Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle and the Ordering Principle

In this paper, we investigate the total coefficient size of Nullstellensatz proofs. We show that Nullstellensatz proofs of the pigeonhole principle on $n$ pigeons require total coefficient size $2^{\Omega(n)}$ and that there exist Nullstellensatz proofs of the ordering principle on $n$ elements with total coefficient size $2^n - n$

On the Construction of Radio Environment Maps for Cognitive Radio Networks

The Radio Environment Map (REM) provides an effective approach to Dynamic Spectrum Access (DSA) in Cognitive Radio Networks (CRNs). Previous results on REM construction show that there exists a tradeoff between the number of measurements (sensors) and REM accuracy. In this paper, we analyze this tradeoff and determine that the REM error is a decreasing and convex function of the number of measurements (sensors). The concept of geographic entropy is introduced to quantify this relationship. And the influence of sensor deployment on REM accuracy is examined using information theory techniques. The results obtained in this paper are applicable not only for the REM, but also for wireless sensor network deployment.Comment: 6 pages, 7 figures, IEEE WCNC conferenc