A necessary and sufficient condition for a connected amenable group to have polynomial growth

A Degree Condition For Graphs to Have Connected (g, f)-Factors

On a Sufficient and Necessary Condition for a Multivariate Polynomial to Have Algebraically Dependent Roots - an Elementary Proof

In the paper we prove that a multivariate polynomial has algebraically dependent roots iff the coefficients are algebraic numbers up to a common proportional term. A complex analytic proof can be found in [2] with applications in the theory of linear functional equations, see also [3, an open problem, section 4.4] and [1]. Here we present an elementary proof involving cardinality properties and...

A necessary and sufficient eigenvector condition for a connected graph to be bipartite

We give a necessary and sufficient condition for a graph to be bipartite in terms of an eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the graph.

A Necessary and Sufficient Condition for Transcendency

As has been known for many years (see, e.g., K. Mahler, /. Reine Angew. Math., v. 166, 1932, pp. 118-150), a real or complex number f is transcendental if and only if the following condition is satisfied. To every positive number co there exists a positive integer n and an infinite sequence of distinct polynomials {p,(z)} = {pr + pr z + • • • + p z } at most of 0 1 n degree n with integral coef...

A Necessary and Sufficient Condition for Peace∗

This paper examines the possibility for two contestants to agree on a peace settlement thereby avoiding a contest, in which each would exert a costly effort given the posterior distributions inferred from the negotiation. I find a necessary and sufficient condition of the prior distributions for there to exist a negotiation mechanism that admits a peace-ensuring perfect Bayesian equilibrium. Th...