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In the direction towards the question when mixed multiplicities are equal to the Hilbert–Samuel multiplicity of joint reductions, this paper not only generalizes Viet et al. (Proc Am Math Soc 142:1861–1873, 2014, Theorem 3.1) that covers the Rees's theorem Rees (J Lond Math Soc 29:397–414, 1984, Theorem 2.4), but also removes the hypothesis that joint reductions are systems of parameters in Viet et al. (2014, Theorem 3.1). The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert–Samuel multiplicity of joint reductions become closer. [ABSTRACT FROM AUTHOR]

An old problem since Leray (J Math Pure Appl (French) 9:1–82, 1933) asks whether homogeneous D-solutions of the 3 dimensional Navier–Stokes equation in R 3 or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two cases: (1) the full 3 dimensional slab case R 2 × 0 , 1 with Dirichlet boundary condition (Theorem 1.1); (2) when the solution is axially symmetric and periodic in the vertical variable (Theorem 1.3). Also, for the slab case, we prove that even if the Dirichlet integral has some growth, axially symmetric solutions with Dirichlet boundary condition must be swirl free, namely u θ = 0 , thus reducing the problem to essentially a "2 dimensional" problem. In addition, a general D-solution (without the axial symmetry assumption) vanishes in R 3 if, in spherical coordinates, the positive radial component of the velocity decays at order -1 of the distance. The paper is self contained comparing with (Carrillo et al. in Funct Anal, 2020. 10.1016/j.jfa.2020.108504) although the general idea is related. [ABSTRACT FROM AUTHOR]

The aim of the paper is to argue for the cognitive unity of the mathematical results ascribed by ancient authors to Thales. These results are late ascriptions and so it is difficult to say anything certain about them on philological grounds. I will seek characteristic features of the cognitive unity of the mathematical results ascribed to Thales by comparing them with Galilean physics. This might seem at a first sight a rather unusual move. Nevertheless, I suggest viewing the process of turning geometry into an axiomatic-deductive science as a process of idealization in mathematics that is parallel to the process of idealization in physics. In Kvasz (Acta Phys Slovaca 62:519–614, 2012) I offered an epistemological reconstruction of the process of idealization in physics during the scientific revolution of the seventeenth century. In the present paper I try to employ these epistemological insights in the process of idealization in physics and propose a reconstruction of the cognitive unity of the mathematical results ascribed to Thales, who can, on the basis of these ascriptions, be seen as one of the initiators of idealization in mathematics. [ABSTRACT FROM AUTHOR]

As argumentation is an activity at the heart of mathematics, (not only) German school curricula request students to construct mathematical arguments, which get evaluated by teachers. However, it remains unclear which criteria teachers use to decide on a specific grade in a summative assessment setting. In this paper, we draw on two sources for these criteria: First, we present theoretically derived dimensions along which arguments can be assessed. Second, a qualitative interview study with 16 teachers from German secondary schools provides insights in their criteria developed in practice. Based on the detailed presentation of the case of one teacher, the paper then illustrates how criteria developed in practice take a variety of different aspects into account and also correspond with the theoretically identified dimensions. The findings are discussed in terms of implications for the teaching and learning about mathematical argumentation in school and university: An emphasis on more pedagogical criteria in high school offers one explanation to the perceived gap between school and university level mathematics. [ABSTRACT FROM AUTHOR]

The metaphors that students form and encounter have been shown to exert a powerful influence on how they think about mathematics. In this paper, we explore the linguistic metaphors about learning and doing mathematics that were prevalent in 11 advanced mathematics lectures. We present four metaphor clusters that were common in the corpus that we analyzed: Learning Mathematics is a Journey, Doing Mathematics is Work, Mathematics is Discovery, and Presenting Mathematics is a Story. We conclude the paper by discussing the implications of the metaphors that mathematicians use and students hear in their mathematics courses. [ABSTRACT FROM AUTHOR]

In a beautiful paper, Deligne and Illusie (Invent Math 89(2):247–270, 1987) proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Kato (in: Igusa (ed) ALG analysis, geographic and numbers theory, Johns Hopkins University Press, Baltimore, 1989) generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections developed in Arinkin et al. (Algebraic Geom 4:394–423, 2017) and the author of this paper to give a new, geometric interpretation of the Hodge theorem for the logarithmic de Rham complex. [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]

