Search Results (833)

For any point P in the Euclidean plane of a triangle $\Delta$, the six parallelians of P lie on a single conic, which shall be called the parallelian conic of P with respect to $\Delta$. We provide a synthetic and an analytic proof of this fact. Then, we studied the shape of this particular conic, depending on the choice of the pivot point P. This led to the finding that the only circular parallelian conic is the first Lemoine circle. Points on the Steiner inellipse produce parabolae, and those on a certain central line yield equilateral hyperbolae. The hexagon built by the parallelians has an inconic $\mathcal{I}$ and the tangents of $\mathcal{P}$ at the parallelians define some triangles and hexagons with several circum- and inconics. Certain pairings of conics, together with in- and circumscribed polygons, give rise to different kinds of porisms. Further, the inconics and circumconics of the triangles and hexagons span exponential pencils of conics in which any pair of subsequent conics defines a new conic as the polar image of the inconic with regard to the circumconic. This allows us to construct chains of nested porisms. The trilinear representations of the centers of the appearing conics, as well as the perspectors of some deduced triangles, depending on the indeterminate coordinates of P, define some algebraic transformations that establish algebraic relations between well- and lesser-known triangle centers. We completed our studies by compiling a list of possible porisms between any pair of conics. Further, we describe the possible loci of pivot points so that the mentioned conics allow for porisms of polygons with arbitrary numbers of vertices. Full article

This study employs a finite element approach to investigate wellbore stability in interbedded weak formations, such as unconsolidated layers, with a focus on the failure-tendency method, which is derived according to the principle of Mohr–Coulomb theory. The numerical model is successfully verified through analytical solutions for stress distributions around a borehole. Through finite element modeling, the method captures critical shear failure thresholds, exemplifying how variations in horizontal stress anisotropy, orientation of interbedded weak layers, and mechanical properties of layered geological formations impact wellbore stability in stratified formations. Results indicate that the potential unstable regions, aligned in the direction of minimum principal stress, and the range of unstable regions gradually enlarge as the internal cohesive strength decreases. By modeling heterogeneous rock sequences with explicit representation of interbedded weak layers and stress anisotropy, the analysis reveals that interbedded weak layers are prone to shear-driven borehole breakouts due to stress redistribution and relatively lower internal cohesive strength. As compressive stresses concentrate at interfaces between stiff and compliant layers, breakouts are induced at those weak layers along the interfaces; this type of failure is also manifested through a field borehole breakout observation. Simulation results reveal the significant influences of the mechanical properties of layered formations and in situ stress on the distribution of instability regions around a borehole. The study underscores the necessity of layer-specific geomechanical models to predict shear failure in complex layered geological formations and offers insights for optimizing drilling parameters to enhance wellbore stability in anisotropic, stratified subsurface environments. Full article

This paper develops an information-theoretic reliability inference framework for consecutive $r$-out-of-$n$:G systems by employing the concept of residual extropy, a dual measure to entropy. Explicit analytical representations are established in tractable cases, while novel bounds are derived for more complex lifetime models, providing effective tools when closed-form expressions are unavailable. Preservation properties under classical stochastic orders and aging notions are examined, together with monotonicity and characterization results that offer deeper insights into system uncertainty. A conditional formulation, in which all components are assumed operational at a given time, is also investigated, yielding new theoretical findings. From an inferential perspective, we propose a maximum likelihood estimator of residual extropy under exponential lifetimes, supported by simulation studies and real-world reliability data. These contributions highlight residual extropy as a powerful information-theoretic tool for modeling, estimation, and decision-making in multicomponent reliability systems, thereby aligning with the objectives of statistical inference through entropy-like measures. Full article

The primary aim of this study is to establish new Wirtinger-type inequalities involving fractional derivatives, which are essential tools in analysis and applied mathematics. We derive generalized Wirtinger-type inequalities incorporating the $(k,\psi )$-Caputo fractional derivatives using Taylor’s expansion. The inequalities are derived in $L_p$ spaces $(p>1)$ through Hölder’s inequality. A detailed analytical discussion is provided to further examine the derived inequalities. The theoretical findings are validated through numerical examples and graphical representations. Furthermore, the novelty and applicability of the proposed technique are demonstrated through the applications of the resulting inequalities to derive new results related to the arithmetic–geometric mean inequality. Full article

