Trees, functional equations, and combinatorial Hopf algebras

One of the main virtues of trees is to represent formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in power series rings. When analyzed in terms of combinatorial Hopf algebras, the simplest examples yield interesting algebraic identities or enumerative results.Comment: 14 pages, LaTE

Algebraic expansions for curvature coupled scalar field models

A late time asymptotic perturbative analysis of curvature coupled complex scalar field models with accelerated cosmological expansion is carried out on the level of formal power series expansions. For this, algebraic analogues of the Einstein scalar field equations in Gaussian coordinates for space-time dimensions greater than two are postulated and formal solutions are constructed inductively and shown to be unique. The results obtained this way are found to be consistent with already known facts on the asymptotics of such models. In addition, the algebraic expansions are used to provide a prospect of the large time behaviour that might be expected of the considered models.Comment: 16 pages, no figures; v2: typos corrected, references adde

Domain Walls in MQCD and Monge-Ampere Equation

We study Witten's proposal that a domain wall exists in M-theory fivebrane version of QCD (MQCD) and that it can be represented as a supersymmetric three-cycle in G_2 holonomy manifold. It is shown that equations defining the U(1) invariant domain wall for SU(2) group can be reduced to the Monge-Ampere equation. A proof of an algebraic formula of Kaplunovsky, Sonnenschein and Yankielowicz is presented. The formal solution of equations for domain wall is constructed.Comment: Latex, 18 pages, section 4.2 modified, typos correcte