The dynamics of nonlinear continuous-discrete (hybrid) systems and its dependence on the sampling step h are studied. Such systems contain phase variables and equations with both continuous and discrete time. The main focus of the work is the issue of local bifurcations during loss of stability of equilibrium points of hybrid systems. Sufficient signs of bifurcations are given, the properties of bifurcations are studied, and possible bifurcation scenarios are determined. The concept of transversal bifurcation is introduced, meaning that the corresponding eigenvalue of the matrix of the linearized problem passes through the unit circle when the parameter h passes through the bifurcation point h₀. It is shown that in a one-parameter formulation, two main scenarios are typical: transversal bifurcation of period doubling and transversal Andronov-Hopf bifurcation, while the scenario of transversal bifurcation of multiple equilibrium, as a rule, is not realized. Examples are given to illustrate the effectiveness of the proposed approaches in the problem of studying bifurcations in hybrid systems.

A continuous map X ^f→ Y and its extension exp_τ X ^f→ exp_τ Y are considered (exp_τ X is the hyperspace (endowed with a topology τ) of the topological space X, ḟ(F) = [f (F)]Y (the closure of a set f(F) in the space Y)). A necessary and sufficient condition (a modification of the Harris’ (WO) condition) of continuity of the map ḟ in the cases when τ = τ_LF (the locally finite topology) and τ = τ_F (the Fell topology) is found. When X and Y are metrizable spaces the topology τ_inf, as the infimum of all Hausdorff metric topologies, is considered. A sufficient condition (TUC) condition) of continuity of the map ḟ in the case when τ = τ_inf is found. It is also shown, that this condition is necessary, when the space Y is locally compact and second countable. The results are commented from the point of view of the category theory.

It is shown that the set of all C-compact orthogonally additive operators from a vector lattice E to an AM-space F is a vector lattice and the lattice operations can be calculated by the Riesz–Kantorovich formulas. Moreover, a positive AM-compact orthogonally additive map defined on a lateral ideal of a vector lattice E and taking values in an AM-space F can be extended to the whole space E.

This work investigates an initial-boundary value and an inverse coefficient problem of determining a time dependent coefficient in the fractional wave equation with the conformable fractional derivative and an integral. In the beginning, the initial boundary value problem (direct problem) is considered. By the Fourier method this problem is reduced to equivalent integral equations. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get apriori estimate for the solution via the unknown coefficient which will be used to study the inverse problem. The inverse problem is reduced to an equivalent integral equation of Volterra type. To show the existence and uniqueness of the solution to this equation, the Banach principle is applied. The local existence and uniqueness results are obtained.

A linear integral equation related to the coefficient inverse problem for a hyperbolic equation is considered. In the inverse problem, based on measurements of scalar wave fields scattered by an inhomogeneity, it is necessary to reconstruct the propagation velocity of the signal on the inhomogeneity. Probing fields are generated by point sources centered on a circle. We prove the unique solvability of the inverse problem with such an arrangement of sources under quite general assumptions on the choice of a variety of detectors. The relationship between the axial symmetry of the scattering data and the symmetry of the desired function relative to the same axis is established.

This article explores the challenges and applications of the octonion Fourier transform, with a focus on wavelet analysis. We extend the Moritoh wavelet transform to the octonionic Besov spaces, the weighted octonionic Besov spaces, the octonionic BMO spaces, and the octonionic weighted BMO spaces. The derived bounds for the octonionic Moritoh wavelet transform in these spaces contribute to a deeper understanding of its behaviour. Our findings pave the way for future research in signal processing, image analysis, and the intersection of octonion wavelet analysis with other mathematical theories.

The problems related to the description of identities that hold in all n-dimensional associative nilpotent algebras over a field (n is fixed) are studied. The author previously formulated the hypothesis that an arbitrary n-dimensional nilpotent algebra over any field satisfies some standard identity of minimal degree, and a number of results were obtained in support of this hypothesis. In this article, it turns out that this hypothesis is also confirmed in the class of 2-algebras, that is, such locally nilpotent algebras over a field that the square of the principal ideal generated by any of the generators of the algebra is equal to zero. Moreover, the ideal of identities of a variety generated by n-dimensional 2-algebras over an arbitrary field (n is fixed) is described.

This paper considers a free boundary problem for a system of quasi-linear parabolic equations in one dimension. Nonlinear problems with a free boundary are studied using a method based on constructing a priori estimates. For the solutions of the problem, apriory estimates of Shauder type are established. On the base of apriory estimates, the existence and uniqueness theorems are proved.

The paper establishes the solvability in the weak sense of the initial-boundary value problem for the second-order Kelvin-Voigt model with smoothed Jaumann time derivative taking into account the memory of fluid motion. For the proof, a problem approximating the original one is considered, and its solvability is established based on a priori estimates of solutions and the Leray-Schauder degree theory. After that, the limit transition is carried out as the approximation parameter tends to zero, and it is shown that the solutions of the approximation problem weakly converge to the solution of the original problem.

The article is devoted to the problem of classification of Calogero-type equations with respect to point transformations. It is known that Calogero equations can be reduced to linear equations using contact transformations, provided that the function f is quadratic. However, for an arbitrary function f the equivalence problem is open. Admissible point transformations that preserve the class of Calogero equations are considered. We construct differential invariants with respect to such transformations and apply them to solve the equivalence problem.

A number of spectral and functional inequalities related to Schrödinger operators defined on antisymmetric functions is presented. Among them are Hardy and Sobolev inequalities, Lieb-Thirring and CLR inequalities.