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The International Sirius Mathematics Center, located on the Black Sea coast, was founded in 2019 by the Talent and Success Foundation for organizing mathematical conferences and other events promoting the development of mathematics in Russia. In 2024, the Center has acquired its own Russian-language journal for publication of original research papers by participants of conferences held at the Center. The authors are invited to submit papers to the journal at https://ef.msp.org/submit/sirius [ABSTRACT FROM AUTHOR]

The paper responds to the need for understanding trends and gaps in extant research related to adult education in mathematics and numeracy, given changing skill demands and skill gaps regarding adults, and related policy, theorizing, and practice trends. This paper presents the results of a scoping review of recent empirical research related to adult education in mathematics and numeracy, published in 22 selected journals from 2019 to 2022, including 15 journals in adult education and seven in mathematics education. The results show that only 39 relevant empirical studies were found among over 2300 research papers reviewed, and that few of those focus on practice-related of adult education in mathematics and numeracy. The results provide quantitative evidence suggesting that the field of adult numeracy education is under-researched, and help to identify gaps in empirical research involving adult numeracy, including on emerging topics such as on modeling and critical interpretation. The results also point to research opportunities that can strengthen theorizing and practice in both mathematics education and adult numeracy education. [ABSTRACT FROM AUTHOR]

This systematic review aims to provide a complementary to existing synopses of the state-of-the-art of mathematics education research on proof and proving in both school and university mathematics. As an organizing framework, we used Cohen et al.'s triadic conceptualization of instruction, which draws attention not only to the main actors of the didactical process (i.e., the Teacher and Students) and the Content around which the actors' work is organized (herein, content related to proof and proving), but also to the relationships among the actors and the content. Out of the 103 papers we reviewed, almost half fell in the Student-Content category, which is consistent with the existence of a substantial number of frameworks, methods, and research findings related to students' engagement with proof and proving. About a quarter of the papers fell in the Student–Teacher-Content category, which reflects an emphasis on viewing instructional practice in proof and proving in a holistic, systemic way. Only few papers fell in the categories that did not include Content in them, namely, the categories of Student, Teacher, and Student–Teacher; this suggests mathematics education research on proof and proving has a strong disciplinary identity, which potentially differentiates it from other mathematics education research strands. About a fifth of the papers were oriented towards 'breaking ground' through making an explicit theoretical and/or methodological contribution (Student–Teacher-Content and Content were the main categories where such contributions appeared), whilst the majority of the papers were focused on 'building ground' through elaborating or employing existing methodological and/or theoretical approaches. [ABSTRACT FROM AUTHOR]

This paper concerns with the dynamical properties of a diffusive viral model with free boundaries, which describes the spread of viruses in space. Unlike the paper (Li et al. in Sci China Math 64:1971–1992, 2021), free boundaries at both ends will extend to infinity, i.e., the habitat eventually expands to the entire space. Moreover, the virus will eventually die out when the Basic Reproduction Number R 0 ≤ 1 , while the virus will stabilize in a positive equilibrium when R 0 > 1 and some further assumptions (Theorem 3.2). [ABSTRACT FROM AUTHOR]

The interrogation of often unintended practices of marginalisation has gained focus in research on mathematics teaching and mathematics teacher education throughout the last decades. In this introductory survey paper, work against marginalisation in these contexts of mathematics education is viewed in terms of work towards equity, diversity and inclusion. Based on this interpretation, we present a framework on awareness and practice of equity, diversity and inclusion in mathematics teaching and mathematics teacher education research. We then use this framework and a survey method of mapping review to identify and comment on a selection of studies. As a result, we illustrate three research moves towards equity, diversity and inclusion, in the form of interconnected themes: (1) Widening the understanding of the mathematics and the mathematics education curricula (2) Improving the practice and discussion of mathematics teaching (3) Unpacking ideologies in mathematics teaching and mathematics teacher education. We finally examine the themes and the special issue papers together to foreground commonalities regarding awareness of discriminatory discourses and practices of creating and distributing opportunities for all groups, including those historically and currently marginalised. Despite the important increase of equity-driven principles of awareness, we conclude that mathematics education research on teaching and on teacher education needs more examples of practices whose development has been proved to challenge marginalisation. [ABSTRACT FROM AUTHOR]

