Roth’s theorem in many variables

We prove that if A ? {1, ..., N} has no nontrivial solution to the equation x ₁ + x ₂ + x ₃ + x ₄ + x ₅ = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \) , c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.     

Factorization of selfadjoint quadratic matrix polynomials with real spectrum

Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.     

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

High-resolution plane-wave decomposition in an auditorium using a dual-radius scanning spherical microphone array

The spatial and temporal distribution of early reflections in an auditorium is considered important for sound perception. Previous studies presented measurement and analysis methods based on spherical microphone arrays and plane-wave decomposition that could provide information on the direction and time of arrival of early reflections. This paper presents recent results of room acoustics analysis based on a spherical microphone array, which employs high spherical harmonics order for improved spatial resolution, and a dual-radius spherical measurement array to avoid ill-conditioning at the null frequencies of the spherical Bessel function. Spatial-temporal analysis is performed to produce directional impulse responses, while analysis based on the windowed Fourier transform is employed to detect direction of arrival of individual reflections at selected frequencies. Experimental results of sound-field analysis in a real auditorium are also presented.     

A structural study of (1RS,2SR,3RS,4SR,5RS)‐2,4‐dibenzoyl‐1,3,5‐triphenylcyclohexan‐1‐ol chloroform hemisolvate and (1RS,2SR,3RS,4SR,5RS)‐2,4‐dibenzoyl‐1‐phenyl‐3,5‐bis(2‐methoxyphenyl)cyclohexan‐1‐ol

(1RS,2SR,3RS,4SR,5RS)‐2,4‐Dibenzoyl‐1,3,5‐triphenylcyclohexan‐1‐ol or (4‐hydroxy‐2,4,6‐triphenylcyclohexane‐1,3‐diyl)bis(phenylmethanone), C₃₈H₃₂O₃, (1), is formed as a by‐product in the NaOH‐catalyzed synthesis of 1,3,5‐triphenylpentane‐1,5‐dione from acetophenone and benzaldehyde. Single crystals of the chloroform hemisolvate, C₃₈H₃₂O₃·0.5CHCl₃, were grown from chloroform. The structure has triclinic (P) symmetry. One diastereomer [as a pair of (1RS,2SR,3RS,4SR,5RS)‐enantiomers] of (1) has been found in the crystal structure and confirmed by NMR studies. The dichoromethane hemisolvate has been reported previously [Zhang et al. (2007). Acta Cryst. E 63 , o4652]. (1RS,2SR,3RS,4SR,5RS)‐2,4‐Dibenzoyl‐3,5‐bis(2‐methoxyphenyl)‐1‐phenylcyclohexan‐1‐ol or [4‐hydroxy‐2,6‐bis(2‐methoxyphenyl)‐4‐phenylcyclohexane‐1,3‐diyl]bis(phenylmethanone), C₄₀H₃₆O₅, (2), is also formed as a by‐product, under the same conditions, from acetophenone and 2‐methoxybenzaldehyde. Crystals of (2) have been grown from chloroform. The structure has orthorhombic (Pca2₁) symmetry. A diastereomer of (2) possesses the same configuration as (1). In both structures, the cyclohexane ring adopts a chair conformation with all bulky groups (benzoyl, phenyl and 2‐methoxyphenyl) in equatorial positions. The molecules of (1) and (2) both display one intramolecular O—H...O hydrogen bond.     

Aza-substitution,benzo-annulation effects and catalytic activity of <Emphasis Type="Italic">β</Emphasis>-octaphenyl-substituted tetrapyrrolic macroheterocyclic cobalt complexes. I. heterogeneous catalysis

The influence of the size of conjugated π-system on catalytic activity of cobalt complex with β-octaphenylporphyrin and its tetraaza-, tetrabenzo and tetrabenzotetraaza derivatives was studied in present work. It is found that catalytic activity for oxidation of sulfur-containing compounds increases under extension of conjugated macrocycle system according to the following series CoP < CoBP ≤ CoPz < CoPPz ? CoPc.     

On a single discrete scale for preventive maintenance with two shock processes affecting a complex system

A new approach to optimal maintenance of systems (networks) is suggested. It is applied to systems subject to two external independent shock processes. A system ‘consists’ of two parts, and each shock process affects only its own part. A new notion of bivariate signature is suggested and used for obtaining survival characteristics of a system and further optimization of the preventive maintenance actions. The preventive maintenance optimization is considered in the univariate discrete scale that counts the overall numbers of shocks of both types. An example of a transportation network is considered. Copyright © 2016 John Wiley & Sons, Ltd.