Computer Science and Mathematics

Abstract: Autonomous driving technology imposes stringent requirements on real-time environmental perception and dynamic distance measurement, which are critical prerequisites for vehicle path planning and driving safety. Traditional ranging methods relying on LiDAR and visual sensors often suffer from poor real-time performance and insufficient robustness in complex and dynamic traffic scenarios. To address this limitation, this paper proposes a novel fixed-time zeroing neural network (ZNN) to achieve high-precision and real-time inter-vehicle distance estimation for autonomous vehicles. First, a time-varying optimization model for inter-vehicle distance calculation is established based on dynamic programming theory, which fully considers the motion trajectories and structural constraints of adjacent vehicles. Subsequently, a dedicated activation function is designed to construct the improved fixed-time ZNN model for solving the above time-varying optimization problem. Rigorous theoretical proofs are presented to verify the fixed-time stability of the proposed network, and a tight upper bound of the convergence time is analytically derived. Moreover, comprehensive parameter selection strategies are discussed to guide practical model deployment. Finally, numerical simulation results demonstrate that the developed ZNN model possesses faster convergence speed and stronger robustness compared with existing methods, and it can accurately complete real-time distance calculation within a fixed time upper bound. The proposed method provides a new effective solution for dynamic ranging tasks in autonomous driving systems.

Abstract: Resolvent Monte Carlo estimates eigenvalues of large matrices by sampling Markov chains and reading the target value off a truncated resolvent quotient, trading exact arithmetic for a stochastic error that the almost-optimal sampling scheme is designed to suppress. We study when that error vanishes outright. We derive an exact closed-form identity for the variance of the moment estimators of a general, possibly signed matrix, and use it to isolate a hierarchy of zero-variance notions ranging from the most local, which constrains only the first draws, through the finite-truncation regime that a practical run can certify, to the global regime in which every moment estimator is deterministic. We separate determinism of the estimator from correctness of the eigenvalue it reports, and exhibit the exact conditions under which each notion holds and the examples that separate them. A single edgewise condition, which we call the eigen-triple condition, forces the truncated quotient to equal the target eigenvalue in finite samples; the associated moment and quotient variances are second order in the maximal edge defect and vanish at the eigen-triple. A linear-time procedure certifies the condition.

Abstract: We define a class of Markov cell processes P_X on a finite set X as a product of conditional probabilities on cells (subsets of X forming a partially directed intersection graph). The class of Markov cell processes includes Bayesian networks and Markov edge processes. A nested conditional independence (NCI) condition is introduced that allows an explicit expression of a joint probability distribution through its marginals. The NCI condition is used in our construction of consistent Markov cell processes P_X whose marginals coincide with P_X′ on smaller sets X′ ⊂ X. We discuss three classes of consistent Markov cell processes on rectangular domains X ⊂ Z³ equipped with ‘cubic’ cells, which include Arak model and 3D Pickard model.

Abstract: We introduce a new class of split inverse problems, termed the Split Pseudomonotone Equilibrium Problem with Multiple Output Sets, which generalizes classical equilibrium formulations to accommodate multiple decision outputs and pseudomonotonicity. To solve this problem, we propose a novel iterative method that employs an inertial technique and self adaptive step sizes to improve the convergence properties. Under suitable conditions, we establish the convergence of the method and provide a detailed theoretical analysis. The proposed framework is then applied to medical diagnosis classification tasks, considering diabetes, chronic kidney disease, heart disease, and breast cancer datasets where decision making involves heterogeneous data. Numerical tests reveal the algorithm’s strength and effectiveness, underscoring its potential for wider use in optimization-based classification and decision-making systems.

