Abstract: The global regularity of solutions to the two-dimensional (2D) incompressible Navier-Stokes equations (NSE) is disproved by demonstrating the existence of singularities in plane Couette flow. Using tools from the Sobolev space theory, the energy dissipation analysis, and the energy gradient theory (Dou, 2025), it is shown that the flow develops "velocity discontinuities" under generic disturbances, where the velocity is not differentiable, forming singularities of the Navier-Stokes equation. Thus, the solution fails to be in C^1(OmegaX[0, T )) for any T > T0 (with T0 is the singularity formation time). Specifically, it is demonstrated that such singularities force the solution to exit the Sobolev space H^1(Omega), violating the smoothness criterion required for global regularity. The result of present study indicates that the classical result of Ladyzhenskaya (1969) on the existence of global smooth solutions to the 2D NSE is incorrect.

Abstract: Rapid advances in artificial intelligence are increasing the rate and steepness of informational and economic gradients experienced by human systems, challenging traditional models of adaptation based on stable identities, static optimization, and long-term professional blueprints. This study proposes a unified dynamical framework connecting thermodynamic entropy, information-theoretic entropy, and a formally defined entropy of the self through a shared stochastic gradient-flow model. Drawing on Langevin dynamics and simulated annealing, physical relaxation, probabilistic learning, and human identity formation are treated as governed by the same principles of regulated exploration followed by gradual stabilization. Within this framework, ambition is reinterpreted as temperature control: the capacity to sustain stochastic exploration in the absence of immediate external pressure. Agency is formalized as a rate-limited process constrained by an information-theoretic channel capacity of the self. Phase-portrait analysis and illustrative case studies show that environments of abundance and safety induce premature cooling, collapsing future possibility spaces and producing locally stable but globally brittle configurations. This effect is especially pronounced in traditionally professional career paths, where early specialization historically conferred robustness but now increases vulnerability under AI-driven task displacement and continuous retraining demands. The results indicate that adaptive human–AI systems should optimize for continuity of agency under accelerating change.

Abstract: This paper presents modified Lagrange-Jacobi functions derived from the sine, exponential, and hyperbolic tangent coordinate transformations. The resulting Lagrange-Jacobi functions and their respective matrix elements for observables can be reduced to their respective Lagrange-Legendre, Lagrange-Chebyshev, and Lagrange-Gegenbauer functions. Furthermore, this paper postulates that the Lagrange-mesh functions form approximate complete set of basis, a property implied by their approximate orthogonality.

Abstract: To ensure the security and confidentiality of various data types (including text, images, audio, and video), this paper proposes a multi-wavelet figure-and-text hiding algorithm (MWFTHA) and its corresponding multi-wavelet figure-and-text restoration algorithm (MWFTRA). These algorithms facilitate the encoding and embedding of text and color images into a one-dimensional signal through multi-wavelet transforms. Text data is encoded using a character dataset, while color images are processed via a linear transformation before being integrated into the signal. Subsequently, the original text and image can be precisely retrieved from the synthesized signal using MWFTRA. An illustrative case demonstrates the efficacy of this approach. The efficiency of MWFTHA and MWFTRA is verified through 1,000 simulations. The results indicate rapid data hiding and recovery, as indicated by the mean execution time and standard deviation. The method's performance is evaluated using the structural similarity index measure (SSIM) and peak signal-to-noise ratio (PSNR), which indicate slight improvements in quality relative to traditional wavelet and integer wavelet transforms. Additionally, the system's security is analyzed, with a focus on private-key mechanisms and resistance to data tampering. This steganography technology provides a robust solution for the secure transmission and storage of sensitive data, thereby reducing the risk of information leakage.

Abstract:

Traditional mean–variance portfolio optimization proves inadequate for cryptocurrency markets, where extreme volatility, fat-tailed return distributions, and unstable correlation structures undermine the validity of variance as a comprehensive risk measure. To address these limitations, this paper proposes a unified entropy-based portfolio optimization framework grounded in the Maximum Entropy Principle (MaxEnt). Within this setting, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) are formally derived as particular specifications of a common constrained optimization problem solved via the method of Lagrange multipliers, ensuring analytical coherence and mathematical transparency. Moreover, the proposed MaxEnt formulation provides an information-theoretic interpretation of portfolio diversification as an inference problem under uncertainty, where optimal allocations correspond to the least informative distributions consistent with prescribed moment constraints. In this perspective, entropy acts as a structural regularizer that governs the geometry of diversification rather than as a direct proxy for risk. This interpretation strengthens the conceptual link between entropy, uncertainty quantification, and decision-making in complex financial systems, offering a robust and distribution-free alternative to classical variance-based portfolio optimization. The proposed framework is empirically illustrated using a portfolio composed of major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—based on weekly return data. The results reveal systematic differences in the diversification behavior induced by each entropy measure: Shannon entropy favors near-uniform allocations, Tsallis entropy imposes stronger penalties on concentration and enhances robustness to tail risk, while WSE enables the incorporation of asset-specific informational weights reflecting heterogeneous market characteristics. From a theoretical perspective, the paper contributes a coherent MaxEnt formulation that unifies several entropy measures within a single information-theoretic optimization framework, clarifying the role of entropy as a structural regularizer of diversification. From an applied standpoint, the results indicate that entropy-based criteria yield stable and interpretable allocations across turbulent market regimes, offering a flexible alternative to classical risk-based portfolio construction. The framework naturally extends to dynamic multi-period settings and alternative entropy formulations, providing a foundation for future research on robust portfolio optimization under uncertainty.

