Discrete and Continuous Models and Applied Computational Science

The siunitx package is designed for typographically correct and consistent typesetting of physical quantities (numbers with units of measurement) in LaTeX documents. It automates formatting according to the rules of the International System of Units (SI), eliminating the need to manually manage spaces, fonts, and separators.

\emph {Introduction} In the modern world of computers and networks the idea of expanding of personal computer resources with the help of cloud storages and computation looks more and more lucrative. However, usage of these resources may endanger data being processed. In last twenty years several algorithms of homomorphic encryption were developed allowing solving of this problem among other applications. However such algorithms are usually constructed as public key systems for long term storage and processing of data. In this article two algorithms of homomorphic encryption optimized for single data processing are proposed. \emph {Purpose} The target of research is development of data coding system which allows safe data processing in public clouds. \emph {Results} Two homomorphic coding systems had been developed, first is based on representation of numbers in the form of polynomials, second based on further representation of polynomials in the form of sets of values. Developed systems allow approximate calculations of coded data without decryption allowing processing of real numbers. System has high level of protection and provides high precision of calculations, comparable with standard personal computer calculation precision. Structure of coded data allows parallel computing. Proposed system allows safe data processing in public networks. Question of finding of optimal parameters for the system stands open both for high precision calculation of limited sets of operations and repeatedly good precision for big sets of operations.

In this paper, we develop a new family of high-order derivative-free iterative methods for solving systems of nonlinear equations. Specifically, we propose four two-step derivative-free schemes with convergence orders four and five, together with twelve three-step derivative-free schemes achieving convergence orders six, seven, and eight. The main specific of these iterations is that they include a vector or even a scalar iteration parameter instead of the matrix parameter inherent to other existing iterative methods. This structural simplification significantly reduces computational cost, storage requirements, and matrix operations, thereby improving overall computational efficiency. A convergence analysis is presented, establishing the theoretical order of convergence of the proposed methods. The efficiency indices of the proposed schemes are derived and compared with those of several well-known derivative-free iterative methods. The numerical experiments on standard academic problems confirm the theoretical results and demonstrate that the proposed methods are competitive and, in many cases, superior in terms of efficiency and robustness.

\emph {Background} In a previous article we discussed the use of dual quaternions for modeling points, lines and planes and solving standard geometric problems. This article is a logical continuation and reveals the use of dual quaternions to describe isometries of three-dimensional space. \emph {Purpose} The derivation of all necessary formulas for the screw motion of points, straight lines and planes, as well as reflection relative to the plane. Refinement of notation and formalism. \emph {Method} The algebra of dual numbers, quaternions and dual quaternions is used, as well as elements of the theory of screws and sliding vectors. \emph {Results} Formulas for rotation, translation, reflection, helical motion, and mirror rotation are obtained and systematized. \emph {Conclusions} Dual quaternions can serve as a full-fledged tool for describing helical motion in space. Due to the possibility of expressing dual quaternion operations in terms of standard vector and scalar products, the formulas obtained allow for effective software implementation.

The rigorous theory and characterization of charged-particle dynamics in high-intensity electromagnetic fields are fundamental for the development of advanced plasma-based applications. Accurate analytical models must bridge the gap between smoothed trajectories and exact particle motion to predetermine injection and energy gain. The main objective of this review is to establish a rigorous framework for the averaged relativistic motion of electrons, focusing on the strict derivation of ponderomotive forces and the impact of fast-oscillating periodic additions on dynamical variables. By making use of the Krylov--Bogoliubov--Mitropolsky averaging method to obtain the equations of motion, the study analyzes relativistic effects in laser beams and waveguides. These theoretical predictions are substantiated through numerical validation, including test-particle simulations and three-dimensional particle-in-cell simulations (PIC) of relativistic self-trapping regimes such as “laser bullet” and “bubble” structures. The review details the independence of the results on the formulation framework, the strict dependence on wave polarization, and the non-strict potential character of the relativistic ponderomotive force. The analysis demonstrates that periodic fast-oscillating additions are essential for a complete description, accurately setting initial conditions in averaged equations and enabling precise predictions of electron reflection and refraction. Simulations confirm that these fast-oscillating corrections determine electron injection and beam charge in realistic laser–plasma acceleration scenarios. The present review clearly shows that the dual framework of test-particle and PIC models is vital for probing the limits of averaged motion theory. The findings are of direct practical relevance for the optimization of radiation sources and guide the development of future theories incorporating non-adiabatic and field topology dependent effects.

In the work by the power series method the one-dimensional Schrödinger equation is solved with a triangular potential function which is applied in various modern heterostructures, in particular for GaAs and the others. By varying available parameters it is possible to obtain the desired precision of the numerical solution of the Schrödinger equation with any type of potential function for modern heterostructures. For the original Schrödinger equation are obtained wave functions in the form Airy functions and the analytical formula for the energy levels through the zeros of the Airy function. The values energy levels from this analytical formula agree with its results obtained by direct power series method with precision up to $10^{{-4}}$ percents, that is, up to 5 decimal signs. However, it is more rational and easier to use the Schrödinger equation solution, because the numerical calculations zeros of Airy function present separate complex and complicated numerical problem. But in order to achieve high numerical accuracy, it is necessary to set the Digits flag to several dozen significant digits and increasing the number of power series, that leads to an increasing in the time spent on the computer.

This paper attempts to modify the standard Verhulst model to describe the dynamics of scientific conferences taking into account cumulative advantage.