Abstract In the recent literature, moderate naturalistic metaphysicians have been attempting to justify the existence of ‘free range’ analytic metaphysics by employing an analogy with pure mathematics: just as pure mathematics is justified by its potential applications to science, so too, they argue, is analytic metaphysics justified by its potential applications in philosophy of science. Employing standard textbook logical tools to evaluate analogies, we argue that the analogy doesn’t hold: there are relevant dissimilarities between the two disciplines. The grounds and domain of application of metaphysics and mathematics to science are ultimately different, and arguments intended to justify the epistemic value of metaphysics for science often presuppose its value rather than demonstrate it. This is why metaphysics is not like mathematics.

This study examines the Apostol Euler-Padovan polynomials and Apostol Bernoulli-Padovan numbers, explaining their definitions and exploring their respective properties. It introduces harmonic-based P-exponential generating functions tailored for both sequences, enhancing their study. Additionally, a section is dedicated to the concept of harmonic-based Apostol Euler-Padovan and Apostol Bernoulli-Padovan numbers, integrating harmonic numbers with P-exponential generating functions for these sequences. This integration provides valuable insights into the relationship between harmonic series and these families of numbers, enriches our understanding of their mathematical properties, and contributes to the understanding of special functions and their mathematical applications.

Artificial Intelligence (AI) has become integral to education, transforming the way students approach mathematics. This qualitative study examines the perceptions, experiences, and usage patterns of BS Applied Mathematics students in using AI tools such as ChatGPT, Gemini, Meta AI, and Cici for solving mathematical problems within the context of Technology-Enhanced Learning (TEL). Grounded in the Technology Acceptance Model (TAM), it explores five dimensions: usefulness, ease of use, attitude toward use, behavioral intention, and actual use. Data was analyzed using both phenomenological and thematic approaches, incorporating interviews, focus group discussions, and observations. Students primarily used AI tools for speed, convenience, and quick solutions under time pressure. While most preferred AI as a complement to traditional learning, they expressed concerns about overreliance, limited understanding, and reduced motivation. Nonetheless, students demonstrated digital literacy by validating AI outputs and using multimodal resources, such as YouTube tutorials, to enhance comprehension. The study advocates for balanced AI integration in education, emphasizing digital literacy, responsible use, and the development of improved AI-based platforms to strengthen multimodal and explainable learning within TEL frameworks.

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Editorial: 10th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2024)

The present study aimed to assess the contribution of Dr. C. Manuel Eduardo Cortés Cortés to the teaching of Applied Mathematics in Cuban Higher Education, considering his influence on professional training and the scientific-methodological development in the province of Cienfuegos. The research was conducted from a qualitative approach, grounded in the dialectical-materialist method and employing the systematization of experiences as the main methodological strategy. This approach allowed for a critical reconstruction and analysis of Dr. Cortés’s professional trajectory between 1971 and 2000. Data collection was carried out through documentary analysis of publications and teaching materials, semi-structured interviews with colleagues and students, review of institutional archives, and indirect observation, prioritizing the authenticity and historical relevance of the sources. The findings indicate that Dr. Cortés’s educational work was characterized by a continuous integration of theory, practice, and research, facilitating the alignment of teaching with the productive and social needs of the region. His contributions in mathematical modeling, operations research, and applied computing enabled the resolution of real industrial problems and the incorporation of technological experiences into the university curriculum. The systematization revealed that his career not only produced formal knowledge but also constitutes a methodological and pedagogical reference for Higher Education, promoting critical reflection on teaching practice and the development of contextualized educational proposals. In this sense, the study underscores the importance of valuing significant teaching experiences as a source of innovation and continuous improvement.