Abstract: We establish the estimation of solutions of two classes of weakly singular Wendroff-type integral inequalities of multiple variables with multiple nonlinear terms, apply them to fractional partial differential equations, and give the proof procedures for uniqueness, boundedness and continuous dependence.

Abstract: In the current study, we propose a novel reduced-order model for the drag coefficient of a circular cylinder model that can be either fixed or undergoing an oscillatory linear motion in the cross-flow direction, the streamwise direction, or at an arbitrary tilt angle. Thus, the proposed model is not restricted to a single geometric setting of the cylinder. The model establishes a proper nonlinear coupling between the drag coefficient and the lift coefficient, such that the drag coefficient can be restructured using the simple reduced-order model, given the time signal of the lift coefficient. The proposed model is able to capture both the mean component of the drag coefficient, as well as the oscillatory component of it. We derived closed-form expressions to estimate the model parameters from a training dataset. The model was tested and found to be performing satisfactorily under different motion modes. We generated the training data using computational fluid dynamics simulation for a circular cylinder at a low Reynolds number of 300. The computational fluid dynamics solver used was successfully validated by comparison against independent published data. The current study is viewed as a contribution to the fields of nonlinear dynamics, fluid mechanics, and computational mathematics.

Abstract: The resummation of Stieltjes series remains a key challenge in mathematical physics, especially when Pad\'e approximants fail, as in the case of superfactorially divergent series. Weniger’s $\delta$-transformation, which incorporates a priori structural information on Stieltjes series, namely the inverse factorial series representation of their converging factors, offers a superior framework with respect to Pad\'e. Here, the problem of the pole distribution of the $\delta$-transformation is addressed. We show that the algebraic structure of the transformation, together with the intrinsic log-concavity of Stieltjes moments, satisfy the necessary conditions for having real poles. Moreover, by recasting the denominator of the $\delta$-transformation rational approximant as a high-order derivative of a log-concave polynomial and invoking the Gauss-Lucas theorem, a possible geometrical justification of the pole positioning along the negative real axis is proposed. While a fully rigorous proof remains an open challenge, our conjecture is substantiated by a comprehensive numerical investigation across an extensive catalog of Stieltjes series. In particular, our results provide systematic evidence that the mandatory branch cut conditions are respected even in the more delicate case of superfactorial growth, recently addressed from a converging factor perspective.

Abstract: This paper introduces MealMind, an integrated AI framework designed to address two interconnected challenges in household management: Decision Fatigue in meal planning and significant financial losses from preventable food waste. The system implements a three-component architecture: (1) Computer Vision (CV) for auto mated fridge inventory via smartphone scanning, eliminating the 59.3% failure rate of manual entry; (2) a Stock Optimization Algorithm using a Spoilage Proximity Index (S) to prioritize soon-to-expire items; and (3) a Hybrid LLM Planner gen erating personalized meal plans from available inventory, including comprehensive 7-day menus and special event planning for guests. Our mixed-methods study (survey: N=82 complete responses; interviews: N=5) quantifies the problem: 57.3% of households face daily "what to cook?" stress, 52.4% discard food due to forgetfulness, 78% of waste comprises expired items, and 58.5% demand better tools for weekly meal planning. We present a functional mobile prototype demonstrating technical feasibility and propose two testable hypotheses: MealMind reduces weekly planning time by >40% (H1) and decreases financial waste by >30% (H2). The paper concludes with a rigorous experimental design for val idation, positioning MealMind as a foundational layer for sustainable, intelligent kitchen ecosystems.

