The Landscape of the Spiked Tensor Model

We consider the problem of estimating a large rank-one tensor u ^⊗k ∈ (ℝⁿ)^⊗k , k ≥ 3 , in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio λ  _Bayes = O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial-time algorithm is known that achieved this goal unless λ ≥ Cn^{(k − 2)/4} , and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n^{(k − 2)/4} . In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n , and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ(λ^{−1/(k − 1)}) around the maximum circle that is orthogonal to u . For local maxima, this band shrinks to be of size Θ(λ^{−1/(k − 2)}) . These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc.     

Preparation and Chemical and Biological Studies of the Novel Complex μ-1,4,5-triphenyl-1,3,4-triazole-2-thiolate-bis[dichlorotriethylphosphineplatinum(II)]a

The mesoionic compound, 1,4,5-triazole-2-thiolate, is a versatile ligand, capable of binding metal centres in different ways. The present study shows how it reacts with the bridging platinum(II) complex μ-dichlorobis[(chlorotriethylphosphineplatinum(II)], [Pt₂(PEt₃)₂Cl₄], originating a bimetallic complex. Both the sulphur atom and nitrogen 3 behave as basic sites, coordinating different platinum centres. An interesting isomerization process, shown by ³¹P{¹H} n.m.r. spectroscopy, occurs at the S-bonded platinum moiety, resulting in a fluxional behaviour of the complex. It is also shown to target CT-DNA with moderate binding affinity.To the memory of Dr. Wagner M. Teles (1971–2004).     

Influence of the exchange‐correlation functional in all‐electron calculations of the vibrational frequencies of corundum (α‐Al2O3)

The equilibrium structural parameters, high‐ and low‐frequency dielectric tensors, Born effective charges, and Γ‐point vibrational frequencies of bulk Al₂O₃ corundum are calculated by using the periodic, ab initio program CRYSTAL, which adopts an all‐electron Gaussian‐type basis set. The effect of basis set and the performance of three different functionals, i.e., LDA, PW91, and B3LYP, are discussed. The mean absolute deviation from the measured frequencies is as small as 7 cm^?1 for both the LDA and B3LYP functionals, indicating that these functionals perform extremely well in this case. The mean absolute deviation increases to 18 cm^?1 when the PW91 functional is used. All three functionals reproduce the equilibrium geometry of corundum to a high level of accuracy, with LDA and B3LYP outperforming PW91 slightly. The comparison of the current all‐electron calculations with previous plane‐wave, pseudo‐potential calculations shows an overall similar performance. The results of isotopic substitution for both Al and O are also presented. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Molecular spectra and molecular structure of heterocyclic compounds. Infrared and near ultraviolet absorption spectra of pyridine N-oxide derivatives

The infrared and ultraviolet spectra of many pyridine N-oxide derivatives have been studied. Ultraviolet spectra show a strong solvent effect in going from aprotic to hydroxylic solvents due to strong hydrogen bonding. Values of δ H° for several systems CH₃OH/XPyNO in tetra-chloroethylene solution were evaluated by infrared techniques and a linear correlation δ H° vs. δ v OH has been found. Features of the infrared and ultraviolet characteristic absorption bands were also investigated.     

Laser-spectrochromatographic identification of isomer in mixture

The combined “Gas Chromatograph — Laser Optoacoustic Spectrometer” (GC-LOAS) has been used to perform selective analysis of a mixture of fatty acid esters stereocisomers and a mixture of methylcyclopentadiene positional isomers. The GCLOAS system allows simultaneous recording of the retention time of the components in the mixture under analysis and their IR absorption spectra in the spectral operating range of the ¹²CO₂, ¹³CO₂ and HeNe lasers. It has been shown that the hydrogenated sample of fatty acids contains a lot of trans-isomer. Its content has been quantitatively estimated. The identification of methylcyclopentadiene isomers (both monomers and dimers) has been carried out, too.     