In the literature different concepts of compatibility between a projective structure P and a conformal structure C on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. In an often cited paper (Ehlers et al. in: O'Raifertaigh (ed) General Relativity, Papers in Honour of J.L. Synge, Clarendon Press, Oxford, 2012) Ehlers/Pirani/Schild introduce still another criterion which is natural from the physical point of view: every light like geodesics of C is a geodesics of P . Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned (Trautman in Gen Relativ Gravit 44:1581–1586, 2012); (Vladimir in Commun Math Phys 329:821–825, 2014); as reported by Scholz (in: A scalar field inducing a non-metrical contribution to gravitational acceleration and a compatible add-on to light deflection, 2019). Here it is proved that the conjecture of EPS is correct. [ABSTRACT FROM AUTHOR]

This paper discusses the place of precision in mathematics education by exploring its role in curricular guidelines and in classroom life. By means of a joke on precision delivered by a school student in South Sweden, our study focuses on student participation in mathematical tasks that require precision in processes of measuring and reasoning. The paper uses theories on humour and inferentialism to revisit the normative place of "precision" in mathematics classroom discourse. [ABSTRACT FROM AUTHOR]

Recent evidence demonstrates that mathematics knowledge in the early years is an important contributor to students' learning and later achievement in mathematics. As early mathematics education has assumed heightened importance, the quality of early mathematics teaching and learning experiences has attracted global attention. Several researchers have designed and used a number of instruments to measure and portray the quality of mathematics teaching in early childhood classrooms. As this body of work is growing fast, it is imperative to explore similarities and differences among the existing research tools that aim to measure the quality of early mathematics instruction. This paper takes a step towards exploring such similarities and differences by reviewing six observation instruments designed to measure early mathematics teaching quality. Therefore, the aim of this paper is twofold: (1) to provide an up-to-date overview by examining these measures' theoretical bases, foci, and psychometrics; and (2) to examine the similarities and differences between the reviewed tools. [ABSTRACT FROM AUTHOR]

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation { u t = ∇ ⋅ ( u p ∇ u u 2 + | ∇ u | 2 ) − χ ∇ ⋅ ( u q ∇ v 1 + | ∇ v | 2 ) , x ∈ Ω , t > 0 , 0 = Δ v − μ + u , x ∈ Ω , t > 0 , where Ω : = B R (0) ⊂ R n (n ∈ N ) is a ball with some R > 0 , and χ > 0 , p , q ≥ 1 , μ : = 1 | Ω | ∫ Ω u 0 and u 0 is an initial data of an unknown function u . Bellomo–Winkler (Trans. Am. Math. Soc. Ser. B 4, 31–67, 2017) established existence of an initial data such that the corresponding solution blows up in finite time when p = q = 1 . This paper gives existence of blow-up solutions under some condition for χ and u 0 when 1 ≤ p ≤ q . [ABSTRACT FROM AUTHOR]

In this paper, some results on the complete moment convergence of extended negatively dependent (END) random variables are established. The results in the paper improve and extend the corresponding ones of Qiu et al. (Acta Math. Appl. Sin. 40(3):436–448, 2017) under some weaker conditions. Our results also improve and extend the related known works in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we reflect on our experiences teaching and working with mathematics teachers in Canada, Ghana, India and Swaziland to explore challenges and opportunities for creating mathematics tasks for peace and sustainability. Our exploration of these experiences is oriented around our interest in embedding peace and sustainability into mathematics education following the Sustainable Development Goals identified by the United Nations. We claim that attention to local contexts affords mathematics educators a medium for engaging in authentic, meaningful and context-driven mathematics tasks that address issues of local environmental, cultural and societal concerns. However, we argue that globalisation has already colonised local communities. The associated "technoscientificity", along with the conservative nature of textbooks, time constraints and the dominant force of poverty remain hindrances in creating mathematics tasks that are issue centric and socially relevant to address concerns of the local community. We end this paper by suggesting a development of a mathematics task to illustrate the possibility of creating what we call a "situated mathematics task" for students that respond to issues of local community using rural Ghana as a context. [ABSTRACT FROM AUTHOR]

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector—WiiGraph—with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A – B = C. We examine the notion of abstraction on its own, without assuming a dualism abstract-concrete according to which more of one is less of the other. We propose a distinct path for the attainment of abstraction, which involves navigating a surplus of sensible qualities. The work described in this paper belongs to early algebra, we suggest, because it involves the elementary symbolic treatment of unknowns and generals. More broadly, it advances a perspective on the nature of mathematical abstraction. [ABSTRACT FROM AUTHOR]

In a series of papers, Dartyge and Sárközy (partly with other coauthors) studied pseudorandom subsets. In this paper, we study the minimal values of correlation measures of pseudorandom subsets by extending the methods introduced in Alon et al. (Comb Probab Comput 15:1–29, 2006), Anantharam (Discrete Math 308:6203–6209, 2008), Gyarmati (Stud Sci Math Hung 42:79–93, 2005) and Gyarmati and Mauduit (Discrete Math 312:811–818, 2012). [ABSTRACT FROM AUTHOR]

In his seminal paper introducing the fine structure of L, Jensen (Ann Math Log 4:229–308, 1972) proved that under V = L any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen's result that is straight-forward and accessible to those without a knowledge of Jensen's fine structure theory. The proof here instead uses hyperfine structure, a very natural and simpler alternative to fine structure theory introduced by Friedman and Koepke (Bull Symb Log 3:453–468, 1997). [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]

Attempts to better conceptualize and measure teacher knowledge, and through that examine its association to instructional quality have been intensified over the past decades. Despite the progress made, a review of pertinent literature points to four challenges related to studying this association: proximity (i.e., the extent to which the measures of teacher knowledge and instructional quality are proximal to the actual work of teaching); alignment (i.e., the degree to which these measures are aligned to each other); comparability (i.e., the extent to which the instructional interactions and milieu are comparable across the lessons investigated and the degree to which the conceptualization, operationalization, and measurement of teacher knowledge are comparable across studies); and occurrence (i.e., the degree to which instructional settings provide the possibility for certain events to occur to study how teacher knowledge plays out in instruction). Teaching simulations that allow for the investigation of teacher action-related competence—a construct occupying the intermediate space in between static aspects of teacher knowledge and instructional quality—can partly address these challenges. In this paper, I present such a teaching simulation and explain how it can support the capture of teacher action-related competence, by manipulating instructional complexity and providing a productive platform for performing the work of teaching in controlled environments. I partly illustrate the potential of this approach for disentangling and unpacking the focal association, by presenting results from an exploratory study examining the association between preservice teachers' paper-and-pencil mathematical knowledge for teaching (MKT) and their action-related competence yielded by studying their performance in the simulation. [ABSTRACT FROM AUTHOR]

This paper is concerned with a time-periodic and nonlocal system arising from the spread of a deterministic epidemic in multi-types of population by incorporating a seasonal variation. The existence of the critical wave speed of the periodic traveling wavefronts and its coincidence with the spreading speed were proved in Wu et al. (J Math Anal Appl 463:111–133, 2018). In this paper, we prove the uniqueness and stability of all non-critical periodic wavefronts. Of particular interest is the influences of time-periodicity on the spreading speed in one-dimensional case. It turns out that, in comparison with the autonomous case, the periodicity of the infection rate increases the spreading speed, while the periodicity of the combined death/emigration/recovery rate for infectious individuals decreases the spreading speed. We also find that the contact distribution increases the spreading speed. [ABSTRACT FROM AUTHOR]

Fifty years ago, a small one and a half page paper without a single reference was published in the leading journal Nature. The paper laid out the most general mathematical formulation of natural selection that would work for all kinds of selection processes and under any form of inheritance (not just biological evolution and Mendelian genes), although the paper discussed the issue in a genetical framework. Written by a maverick American expatriate in England, with no prior background of studying evolution or genetics, the paper had initially been turned down by the editor of Nature as too difficult to understand. Largely ignored by the evolutionary biology community till the 1990s, the Price Equation is now widely recognized as an extremely useful conceptualization, permitting the incorporation of non-genetic inheritance into evolutionary models, serving to clarify the relationship between kin-selection and group-selection, unifying varied approaches used in the past to model evolutionary change, and forming the foundation of multi-level selection theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonparametric regression model based on extended negatively dependent errors. Some consistency results for the estimator of the regression function g(x) are presented, including the rates of strong consistency and complete consistency, and the mean convergence. The results obtained in this paper improve and extend the corresponding ones of Yang and Wang (Acta Math Appl Sin 22(4):522–530, 1999) and Priestley and Chao (J R Stat Soc B 34:385–392, 1972). Finally, we present a numerical simulation study to verify the validity of the results established here. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with resolvent estimates for the Laplacian Δ in Euclidean spaces. Uniform resolvent estimates for Δ were shown by Kenig et al. (Duke Math J 55(2):329–347, 1987) who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp L p – L q bounds depending on z has not been considered in a general framework which admits all possible p, q. In this paper, we present a complete picture of sharp L p – L q resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner–Riesz operators of negative index. [ABSTRACT FROM AUTHOR]

