This can of course be solved using algebra. The rst treatise on algebra was written by Diophantus of Alexandria in the 3rd century AD. The term derives from the Arabic al-jabr or literally \the reunion of broken parts." As well as its mathematical meaning, the word also means the surgical treatment of fractures. It gained widespread use through the title of a book ilm al-jabr wa'l-mukabala|the ...

We use Brownian motion ideas to study Schriidinger operators H = +A + V on Lp(R”). In particular: (a) We prove that limt+m t-l In /I e&H j111.9 is p-independent for a very large class of V’s where 11 A ll,,,n = norm of A as an operator from L’ to L”. (b) For Y > 3 and V E LY/~--E n L”Ip+c, we show that sup jl eetH Ilm,m -: (D if and only if H has no negative eigenvalues or zero energy resonance...

The ‘Abb®sid caliphs al-Ma’m‚n and al-Mu‘ta◊im, who between them reigned from 198 to 227 A.H. (813-842 A.D.) are best-known to historians of philosophy for two things. First, they supported a massive translation project that had already begun in the previous century with al-Ma’m‚n’s grandfather, al-Man◊‚r. The result was that many of the most important works of Greek philosophy and science were...

The Frege-Russell view is that existence is a second-order property rather than a property of individuals. One of the most compelling arguments for this view is based on the premise that there is an especially close connection between existence and number. The most promising version of this argument is by C.J.F. Williams (1981, 1992). In what follows, I argue that this argument fails. I then de...