Hilbert's 16th Problem for Quadratic Systems. New Methods Based on a Transformation to the Lienard Equation

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained

Correlated electronic structure, orbital-dependent correlations, and Lifshitz transition in tetragonal FeS

Using density functional plus dynamical mean-field theory method (DFT+DMFT) with full self-consistency over the charge density, we study the effect of electronic correlations on the electronic structure, magnetic properties, orbital-dependent band renormalizations, and Fermi surface of the tetragonal phase of bulk FeS. We perform a direct structural optimization of the $P_4/nmm$ crystal structure of paramagnetic FeS, with respect to the lattice constant $a$ and the internal coordinate $z_\mathrm{S}$ of atom S. Our results show an anomalous sensitivity of the electronic structure and magnetic properties of FeS to fine details of its crystals structure. Upon expansion of the lattice volume, we observe a remarkable change of the electronic structure of FeS which is associated with a complete reconstruction of the Fermi surface topology (Lifshitz transition). This behavior is ascribed to a correlation-induced shift of the Van Hove singularity associated with the Fe $t_2$ orbitals at the $M$ point across the Fermi level. The Lifshitz phase transition is accompanied by a significant growth of local magnetic moments and emergence of strong orbital-selective correlations. It is seen as a pronounced anomaly (`kink') in the total energies upon expansion of the lattice, associated with a remarkable enhancement of compressibility. This behavior is accompanied by an orbital-dependent formation of local moments, a crossover from itinerant to localized orbital-selective moment behavior of the Fe $3d$ electrons. While exhibiting weak effective mass enhancement of the Fe $3d$ states $m^*/m \sim 1.3-1.4$, correlation effects reveal a strong impact on a position of the Van Hove singularity at the $M$ point, implying a complex interplay between electronic correlations and band structure effects in FeS

The Theory from Large Deviations for Random Processes and Strong Convergence of Stochastic Approximation Procedures

This paper deals with the application of "large deviation" theory to the analysis of stochastic approximation procedures. The approach allows to get new results in the asymptotical behaviour of stochastic procedures under very mild assumption about the "noise". The paper contains a short but illuminative survey of these results together with some new author's findings. For applications the last section seems to be interesting presenting some new ideas in multiobjective optimization

The neural network art which uses the Hamming distance to measure an image similarity score

This study reports a new discrete neural network of Adaptive Resonance Theory (ART-1H) in which the Hamming distance is used for the first time to estimate the measure of binary images (vectors) proximity. For the development of a new neural network of adaptive resonance theory, architectures and operational algorithms of discrete neural networks ART-1 and discrete Hamming neural networks are used. Unlike the discrete neural network adaptive resonance theory ART-1 in which the similarity parameter which takes into account single images components only is used as a measure of images (vectors) proximity in the new network in the Hamming distance all the components of black and white images are taken into account. In contrast to the Hamming network, the new network allows the formation of typical vector classes representatives in the learning process not using information from the teacher which is not always reliable. New neural network can combine the advantages of the Hamming neural network and ART-1 by setting a part of source information in the form of reference images (distinctive feature and advantage of the Hamming neural network) and obtaining some of typical image classes representatives using learning algorithms of the neural network ART-1 (the dignity of the neural network ART-1). The architecture and functional algorithms of the new neural network ART which has the properties of both neural network ART-1 and the Hamming network were proposed and investigated. The network can use three methods to get information about typical image classes representatives: teacher information, neural network learning process, third method uses a combination of first two methods. Property of neural network ART-1 and ART-1H, related to the dependence of network learning outcomes or classification of input information to the order of the vectors (images) can be considered not as a disadvantage of the networks but as a virtue. This property allows to receive various types of input information classification which cannot be obtained using other neural networks

Correlation strength, Lifshitz transition and the emergence of a two- to three-dimensional crossover in FeSe under pressure

We report a detailed theoretical study of the electronic structure, spectral properties, and lattice parameters of bulk FeSe under pressure using a fully charge self-consistent implementation of the density functional theory plus dynamical mean-field theory method (DFT+DMFT). In particular, we perform a structural optimization and compute the evolution of the lattice parameters (volume, $c/a$ ratio, and the internal $z$ position of Se) and the electronic structure of the tetragonal (space group $P4/nmm$) paramagnetic FeSe. Our results for the lattice parameters are in good quantitative agreement with experiment. The $c/a$ ratio is slightly overestimated by about $3$~\%, presumably due to the absence of the van der Waals interactions between the FeSe layers in our calculations. The lattice parameters determined within DFT are off the experimental values by a remarkable $\sim$$6$-$15$~\%, implying a crucial importance of electron correlations. Upon compression to $10$~GPa, the $c/a$ ratio and the lattice volume show a decrease by $2$ and $10$~\%, respectively, while the Se $z$ coordinate weakly increases by $\sim$$2$~\%. Most importantly, our results reveal a topological change of the Fermi surface (Lifshitz transition) which is accompanied by a two- to three-dimensional crossover. Our results indicate a small reduction of the quasiparticle mass renormalization $m^*/m$ by about $5$~\% for the $e$ and less than $1$~\% for the $t_2$ states, as compared to ambient pressure. The behavior of the momentum-resolved magnetic susceptibility $\chi({\bf q})$ shows no topological changes of magnetic correlations under pressure, but demonstrates a reduction of the degree of the in-plane $(\pi,\pi)$ stripe-type nesting. Our results for the electronic structure and lattice parameters of FeSe are in good qualitative agreement with recent experiments on its isoelectronic counterpart FeSe$_{1-x}$S$_x$.Comment: 10 pages, 6 figure

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered