The algebraic properties of difference approximations of Hamiltonian systems are investigated. Symplectic schemes exactly preserve linear and quadratic integrals by virtue of Cooper's theorem, but not the total mechanical energy of nonlinear systems. However, it is known that instead of energy, symplectic difference schemes preserve with a given order of approximation a quantity that goes over into the Hamiltonian as the time step tends to zero. The paper presents an algorithm for calculating such a modified Hamiltonian and its implementation in the Sage computer algebra system for a given symplectic difference scheme, the required order of energy conservation, and the Hamiltonian of the original mechanical system. The program successfully reproduces formulas previously derived manually, which confirms its consistency. Numerical experiments show that solutions obtained using symplectic schemes closely coincide with the level lines of the modified Hamiltonian, which emphasizes its role in preserving the qualitative behavior of the system during numerical integration over large time intervals
The importance of studying quadratic Cremona transformations over algebraically non-closed fields for the theory of Kahan difference schemes for dynamical systems with a quadratic right-hand side is discussed. Cremona transformations of the projective plane over a Galois field of size 2, i.e., the Fano plane, are considered. The notation proposed by J. Rosanes is used to describe quadratic Cremona transformations. The relationship between quadratic transformations and matrix pencils is described. Quadratic transformations without singular points are called regular. The Sage system is used to implement procedures that convert a quadratic transformation into a permutation of 7 points of the Fano plane (an element of the symmetric group $S_7$) and a permutation of 7 points of the plane into a quadratic transformation. By enumerating all quadratic transformations, it is proved that regular quadratic transformations generate the entire permutation group of the Fano plane. This theorem is analogous to Noether's theorem over an algebraically non-closed field. It is proved that regular quadratic Cremona transformations have even order, and a description of the corresponding permutations of the group $S_7$ is given. It is shown that a permutation always corresponds to some quadratic Cremona transformation, but this transformation is not always regular. The resulting quadratic transformations can be supplemented by a rule that resolves ambiguities at fundamental points. An example of a quadratic transformation for which such resolution is impossible is given. This leads to a natural classification of quadratic transformations of the Fano plane.
Background This article focuses on a rapidly developing area of fractional calculus that describes anomalous diffusion. Since the correct form of the fractional-order equation describing anomalous diffusion - remains an open question, the authors use examples of models based on fractional diffusion equations to examine the advantages of various fractional-order equations proposed for modelling the advection-diffusion process in media with a fractal structure. Purpose This paper examines boundary value problems for the advection-diffusion equation with Caputo and Riemann-Liouville fractional operators. Given that many authors consider in their work models with fractional-order operators obtained simply by replacing ordinary derivatives with fractional ones, the authors of this paper consider it necessary to compare fractional differential equations with various operators. We also note that one of the aims of this work is to construct an effective approximate method for solving the first initial-boundary value problem for a homogeneous fractional differential equation. Method A fractional calculus approach is used. The approximate method under consideration is based on the analytical method of separation of variables (the Fourier method). The proposed approximate method and all the necessary calculations are implemented using the Matlab programming language. Results Approximate solutions have been obtained for boundary value problems for the advection-diffusion equation with fractional operators of Caputo and Riemann-Liouville. The solutions obtained have been compared both with one another and with the classical model based on ordinary derivatives. Conclusions Based on the results obtained, the authors recommend using an equation containing the Riemann-Liouville fractional operator to model anomalous diffusion processes.
Background: The rapid increase in cyberattacks targeting infrastructure, enterprises, and individual users through mobile devices has created an urgent need for effective intrusion detection systems. Traditional signature-based methods are inadequate against zero-day vulnerabilities and advanced persistent threats, prompting the exploration of machine learning approaches for network traffic analysis. Purpose: This study aims to develop and evaluate a privacy-preserving, locally operating Android application for real-time malicious traffic detection using machine learning models, addressing the limitations of cloud-dependent security architectures that compromise user privacy and availability. Methods: We implemented five machine learning algorithms - XGBoost, LightGBM, Random Forest, Decision Tree, and Logistic Regression - trained on the Network Traffic Android Malware dataset. The models were exported to the ONNX format for local deployment on Android devices. Performance was evaluated using accuracy, precision, recall, AUC score, training time, and model size metrics, with multi-criteria decision analysis (MCDA) applied for comprehensive comparison. Results: Gradient boosting algorithms demonstrated superior performance, with LightGBM achieving the highest MCDA score (0.9759), fastest training time (0.32 s), and smallest model size (0.28 MB), while XGBoost attained the highest AUC-score (0.9627). Both models significantly outperformed Random Forest (MCDA: 0.7532), Decision Tree (0.6743), and Logistic Regression (0.1407). Conclusions: LightGBM provides an optimal balance between detection accuracy and mobile resource constraints, making it suitable for on-device deployment. The proposed architecture demonstrates that fully local, ML-based traffic analysis is feasible without compromising detection quality, offering a privacy-centric alternative to server-dependent solutions for next-generation mobile intrusion detection systems.
Using ISO standards for mathematical notation is important for improving the quality of scientific paper presentation. This article describes the key requirements of ISO standards for mathematical notation. It also provides recommendations for the notation of differential operators.
