Abstract: The standard $\varepsilon$--$\delta$ definition of continuity is inherently quantitative, yet the precise dependence of the admissible radius $\delta$ on the accuracy $\varepsilon$ and the base point $x_0$ is rarely treated as an independent mathematical object. In this paper, we introduce the \textit{radius of continuity} through two variants: the radius of pointwise continuity and the radius of uniform continuity, defined as explicit numerical invariants that capture the maximal symmetric neighborhood on which a real-valued function maintains a prescribed tolerance. We establish the fundamental structural properties of these radii, including their behavior under algebraic operations such as sums, products, and compositions, and demonstrate their inverse relationship to the classical modulus of continuity. Furthermore, we prove that the finiteness pattern of these radii characterizes constant versus non-constant functions. To illustrate the utility of this framework, we derive closed-form expressions for the pointwise radius of quadratic polynomials and the uniform radius of the normal probability density function. These examples highlight how the radius of continuity encodes geometric and probabilistic features, such as local curvature and global scale parameters. Ultimately, this perspective bridges the gap between real analysis and quantitative methods in metric geometry, offering a concrete measure of the stability of a function's continuity.

Abstract: We ask for C*-metric version of following three: (1) Bourgain-Figiel-Milman Theorem, (2) Enflo Type, (3) Mendel-Naor Cotype.

Abstract: Massera and Schaffer [Ann. Math. (2), 1958] established a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space. Maligranda [Am. Math. Mon., 2006] improved Massera-Schaffer upper bound for the Clarkson angle. Pecaric and Rajic [Math. Inequal. Appl., 2007] extended Maligranda's inequality to finitely many nonzero vectors. We derive a finite field version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.

Abstract: This study presents a multi-metric analysis of the QS World University Rankings by Subject 2026: Engineering & Technology, examining ranking structure, temporal movement between 2025 and 2026, country-level concentration, and indicator-level performance across the leading universities. The dataset contains 2025 and 2026 ranks, overall score, and five core QS indicators: academic reputation, employer reputation, citations, H-index, and international research network (IRN). The results show a highly concentrated global landscape, with the United States contributing 22 universities, while China, the United Kingdom, and Australia each contribute 8 universities. At the institutional level, the score distribution is steep at the top and compressed in the lower part of the list, led by MIT (95.9), Stanford University (93.5), and ETH Zurich (92.7), indicating that relatively small score differences can separate many lower-ranked institutions.The comparative analysis shows strong temporal stability, with a Pearson correlation of r = 0.946 between the 2025 and 2026 rankings, despite visible upward and downward mobility among a subset of institutions. Country-level results further reveal asymmetric national profiles: Switzerland records the strongest average ranking position (7.5) and highest average score (90.4), while Singapore achieves the highest employer reputation (92.8), citations (95.8), and H-index (94.8) among the top-performing countries. Volatility analysis indicates that China (STD = 16.2) and Australia (STD = 14.9) exhibit the largest cross-institutional rank fluctuations. Correlation analysis confirms that overall score is most strongly associated with 2026 rank (r ≈ -0.93), followed by academic reputation (r ≈ -0.80) and employer reputation (r ≈ -0.75), whereas citations (r ≈ -0.34), H-index (r ≈ -0.53), and IRN show weaker direct effects. Overall, the findings demonstrate that elite university performance is shaped by a combination of strong reputation, research impact, and national concentration, rather than by a single dominant metric.