Influence of quaternary onium salts,crown ethers and cryptands on olefin epoxidations promoted by HOCl/ClO− in the Presence of Mn(III)-tetrakis(2,6-dichlorophenyl)porphyrin chloride

Reaction rates of alkene epoxidations, promoted by aqueous NaOCl and catalyzed by Mn(III)-tetrakis(2,6-dichlorophenyl)porphyrin chloride1 (P) in the presence of a lipophilic axial ligand (L) (e.g.N-hexylimidazole) and carried out under H₂O/CH₂Cl₂ two phase conditions at 0°C, are strongly enhanced by lowering the pH of the aqueous phase from 12.7 to 9.5. Under these conditions, a further relevant increase in the reaction rates can be achieved by adding phase-transfer catalysts (PT), e.g. quaternary ammonium salt 3, lipophilic crown ether4 or cryptand5, provided that the amount of L is very small (L/P=1 for very reactive alkenes, e.g. cyclooctene, and 10 for poorly reactive ones, e.g. 1-dodecene). In the case of cyclooctene epoxidation, the use of 0.006–0.03 mol. equiv. of PT completes the reaction in 1–10 min., the initial rates being up to 600 turnovers/min. with (2.2.2,C₁₀) cryptand. In the absence of the axial ligand, the quaternary ammonium salt3 and cryptand5 show an inhibitory effect. Such an effect is due to the formation of the poorly reactive Mn(P)CI species, by Cl– extraction to the organic phase. However, dibenzo crown ether 4 does not show this effect. In the presence of 4, and with L/P =1, the 1-dodecene epoxidation reaches 94% in 1 min. The unique behavior of crown ethers can be explained by their ineffectiveness in extracting alkali chlorides, providing a very low concentration of Cl– in the organic phase and thus avoiding the Mn(III)-porphyrin deactivation.This paper is dedicated to the memory of the late Dr C. J. Pedersen.     

Model-based clustering of probability density functions

Complex data such as those where each statistical unit under study is described not by a single observation (or vector variable), but by a unit-specific sample of several or even many observations, are becoming more and more popular. Reducing these sample data by summary statistics, like the average or the median, implies that most inherent information (about variability, skewness or multi-modality) gets lost. Full information is preserved only if each unit is described by a whole distribution. This new kind of data, a.k.a. “distribution-valued data”, require the development of adequate statistical methods. This paper presents a method to group a set of probability density functions (pdfs) into homogeneous clusters, provided that the pdfs have to be estimated nonparametrically from the unit-specific data. Since elements belonging to the same cluster are naturally thought of as samples from the same probability model, the idea is to tackle the clustering problem by defining and estimating a proper mixture model on the space of pdfs. The issue of model building is challenging here because of the infinite-dimensionality and the non-Euclidean geometry of the domain space. By adopting a wavelet-based representation for the elements in the space, the task is accomplished by using mixture models for hyper-spherical data. The proposed solution is illustrated through a simulation experiment and on two real data sets.     

La distribution on the crater surface of W-1%La2O3 produced by a single laser pulse

The W-1%La₂O₃ alloy has been irradiated by a single laser pulse (λ = 1064 nm) to simulate transient thermal loads of high energy occurring in a tokamak under operative conditions. A zone with a diameter of ~2 mm, namely, much larger than the focal spot, results to be affected by the pulse, and a crater of about 300 μm is observed in its center. La₂O₃ particles are not present inside the crater. The change of surface morphology is accompanied by elemental redistribution. Multipoint XPS analysis evidenced that the concentration of La is very low in the crater and increases moving toward the border of the affected zone while that of W shows an opposite trend. The composition changes involve only the outmost 5 nm of the sample: through depth profiling, no differences of chemical composition were detected deeper in the alloy between the center and external border of the affected area.     

XPS study of Cr segregation in a martensitic stainless steel

Cr martensitic steels are promising materials for structural applications in future nuclear fusion reactors. Because the embrittlement after tempering treatments can be a serious problem, the fracture mode of a steel with 10.5 wt% of Cr treated at 700°C for 18 h has been investigated through Charpy tests in the temperature range from −100°C to +150°C. X-ray photoelectron spectroscopy (XPS) analyses carried out on the fracture surfaces evidenced the segregation of Cr in both ductile and brittle (quasicleavage) fields. The unexpected result indicates that Cr segregation weakens the atomic bonds; thus, the fracture path in both the cases corresponds to the zones with higher Cr content.