The competence-based transformation of primary mathematics education requires instructional models that connect conceptual understanding, procedural fluency, reasoning, and communication within meaningful learning situations. Geometry provides a particularly powerful pathway for competence formation because it naturally integrates visualisation, measurement, spatial reasoning, and argumentation, while also supporting the development of mathematical language. This article proposes and justifies a didactic model for forming mathematical competence in primary education through fundamental geometric concepts. The model is grounded in documentary analysis of international frameworks that emphasise mathematical reasoning and application, including the cognitive domain logic of TIMSS (knowing, applying, reasoning) and the mathematical literacy orientation of PISA, alongside research on developmental progressions in geometric thinking (notably the van Hiele theory). The model is described as a coherent system linking learning outcomes, content selection, instructional design, formative assessment, and feedback loops. It specifies how core geometric ideas—point, line, segment, ray, angle, plane figures, symmetry, perimeter, area, and spatial relationships—can function as “conceptual organisers” for broader mathematical competence. The proposed design foregrounds representational transitions (from concrete manipulation to schematic drawings and symbolic notation), discourse routines, and criterion-referenced assessment. The article argues that the model is feasible for teacher education and school practice because it offers operational design principles for lesson planning, task construction, and assessment. It also defines evaluation indicators for monitoring competence growth, including shifts in students’ reasoning quality, use of geometric language, and ability to model real situations with geometric representations.

The foundation of Artificial Intelligence (AI) and Machine Learning is Mathematics. Core mathematical disciplines such as linear algebra, calculus, probability, statistics, and optimization provide the structural framework that enables machines to learn from data and make intelligent predictions. Linear algebra plays a vital role in representing and manipulating large datasets, powering neural networks, and handling multidimensional computations. Calculus enables optimization processes by calculating gradients and updating parameters to minimize loss functions in deep learning models. Probability and statistics serve as the backbone for modelling uncertainty, developing predictive algorithms, and assessing data-driven inferences. Machine Learning (ML) and Artificial Intelligence are changing the world in which we live. AI and ML are most popular topics in Technology filed. But behind all Machine learning and AI algorithms are good Mathematics foundation. Behind the remarkable advancements and capabilities of AI lies the foundational role of mathematics. Mathematics provides the framework that enables AI systems to learn, reason, and make intelligent decisions. In this paper, we explore the applications of Mathematics in AI and Machine Learning and sub fields of AI.

The historiographical treatment of Albania often depicts isolation. However, its computer network history reveals the flows of information, commodities and individuals between Albania and the world. It reflects Albania’s fluctuating international relations with the USSR and China, the United Nations and the role of Soros Foundation in the 1990s. This study focuses on these last two little-known periods. Between 1981 and 1989, Albania’s Institute of Informatics and Applied Mathematics (INIMA) benefitted from the establishment of a “computer network for the development of scientific and technical activities” under the aegis of UNESCO, UNDP and implemented by the French company Bull. Based on direct testimonies from Bull project managers and Albanians trained in France, this study explores the expectations of the Albanian regime, the relationship developed between Albanian and foreign technicians, the effects of the training of Albanians abroad, and the impact of applied informatics in Albania. We also draw on UNESCO reports, newspapers, public speeches and archives available in Albania. While this project consolidated INIMA’s capacities during the 1980s, its survival was at stake in the 1990s. We then look in particular at the role of Soros Foundation, a unique case of an international NGO providing Internet access between 1996 and 2000. We analyse testimonies via the AlbNet mailing list archives. This enables us to see the socio-technical choices made by UNDP, the marginalisation of INIMA and the importance of Soros Foundation.

Objective: To analyze the effect of mathematical games application as a didactic strategy to strengthen motivation and learning in elementary level students at María Teresa Dávila de Rosania Basic Education School in Quito, Ecuador. Method: A quasi-experimental design with pretest-posttest in a single group was employed using a mixed-methods approach. Forty-five elementary level students participated, selected through non-probabilistic convenience sampling. A Likert-type scale measuring mathematics motivation and an academic achievement test were administered before and after an eight-week intervention based on mathematical games. Semi-structured teacher interviews complemented the data collection. Quantitative analysis included paired samples t-test with effect size calculation (Cohen's d), while qualitative data were processed through thematic content analysis. Results: Statistically significant differences were found between pretest and posttest in both motivation (t = 5.82, p < .001, d = 0.87) and academic achievement (t = 6.14, p < .001, d = 0.92). Teachers reported substantial improvements in student participation, interest, and attitude toward mathematics. Conclusions: Mathematical games constitute an effective didactic strategy for increasing intrinsic motivation and improving mathematics learning at the elementary level, with practical implications for pedagogical transformation in Latin American educational contexts.