In this paper, the monotonicity is investigated with respect to Orlicz sequence space l Φ , p equipped with the p-Amemiya norm, and the necessary and sufficient condition is obtained to guarantee the uniform monotonicity, locally uniform monotonicity, and strict monotonicity for l Φ , p . This completes the results of the paper (Cui et al. in J. Math. Anal. Appl. 432:1095–1105, 2015) which were obtained for the non-atomic measure space. Local upper and lower coefficients of monotonicity at any point of the unit sphere are calculated, l Φ , p is calculated. [ABSTRACT FROM AUTHOR]

Background: There is anecdotal evidence that many elementary teachers integrate mathematics lessons and art activities by having students first make colorful, rich material that is subsequently used in an instructional activity. However, it is unclear whether such activities effectively promote learning and transfer of mathematical concepts. The goal of the present research was to examine the use and effectiveness of such "math-and-art" activities on children's ability to acquire basic fraction knowledge. We report the results of a survey of practicing elementary school teachers in the United States, their use of activities involving physical material, and the resources they use for ideas to supplement the standard curriculum. Two experiments examined first-grade students' learning, transfer, and recognition of fraction knowledge from rich, contextualized material versus simple, generic material. Results: The survey results confirm that many U.S. teachers use math-and-art activities and are often inspired by informal sources, such as Pinterest and YouTube. Experiment 1 examined the effectiveness of colorful, contextualized student-constructed material (paper pizzas) versus simple, pre-made material (monochromatic paper circles) in an instructional activity on fractions. Students who used the pre-made circles scored higher than those who used the student-made pizzas on pre-instruction tests of basic fraction knowledge, immediate tests of learning, and delayed tests of transfer. Experiment 2 tested students' ability to spontaneously write fractions to describe proportions of pizzas and circles. Students who answered generic circle questions first were markedly more accurate than those who answered pizza questions first. Conclusions: These findings suggest that rich, contextualized representations, including those made by the student, can hinder students' learning and transfer of mathematical concepts. We are not suggesting that teachers never integrate mathematics and colorful, contextualized material, and activities. We do suggest that elementary students' mathematics learning can benefit when initial instruction involves simple, generic, pre-made material and opportunities for students to make and use colorful, contextualized representations come later. [ABSTRACT FROM AUTHOR]

In this paper, we study the value distribution of L-functions in the Selberg class and concentrate on the uniqueness questions of L-functions that share one or two sets with an arbitrary meromorphic function. The results obtained in this paper improve the recent results due to Q.-Q. Yuan, X.-M. Li, and H.-X. Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J., 58(2):249–262, 2018] and also extend a result due to B.Q. Li [A result on value distribution of L-functions, Proc. Am. Math. Soc., 138(6):2071–2077, 2010]. [ABSTRACT FROM AUTHOR]

This paper will summarize the Newton-Leibniz controversy and how it relates to Green's [24] paper, Green's background and the influence of John Toplis, and the methods we used in exploring the Nottingham Subscription Library archives. Newton and Leibniz each claimed to have discovered calculus first, with Newton publicly accusing Leibniz of copying his ideas in 1711, although Leibniz managed to prevent criminal charges being brought against him (Sastry, [45]). The inclusion of Cavendish suggests very strongly that Green was able to understand I both i Newtonian fluxions and Leibnizian calculus, as well as the variational calculus of Lagrange. Green's use of European calculus in his 1828 essay signaled a bigger change in British mathematics away from Newton's fluxion methods, embracing European calculus to the ultimate benefit of the British mathematical and scientific communities. [Extracted from the article]

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination of context and syntax nuances of relevant sections from the grade 1 Greek series of textbooks and workbooks. Using a conceptual framework of context variation, the analysis shows qualitative differences between equations of similar syntax and provides a nuanced determination of contextual and structural aspects of 'variation' in how the equal sign is presented in elementary mathematics. The paper proposes that since equations have context-specific meanings, context variations should constitute a separate element of analysis when investigating how the equal sign is presented. The implication for practice and future research is that nuanced considerations of equation syntax within varied contexts are needed for elaborating analyses of the equal sign presentation that move beyond dichotomized categorizations of canonical/non-canonical syntaxes. [ABSTRACT FROM AUTHOR]