This paper describes a Lesson Study in which in-service mathematics secondary-school teachers, collaborating with researchers, involve grade 10 students in tessellation problems. The data are collected by an experiment carried out in the context of the "Liceo Matematico" project, with three volunteer teachers. The experiment goal was to craft a collaborative design of the research lesson between teachers and researchers. The research aim of the paper is to examine the use of Lesson Study in the institutional and cultural context of Italian secondary school with the use of Cultural Transposition as a theoretical framework. The research is qualitative with idiographic aims, based on video research. The educational aim of the research is to provide a solid basis for a revamped in-service teacher education first in the context of the project, then in curricular context. Semiotic mediation is used to provide, within Lesson Study, the conceptual framework for teachers and researchers collaborative design of the research lesson. The results show that Lesson Study, as a foreign practice, is an opportunity for teachers to confront their teaching practices, to enrich their professional development, resulting in more awareness on their didactical action in and outside the classroom. [ABSTRACT FROM AUTHOR]

In this paper, we study the scattering theory for the cubic inhomogeneous Schrödinger equations with inverse square potential i u t + Δ u - a | x | 2 u = λ | x | - b | u | 2 u with a > - 1 4 and 0 < b < 1 in dimension three. In the defocusing case (i.e. λ = 1 ), we establish the global well-posedness and scattering for any initial data in the energy space H a 1 (R 3) . While for the focusing case(i.e. λ = - 1 ), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson and Murphy (Proc Am Math Soc 145:4859–4867, 2017) and Campos and Cardoso (Proc Am Math Soc 150:2007–2021, 2022) that avoids the use of interaction Morawetz estimate. We also address the existence and the non-existence of normalized solutions of the above Schrödinger equation in dimension N for the focusing and defocusing cases. [ABSTRACT FROM AUTHOR]

Several notions of topological entropy for non necesarily continuous semiflows were introduced because the classical notion of topological entropy does not work cause the discontinuity of these systems. In particular, in Jaque and San Martin (J Differ Equ 266:3580–3600, 2019) was studied the definition of topological entropy by using separated and spanned sets, respectively, associated to a certain pseudosemimetric. Besides, it was proved that for regular impulsive semiflows the notion obtained by using separated sets can be computed trough a suitable conjugacy with a continuous system. In this paper, we will show that the same result holds for the entropy defined by spanning sets by adding a mild condition on the semiflow. This result extends, for this kind of semiflows, those obtained by Bowen (Trans Am Math Soc 153:401–414, 1971) and Dinaburg (Dokl Akad Nauk SSSR 190:19–22, 1970) respectively for the continuous case. [ABSTRACT FROM AUTHOR]

This paper provides a brief overview of findings in mathematical cognition and how a human-like AI in mathematics may look like. Then, it provides six reasons in favor of a human-like AI for mathematics: (1) human cognition, with all its limits, creates mathematics; (2) human mathematics is insightful, not merely deductive steps; (3) human cognition detects structure in the real world; (4) human cognition can tackle and detect complex problems; (5) human cognition is creative; (6) human cognition considers ethical issues. The paper provides a tentative frame to the question whether human mathematical cognition is relevant for designing an artificial intelligence that works on and creates mathematics. [ABSTRACT FROM AUTHOR]

In Kim et al. (Nagoya Math J 178: 37-53, 2005), Lee and Lee (J Geometry Phys 56(10): 2011-2023, 2006), the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to n-valued maps in Deconinck and Dekimpe (J Fixed Point Theory Appl 25(4): Paper No. 84, 29, 2023). In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any n-valued affine map on an infra-nilmanifold. [ABSTRACT FROM AUTHOR]