Abstract: We study a delayed time-fractional semilinear evolution equation driven by the spectral fractional Laplacian. The model combines a Caputo time derivative, a source evaluated at the past state u(t−τ), and an absorption term evaluated at the present state u(t). The fractional structure is used throughout the analysis through the Caputo Volterra kernel, the spectral fractional energy form, and Mittag–Leffler resolvent estimates. We prove local existence, positivity and an L∞(Ω) continuation criterion for bounded mild solutions, and show that these solutions satisfy the weak formulation before the maximal time. In the pure delayed-source case μ=0, we prove global continuation and boundedness on every finite time interval, together with a stepwise propagation of positive lower bounds for the first Dirichlet mode. When μ>0 and the initial history is sufficiently small, the present absorption and the spectral damping dominate the delayed feedback. In this regime, the solution is globally bounded and satisfies u(t)L2(Ω)→0 as t→+∞, with a Mittag–Leffler type decay estimate for the L2-energy. The numerical section is restricted to reduced first-mode comparison computations and illustrates the scalar comparison estimates derived from the first-mode analysis.

Abstract: The current paper suggests a new image encrypting method, which combines an original and designed Block Cellular Connection (BCC) algebra with the Advanced Encryption Standard (AES) in Cipher Block Chaining (CBC) mode. The strong validity of the suggested approach is proved by experimental assessment of conventional grayscale images with high NPCR (99.78%), UACI (33.85%), and entropy (7.99), and low correlation coefficients in horizontal, vertical, and diagonal scan directions. These findings demonstrate the possibilities of the BCC-AES-CBC hybrid to be used in secure and efficient image encryption applications.

Abstract: This article develops duality principles and numerical results for a large class of non-convex variational models. The main results are based on fundamental tools of convex analysis, duality theory and calculus of variations. More specifically the approach is established for a class of non-convex functionals similar as those found in some models in phase transition. Moreover, we develop a general duality principle for quasi-convex relaxed formulations for some models in the vectorial calculus of variations. Concerning applications of such results are presented for a non-linear model of plates and for nonlinear elasticity. Finally, in some sections we present concerning numerical examples and the respective softwares.

Abstract: Ancient manuscripts are cultural objects encoding physical, chemical, linguistic, historical, philosophical and social information. To unify heterogeneous disciplinary evidence within a single relational and geometric formalism, we propose a Grothendieckian framework inspired by category theory, sheaf theory, topoi and algebraic geometry. Each discipline is represented as an independent graph whose nodes and edges encode its own entities and relationships. Local observations are attached to graph neighborhoods through sections of sheaves, while structure-preserving functors align corresponding entities across disciplinary domains. Natural transformations evaluate the consistency of alternative mappings, while cohomological analysis identifies hidden structural constraints. Subsequently, compatible local information is assembled using descent theory, whereas stacks and derived categories incorporate multiple manuscript versions and their historical evolution. The integrated relational structure is represented geometrically through schemes, with the common invariant subgraph interpreted as the manuscript’s motive, namely the maximal relational architecture preserved across all disciplinary representations. This approach could provide a mathematical strategy for integrating heterogeneous evidence without privileging any single disciplinary viewpoint. Potential applications include manuscript authentication, conservation planning, textual criticism, comparative philology, cultural heritage analytics, historical knowledge integration and development of interoperable artificial intelligence systems for multidisciplinary manuscript research.

Abstract: A reliable sign of the absence of a document presentation attack, when a fake copy is presented instead of the original document, is the presence of such security features as OVDs (Optical Variable Devices). To verify document authenticity on a scanner, it is sufficient to use the visible light and a series of document scans obtained with changing the illumination position. This work proposes a method for detecting security OVDs on identity documents using a scanner with controlled illumination. The method is based on obtaining a series of document images in various illumination modes and identifying features characteristic of OVDs. To test the method, a dataset MIDV-Holo-Scan (https://zenodo.org/records/20758652) was collected by scanning physical documents used in the creation of the open dataset MIDV-Holo. It includes both documents with OVDs, accepted in this work as originals, and documents without OVDs, simulating an attack on document presentation. The proposed method for detecting attacks on document presentation achieves a quality of TPR=100% and FPR=0%, which surpasses the quality of the baseline method published with the MIDV-Holo dataset.

