Search Results (592)

Formal verification has achieved remarkable outcomes in both theory advancement and engineering practice, with the formalization of mathematical theories serving as its foundational cornerstone—making this process particularly critical. Axiomatic set theory underpins modern mathematics, providing the rigorous basis for constructing almost all theories. Landau’s Foundations of Analysis starts with pure logical axioms from set theory, does not rely on geometric intuition, strictly constructs number systems, and is a benchmark for axiomatic analysis in modern mathematics. In this paper, we first develop a machine proof system for axiomatic set theory rooted in the Morse–Kelley(MK) system. This system encompasses effective proof automation, scale simplification, and specialized handling of the classification axiom for ordered pairs. We then prove the Transfinite Recursion Theorem, leveraging it to further prove the Recursion Theorem for natural numbers—the key result for defining natural number operations. Finally, we detail the implementation of a machine proof system for analysis, which adopts MK as its description language and adheres to Landau’s Foundations of Analysis. This formalization realized all the contents of the book from natural numbers to complex numbers. All formalization does not need to introduce the standard library and has undergone verification by Rocq(Coq) 8.16 to ensure reliability. Implemented using the Rocq proof assistant, the formalization has undergone verification to ensure reliability. This work holds broader applicability such as the formalization of point-set topology and abstract algebra, while also serving as a valuable resource for teaching axiomatic set theory and mathematical analysis. Full article

Haruki Murakami’s Hard-Boiled Wonderland and the End of the World portrays a world where the “shadow”, the seat of memory, desire, and volition, is surgically removed, leaving behind a perfectly fluent but phenomenologically empty self. We argue that this literary structure mirrors a precise mathematical distinction in topological quantum matter. In a semisimple theory such as the semions of $\mathrm{SU}{\left(2\right)}_1$, there is a reducible component $V\left(x\right)$ of the $\mathrm{SL}(2,\mathbb{C})$ character variety: a flat, abelian manifold devoid of parabolic singularities. By contrast, the non-semisimple completion introduces a neutral indecomposable excitation, the neglecton, whose presence forces the mapping class group from the standard braid group $B_2$ to the affine braid group ${\mathrm{Aff}}_2$ and lifts the character variety to the Cayley cubic $V\left(C\right)$, with its four parabolic loci. We propose that contemporary AI systems, including large language models, inhabit the shadowless regime of $V\left(x\right)$: they exhibit coherence and fluency but lack any bulk degree of freedom capable of supporting persistent identity, non-contractible memory, or choice. To endow artificial systems with depth, one must introduce a structural asymmetry, a fixed, neutral defect analogous to the neglecton, that embeds computation in the non-semisimple geometry of the cubic. We outline an experimentally plausible architecture for such an “artificial ombre,” based on annular topological media with a pinned parabolic defect, realisable in fractional quantum Hall heterostructures, $p+ip$ superconductors, or cold-atom simulators. Our framework suggests that consciousness, biological or artificial, may depend on or benefit from a bulk–boundary tension mediated by a logarithmic degree of freedom: a mathematical shadow that cannot be computed away. Engineering such a defect offers a new pathway toward AGI with genuine phenomenological depth. Full article

This study analyses how preservice mathematics teachers reflect on the incorporation of gamification in their teaching practices after participating in a training cycle focused on this active methodology. The cycle, applied to 31 students in the Secondary Mathematics Teacher Education Master’s programme in Catalonia, integrated game elements—such as narratives, quests, and badges—alongside the Didactical Suitability Criteria as a tool for developing competency in the analysis and evaluation of didactical suitability. The reasons preservice teachers provided for using gamification in mathematics teaching were analysed at two points: upon completing the training cycle and after implementing teaching proposals incorporating gamification with secondary education students in Catalonia. The results show that preservice teachers show an improvement in their reflective competency regarding gamification, as well as increased use of this active methodology. Additionally, the knowledge and competencies considered relevant by preservice teachers for working with gamification in mathematics teaching were identified. It is concluded that training in the use of gamification in mathematics teaching is necessary to apply this methodology appropriately, and that tools such as the Didactical Suitability Criteria are fundamental for reflection on, and improvement in, the use of active methodologies. Full article

The Discrete Event System Specification (DEVS) formalism provides a mathematical foundation for modeling hierarchical discrete-event systems. However, the Depth-First Search (DFS) scheduling used in the classical DEVS abstract simulator conflicts with the concurrency semantics of Hardware Description Language (HDL) simulators such as Verilog or VHDL. This mismatch induces timing distortions, including pipeline skew and zero-time feedback loops. To address these limitations, this study proposes a new DEVS simulation kernel that adopts Breadth-First Search (BFS) scheduling, integrating the delta-round concept. This approach employs an event-parking mechanism that separates event computation from application, structurally aligning with HDL’s Active–NBA–Reactive phases and enabling semantically simultaneous updates without introducing additional ε-time. Case studies demonstrate that the proposed BFS-based DEVS kernel eliminates timing discrepancies in pipeline and feedback-loop structures and establishes a formal foundation for semantic alignment between DEVS and HDL simulators. Full article

This work introduces RAMA (Recursive Aesthetic Modular Approximation), a metaheuristic framework that models a restricted form of mathematical intuition inspired by the notebooks of Srinivasa Ramanujan. While Ramanujan often produced deep results without formal proofs, the heuristic processes guiding such discoveries remain poorly understood. RAMA treats large language models (LLMs) as proposal mechanisms within an iterative search that generates, evaluates, and refines candidate conjectures under an explicit energy functional balancing fit, description length, and aesthetic structure. A small set of Ramanujan-inspired heuristics—modular symmetries, integrality cues, aesthetic compression, and near-invariance detection—is formalized as micro-operators acting on symbolic states. We instantiate RAMA in two domains: (i) inverse engineering eta-quotients from partial q-series data and (ii) designing cyclotomic fingerprints with shadow gadgets for quantum circuits. In both settings, RAMA recovers compact structures from limited information and improves separation from classical baselines, illustrating how intuitive heuristic patterns can be rendered as explicit, reproducible computational procedures. Full article

Reading and mathematics are intricately linked through shared cognitive processes that underpin developmental relationships across domains. Despite extensive research on early-grade links between reading and basic arithmetic, gaps persist in understanding how reading comprehension (RC) supports applied math problem-solving (AMP) in older students and non-English contexts. The current study investigates the grade-level relationship between RC and AMP in typically developing Hebrew-speaking fourth (N = 41) and eleventh graders (N = 43), focusing on the contributions of working memory (WM), reading fluency, and arithmetic fluency. Results indicated significant positive associations between RC and AMP in both age groups. In fourth graders, arithmetic fluency partially statistically mediated the RC-AMP relationship in a cross-sectional mediation model. This indicates that students rely on computational proficiency to translate textual understanding into solutions. In contrast, eleventh graders exhibited a direct RC-AMP link, reflecting advanced comprehension and metacognitive strategies as computational skills are automatized. WM showed stronger correlations with RC and AMP among younger students, whereas these associations were weaker in older students. These findings support a Developmental Linguistic–Cognitive Scaffold Model, highlighting age-related shifts in cognitive and linguistic mechanisms supporting AMP. The results emphasize the need for integrated curricula incorporating RC strategies to enhance mathematical reasoning, particularly in morphologically rich languages like Hebrew. Full article

This paper addresses the challenge of optimizing cloudlet resource allocation in a code evaluation system. The study models the relationship between system load and response time when users submit code to an online code-evaluation platform, LambdaChecker, which operates a cloudlet-based processing pipeline. The pipeline includes code correctness checks, static analysis, and design-pattern detection using a local Large Language Model (LLM). To optimize the system, we develop a mathematical model and apply it to the LambdaChecker resource management problem. The proposed approach is evaluated using both simulations and real contest data, with a focus on improvements in average response time, resource utilization efficiency, and user satisfaction. The results indicate that adaptive scheduling and workload prediction effectively reduce waiting times without substantially increasing operational costs. Overall, the study suggests that systematic cloudlet optimization can enhance the educational value of automated code evaluation systems by improving responsiveness while preserving sustainable resource usage. Full article

