We suggest an explanation based on the Blume-Capel model of why some layered compounds of the iron-intercalated transition metal dichalcogenides TaS2(Se2) exhibit spin-glass behavior, while another group of this family demonstrates low-temperature paramagnetism. In these materials, the doped Fe atoms either substitute the Ta atoms with losing their magnetic moments or sit between the TaS2(Se2) layers keeping their spin states. The Blume-Capel model allows us to introduce a chemical potential to control a balance of the intercalated elements of both types. The Ghatak-Sherrington theory of spin-glass behavior of this model predicts an existence of a tricritical point that means that there is a concentration threshold of Fe ions retaining their magnetic moments, above which spin-glass ordering occurs. Below the threshold, Fe ions behave as independent paramagnetic centers. We build temperature dependencies of magnetic susceptibility and field dependencies of magnetization to highlight specific features of the model related with a variable content of Fe ions in the high-spin state. A specific crystal structure of the layered transition metal dichalcogenides gives an opportunity to increase the concentration of ions with nonzero magnetic moments by co-intercalating non-Kramers 3d ions into the van der Waals gaps. This process may trigger spin-glass ordering in the initially paramagnetic Fe-doped TaS2(Se2) polytype complexes.

Based on the notion of emergent elasticity, we theoretically demonstrate that wavelike to particle-like crossover occurs in the chiral soliton lattice (CSL) formed in a monoaxial chiral helimagnet. We use the principle of virtual work and analyze the reaction of the CSL to the external disturbances, which is described in terms of the elastic continuum with the Voigt constant c(11). We also analyze dependence of "spring constant" of the CSL on the magnetic field strengths, when the magnetic field is applied perpendicular to the chiral axis. As a result, we found that the CSL exhibits crossover phenomena from a ductile-material-like behavior at small and intermediate magnetic field strengths to an elastic-medium-like behavior at higher magnetic fields for 0.65(c) less than or similar to H < H-c, with H-c being the critical field above which the forced ferromagnetic state appears. This ductile-to-elastic crossover is characterized by the emergence of the restoring elastic force and corresponds to the crossover from the wavelike collective soliton lattice dynamics to the particle-like dynamics of the individual kinks. This result may explain the crossover from "weakly nonlinear" to "highly nonlinear regimes" in the CSL formation, found in CrNb3S6, and the nature of which has remained unclear as yet.

A 1D model of a finite-length spin chain with the Dzyaloshinskii–Moriya interaction is considered. An algorithm for simulation of the problem is developed, and calculations that demonstrate the existence of discrete magnetic breathers of various types in the phase of forced ferromagnetism of a chiral helimagnet are presented. It is shown that the analysis with allowance for the Dzyaloshinskii–Moriya interaction yields a decrease in the amplitude of discrete magnetic breathers.

We generalize the theory of extending breather solutions in the anticontinuous limit to the case of discrete spin systems. We formulate necessary conditions and determine the upper bound for the intersite coupling constant for which the extension procedure is possible. Using a numerical algorithm, we obtain breather modes of a discrete spin chain related to single-site and two-site excitations of the anticontinuous limit and show their linear stability.

Today online technologies became an integral part of educational system. That created new challenges and arose new problems that concern all participants of the educational process. One of the challenges is to balance massive implementation of online learning with benefits of traditional education models, creating an individual approach to every student. That requires monitoring the results of current academic activities, reflected in passing the massive open online course (MOOC) milestones. Monitoring allows timely reaction on the emerging negative trends and strengthens position of testing as an important mean for monitoring student knowledge and determines requirements for the tools of such monitoring. In this paper, authors attempt to describe a mathematical model for the academic performance evolution used as a base for scrutinizing various aspects of learning success monitoring and prediction. Such model may be also used to validate internal testing systems of MOOCs. Analysis based on this model provide useful insights for all participants of educational process.In the paper authors derive equations that allow calculation of probabilities for student falling into academic performance groups (unsuccessful, successful, very successful) both in terms of final test in MOOC, and at any checkpoint. In this case, model parameters are determined either according to data on current performance of the students at milestone preceding the analyzed one or based on student performance in previous instances of a course.It is demonstrated that probability of transition among the academic performance groups in course of passing a milestone depends upon the number of students in each group before going through a checkpoint. These dependencies were defined for the courses conducted on Edex platform of a Ural Federal University. The obtained results were used to examine temporal evolution of distributions for academic performance.It is demonstrated that distributions can be rather complex, for example, displaying multimodal behavior for certain combinations of inter-group transition probabilities. Authors demonstrate possibility of dynamic chaos type behavior due to feedback influence of information about learning outcomes on academic activities of the students. Described algorithms allow assessing information contents and quality for tests in course materials, or the entire course. That allows introducing quality indicator for an online course. Indication describes the level of uncertainty related with students achieving overall MOOC study goals, and the rate this uncertainty decreases while students go through checkpoints. In order to illustrate the proposed approach, authors clustered different UrFU online courses using k-means algorithm based on the said quality parameter, and levels of academic activity in course of studies. Authors also analyzed possible reasons for differences in students’ academic performance distribution for representatives from different clusters.Authors analyzed potential applicability of the proposed mathematical probabilistic model for predicting both group and individual student progress and demonstrated use of the algorithms within the "Online Tutor" special information service framework developed by the authors and implemented in the Ural Federal University.

