Abstract: The paper investigates the randomly stopped sums. Primary random variables are supposed to be nonnegative, independent, and identically distributed, whereas the stopping moment is supposed to be a~ nonnegative, integer-valued, and nondegenerate at zero random variable, independent of primary random variables. We find the conditions under which dominated variation or extended regularity of randomly stopped sum determines the stopping moment to belong to the class of dominatedly varying distributions. In the case of extended regularity, we derive the asymptotic inequalities for the ratio of tails of the distributions of randomly stopped sums and a stopping moment. The obtained results generalize analogous statements recently obtained for a narrower class of regularly varying distributions. Compared with the previous studies, we apply new methods to the proofs of the main statements whereas methods applied to regularly varying functions are not suitable for the broader class of generalized regularly varying distributions. At the end of the paper, we provide one example that illustrates the theoretical results.

Abstract: This paper describes methods for optimal filtering of random signals that involve large matrices. We develop a procedure that allows us to significantly decrease the computational load needed to numerically realize the associated filter and increase the associated accuracy. The procedure is based on the reduction of a large covariance matrix to a collection of smaller matrices. It is done in such a way that the filter equation with large matrices is equivalently represented by a set of equations with smaller matrices. The filter Fp we develop is represented by Fp(v1,…vp)=∑j=1pMjvj and minimizes the associated error over all matrices M1,…,Mp. As a result, the proposed optimal filter has two degrees of freedom to increase the associated accuracy. They are associated, first, with the optimal determination of matrices M1,…,Mp and second, with the increase in the number p of components in the filter Fp. The error analysis and results of numerical simulations are provided.

Abstract: The article investigates the boundary value problem for an extended stationary system of Nernst-Planck-Poisson equations, corresponding to a mathematical model of the influence of changes in the equilibrium coefficient on the transport of ions of a binary salt in the diffusion layer. Dimensionless variables were introduced using characteristic parameter values. As a result, a dimensionless boundary value problem was obtained, which is singularly perturbed, containing a small parameter in the derivative of the Poisson equation and, additionally, another regular small parameter. A similarity theory was developed: trivial and non-trivial similarity criteria and their physical meaning were determined, which allowed for the identification of general properties of the solutions. A numerical investigation of the boundary value problem was conducted using the finite element method. With an increase in the initial solution concentration, the value of the small parameter entering singularly decreases, reaching values on the order of 10−12 and below, leading to computational difficulties that prevent a comprehensive analysis of the influence of changes in the equilibrium coefficient on salt ion transport. In this regard, an analytical solution to the problem was constructed, based on dividing the solution domain into several subdomains (regions of electroneutrality, extended space charge region, quasi-equilibrium region, recombination region, intermediate layer), in each of which the problem is solved differently, followed by matching these solutions. Verification of the analytical solution was carried out by comparing it with the numerical solution. The advantage of the obtained analytical solution is the possibility of a comprehensive analysis of the influence of the dissociation/recombination reaction of water molecules on salt ion transport over a wide range of real changes in the concentration and composition of the electrolyte solution and other input parameters. This boundary value problem serves as a benchmark for constructing asymptotic solutions for other singularly perturbed boundary value problems in membrane electrochemistry.

Abstract: The market risk measurement of a trading portfolio in banks, specifically the practical implementation of the value-at-risk (VaR) and expected shortfall (ES) models, involves intensive recalls of the pricing engine. Machine learning algorithms may offer a solution to this challenge. In this study, we investigate the application of the Gaussian process (GP) regression and multi-fidelity modeling technique as approximation for the pricing engine. More precisely, multi-fidelity modeling combines models of different fidelity levels, defined as the degree of detail and precision offered by a predictive model or simulation, to achieve rapid yet precise prediction. We use the regression models to predict the prices of mono- and multi-asset equity option portfolios. In our numerical experiments, conducted with data limitation, we observe that both the standard GP model and multi-fidelity GP model outperform both the traditional approaches used in banks and the well-known neural network model in term of pricing accuracy as well as risk calculation efficiency.