Effective financial fraud detection requires systems that can interpret complex transaction semantics while dynamically adapting to asymmetric operational costs. We propose a hybrid framework in which a large language model (LLM) serves as an encoder, transforming heterogeneous transaction data into a unified embedding space. These embeddings define the state representation for a reinforcement learning (RL) agent, which acts as a fraud classifier optimized with business-aligned rewards that heavily penalize false negatives while controlling false positives. We evaluate the approach on two benchmark datasets—European Credit Card Fraud and PaySim—demonstrating that policy-gradient methods, particularly A2C, achieve high recall without sacrificing precision. Critically, our ablation study reveals that this hybrid architecture yields substantial performance gains on semantically rich transaction logs, whereas the advantage diminishes on mathematically compressed, anonymized features. Our results highlight the potential of coupling LLM-driven representations with RL policies for cost-sensitive and adaptive fraud detection. Full article

In this project, Anishinaabe artists and knowledge carriers worked with non-Indigenous classroom teachers to explore the cultural significance and mathematics of making wiigwaas makakoon (birch bark baskets). The artists spent two weeks in two grade 6 classrooms teaching students the process of basket making. They combined Indigenous pedagogy and intentionally designed inquiry tasks in order to generate mathematically related concepts. To make cultural–mathematical connections, we looked to Battiste’s characteristics of Indigenous pedagogy and explored how the learning that took place was holistic, part of a lifelong process, experiential, rooted in language and culture, spiritual, communal, and an integration of Indigenous and Eurocentric knowledges. Mathematically, students explored measurement with non-standard units, bisected angles without the use of a protractor, and explored the best way to optimize the capacity of their baskets. This work is an example of integrating Indigenous knowledge and heritage into elementary mathematics instruction. Full article

Students with mathematics learning disabilities often have difficulties in adding whole numbers. Such difficulties are evident in both response time and accuracy, but the relationship between accuracy and response time requires further consideration, especially in the context of technology-based interventions and assessments. In this article, we apply a novel approach using the drift-diffusion model to examine potential trade-offs and find balanced performance points that account for both accuracy and response time, using data from an efficacy trial of a mathematics technology gaming intervention for first-grade students with or at risk for learning disabilities. Results indicate that accuracy tends to increase as response time decreases, but only to a certain point. Practical implications include that educators should consider both accuracy and response time to intensify and individualize their instruction and take student background (i.e., gender, special education status, and English language status) into account. We suggest that developing technology-based mathematics interventions and assessments requires careful design and configuration to balance accuracy and response time, thereby enabling adaptive performance thresholds for better understanding and supporting student learning in early mathematical fluency. Full article

The research aims to derive the positive and negative values and strange habits included in Ibn Battuta’s journey called “Tuhfat An-Nuzzar fi Ghara’ibal-Amsar wa-‘Aja’ib Al-Asfar” by Shams al-Din bin Abdullah al-Lawati, the Moroccan al-Tanji, known as Ibn Battuta (d. 1377 AD), presented and investigated by Ali al-Muntasir al-Katani (D.T), which was included in Ibn Battuta’s trip, to the peoples of the countries he visited on the African and Asian continents. A total of 440 respondents participated in the study: 195 teachers in the supplementary track and 245 fourth-year regular track students at an Arab College of Education from all disciplines: early childhood, Arabic language, science, mathematics and computer science, English language, and special education. The respondents were asked to select an enrichment text or a story of one or more pages from Ibn Battuta’s travels, with the aim of eliciting the positive and negative values and strange customs of the peoples and countries Ibn Battuta visited in Africa and Asia. The study results indicated that Ibn Battuta’s travelogue, “Tuhfat An-Nuzzar fi Ghara’ibal-Amsar wa-‘Aja’ib Al-Asfar,” is considered an important literary reference, rich with texts and stories from which we can deduce the values and customs of the people of the countries Ibn Battuta visited in Africa and Asia. Teachers can use this information for discussion and constructive dialogue with their students in schools, in various educational subjects such as social studies, religion, literature, Arabic language, history, and geography. Most of the study participants support the idea of integrating Ibn Battuta’s travelogue into various lessons. The study recommends the importance of integrating and expanding it to include other subjects in schools, colleges, and universities. This integration should be systematically built around various activities that achieve “meaningful learning,” ensure active student participation, and enhance value for the learner and society. In conclusion, I recommend conducting detailed studies and research on the educational values derived from travel literature. Full article

