In this article, the properties of quantum hash functions are further explored. Previous findings show that so-called small-bias sets (special subsets of the set of elements of a cyclic group) generate a “phase” quantum hash function. Here, it was proved that they also generate an “amplitude” quantum hash function. Namely, it turned out that constructing small-bias sets while generating amplitude quantum functions yields a well-balanced combination of the cryptographic properties of unidirectionality and collision resistance. As a corollary of the obtained theorem, a general statement about the generation of new amplitude quantum hash functions based on universal hash families and small-bias sets was proved.

This review summarizes the findings of some published studies that have explored the influence of various delays (elasticity, damping, and self-oscillatory mechanism of excitation) on the dynamics of classes (or types) of mixed oscillations (MO) without and with consideration of the interaction between the oscillating system and the energy source. A general holistic framework was provided for how such delays, both separately and in combination, affect the dynamics of MOs. A unified computational scheme (model) used in the works studied made it easy to understand and compare the results of this influence on different types of MOs. With the account of the interaction with the energy source, the known calculation scheme (or model) of a mechanical frictional self-oscillating system serves as a unified basis for considering all types of MOs. Nonlinear differential equations of motion valid for all types of MOs with their respective solutions were presented, from which the relations for any certain type of MO are derived as special cases. Equations of unsteady motion and relations to calculate the amplitude and phase of stationary oscillations, the velocity of the energy source and the load of the oscillating system on it, as well as the stability conditions of stationary oscillations were given. The results of the calculations carried out to gain insight into the influence of delays on the system dynamics were discussed. Overall, the calculations show that the interaction between the forces with delay and the forces in the energy source is at the core of a variety of phenomena. Different delays in the same system change the shape of the amplitude-frequency curves, shift them, and influence the stability of motion.

This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out.

This article describes an algorithm developed for the finite element analysis of the stressstrain state of a shell that takes the shape of a triaxial ellipsoid with varying parameterization of its mid-surface. A quadrangular fragment of the shell mid-surface with nodal unknowns in the form of displacements and their first derivatives along the curvilinear coordinates was used as the discretization element.
When approximating the displacements through the nodal values, two variants were considered. In the first variant, the known approximating functions were applied to each component of the displacement vector of the internal point of the finite element through the nodal values of the same component. In the second variant, the approximating expressions were used directly for the expression of the displacement vector of the internal point of the finite element through the vector unknowns of the nodal points. After the coordinate transformations, each component of the displacement vector of the internal point of the finite element was expressed through the nodal values of all components of the nodal unknowns. The approximating expressions of the required displacements of the internal point of the finite element also include the parameters of the curvilinear coordinate system used in the calculations.
The high efficiency of the developed algorithm was confirmed by the results of the numerical experiments.

In our previous articles, we introduced and explored the notion of φB -distributions with values in the Banach space. This offers a new perspective on the theory of solvability of linear problems, which is important for solving partial differential equations, especially equations with deviating arguments. Here, we provide an overview of the theory of such distributions, propose a new approach to justify the use of the Fourier method for solving linear problems, and write out a correctly solvable problem for a system of partial differential equations with deviating arguments.

This article analyzes the approaches to macro- and mesoscale computational modeling of the dynamics of metal powder particles motion in the condensation chamber of a plasma reactor, spheroidization, coagulation, and phase transitions in particles. The features of different regimes of vaporization and condensation were described. The influence of phenomena such as Brownian motion and thermophoresis on the process was explored. The parameters of the process at which the formation of core-shell particles occurs were determined. The model can be used to optimize and select the effective regimes for the processing and synthesis of powder materials in inductively coupled plasma.