This paper proposes a method for assessing societal risk along a traffic link by integrating a stochastic formulation of the fundamental diagram. The approach accounts for uncertainty in vehicle speed due to user heterogeneity, vehicle characteristics, and environmental conditions. The risk index is decomposed into occurrence, vulnerability, and exposure components, with the occurrence probability modeled as a function of stochastic speed. The inverse gamma distribution is adopted to represent speed variability, enabling analytical tractability and control over dispersion. Numerical results show that urban and suburban environments exhibit distinct sensitivity to model parameters, particularly the gamma shape parameter η and the composite parameter c = β · v₀ obtained by the product of the occurrence parameter β and the free speed flow v₀. Graphical representations illustrate the impact of uncertainty on risk estimation. The proposed framework enhances existing deterministic methods by incorporating probabilistic elements, offering a foundation for future applications in traffic safety management and infrastructure design. Full article

Building Energy Modelling (BEM) practitioners play a crucial role in delivering energy-efficient buildings by analysing building performance using simulation tools. However, their experiences while using BEM software to predict building energy performance are understudied. In addition, research that directly engages with practitioners and stakeholders is particularly lacking in the Global South (GS), where the bulk of new building construction takes place. This study explores the implementation challenges and opportunities associated with BEM software among South African industry practitioners, focusing on their experiences in utilising BEM tools. Structured interviews were conducted with 19 South African industry specialists, supplemented by quantitative data collected through a questionnaire. Qualitative data from the interviews were analysed using MAXQDA 24 Analytics Pro to identify key themes, while quantitative data were visualised to compare software preferences. The analysis indicated that DesignBuilder is widely used, followed by BSIMAC. These tools highlight the largest opportunities for supporting active South African practitioners. The respondents highlighted the need for user-friendly interfaces, standardised methodologies, and improved training to address entry barriers and inconsistent simulation outcomes. Mixed opinions exist regarding the preference for tools with visual representations of 3D geometry, primarily influenced by the field of specialisation and how it impacts client engagement. The research concludes that while BEM software is critical for advancing sustainable design, its effective implementation is hindered in South Africa and potentially in the GS. Recommendations include developing more intuitive software interfaces, establishing standardised modelling approaches, and creating structured training programmes and professional forums to enhance practitioner proficiency, knowledge transfer across contexts, and industry-wide adoption. Full article

The primary objective of this study is to provide a more precise and beneficial mathematical model for assessing corruption dynamics by utilizing non-local derivatives. This research aims to provide solutions that accurately capture the complexities and practical behaviors of corruption. To illustrate how corruption levels within a community change over time, a non-linear deterministic mathematical model has been developed. The authors present a non-integer order model that divides the population into five subgroups: susceptible, exposed, corrupted, recovered, and honest individuals. To study these corruption dynamics, we employ a new method for solving a time-fractional corruption model, which we term the q-homotopy analysis transform approach. This approach produces an effective approximation solution for the investigated equations, and data is shown as 3D plots and graphs, which give a clear physical representation. The stability and existence of the equilibrium points in the considered model are mathematically proven, and we examine the stability of the model and the equilibrium points, clarifying the conditions required for a stable solution. The resulting solutions, given in series form, show rapid convergence and accurately describe the model’s behaviour with minimal error. Furthermore, the solution’s uniqueness and convergence have been demonstrated using fixed-point theory. The proposed technique is better than a numerical approach, as it does not require much computational work, with minimal time consumed, and it removes the requirement for linearization, perturbations, and discretization. In comparison to previous approaches, the proposed technique is a competent tool for examining an analytical outcomes from the projected model, and the methodology used herein for the considered model is proved to be both efficient and reliable, indicating substantial progress in the field. Full article

Artificial intelligence and machine learning have increasingly transformed optometry, enabling automated classification and predictive modeling of eye conditions. In this study, we introduce Optometry Random Forest, an artificial intelligence-based system for automated classification and forecasting of optometric data. The proposed methodology leverages Random Forest models, trained on academic optometric datasets, to classify key diagnostic categories, including Contactology, Dry Eye, Low Vision, Myopia, Pediatrics, and Refractive Surgery. Additionally, an autoRegressive integrated moving average based forecasting model is incorporated to predict future research trends in optometry until 2030. Comparing the one-shot and epoch-trained Optometry Random Forest, the findings indicate that the epoch-trained model consistently outperforms the one-shot model, achieving superior classification accuracy (97.17%), precision (97.28%), and specificity (100%). Moreover, the comparative analysis with Optometry Bidirectional Encoder Representations from Transformers demonstrates that the Optometry Random Forest excels in classification reliability and predictive analytics, positioning it as a robust artificial intelligence tool for clinical decision-making and resource allocation. This research highlights the potential of Random Forest models in medical artificial intelligence, offering a scalable and interpretable solution for automated diagnosis, predictive analytics, and artificial intelligence-enhanced decision support in optometry. Future work should focus on integrating real-world clinical datasets to further refine classification performance and enhance the potential for artificial intelligence-driven patient care. Full article