Abstract: In this paper the uniform approximation of continuous functions on \( [0,1] \) by rational functions with prescribed poles and bounded multiplicities is studied. A classical theorem of Fichera characterizes density in \( C([0,1]) \) through the divergence of a conformally invariant series involving the pole distribution. A modern reformulation of this result is developed and it is given an operator-theoretic interpretation in which the approximation property is equivalent to cyclicity and to the absence of nontrivial invariant subspaces in an associated Hardy-space model. In this framework, the classical Blaschke condition emerges as the fundamental obstruction to density, linking rational approximation to the structure of model spaces and non-selfadjoint operator algebras. The density criterion is interpreted in terms of symmetry: divergence corresponds to a balanced distribution of poles compatible with the conformal geometry of the slit domain, while convergence induces symmetry breaking and the emergence of invariant structures. Numerical models illustrate the sharpness of the criterion and provide a concrete manifestation of the Blaschke obstruction and cyclicity mechanism. This new approach places Fichera’s theorem within a broader operator-theoretic and spectral framework, connecting classical approximation theory with Hardy spaces, invariant subspace theory, and modern rational approximation methods.

Abstract: We study the incompressible Navier–Stokes equations on the three-dimensional torus for smooth mean-zero divergence-free periodic data. The argument begins with an exact shellwise energy identity in which the nonlinear contribution is decomposed into four explicit channels. Among these, the only shellwise contribution at the critical scale is the balanced high--high to low quadrupole transfer, while the remaining side channels are shown to be strictly subcritical. The main new mechanism is a finite packet-exhaustion theorem for the full shell obstruction: on every finite time horizon, the obstruction is decomposed into finitely many standard radial--angular packet channels together with a remainder satisfying a quantitative bound of the form \( \int_0^T |R_j(t)|\,d t \le \|u_0\|_{L_x^2}\bigl(\kappa 2^{2j}+C_\lambda 2^{\lambda j}\bigr)B_j(T)^2, \qquad \lambda<2, \qquad \kappa \|u_0\|_{L_x^2} < v. \) The packet terms satisfy a uniform endpoint estimate with exponent 7/4<2, the critical remainder is absorbed by viscosity, and the exact shell bridge closes on every finite interval. This yields strong continuation and therefore global smoothness for the periodic Navier--Stokes flow issuing from smooth data.

Abstract: Five machine learning algorithms — Linear Regression (LR), Multilayer Perceptron (MLP), Support Vector Machine for regression (SMOreg), RandomTree, and Reduced Error Pruning Tree (REPTree) — were trained and compared for predicting durum wheat (Triticum durum Desf.) grain yield simulated by AquaCrop-GIS across the Capitanata plain (Southern Italy). A dataset of 342 instances was constructed by crossing 25 soil profiles, three sowing dates, and two irrigation regimes over 15 climatic grid cells (2014–2023), validated by stratified 10-fold cross-validation. MLP achieved the highest accuracy (R = 0.983; MAE = 0.059 t ha-1; RMSE = 0.083 t ha-1); the four interpretable models clustered at R = 0.891–0.907 (RMSE = 0.192–0.203 t ha-1). All models converged on consistent agronomic signals: standard sowing (1 November) yielded +0.53 t ha-1 over late sowing (15 November); supplemental irrigation added +0.17 t ha-1; high-silt and clay soils produced superior yields. The SMOreg normalised weight vector identified autumn temperature (Tmin_oct_nov: −0.462; Tmax_oct_nov: −0.405) as the dominant climate predictor, reflecting the AquaCrop phenological mechanism whereby elevated early-season thermal loads curtail tillering. The convergence of directional signals across fundamentally different algorithmic architectures — linear, kernel-based, and tree-based — confirms that ML surrogates can efficiently emulate AquaCrop response surfaces for scenario analysis and decision-support in Mediterranean dryland farming systems.

Abstract: Missing data remains a pervasive challenge in air quality data analysis, where inappropriate imputation techniques can introduce hidden biases and compromise the reliability of time-series models such as AutoRegressive Integrated Moving Average (ARIMA). This paper examines the impact of linear interpolation and mean/median imputation on the performance of the ARIMA model and biases in the prediction of particulate matter 2.5 (PM2.5) concentration, together with a detailed analysis of ARIMA generated error metrics and their implications for the accuracy and reliability of the prediction. The findings reveal that package-default imputation significantly influences ARIMA forecasts, while mean/median imputation consistently delivers superior predictive performance, highlighting its robustness for handling missing environmental data. Moreover, imputation during the data transformation stage exerts a greater influence on model outcomes than methods applied at later analysis stages.