Abstract: There is an intimate theoretical relationship between Zipf’s law and the expected number of hapax legomena, dis-legomena, and n-legomena in general, as established in recent papers. The relationship was confirmed by the empirical analysis for very large texts. However, the known theoretical relationship was established under the assumption of very large vocabularies and very large texts, where the dimension of the vocabularies used does not limit the maximal rank value and thus the hapax legomena, or the dis-legomena etc. (n-legomena for small n values). In addition, the theoretical results were established under the hypothesis that the probabilities of the words are context-independent, which is not satisfied for small vocabularies and small texts, as typically used in fake news and in other classes of small texts over the Internet. We provide a theoretical analysis under rectified hypotheses, prove several results under relaxed hypotheses, and show examples of practical results.

Abstract: We construct a k-shadow complex Δk(N) from the nerve N of a convex sensor cover and prove that the region covered by at least k sensors is homotopy equivalent to Δk(N), recovering the usual nerve-based coverage test at k = 1. We test the construction on five independent synthetic sensor fields. Two confirm exact agreement with direct geometric sampling. A third, an unstructured random field, exposes instability in the raster-based ground truth at near-tangent sensor contacts, which we trace to twenty disk pairs within 0.05 length units of tangency; the exact nerve classifies every one correctly. A fourth, a densely clustered field, shows that Δ2(N) itself can be far larger than its underlying sensor count suggests: removing an initial clique-size obstruction in the construction reveals an object with over 800,000 vertices, against under 500 for the other fields tested. All scripts, data, and logs are supplied for reproduction.

Abstract: Dynamic risk prediction is an important statistical technique for detecting temporal changes in risk and provides quantitative support for early risk identification in clinical decision-making, industrial process monitoring, and financial anomaly detection. This study proposes a Temporal Convolutional Network Deep Cox Mixtures model (TCN-DCM) for longitudinal survival data by integrating a Temporal Convolutional Network, which learns temporal patterns from longitudinal covariates, with a Deep Cox Mixture framework that relaxes the conventional proportional hazards assumption. Simulation studies were conducted to compare the proposed model with existing deep learning-based methods, including Recurrent Deep Survival Machines and Dynamic-DeepHit, as well as the traditional joint model. The results showed that, when the proportional hazards assumption held, TCN-DCM outperformed the existing deep learning-based models. When the proportional hazards assumption was violated, TCN-DCM achieved predictive performance comparable to that of Recurrent Deep Survival Machines and yielded superior results for some evaluation metrics. The proposed model was further applied to a primary biliary cholangitis dataset, where it achieved the best overall predictive performance and illustrated dynamic individualized survival risk prediction. These findings indicate that TCN-DCM provides a flexible and broadly applicable approach for dynamic risk prediction in longitudinal survival analysis.

Abstract: A novel type of endogenous convex transformations of probability distributions is proposed for the controlled modification of a principal aggregate parameter while automatically preserving the properties of a probability distribution. Unlike conventional weighted distributions and mixture models, where the transformation mechanism is specified externally, the proposed approach determines the direction of evolution endogenously from the current distribution through a chosen outcome function and its associated principal moment. The resulting operator is constructed as a convex combination of the original density and its normalized weighted counterpart, ensuring non-negativity and unit normalization for all admissible transformation parameters. Analytical expressions describing the evolution of the principal moment are obtained, and the continuous limit of infinitely small transformations is investigated. An exact solution of the corresponding evolution equation is derived, showing that the continuous dynamics remains within the exponential family generated by the initial distribution and the outcome function. The gamma distribution is studied as a representative example. Explicit formulas are obtained for both finite and continuous gamma transformations. It is shown that repeated finite transformations naturally generate increasingly complex mixtures of gamma distributions, whereas the continuous dynamics preserves the gamma family by changing only its scale parameter. These results reveal a fundamental structural dichotomy between discrete mixture-generating transformations and continuous exponentially closed evolution.