Physics-Informed Neural Networks (PINNs) represent a transformative approach to solving partial differential equation (PDE)-based boundary value problems by embedding physical laws into the learning process, addressing challenges such as non-physical solutions and data scarcity, which are inherent in traditional neural networks. This review analyzes critical challenges in PINN development, focusing on loss function design, geometric information integration, and their application in engineering modeling. We explore advanced strategies for constructing loss functions—including adaptive weighting, energy-based, and variational formulations—that enhance optimization stability and ensure physical consistency across multiscale and multiphysics problems. We emphasize geometry-aware learning through analytical representations—signed distance functions (SDFs), phi-functions, and R-functions—with complementary strengths: SDFs enable precise local boundary enforcement, whereas phi/R capture global multi-body constraints in irregular domains; in practice, hybrid use is effective for engineering problems. We also examine adaptive collocation sampling, domain decomposition, and hard-constraint mechanisms for boundary conditions to improve convergence and accuracy and discuss integration with commercial CAE via hybrid schemes that couple PINNs with classical solvers (e.g., FEM) to boost efficiency and reliability. Finally, we consider emerging paradigms—Physics-Informed Kolmogorov–Arnold Networks (PIKANs) and operator-learning frameworks (DeepONet, Fourier Neural Operator)—and outline open directions in standardized benchmarks, computational scalability, and multiphysics/multi-fidelity modeling for digital twins and design optimization. Full article

Extracting key features for phenotype classification from high-dimensional and complex mass spectrometry (MS) data presents a significant challenge. Conventional data representation methods, such as traditional peak lists or grid-based imaging strategies, are often hampered by information loss and compromised signal integrity, thereby limiting the performance of downstream deep learning models. To address this issue, we propose a novel data representation framework named MSIMG. Inspired by object detection in computer vision, MSIMG introduces a data-driven, “density-peak-centric” patch selection strategy. This strategy employs density map estimation and non-maximum suppression algorithms to locate the centers of signal-dense regions, which serve as anchors for dynamic, content-aware patch extraction. This process transforms raw mass spectrometry data into a multi-channel image representation with higher information fidelity. Extensive experiments conducted on two public clinical mass spectrometry datasets demonstrate that MSIMG significantly outperforms both the traditional peak list method and the grid-based MetImage approach. This study confirms that the MSIMG framework, through its content-aware patch selection, provides a more information-dense and discriminative data representation paradigm for deep learning models. Our findings highlight the decisive impact of data representation on model performance and successfully demonstrate the immense potential of applying computer vision strategies to analytical chemistry data, paving the way for the development of more robust and precise clinical diagnostic models. Full article

We derive an analytical solution to the one-dimensional linearized Boussinesq equation with mixed boundary conditions (Dirichlet–Neumann), formulated to describe drainage in porous media. The solution is obtained via the unified transform method (Fokas method), extending its previous applications in infiltration problems and illustrating its utility in soil hydrology. An explicit integral representation is constructed, considering different types of initial conditions. Numerical examples are presented to demonstrate the accuracy of the solution, with direct comparisons to the classical Fourier series approach. Full article

The increasing scale and complexity of data mining outputs, such as frequent itemsets, association rules, sequences, and subgraphs have made efficient pattern retrieval a critical, yet underexplored challenge. This review addresses the organisation, indexing, and access strategies, which enable scalable and responsive retrieval of structured patterns. We examine the underlying types of data and pattern outputs, common retrieval operations, and the variety of query types encountered in practice. Key indexing structures are surveyed, including prefix trees, inverted indices, hash-based approaches, and bitmap-based methods, each suited to different pattern representations and workloads. Storage designs are discussed with attention to metadata annotation, format choices, and redundancy mitigation. Query optimisation strategies are reviewed, emphasising index-aware traversal, caching, and ranking mechanisms. This paper also explores scalability through parallel, distributed, and streaming architectures, and surveys current systems and tools, which integrate mining and retrieval capabilities. Finally, we outline pressing challenges and emerging directions, such as supporting real-time and uncertainty-aware retrieval, and enabling semantic, cross-domain pattern access. Additional frontiers include privacy-preserving indexing and secure query execution, along with integration of repositories into machine learning pipelines for hybrid symbolic–statistical workflows. We further highlight the need for dynamic repositories, probabilistic semantics, and community benchmarks to ensure that progress is measurable and reproducible across domains. This review provides a comprehensive foundation for designing next-generation pattern retrieval systems, which are scalable, flexible, and tightly integrated into analytic workflows. The analysis and roadmap offered are relevant across application areas including finance, healthcare, cybersecurity, and retail, where robust and interpretable retrieval is essential. Full article

