Loop Equations for the d-dimensional One-Hermitian Matrix model

We derive the loop equations for the one Hermitian matrix model in any dimension. These are a consequence of the Schwinger-Dyson equations of the model. Moreover we show that in leading order of large $N$ the loop equations form a closed set.Comment: 13 pages, Latex, CERN-TH-6966/9

Non-Abelian Antibrackets

The $\Delta$-operator of the Batalin-Vilkovisky formalism is the Hamiltonian BRST charge of Abelian shift transformations in the ghost momentum representation. We generalize this $\Delta$-operator, and its associated hierarchy of antibrackets, to that of an arbitrary non-Abelian and possibly open algebra of any rank. We comment on the possible application of this formalism to closed string field theory.Comment: LaTeX, 8 pages (minor modification

Generation of interface for an Allen-Cahn equation with nonlinear diffusion

In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property

Fujita blow up phenomena and hair trigger effect: the role of dispersal tails

We consider the nonlocal diffusion equation $\partial \_t u=J*u-u+u^{1+p}$ in the whole of $\R ^N$. We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel $J$ near the origin, which is linked to the tails of $J$. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation $\partial \_tu=\Delta u+u^{1+p}$. On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of $J$. As an application of the result in population dynamics models, we discuss the hair trigger effect for $\partial \_t u=J*u-u+u^{1+p}(1-u)