Abstract: The Number of Pixel Change Rate (NPCR) and the Unified Average Changed Intensity (UACI) are the two canonical statistics for measuring a cipher’s sensitivity to plaintext and key perturbations. Their hypothesis tests compare the measured scores against reference values derived under the ideally-encrypted-image model, in which every ciphertext byte is an independent uniform draw from {0,…,255}. The resulting constants — 99.609375% for NPCR and 33.463541667% for UACI — have been adopted unchanged for 3D-model encryption, yet no prior work has verified that the model holds for the formats involved. We show that this assumption fails for STL files, whose triangle records consist almost entirely of IEEE-754 single-precision floats: structural validity forbids non-finite values, so the uniform law over valid files has non-uniform, position-dependent byte marginals. We formalize this law as the valid-STL (VSTL) distribution and derive in closed form the exact NPCR and UACI reference means, their finite-sample variances, and the resulting critical values. Both references lie strictly below their uniform-byte counterparts — at 99.609190456% for NPCR and 33.455728563% for UACI — implying that tests based on the conventional constants become increasingly prone to falsely rejecting ideal ciphers as file size grows; this effect is observable for UACI at practical file sizes, whereas for NPCR it arises at larger scales. We therefore recommend validating STL ciphers against the format-specific references derived in this paper.

Abstract: This paper presents a numerical-computational analysis of ℓ2-regularized logistic learning for binary intrusion detection under heterogeneous datasets, class imbalance, and cross-dataset shift. Rather than proposing a new intrusion-detection architecture, the study examines how numerical conditioning, feature scaling, feature-set design, threshold selection, and distribution shift affect operational detection behavior. Experiments were conducted on CICIDS2017, UNSW-NB15, and CIRA-CIC-DoHBrw-2020 using reproducible train-validation-test protocols over five fixed random seeds. The numerical audit showed that standard scaling reduced the spectral condition number of traffic-feature matrices by several orders of magnitude across datasets and feature configurations. However, scaling did not produce uniformly monotonic predictive gains: in some cases, raw-feature optimization achieved comparable or higher F1-score, whereas scaled preprocessing produced more controlled false-alarm behavior. In-domain experiments showed that dataset-specific features may improve ranking metrics such as AUROC or AUPR without necessarily improving thresholded operational metrics. Cross-dataset transfer experiments revealed strong source-target asymmetry, with transferred thresholds producing either near-zero positive detection or excessive false alarms. Finally, a Kolmogorov–Smirnov-based distribution-shift analysis showed that in-domain discrepancies were small, whereas cross-dataset discrepancies were consistently large under common standardized traffic features. These findings suggest that numerical stability, ranking quality, thresholded detection performance, false-alarm behavior, and distribution shift should be analyzed jointly when evaluating intrusion-detection models.

Abstract: To illustrate finite-time runaway phenomena and their possible implications for the AIsingularity, we propose a unified cubic dynamical system with two shape parameters fordescribing single-fold and double-fold tipping dynamics. The so-called “AI singularity”may be interpreted as a particular form of tipping-point phenomenon characterized byforcing-induced loss of stability and nonlinear self-amplification. In the limiting single-foldcase, the cubic system reduces to a quadratic saddle-node normal form. In this reducedsetting, post-threshold dynamics can exhibit rapid acceleration and, in an idealized formulation,finite-time singularity. This quadratic model provides a minimal dynamical templatefor describing threshold-induced acceleration in AI systems, including possible runawaybehavior. To examine whether finite-time singularity is intrinsic to such tipping processes,we also analyze the full cubic system. In contrast to the quadratic single-fold model, thecubic system contains a second fold and a nonlinear saturation mechanism, allowing thepost-fold transition to remain bounded. Numerical solutions of the full cubic equationprovide a direct visual illustration of initial slow growth, rapid transition, slowdown, andsaturation. Complementary analytical approximations based on tan and tanh functionsfurther clarify these stages. These results suggest that finite-time singularity is a feature ofthe reduced quadratic single-fold approximation, and that cubic nonlinearity can introducenonlinear saturation that prevents unbounded growth of the modeled intelligence variable.