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function f : R → R which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points (x k) k ∈ N converging to a point x ∗ where f is discontinuous, lower semicontinuous and whose objective function value f (x ∗) is strictly less than lim k → ∞ f (x k) . Moreover the dDSM generates trial points in only one of the continuity sets of f near x ∗ . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample. [ABSTRACT FROM AUTHOR]

The focus of this study is to use the mixture item response theory (MixIRT) model while implementing the no-U-turn sampler as a technique for investigating the presence of latent classes (i.e., subpopulations) among eighth-grade students who were administered TIMSS 2019 mathematics subtest in paper format from the gulf cooperation council (GCC) countries. One-, two-, and constrained three-parameter logistic MixIRT models with one to four classes were used to fit to the data, where the model data fit was assessed using Bayesian fit indices. The results indicate that multiple latent classes or subpopulations can better reflect the mathematical proficiency of eighth graders from the four GCC countries, and specifically the two-class constrained three-parameter MixIRT model provides a relatively better fit to the data. The results also indicate that when a mixture of several latent classes present, the conventional unidimensional IRT model is limited in providing information for multiple latent classes and shall be avoided. In addition to adding to the existing literature on MixIRT models for international large-scale assessments such as TIMSS on its heterogenous subpopulations from a fully Bayesian approach, this study sheds light on the limitation of conventional unidimensional IRT models and subsequently directs attention to the use of the more complex MixIRT model for such assessments. [ABSTRACT FROM AUTHOR]

By introducing two parameters in the splittings of the matrices A and B, this paper presents a parameterized accelerated iteration (PAI) method for solving the matrix equation A X B = C . The convergence property of the PAI method and the choices of the parameters are thoroughly investigated. Additionally, based on some special splittings of the matrices A and B, several variants of the PAI method are established. Furthermore, for some certain cases, the optimal parameters can be determined, and it is demonstrated that the PAI method is more efficient than the gradient-based iteration (GBI) method (Ding et al. Appl. Math. Comput. 197, 41–50 2008). Finally, by comparing it with several existing iteration methods, the effectiveness of the PAI method is verified through four numerical examples. [ABSTRACT FROM AUTHOR]

In 2015, Mao (J. Comput. Appl. Math., 290, 370–384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In 1967, Grünbaum conjectured that any d-dimensional polytope with d + s ⩽ 2 d vertices has at least ϕ k (d + s , d) = d + 1 k + 1 + d k + 1 - d + 1 - s k + 1 k-faces. This conjecture along with the characterization of equality cases was recently proved by the author (A proof of Grünbaum's lower bound conjecture for general polytopes. Israel J. Math. 245(2), 991–1000 (2021)). In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (e.g., abstract polytopes) and d + s ⩽ 2 d atoms have at least ϕ k (d + s , d) elements of rank k + 1 . Furthermore, in the case of face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to 2d vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of k-faces of strongly regular CW complexes representing normal pseudomanifolds with 2 d + 1 vertices are obtained. These bounds are given by the face numbers of certain polytopes with 2 d + 1 vertices. [ABSTRACT FROM AUTHOR]

