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.
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.
Julius Weingarten (1836–1910) made significant contributions to differential geometry, particularly in addressing the second problem of applicability. This paper examines his results on this theme, including the introduction of W-surfaces, his research of complete classes of applicable surfaces in the 1980s and the 1890s, and his award-winning 1894 memoir recognised by the Académie des Sciences de Paris. Besides showing how these achievements marked two important stages in Weingarten’s career, by analysing both his publications and correspondence with Luigi Bianchi the paper reconstructs Weingarten’s evolving methods and situates his work within the broader nineteenth-century pursuit of explicit solutions to differential equations.
We classify Jordan derivations and Lie derivations of some important classes of alternative rings, namely alternative division rings and split octonion algebras. For these rings, every Jordan derivation that also satisfies a certain additional condition is a derivation. This additional condition is not necessary for associative rings, but is necessary for alternative rings. Also, for these rings, we show that every Lie derivation has the expected form, being the sum of a derivation and an additive function that has image in the center and vanishes on commutators.
This paper focuses on the combinatorial properties of plus–one generated line arrangements. We provide combinatorial constraints on such arrangements and construct a polynomial with a shape similar to the well–known Poincaré polynomial that decodes the plus–one generatedness property. We demonstrate how to create new plus–one generated arrangements using classical Klein and Wiman reflection arrangements. Furthermore, among all known sporadic simplicial arrangements with up to 27 lines, we identify nine minimal plus–one generated arrangements.
Let a and b be two positive integers such that a, b < n . We denote the inclusion ΣℂP^a→ SU(n) by ε _a,n . In this article, first, we study the order of the Samelson product ⟨ε _a,n, ε _b,n⟩ where a+b=n+k , for k ≥ 0 . Then, localized at an odd prime p, we will calculate the order of the commutator map SU(n)∧ SU(n) → SU(n) for n=4,5 , and continue to give an upper bound on the number of homotopy types of gauge groups for principal SU(4)-and SU(5)-bundles over a n-sphere.
Besides a self-contained introduction into well quasi-orders and better quasi-orders, the focus is on how these concepts lead in a neat and quick way to some well-known facts regarding monomials and monomial ideals of a polynomial ring over a field. Not uninteresting connections emerge, such as a direct path to Dickson’s lemma for monomial ideals from Higman’s theorem for monoids of terms in the vein of Hilbert’s basis theorem in iterative form.
Mathematical Induction (MI) plays a central role in mathematics, both for its theoretical and foundational implications and for the variety of its applications as a proving or defining scheme. Over the last decades, research in Mathematics Education has investigated MI from a didactic perspective, producing numerous studies with different foci of analysis. This paper presents a review of this body of research. Relevant studies have been firstly searched on the online database SCOPUS and then snowballing was applied to include other not yet found papers. This process resulted in a corpus of 30 studies, whose analysis is presented in this paper. Three main directions of discussion are identified: logical-epistemological aspects involved in MI, students’ difficulties with MI, and effective teaching experiences for the learning of MI. Overall, this review underscores the multifaceted complexity of MI as an educational issue, spanning logical-epistemological and didactic dimensions. Hopefully the presented discussion can provide a useful starting point for educators and researchers looking to delve into this theme.
In this article, the convolution theorems and continuity results for the linear canonical curvelet transform (LCCT) are established. Using the generalized translation operator, the convolution theorem associated with the LCCT is formulated, and some new results are presented. Spectral and spatial convolution theorems are also derived. Furthermore, the LCCT is extended to function spaces such as generalized Sobolev, L^p -Sobolev, and Besov spaces. The approximation property of the LCCT is examined in the generalized Sobolev space B^w,M_p,k(ℝ^2) . It is shown that the LCCT is a continuous linear operator on Sobolev spaces H^M_s(ℝ^2) and H^M_s,p(ℝ^2) . Additionally, continuity and boundedness results for the LCCT within the weighted Besov space are presented.
We prove a result on stochastic homogenisation of integral functionals of the form ∫ _U f (ω , x/ε , 𝔸u ) d x where ω is a random parameter, ε >0 and 𝔸 is a real elliptic vectorial differential operator. This work is intended to generalise results for the full gradient and to cover the cases of the symmetric gradient and the deviatoric operator. The homogenisation procedure is carried out by employing a variant of the blow-up method in the setting of 𝔸 -Sobolev spaces along with the Akcloglu-Krengel subadditive ergodic theorem.
