Richard E. Crandall

Learn More
Box integrals—expectations | r| s or | r − q| s over the unit n-cube—have over three decades been occasionally given closed forms for isolated n, s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of n = 1, 2, 3, 4 dimensions the box integrals are for any integer s hypergeometrically closed ("(More)
We propose a theory to explain random behavior for the digits in the expansions of fundamental mathematical constants. At the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base-2 normality—namely bit randomness in a specific technical(More)
Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [5], we investigate a complementary, statistical picture in which pseu-dorandom number generators (PRNGs) are central. Some rigorous results are achieved: We establish b-normality for constants of the form i 1/(b m i c n i) for certain sequences (m i), (n i)(More)
From an experimental-mathematical perspective we analyze " Ising-class " integrals. These are structurally related n-dimensional integrals we call C n , D n , E n , where D n is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyze C n := 4 n! ∞ 0 · · · ∞ 0 1 n j=1 (u j + 1/u j) 2 du 1 u 1 · · · du n u n. We(More)
D > 1, alors le nombre #(|y|, N) de 1 dans le développement de |y| parmi les N premiers chiffres satisfait #(|y|, N) > CN 1/D avec un nombre positif C (qui dépend de y), la minorationétant vraie pour tout N suffisamment grand. On en déduit la transcen-dance d'une classe de nombres réels n≥0 1/2 f (n) quand la fonc-tion f , ` a valeursentì eres, crot(More)
We apply experimental-mathematical principles to analyze integrals C n,k := 1 n! ∞ 0 · · · ∞ 0 dx 1 dx 2 · · · dx n (cosh x 1 + · · · + cosh x n) k+1. These are generalizations of a previous integral C n := C n,1 relevant to the Ising theory of solid-state physics [8]. We find representations of the C n,k in terms of Meijer G-functions and nested-Barnes(More)