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process–object transitions and externalist approaches that stress mathematical artifacts—such as physical manipulatives and formulas—as constituting mathematical objects. We search for a view on process–object duality that overcomes the dualism of mind and body. One such approach is commognition that describes mathematical objects as discursive entities. This paper expands the nature of mathematical objects beyond discourse and highlights the role of learners' interaction with the environment by adopting ecological onto-epistemology. We develop a functional dynamic systems perspective on process–object duality in mathematics learning emphasizing embodied actions and the re-invention of artifacts' affordances. As a main result, we reconsider process–object duality as a reification of repetitive actions into a cultural artifact that consists of two steps: (1) forming a new sensory-motor coordination that brings new perception to the fore and (2) crystallizing a new artifact in a mathematical environment that captures this new perception. An empirical example from research on embodied action-based design for trigonometry illustrates our theoretical ideas. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive study of a model called the (2 + 1) q -deformed tanh-Gordon model. This model is particularly useful for studying physical systems with violated symmetries, as it provides insights into their behavior. To solve the (2 + 1) q -deformed equation for specific parameter values, the (H + G ′ G 2) -expansion approach is employed. This technique generates analytical solutions that reveal valuable information about the system's dynamics and behavior. These solutions offer insights into the underlying mathematics and deepen the understanding of the system's properties. To validate the accuracy of the analytical solutions, the finite difference technique is also used to find a numerical solution to the q -deformed equation. This numerical approach ensures the correctness of the solutions and enhances the reliability of the results. Tables and graphics are presented in the publication to aid comprehension and comparison. These visuals improve the clarity and interpretability of the data, allowing readers to better understand the similarities and differences between the analytical and numerical solutions. [ABSTRACT FROM AUTHOR]

In the paper, we consider a system of thermoelasticity of type I with viscoelastic damping and nonlinear sources. By using the Galerkin method and the Banach fixed point theorem, we first prove the local existence and uniqueness of weak solution. Secondly, by extending the potential well method, we prove that the local solution exists globally if its initial position starts inside a family of "potential wells." In particular, we also establish an explicit and optimal decay rate of energy driven by the decay rate of the relaxation function which includes exponential, algebraic, and logarithmic decay rates. Finally, by using the continuation theorem and the concavity arguments due to Levine (Trans Am Math Soc 192:1–21, 1974), we show that the local solution blows up at finite time in the sense of Ball (Q J Math Oxf 28(4): 473–486, 1977) if its initial position starts outside the "potential wells." Further, an upper bound for the blow-up time is also given explicitly. Notice that our results imply a sharp result on the global existence and blow-up of the local weak solution and they also allow a relatively large class of relaxation functions that generalize the existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

Recent rank-based attacks have reduced the security of Rainbow, which is one of the multi-layer UOV signatures, below the NIST security requirements by speeding up iterative kernel-finding operations using classical mathematics techniques. If quantum algorithms are applied to perform these iterative operations, the rank-based attacks may be more threatening to multi-layer UOV, including Rainbow. In this paper, we propose a quantum rectangular MinRank attack called the Q-rMinRank attack, the first quantum approach to key recovery attacks on multi-layer UOV signatures. Our attack is a general model applicable to multi-layer UOV signature schemes, and in this paper, we provide examples of its application to Rainbow and the Korean TTA standard, HiMQ. We design two quantum oracle circuits to find the kernel in consideration of the depth-width trade-off of quantum circuits. One is to reduce the width of the quantum circuits using qubits as a minimum, and the other is to reduce the depth using parallelization instead of using a lot of qubits. By designing quantum circuits to find kernels with fewer quantum resources and complexity by adding mathematical techniques, we achieve quadratic speedup for the MinRank attack to recover the private keys of multi-layer UOV signatures. We also estimate quantum resources for the designed quantum circuits and analyze quantum complexity based on them. The width-optimized circuit recovers the private keys of Rainbow parameter set V with only 1089 logical qubits. The depth-optimized circuit recovers the private keys of Rainbow parameter set V with a quantum complexity of 2 174 , which is lower than the complexity of 2 221 recovering the secret key of AES-192, which provides the same security level as parameter set III. [ABSTRACT FROM AUTHOR]

