Synthesis and characterization of poly(ethylene oxide‐co‐ethylene carbonate) macromonomers and their use in the preparation of crosslinked polymer electrolytes

Methacrylate‐functionalized poly(ethylene oxide‐co‐ethylene carbonate) macromonomers were prepared in two steps by the anionic ring‐opening polymerization of ethylene carbonate at 180 °C, with potassium methoxide as the initiator, followed by the reaction of the terminal hydroxyl groups of the polymers with methacryloyl chloride. The molecular weight of the polymer went through a maximum after approximately 45 min of polymerization, and the content of ethylene carbonate units in the polymer decreased with the reaction time. A polymer having a number‐average molecular weight of 2650 g mol^?1 and an ethylene carbonate content of 28 mol % was selected and used to prepare a macromonomer, which was subsequently polymerized by UV irradiation in the presence of different concentrations of lithium bis(trifluoromethanesulfonyl)imide salt. The resulting self‐supportive crosslinked polymer electrolyte membranes reached ionic conductivities of 6.3 × 10^?6 S cm^?1 at 20 °C. The coordination of the lithium ions by both the ether and carbonate oxygens in the polymer structure was indicated by Fourier transform infrared spectroscopy. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2195–2205, 2006     

Electrical properties of polypyrrole/p‐InP structure

The polypyrrole/p‐InP structure has been fabricated by the electrochemical polymerization of the organic polypyrrole onto the p‐InP substrate. The current–voltage (I–V), capacitance–voltage (C–V), and capacitance–frequency (C–f) characteristics of the PPy/p‐InP structure have been determined at room temperature. The structure showed nonideal I–V behavior with the ideality factor and the barrier height 1.48 and 0.69 eV respectively. C–f measurements of the structure have been carried out using the Schottky capacitance spectroscopy technique and it has been seen that there is a good agreement between the experimental and theoretical values. Also, it has been seen that capacitance almost show a plateau up to a certain value of frequency, after which, the capacitance decreases. The higher values of capacitance at low frequencies were attributed to the excess capacitance resulting from the interface states in equilibrium with the p‐InP that can follow the a.c. signal. The interface state density N_ss and relaxation time τ of the structure were determined from C–f characteristics. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1572–1579, 2006     

Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs

A probabilistic result of Bollobás and Catlin concerning the largest integer p so that a subdivision of K_p is contained in a random graph is generalized to a result concerning the largest integer p so that a subdivision of A_p is contained in a random graph for some sequence A₁, A₂,… of graphs such that A_i+1 contains a subdivision of A_i. A similar result is proved for subdivisions with odd paths or cycles. The result is applied to disprove a conjecture of Chartrand, Geller, and Hedetniemi. The maximum number of edges in a graph without a subdivision of K_p, p = 4, 5, with odd paths or cycles is determined.     

On a diophantine problem of P. Erdös

Research supported in part by Grant 4055 from the Hungarian National Foundation for Scientific Research and by Kereskedelmi és Hitelbank Rt. Universitas Foundation.     

Entropy of Bogoliubov automorphisms of the canonical anticommutation relations

We compute the entropyh _A ( _U ) in the sense of Connes, Narnhofer and Thirring of Bogoliubov automorphisms _U of the CAR-algebra with respect to invariant quasifree states _A with 0A1 having pure point spectrum.Supported in part by a grant from the National Science Foundation     

Rectangular convexity of convex domains of constant width

An E R ² is r-convex if for every x, y E there exists a closed rectangle R such that x, y R and R E. Several results about r-convexity appeared in [1]. Its authors formulated a conjecture about conditions for a compact, convex set in R ² to be r-convex. We prove this conjecture in the case of convex domains of constant width.     

A combinatorial property of points and ellipsoids

For eachd1 there is a constantc _d>0 such that any finite setXR ^d contains a subsetYX, |Y|[1/4d(d+3)]+1 having the following property: ifEY is an ellipsoid, then |E X|c _d |X|.On leave from the Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest, P.O. Box 127, Hungary. Supported by a research fellowship from the Science and Engineering Research Council, U.K., and by Hungarian National Foundation for Scientific Research Grant No. 1812.