Modelling physical phenomena and acquiring mathematical formalisation can be challenging in physics and math education at all grades. It is recommended that interpretative and mathematical models be developed gradually from an early age, starting from kindergarten. Although innate visuospatial representation and numeracy skills are present in individuals from an early age, numerous socio-cultural factors encountered during the educational journey impede the realisation of an individual's full potential. This study explored how to harness the transformative power of transduction and measurement in mathematics education. The key connection between these activities is the shift from using the entire body for large macro movements to manipulating small objects at a micro level. This progression from bodily transformations to fine motor skills helps students link significant movements with precise tasks.
Our object is the Poisson integral for the upper half-space ℝ^n × (0,+∞ ) such that its boundary datum is integrable, non-negative, non-trivial and compactly supported. We show that the function is strictly -1/(n+1) -concave in the variable of ℝ^n if the variable of (0,+∞ ) is large enough compared with the diameter of the support of the boundary datum. For each α∈ (-1/(n+1),+∞ ) , we give a region in which the function is strictly α -concave in the variable of ℝ^n if the variable of (0,+∞ ) is large enough. For each α∈ (-1/(n+1),1] , we also give a region in which the function is not α -concave in the variable of ℝ^n .
The purpose of this paper is to prove the global L^∞ -boundedness of solutions for a Dirichlet nonlinear anisotropic elliptic systems with non standard growth conditions. The functional requirements involves anisotropic Lebesgue–Sobolev spaces in the scalar case and their ℝ^m -valued versions.
This paper presents a model for studying the processes involved in mathematics teacher professional development. Specifically, it focuses on professional development programs in which researchers in mathematics education act as teacher educators and collaborate with teachers in designing classroom activities and mathematical tasks for students. The model is constructed by combining the meta-didactical transposition framework, the documentational approach to didactics, and the construct of beliefs. Its evolution is described through an account of the subsequent developments and refinements. A general methodology for data collection and analysis is proposed, aimed at providing a nuanced understanding of the complex processes characterizing teacher professional development. This includes the examination of teachers’ and researchers’ meta-didactical praxeologies, documentation work, and beliefs, as well as the interrelations among these elements. For each version of the model, an example of application is provided to illustrate its modularity and the possibility of partial use. Finally, the application of the complete model in its final version is presented, showing how it can be fruitfully adopted to shed light on different yet complementary aspects of the processes under investigation.
In this paper, we contribute to this growing body of research by studying a class of nonlinear anisotropic elliptic equations with drift terms, where both the diffusion and the lower-order terms exhibit non-standard directional growth. We aim to establish existence, regularity, and integrability results for weak or distributional solutions, even when the source term f belongs to a low Lebesgue space. This level of generality is essential for applications involving irregular data.
This note is devoted to the failure of the Calderon-Zygmund theory for linear differential operators with discontinuous coefficients. It is known that the theory holds if the data belong to L^m(Ω ) , with 1 < m ≤2N/N+2 (see Boccardo and Gallouët in Commun Partial Diff Equ 17:641–655, 1992). In this paper we prove that the theory fails if 2N/N+2< m < N , thus extending to the case 2N/N+2< m < N/2 the results of Boccardo (Atti Accad Naz Lincei Rend Lincei Mat Appl 26:215–221, 2015).
Quasi-Newton algorithms, particularly the BFGS technique, are well-defined methods for solving unconstrained single-objective optimization problems that have recently been extended to multiobjective optimization. However, implementing these methods for large-scale problems proves challenging due to the need to store Hessian approximations. To address this challenge, we suggest a scaled memoryless BFGS quasi-Newton technique designed for large-scale multiobjective optimization problems (MOPs), which eliminates the requirement for matrix storage while preserving convergence properties. Our approach applies a novel scaling technique to enhance efficiency and approximates the Pareto front uniformly through an iterative, memory-efficient approach. To validate the procedure, we investigate numerical test problems using well-known performance metrics and compare the results with some existing algorithms. The findings show that the suggested approach attains superior computational efficiency and solution quality, specially in high-dimensional settings. This study bridges the deep gap in large-scale multiobjective optimization, by suggesting a scalable and practical alternative to standard quasi-Newton techniques while maintaining theoretical rigor.
Protesting activity often displays spatio-temporal patterns, previously modeled with reaction–diffusion systems. With the global rise of social media, it is important to understand how online networks shape these dynamics. In this work, we extend the reaction–diffusion model of Berestycki et al. (Netw Heterog Media 10:443–475, 2015) to regular network structures, capturing the interplay between activity levels and social tension. Depending on parameters, the system exhibits Fisher–KPP, weak Allee, or strong Allee growth. We analyze spreading speeds in the Fisher–KPP and weak Allee regimes, where a phase transition occurs between pushed and pulled waves. Numerical and asymptotic analyses characterize this transition and its limiting behavior. In the strong Allee (bistable) regime, we identify conditions under which small diffusion leads to pinned waves.
We investigate the isoperimetric problem for the Voronoi cells of three-dimensional lattices. Using Selling parameters, we derive an explicit closed formula for the scale-invariant isoperimetric quotient F in terms of six non-negative variables. We then analyse the local behaviour of F at the most relevant lattice configurations: we prove that the body-centered cubic lattice (BCC) is a strict local minimiser of F at fixed volume, whereas the face-centered cubic lattice (FCC) and the simple cubic lattice (SC) are not local minimisers. Then, we consider a family of lattices which interpolates between BCC and FCC, showing that BCC is the global minimiser of F restricted to this family.