Tourist reviews provide essential insights into travellers experiences and public perceptions of destinations. In Kazakhstan, however, sentiment analysis, particularly using ensemble learning, remains underexplored for evaluating such reviews. This study proposes a hybrid stacking ensemble for sentiment analysis of English-language tourist reviews about Kazakhstan, integrating four complementary approaches: VADER, TextBlob, Stanza, and Local Context Focus Mechanism with Bidirectional Encoder Representations from Transformers (LCF-BERT). Each model contributes distinct analytical capabilities, including lexicon-based polarity detection, rule-based subjectivity evaluation, generalised star-rating estimation, and contextual aspect-oriented sentiment classification. The evaluation utilised a cleaned dataset of 11,454 TripAdvisor reviews collected between February 2022 and June 2025. The ensemble aggregates model outputs through majority and weighted voting strategies to enhance robustness. Experimental results (accuracy 0.891, precision 0.838, recall 0.891, and F1-score 0.852) demonstrate that the proposed method KazSATR outperforms individual models in overall classification accuracy and exhibits superior capacity for aspect-level sentiment detection. These findings underscore the potential of the hybrid ensemble as a practical and scalable tool for the tourism sector in Kazakhstan. By leveraging multiple analytical paradigms, the model enables tourism professionals and policymakers to better understand traveller preferences, identify service strengths and weaknesses, and inform strategic decision-making. The proposed approach contributes to advancing sentiment analysis applications in tourism research, particularly in underrepresented geographic contexts. Full article

The rapid expansion of AI applications in various domains demands models that balance predictive power with human interpretability, a requirement that has catalyzed the development of hybrid algorithms combining high accuracy with human-readable outputs. This study introduces a novel neuro-symbolic framework, OIKAN (Optimized Interpretable Kolmogorov–Arnold Network), designed to integrate the representational power of feedforward neural networks with the transparency of symbolic regression. The framework employs Gaussian noise-based data augmentation and a two-phase sparse symbolic regression pipeline using ElasticNet, producing analytical expressions suitable for both classification and regression problems. Evaluated on 60 classification and 58 regression datasets from the Penn Machine Learning Benchmarks (PMLB), OIKAN Classifier achieved a median accuracy of 0.886, with perfect performance on linearly separable datasets, while OIKAN Regressor reached a median R² score of 0.705, peaking at 0.992. In comparative experiments with ElasticNet, DecisionTree, and XGBoost baselines, OIKAN showed competitive accuracy while maintaining substantially higher interpretability, highlighting its distinct contribution to the field of explainable AI. OIKAN demonstrated computational efficiency, with fast training and low inference time and memory usage, highlighting its suitability for real-time and embedded applications. However, the results revealed that performance declined more noticeably on high-dimensional or noisy datasets, particularly those lacking compact symbolic structures, emphasizing the need for adaptive regularization, expanded function libraries, and refined augmentation strategies to enhance robustness and scalability. These results underscore OIKAN’s ability to deliver transparent, mathematically tractable models without sacrificing performance, paving the way for explainable AI in scientific discovery and industrial engineering. Full article

This work establishes definitive conditions for the inheritance of categorical completeness and cocompleteness by categories of internal group objects. We prove that while the completeness of $\mathbf{Grp}\left(\mathcal{C}\right)$ follows unconditionally from the completeness of the base category $\mathcal{C}$, cocompleteness requires $\mathcal{C}$ to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit limit and colimit constructions are provided, with colimits realized via coequalizers of relations induced by group axioms over free group objects. Applications demonstrate cocompleteness in topological groups, ordered groups, and group sheaves, while Lie groups serve as counterexamples revealing necessary analytic constraints—particularly the impossibility of equipping free groups on non-discrete manifolds with smooth structures. Further results include the inheritance of regularity when the free group functor preserves finite products, the existence of internal hom-objects in locally Cartesian closed settings, monadicity for locally presentable $\mathcal{C}$, and homotopical extensions where model structures on $\mathbf{Grp}\left(\mathcal{M}\right)$ reflect those of $\mathcal{M}$. This framework unifies classical category theory with geometric obstruction theory, resolving fundamental questions on exactness transfer and enabling new constructions in homotopical algebra and internal representation theory. Full article