Exact first and second derivatives by automatic differentiation

Automatic differentiation (AD) is a method for computing derivatives of outputs of numerical programs with respect to its inputs both accurately (with machine precision) and efficiently. The two basic modes of AD – forward and reverse – and combinations thereof yield products of a vector with the Jacobian, its transposed, or the Hessian, respectively. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the irreducibility of the two variable zeta-function for curves over finite fields

R. Pellikaan (Arithmetic, Geometry and Coding Theory, Vol.  4, Walter de Gruyter, Berlin, 1996, pp. 175–184) introduced a two variable zeta-function Z(t,u) for a curve over a finite field $\text{F}{}_{\mathrm{q}}$ which, for u=q, specializes to the usual zeta-function and he proved rationality: Z(t,u)=(1?t)^?1(1?ut)^?1P(t,u) with $\text{P(t,u)∈}\text{Z}\text{[t,u]}$. We prove that P(t,u) is absolutely irreducible. This is motivated by a question of J. Lagarias and E. Rains about an analogous two variable zeta-function for number fields. To cite this article: N. Naumann, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Trifluoromethylselenate(0), [SeCF3]—, and Trifluoromethyltellurate(0), [TeCF3]— — Crystalline Products of the Reduction of the Bis(trifluoromethyl)dichalkogen Compounds,E2(CF3)2 (E = Se,Te), with Tetrakis(dimethylamino)ethene (TDAE)

[Bis(dimethylamino)ethanediylidene]bis(dimethylammonium) bis(trifluoromethylchalcogenates(0)), (TDAE)[ECF₃]₂ (E = Se, Te), are quantitatively formed from the reductions of E₂(CF₃)₂ with tetrakis(dimethylamino)ethene, TDAE. Both compounds are bright yellow to orange solids which crystallize isostructurally with the corresponding sulfur derivative in the orthorhombic space group Pbca (No. 61). (TDAE)[SeCF₃]₂ has alternatively been prepared by cation exchange from [NMe₄]SeCF₃ and (TDAE)Br₂.     

Fluoride-mediated selective cross-coupling reactions of alkyl halides and trimethyl(perfluoroalkyl)silanes, Me3SiRf (Rf = CF3, C2F5) in the absence of any catalysts

A temperature range of −18 °C to room temperature was found to be effective for selective fluoride-mediated cross-coupling reactions of trimethyl(perfluoroalkyl)silanes, Me₃SiCF₃ and Me₃SiC₂F₅, and alkyl halides, RX (X = Br, I) in the absence of any catalyst.