We demonstrate that magnetic discrete breathers (DBs) may appear in the monoaxial chiral helimagnet and describe the numerical algorithm of their search. We find out under what conditions these modes occur and examine their stability with the help of the Floquet analysis.

The worldwide spread of a new infection SARS-CoV-2 makes relevant the analysis of the different factors that lead to the vulnerability of modern civilization to previously unknown diseases. In this regard, the development of mathematical models describing the evolution of epidemics like COVID-19 and the identification of socio-economic factors affecting the epidemiological situation in regions is an important research task. The paper proposes a probabilistic mathematical model for the spread of the COVID-19 epidemic, which allows to analyze the evolution of the main characteristics of the disease and to assess main factors influencing them. The study is based on the official statistical data on the spread of the COVID 19 presented on coronavirus sites in the Russian Federation and other countries, the Yandex Data Lens dataset service, as well as the data from the Federal State Statistics Service. In the research some data mining methods were used for evaluation the model’s parameters. The model equations allow to predict the evolution of the disease and estimate the confidence interval of such prognosis. We estimated the ratio of detected and hidden cases of the disease, the distribution of the disease’s duration probability and its average value for different regions. It has been mathematically proven that the vaccination is the necessary and sufficient condition of achievement a stationary stable state - the cessation of a pandemic. The regions of Russian Federation were clustered by the course of the disease COVID-19 on the base of k-means method. The analysis of the most important socio-economic factors affecting the epidemiological situation was provided separately for each cluster.

We analyze spatially localized breather excitations for the model of a discrete Heisenberg spin chain, which includes the antisymmetric exchange interaction and single-ion anisotropy of the easy-plane type. In a finite size chain the breather modes may be indexed by number of embedded kink-antikink pairs forming the regular breather lattice. The influence of the antisymmetric exchange on properties of the discrete breather modes is examined.

By using numerical methods, we consider possibility of the spatially localized breather-type excitations for the model of the discrete Heisenberg spin chain, which includes the antisymmetric exchange interaction, the single-ion anisotropy of the easy plane type, and the Zeeman interaction with an external magnetic field, which exceeds a critical field of the transition to the state of forced ferromagnetism. To find solutions, we used equations of motion for spin operators. The case is considered when the frequency of discrete magnetic breathers lies above the upper edge of the spin wave spectrum. The chain length used in the calculations had been taken as 100 and 101 nodes, and the open boundary conditions were used. To carry out numerical calculations, an original program was written which simplifies maximally calculation of spin deviations inside the chain and enables us to use parallel computing technologies. A classification of symmetric and antisymmetric solutions was established, which made it possible to halve a number of calculations for spin deviations. An algorithm was elaborated to specify the amplitudes of spin deviations, that makes possible to construct a desired breather solution in a reasonable amount of time. The numerical calculations of the spin spatial distribution show that it has an antisymmetric ordering with respect to the center of the chain in the presence of the Dzyaloshinskii-Moriya interaction. The center of the solution can be located either between the lattice nodes (Page mode) in the case of an even number of lattice sites, or directly at the node in the case of the odd number (Takeno-Sievers mode). In the first case, the breather mode contains an odd number of pairs of magnetic kinkantikinks with a maximum of the envelope function at the center. Breather modes include an even number of these pairs for an odd number of lattice nodes.

