ar X iv : m at h / 05 09 01 6 v 1 [ m at h . PR ] 1 S ep 2 00 5 CALCULATION OF GREEKS FOR JUMP - DIFFUSIONS

ar X iv : m at h / 05 05 01 1 v 3 [ m at h . PR ] 8 S ep 2 00 5 EXCHANGEABLE , GIBBS AND EQUILIBRIUM MEASURES FOR MARKOV

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of “symmetric measure”: exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.

ar X iv : h ep - t h / 01 03 09 3 v 1 1 3 M ar 2 00 1 Physical Aspects of the Space - Time Torsion

ar X iv : h ep - p h / 02 05 01 6 v 1 2 M ay 2 00 2 Scalar a 0 ( 980 ) meson in φ → π 0 ηγ decay

ar X iv : m at h / 05 05 01 1 v 4 [ m at h . PR ] 5 J un 2 00 6 EXCHANGEABLE , GIBBS AND EQUILIBRIUM MEASURES FOR MARKOV

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of “symmetric measure”: exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.

ar X iv : h ep - t h / 05 01 16 3 v 2 2 1 A pr 2 00 5 Why do we live in 3 + 1 dimensions ?

In the context of string theory we argue that higher dimensional Dp-branes unwind and evaporate so that we are left with D3-branes embedded in a (9+1)-dimensional bulk. One of these D3-branes plays the rôle of our Universe. Within this picture, the evaporation of the higher dimensional Dp-branes provides the entropy of our Universe.