Zeros of the Davenport-Heilbronn counterexample

We compute zeros off the critical line of a Dirichlet series considered by H. Davenport and H. Heilbronn. This computation is accomplished by deforming a Dirichlet series with a set of known zeros into the Davenport-Heilbronn series.

     

Some New Results for McKean’s Graphs with Applications to Kac’s Equation

The main goal of the present paper is to sharpen some results about the error made when the Wild sums, used to represent the solution of the Kac analog of Boltzmann’s equation, are truncated at the n-th stage. More precisely, in Carlen, Carvalho and Gabetta (J. Funct. Anal. 220: 362–387 (2005)), one finds a bound for the above-mentioned error which depends on (an ^Λ+ε). On the one hand, it is shown that Λ, the least negative eigenvalue of the linearized collision operator, is the best possible exponent. On the other hand, ε is an extra strictly positive number and a a positive coefficient which depends on ε too. Thus, it is interesting to check whether ε can be removed from the above bound. According to the aforesaid reference, this problem is studied here by means of the probability distribution of the depth of a leaf in a McKean random tree. In fact, an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with (a′ n ^Λ).     

On mixtures of distributions of Markov chains

Let X be a chain with discrete state space I, and V be the matrix of entries V_i,n, where V_i,n denotes the position of the process immediately after the nth visit to i. We prove that the law of X is a mixture of laws of Markov chains if and only if the distribution of V is invariant under finite permutations within rows (i.e., the V_i,n's are partially exchangeable in the sense of de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of V and Markov exchangeability in the sense of Diaconis and Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the Diaconis and Freedman basic result for mixtures of Markov chains.     

Conversion of n,n-dimethylhydrazones to carbonyl compounds by clay-supported ferric nitrate

Cleavage of N,N-dimethylhydrazones is cleanly effected with good yield in dichloromethane using the extremely inexpensive clay-supported ferric nitrate.     

An investigation of the dark formation of nitrous acid in environmental chambers

The formation of nitrous acid (HONO) in the dark from initial concentrations of NO₂ of 0.1–20 ppm in air, and the concurrent disappearance of NO₂, were monitored quantitatively by UV differential optical absorption spectroscopy in two different environmental chambers of ca.4300- and 5800-L volume (both with surface/volume ratios of 3.4 m^?1). In these environmental chambers the initial HONO formation rate was first order in the NO₂ concentration and increased with the water vapor concentration. However, the HONO formation rate was independent of the NO concentration and relatively insensitive to temperature. The initial pseudo-first-order consumption rate of NO₂ was (2.8 ± 1.2) × 10^?4 min^?1 in the 5800-L Teflon-coated evacuable chamber and (1.6 ± 0.5) × 10^?4 min^?1 in a 4300-L all-Teflon reaction chamber at ca.300 K and ca.50% RH. The initial HONO yields were ca.40–50% of the NO₂ reacted in the evacuable chamber and ca.10–30% in the all-Teflon chamber. Nitric oxide formation was observed during the later stages of the reaction in the evacuable chamber, but ca.50% of the nitrogen could not be accounted for, and gas phase HNO₃ was not detected. The implications of these data concerning radical sources in environmental chamber irradiations of NO_x? organic-air mixtures, and of HONO formation in polluted atmospheres, are discussed.