This article proposes the conventional definition and real life based applications of linear Diophantine type-2 fuzzy sets (LDT2FS), manifesting its supremacy over common fuzzy models. LDT2FS is a new mathematical framework that can address the drawbacks of existing fuzzy sets. As common fuzzy systems, however, these models hold restrictions on acceptance and rejection grades, limiting the flexibility of decision-makers in managing uncertainty. LDT2FS extends the previous study by relaxing these limitations, as decision-makers can specify grades-freely with reference parameters in practice in cases where decision making under uncertainty is necessary when functional relationships are unknown, or when data contain high levels of imprecision. To this end, the less demanding structure of LDT2FS allows for establishing the fit to allow dealing with these uncertainties in an efficient way. It also describes basic arithmetic operations on LDT2FS such as union, intersection, complement, containment, etc. along with their algebraic characteristics. Furthermore, two new operators, the Certainty Operator, and Feasibility Operator, are proposed to convert an LDT2FS into a conventional fuzzy set, simplifying mathematical processing and applications. This article proposes Haming Distance and Euclidean Distance commonly used in pattern recognition, clustering, and classification problems to quantify the differences or similarity between LDT2FS instances.

We study optimal stopping for diffusion processes with unknown model primitives within the continuous-time reinforcement learning (RL) framework Wang, Zariphopoulou and Zhou [Wang H, Zariphopoulou T, Zhou XY (2020), J. Machine Learn. Res. 21(198):1-34], and present applications to option pricing and portfolio choice. By penalizing the corresponding variational inequality formulation, we transform the stopping problem into a stochastic optimal control problem with two actions. We then randomize controls into Bernoulli distributions and add an entropy regularizer to encourage exploration. We derive a semianalytical optimal Bernoulli distribution, based on which we devise RL algorithms using the martingale approach Jia and Zhou [Jia Y, Zhou XY (2022a) J. Machine Learn. Res. 23(154):1–55]. We establish a policy improvement theorem and prove the fast convergence of the resulting policy iterations. We demonstrate the effectiveness of the algorithms in pricing finite-horizon American put options, solving Merton’s problem with transaction costs, and scaling to high-dimensional optimal stopping problems. In particular, we show that both the offline and online algorithms achieve high accuracy in learning the value functions and characterizing the associated free boundaries. This paper has been accepted by Kay Giesecke for the Special Issue on AI for Finance and Business Decisions. Funding: M. Dai was supported by the Hong Kong Research Grants Council [15212324, 15217123, 15213422, and T32-615/24-R], the Hong Kong Polytechnic University [P0042708, P0042456, and P0039114], and the Key Project of National Natural Science Foundation of China [72432005]. Y. Sun acknowledges financial support from a start-up fund at Peking University HSBC Business School. Z. Xu acknowledges financial support from The Hong Kong RGC [GRF 15204622, 15203423], NSFC 12571517, the PolyU-SDU Joint Research Center on Financial Mathematics, the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, the Research Centre for Quantitative Finance (1-CE03), and internal grants from The Hong Kong Polytechnic University. X. Zhou was supported by the Nie Center for Intelligent Asset Management at Columbia University. This work was also part of a Columbia-CityU/HK collaborative project that was supported by the InnoHK Initiative, the Government of the HKSAR, and the AIFT Lab. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2024.07614 .

Mathematics was used to build and organize the world during the Mughal Empire, not just as a topic for academics to debate. The Mughals were masters of "applied mathematics" despite not inventing any famous new formulas. They combined Islamic concepts, such as algebra and advanced geometry, with the best ideas from ancient India, such as zero and decimals. They were able to manage such a large empire thanks to this combination of knowledge. For example, they used geometry to ensure that each dome and tower was perfectly balanced and symmetrical when they built the Taj Mahal. Mathematics was also used in the "business side" of the empire. They devised a method to precisely determine how much grain or money a farmer should pay in taxes by measuring millions of acres of farmland with specific units and long ropes. They made sure that these skills were passed down from one generation to the next by opening schools and translating old science books into Persian. In a nutshell, the Mughals were the greatest "integrators" of mathematics, making it a useful tool for science, architecture, and money.

