Quantum chemical DFT study of 4-azatricyclo [5.2.2.02,6] undecane-3,5,8-trione

Fourier-transform-Raman and infrared spectrum of 4-azatricyclo [5.2.2.0^2,6] undecane-3,5,8-trione were recorded and analyzed. The vibrational wavenumbers were examined theoretically using the Gaussian03 set of quantum chemistry codes, and the normal modes were assigned by potential energy distribution (PED) calculations. The first hyperpolarizability, predicted infrared intensities and Raman activities are reported. The calculated first hyperpolarizability is comparable with reported values of similar structures which makes this compound an attractive object for future studies of nonlinear optics. Optimized geometrical parameters of the compound are in agreement with similar reported structures. The red shift of the NH stretching wavenumber in the infrared spectrum from the computational wavenumber indicates the weakening of NH bond.     

La somme des chiffres des carrés

In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of squares written in a basis q ⩾ 2 is equidistributed in arithmetic progressions.     

Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel. These scenarios imply that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.