This paper illustrates how years 1 and 2 students were guided to engage in data modelling and statistical reasoning through interdisciplinary mathematics and science investigations drawn from an Australian 3-year longitudinal study: Interdisciplinary Mathematics and Science Learning (https://imslearning.org/). The project developed learning sequences for 12 inquiry-based investigations involving 35 teachers and cohorts of between 25 and 70 students across years 1 through 6. The research used a design-based methodology to develop, implement, and refine a 4-stage pedagogical cycle based on students' problem posing, data generation, organisation, interpretation, and reasoning about data. Across the stages of the IMS cycle, students generated increasingly sophisticated representations of data and made decisions about whether these supported their explanations, claims about, and solutions to scientific problems. The teacher's role in supporting students' statistical reasoning was analysed across two learning sequences: Ecology in year 1 and Paper Helicopters in year 2 involving the same cohort of students. An explicit focus on data modelling and meta-representational practices enabled the year 1 students to form statistical ideas, such as distribution, sampling, and aggregation, and to construct a range of data representations. In year 2, students engaged in tasks that focused on ordering and aggregating data, measures of central tendency, inferential reasoning, and, in some cases, informal ideas of variability. The study explores how a representation-focused interdisciplinary pedagogy can support the development of data modelling and statistical thinking from an early age. [ABSTRACT FROM AUTHOR]

This research paper delves into the profound interplay between mathematics and visual art, revealing a fundamental correlation rooted in the concept of order. By examining historical periods, particularly the Renaissance and Enlightenment, we illuminate how mathematics catalyzed intellectual and cultural transformations. The Renaissance witnessed a revival of ancient mathematical knowledge, intertwining with the prevailing humanist philosophy. Artists of this era employed mathematical principles, such as perspective and proportion, to bridge the abstract and tangible. In the Enlightenment, mathematical innovation was driven by a commitment to rationality and empirical evidence, mirroring the era's quest for universal truths. Engaging with theoretical and historical literature, we decode the intricate threads connecting mathematics to the pulse of human knowledge and progress. [ABSTRACT FROM AUTHOR]

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Nexus Network Journal guest editor Roberta Spallone introduces papers selected from the Nexus Conference 2023 in Turin for detailed review, expansion and development. These papers identify close links between architecture, mathematics, and specific areas of interest related to historical periods, architectural cultures, typological elements, and analysis tools. [ABSTRACT FROM AUTHOR]

As a quantum affinization, the quantum toroidal algebra U q , q ¯ (gl ¨ n) is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation U q (gl ˙ n) ↷ C n (z) and its usual R-matrix R (z) ∈ End (C n ⊗ C n) (z) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on U q , q ¯ (gl ¨ n) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra U q (gl ˙ n) ⊂ U q , q ¯ (gl ¨ n) . [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading behavior of a two-species strong-weak competition system with two free boundaries. The model may describe how a strong competing species invades into the habitat of a native weak competing species. The asymptotic spreading speed of invading fronts has been determined by making use of semi-wave systems in Du et al. (J Math Pures Appl 107:253–287, 2017). Here we give a sharp estimate for the asymptotic spreading speed of invading fronts. Moreover, we prove that the solution of the free boundary problem evolves eventually into a semi-wave solution when the spreading happens, while the solution of the free boundary problem exponentially converges to a semi-trivial solution of such system when the vanishing happens. [ABSTRACT FROM AUTHOR]

In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO. [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

Intelligent Tutoring Systems (ITS) have been widely used to enhance math learning, wherein teacher's involvement is prominent to achieve their full potential. Usually, ITSs depend on direct interaction between the students and a computer. Recently, researchers started exploring handwritten input (e.g., from paper sheets) aiming to provide equitable access to ITSs' benefits. However, research on math ITSs ability to handle handwritten input is limited and, to our best knowledge, no study has summarized its state of the art. This article fulfills that gap with a scoping review of handwritten recognition methods, characteristics, and applications of math ITSs compatible with handwritten input. Based on a search of 11 databases, we found eight primary studies that met our criteria. Mainly, we found that all ITSs depend on receiving handwritten input from a touchscreen interface, in contrast to recognizing solutions developed on paper. We also found that most ITSs focus on similar audiences (e.g., English speakers students), subjects (e.g., algebraic questions), and applications (e.g., in-class to understand student perceptions). Thus, towards enabling equitable access to ITSs, we propose ITS Unplugged (i.e., ITSs that i) run on low-cost, resource-restricted devices with little to no internet connection and ii) receive as well as return information in the format target users usually use) and contribute a research agenda concerning challenges of developing such ITSs. [ABSTRACT FROM AUTHOR]

