On a conjecture of Erdös and Rényl

We show that for A ranging over n×n circulants with three ones in each row, where n is prime, $\text{lim inf}\text{|}\text{det}\text{A|}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{>1}$. For a subfamily containing almost all A's this lim inf is in fact 1.38…. We also compute the permanents of a certain family of matrices.     

On <Emphasis Type="Italic">g</Emphasis>-functions for Laguerre function expansions of Hermite type

We examine weighted L ^p boundedness of g-functions based on semigroups related to multi-dimensional Laguerre function expansions of Hermite type. A technique of vector-valued Calderón–Zygmund operators is used.     

Enantioselective syntheses of syn- and anti-β-hydroxyallylsilanes via allene hydroboration-aldehyde allylboration reactions

The kinetic hydroboration of allenylsilane 5 with ((d)Ipc)(2)BH at -40 °C provides allylborane 9Z with ≥12:1 selectivity. When the hydroboration is performed at temperatures above -40 °C, 9Z isomerizes to the thermodynamically more stable allylborane 9E with >20:1 selectivity. Subsequent treatment of 9Z or 9E with aldehydes at -78 °C provides syn- or anti-β-hydroxyallylsilanes, 7 or 8, respectively.     

Studies on the Synthesis of Reidispongiolide A: Stereoselective Synthesis of the C(22)-C(36) Fragment

A highly stereoselective synthesis of the C(22)-C(36) fragment 2 of reidispongiolide A is described. This synthesis features the highly stereoselective mismatched double asymmetric crotylboration reaction of the aldehyde derived from 5 and the new chiral reagent (S)-(E)-7 that provides 12 with >15:1 dr. Subsequent coupling of the derived vinyl iodide 3 with aldehyde 16 provided allylic alcohol 17, that was elaborated by three steps into the targeted reidispongiolide fragment 2.     

The single-mode rate equations for semiconductor lasers with thermal effects

A mathematical model describing the coupling of electrical,optical and thermal effects in semiconductor lasers is introduced.Numerical and asymptotic solutions are derived, including expressionsfor key physical quantities such as the initial time delay,the frequency of spike oscillation and the temperature rise,together with its influence on the photon density, the electronconcentration and the threshold current. The consequences ofthermal effects in reducing efficiency are thus quantified.