The weighted Ramsey number, wR(n, k), is the minimum q such that there is an assignment of nonnegative real numbers (weights) to the edges of Kn with the total sum of the weights equal to ( n 2 ) and there is a Red/Blue coloring of edges of the same Kn, such that in any complete k-vertex subgraph H, of Kn, the sum of the weights on Red edges in H is at most q and the sum of the weights on Blue ...

This article defines a new minimization problem, the k-Compartment Problem, and presents its solution. The k-Compartment Problem is to determine the minimum sum of k + 1 line segments that intersect two parallel lines which form k compartments.

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A ⊂ Z such that |A+A| < |A−A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O’Bryant [MO] proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We p...

Let f(N) and xi(-1)(N) represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a N x infinity lattice, with f(N)-->f(infinity) as N-->infinity. We obtain analytic expressions for a(k) and b(k) in the expansions N( f(N)-f(infinity)) = SUM (k = 1)(infinity)a(k)/N(2k-1) and xi(-1)(N) = SUM (k = 1)(infinity)b(k)/N(2k-1) for squar...

An overpartition of the nonnegative integer n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. Let k ≥ 1 be an integer. An overpartition k-tuple of a positive integer n is a k-tuple of overpartitions wherein all listed parts sum to n. Let pk(n) be the number of overpartition k-tuples of n. In this paper, we will give a sh...

We consider two-person zero-sum stochastic games with perfect information and, for each k ∈ Z+, introduce a new payoff function, called the k-total reward. For k = 0 and 1 they are the so called mean and total rewards, respectively. For all k, we prove solvability of the considered games in pure stationary strategies, and show that the uniformly optimal strategies for the discounted mean payoff...

We call a subset A of the (additive) abelian group G t-independent if for all non-negative integers h and k with h + k ≤ t, the sum of h (not necessarily distinct) elements of A does not equal the sum of k (not necessarily distinct) elements of A unless h = k and the two sums contain the same terms in some order. A weakly t-independent set satisfies this property for sums of distinct terms. We ...

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1, 2, . . . , n} are B2[g]-s...

In cognitive multiple access networks, feedback is an important mechanism to convey secondary transmitter primary base station (STPB) channel gains from the primary base station (PBS) to the secondary base station (SBS). This paper investigates the optimal sum-rate capacity scaling laws for cognitive multiple access networks in feedback limited communication scenarios. First, an efficient feedb...

Let q be an odd number and Sq,0(n) the difference between the number of k < n, k ≡ 0 mod q, with an even binary digit sum and the corresponding number of k < n, k ≡ 0 mod q, with an odd binary digit sum. A remarkable theorem of Newman says that S3,0(n) > 0 for all n. In this paper it is proved that the same assertion holds if q is divisible by 3 or q = 4N + 1. On the other hand, it is shown tha...