Throughout this section assume [beta] > [alpha] > -1/2 and m, n two

non-negative integers such that m [greater than or equal to] n.

l, [m.sub.1], ..., [m.sub.r] are

non-negative integers, y [member of] C, [b.sub.1] ...

This fact allowed them to define for any given hypergraph F the monotone threshold bias [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the largest

non-negative integer for which Enforcer wins the (1 : b) game F under the new set of rules if and only if b [less than or equal to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A

non-negative integer mapping [phi]: E(D) [right arrow] Z is w-consistent if 0 [less than or equal to] [phi](e) [less than or equal to] 2w(e) - 1 for all e [member of] E(D).

Let R := (E, [([[rho].sub.i]).sub.i[member of]I]) be a relational structure of signature [mu] := [([m.sub.i]).sub.i[member of]I] and let n be a

non-negative integer.

a sequence of

non-negative integers [lambda] = ([[lambda].sub.1], ..., [[lambda].sub.d]) such that [[lambda].sub.i] [greater than or equal to] [[lambda].sub.i+1] for each i = 1, ..., d - 1.

This article is concerned with finite graphs with n vertices labelled 1,..., n such that every vertex has degree at most R, where R will be a fixed

non-negative integer. By graph we always mean finite undirected graph.

The descent of T denoted by q(T) is the least

non-negative integer n such that ran([T.sup.n]) = ran([T.sup.n+1]).

This distribution yields an integral for each

non-negative integer m:

(a) Is it true that if for some

non-negative integer c and for all v, we have [r.sub.v] = c, then r = c?

Theorem 1.1 Let P be a ranked poset and m a

non-negative integer. With the notations above,

Below we denote by [[zeta].sup.(k)](s) the k-th derivative of the Riemann zeta function [zeta](s) for

non-negative integer k, where, [[zeta].sup.(0)](s) denotes the Riemann zeta function [[zeta].sup.(s)] itself.