Discrete and Continuous Models and Applied Computational Science

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.

Attributing authorship of short texts is a complex task, and its accuracy heavily depends on the quality and quantity of training data. In many practical scenarios, however, this data is subject to degradation. This study examines how two specific types of degradation - textual corruption and reduction in corpus size - affect the performance of six standard classifiers for authorship attribution of English-language tweets. We model textual corruption as random and independent character-level distortions applied to the texts in the training corpora. The test set (the texts whose authorship is to be attributed) remains unchanged. The experimental setup involves 50 authors, with 200 test texts. The volume of training texts per author varies from 800 to 50 (across 16 incremental steps), and the distortion level ranges from 0 to 20\% (6 gradations). For each combination of corpus size and distortion level, we train and test six classifiers: Support Vector Machines (SVM), Logistic Regression, Naive Bayes, Random Forest, Decision Tree, and $k$-Nearest Neighbors (kNN). Authorial style is represented using Bag of Words (BOW), TF-IDF, word token $N$-grams, and their combinations. The results show that for most classifiers, degradation in the training sets leads to comparable reductions in attribution accuracy. The highest overall performance was achieved by the Support Vector Machine with TF-IDF features. It proved to be the most robust method both when the number of training texts was reduced and under high distortion levels. In the absence of distortions, SVM with TF-IDF achieved an accuracy of 0.92; at distortion levels of 5 and 20\%, its accuracy reached 0.89 and 0.75, respectively. Logistic Regression with Bag of Words ranked second, delivering accuracies of 0.9, 0.87, and 0.74 under the same conditions. The kNN method demonstrated the lowest accuracy among the classifiers examined.

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.

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.

In the 1990s, P. Vanhecke described a Hamiltonian system with two degrees of freedom and a polynomial Hamiltonian integrable in Abelian functions of two variables. This system provides a convenient example of an integrable system (in the sense of Liouville) in which integral curves are wound on a two-dimensional manifold, an algebraic surface in a 4-dimensional phase space. We show that all necessary calculations can be performed in the Sage system. The results of numerical experiments performed in our package FDM for Sage are presented. It is shown that orbits of Vanhecke system can be divided into two classes, periodic and non-periodic. The points of the periodic orbits corresponding to solutions with the same period form an equiperiodic surface. There are infinitely many different algebraic equiperiodic surfaces. The behavior of nonperiodic orbits in numerical experiments is determined not so much by the rationality or irrationality of the ratio of two periods of Abelian functions, but by the possibility of approximating this number with a rational fraction with high accuracy.

Background Using the Einstein's idea on particles-solitons, the stochastic representation of the wave function as a large sum of solitons with random phases is suggested. Purpose To illustrate the main idea of the stochastic representation, let us recall the lectures by N. Wiener on nonlinear problems in random theory. Method In these lectures Wiener used the so-called $\alpha$-representation of the wave function for a nonrelativistic particle in three-dimensional space, the wave function being considered as an element of the random Hilbert space with the Gaussian dispersion. Results This representation appears to be equivalent, in accordance with the central limiting theorem, to the sum of many complex solitons with random phases. Conclusions On the basis of this representation it is possible to substantiate the Born's rule for measuring physical observables and also to explain the ``spin-statistics'' correlation in quantum mechanics.

Purpose To develop and experimentally evaluate an autonomous computer-aided diagnosis (CADx) system for colonoscopy that integrates polyp segmentation with binary histological classification (adenoma/non-adenoma) while maintaining computational efficiency for near real-time clinical deployment. Materials and Methods We propose a cascaded YOLO11-SAM3-LoRA architecture wherein a YOLO11-medium detector generates region proposals (bounding boxes), followed by SAM 3 performing box-prompted segmentation with parameter-efficient domain adaptation via Low-Rank Adaptation (LoRA) and a semantic anchoring mechanism based on text prompt ensembling. Histological classification is implemented through a dual-stream classifier that fuses region-of-interest (ROI) features extracted by EfficientNet-B0 with morphological mask descriptors. Training was conducted on the combined Kvasir-SEG and CVC-ClinicDB datasets (detection/segmentation) and CVC-HDClassif (classification). Generalization was evaluated on the external ETIS-Larib dataset under an external zero-shot protocol. Results On the primary test set, the system achieved a mean Dice coefficient (mDice) of 0.962 for segmentation, adenoma sensitivity of 92.3\%, and area under the receiver operating characteristic curve (AUROC) of 0.942 for binary classification. Under external zero-shot evaluation on ETIS-Larib, the system maintained segmentation quality (mDice = 0.884), indicating robustness to domain shift. End-to-end processing throughput reached 38.2 FPS on an NVIDIA A100 GPU. Conclusions The results demonstrate that the combination of cascaded localization, foundation model segmentation with LoRA adaptation, and text prompt ensembling achieves high segmentation quality and competitive CADx accuracy while preserving computational efficiency. Clinical deployment requires further prospective multicenter validation and error analysis stratified by lesion subtypes.

We study the characterizations of the classical Volterra operator based on a simple relation between its singular numbers and the eigenvalues of its imaginary part on \( L^2[0,1] \).