1) is established on the linearity, it is in fact truly for the linear and non-liner cases because the

underlying graph of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-flow can be decomposed into bouquets and dipoles, hold with conditions of Theorem 4.

SIGMA]) consists of an

underlying graph S together with a set of atomic constraints [C.

alpha]W), where A is the maximum degree of the

underlying graph of W.

n] = (G, [micro])), where G = (V, E) is a graph called the

underlying graph of [S.

A sigraph is an ordered pair S = (G, [sigma]), where G = (V,E) is a graph called

underlying graph of S and [sigma]: E [right arrow] {+, -} is a function.

7 The polynomial N associated to the

underlying graph G of a rooted map M is given by the following combinatorial formula:

In our paper we showed that the

underlying graph theoretic model of hierarchical opposition is a twin binary tree in which the two 'son' nodes of every 'father' node have the same pair of contrastive labels: pure and impure in the Indian caste system, male and female in Indonesia, and land and noble in Fiji.

n] = (G, [mu])), where G = (V, E) is a graph called the

underlying graph of [S.

algorithm, the feeler and the

underlying graph, characterized respectively by the two parameters l [less than or equal to] L and k [less than or equal to] Q, make it possible to make a trade-off between the computational cost and the success rate.

We propose these techniques for approximating Schur complements that have entries which decay with distance on some

underlying graph.

Vaidya's method constructs a preconditioner M whose

underlying graph [G.

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = ([G, [sigma]) (S = (G,[mu])) where G = (V, E) is a graph called

underlying graph of S and [sigma] : E [right arrow] ([bar.