In the large N limit, we show that the Local Potential Approximation to the flow equation for the Legendre effective action, is in effect no longer an approximation, but exact – in a sense, and under conditions, that we determine precisely. We explain why the same is not true for the Polchinski or Wilson flow equations and, by deriving an exact relation between the Polchinski and Legendre effec...

The group Sn, n ≥ 1, is usually defined as the group of all permutations of the set Xn = {1, 2, . . . , n}. So, an element s ∈ Sn is a bijection (i.e., one-to-one map) from Xn to Xn : k 7→ s(k). The composition of two such map is denoted by s1 ◦ s2 : k 7→ s1 ( s2(k) ) . It is convenient to depict the permutation s as follows. Take two copy of Xn represented as column vectors and draw a system o...

Let A be a non-trivial Abelian group. We call a graph G = (V, E) A-magic if there exists a labeling f : E → A∗ such that the induced vertex set labeling f : V → A, defined by f(v) = ∑ uv∈E f(uv) is a constant map. In this paper, we show that Kk1,k2,...,kn (ki ≥ 2) is A-magic, for all A where |A| ≥ 3.