Symbolism is commonplace in mathematics text. This case study investigated primary preservice teachers’ knowledge and understanding of selected mathematical codes and conventions, including symbolism used in middle years of schooling. Data were gathered from 23 primary preservice teachers who were enrolled in the Graduate Diploma program at one Australian university. The preservice teachers volunteered to participate in a series of learning sessions focused on their understanding of mathematical text. The paper argues that understanding of mathematical codes and conventions is a necessary component of mathematical literacy. Mathematics teachers, and teachers of many other STEM subjects, require knowledge of the complexities that mathematical symbolism introduces into learning, to enable them to aid their students’ learning. With reference to curricula in United States and Australia, the article draws readers’ attention to the diversity of codes and conventions used in mathematics, and in various other STEM subjects. Related difficulties that learners may experience are discussed, together with several teaching and learning suggestions.
Water Distribution Networks (WDNs) are large-scale, spatially irregular systems composed of interconnected nodes and pipes shaped by urban infrastructure. Despite their critical role in public health and urban resilience, WDNs often lag behind other utilities in terms of digitization and integration into smart city frameworks. This work presents a generative, data-driven method for reconstructing the full hydraulic state—pressures at nodes and flows in pipes—using a feedforward neural network with radial basis function (RBF) activations. The model is trained on synthetic data generated via hydraulic simulations and uses sparse real-time measurements from a limited number of strategically placed pressure and flow sensors. Unlike classical RBF Neural Networks, which rely on fixed spatial centers and local interpolation, the proposed architecture enables full-field inference through a single forward pass. The RBF activations capture spatial dependencies while allowing the network to generalize across the entire topology of the WDN. The approach achieves high-precision reconstruction performance, with Mean Squared Errors on the order of 10^-13 and Mean Absolute Errors around 10^-7 , confirming the method’s accuracy and stability. So, RBF-NNs emerge as a particularly effective solution, as they efficiently capture local-to-global spatial dependencies inherent in the dynamic behavior of WDNs, contributing to the broader integration of AI-driven solutions in infrastructure management.
We provide local bounds for positive viscosity sub and supersolutions to a class of doubly nonlinear parabolic equations, H(Du,D^2u)-u^α u_t=0, 0≤α≤ k-1, k> 1, in Ω× [0,T), where Ω⊂ℝ^n is a bounded domain and 0
In this note, we show that for a smooth algebraic variety Y and a smooth m-secant section X of the P-1-bundle f : P(O-Y circle plus O-Y(E)) -> Y, where E is an effective divisor on Y satisfying H-1(Y,O-Y(kE)) = 0 for all k=1, . . . . , m-1, the Tschirnhausen module of the induced covering f vertical bar X : X -> Y is completely decomposable. We then apply it to coverings of curves arising in such a way.
In this short note we show an equivalence between Sobolev type inequalities and so called isocapacitary inequalities in the context of a large class of nonlinear Dirichlet forms, their associated Dirichlet spaces and their associated capacities.
Whether the successor of a singular cardinal can be Jónsson is a very old and famous open problem in set theory. Here, we collect necessary conditions for an affirmative answer, and put forward a list of closely-related questions in singular cardinals combinatorics that could eventually lead to settling the main problem at hand.
The computation of the index ideal of a finite algebraic extension over a number field, is a deep task. Early, Hall gives a nice decomposition of the absolute index of cubic field (see Hall in Bull Am Math Soc 43(2):104–108, 1937). In this paper we prove a new version of the well-known structure Theorem of finitely generated modules over a PID which allows us to generalize the Hall’s result for cubic numbers to arbitrary extension of a fraction field of principal ideal domain. More precisely we show the existence of a monic triangular basis such as the sequence of its denominators is a factorial sequence. Further we express explicitly the ideal index as a product of a such factorial sequence and hence obtain an interesting decomposition of the ideal index over a PID. Some useful examples are also given.
This paper presents a comprehensive study of right S -maximal ideals in noncommutative rings, providing a natural extension of classical ideal theory through the framework of m -systems. We establish deep connections among right S -maximal, S-comaximal, and S -prime ideals, and introduce new concepts such as the S -Jacobson radical and S -invertible elements to further develop the structural theory of rings. A key contribution is the introduction of S -local rings and the formulation of an S -version of Nakayama’s Lemma. In addition, we propose two complementary definitions of left S -primitive ideals–one ideal-theoretic and the other annihilator-based–and demonstrate their intrinsic relationship to S -prime ideals. Our results not only unify and generalize existing notions but also open promising avenues for further research, particularly in exploring the interplay between different forms of S -primitivity.
In this note, we show that for a smooth algebraic variety Y and a smooth m-secant section X of the ℙ^1 -bundle f : ℙ(𝒪_Y ⊕𝒪_Y(E)) ⟶ Y, where E is an effective divisor on Y satisfying H^1(Y, 𝒪_Y(kE)) = 0 for all k = 1, … , m-1 , the Tschirnhausen module of the induced covering f|_X : X ⟶ Y is completely decomposable. We then apply it to coverings of curves arising in such a way.
This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising unrefinable partitions are established. A correspondence between missing parts and the gaps of numerical semigroups is developed, extending previous classifications and enabling the characterisation of partitions with maximal numbers of missing parts. In particular, the results show that certain families of unrefinable partitions correspond precisely to symmetric numerical semigroups when the maximal part is prime. Further structural consequences, examples, and a decomposition of unrefinable partitions by minimal excludant are discussed, together with implications for the study of maximal unrefinable partitions.