Proportional reasoning is the ability to use multiplicative thinking and make multiple comparisons. It is known to be challenging for many students and at the same time, many teachers require support to develop sufficient subject matter knowledge and pedagogical content knowledge to teach the diverse concepts that underpin proportional reasoning. The data reported in this paper are drawn from the first year of a broader study aiming to promote the teaching and learning of elements of proportional reasoning across the curriculum by engaging primary school teachers in ongoing professional development that includes the implementation of a series of mathematics interventions, each of which included a research component. This paper focuses on the impact of implementing the interventions on the teachers' mathematical knowledge for teaching. Three structured interventions were implemented by eight teachers (Years 3, 4, 5) during each of three school terms. Data collected showed that engaging in this scaffolded type of practitioner research, the structured nature of the interventions, and reflection on the outcomes of each intervention promoted teachers' subject matter knowledge and pedagogical content knowledge. [ABSTRACT FROM AUTHOR]

Assessment is a crucial factor in the implementation of curriculum reform. Little is known, however, on how curriculum changes can be reflected adequately in assessment, particularly if the reform concerns process skills. This issue was investigated for the case of assessing mathematical thinking in a mathematics curriculum reform for 15–18-year-old students in the Netherlands. From 2011 until 2017, these reform curricula were field tested in pilot schools, while other schools used the regular curricula. The research question is how this reform was reflected in national examination papers and in student performance on corresponding assignments. To address this question, we developed a theory-based model for mathematical thinking, analyzed pilot and regular examination papers, and carried out a quantitative and qualitative analysis of students' work on assignments that invite mathematical thinking. The results were that the pilot examination papers did address mathematical thinking to a greater degree than the regular papers, but that there was a decrease over time. Pilot school students outperformed their peers in regular schools on assignments that invite mathematical thinking by 4–5% on average and showed more diversity in problem-solving strategies. To explain the decreasing presence of mathematical thinking in examination papers, we conjecture that conservative forces within the assessment construction process may push back change. [ABSTRACT FROM AUTHOR]

This paper is devoted to the qualitative properties of discretized parabolic operators, such as nonnegativity and nonpositivity preservation, maximum/minimum principles and maximum norm contractivity. In the linear case, earlier papers of the authors (Faragó and Horváth in SIAM Sci Comput 28:2313–2336, 2006, IMA J Numer Anal 29:606–631, 2009) have established the connections between the above qualitative properties and have given sufficient conditions for their validity. The present paper extends the above results to nonlinear discretized parabolic operators, also motivated by the authors' recent paper (Faragó and Horváth in J Math Anal Appl 448:473–497, 2017), which has given related results on the continuous PDE level. A systematic study is presented, ranging from general discrete mesh operators to proper finite element applications. [ABSTRACT FROM AUTHOR]

The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of propositions and their proofs are considered. In addition I use part of Peirce's semiotics to characterise different types of signs used in mathematical reasoning, distinguishing between symbolic expressions and 2-dimensional diagrams. As a starting point I examine a proposal by Macbeth (Realising reason. A narrative of truth & knowing, Oxford University Press, Oxford, 2014). Macbeth explains how it can be that objects "pop up", e.g., as a consequence of the constructions made in the diagrams of Euclid, that is, why they are fruitful. It turns out, however, that diagrams are not exclusively fruitful in this sense. By analysing the proofs given in the paper I introduce the notion of a 'faithful representation'. A faithful representation represents as either an image (resembling what it stands for) or as a metaphor (sharing some underlying structure). Secondly it represents certain relevant relations (that is, as an iconic diagram in Peirce's terminology). Thirdly manipulations on the representations respect manipulations on the objects they represent, so that new relations may be found. The examples given in the paper illustrate how such representations can be fruitful. These examples include proofs based on both symbolic expressions as well as diagrams and so it seems diagrams are not special when it comes to fruitfulness. Having said this, I do present two features of diagrams that seem to be unique. One consists of the possibility of exhibiting the type of relation in a diagram—or simply showing that a relation exists—as a contrast to stating in words that it exists. The second is the spatial configurations possible when using diagrams, e.g., allowing to show multiple relations in a single diagram. [ABSTRACT FROM AUTHOR]

How to pedagogically integrate the four STEM (science, technology, engineering, and mathematics) disciplines is a main discussion issue in STEM education. Inquiry-based learning, mathematical modeling, and tool-based pedagogy could go hand in hand to provide a possible integrated STEM pedagogical framework for mathematics and science. This discussion paper explores such a boundary pedagogy via the proposal of an inquiry-based modeling pedagogical cycle. The cycle is used as a lens to analyze a sequence of tool-based mathematics school lessons conducted as a science experiment on the topic of estimation. The lesson design is seen through the proposed pedagogical cycles and selected students' work is presented to illustrate students' different problem-solving tracks in a minimalist instruction environment. Pedagogical features that characterize a STEM boundary pedagogical approach are identified. The paper concludes with a discussion on some core values of STEM education. [ABSTRACT FROM AUTHOR]

Presents several Origami-based mathematical problems. Application of the activity of paper folding in solving math puzzles using the angles, geometric shapes and creases of the paper; Construction of paper shapes based on mathematical rules and parameters provided by the puzzle.

