Suprema of Chaos Processes and the Restricted Isometry Property

We present a new bound for suprema of a special type of chaos process indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and time‐frequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m ? s log² s log² n. © 2014 Wiley Periodicals, Inc.     

Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X\in {\mathbb{C}}^{n\times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j{a}_j^{?}$ for some measurement vectors $a_1,\dots ,a_m$, i.e., the measurements are given by $b_j=\mathrm{tr}(X{a}_j{a}_j^{?})$. The case where the matrix $X=x{x}^{?}$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements $b_j=|〈x,a_j〉{|}^2$) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors $a_j$, $j=1,\dots ,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective t-design with $t=4$. In the Gaussian case, we require $m\ge Crn$ measurements, while in the case of 4-designs we need $m\ge \mathit{Cr}n\log ?(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.     

Double proton transfer reactions with plateau-like transition state regions: pyrazole-trifluoroacetic acid clusters

The double proton transfer reactions between carboxylic acids and pyrazole were studied by computational methods up to the coupled-cluster level. Introduction of substituents allowed for a systematic modulation of the reaction profile, resulting in imaginary frequencies of the associated transition states between -1180 and -45 cm(-1). In the latter case, a local transition state is replaced by an extremely flat (plateaulike) transition region, which constitutes the transition from a concerted toward a stepwise mechanism. Vertical excitation energies along the reaction path reveal that the feature of a plateau in the ground state is mirrored in the excited states.     

Phosphorus Analogues of Methyl Nitrite and Nitromethane: CH3OPO and CH3PO2

Methoxyphosphinidene oxide (CH₃OPO) and isomeric methyldioxophosphorane (CH₃PO₂) are key intermediates in the degradation of organophosphorus compounds (OPCs). Unlike the nitrogen analogues CH₃ONO and CH₃NO₂, the experimental data for these two prototypical OPCs are scarce. By high‐vacuum flash pyrolysis (HVFP) of the diazide CH₃OP(O)(N₃)₂ at 1000 K, the cis and trans conformers of CH₃OPO have been generated in the gas phase and subsequently isolated in cryogenic Ar and N₂ matrices for IR spectroscopic characterization. Upon 266 nm laser irradiation of CH₃OPO, cis→trans conformational conversion occurs with concurrent isomerization to CH₃PO₂. The spectroscopic identification of CH₃OPO and CH₃PO₂ is supported by D‐, ¹³C‐, and ¹⁸O‐isotope labeling and quantum chemical calculations at the CCSD(T)‐F12a/cc‐pVTZ‐F12 level using configuration‐selective vibrational configuration interaction (VCI).     

Meta- and para-difunctionalization of arenes via a sulfoxide-magnesium exchange reaction

The aryl sulfoxide moiety (ArSO) allows an expedient two-step meta-, para-difunctionalization of readily available diaryl sulfoxides. In the first step, the sulfoxide plays the role of a directing metalation group. In the second step, triggered by i-PrMgCl x LiCl, it becomes a leaving group and undergoes a regioselective sulfoxide-magnesium exchange.     

Photolysis of Carbonyl Diisocyanate: Generation of Isocyanatocarbonyl Nitrene and Diazomethanone

The stepwise decomposition of carbonyl diisocyanate, OC(NCO)₂, has been studied by using IR spectroscopy in solid argon matrices at 16 K. Upon irradiation with an ArF laser (λ=193 nm), carbonyl diisocyanate split off CO and furnished a new carbonyl nitrene, OCNC(O)N, in its triplet ground state. Two conformers of the nitrene, syn and anti, that were derived from the two conformers of OC(NCO)₂ (62 % syn–syn and 38 % syn–anti) were identified and characterized by combining IR spectroscopy and quantum chemical calculations. Subsequent irradiation with visible light (λ>395 nm) caused the Curtius rearrangement of the nitrene into OCNNCO. In addition to the expected decomposition products, N₂ and CO, further photolysis of OCNNCO with the ArF laser yielded NOCN, through a diazomethanone (NNCO) intermediate. To further validate our proposed reaction mechanism, ArF‐laser photolysis of the closely related NNNNCO and cyclo‐N₂CO in solid argon matrices were also studied. The observations of NOCN and in situ CO‐trapped product OCNNCO provided indirect evidence to support the initial generation of NNCO as a common intermediate during the laser photolysis of OCNNCO, NNNNCO, and cyclo‐N₂CO.     

Sparse Legendre expansions via ℓ1-minimization

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre $s$-sparse polynomial of maximal degree $N$ can be recovered from $m?s{log}^4\left(N\right)$ random samples that are chosen independently according to the Chebyshev probability measure $d\nu \left(x\right)={\pi }^{?1}{(1?x^2)}^{?1/2}dx$. As an efficient recovery method, ${?}_1$-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces.     

Cooperative Al(salen)-pyridinium catalysts for the asymmetric synthesis of trans-configured β-lactones by [2+2]-cyclocondensation of acylbromides and aldehydes: investigation of pyridinium substituent effects

The trans-selective catalytic asymmetric formation of β-lactones constitutes an attractive surrogate for anti-aldol additions. Recently, we have reported the first catalyst which is capable of forming trans-β-lactones with high enantioselectivity from aliphatic (and aromatic) aldehyde substrates by cyclocondensation with acyl bromides. In that previous study the concepts of Lewis acid and organic aprotic ion pair catalysis were combined in a salen-type catalyst molecule. Since a pyridinium residue on the salen periphery is essential for high trans- and enantioselectivity, we were interested in the question of whether substituents on the pyridinium rings could be used to further improve the catalyst efficiency, as they might have a significant impact on the effective charges within the heterocycles. In the present study we have thus compared a small library of aluminum salen/bispyridinium catalysts mainly differing in the substituents on the pyridinium residues. As one result of these studies a new catalyst was identified which offers slightly superior stereoselectivity as compared to the previously reported best catalyst. NBO calculations have revealed that the higher stereoselectivity can arguably not be explained by the variation of the effective charge.     

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