Search Results (1,611)

Mathematical modeling is a cyclical process in which mathematics is used to represent, explore, and better understand real-world situations by mathematizing a problem and validating the results. Unlike traditional word problems, modeling tasks require learners to make assumptions, define quantities, apply mathematics, interpret results, and revise solutions within authentic contexts. Mathematical modeling is increasingly recognized as essential in mathematics education, highlighted in standards such as the Common Core State Standards for Mathematics through mathematical practice 4 (MP4): model with mathematics. This entry explores the evolving landscape of mathematical modeling education. First, it reviews the diverse conceptions of modeling and the various frameworks used to describe the modeling process, including distinctions between holistic and atomistic approaches. Second, it examines the current state of research, including the documented benefits of modeling for student engagement and access, as well as the challenges teachers face in enacting modeling tasks. Finally, the entry discusses implications for future research and practice, emphasizing the need for targeted teacher preparation and continued theoretical refinement to better support the integration of modeling into mathematics instruction. Full article

In the context of Industry 4.0 and the proliferation of smart buildings, elevators represent critical assets whose performance is often inadequately measured by traditional indicators that overlook energy consumption. This study addresses the need for a more holistic Key Performance Indicator (KPI) by developing the Overall Equipment Effectiveness for Elevators (OEEE), an index designed to integrate operational effectiveness with energy efficiency. The methodology involves adapting the classical OEE framework through a comprehensive literature review and an analysis of elevator energy standards. This leads to a novel structure that incorporates a dedicated energy efficiency dimension alongside the traditional pillars of availability, performance, and quality. The framework further refines the performance and energy efficiency dimensions, resulting in six distinct sub-indicators that specifically measure operational uptime, speed adherence, electromechanical conversion, fault-free cycles (as a proxy for operational quality), and energy use during both movement and standby modes. The primary result is the complete mathematical formulation of the OEEE, a single, integrated KPI derived from these six metrics and designed for implementation using data from modern IoT-enabled elevators. The study concludes that the OEEE provides a more accurate and comprehensive tool for asset management, enabling data-driven decisions to enhance reliability, optimise energy consumption, and reduce operational costs in smart vertical transportation systems. Full article

In this paper a new single-input independent multiple-output universal tunable filter employing second-generation current conveyors (CCII) and second-generation voltage conveyors (VCII) as active elements is presented. The proposed filter has been analyzed at transistor level, using a CMOS standard AMS 0.35 μm technology, and implemented using discrete components based on the commercially available AD844. A detailed mathematical analysis is carried out, considering also parasitic impedances and non-ideal parameters. The low-pass, band-pass, and high-pass responses are simultaneously obtained and experimentally verified at 10 kHz central frequency where the voltage gain is about 27 dB for each output. THD analysis has been performed to evaluate the proposed work. Full article

In this paper, we investigate the Malliavin differentiability and density smoothness of solutions to stochastic differential equations (SDEs) with non-Lipschitz coefficients. Specifically, we consider equations of the form $d{X}_t= b\left(X_t\right)dt + \sigma \left(X_t\right)d{W}_t, X_0= x_0$ where the drift b(·) and diffusion σ(·) may violate the global Lipschitz condition but satisfy weaker assumptions such as Hölder continuity, linear growth, and non-degeneracy. By employing Malliavin calculus theory, large deviation principles, and Fokker–Planck equations, we establish comprehensive results concerning the existence and uniqueness of solutions, their Malliavin differentiability, and the smoothness properties of density functions. Our main contributions include (1) proving the Malliavin differentiability of solutions under the standard linear growth condition combined with Hölder continuity; (2) establishing the existence and smoothness of density functions using Norris lemma and the Bismut–Elworthy–Li formula; and (3) providing optimal estimates for density functions through large deviation theory. These results have significant applications in financial mathematics (e.g., CIR, CEV, and Heston models), biological system modeling (e.g., stochastic population dynamics and neuronal and epidemiological models), and other scientific domains. Full article