Abstract: Diffusion within porous media, such as biological tissues, often deviates from conventional Fick’s laws that may be described by space-fractional diffusion equations. Microscale tissue heterogeneity can be represented by a space-fractional Riesz Laplacian operator acting on the concentration. We consider a reaction-diffusion system with two spatial compartments – a proximal one of finite radius having a source, and an outer one extending to infinity where the source is absent but first-order decay takes place. The steady state is derived using Hankel and Mellin transforms, resulting in an integral kernels containing Bessel functions. We develop and compare three numerical quadrature methods for the Hankel transform: sinc quadrature, Ogata quadrature (based on Bessel zeros), and a hybrid asymptotic-numerical scheme. Numerical results and plots are presented for fractional exponents β=1/2,2/3,3/4 (2α=1+β). The integer-order case (α=1) is recovered as a limiting case. The hybrid method is about five times faster than the global quadratures for the same accuracy.

Abstract: Nonlinear systems of equations arise in a wide range of applications in engineering, physics, and biological modeling; however, classical Newton-type methods may fail when the initial approximation lies outside the basin of attraction of the desired solution. This work proposes a two-stage hybrid framework that couples Particle Swarm Optimization (PSO) for global exploration with the fifth-order Newton-Jarratt (NJN) iterative method for local refinement. The fifth-order convergence of the NJN phase, established through a complete Fréchet-derivative Taylor expansion with explicitly computed error constants, guarantees rapid local convergence once PSO delivers a sufficiently close starting point. The framework is validated on four test problems of increasing dimension (n=2,5,20,40): a two-dimensional benchmark algebraic system, a five-dimensional metabolic network model for ethanol production in Saccharomyces cerevisiae, and two large-scale systems arising from the discretization of a Hammerstein nonlinear integral equation. Over 30 independent runs per method, PSO-NJN achieves a 100% convergence rate in all four problems, with mean final residuals on the order 10⁻¹⁴-10⁻¹⁶. In comparison, pure PSO fails completely (0% success) on the high-dimensional Hammerstein cases (n=20,40) and achieves only 10% success on the metabolic model. These results confirm that combining global metaheuristic search with high-order local refinement yields a robust, scalable solver suitable for complex biological and engineering nonlinear systems.

Abstract: This paper presents a novel formulation of Vieta–Pell wavelets functions (VPW) constructed from classical Vieta–Pell polynomials through systematic scaling and translation techniques. The transformation from a global polynomial basis to a localized wavelet basis enables efficient representation of functions on the semi-open interval [0, 1), preserving orthogonality and compact support properties. The proposed wavelet system is employed to develop an operational matrix framework, which converts differential equations into algebraic systems. Based on this formulation, two numerical approaches are adopted for solving optimal control problems. The direct method is applied by transforming the performance index into a quadratic programming problem, which is solved using the Lagrange multiplier technique. This approach is utilized in engineering applications such as solar energy systems and power system protection, where optimal performance and stability are achieved. In addition, an indirect method based on Pontryagin’s Minimum Principle combined with spectral techniques is employed to solve a practical control problem related to dishwasher systems. The spectral formulation enhances the accuracy of the solution while maintaining computational efficiency. Theoretical properties of the proposed wavelets, including orthogonality, convergence, and accuracy, are rigorously established. Numerical results demonstrate that the Vieta–Pell wavelet approach provides high accuracy, flexibility, and efficiency in solving modern optimal control problems across various engineering domains.

Abstract: Claim reserving is one of the most important tasks in non-life insurance, as it directly affects solvency assessment, financial reporting, and risk management. Traditional reserving methods often assume a relatively homogeneous claim-development process and may fail to capture hidden structures within complex insurance portfolios. This paper introduces a novel reserving framework that integrates Topological Data Analysis (TDA) with both aggregate and micro-level reserving methodologies. Using a portfolio of motor insurance claim payments, we employ topological techniques to identify latent claim-development regimes and portfolio heterogeneity. The extracted topological information is subsequently incorporated into an Inverse Probability Weighting (IPW) reserving framework and a TDA-enhanced Chain-Ladder (CL) methodology. The empirical results suggest that the proposed TDA-based approaches may improve reserve estimation accuracy relative to their traditional counterparts. Both TDA-IPW and TDA-CL produce reserve estimates that are remarkably close to realized future claim payments. The findings suggest that topological structures contain valuable information for actuarial reserving and that Topological Data Analysis may provide a promising new direction for the development of reserving methodologies.