Abstract: The digital information ecosystem is increasingly characterized by the rapid proliferation of misinformation and the spontaneous emergence of polarized communities known as echo chambers. While classical epidemiological models have been widely adapted to describe information diffusion, they frequently fail to account for the unique sociotechnical mechanisms governing digital interaction: specifically, the interplay between individual cognitive resistance and algorithmic curation. This paper introduces a modified Susceptible-Critical-Infected-Recovered (SCIR) compartmental model designed to analyze the stability of echo chambers within scale-free networks. We explicitly incorporate two novel parameters: Cognitive Immunity (σ), representing the psychological capacity to critically evaluate and reject misinformation, and Algorithmic Bias (α), quantifying the platform-driven amplification of homophilic interactions. By applying heterogeneous mean-field theory, we derive the exact basic reproduction number, R₀, for this system. Our stability analysis reveals that while scale-free topologies typically exhibit a vanishing epidemic threshold, high levels of algorithmic bias can stabilize endemic states of misinformation (echo chambers) even in the presence of moderate cognitive immunity. Conversely, we demonstrate that a critical threshold of cognitive immunity exists, beyond which the information cascade collapses, regardless of algorithmic amplification. These findings provide a rigorous mathematical framework for evaluating the efficacy of ”prebunking” interventions versus algorithmic regulation in mitigating the spread of digital disinformation.

Abstract: An Uncertain Set assigns to each element a generalized uncertainty value, providing a unified language that encompasses fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and related models [1]. In this paper, we extend the notion of a membership function from scalar degrees to structured degrees represented by vectors, matrices, and higher-order tensors. We introduce vector-valued, matrix-valued, and tensor-valued Uncertain Sets (including the corresponding fuzzy, neutrosophic, and related special cases) and investigate their fundamental properties.

Abstract: Graph-based energy functionals constitute a fundamental tool in the mathematical modeling of networked systems, numerical schemes, and data-driven variational methods. In many applications, such energies are defined on discrete structures that serve as approximations of an underlying continuum domain. While the discrete-to-continuum behavior of such functionals has been extensively studied for regular graph sequences (e.g., uniform lattices, quasi-uniform point clouds), substantially less is known in the presence of graph irregularity. In this work, we analyze a class of graph-regularized energy functionals defined on highly irregular discrete domains. We introduce structural assumptions that allow for non-uniform connectivity, heterogeneous weights, and varying vertex densities, and we study their impact on compactness and Γ-convergence. Our main results establish sufficient conditions for convergence toward a continuum variational problem involving a p-Laplacian type energy with an Lq fidelity term. We demonstrate that irregular graph geometry may both preserve and destroy convergence depending on specific scaling regimes and connectivity patterns. Several illustrative examples and counterexamples are provided to show the sharpness of our assumptions. The analysis reveals a delicate interplay between local connectivity properties and global geometric constraints in determining the limiting behavior.

Abstract: Newton’s method is traditionally regarded as most effective when exact derivative information is available, yielding quadratic convergence near a solution. In practice, however, derivatives are frequently approximated numerically due to model complexity, noise, or computational constraints. This paper presents a comprehensive numerical and analytical investigation of how numerical differentiation precision influences the convergence and stability of Newton’s method. We demonstrate that, for ill-conditioned or noise-sensitive problems, finite difference approximations can outperform exact derivatives by inducing an implicit regularization effect. Theoretical error expansions, algorithmic formulations, and extensive numerical experiments are provided. The results challenge the prevailing assumption that exact derivatives are always preferable and offer practical guidance for selecting finite difference step sizes in Newton-type methods. Additionally, we explore extensions to multidimensional systems, discuss adaptive step size strategies, and provide theoretical convergence guarantees under derivative approximation errors.

Abstract: Fuzzy set theory enriches classical sets by assigning to each element a graded membership in [0,1], thereby capturing partial inclusion and uncertainty. The notion of an Uncertain Set further abstracts this idea by allowing membership to take values in a general degree-domain, providing a unified language that subsumes fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and related models. On the algebraic side, a hyperlattice replaces one lattice operation by a multivalued hyperoperation, enabling the representation of ambiguous or non-deterministic combinations, while a superhyperlattice iterates this structure through powerset lifting to obtain higher-order layers of interaction. Motivated by these developments, we introduce HyperLattice-valued and SuperHyperLattice-valued Uncertain Sets as lattice valued uncertainty frameworks whose degrees range over hyperlattices and their superextensions. Weestablish basic definitions, show that the proposed formalisms generalize existing lattice-valued models(including L-fuzzy, L-neutrosophic, and L-plithogenic sets), and discuss fundamental structural properties and canonical embeddings between the resulting classes.