The use of digital technology has the potential to support students' understanding in the mathematics classroom with the teacher playing a vital role. However, teaching with digital technology is not trivial, especially for teachers who are new to this. In this paper, we present an analysis of the enactment of a function lesson of a Sri Lankan mathematics teacher who used digital technology for the first time in her teaching. We combined the instrumental orchestration and ROG (resources, orientations and goals) frameworks into a conceptual framework to analyse her teaching. In particular, we used instrumental orchestration to identify how the teacher orchestrated the resources in her technology-rich classroom. This was combined with ROG theory to understand the reasons underpinning the decisions involved in moving from one orchestration to another. We demonstrate that this teacher showed diverse orchestrations and use the ROG framework to present these in the sequences in which they were used, formed into chains of orchestrations linked by goals. We propose that her didactical performance is a function of orchestration types over in-the-moment decision-making. [ABSTRACT FROM AUTHOR]

This study aims to analyze possible connections among the noticing skills of pre-service mathematics teachers, and specifically the skills of attending to students' thinking, interpreting students' mathematical understanding, and proposing teaching alternatives. We characterized these connections in terms of coherency, i.e., the extent to which pre-service teachers take into account (a) an identified critical event—including its mathematics and context—when interpreting students' statements, (b) their own interpretation of students' statements when interpreting the teacher's response, and (c) their own interpretation of students' statements when proposing alternative teaching strategies. The construct of coherency evolved during our attempts to analyze pre-service teachers' interpretations of the critical events they identified while observing mathematics lessons as part of their preparation program. In this paper, we elucidate the construct of coherency and its different levels and demonstrate how this construct relies upon and expands existing ideas from the literature on noticing. We propose a framework for identifying different coherency levels and the different patterns identified when using the framework to noticing among the pre-service teachers. Some of these patterns are supported by previous research, while others were unanticipated. We explore possible explanations for the emergence of these unanticipated patterns. [ABSTRACT FROM AUTHOR]

Recent research syntheses have identified several potentially high-leverage teaching strategies for improving low-attaining secondary school students' learning of mathematics. These strategies include the structured use of representations and manipulatives and an emphasis on derived facts and estimation. This paper reports on 70 semi-structured interviews conducted with low-attaining students in Years 9–10 (ages 13–15) in England. The interviews addressed the students' perceptions of learning mathematics and the teaching strategies that they experienced and believed were most helpful. Many students reported rarely using number lines, not spontaneously estimating answers and being unfamiliar with derived facts. During the interviews, with minimal direction, students often showed that they were well able to make use of these strategies; however, they did not report making spontaneous use of them independently. We conclude that many of the most well-evidenced and recommended strategies to support low-attaining students in mathematics appear to be unfamiliar and unvalued, and we discuss how this might be addressed. [ABSTRACT FROM AUTHOR]

Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less consideration. This paper provides a novel account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting framework captures many important aspects of the use of analogical inference in the domain of pure mathematics. [ABSTRACT FROM AUTHOR]

In the present paper, we have obtained common best proximity point theorems of nonself maps in Hausdorff topological space. Further, our results extend the results due to Gerald F. Jungck, thereby proving a generalized version of Kirk's theorem (J. London Math. 1(1):107–111, 1969). [ABSTRACT FROM AUTHOR]

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

Reference (Ericksen in Q. J. Mech. Appl. Math. 45:545–554, 1992), a short and dense paper by J.L. Ericksen, is taken as a primary source of terms frequently used in continuum thermodynamics, in an attempt to make their significance as precise as possible. [ABSTRACT FROM AUTHOR]

Telescopers for a function are linear differential (resp. difference) operators annihilating the definite integral (resp. definite sum) of this function. They play a key role in Wilf–Zeilberger theory and algorithms for computing them have been extensively studied in the past 30 years. In this paper, we introduce the notion of telescopers for differential forms with D-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given. [ABSTRACT FROM AUTHOR]

Semidefinite programming (SDP) problems typically utilize a constraint of the form X ⪰ x x T to obtain a convex relaxation of the condition X = x x T , where x ∈ R n . In this paper, we consider a new hyperplane branching method for SDP based on using an eigenvector of X - x x T . This branching technique is related to previous work of Saxeena et al. (Math Prog Ser B 124:383–411, 2010, https://doi.org/10.1007/s10107-010-0371-9) who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the two-trust-region subproblem. [ABSTRACT FROM AUTHOR]

The exploration of children's drawings as mathematical representations is a current focus in early years mathematics education research. This paper presents a qualitative analysis of 72 kindergarten to Grade 3 (5 to 8 years old) children's drawings produced during problem-solving tasks centred on multiplicative strategies. Existing frameworks for the developmental sequence of mathematical drawings and the progression of children's strategies for multiplicative situations were an interpretive lens through which to analyse the drawings. Children used pictographic and iconic drawing types to represent the "story" in the problem and the multiplicative strategies employed to solve the tasks. Exploration of the children's drawings suggested that as children's drawings become more structural, schematic in nature, it may be easier for children to show their understanding of the structural elements of multiplicative relationships. Results revealed that structural elements of multiplicative relationships were more easily seen in iconic representations; however, both pictographic and iconic drawings were useful to observe counting, additive, and multiplicative strategies when mathematical elements of the problem were visible. Additional representations attached to the drawings (e.g. numerical) were needed to confirm children's strategies when their drawings lacked structure. These findings have implications for how young children's drawings are interpreted by classroom teachers. The interpretation of these drawings suggested that some children may not yet realise how their drawings in mathematics need to shift from illustrations of the problem's story context to representing mathematical ideas and processes — which requires intentional teaching of the purpose of drawings for mathematical contexts. [ABSTRACT FROM AUTHOR]

This survey describes the structure of the field of research on curriculum resources in mathematics education in the period from 2018 till 2023. Based on the procedures of a systematic review relevant literature was identified using Web of Science as a database. The included literature was analyzed and categorized according to the type of curriculum resource and the area of study. Seven areas of studies were identified: studies on the role of curriculum resources, content analysis, user studies, studies on the effects of curriculum resources, studies on curriculum resource design, curriculum resources as data, and reviews. The areas were further subdivided into different subcategories based on the research questions of the included papers. The findings show that research on mathematics textbooks is still predominant in the field. The most popular areas of research are content analysis, user studies, studies on design, and studies on effects. Emerging areas are research on students' use of curriculum resources and the employment of user data from digital curriculum resources as data basis in mathematics education research. [ABSTRACT FROM AUTHOR]

In this paper we review selected significant developments in the use of digital technology in the teaching and learning of mathematics over the last five years. We focus on a number of important topics in this field, including the evolvement of STEAM and critical making as well as the process of redefining learning spaces in the transformation of the mathematics classroom. We also address the increasing use of computer algebra systems and dynamic geometry packages; and the issue of student collaboration online, especially using learning environments and social media. We briefly touch on artificial intelligence systems, including hyper-personalisation of learning, multimodality and videos. We include a brief discussion on the impact of COVID-19 on mathematics education, and lastly on the more theoretical perspective of the epistemology of digital technology and the construct of humans-with-media. We conclude the discussion with some possible concerns and mentioning some possible new topics for research in the field. [ABSTRACT FROM AUTHOR]

