Gradient methods for minimizing composite functions

In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two terms: one is smooth and given by a black-box oracle, and another is a simple general convex function with known structure. Despite the absence of good properties of the sum, such problems, both in convex and nonconvex cases, can be solved with efficiency typical for the first part of the objective. For convex problems of the above structure, we consider primal and dual variants of the gradient method (with convergence rate $O\left({1 \over k}\right)$ ), and an accelerated multistep version with convergence rate $O\left({1 \over k^2}\right)$ , where $k$ is the iteration counter. For nonconvex problems with this structure, we prove convergence to a point from which there is no descent direction. In contrast, we show that for general nonsmooth, nonconvex problems, even resolving the question of whether a descent direction exists from a point is NP-hard. For all methods, we suggest some efficient “line search” procedures and show that the additional computational work necessary for estimating the unknown problem class parameters can only multiply the complexity of each iteration by a small constant factor. We present also the results of preliminary computational experiments, which confirm the superiority of the accelerated scheme.     

Graphene supports the propagation of subwavelength optical solitons

Nonlinear propagation of light in a graphene monolayer is studied theoretically. It is shown how the large intrinsic nonlinearity of graphene at optical frequencies enables the formation of quasi one‐dimensional self‐guided beams (spatial solitons) featuring subwavelength widths at moderate electric‐field peak intensities. A novel class of nonlinear self‐confined modes resulting from the hybridization of surface plasmon polaritons with graphene optical solitons is also demonstrated.     

Asymptotic summation of perturbed linear difference systems in critical case

We propose an asymptotic summation method for certain class of linear difference systems. Both the ideas of the centre manifold theory and the averaging method are used to construct the asymptotics for solutions. We illustrate the asymptotic summation method by constructing the asymptotics for solutions of certain higher order scalar difference equation with an oscillatory decreasing coefficient.     

An unprecedented heterotrimetallic Fe/Cu/Co core for mild and highly efficient catalytic oxidation of cycloalkanes by hydrogen peroxide

An unprecedented hexanuclear heterotrimetallic Fe/Cu/Co complex bearing two Cu(mu-O)2Co(mu-O)2Fe cores is easily prepared by self-assembly and acts as a remarkable catalyst for the peroxidative oxidation of cycloalkanes under mild conditions.     

Heterometallic Co(III)(4)Fe(III)(2) Schiff Base Complex: Structure, Electron Paramagnetic Resonance, and Alkane Oxidation Catalytic Activity

The heterometallic complex [Co(4)Fe(2)OSae(8)]·4DMF·H(2)O (1) was synthesized by one-pot reaction of cobalt powder with iron chloride in a dimethylformamide solution of salicylidene-2-ethanolamine (H(2)Sae) and characterized by single crystal X-ray diffraction analysis, magnetic measurements, high frequency electron paramagnetic resonance (HF-EPR), and M?ssbauer spectroscopies. The exchange coupling in the Fe(III)-Fe(III) pair is of antiferromagnetic behavior with J/hc = -190 cm(-1). The HF-EPR spectra reveal an unusual pattern with a hardly detectable triplet signal of the Fe(III) dimer. The magnitude of D (ca. 13.9 cm(-1)) was found to be much larger than in related dimers. The catalytic investigations disclosed an outstanding activity of 1 toward oxidation of cycloalkanes with hydrogen peroxide, under mild conditions. The most efficient system showed a turnover number (TON) of 3.57 × 10(3) with the concomitant overall yield of 26% for cyclohexane, and 2.28 × 10(3)/46%, respectively, for cyclooctane. A remarkable turnover frequency (TOF) of 1.12 × 10(4) h(-1) (the highest initial rate W(0) = 3.5 × 10(-4) M s(-1)) was achieved in oxidation of cyclohexane. Kinetic experiments and selectivity parameters led to the conclusion that hydroxyl radicals are active (attacking C-H bonds) species. Kinetic and electrospray ionization mass spectrometry (ESI-MS) data allowed us to assume that the trinuclear heterometallic particle [Co(2)Fe(Sae)(4)](+), originated from 1 in solution, could be responsible for efficient generation of hydroxyl radicals from hydrogen peroxide.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Universal Method for Stochastic Composite Optimization Problems

A fast gradient method requiring only one projection is proposed for smooth convex optimization problems. The method has a visual geometric interpretation, so it is called the method of similar triangles (MST). Composite, adaptive, and universal versions of MST are suggested. Based on MST, a universal method is proposed for the first time for strongly convex problems (this method is continuous with respect to the strong convexity parameter of the smooth part of the functional). It is shown how the universal version of MST can be applied to stochastic optimization problems.     

Fission studies of highly excited nuclei

The binary fission process in Bi, Au, W, Sb, Ag and Ni induced by 1 GeV protons has been investigated. The energies, time-of-flight and angular correlations of coincident fragments were measured by using a double arm spectrometer permitting a direct mass determination. Mass distributions are found to change their shape for nuclei lighterW. The interaction of protons with nuclei followed by fission is described in the framework of the cascade-evaporation model and the statistical approach. The observed mass distributions are in a good agreement with the results of the calculation. The experimental results present evidence for existence of an instability region with respect to the mass asymmetry coordinate in nuclear fission.