring of real numbers all of whose nonzero elements are normal to baseb and nonnormal to baseab, wherea, b are and pair of integers greater than one with (a,b)=1. To the memory of Gerold Wagner     

Synthesis and properties of polyimide ionomers containing sulfoalkoxy and fluorenyl groups

A series of sulfopropylated and sulfobutylated polyimide copolymers containing fluorenyl groups, SPI‐4, were synthesized to investigate the effect of alkyl side chains on the properties (stability, mechanical strength, water uptake, and proton conductivity) of the polymimide electrolyte membranes. SPI‐4 showed much better hydrolytic stability (in 10% MeOH aq at 100 °C) than the main chain sulfonated polyimide, SPI‐1. Tough, flexible, and ductile membranes were obtained from these copolymers. At high relative humidity all the SPI‐4 membranes showed high mechanical properties (>34 MPa of the maximum stress) and proton conductivity (>0.1 Scm^?1). These properties are comparable to or even better than those of the perfluorosulfonic acid ionomer (Nafion 112). The new polyimide ionomers have proved to be a possible candidate as polymer electrolyte membrane for PEFCs and DMFCs. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 4439–4445, 2005     

Some results on odd factors of graphs

A {1, 3, …,2n ? 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, …, 2n ? 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, …, 2n ? 1}-factor.