Let M(x) = ∑ 1≤n≤x μ(n) where μ is the Möbius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that M(x) = O(x1/2+ ) for all > 0. There has been much interest and progress in further bounding M(x) under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of M(x) √ x exp ( (log x)(log log x) ) (setting c to 14, ...

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number independent sets in bipartite graphs. Our first algorithm applies to d $$ -regular, graphs satisfying a weak expansion condition: when is constant, and Ω ( log 2 / ) \Omega \left({\log}^2d/d\right) -expander, obtain an FPTAS sets. Previously such result > 5 d>5 was k...

Could calculus on graphs have emerged by the time of Archimedes, if function, graph theory and matrix concepts were available 2300 years ago? 1. Single variable calculus Calculus on integers deals with functions f (x) like f (x) = x 2. The difference Df (x) = f (x) = f (x + 1) − f (x) = 2x + 1 as well as the sum Sf (x) = x−1 k=0 f (k) with the understanding Sf (0) = 0, Sf (−1) = f (−1) are func...

We present the best known separation between tree-like and general resolution, improving on the recent $exp(n^{\varepsilon}$ separation of [BEGJ98]. This is done by constructing a natural family of contradictions, of size $n$, that have $O(n)$-size resolution refutations, but only $exp(\Omega(n / log n))$size tree-like refutations. This result implies that the most commonly used automated theor...

Abstract We show that for all $n\leq X$ apart from $O(X\exp (-c(\log X)^{1/2}(\log \log X)^{1/2}))$ exceptions, the alternating group $A_{n}$ is invariably generated by two elements of prime order. This answers (in a quantitative form) question Guralnick, Shareshian, and Woodroofe.

WinBUGS code to fit the multivariate Bayesian relative risk model: model{ for( i in 1 : N ) { for( j in 1 : K ) { Y[i, j] ~ dpois(lambda[i, j]) # distribution of observations lambda[i, j] E[i, j] * theta[i, j] theta[i, j] exp(phi[ i, j]) # log parametrization } phi[i, 1:K ] ~ dmnorm(mu[ ], Omega[, ]) } for(j in 1:K){ mu[j] ~ dunif( 2,2) } Omega[1:K, 1:K ] ~ dwish(R[, ], 12) # Wishart on prec. m...