Abstract: This paper considers the difference equations with continuous time, piecewise linear delay functions, and oscillatory coefficients. We present new conditions on coefficients that provide the oscillatory property of solutions of the considered difference equation. The given criteria are compared to existing oscillatory conditions in literature through examples.

Abstract: Fractional variable order systems with complex dynamics in the order is a little studied topic. In this research, we present three examples of a very simple fractional system with complex dynamics in the order of the derivative. These cases involve different approaches to define the variable order dynamics: 1) an integer-order differential equation that includes the state variable, 2) a differential equation that incorporates the state variable and features both integer and fractional-order derivatives, and 3) fractional variable-order differential equations nested in the orders of the derivatives. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems with complex behavior can reveal through the order of their derivatives.

Abstract: In this paper, it analyses the phenomena of Casson fluid flow, including the complex interactions between permeation, viscous dispersion, Darcy-Forchheimer implications, heat source, chemical reaction, and the heat boundary layer. The behaviour of a two-dimensional continuous stream comprising gyrotactic microbes of bioconvection Casson nanofluid through a stretchy membrane that is warmed and permeable are investigated in this study. A collection of independent partial differential equations is converted into a set of non-linear ordinary differential equations by using the proper conversion system. The analytical assessment of the current work is done using the homotopy analytic strategy. The relevant parameters are visually demonstrated to impact the concentration of nanoparticles, temperature, velocities, and gyrotactic microorganism profiles. Mathematica tool is used to calculate findings and visuals. The results of this research are extremely important regarding real-world uses for chilling and also for academic advances in the mathematical modeling of Casson liquid motion with thermal exchange in engineering structures. The body's friction coefficient, mobile microbes, Sherwood number, and Nusselt coefficient are calculated. A comparison study between the shooting and HAM findings is carried out as well.

Abstract: In this study, we extend the Simple Equations Method (SEsM) and adapt it for finding exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). In accordance with the extended SEsM, the analytical solutions to the studied FNPDEs are constructed as complex composite functions which combine two single composite functions, comprising power series of solutions of two simple equations or two special functions. The novel and innovative aspect of the proposed extended methodology is that the simple equations or the special functions used have different independent variables (different wave coordinates). In this context, the proposed approach is mostly suitable for application to systems of FNPDEs (or NPDEs) that describe the complex wave dynamics of real processes in different scientific fields such as fluid mechanics, plasma physics, optics, etc. The extended SEsM is applied to the time–fractional Boussinesq-like system, assuming that each system variable supports multi-wave dynamics, which involves the combined propagation of two distinct waves traveling at different wave speeds. As a result, numerous complex multi-wave solutions including combinations of different hyperbolic, elliptic and trigonometric functions are derived. In order to visualize the complex wave dynamics obtained through the applied analytical approach, some analytical solutions have been numerically simulated.

Abstract: Background: As global trade networks continue to evolve, China has emerged as one of the most critical logistics epicenters worldwide. Rapid developments in infrastructure, technological innovation, and e-commerce have made efficient logistics a cornerstone of competitive advantage. This paper presents a methodological framework that integrates auction-based game theory mechanisms with the Traveling Salesman Problem (TSP) to address carrier selection and cargo-consolidation strategies in China’s freight sector. Special emphasis is placed on the year 2025, where continued urbanization, digitization, and heightened consumer expectations demand novel optimization techniques. Methods: Six major logistics centers in China (Shanghai, Beijing, Guangzhou, Shenzhen, Chengdu, and Wuhan) are analyzed as potential consolidation hubs for 12 designated supplier locations spread across multiple provinces. We employ a dual approach: (1) a game-theoretic auction mechanism ensures cost-competitive carrier selection based on dynamic regional tariffs, and (2) a TSP-based algorithm is used to optimize routing in scenarios with heterogeneous fleet characteristics and variable fuel costs. Results: Using distance data and region-specific cost parameters, we demonstrate significant monthly savings (over 15% reduction) when shifting to an optimized, integrated system. Calculations show how synergy between proper hub assignment and combinatorial route optimization yields robust improvements in both cost-efficiency and service speed. Beyond numerical gains, the model exhibits adaptability to real-world constraints such as capacity limits, seasonality, and fluctuating energy prices. Conclusions: The findings underscore the transformative potential of combining game theory with classical optimization for China’s logistics sector in 2025. By choosing strategically located consolidation centers and leveraging auction-driven pricing, stakeholders reduce redundant routing, thus lowering carbon emissions and operational expenditure. This integrated blueprint can be generalized to other rapidly expanding markets, supporting data-driven managerial decisions in a complex, evolving logistics landscape.