The teacher noticing construct is widely recognized in teacher competence and education research, particularly in the field of mathematics education. This paper surveys recent research on mathematics teacher noticing published between July 2019 and 2022, following an earlier literature review on teacher noticing across different disciplines. The study presented here analyzed 118 English-language articles published in peer-reviewed journals, focusing on conceptualizations, research methods, and relationships with other constructs, including teacher knowledge and beliefs. The findings suggest that the cognitive-psychological perspective on noticing, which emphasizes a set of cognitive processes, remains the predominant conceptualization. Recent research on noticing is characterized by a high proportion of studies based on small samples and qualitative research methods. While several studies have demonstrated the interrelatedness of noticing and professional knowledge, the relationship between noticing and beliefs and between noticing and instructional quality has rarely been addressed. Based on these findings, we highlight noteworthy contributions and critical shortcomings, and suggest directions for future research. [ABSTRACT FROM AUTHOR]

This paper provides sets of sufficient conditions for mean convergence theorems for arrays of dependent random variables. We expand and improve a number of particular cases in the literature including Theorem 2.1 in Sung (Appl Math Lett 26(1):18–24, 2013), Theorems 3.1–3.3 in Wu and Guan (J Math Anal Appl 377(2):613–623, 2011), and Theorem 3 in Lita da Silva (Results Math 74(1):1–11, 2019), among others. The proof is different from those in the aforementioned papers and the main results can be applied to obtain mean convergence results for arrays of functions of non-homogeneous Markov chains and dependent bootstrap. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

Successful early mathematical development is vital to children’s later education, employment, and wellbeing outcomes. However, established measurement tools are infrequently used to (i) assess children’s mathematical skills and (ii) identify children with or at-risk of mathematical learning difficulties. In response, this pre-registered systematic review aimed to provide an overview of measurement tools that have been evaluated for their psychometric properties for measuring the mathematical skills of children aged 0–8 years. The reliability and validity evidence reported for the identified measurement tools were then synthesised, including in relation to common acceptability thresholds. Overall, 41 mathematical assessments and 25 screeners were identified. Our study revealed five main findings. Firstly, most measurement tools were categorised as child-direct measures delivered individually with a trained assessor in a paper-based format. Secondly, the majority of the identified measurement tools have not been evaluated for aspects of reliability and validity most relevant to education measures, and only 15 measurement tools met the common acceptability thresholds for more than two areas of psychometric evidence. Thirdly, only four screeners demonstrated an acceptable ability to distinguish between typically developing children and those with or at-risk of mathematical learning difficulties. Fourthly, only one mathematical assessment and one screener met the common acceptability threshold for predictive validity. Finally, only 11 mathematical assessments and one screener were found to concurrently align with other validated measurement tools. Building on this current evidence and improving measurement quality is vital for raising methodological standards in mathematical learning and development research. [ABSTRACT FROM AUTHOR]

The mechanism of cytoplasmic incompatibility (CI) is important in the study of Wolbachia invasion in wild mosquitoes. Su et al. (Bull Math Biol 84(9):95, 2022) proposed a delay differential equation model by relating the CI effect to maturation delay. In this paper, we investigate the dynamics of this model by allowing the same density-dependent death rate and distinct density-independent death rates. Through analyzing the existence and stability of equilibria, we obtain the parameter conditions for Wolbachia successful invasion if the maternal transmission is perfect. While if the maternal transmission is imperfect, we give the ranges of parameters to ensure failure invasion, successful invasion and partially suppressing, respectively. Meanwhile, numerical simulations indicate that the system may exhibit monostable and bistable dynamics when parameters vary. Particularly, in the bistable situation an unstable separatrix, like a line, exists when choosing constant functions as initial values; and the maturation delay affects this separatrix in an interesting way. [ABSTRACT FROM AUTHOR]

In this paper, we study the Z / 2 action on complex Grassmann manifolds G n (C 2 n) given by taking orthogonal complement. We completely compute the associated Z / 2 Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds G n (R 2 n) by Baralić et al. [Forum Math., 30 (2018), pp. 1539–1572]. [ABSTRACT FROM AUTHOR]

Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students' development of arithmetic skills. [ABSTRACT FROM AUTHOR]