Abstract: We ask for non-Archimedean version of following four: (1) John Theorem, (2) Dvoretzky-Milman Theorem, (3) Type-Cotype of Banach space, (4) Kwapien Theorem.

Abstract: We study the long-time behavior of nonlinear stochastic evolution equations in a separable Hilbert space driven by a Q-Wiener process. The linear part of the equation is generated by a strongly continuous semigroup with an exponential dichotomy, which provides fixed rates of decay and growth. The nonlinear drift and diffusion terms are globally Lipschitz and become small as time tends to infinity. Our main result shows that under these conditions, the mean-square Lyapunov exponents of the nonlinear system coincide with those of the linear part. In other words, nonlinear stochastic perturbations that decay in time do not change the main growth or decay rates of solutions in the mean-square sense. This result provides simple and verifiable criteria ensuring that the long-time Lyapunov behavior of the nonlinear stochastic equation is fully determined by the linear semigroup, even in the presence of time-dependent stochastic perturbations.

Abstract: Construction progress monitoring is vital for effective project management, as it provides essential information that empowers managers to make timely and informed decisions, thereby ensuring successful project completion while preventing delays and cost over-runs. The integration of advanced technologies, such as drones, the Internet of Things, and artificial intelligence (AI), offers promising techniques for automated monitoring, trans-forming traditional manual processes into efficient, data-driven systems that enhance ac-curacy and reliability in tracking project progress. This paper comprehensively analyzes the key research topics within the field of construction progress monitoring by integrating machine learning with large language models. We utilize AI techniques to extract, inter-pret, and synthesize key thematic clusters from a corpus of scientific publications span-ning the years 2000 to 2024. Our findings reveal three major thematic areas, underscoring the interdisciplinary nature of construction research. The study demonstrates the dynamic evolution of topics, reflecting shifts in research focus and the growing influence of techno-logical innovations. This research not only advances understanding in construction pro-gress monitoring but also showcases the transformative potential of AI-driven methods in uncovering insights from large-scale data.

Abstract: In this paper, we introduce and systematically investigate novel classes of vector-valued multiplier spaces associated with operator-valued series, utilizing the concepts of \( f \)-statistical and weak \( f \)-statistical convergence. We begin by studying the topological properties of these newly defined spaces, establishing that their completeness is completely characterized by the \( c_0(X)- \) multiplier convergence of the underlying series. Building upon this structural foundation, we then explore the precise relationships between these \( f \)-statistical spaces and classical statistical multiplier spaces, proving that they perfectly coincide under the assumption of a compatible modulus function. Furthermore, we define a natural summing operator acting on these spaces and conduct a detailed analysis of its mapping properties. By establishing necessary and sufficient conditions for the continuity and (weak) compactness of this summing operator, we obtain new characterizations for both \( c_0(X)- \) and \( \ell_\infty(X)- \) multiplier convergent series.

Abstract: In this paper, we move beyond the classical setting by redefining the Chebyshev functional in the context of q-circles situated within Minkowski space, rather than the standard Euclidean circles in R². This approach introduces a new theoretical framework suitable for non-Euclidean geometries. We derive sharp estimates for the functional when applied to functions on q-circles that adhere to Hölder-type continuity conditions.