This study presents a general 3 × 5 × 5 factorial experimental design to maximize the Power Coefficient (Cp) of an H-Darrieus hydrokinetic turbine equipped with external accessories. Five accessory configurations (standard, cycloidal, flat plate, curve, and blocking plate), three solidity levels, and five Tip-Speed Ratio (TSR) levels were evaluated as main factors under the hypothesis that these factors significantly influence Cp. The data analyzed were obtained from numerical simulations, and their processing was conducted using Analysis of Variance (ANOVA), linear regression models, and response surfaces in the software programs Minitab 21 and RStudio V4.4.2. ANOVA makes it possible to determine the statistical significance of the effect of each factor and their interactions on the obtained Cp, identifying the accessories, TSR, and solidity that have the greatest impact on turbine performance. The results indicate that the optimal configuration to maximize Cp includes the flat-plate accessory, a solidity of 1.0, and a TSR of 3.2. From the linear regression models, mathematical relationships describing the system’s behavior were established, while the response surface analysis identified optimal operating conditions. These findings provide an effective tool for optimizing H-Darrieus turbine designs, highlighting the positive impact of accessories on performance improvement. Full article

This paper presents a novel ontology-driven digital twin framework specifically designed for aviation maintenance and operations that addresses these challenges through semantic reasoning and explainable decision support. The proposed framework integrates seven interconnected ontologies—structural, functional, behavioral, monitoring, maintenance, lifecycle, and environmental. It collectively provides a comprehensive semantic representation of aircraft systems and their operational context. Each ontology is mathematically formalized using description logics and graph theory, creating a unified knowledge graph that enables transparent, traceable reasoning from sensor observations to maintenance decisions. The digital twin is formally defined as a 6-tuple that incorporates semantic transformation engines, cross-ontology mappings, and dynamic reasoning mechanisms. Unlike traditional data-driven approaches that operate as black boxes, the ontology-driven framework provides explainable inference capabilities essential for regulatory compliance and safety certification in aviation. The semantic foundation enables causal reasoning, rule-based validation, and context-aware maintenance recommendations while supporting standardization and interoperability across manufacturers, airlines, and regulatory bodies. The research contributes a mathematically grounded, semantically transparent framework that bridges the gap between domain knowledge and operational data in aviation maintenance. This work establishes the foundation for next-generation cognitive maintenance systems that can support intelligent, adaptive, and trustworthy operations in modern aviation ecosystems. Full article

Maxwell’s equations epitomize our knowledge of standard electromagnetic theory in vacuums and matter. Here, we report the clearcut results of an extensive, ongoing investigation aiming to mathematically digest Maxwell’s equations in virtually all problems based on the three standard building units, dielectric and magnetic, found in practice (i.e., spheres, cylinders and plates). Specifically, we address the static/quasi-static case of a linear, homogeneous and isotropic dielectric and magnetic sphere subjected to a DC/low-frequency AC external scalar potential, (vector field, ), of any form, produced by a primary/free source residing outside the sphere. To this end, we introduce an expansion-based mathematical strategy that enables us to obtain immediate access to the response of the dielectric and magnetic sphere, i.e., to the internal scalar potential, (vector field, ), produced by the induced secondary/bound source. Accordingly, the total scalar potential, = + (vector field, = + ), is immediately accessible as well. Our approach provides ready-to-use expressions for and ( and ) in all space, i.e., both inside and outside the dielectric and magnetic sphere, applicable for any form of (). Using these universal expressions, we can obtain and ( and ) in essentially one step, without the need to solve each particular problem of different () every time from scratch. The obtained universal relation between and ( and ) provides a means to tailor the responses of dielectric and magnetic spheres at all instances, thus facilitating applications. Our approach surpasses conventional mathematical procedures that are employed to solve analytically addressable problems of electromagnetism. Full article