Nonlinearity and discreteness are two pivotal factors for an emergence of discrete breather excitations in various media. We argue that these requirements are met in the forced ferromagnetic phase of the monoaxial chiral helimagnet CrNb$_3$S$_6$ due to specific domain structure of the compound. The stationary, time-periodic breather modes appear as the discrete breather lattice solutions whose period mismatches with a system size. Thanks to easy-plane single-ion anisotropy intrinsic to CrNb$_3$S$_6$, these modes are of the dark type with frequencies lying within the linear spin-wave band, close to its bottom edge. They represent cnoidal states of magnetization, similar to the well-known soliton lattice ground state, with differing but limited number of embedded $2\pi$-kinks. Linear stability of these dark breather modes is verified by means of the Floquet analysis. Their energy controlled by two parameters, namely the breather lattice period and amplitude, falls off linearly with a growth of the kink number. These results may pave a new path to design spintronic resonators on the base of chiral helimagnets.

We present a theory of standing spin wave (SSW) in a monoaxial chiral helimagnet. Motivated by experimental findings on the magnetic field-dependence of the resonance frequency in thin films of Cr${}$Nb$_{3} $S${}_{6}$[Goncalves et al., Phys. Rev. B95, 104415 (2017)], we examine the SSW over a chiral soliton lattice (CSL) excited by an ac magnetic field applied parallel and perpendicular to the chiral axis. For this purpose, we generalize Kittel-Pincus theories of the SSW in ferromagnetic thin films to the case of non-collinear helimagnet with the surface end spins which are softly pinned by an anisotropy field. Consequently, we found there appear two types of modes. One is a Pincus mode which is composed of a long-period Bloch wave and a short-period ripple originated from the periodic structure of the CSL. Another is a short-period Kittel ripple excited by space-periodic perturbation which exists only in the case where the ac field is applied perpendicular the chiral axis. We demonstrate that the existence of the Pincus mode and the Kittel ripple is consistent with experimentally found double resonance profile.

We discuss how the collective magnetic resonance response of the monoaxial chiral helimagnetic crystal ${\mathrm{CrNb}}_{3}{\mathrm{S}}_{6}$ can be tailored by changing the area of its magnetization plane. Micrometer-sized samples of this crystal yield a number of resonance modes occurring at frequencies ranging from 15 to 20 GHz, even in the absence of a magnetic field. Changes in the resonance bandwidth, of the order of several GHz, are attributed to the effect of the spatially nonuniform demagnetization fields on standing spin wave modes. This material hosts a chiral spin soliton lattice phase, whose field robustness, degree of controllability, and the relatively unexplored polarization-dependent microwave absorption make way for novel microwave applications.

The magnetic resonance properties of microsized monoaxial chiral crystals of CrNb3S6 are investigated. We observed that the resonance of the chiral soliton lattice is sensitive to the polarization of the driving microwave field. When the microwave field is parallel to the helical axis, the resonance is symmetric with regards to the magnetic field direction. In contrast, asymmetric field dependence emerges when the microwave field is perpendicular to the helical axis. The robustness of the chiral magnetic order, due to topological protection, allows tuning the resonance frequency in ways hardly accessible using nanopatterned films.

We analyze the critical behavior of magnetically ordered phases appearing in a monoaxial chiral helimagnet in a weak external magnetic field. Using the formalism of the equations of state in the critical region, we determine the temperature dependence of the order parameters for the conical phase and the soliton-lattice phase. We calculated the critical exponents and show that they coincide with those in the three-dimensional Heisenberg model.

We study the spin dynamics of a confined chiral soliton lattice whose ends are weakly held. We demonstrate that in this case the system possesses its own resonant frequency. To study features of the resonant dynamics, we analyze the collective motion of the system driven by an oscillating magnetic field directed along the chiral axis. By using the method of collective coordinates we find analytically the resonant frequency and verify the result by numerical simulation of the spin dynamics with the aid of Landau-Lifshitz-Gilbert equations. The numerical simulation shows an appearance of the asymmetric profile of the frequency response function with increasing ac field, which is typical for a nonlinear resonance. To give an explanation of this behavior, we invoke the multiple-time-scale method and predict an emergence of hysteresis phenomena. We also demonstrate that the spin-motive force is strongly amplified by the resonant oscillations.

We present how macroscopically coherent ordering within a chiral state can be manifested in the physical properties within the context of an archetypical system-the chiral spin soliton lattice in a monoaxial chiral magnet CrNb3S6. Using magnetotransport measurements and state-of-the-art Lorentz electron microscopy, we demonstrate spin soliton confinement in 1-mu m-wide grains with different crystalline chirality and discretized magnetoresistance in 10-mu m-wide crystals. Discretization effects are found to be prominent when the system size is reduced to the order of 10 mu m along the chiral axis. A consequence that we identify is a robust coherence of the chiral soliton lattice against deformation. The spin configuration at the grain boundaries, which leads to soliton confinement, is experimentally clarified.