In recent years, many researchers have explored the use of fractional partial differential equations (FPDEs) to model real-world phenomena through transport of porous media in engineering and applied sciences. Investigating nonlinear time-fractional Telegraph equation with quadratic nonlinearity, we transform the partial differential equation (PDE) into ordinary differential equation (ODE) by using the stability analysis. The obtained solutions con- tain different ways of exponential functions. Our method uses stability analysis to solve the complex equations easily and their applications in physics, mathematics and engineering. This indicates how system behaves near an equilibrium point and underlying the models are stable and unstable using the concept of stability. Signifficant outcomes are obtained from the various solution families. To examine the impact of several gained parameters, the propagation behavior of the obtained data is shown in 2D, 3D, and contour representations. The signifficance of physical situations will be better understood by the researchers to these findings. The nonlinear time-fractional Telegraph equations play a central role in many areas of electrical engineering for transport in porous media. This paper reports that our research contributes to the existing litera- ture, offering a fresh perspective on these equations and concrete the way for future research for transport in porous media. Our approach provides stability analysis with a high accuracy for two-dimensional plane autonomous systems.

Purpose This study aims to investigate how a single, well-designed instructional video can influence students’ perceptions, learning outcomes and response heterogeneity in an Applied Mathematics course, focusing on the topic “domain of a function.” It aims to quantify short-term learning gains and identify learner profiles based on attitudes, prior difficulties and study habits. Design/methodology/approach A pre/post experimental design was implemented with 81 first-year Business students enrolled in a Applied Mathematics course. Students completed a pretest, watched an instructional video and then solved three posttasks of increasing complexity. A stepwise scoring rubric measured learning gains, and Latent Class Analysis (LCA) was used to identify learner subgroups. Wilcoxon signed-rank and nonparametric tests examined overall and subgroup differences. Findings Results show statistically significant improvements across all posttasks (p < 0.001), including among students who reported frequent difficulties with Mathematics. Nevertheless, higher-complexity items continued to reveal persistent gaps in solving inequalities and applying set operations. Students highly valued the video’s clarity, audiovisual quality and usefulness but consistently indicated that it cannot replace face-to-face instruction. Research limitations/implications Results derive from one institution, one topic and a short-term intervention without delayed posttest; LCA indicators were self-reported. External validity is limited. Future work should include multiple topics/institutions, link to clickstream measures and assess retention/transfer with follow-up testing (Bergdahl et al., 2024). Practical implications Actionable guidance for Technology-Enhanced Learning (TEL): concise, segmented videos with signaling and worked examples; high audiovisual quality; prompt-and-pause moments; and immediate practice with feedback. Pair videos with targeted activities addressing inequalities and set operations, and deploy in flipped/blended models to support both recent and long-gap learners. Social implications Scalable microvideos can expand flexible access for diverse higher education (HE) cohorts, including working students and those resuming Mathematics after long gaps, helping narrow opportunity gaps. Equitable deployment requires accessibility features (e.g. captions) and preservation of teacher–student interaction to sustain motivation and inclusion. Originality/value While video-based learning in HE and Mathematics Education has been widely investigated, most empirical studies focus on full flipped-course implementations, video series or multiresource learning environments. This study adopts a fine-grained, microanalytic perspective by isolating the learning impact of a single, carefully designed instructional video on a specific mathematical topic (“Domain of a Function”). Methodologically, it combines a transparent stepwise scoring rubric with pre/post assessment and LCA to capture learner heterogeneity beyond average performance effects. This allows identification of distinct learner profiles (“Casual Learners” and “Struggling Video Users”) and examination of how short-term procedural gains vary across profiles. By demonstrating that a single procedural video can generate statistically significant short-term learning gains across heterogeneous student profiles while simultaneously revealing persistent conceptual difficulties, this study contributes empirical evidence that refines current understandings of the instructional scope and pedagogical limits of microvideo use in Higher Education Mathematics. It further distills evidence-based design principles for integrating short instructional videos into blended and flipped learning environments.

This Special Issue comprises the fifth collection of papers submitted by both the Editorial Board Members (EBMs) of the journal Mathematical and Computational Applications (MCA) and the outstanding scholars working in the core research fields of MCA [...]

