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

Universal constants and natural systems of units in a spacetime of arbitrary dimension

We study the properties of fundamental physical constants using the threefold classification of dimensional constants proposed by J.-M. L{\'e}vy-Leblond: constants of objects (masses, etc.), constants of phenomena (coupling constants), and "universal constants" (such as $c$ and $\hbar$). We show that all of the known "natural" systems of units contain at least one non-universal constant. We discuss the possible consequences of such non-universality, e.g., the dependence of some of these systems on the number of spatial dimensions. In the search for a "fully universal" system of units, we propose a set of constants that consists of $c$, $\hbar$, and a length parameter and discuss its origins and the connection to the possible kinematic groups discovered by L{\'e}vy-Leblond and Bacry. Finally, we give some comments about the interpretation of these constants.Comment: 18 pages, pedagogical article. v3: small corrections and extensions, some references added. This version matches the published on

Diffusion constants and martingales for senile random walks

We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be derived from martingales associated with the time-changed walks. Using an inverse time-change, the diffusion constants for senile random walks are then obtained via these martingales. When the walks are diffusive, weak convergence to Brownian motion can be shown using a martingale functional limit theorem.Comment: 17 pages, LaTeX; the proof of Proposition 2.3 has been simplified, and an error in the proof of Theorem 2.4 has been correcte