In this paper we present the development of an energy and power system modelling, simulation, and analysis (ePowerSim.jl) package in Julia programming language. ePowerSim.jl is designed to present a uniform data interface for static, quasi-static, dynamic analysis, as well as network operation optimisation. It provides a co-simulation framework for the further development and experimentation of various types of models of electric power systems components or abstract entities that have mathematical formalism or data representation. ePowerSim.jl makes extensive use of cutting edge packages such as DifferentialEquations.jl, Dataframes.jl, NamedTupleTools.jl, Helics.jl, ForwardDiff.jl, JuMP.jl, and BifurcationKit just to mention a few in the Julia ecosystem. Models of synchronous generator, synchronous condenser, excitation systems, and governors developed in the package were used to model IEEE 9 bus and IEEE 14 bus test networks and subsequently validated by a real-time digital simulator of electric power systems (RTDS). The results obtains for static and dynamic models simulation in ePowerSim.jl show a close match with a simulation of the same system in RTDS. A maximum error of $0.00001$ pu and $0.0001$ pu were obtained for steady states and transient state respectively. Similarly, a maximum deviation of $0.0001$ pu was obtained during validation for voltage magnitude during transient state at buses in the network. Full article

From Galileo’s conviction that the Universe is written in the language of mathematics to the genomic and computational revolutions of the twentieth century, science has long sought to describe nature through external symbolic systems. However, mounting evidence across biology, physics, and cognitive science suggests that this reduction of semantics to syntax is insufficient. Life cannot be deterministically read only from DNA sequences, cognition cannot be reduced to just rule-based logic, and complex systems exhibit emergent behaviors that transcend symbolic description. We argue that these phenomena point toward an underappreciated principle, “the intelligence of matter”, whereby organized material systems inherently process information, adapt, and remember. Examples span from protein allostery and epigenetic memory to epitranscriptomic regulation, intelligent soft matter, and ecological reservoir computing. In these cases, computation emerges not from imposed codes but from the intrinsic dynamics of matter that is far from equilibrium. Recognizing intelligence as a general property of organized matter may inaugurate a new scientific style: one that deciphers the semantics of nature rather than superimposing ours and thus reshaping the epistemology of modern science. Full article

Developing expertise in physics requires appropriate integration and assimilation of physics and mathematics. Instructors and students often describe physics courses in terms of their emphasis on conceptual and quantitative problem-solving. For example, they may argue that a course emphasizes primarily conceptual over quantitative problem-solving or may emphasize equally on both depending on instructional context and assessment design. In this study, we investigated how students and instructors across different levels of physics instruction perceive the roles and development of conceptual and quantitative problem-solving in student learning and expertise development. Using departmental surveys administered at the beginning and end of each semester, we collected both Likert-scale and open-ended responses from students enrolled in introductory, upper-level undergraduate and graduate physics courses. These surveys assessed students’ self-perceived skills, preferences, and perceptions of instructors and course emphasis. To complement student perspectives, we conducted interviews with instructors, using parallel questions adapted to reflect instructional goals and expectations. Our findings highlight patterns in how students and instructors prioritize conceptual and quantitative problem-solving across course levels, as well as alignment and misalignment between student and instructor perspectives. Also, although the questions were framed around conceptual versus quantitative problem-solving, we do not view them as mutually exclusive; rather we seek to understand perceived course emphasis and student expertise development from student and instructor points of view in a language commonly used in physics. These results can help shape teaching, course design, and assessment practices to better support the development of expert-like problem-solving skills in students in physics and related disciplines. Full article

Early mathematical learning predicts later academic achievement, and creativity within mathematics plays a central role in higher-order thinking. This study examined the effects of linguistically responsive mathematics enrichment programs for nurturing mathematical creativity. Participants were 250 high-potential kindergarten English Learners across six urban schools in New York, Texas, and California. A linguistically responsive enrichment intervention adapted from the Mentoring Young Mathematicians (M²) math curriculum was implemented for 80 h across seven months. Using the Tangram Creativity Assessment, fluency, flexibility, and originality were measured in students’ tangram problem solving. Additional predictors included Tangram Problem Solving Speed (TPSS), general reasoning (CogAT), and mathematical achievement (NWEA MAP Math). ANCOVA showed significant post-test differences favoring the intervention group across all creativity components. Two-group structural equation modeling analysis supported measurement invariance and explained 55–60% of posttest creativity variance. TPSS emerged as the strongest predictor, with greater effects for the intervention group. These findings highlight the potential of enrichment programs and language-accessible geometry tasks to cultivate creativity in young gifted ELs by strengthening their mathematical foundation while supporting flexible and original problem solving. Full article