Abstract: Time series data, characterized by temporal dependencies, seasonality, and noise, are prevalent in domains such as healthcare, finance, energy, and transportation. Effective clustering of time series enables the discovery of patterns, supports forecasting, and facilitates data-driven decision-making. This paper provides a comprehensive review of time series clustering techniques, including conventional methods (e.g., k-means, hierarchical, and fuzzy clustering), similarity-based approaches (e.g., Dynamic Time Warping), feature-based methods, and deep learning models (e.g., autoencoders, convolutional and recurrent neural networks). The review analyzes tasks, application domains, performance outcomes, and key limitations, highlighting common challenges such as computational complexity, sensitivity to noise, and scalability issues. A particular focus is given to transport-related time series, including traffic flow, travel time, and congestion patterns, demonstrating how clustering can support traffic state classification, anomaly detection, and infrastructure planning. The analysis reveals a trade-off between accuracy, interpretability, and computational efficiency, emphasizing the need for scalable, robust, and domain-aware clustering frameworks. Finally, practical directions for future research are discussed, including lightweight hybrid approaches and transport-specific feature engineering to enhance clustering performance in real-world applications.

Abstract: This study presents the development and verification of CUPID-MSR, an extended thermal-hydraulics code incorporating temperature-dependent properties of two chloride-based molten salts, KCl–UCl3 and NaCl–MgCl2–TRUCl3. Verification against the de Vahl Davis natural-convection benchmark across Rayleigh numbers 1,000–1,000,000 showed agreement within 0.4–3.9%, accurately capturing reference Nusselt numbers, flow structures, and thermal boundary layers. Additional temperature-variation studies confirmed stable and consistent performance of the implemented material correlations. The applicability of the Boussinesq approximation was assessed by comparing full variable-density and Boussinesq formulations, revealing that the approximation remains accurate for βΔT ≲ 0.1. Since this threshold depends only on relative density change, it is broadly relevant for natural-convection flows in Newtonian fluids.

Abstract: This paper addresses the failure of major digital investments to achieve sustained technology adoption in developing countries, hindering their business growth. While existing research identifies institutional drawbacks as a key problem, it offers limited guidance on progress within these constraints. To address this gap, the new Institutional Framework for Smart Technology Adoption (IFSTA), pronounced Eye-f-sta, is developed as a contingent institutional framework connecting digital transformation theory with practical assessment tools. IFSTA argues that adoption success depends not on technology alone, but on strategic alignment with specific institutional contexts. Built around three core pillars, governance, socio-technical infrastructure, and adaptive capacity, the framework explains how their interactions shape adoption. Three questions are addressed: (1) how local conditions moderate infrastructure impact; (2) what workarounds enable progress amid fragile systems; and (3) how digital investments can be sequenced based on institutional starting points. A central insight is the critical role of localization, adapting standards, platforms, and partnerships to local context as a fundamental mechanism. Contributions are threefold: addressing the gap between diagnosis and implementation by developing effective guidance for developing economies; methodologically bridging static assessments with actionable diagnostics; and practically providing a structured framework and Performance-Knowledge Index (PKI) tool to diagnose contexts and prioritize interventions, moving from agnostic best practices to local strategies.

Abstract: Topic modeling plays an essential role in extracting latent structures from large text corpora. The choice of model and the number of topics which can strongly influence the performance and interpretability of the outcomes. In this work, I compare three widely used models in topic modeling: Latent Dirichlet Allocation, Non-Negative Matrix Factorization, and Bidirectional Encoder Representations from Transformers. The outcomes of the models are studied using Entropy, Jaccard similarity, Coherence, and Silhouette over a wide number of topics. The results show that NMF consistently produces the most interpretable and distinct topics, achieving the highest coherence score, with optimal performance observed at k = 15. LDA yields broader and less coherent topics. In contrast, BERT-based clustering shows low Silhouette scores, indicating weak cluster separation.