Abstract In this study the finite element implementation of the hyperelastic-plastic constitutive relationship with isotropic hardening as proposed by Simo and Hughes (Computational Inelasticity, Interdisciplinary Applied Mathematics Vol. 7, Springer-Verlag), Simo (Comput Meth in Appl Mech and Eng 66:199–219. https://doi.org/10.1016/0045-7825(88)90076-X , 1988), Simo (Comput Meth Appl Mech Eng 68:1–31. https://doi.org/10.1016/0045-7825(88)90104-1 , 1988) is discussed. The major novel contributions of this work focus around the development of a general purpose UMAT code. The code enables to implement a family of hyperelastic-plastic constitutive relationships into the non-commercial finite element (FE) program CalculiX. The built-in implementation of hyperelasto-plasticity in this program permits only to use piecewise-linear isotropic hardening curves and a single selected formulation of volumetric stored-energy function. The present implementation of the model via subroutine UMAT allows the usage of nonlinear hardening curves and various formulations of the volumetric elastic energy. Thus, a whole family of hyperelastic-plastic models can be implemented into CalculiX using the developed UMAT code. What is more, several questions regarding the derivation of the flow rule, the radial return mapping algorithm and the consistent tangent operator have been more extensively discussed than previously in the literature. The presented relations will be useful for implementing the hyperelastic-plastic constitutive model into other FE packages. Numerous benchmark test problems have been solved and compared with both analytical results and experimental measurements which validated the performance of the developed code. The UMAT subroutine is attached to this work as supplementary material.

General Background The integration of digital learning media has become increasingly important in elementary mathematics instruction to create engaging and student-centered learning environments. Specific Background Many classrooms still rely on conventional methods that limit students’ participation and contribute to low learning outcomes in geometry topics. Knowledge Gap Although interactive platforms such as Wordwall are widely available, empirical evidence regarding their classroom application in primary mathematics remains limited. Aims This study aimed to examine the use of the Wordwall application in supporting students’ mathematics learning outcomes on plane geometry materials. Results Classroom implementation showed higher post-learning scores, improved participation, and more active student responses compared with previous conventional instruction. Novelty The study provides contextual evidence of integrating a game-based digital platform directly into daily classroom practice at the elementary level. Implications The findings suggest that structured use of interactive digital media can support more engaging mathematics lessons and contribute to better learner achievement and classroom interaction. Keywords : Wordwall, Elementary Mathematics, Digital Learning Media, Learning Outcomes, Interactive Instruction Key Findings Highlights: Post-test scores exceeded initial performance levels Learners showed stronger classroom participation Game-based tasks supported concept understanding

The paper is devoted to the origin and development in the period from 1930 to 1980 of the scientific school of mechanics and applied mathematics of NTU «KhPI». This school made a significant contribution both to the development of deformable solid mechanics and to its practical application to the most advanced examples of technology, such as steam and gas turbines, tank and locomotive power plants, etc. It is proven that KhPI mechanical scientists have always sought to solve the most important, prestigious tasks dictated by new directions in the development of technology. KhPI mechanical scientists have always sought to solve the most important, prestigious tasks dictated by new directions in the development of technology. Graduates of the department create unique control systems not only for aerospace facilities, but also for nuclear power, turbomachinery, and other science-intensive industries. In the future, NTU "KhPI" will develop in accordance with the requirements of the labor market and applied sciences.

Dynamic applications in mathematics, supports interactive textbooks recently. These products enhance doing and making inquiries in mathematics for students, and they include functional dependency. 12th grade dynamic applications from OGM web page of MoNE, were classified on van Hiele theory and Bloom’ s revised taxonomy. These applications were analyzed for functional dependency afterwards. The ones which consist of functional dependency were analyzed based on number of variables, topology, if they include compulsory choices, and slider and dragging, via draw functions. Data analysis was done by three experts for selected applications and then w.r.t. the agreed characteristics, one expert finished the analysis after full expert agreement on decisions. Regarding all these variables, high school mathematics curriculum dynamic applications are discussed in depth for functional dependency. Qualitative research study (case analysis) was carried out by content analysis. Content analysis is supported by descriptive statistics. As a result, how these mathematical dynamic applications should be established regarding their functional dependency measures were discussed. All dynamic applications were analyzed for van Hiele and Bloom taxonomy and then selected applications (with high levels from both) were analyzed for functional dependency and related variables. It is hoped that the results could shed light to dynamic mathematics applications producers to develop interactive diagrams with functional dependency based on the instructional theories and mathematics teachers in turn to use them appropriately.