Abstract: A method is introduced for the numerical continuation of the lunar-type, near-circular periodic orbits, with the background of middle-altitude lunar orbiters. In the Moon-Earth elliptic restricted three-body problem, some near-polar near-circular lunar-type periodic orbits are numerically continued from the Keper circular orbits by Broyden’s method. The integer ratio j/k of the mean motions between the inner and outer orbits can be in the range [9,150]. For the ratio j/k∈[38,70], the J2,C22 perturbations are added, and some near-polar periodic orbits are calculated. The orbital dynamics are well explained via the first-order double-averaged system. The linear stability can be studied by the characteristic multipliers, which are calculated from the linear variational system.

Abstract: The values of a measured, derived or estimated variable often differ from the “true”, “undistorted” values of a desired dimension. Output values of non-calibrated measuring instruments, misspecification of analysis models or too inflexible activation functions can lead to inappropriate decisions in all situations. Therefore, a highly flexible mathematical function for the isotonic transformation of a variable X of the value space [0-1] to a variable Y of the same value space [0-1] is presented here. With four or six parameters, almost all conceivable function curves can be represented. This allows restrictions of other functions, e.g. linearity or constant curvature, to be overcome.

Abstract: In this paper, the author presents the generalized form of the Median-Based Unit Rayleigh (MBUR) distribution, a novel statistical distribution that is specifically defined within the interval (0, 1) expressing oscillating hazard rate function. This generalization adds a new parameter to the MBUR distribution that significantly addresses the unique characteristics of data represented as ratios and proportions, which are commonly encountered in various fields of research. The establishment of this generalization aims to deepen our understanding of these phenomena by providing a robust framework for analysis. The paper offers a thorough and meticulous derivation of the probability density function (PDF) for the MBUR distribution, illuminating each phase of the process with clarity and precision. It delves deep into the intricacies of the MBUR distribution's properties, presenting a rigorous examination of the accompanying functions that are vital for robust statistical evaluation. These functions—comprising the cumulative distribution function (CDF), survival function, hazard rate and reversed hazard rate function. The paper discusses real data analysis and how the generalization improves such analysis.

Abstract: Background: In the competitive realm of freight transportation, optimizing carrier selection and consolidation strategies is paramount to reducing operational costs and enhancing supply chain efficiency. Methods: This paper presents an integrated framework that combines game theoryvia an auction methodand the Traveling Salesman Problem (TSP) for route optimization, applied to U.S. freight logistics. New supplier cities across the Eastern, Central, and Western regions of the United States are considered, and two optimal consolidation warehouses are determined through weighted cluster analysis. Distinct transportation tariffs, determined by regional characteristics, are employed to calculate delivery costs. Results: The analysis demonstrates that by reassigning consolidation hubs and leveraging competitive bidding, monthly transportation costs can be reduced by approximately $20,000. Conclusions: The synergy between game-theoretic auctions and TSP-based route optimization significantly enhances logistical efficiency and cost-effectiveness, offering practical benefits for supply chain management in the freight industry.

Abstract: After the first COVID-19 patient was diagnosed in China, in December 2019, the market's reaction to the pandemic evolved from initial unawareness and minimal response to rapid panic among investors as the outbreak escalated. As the pandemic continued to develop, the market gradually absorbed the impact, and stock market performance experienced a process of decline from high to low points. From the perspective of behavioral finance, the development of the pandemic suppressed the stock market through both profitability and risk preferences. The initial lack of understanding of the situation further exacerbated market volatility. This paper examines the correlation between the development of the pandemic and daily stock returns from March to June 2020, as well as the mechanism through which the pandemic affected investor sentiment. It reveals a negative correlation between the growth rate of infections and daily stock returns, and how the pandemic increased investors' pessimism, influencing their trading behavior by increasing turnover rates, thereby exacerbating the negative impact on stock returns. By empirically revisiting the impact of the 2020 COVID-19 pandemic on stock market investor behavior, this study provides insights for predicting and responding to the impact of the 2019 novel coronavirus (2019-nCoV) on the stock market.