Abstract: We prove that the Riemann Hypothesis (RH) admits a theorem-level stagewise arithmetical normal form of type \( \Pi^0_2 \), obtained from a single fixed terminating certificate calculus for the Riemann \( \Xi \)–function. Let \( \xi(s):=\tfrac12\,s(s-1)\,\pi^{-s/2}\Gamma\!\Bigl(\frac{s}{2}\Bigr)\zeta(s), \qquad \Xi(z):=\xi\!\left(\tfrac12+i z\right), \), and let U := {z = x + iy ∈ C : x > 0, 0 < y < 1/2}. Then RH is equivalent to Z(Ξ; U) = ∅. We construct a countable family of rational stage rectangles {Ωj,k}j≥1,k∈Z with Ωj,k ⊂ U and U ⊆ S j,k Ωj,k, and we define an explicit predicate Cert(j, k, c) ⊆ N≥1 × Z × N whose truth asserts that the code c is a mechanically checkable certificate that Ξ is zero-free on Ωj,k. Soundness is proved via certified boundary nonvanishing, a certified winding computation, and the argument principle. Decidability of Cert is proved by a terminating verifier based on rational disk arithmetic together with explicit rational remainder bounds for special-function evaluations (Euler–Maclaurin for ζ, ζ′, ζ′′ and Stirling-type bounds for Γ, ψ, ψ′). The verifier uses only rational computations and certified rational upper bounds; external libraries (e.g. Arb) may be used to discover certificates but are not trusted by the formal predicate. Define the sweep sentence CS :⇐⇒ ∀j ≥ 1 ∀k ∈ Z ∃c ∈ N Cert(j, k, c). We prove RH ⇐⇒ CS. Since Cert is decidable, CS is a Π02 sentence; thus RH is \( \Pi^0_2 \) .

Abstract: We develop a comprehensive Lagrangian framework for the analysis of singularities in the threedimensional chemotaxis–Navier–Stokes system. Focusing on suitable weak solutions, we introduce the notion of Lagrangian singular trajectories and establish a geometric characterization of the space–time blow-up set. Our main theoretical advance shows that singularities are confined to a low-dimensional Lagrangian structure transported by the flow. More precisely, we prove that the space–time singular set S is contained in a countable union of Lagrangian trajectories associated with the velocity field and satisfies the sharp estimate dim_H(S) ≤ 1. This result constitutes a substantial refinement of classical Eulerian partial regularity bounds of Caffarelli–Kohn–Nirenberg type and provides a genuinely geometric interpretation of singularity formation in coupled fluid–chemotaxis models. The proof combines global energy inequalities, compactness methods, and partial regularity theory with a refined analysis of the Lagrangian flow map in the DiPerna–Lions–Ambrosio setting. A key feature of our approach is the propagation of regularity along particle trajectories, which allows singularities to be tracked dynamically and yields improved dimensional estimates via tools from geometric measure theory. Beyond the dimensional bound, the proposed Lagrangian formulation clarifies the mechanism by which chemotactic forcing interacts with fluid transport to produce potential blow-up and establishes a direct connection between singularity formation and low-dimensional invariant structures. These results open new perspectives for the geometric analysis of singularities in active fluid systems and related nonlinear PDEs.

Abstract: We consider the Fisher–KPP equation with Neumann boundary conditions on the real half line. We claim that the Fisher-KPP equation with Neumann boundary conditions is well-posed only for odd positive stationary solutions. We begin by proving that the Fisher-KPP equation with a Dirichlet boundary condition is stable, and with a Robin condition is stable only for odd positive stationary solutions. Then we inferred and proved that the Fisher-KPP equation with Neumann boundary conditions is stable only for odd positive stationary solutions. We solved the Fisher–KPP equation with Neumann boundary conditions to demonstrate the existence of the solution. In addition, we proved the uniqueness of the solution. Moreover, we proved the the solution of Fisher-Kpp equation with Dirichlet condition is stable. We also showed that the Fisher-KPP equation with Robin boundary conditions is stable only for odd positive stationary solutions. The uniqueness and existence proof of the Fisher-KPP equation with Robin condition are similar to the Neumann condition. Hence, we conclude that the Fisher-KPP equation on the real line is well-posed for the Dirichlet condition, and well-posed only for odd positive stationary solutions for both the Neumann condition and the Robin condition.