Nozzle blockage has been a critical issue for productivity and product quality since the introduction of continuous casting. Despite numerous studies on the subject, the problem persists, affecting steel production. This detrimental phenomenon causes changes in the internal nozzle geometry and severe wall irregularities that are neither symmetrical nor uniform. A common approach to studying the complex internal shape of clogged nozzles is considering nozzles with symmetrical transversal area reductions. Therefore, this study aims to quantitatively evaluate the effects of using realistic submerged entry nozzle (SEN) clogging geometries on the fluid dynamic behavior of molten steel inside the SEN and the mold and is compared to simplified symmetric reductions. A three-dimensional mathematical simulation based on the Navier–Stokes equations, the standard k − ε turbulence model, and the Volume of Fluid (VOF) method was used. The main findings indicate that symmetric reductions can only provide a qualitative prediction of the results, such as increased velocity and asymmetries at the meniscus bath level, but with errors that can reach up to 25%. Symmetric reductions fail to accurately capture the fluid dynamics inside the nozzle and the mold and should therefore be used with caution in studies that require precise flow characterization near the nozzle walls. Full article

With the rapid development of China’s economy, the number and scale of infrastructure projects in energy, water conservancy, and transportation have expanded significantly. Anchoring technology has been widely applied, resulting in the formation of numerous rock slope–anchoring systems. This study proposes a novel method for evaluating the stability of rock slope–anchoring systems by introducing catastrophe theory into the stability assessment framework. Based on the characteristics of the rock slope–anchoring system and its stability-influencing factors, a hierarchical analytic structure for catastrophe-level evaluation is constructed, and relevant indicator data are collected. Catastrophe models are selected according to the identified state and control variables, and catastrophe levels are computed to establish a sample dataset. The relationship between catastrophe levels and the stability coefficients of rock slope–anchoring systems is verified to define stability grade intervals. Stability evaluation is then performed by calculating the catastrophe level of each system. The results indicate that: (1) the proposed method effectively considers the influence of multiple factors on the stability of rock slope–anchoring systems, ensuring high accuracy in the evaluation. (2) The method allows for the automatic quantification of the relative importance of indicators within the same hierarchy, reducing subjectivity caused by manual weighting. (3) By standardizing state variables and computing catastrophe levels, the method couples qualitative descriptions with mechanical parameters, enhancing the objectivity of the assessment. (4) The stability evaluation method for rock slope–anchorage systems based on mathematical catastrophe theory determines system stability through catastrophe-order analysis, featuring a concise process and clear results. It enables rapid evaluation of the stability of similar rock slope–anchorage systems and offers high efficiency for cluster assessments. Full article

Human activity recognition (HAR) is an automated technique for identifying human activities using images and sensor data. Although numerous studies exist, most of the models proposed are highly complex and rely on deep learning. This research utilized two novel frameworks based on the Extreme Gradient Boosting (XGB) classifier, also known as the XGBoost classifier, enhanced with metaheuristic algorithms: Golden Jackal Optimization (GJO) and War Strategy Optimization (WARSO). This study utilized the KU-HAR dataset, which was collected from smartphone accelerometer and gyroscope sensors. We extracted 48 mathematical features to convey the HAR information. GJO-XGB achieved a mean accuracy in 10-fold cross-validation of 93.55% using only 23 out of 48 features. However, WARSO-XGB outperformed GJO-XGB and other traditional classifiers, achieving a mean accuracy, F-score, precision, and recall of 94.04%, 92.88%, 93.47%, and 92.40%, respectively. GJO-XGB has shown lower standard deviations on the test set (accuracy: 0.200; F-score: 0.285; precision: 0.388; recall: 0.336) compared to WARSO-XGB, indicating a more stable performance. WARSO-XGB exhibited lower time complexity, with average training and testing times of 30.84 s and 0.51 s, compared to 39.40 s and 0.81 s for GJO-XGB. After performing 10-fold cross-validation using various external random seeds, GJO-XGB and WARSO-XGB achieved accuracies of 93.80% and 94.19%, respectively, with a random seed = 20. SHAP identified that range_gyro_x, max_acc_z, mean_gyro_x, and some other features are the most informative features for HAR. The SHAP analysis also involved a discussion of the individual predictions, including the misclassifications. Full article

