Lipschitz extension constants equal projection constants

For a Banach space $V$ we define its Lipschitz extension constant, \cL\cE(V), to be the infimum of the constants $c$ such that for every metric space $(Z,\rho)$, every $X \subset Z$, and every $f: X \to V$, there is an extension, $g$, of $f$ to $Z$ such that $L(g) \le cL(f)$, where $L$ denotes the Lipschitz constant. The basic theorem is that when $V$ is finite-dimensional we have \cL\cE(V) = \cP\cC(V) where \cP\cC(V) is the well-known projection constant of $V$. We obtain some direct consequences of this theorem, especially when V = M_n(\bC). We then apply techniques for calculating projection constants, involving averaging projections, to calculate \cL\cE((M_n(\bC))^{sa}). We also discuss what happens if we also require that $\|g\|_{\infty} = \|f\|_{\infty}$.Comment: 16 pages. Three very minor mathematical typos corrected. Intended for the proceedings of GPOTS0

Generalized Euler constants

We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent monotonicity is investigated. We also prove that a certain property of these numbers is equivalent to the Riemann Hypothesis.Comment: v2. 13 pages. Minor changes suggested by the referee. To appear in Math. Proc. Cambridge Phil. So

Chaotic Coupling Constants

We examine some novel physical consequences of the general structure of moduli spaces of string vacua. These include (1) finiteness of the volume of the moduli space and (2) chaotic motion of the moduli in the early universe. To fix ideas we examine in detail the example of the (conjectural) dilaton-axion ``$S$-duality'' of four-dimensional string compactifications. The facts (1) and (2) together might help to solve some problems with the standard scenarios for supersymmetry breaking and vacuum selection in string theory.Comment: 18 pages (4 figs), YCTP-P2-94, RU-94-2

Dyes with high dielectric constants

The dielectric constants of perylene dyes, perylene-3,4: 9,10-tetracarboxylic bisimides, are reported. With aromatic substituents, dielectric constants up to 110 are obtained. With polymeric dyes, the dielectric constants rise to 260. Mechanisms and applications are discussed