Estimates of general Mayer graphs III: Upper bounds obtained by means of spanningn-trees

Abstract

We obtain computable upper bounds for any given Mayer graph withn root-points (orn-graph). These are products of integrals of the type$\left( {\int {\left| {f_L } \right|^{z_{iL} y_i^{ - 1} } dx} } \right)^{yi}$, where thez _iL andy _i are nonnegative real numbers whose sum overi is equal to 1. As a particular case, we obtain the canonical bounds (see their definition in Section 2.2):

$$\left| {\int {\prod\limits_L {f_L \left( {x_i ,x_j } \right)dx_{n + 1} \cdot \cdot \cdot dx_{n + k} } } } \right| \leqslant \prod\limits_L {\left( {\int {\left| {f_L } \right|^{\alpha _L } dx} } \right)^{\alpha _L^{ - 1} } } $$

where theα _L 's satisfy the conditionα _L ≥1 for anyL, and ∑ _L α ⁻¹_L =k (k is the number of variables that are integrated over). These bounds are finite for alln-graphs of neutral systems. We obtain also finite bounds for all irreduciblen-graphs of polar systems, and for certainn-graphs occurring in the theory of ionized systems. Finally, we give a sufficient condition for an arbitraryn-graph to be finite.