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering. Full article

This research investigates the impact of incorporating montmorillonite-based nanoclay additives on the anti-corrosive properties of a polyester/triglycidyl isocyanurate (polyester/TGIC) powder coating on phosphated steel. The self-repairing capability facilitated by the swelling and expansion of nanoclay was demonstrated to enhance the corrosion resistance of the coatings significantly. A statistical Mixture Design methodology was employed to establish the optimal combination of nanoclay dosage and coating film thickness. Nineteen experiments were conducted using Design of Experiments, and two regression models were developed using the measured polarization resistance (R_p) and specular gloss values as responses. The mathematical maximization of the R_p value predicted an optimal nanoclay dosage of 4.1% with a corresponding film thickness of 80 µm. Statistical and experimental verification validated the results obtained from the regression models. Notably, the optimized coating demonstrated an R_p value one order of magnitude higher than the coating with 4% nanoclay and a standard film thickness of 60 µm. The behavior of the newly developed coatings was analyzed and compared through measurements of open circuit potential, polarization resistance, and electrochemical impedance spectroscopy. The findings confirm the substantial improvement in the anti-corrosive and self-repairing properties of the polyester/TGIC powder coating with the incorporation of montmorillonite-based nanoclay additives. Full article

This paper presents a machine verification of a real number theory where real numbers are constructed using concepts related to filters. The theory encompasses a special filter, namely the non-principal arithmetical ultrafilter whose existence can be proven with the Continuum Hypothesis, to establish several non-standard number sets: *N, *Z and *Q. The set of real numbers, $\mathbf{R}$, is subsequently obtained by the equivalence classification of a specific subset of *Q. The entire theory is thoroughly formalized, with each detail verified to ensure rigor and precision. The verification is implemented using the Coq proof assistant and is grounded in the Morse–Kelley axiomatic set theory. This work contributes a new selection of foundational material for the formalization of mathematical theories. Full article

Seagoing marine propulsion analysis in terms of main engine performance and fixed-pitch propeller hydrodynamics is an engineering problem that has not been exactly defined to date. This study utilizes an original and comprehensive mathematical approach—involving the approximate representation of one function by another—to define this problem in mathematical terms and solve it. This is achieved by imperatively applying an original and sophisticated hybrid combination of an existing, formidable and ingenious, mathematical methodology with different original comprehensive functional systems. These original functional systems approximately represent the operations of vessels under seagoing conditions, including the thermo-fluid and frictional processes of vessels’ main engines in terms of fuel oil consumption, as well as the hydrodynamic performance of the respective vessels in terms of the shaft propulsion power and the rotational speed of the fixed-pitch propellers driven by these engines. Based on the least-squares criterion, this original and sophisticated hybrid combination systematically attains remarkably close approximate representations under seagoing conditions. Apart from this novel exact definition in mathematical terms and the significance of the above original representations, this combination is also applicable for the approximation of the baselines demarcating the standard engineering context representing the ideal reference (sea trials) conditions, from the seagoing conditions. Full article

Robust, reliable, and secure communications are essential for efficient railway operation and keeping employees and passengers safe. The Future Railway Mobile Communication System (FRMCS) is a global standard aimed at providing innovative, essential, and high-performance communication applications in railway transport. In comparison with the legacy communication system (GSM-R), it provides high data rates, ultra-high reliability, and low latency. The FRMCS architecture will also benefit from cloud computing, following the principles of the cloud-native 5G core network design based on Network Function Virtualization (NFV). In this paper, an approach to the management of virtualized FRMCS applications is presented. First, the key management functionality related to the virtualized FRMCS application is identified based on an analysis of the different use cases. Next, this functionality is synthesized as RESTful services. The communication between application management and the services is designed as Application Programing Interfaces (APIs). The APIs are formally verified by modeling the management states of an FRMCS application instance from different points of view, and it is mathematically proved that the management state models are synchronized in time. The latency introduced by the designed APIs, as a key performance indicator, is evaluated through emulation. Full article