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.     

Finsler vector bundles -- Metrizable connections

Periodica Mathematica Hungarica -     

Mathematical modeling of two-photon thermal fields in laser–solid interaction

In this paper, we have developed an analytical model to study the temperature distributions in IR optical materials heated by laser pulses. Our model takes into account the two-photon absorption (TPA). The calculations are based on a three-dimensional model of heat diffusion in solids using the integral transform method. We find out the rigorous analytical expression of the thermal field when one considers both one- and two-photon absorption. The model is valid for any laser–solid system whose interaction can be described by the generalized Beer–Lambert law. Specific results are presented for an application of the model to ZnSe sample. We find out that TPA can produce detectable temperature variation.     

Systematic formulations for electronegativity and hardness and their atomic scales within density functional softness theory

A unified Mulliken valence with Parr ground‐state electronegativity picture is presented. It provides a useful analytical tool on which the absolute hardness as well ionization potential and electron affinity functionals are based. For all these chemical reactivity indices, systematic approximate density functionals are formulated within density functional softness theory and are applied to atomic systems. For the absolute hardness, a special relationship with the new electronegativity ansatz and a particular atomic trend paralleling the absolute electron affinity are established that should complement and augment the earlier finite‐difference energetic approach. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Chelate polymers: II. Some novel transition metals complexes with azomethine‐containing siloxanes and their polyesters

New Schiff bases of 2,4‐dihydroxybenzaldehyde with siloxane‐α,ω‐diamines having different numbers of siloxane units in the chain have been synthesized and characterized by spectroscopy, elemental and thermal analyses. These azomethines were found to form complexes readily with copper(II), nickel(II), cobalt(II), cadmium(II) and zinc(II). From IR and UV–Vis studies, the phenolic oxygen and imine nitrogen of the ligand were found to be the coordination sites. Thermogravimetric analysis (TGA) data indicate the chelates to be more stable than the corresponding ligands. The melting points increase with shortening of the siloxane segment from azomethine, as well as the result of complexation. The chelates obtained were covalently inserted in polymeric linear structures by polycondensation through the OH‐difunctionalized ligand with 1,3‐bis(carboxypropyl)tetramethyldisiloxane. Direct polycondensation, assisted either by acetic anhydride or N,N′‐dicyclohexylcarbodiimide as dehydrating agent and the complex 4‐(dimethylamino)pyridinium 4‐toluenesulfonate as catalyst, was used for the synthesis of these compound types. The structures of the polymers obtained were confirmed by IR, UV and ¹H NMR. Characterization was undertaken by TGA, solubility tests and viscosity measurements. Copyright © 2003 John Wiley & Sons, Ltd.     

Limits of tangents and minimality of complex links

We show that the complex link of a large class of space germs (X,x₀) is characterized by its “simplicity”, among the Milnor fibres of functions with isolated singularity on X. This amounts to the minimality of the Milnor number, whenever this number is defined. Such a phenomenon has been first pointed out in case (X,x₀) is an isolated hypersurface singularity, by Teissier (Cycles évanescents, sections planes et conditions de Whitney, in: Singularités à Cargèse 1972, Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285-362).     

Thermochemically induced binding of methyl orange to polycations containing primary amine groups

The thermochemical transformation of electrostatically formed complexes of methyl orange (MO) with polycations containing primary amine groups such as ammonium salts afforded new polymers with a high concentration of covalently bound 4‐N,N‐dimethylaminoazobenzene groups in the side chain. Poly(allylamine hydrochloride) and poly(β‐aminoethylene acrylamide hydrochloride) were employed as support polycations for MO. The transformation of sulfonate–ammonium ion pairs into sulfonamide bonds, via heating at an elevated temperature, was supported by the polymer properties before and after the thermal treatment. The polymer structure changes were monitored with elemental analysis, Fourier transform infrared, ¹H NMR, and ultraviolet–visible absorption spectroscopy, and thermogravimetric analysis. The spacer length between the backbone and azobenzene structures used as side chains strongly influenced the polymer properties before and after the heat‐induced reaction. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5898–5908, 2006     

On hermitian polynomial optimization

We compare three levels of algebraic certificates for evaluating the maximum modulus of a complex analytic polynomial, on a compact semi-algebraic set. They are obtained as translations of some recently discovered inequalities in operator theory. Although they can be stated in purely algebraic terms, the only known proofs for these decompositions have a transcendental character. Received: 27 June 2005     

Tensor product surfaces of euclidean plane curves

Recently B.Y. CHEN initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold [3]. In [6] the particular case of tensor product of two Euclidean plane curves was studied. The minimal one were classified, and necessary and sufficient conditions for such a tensor product to be totally real or complex or slant were established. In the present paper we study for tensor product of Euclidean plane curves the problem of B.Y. CHEN: to what extent do the properties of the tensor product immersion f ? h of two immersions f, h determines the immersions f, h ? [3]