The numerical integration of relative equilibrium solutions. The nonlinear Schrodinger equation

We analyse different error propagation mechanisms for conservativeand nonconservative time-integrators of nonlinear Schrödingerequations. We use a geometric approach based on interpretingwaves as relative equilibria.     

Products of Ratios of Consecutive Integers

Take the product of the numbers (n/(n+1))^∊_n for 1≤ n < N, where each ∊_n is ± 1. Express the product as a/b in lowest terms. Evidently the minimal possible value for a over all choices for ∊_n is 1; just take each ∊_n = 1, or each ∊_n = 0. Denote the maximal possible value of a by A(N). It is known from work of Nicolas and Langevin that (log 4+o(1))N≤ log A(N)≤(2/3+o(1))Nlog N. Using the Rosse–Iwaniec sieve, we improve the lower bound to the same order of magnitude as the upper bound.For Jean-Louis Nicolas, on his sixtieth birthday2000 Mathematics Subject Classification: Primary—11N56; Secondary—11N36     

Correction to: Moyennes effectives de fonctions multiplicatives complexes

The Ramanujan Journal - The original version of the article unfortunately contained a number of inaccuracies in equations and statement which however do not essentially scrutinize the validity of...     

A spark chamber measurement of the reactions π + p → ϱ + p at 15 GeV/c

A new method, using spark chambers, for the study of the reactions π^± + p → ?^± + p is described. The charged pion and both γ rays from the π^± decay are detected. Differential and integrated cross sections σ_π⁺=50 ± 9 μb, σ_π⁻=47 ± 9 μb) for 0.0 ?|t|?1. (GeV/c)² and a laboratory momentum (p_Lab) of 15 GeV/c are presented. The momentum dependence of σ_γ^± is well fitted from 2.7 to 16 GeV/c by σ = Kp_Lab⁻ with n_γ⁺ = 1.80 ± 0.80 and n_γ⁻ = 1.87 ± 0.15.     

A note on the distribution of some additive functions

Let f denote an additive arithmetical function with continuous limiting distribution F on the integers. Then f also has a limiting distribution G on shifted primes. Under some growth conditions on the values of f at primes, we provide optimal lower bounds for the modulus of continuity of F and G, at all points from a specified infinite set.     

New blow-up rates for fast controls of Schrödinger and heat equations

We consider the null-controllability problem for the Schrödinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates (see [L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations 204 (2004) 202-226; L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal. 172 (2004) 429-456; L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, J. Funct. Anal. 218 (2005) 425-444; L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrödinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in [T.I. Seidman, The coefficient map for certain exponential sums, Nederl. Akad. Wetensch. Indag. Math. 48 (1986) 463-478] and in [T.I. Seidman, S.A. Avdonin, S.A. Ivanov, The “window problem” for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000) 233-254]. These results are used, following [L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772], to deal with the case of several space dimensions.