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.     

Highly Efficient Copper(II) Ion Sorbents Obtained by Calcium Carbonate Mineralization on Functionalized Cross‐Linked Copolymers

A new type of Cu^II ion sorbents is presented. These are obtained by CaCO₃ mineralization from supersaturated solutions on gel‐like cross‐linked polymeric beads as insoluble templates. A divinylbenzene–ethylacrylate–acrylonitrile cross‐linked copolymer functionalized with weakly acidic, basic, or amphoteric functional groups has been used, as well as different initial inorganic concentrations and addition procedures for CaCO₃ crystal growth. The morphology of the new composites was investigated by SEM and compared to that of the unmodified beads, and the polymorph content was established by X‐ray diffraction. The beads, before and after CaCO₃ mineralization, were tested as sorbents for Cu^II ions. The newly formed patterns on the bead surface after Cu^II sorption were observed by SEM, and the elemental distribution on the composites and the chemical structure of crystals after interaction with Cu^II were investigated by EDAX elemental mapping and by FTIR‐ATR spectroscopy, respectively. The sorption capacity increased significantly after CaCO₃ crystals growth on the weak anionic bead surface (up to 1041.5 mg Cu^II/g sample) compared to that of unmodified beads (491.5 mg Cu^II/g sample).     

Metal-Free Arylation of Benzothiophenes at C4 by Activation as their Benzothiophene S-Oxides

Benzothiophenes, activated by oxidation to the corresponding S-oxides, undergo C−H/C−H-type coupling with phenols to give C4 arylation products. While an electron-withdrawing group at C3 of the benzothiophene is important, the process operates without a directing group and a metal catalyst, thus rendering it compatible with sensitive functionalities—e.g. halides and formyl groups. Quantum chemical calculations suggest a formal stepwise mechanism involving heterolytic cleavage of an aryloxysulfur species to give a π-complex of the corresponding benzothiophene and a phenoxonium cation. Subsequent addition of the phenoxonium cation to the C4 position of the benzothiophene is favored over the addition to C3; Fukui functions predict that the major regioisomer is formed at the more electron-rich position between C3 and C4. Varied selective manipulation of the benzothiophene products showcase the synthetic utility of the metal-free arylation process.     

A Buzano type inequality for two Hermitian forms and applications

A Buzano type inequality for two nonnegative Hermitian forms is obtained. Applications to inequalities for norm and numerical radius of bounded linear operators in complex Hilbert spaces are given.     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

A Generalization of the Univalence on the Boundary Theorem with Applications to Sobolev Mappings

We give a generalization of a known theorem from classical complex analysis, namely the univalence on the boundary theorem. We apply this result to obtain some univalence conditions for Sobolev mappings \(f\in C({\overline{D}},{\mathbb {R}}^n)\bigcap W_{loc}^{1,q}(D,{\mathbb {R}}^n)\) which are injective on \(\partial D\), in connection with a known result of Ball from (Proc R Soc Edinb Sect A 88(3–4):315–328, 1981) modeling nonlinear elasticity.     

A Combined Experimental and Computational Study of Chrysanthemin as a Pigment for Dye-Sensitized Solar Cells

The theoretical study of chrysanthemin (cyanidin 3-glucoside) as a pigment for TiO₂-based dye-sensitized solar cells (DSSCs) was performed with the GAUSSSIAN 09 simulation. The electronic spectra of neutral and anionic chrysanthemin molecules were calculated by density functional theory with B3LYP functional and DGDZVP basis set. A better energy level alignment was found for partially deprotonated molecules of chrysanthemin, with the excited photoelectron having enough energy in order to be transferred to the conduction band of TiO₂ semiconductor in DSSCs. In addition, we used the raw aqueous extracts of roselle (Hibiscus sabdariffa) calyces as the source of chrysanthemin and the extracts with various pH values were tested in DSSCs. The extracts and photosensitized semiconductor layers were characterized by UV-Vis spectroscopy, and DSSCs based on raw extracts were characterized by current density-voltage measurements.     

Isoflavonoid-Antibiotic Thin Films Fabricated by MAPLE with Improved Resistance to Microbial Colonization

Staphylococcus aureus (Gram-positive) and Pseudomonas aeruginosa (Gram-negative) bacteria represent major infectious threats in the hospital environment due to their wide distribution, opportunistic behavior, and increasing antibiotic resistance. This study reports on the deposition of polyvinylpyrrolidone/antibiotic/isoflavonoid thin films by the matrix-assisted pulsed laser evaporation (MAPLE) method as anti-adhesion barrier coatings, on biomedical surfaces for improved resistance to microbial colonization. The thin films were characterized by Fourier transform infrared spectroscopy, infrared microscopy, and scanning electron microscopy. In vitro biological assay tests were performed to evaluate the influence of the thin films on the development of biofilms formed by Gram-positive and Gram-negative bacterial strains. In vitro biocompatibility tests were assessed on human endothelial cells examined for up to five days of incubation, via qualitative and quantitative methods. The results of this study revealed that the laser-fabricated coatings are biocompatible and resistant to microbial colonization and biofilm formation, making them successful candidates for biomedical devices and contact surfaces that would otherwise be amenable to contact transmission.     

An Extremal Problem Arising in the Dynamics of Two-Phase Materials That Directly Reveals Information about the Internal Geometry

In two-phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc.