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In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. 'Al.I. Cuza' Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above. [ABSTRACT FROM AUTHOR]

First of all, we indicate a severe error in the analysis of the main results of both Chakraborty [Ukr. Math. J., 72, No. 11, 1794–1806 (2021)] and Chakraborty–Chakraborty [Ukr. Math. J., 72, No. 7, 1164–1174 (2020)] to show that both these papers cease to be true. Further, pertinent to the results of these two papers, we consider the unique range set of a meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1 under the aegis of its most general form to improve the existing result. [ABSTRACT FROM AUTHOR]

Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

Promoting engagement is crucial for encouraging student participation, interest, and learning in mathematics. Student engagement has been conceptualized as interrelated types comprising behavioural, emotional, and cognitive characteristics. Cognitive engagement, our focus in this paper, relates to students' psychological investment in learning and practices used to enhance learning, such as self-regulatory strategies and metacognitive processes. Although crucial for students' learning, research suggests that teachers' practices for promoting students' cognitive engagement are not well understood. In this qualitative study, we investigated the beliefs of 40 secondary mathematics teachers across eight English schools concerning promoting cognitive engagement in mathematics classrooms, and whether teachers with different cognitive engagement beliefs differ in the features of classroom practice they attend to in relation to promoting student self-regulation and metacognition. We developed a Cognitive Engagement Framework (CEF) for the following purposes: (1) to develop vignettes that described the practices of two contrasting teachers (Teacher A and Teacher B), who differed in their use of specific self-regulation and metacognitive processes; and (2) to use as a tool for analysis. 17 participants identified with Teacher A who favoured a controlling style towards student strategy use such as activating knowledge, planning, and enacting and regulating strategies, and a passive approach towards students' use of self-reflection. 14 participants identified with Teacher B who favoured promoting student autonomy for planning and enacting and regulating strategies, self-reflection, and acknowledged affective elements. In addition to its findings, the paper makes a methodological contribution by using 'vignettes' as a new way of investigating teachers' beliefs, and a theoretical contribution through the development of the CEF. [ABSTRACT FROM AUTHOR]

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of Polyak (USSR Comput Math Math Phys 4(5):1–17, 1964), it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in Aujol et al. (Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization, 2020). Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Łojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Łojasiewicz property. [ABSTRACT FROM AUTHOR]

This is a conceptual paper based on existing literature aiming to provide practical information on designing and implementing activities to promote children's computational thinking. Computational thinking is a relatively new term in early childhood education that refers to a specific problem-solving thinking process involving various logical and analytical thinking skills. Four foundational skills have been identified as core thinking skills of computational thinking: decomposition, abstraction, pattern recognition, and algorithm. We explain these four skills in this paper and their practical applications to teaching and learning in early childhood education. Early computational thinking skills are found in common early childhood activities. This paper identifies activities teachers can use in the classroom to explicitly promote children's computational thinking and provides a new perspective on how to adapt classroom activities to integrate computational thinking. In particular, we emphasize the need to vary the demands of the content in the activity and incorporate computational thinking based on children's needs and development to ensure that children progress through the thinking process. [ABSTRACT FROM AUTHOR]

The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of Criado and Santos (Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1641766), about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them. [ABSTRACT FROM AUTHOR]

Growing use of the internet in educational contexts has been prominent in recent years. In this survey paper, we describe how the internet is transforming the mathematics classroom and mathematics teacher education. We use as references several reviews of use of the internet in mathematics education settings made in recent years to determine how the field has evolved. We identify three domains in which new approaches are being generated by mathematic educators: principles of design of new settings; social interaction and construction knowledge; and tools and resources. The papers in this issue reflect different perspectives developed in the last decade in these three domains, providing evidence of the advances in theoretical frameworks and support in the generation of new meanings for old constructs such as 'tool', 'resources' or 'learning setting'. We firstly highlight the different ways in which the use of digital technologies generates new ways of thinking about mathematics and the settings in which it is learnt, and how mathematics teacher educators frame the new initiatives of initial training and professional development. In this survey paper, we identify trends for future research regarding theoretical and methodological aspects, and recognise new opportunities requiring further engagement. [ABSTRACT FROM AUTHOR]

This paper reports an overview of contemporary research on early childhood mathematics teaching and learning presented at recent mathematics education research conferences and papers included in the special issue (2020–4) of ZDM Mathematics Education. The research covers the broad spectrum of educational research focusing on different content and methods in teaching and learning mathematics among the youngest children in the educational systems. Particular focus in this paper is directed to what lessons can be drawn from teaching interventions in early childhood, what facilitates children's mathematical learning and development, and what mathematical key concepts can be observed in children. Together, these themes offer a coherent view of the complexity of researching mathematical teaching and learning in early childhood, but the research also brings this field forward by adding new knowledge that extends our understanding of aspects of mathematics education and research in this area, in the dynamic context of early childhood. This knowledge is important for future research and for the development of educational practices. [ABSTRACT FROM AUTHOR]

Based upon my experience in research, teaching, writing textbooks, and editing handbooks and journals, this review paper discusses how financial econometrics, mathematics, statistics, and financial technology can be used in research and teaching for students majoring in quantitative finance. A major portion of this paper discusses essential content of Lee and Lee (Handbook of financial econometrics, mathematics, statistics, and machine learning, World Scientific, Singapore, 2020). Then Lee (From east to west: memoirs of a finance professor on academia, practice, and policy, World Scientific, Singapore, 2017), Lee et al. (Financial econometrics, mathematics and statistics, Springer, New York, 2019a; Machine learning for predicting default of credit card holders and success of kickstarters. Working paper, 2019b), and Lee and Lee (Handbook of financial econometrics and statistics, Springer, New York, 2015) are used to enhance the content of this paper. In addition, important and relevant papers, which have been published in different journals are also used to support the issues discussed in this paper. I have found the applications of financial econometrics, mathematics, statistics, and technology have improved drastically over the last five decades. Therefore, both practitioners and academicians need to update their skills in this area to compete in both financial market and academic research. [ABSTRACT FROM AUTHOR]

This paper aims at contributing to the debate in Mathematics Education about the understanding of the dynamics of students' group interactions by proposing an interpretative lens, which defines four modes of participating in a group on the basis of different kinds of utterances, gestures, postures, and glances that each student makes. We apply this lens to two selected cases of students working in a small group, and, by comparing and contrasting similarities and differences observed through our interpretative lens, we attempt to understand how, and under which circumstances, the students reach mathematical understanding as a group, or not. [ABSTRACT FROM AUTHOR]

This is a commentary on the ESM 2021 Special Issue on Innovations in Measuring and Fostering Mathematical Modelling Competencies. We have grouped the ten studies into three themes: competencies, fostering, and measuring. The first theme and the papers therein provide a platform to discuss the cognitivist backgrounds to the different conceptualizations of mathematical modelling competencies, based on the modelling cycle. We suggest theoretical widening through a competence continuum and enriching of the modelling cycle with overarching, analytic dimensions for creativity, tool use, metacognition, and so forth. The second theme and the papers therein showcase innovative ideas on fostering and on the definition and analysis thereof. These reveal the need for a social turn in modelling research in order to capture aspects of student collaboration and agency, as well as tensions in fostering when tasks are derived from real-world scenarios, but socio-mathematical norms come from the (pure) mathematics classroom. The third theme, measuring, and the papers therein offer insights into the challenges of positivist research that aims to develop innovative measurement instruments that are both reliable and valid, particularly in light of student group work, cultural background, and other socio-cultural aspects. Drawing on the three discussions, we go on to make recommendations for further research. [ABSTRACT FROM AUTHOR]

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish a complete theory on the HUM method and give its applications to first order hyperbolic systems. Thus, a deeper relationship between the controllability and the observability can be revealed. Moreover, at the end of the paper, a comparison will be made between these two methods. [ABSTRACT FROM AUTHOR]

This paper introduces an anthropological approach to the foundations of mathematics. Traditionally, the philosophy of mathematics has focused on the nature and origins of mathematical truth. Mathematicians, however, treat mathematical arguments as determining mathematical truth: if an argument is found to describe a witnessably rigorous proof of a theorem, that theorem is considered—until the need for further examination arises—to be true. The anthropological question is how mathematicians, as a practical matter and as a matter of mathematical practice, make such determinations. This paper looks first at the ways that the logic of mathematical argumentation comes to be realized and substantiated by provers as their own immediate, situated accomplishment. The type of reasoning involved is quite different from deductive logic; once seen, it seems to be endemic to and pervasive throughout the work of human theorem proving. A number of other features of proving are also considered, including the production of notational coherence, the foregrounding of proof-specific proof-relevant detail, and the structuring of mathematical argumentation. Through this material, the paper shows the feasibility and promise of a real-world anthropology of disciplinary mathematical practice. [ABSTRACT FROM AUTHOR]

The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach's fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019]. [ABSTRACT FROM AUTHOR]

This paper belongs to a field of research that has appeared comparatively recently, namely, the study of mathematics competitions. The paper utilizes historical-theoretical methodology and is devoted to changes in the way in which mathematics competitions have been conducted in Russia in recent decades. Mathematics competitions, like mathematics education as a whole, are seen as part of a social process, and changes in them are seen as expressions of new social needs and possibilities. The analysis of the system of competitions, as it evolved in the USSR, and the literature devoted to them, makes it possible to identify certain challenges recognized by the mathematics community. In this paper I discuss examples of new competitions that have emerged to address these challenges. It is noted that new types of competitions have emerged, including team competitions and competitions involving mass participation by students. Also discussed is the emergence of new competitions with subject matter closer to that of ordinary school mathematics than that of old, traditional competitions. An analysis of the situation in Russia reveals the existence of a significant group of mathematics educators, parents, and children interested in the development of mathematics competitions. Also, in the paper I call attention to certain questions that I consider worthy of further study. [ABSTRACT FROM AUTHOR]

There is a dearth of research comparing the effects of iPad technology and paper-and-pencil-delivered interventions on student mathematics outcomes. Nine studies have compared intervention modalities such as computer-mediated instruction and teacher-mediated instruction to examine differences in student performance, but only two of these have used iPad technology to do so and neither examined student fact fluency as the outcome. The purpose of this study was to examine the effects of an iPad-based versus a paper-and-pencil-based flashcard intervention on the basic fact fluency of four second-grade boys within a Midwestern US charter school. Using an adapted alternating treatments design, three conditions were compared: iPad-delivered flashcards, paper-and-pencil-based flashcards, and a control condition. The results suggest that for three of four students, there was no difference in gains between treatment conditions, and both were more successful than the no-treatment control. Descriptive data on the acceptability and number of opportunities to respond between intervention modalities are described. [ABSTRACT FROM AUTHOR]

The goal of this paper is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artés et al. (Mem Am Math Soc 134:639, 1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincaré disc modulo limit cycles in this family, and Artés et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification of planar quadratic systems of codimension two in the structural stability. Combining the sets of codimension-one quadratic vector fields one to each other, we obtain ten new sets. Here we consider set AA obtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of set AA and prove their realization. We got 34 new topologically distinct phase portraits in the Poincaré disc modulo limit cycles. Moreover, in this paper we correct a mistake made by the authors in the book of Artés et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) and we reduce to 203 the number of topologically distinct phase portrait of codimension one modulo limit cycles. [ABSTRACT FROM AUTHOR]

This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, ID ^ 1 i , over Heyting arithmetic (HA ), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds ID ^ 1 i into the corresponding theory over Beeson's logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten's result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over HA . [ABSTRACT FROM AUTHOR]

This narrative review of current research on gender and mathematics covers the years 2020 to 2022. The number of exemplary publications within these 3 years and the diversity of topics, theoretical frameworks, subjects, and authors are indications of gender and mathematics remaining a robust and evolving area of study. Of particular interest are studies of lived experiences of students in secondary school and university programs, many of which featured continuing and troubling discriminatory incidents. In addition, the focus in several studies on the intersectionality of gender with race and other identity features is a commendable trend that should be continued. Similar to the transition from sex differences to gender differences that emerged in the 1990s and 2000s, in the papers examined, some authors have reconceptualized gender as a fluid, non-binary construct, although most gender data are reported in binary ways. Gender remains a focus of research in mathematics education because of persistent inequities in achievement, attitudes, representation, and lived experiences. [ABSTRACT FROM AUTHOR]

Lesson planning is of central importance to the teaching of all subjects in school. However, despite its high relevance, there is still a substantial need for a comprehensive review of factors affecting lesson planning. Empirical evidence on how teachers' competence in lesson planning can be developed, what challenges may be encountered during the lesson planning process, and successful lesson planning designs and practices should come to light on. To close this gap the current paper presents the results of a systematic review of 20 empirical research studies on teacher competence in mathematics lesson planning. For detailed insight into the most recent contributions of the reviewed studies on mathematics lesson planning, we analyzed studies conducted during the past decade, adapting the "process model of lesson planning" and the model of "competence as continuum" as a heuristic for approaching lesson planning. We present key results of the studies related to four major themes: (1) dispositions and their influence on developing and implementing lesson plans, (2) quality aspects of lesson plans and the development of lesson planning skills, (3) difficulties in lesson planning, and (4) the relationship between lesson planning skills and performance in implementation of lesson plans. The results of our literature review reveal that teachers (especially novice teachers) face difficulties in lesson planning and their overall competence (and knowledge) are not at an expert level. However, as the results of the examined studies pointed out, teachers can acquire such competence and knowledge through training within initial teacher education and professional development. Overall, teachers need support in planning mathematics lessons by delineating their lesson plan to improve their awareness of students' thinking, possible learning trajectories, effective usage of the curriculum and teaching resources, and the potential of innovative pedagogies that incorporate new technology. [ABSTRACT FROM AUTHOR]

In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed. [ABSTRACT FROM AUTHOR]

In the paper, we will present some important results from the theory of vector-valued invariant means. Discussing the results contained in the author's paper Badora (Ann Pol Math 58:147–159, 1993), we will add new facts that were previously not published. [ABSTRACT FROM AUTHOR]

Advocacy of STEM curricular approaches is based on a concern to engage students in authentic disciplinary and interdisciplinary practices in the STEM (science, technology, engineering, mathematics) disciplines, and the need to promote participation in STEM pathways. The STEM Academy professional learning program was developed to support teachers to engage and motivate students by creating real-world, challenging problems. The initiative involved interdisciplinary teams of secondary STEM teachers attending workshops and working with university experts to design, implement and evaluate STEM curricular experiences. This paper focuses on case studies undertaken in two of the twelve schools involved in the initiative, using interview data from teachers and students, to explore the nature of their programs and their experiences. We investigate key features of these two schools' approaches, using a conceptual framework for integrated STEM, and explore the challenges and benefits of different features of integrated STEM that promote different dimensions of engagement in learning mathematics. We explore how the framework characteristics can be reframed into four dimensions that promote mathematics engagement in integrated STEM education - design thinking in authentic contexts, content integration, STEM practices and 21st Century skills, and exposure to professional practice. [ABSTRACT FROM AUTHOR]

Programming is an interdisciplinary practice with applications in both mathematics and computer science. Mathematics concerns rigor, abstraction, and generalization. Computer science predominantly concerns efficiency, concreteness, and physicality. This makes programming a medium for problem solving that mediates between mathematics and computer science in intriguing ways. Behind programming practices is computational thinking (CT), a mode of thinking involved in formulating and solving problems so that the solutions could be represented and carried out by computing means. In this paper, CT is seen as a boundary object connecting mathematics and computer science in a school problem-solving context. In particular, we examine and analyse middle school students' work upon engaging in mathematical problem solving-in a programming environment, taking CT as a boundary object embedded in the block-based programming environment, Scratch. The analysis is guided by observing boundary crossing features of CT in the students' artefacts produced in Scratch while solving mathematical problems related to symmetry and arithmetic sequence. The findings of this study open up new dimensions to explore CT as a boundary object in integrated STEM pedagogy. [ABSTRACT FROM AUTHOR]

This paper proposes a new data clustering method using the advantages of metaheuristic (MH) optimization algorithms. A novel MH optimization algorithm, called arithmetic optimization algorithm (AOA), was proposed to address complex optimization tasks. Math operations inspire the AOA, and it showed significant performance in dealing with different optimization problems. However, the traditional AOA faces some limitations in its search process. Thus, we develop a new variant of the AOA, namely, Augmented AOA (AAOA), integrated with the opposition-based learning (OLB) and Lévy flight (LF) distribution. The main idea of applying OLB and LF is to improve the traditional AOA exploration and exploitation trends in order to find the best clusters. To evaluate the AAOA, we implemented extensive experiments using twenty-three well-known benchmark functions and eight data clustering datasets. We also evaluated the proposed AAOA with extensive comparisons to different optimization algorithms. The outcomes verified the superiority of the AAOA over the traditional AOA and several MH optimization algorithms. Overall, the applications of the LF and OLB have a significant impact on the performance of the conventional AOA. [ABSTRACT FROM AUTHOR]

Let K and L be two convex bodies in R n , n ≥ 2 , with L ⊂ int K . We say that L is an equichordal body for K if every chord of K tangent to L has length equal to a given fixed value λ . Barker and Larman (Discrete Math. 241(1–3), 79–96 (2001)) proved that if L is a ball, then K is a ball concentric with L. In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. [ABSTRACT FROM AUTHOR]

Due to rapid immigration, many children worldwide are learning mathematics in a second or additional language. This language diversity can be challenging for both teachers and students and carries profound implications for mathematics educators. Research shows that teachers use various ways to support English Language Learners. Research on multilingualism in mathematics classrooms has often focused on qualitative research. This meta-analysis aims to explore the statistically effective successful teaching practices from the studies using quantitative or mixed-method research approaches and aims to inform the research field in a cumulative manner. The specific research question that guided this meta-analysis is: What is the evidence regarding successful teaching of mathematics for Year 1–10 English Language Learners from 2009–2019 in countries where curricula are delivered predominantly in English? Four successful intervention categories were identified: Dual Language Programmes, Curriculum integration, Teacher Professional Development, and Cognitively Focused Interventions. The paper concludes with recommendations for practice and further research in this area. [ABSTRACT FROM AUTHOR]

In this paper, we establish local well-posedness for the Zakharov system on T d , d ≥ 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to ε -loss of regularity when d = 3 and d ≥ 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter's range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies supp f ^ ⊂ { (ξ , τ) ∈ R d + 1 : 1 - 1 N ≤ | ξ | ≤ 1 + 1 N , | τ - | ξ | 2 | ≤ 1 N 2 } for large N ≥ 1 . [ABSTRACT FROM AUTHOR]

In this paper we study a chemotaxis–haptotaxis model with p-Laplacian diffusion and non-flux boundary condition u t = ∇ · (| ∇ u | p - 2 ∇ u) - χ ∇ · (u ∇ v) - ξ ∇ · (u ∇ w) + μ u (1 - u - w) , (x , t) ∈ Ω × (0 , T) , v t = Δ v - v + u , (x , t) ∈ Ω × (0 , T) , w t = - v w , (x , t) ∈ Ω × (0 , T) , where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂ Ω . We prove that when the diffusion parameter is appropriately large, i.e., p larger than some certain value, the global boundedness of the solutions does not depend on the relationship between the coefficient of logistic source term μ and the coefficient of chemotaxis term χ . And we find that the haptotaxis term does not affect the global boundedness of the solutions, in the other words, our result is consistent with the conclusion with no haptotaxis term (Zhuang et al. in Z Angew Math Phys 72:161, 2021). [ABSTRACT FROM AUTHOR]

After about two years of emergency remote teaching during the pandemic, the teaching of mathematics is slowly returning to (what used to be called) normal. However, after the period of mostly teaching online, there is uncertainty about the extent to which we will return to the way we were teaching before. In this survey paper we attempt to give some background to the impact that emergency remote teaching may have had on teaching mathematics. We examine the possible social implications and then focus on the changing mathematics classroom, focusing on the actual mathematics curriculum, learning design and assessment, the role of collaborative activities and social media, educational videos, and the role of family and parents in future. There are indicators from the literature that educators may not return to the traditional way of teaching entirely, especially in secondary and higher education. We conclude with describing some possible new research areas that have developed through emergency remote teaching, including online education for younger learners, local learning ecosystems, the role of family and parents, instructional design, and the mathematics content of curricula. [ABSTRACT FROM AUTHOR]

Birkhoff's representation theorem (Birkhoff, Duke Math J 3(3):443–454, 1937) defines a bijection between elements of a distributive lattice and the family of upper sets of an associated poset. Although not used explicitly, this result is at the backbone of the combinatorial algorithm by Irving et al. (J ACM 34(3):532-543, 1987) for maximizing a linear function over the set of stable matchings in Gale and Shapley's stable marriage model (Gale and Shapley, Am Math Monthly 69(1):9–15 1962). In this paper, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements. We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. To the best of our knowledge, this model generalizes all those for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm. [ABSTRACT FROM AUTHOR]

Let X be a projective variety and D be an effective Cartier divisor on X. In Int J Math 32(12), Paper No. 2140015, 2021, He and Ru author introduced the notion of the Nevanlinna hyperbolicity for the pair (X, D). In this paper, we derive some further properties for the Nevanlinna hyperbolic pairs, as well as provide multiple examples of the Nevanlinna hyperbolic pairs (X, D), mainly for certain blowup varieties. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimal time decay rates for the higher order spatial derivatives of solutions to a diffusive Oldroyd-B model. As pointed out in the Sect. 1.2 of Huang et al. (J Differ Equ 306:456–491, 2022), how to establish the optimal decay estimate for the highest-order spatial derivatives of the solution to this model is still an open problem. Motivated by Wang and Wen (Sci China Math 65:1199–1228, 2022), we give a positive answer to this problem via some delicate analyses on the low- and high-frequency parts of the solution. [ABSTRACT FROM AUTHOR]

Engaging students in processes of mathematizing and active knowledge organization (instead of telling and rehearsing ready-made mathematics) is a key demand for high quality mathematics instruction. Although many mathematically rich context problems have been designed and published in textbooks, their impact on regular mathematics classrooms remains limited, as teachers face challenges in shaping and enacting the important phase of knowledge organization. The 15-year project KOSIMA presented in this paper was aimed at supporting teachers in really enhancing these processes by means of a textbook for Grades 5–10. The paper provides an overview of different studies conducted during the project in order to show how the project developed a research base for the support of teachers. The paper also shows that different research approaches need to be combined to establish an appropriate research base: In iterative design research cycles of topic-specific didactical design research, learning environments were designed and investigated with the aim of understanding the obstacles encountered in implementing the desired processes. In this paper, we focus on an example of one particular obstacle, namely, teachers' need of support in engaging students in processes of active knowledge organization. This obstacle was overcome by developing new types of tasks for active knowledge organization. The theoretical outcome of the design research resulted in two hypotheses for improving relevant features of the design of these tasks, which were tested in two controlled trials. Finally, a longitudinal field study of 312 students gave first indications that the classes that worked with the textbook had significantly higher learning gains than classes using other textbooks. In sum, we can show that textbooks can support changes in classroom practices when the obstacles that occur for both teachers and students are carefully investigated and treated. [ABSTRACT FROM AUTHOR]

This paper concerns the relationship between research and governance policy in three Swedish major development projects in mathematics education: the New Math project (1960–1975), the PUMP project (1970–1980), and the Boost for Mathematics project in (2012–2016). All three projects were driven or financed by the Swedish central school authorities. Using a historical comparative method, this study deepens the understanding of how research co-exists with governance policy when preparing innovations in mathematics education. The main historical sources are official reports and governmental decisions concerning the three projects. The analysis is focused on the nature of the innovations of each project and the role of researchers in the process of creating the innovations. The analysis highlights the theories and the methods involved in those processes. The three projects are also positioned in a context of school governance policy. In Sweden, the prevailing school governing policy changed from a highly centralised governance in the 1960s to a highly decentralised governance in the 2010s. The paper concludes by discussing to what degree the researchers adhered to principles of research or school governance; in particular, the Boost for Mathematics project is considered in this regard. The relevance of the paper in relation to the emerging field of implementation research in mathematics education concerns how historical studies can give new insights about contemporary development projects in mathematics education. [ABSTRACT FROM AUTHOR]

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler. [ABSTRACT FROM AUTHOR]

Studies on conservation of quantum symmetries have been initiated by recent papers (Brannan et al. in Commun Math Phys 376(2):795–839, 2020; Lee et al. in Quantum channels with quantum group symmetry, 2020). We, in this paper, investigate how quantum symmetries can be applied to study additivity questions for covariant quantum channels, particularly for the so-called O N + -Temperley–Lieb channels. Their information-theoretic properties were analyzed in Brannan et al. (Commun Math Phys 376(2):795–839, 2020), but the associated additivity questions remained open due to highly nontrivial structures. We present a new approach to analyze O N + -Temperley–Lieb channels with small perturbations in terms of Bures distance within the asymptotic regime N → ∞ and apply continuity bounds of classical capacity, private classical capacity and quantum capacity to study strong additivity questions. [ABSTRACT FROM AUTHOR]

Boys in the Kingdom of Saudi Arabia consistently and significantly underperform compared to girls across different grades and subjects, forming one of the largest gender gaps in student achievement in the world. Saudi Arabia offers a unique setting in which boys and girls attend separate schools on a universal basis starting from grade 1. This means that boys and girls are educated only by male and female teachers, respectively, in effect inhabiting parallel education systems. In this context, this study examines the factors that are associated with student achievement in mathematics and science in grades 4 and 8 and the extent to which these associations are different for boys and girls, in an effort to gain insights into boys' underachievement in mathematics and science in Saudi Arabia. The paper employs data from two recent large-scale assessments of education: Trends in International Mathematics and Science Study (TIMSS) 2019 and Saudi Arabia's National Assessment of Learning Outcomes (NALO) 2018. A series of hierarchical two-level linear regression models showed that in grade 4, school climate was more strongly associated with boys' compared with girls' achievement in both mathematics and science, with boys attending schools of poorer school climate having a considerably lower performance compared with girls attending such schools. The findings also indicated that although greater literacy and numeracy readiness was linked with higher science achievement among boys and girls, grade 4 boys tended to benefit more from this readiness than girls. In addition, the results show that student absenteeism in grade 4 is particularly strongly associated with decreases in mathematics achievement among boys. In grade 8, significant interactions between gender and the extent to which students feel confident in science, the degree of schools' emphasis on academic success, and teachers' age are observed. The paper concludes by discussing some of the implications of these findings for educators and policy makers in Saudi Arabia. [ABSTRACT FROM AUTHOR]

It is commonly known that the letter M in the two acronyms HPM and STEM stands for mathematics, and it is natural to regard mathematics as playing a significant role in science education. However, it seems that appropriate attention is not usually accorded to the role of M in the area of STEM where mathematics tends to be marginalized. In fact, throughout history one witnesses STEM at work so that the discussion of HPM and STEM in parallel will be beneficial to both. In both the historical and pedagogical aspects, it may be worthwhile to note that history of mathematics affords some means to mirror several different mathematical worlds in the context of HPM in a pluralistic way in order to offer a fuller view of the subject. In this paper I examine this issue through sampling many examples gleaned from the history of mathematics, particularly the role of M in the world of science and technology. [ABSTRACT FROM AUTHOR]

The inclusion of games in mathematics programmes is widely believed to foster the enjoyment of mathematics. The focus of this paper is on fluctuations in emotional climate during the playing of whole-class mathematics games. A multimethod approach drawing on the sociology of emotions was employed to explore changes in the classroom emotional climate that were associated with game playing. The event-oriented inquiry was conducted with two teachers and a class of 10- to 13-year-olds in a New Zealand classroom during mathematics sessions. Over a series of eight mathematics lessons, there were three noticeable fluctuations in emotional climate, all of which occurred during whole-class games. Our analysis of these three events identified a successful interaction with dramatic emotional energy associated with a positive emotional climate, a successful interaction with undramatic emotional energy associated with positive emotional climate, and an unsuccessful interaction associated with negative emotional climate with interactional repair. The third event also illustrated how the incomplete nature of a game's rules can provide an opportunity for a negative emotional climate to be associated with game playing. The taken-for-granted wisdom that whole-class mathematics games can enhance emotional aspects of a classroom learning environment is supported by some of our evidence. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to accurately solve the singular value problem and the linear system for a class of Polynomial-Vandermonde-type (PVT) matrices, which belongs to the class of negative matrices introduced by Huang and Xue (Adv Comput Math 47:73, 2021), and these negative matrices arise in some applications such as interpolation problems. In order to parameterize PVT matrices, we present the explicit expressions of initial minors of such matrices. An algorithm is designed to accurately compute the parametrization matrix for PVT matrices. Based on the accurate parametrization algorithm, all the singular values, both large and small, of PVT matrices are computed, and the linear systems associated with PVT matrices are solved to high relative accuracy. Numerical experiments are performed to confirm the claimed high relative accuracy. [ABSTRACT FROM AUTHOR]