Abstract: This paper present seven uniformly smooth approximating function for absolute value function: five of them approximate absolute value function from above, and the others approximate absolute value function from below. The properties of these uniformly smooth approximating functions are studied, and approximation degree are analyzed in theory and demonstrated by images. Finally, application prospect of uniformly smooth approximating function is pointed out.

Abstract: This study explores the use of Shannon entropy as a filtering mechanism to enhance the trading signals produced by the LVQ (Learning Vector Quantization) machine learning algorithm in algorithmic trading. The integration of Shannon entropy aims to improve trade entry accuracy by reducing market noise and identifying stronger trends. A trading bot was developed and tested on Bitcoin using a three-minute timeframe on the TradingView platform, with backtesting conducted from February 1 to February 18, 2025. The fully automated strategy used 100% of available capital per trade, reinvesting profits for compounding. Positions were closed when an opposite signal was generated. A comparative analysis revealed that incorporating Shannon entropy outperformed a baseline strategy. These findings demonstrate that entropy-based filtering improves trade selection and profitability by reducing market noise and focusing on reliable trends, suggesting its potential for broader application in algorithmic trading.

Abstract:

This paper presents a robust fitted mesh finite difference method for solving a system of n singularly perturbed two parameter convection-reaction-diffusion delay differential equations defined on the interval [0,2]. Leveraging a piecewise uniform Shishkin mesh, the method adeptly captures the solution’s behavior arising from delay term and small perturbation parameters. The proposed numerical scheme is rigorously analyzed and proven to be parameter-robust, exhibiting nearly first-order convergence. Numerical Illustration is included to validate the method’s efficiency and to confirm the theoretical predictions.

Abstract: This study examines the intricate interplay between Hamiltonian dynamics and spectral analysis within the context of algebraic structures and symmetry groups. By integrating perspectives from symplectic geometry, Lie theory, and nonlinear dynamics, we introduce a framework that aims to unify classical conservation laws with advanced spectral decomposition techniques. Our approach synthesizes geometric mechanics with adaptive numerical strategies, revealing novel relationships between phase-space invariants and multi-scale Fourier analysis. Applications in quantum mechanics, signal processing, and machine learning illustrate the potential of Hamiltonian-preserving algorithms to maintain spectral integrity while ensuring long-term stability. These findings assist in detecting chaotic transitions and demonstrate compatibility with established FFT methodologies, thereby offering new insights into the synergy between geometric structures and spectral methods.

Abstract: The propagation of membrane tension perturbations is a potentially critical mechanism in the mechanical signal transduction across the surfaces of living cells. Tethered proteins in the cell membrane play a crucial role in the propagation of the membrane tension. Intact cell membranes in eukaryotic cells possess unique characteristics, such as transmembrane proteins bound to the underlying cortex, rendering them immobile over timescales of minutes to hours. These immobile obstacles significantly alter the dynamics of lipid flow. While existing simplified lipid flow models provide fundamental insights into membrane tension dynamics, they fall short in accounting for complex phenomena like vesicle crumpling. To address this, we propose a more sophisticated model of lipid bilayers, solving the Stokes equations in a two-dimensional framework with embedded obstacles. We employ the finite element method and the FEniCS library to solve the weak form of the Stokes equations, providing a more accurate representation of membrane behaviour under physiological conditions.

Abstract: This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of n singularly perturbed convection-reaction-diffusion differential equations with two small parameters. Defined on the interval [0; 1], this system exhibits boundary layers due to the presence of small parameters, making accurate numerical approximations challenging. The method employs a piecewise uniform Shishkin mesh that adapts to layer regions and efficiently captures the solutions behavior. The scheme is proven to be uniformly convergent with respect to the perturbation parameters, achieving nearly first-order accuracy. Comprehensive numerical experiments validate the theoretical results, illustrating the method’s robustness and efficiency in handling parameter-sensitive boundary layers.