Abstract: We establish a version of Pontryagin’s maximum principle for optimal control problems with impulses and phase constraints. Using the Dubovitskii-Milyutin theory, we construct a conic variational framework that handles impulsive dynamics and general state constraints. The main difficulty lies in working with piecewise continuous functions, required by the impulsive nature of the system. This setting also demands an extension of the classical result on the existence of nonnegative Borel measures, which leads to an adjoint equation formulated as a Stieltjes integral. Theoretical results are illustrated with examples, and key results by I. Girsanov are extended to the impulsive context.

Abstract: Public transport reliability is strongly influenced by the regularity of vehicle headways, defined as the time intervals between consecutive vehicles serving the same route. Irregular headways increase passenger waiting times, cause vehicle bunching, and reduce overall system efficiency. This paper presents a graph-based approach to the analysis and optimization of public transport headways, using the city of Bishkek, Kyrgyzstan, as a case study. The public transport network is modeled as a weighted graph, where stops are represented as vertices and route segments as edges. Headways are incorporated as temporal attributes associated with routes and vehicle movements. An optimization objective is formulated to minimize headway variability across selected routes. Using simulated operational data, the proposed approach demonstrates that graph-based modeling provides a flexible and effective framework for analyzing headway irregularities and evaluating optimization strategies. The results highlight the potential of graph-based methods to support planning and operational decision-making in urban public transport systems.

Abstract: We investigate an optimal transport problem augmented with a total variation regularization term that penalizes deviations of a transport plan from the inde- pendent product of the marginals. This approach yields a convex but non-smooth optimization problem and provides an alternative to entropy-based regularization. We establish existence of minimizers and prove that for any positive regularization parameter, the resulting functional defines a metric on the space of probability mea- sures. Detailed analysis of the triangle inequality and other metric properties is provided. We study limiting regimes as the regularization parameter tends to zero (recovering the Wasserstein distance) and to infinity (yielding a multiple of the total variation distance). A discrete formulation leading to a linear programming problem is presented, along with qualitative examples illustrating the sparsity-promoting na- ture of the model. Comparisons with entropic regularization highlight the trade-offs between computational efficiency and structural properties of optimal couplings.

Abstract: This paper presents a numerical comparison of two classical root-finding algorithms: the bisection method and Newton’s method. Both methods are applied to a selected nonlinear equation in order to analyze their convergence behavior, numerical stability, and practical efficiency under identical conditions. The comparison is based on a simple numerical experiment using a fixed stopping criterion and well-defined initial conditions. The results demonstrate the guaranteed but relatively slow linear convergence of the bisection method, as well as the fast quadratic convergence of Newton’s method when a suitable initial approximation is available. The study highlights the fundamental trade-off between robustness and efficiency in numerical root-finding and provides a clear and accessible illustration of the practical differences between these widely used methods.

Abstract: Abstract An explicit radial velocity relation is evaluated numerically against a large sample of disk galaxies using directly measured rotation curve data. For each radial sample, the predicted circular velocity is computed from the baryonic velocity and radial distance using a fixed algebraic expression with explicitly defined numerical constants applied uniformly across all galaxies. Agreement between predicted and observed velocities is quantified using root mean squared error, Pearson correlation, and Lin’s concordance correlation coefficient computed directly from the paired numerical values across the full dataset [1,2].

Abstract: Empirical debates about a "crisis of trust'' highlight long-lived pockets of high trust and deep distrust in institutions, as well as abrupt, shock-induced shifts between them. We propose a probabilistic model in which such phenomena emerge endogenously from social learning on hierarchical networks. Starting from a discrete model on a directed acyclic graph, where each agent makes a binary adoption decision about a single assertion, we derive an effective influence kernel that maps individual priors to stationary adoption probabilities. A continuum limit along hierarchical depth yields a degenerate, non-conservative logistic--diffusion equation for the adoption probability u(x,t), in which diffusion is modulated by (1-u) and increases the integral of u rather than preserving it. To account for micro-level uncertainty, we perturb this dynamics by multiplicative Stratonovich noise with amplitude proportional to u(1-u), strongest in internally polarised layers and vanishing at consensus. At the level of a single depth layer, Stratonovich–Itô conversion and Fokker–Planck analysis show that the noise induces an effective double-well potential with two robust stochastic phases, u ≈ 0 and u ≈1, corresponding to persistent distrust and trust. Coupled along depth, this local bistability and degenerate diffusion generate extended domains of trust and distrust separated by fronts, as well as rare, Kramers-type transitions between them. We also formulate the associated stochastic partial differential equation in Martin–Siggia–Rose–Janssen–De Dominicis form, providing a field-theoretic basis for future large-deviation and data-informed analyses of trust landscapes in hierarchical societies.