The multi-step inertial randomized Kaczmarz (MIRK) method is an iterative method for solving large-scale linear systems. In this paper, we enhance the MIRK method by incorporating the greedy probability criterion, coupled with the introduction of a tighter threshold parameter for this criterion. We prove that the proposed greedy MIRK (GMIRK) method enjoys an improved deterministic linear convergence compared to both the MIRK method and the greedy randomized Kaczmarz method. Furthermore, we exhibit that the multi-step inertial extrapolation approach can be geometrically interpreted as an orthogonal projection method, and establish its relationship with the sketch-and-project method in Gower and Richtárik (SIAM J Matrix Anal Appl 36(4):1660–1690, 2015) and the oblique projection technique in Li et al. (Results Appl. Math. 16:100342, 2022). Numerical experiments are provided to confirm our results. [ABSTRACT FROM AUTHOR]

In this paper, a subgradient projection method for quasiconvex minimization problems is provided. By employing strong subdifferentials, it is proved that the generated sequence of the proposed algorithm converges to the solution of the minimization problem of a proper, lower semicontinuous, and strongly quasiconvex function (in the sense of Polyak in Soviet Math 7:72–75, 1966), under the same assumptions as those required for convex functions with the convex subdifferentials. Furthermore, a quasi-linear convergence rate of the iterates, extending similar results for the general quasiconvex case, is also provided. [ABSTRACT FROM AUTHOR]

The exchange of medical images and patient data over the internet has attracted considerable attention in the past decade, driven by advancements in communication and health services. However, transferring confidential data through insecure channels, such as the internet, exposes it to potential manipulations and attacks. To ensure the authenticity of medical images while concealing patient data within them, this paper introduces a high-capacity and reversible fragile watermarking model in which an authentication watermark is initially generated from the cover image and merged with the patient's information, photo, and medical report to form the global watermark. This watermark is subsequently encrypted using the chaotic Chen system technique, enhancing the model's security and ensuring patient data confidentiality. The cover image then undergoes a Discrete Fourier Transform (DFT) and the encrypted watermark is inserted into the frequency coefficients using a new embedding technique. The experimental results demonstrate that the proposed method achieves great watermarked image quality, with a PSNR exceeding 113 dB and an SSIM close to 1, while maintaining a high embedding capacity of 3 BPP (Bits Per Pixel) and offering perfect reversibility. Furthermore, the proposed model demonstrates high sensitivity to attacks, successfully detecting tampering in all 18 tested attacks, and achieves nearly perfect watermark extraction accuracy, with a Bit Error Rate (BER) of 0.0004%. This high watermark extraction accuracy is crucial in our situation where patient data need to be retrieved from the watermarked images with almost no alteration. [ABSTRACT FROM AUTHOR]

This paper studies the Liouville type theorems for stationary the tropical climate model on the whole space R 3 . It shows that if (u , v , θ) satisfies certain anisotropic integrability conditions on the components of the u or (u , v) , also θ satisfies certain isotropic integrability conditions, there is only a trivial solution to the stationary tropical climate model. The results are a further extension of the recent work by Chae (Appl. Math. Lett. 142:108655, 2023). [ABSTRACT FROM AUTHOR]

It is known that the so-called Beck's conjecture, i.e. the equality of the finite clique and chromatic numbers of a zero-divisor graph, holds for partially ordered sets Halaš and Jukl (Discrete Math 309(13):4584–4589, 2009). In this paper we present a simple direct proof of this fact. Also, the case when the finiteness assumption of the clique number is omitted is investigated. We have shown that the conjecture fails in general and a bunch of counterexamples is presented. [ABSTRACT FROM AUTHOR]

From a physiological perspective, the delineation between steady-state and non-steady-state exercise, also referred to as the maximal metabolic steady state, holds paramount importance for evaluating athletic performance and designing and monitoring training programs. The critical power and the maximal lactate steady state are two widely used indices to estimate this threshold, yet previous studies consistently reported significant discrepancies between their associated power outputs. These findings have fueled the debate regarding the interchangeability of critical power and the maximal lactate steady state in practice. This paper reviews the methodological intricacies intrinsic to the determination of these thresholds, and elucidates how inappropriate determination methods and methodological inconsistencies between studies have contributed to the documented differences in the literature. Through a critical examination of relevant literature and by integration of our laboratory data, we demonstrate that differences between critical power and the maximal lactate steady state may be reconciled to only a few Watts when applying appropriate and strict determination criteria, so that both indices may be used to estimate the maximal metabolic steady-state threshold in practice. To this end, we have defined a set of good practice guidelines to assist scientists and coaches in obtaining the most valid critical power and maximal lactate steady state estimates. [ABSTRACT FROM AUTHOR]