Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et?al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S ₄-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.     

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.     

Topological properties of strongly monotone planar vector fields

We consider strongly monotone continuous planar vector fields with a finite number of fixed points. The fixed points fall into three classes, attractors, repellers and saddles. Naturally, the relative positions of the fixed points must obey a set of restrictions imposed by monotonicity. The study of these restrictions is the main goal of the paper. With any given vector field, we associate a matrix describing the arrangement of the fixed points on the plane. We then use these matrices to formulate simple necessary and sufficient conditions which allow one to determine whether a finite set of attractors, repellers and saddles at given positions on the plane can be realized as the fixed point set of a strongly monotone vector field.     

High-frequency generation in two coupled semiconductor superlattices

We theoretically show that two semiconductor superlattices arranged on the same substrate and coupled with the same resistive load can be used for a generation of high-frequency periodic and quasiperiodic signals. Each superlattice involved is capable to generate current oscillations associated with drift of domains of high charge concentration. However, the coupling with the common load can eventually lead to synchronization of the current oscillations in the interacting superlattices. We reveal how synchronization depends on detuning between devices and the resistance of the common load, and discuss the effects of coupling and detuning on the high-frequency power output from the system.     

Electronic vs. paper textbook presentations of the various aspects of mathematics

Based in part on our work in adapting existing paper textbooks for secondary schools for a digital format, this paper discusses paper form and the various electronic platforms with regard to the presentation of five aspects of mathematics that have roles in mathematics learning in all the grades kindergarten-12: symbolization, deduction, modeling, algorithms, and representations. In moving to digital platforms, each of these aspects of mathematics presents its own challenges and opportunities for both curriculum and instruction, that is, for the content goals and how they connect with students for learning. A combination of paper and electronic presentations may be an optimal solution but some difficulties with such a complex solution are presented.     

Electric-field distribution in a quantum superlattice with an injecting contact: Exact solution

A very simple model describing steady-state electron transport along a quantum superlattice of a finite length taking into account an arbitrary electrical characteristic of the injecting contact is considered. In the singleminiband approximation, exact formulas for the spatial distribution of the electric field in the superlattice are derived for different types of contact. Conditions under which the field is uniform are identified. Analytical expressions for the current–voltage characteristics are obtained. In the context of the developed theory, the possibility of attaining uniform-field conditions in a diode structure with a natural silicon-carbide superlattice is discussed.     

The effect of temperature on the nonlinear dynamics of charge in a semiconductor superlattice in the presence of a magnetic field

The space-time dynamics of electron domains in a semiconductor superlattice is studied in a tilted magnetic field with regard to the effect of temperature. It is shown that an increase in temperature substantially changes the space-time dynamics of the system. This leads to a decrease in the frequency and amplitude of oscillations of a current flowing through the semiconductor superlattice. The quenching of oscillations is observed, which is attributed to the change in the drift velocity as a function of electric-field strength under the variation of temperature.     

Investigation of Sr uptake by birnessite-type sorbents from seawater

Sorption of micro- and microamounts of Sr from seawater has been studied using granulated Na-birnessite. Distribution coefficients of ⁹⁰Sr in the natural seawater are 0.8–1.2?×?10³ ml g^?1, in the model seawater they are 1.6–1.8?×?10³ ml g^?1. Application of Na-birnessite was shown to be prospective in sorption–desorption–regeneration regime. In dynamic sorption conditions, over 150 bed volumes of seawater can be purified till 5% breakthrough occurs at feed rate 10 BV h^?1. Na-birnessite can be used for ⁹⁰Sr radionuclide removal from liquid radioactive wastes containing seawater.

     

Synthesis of polymetallophenylsiloxanes with the predicted metal content

The possibility of the synthesis of polymetallophenylsiloxanes with the predicted metal content in the synthesized polymers is demonstrated. Variation of the spectral luminescent properties of polyterbium- and polyeuropiumphenylsiloxanes as a function of the metal content is studied.     

Upper estimates for the number of periodic solutions to multi-dimensional systems

The maximal number of zeros of multi-dimensional real analytic maps with small parameter is studied by means of the multi-dimensional generalization of Rouché's theorem. The obtained result is applied to study the maximal number of periodic solutions to multi-dimensional differential systems. An application to a class of three-dimensional autonomous systems is given.