Let D be a closed disk in the complex plane centered at the origin and f, g be a complex-valued continuous function on D. Let P[f, g; D] (res. R[f, g; D])) be the uniform closure on D of polynomials (res. rational functions) in variables f and g. In O'Farrell et al. (Bull Lond Math Soc 34(4):490–494, 2002), using complex dynamical systems, O'Farrell and Sanabria-Garcia proved that (z 2 , z ¯ 1 + z ¯ ) : z ∈ D is not polynomially convex when D is sufficiently small and that P z 2 , z ¯ 1 + z ¯ ; D ≠ C (D) if D is sufficiently small. In this paper, we first give certain conditions for rational convexity for the union of two compact sets of C n and apply them to show that R z 2 , z ¯ 1 + z ¯ ; D = C (D) for all sufficiently small Ds. [ABSTRACT FROM AUTHOR]

Following the popularity of the core-EP (c-EP) and weighted core-EP (w-c-EP) inverses, so called one-sided versions of the w-c-EP inverse are introduced recently in Behera et al. (Results Math 75:174 (2020). These extensions are termed as E-w-c-EP and F-w-d-c-EP g-inverses as well as the star E-w-c-EP and the F-w-d-c-EP star classes of g-inverses. The applicability of these g-inverses in solving certain restricted matrix equations has been verified. Several additional results on these classes of g-inverses are established in this paper. In addition, the Moore–Penrose E-w-c-EP inverse and the F-w-d-c-EP Moore–Penrose inverse are proposed using proper expressions that involve the Moore–Penrose inverse and the E-w-c-EP or F-we-d-c-EP inverse. Further, the W-weighted Moore–Penrose c-EP and the W-weighted c-EP Moore–Penrose g-inverses are considered with the aim to extend the considered w-c-EP generalized inverses to rectangular matrices. Characterizations, properties, representations and applications of these inverses are considered. [ABSTRACT FROM AUTHOR]

A challenging open problem in the 3D Hall-MHD theory is to ask whether or not the global weak solutions are smooth. In this paper, we prove that a weak solution is smooth if the diagonal part of the velocity gradient tensor and the non-diagonal part of the magnetic gradient tensor satisfy Ladyzhenskaya–Prodi–Serrin-type conditions. It is physically interesting since the diagonal part of a gradient tensor is related to the deformation while the non-diagonal part is related to the rotation. Moreover, our main theorems improve significantly a criterion in Ye (Comput Math Appl 70(8):2137–2154, 2015) where all entries of the velocity gradient tensor and the magnetic gradient tensor are needed. [ABSTRACT FROM AUTHOR]

Airy and Bessel functions are one of the most popular and important special functions in various branches of physics, mathematics, and engineering. An observation to their behavior for the real argument suggest that they are related. This relation was studied earlier, but were accompanied by a number of assumptions, approximations, and sometimes even misconceptions. This motivated us to develop a fresh and transparent method to establish these relations. As the continuation of our study of the two papers published in resonance already, here we have used the general asymptotic series and the convergent series of these functions and thereby developed two new methods which throw light on the subtle interrelationships between these functions. Numerical evidences of our claims are provided for better clarity and understanding. [ABSTRACT FROM AUTHOR]

Mathematics education is seen as a right for all children, and important to ensure a prosperous future. However, in Australia and other nations, rural students and students from low socioeconomic backgrounds both perform less well in mathematics and are less likely to pursue advanced mathematics. This paper presents a case study of a low socioeconomic status, rural government school that has high engagement and achievement in senior mathematics, despite its setting. The study uses a conceptual framework informed by Appreciative Inquiry and the theory of Practice Architectures to explore the activities and facilitatory elements that have likely contributed to the school's mathematics success. Rather than being attributed to one or two key programmes, the school's mathematics success seemed associated with a collection of whole-school factors. Setting high expectations while providing proactive learning support, differentiating instruction, emphasising the value of mathematics, linking mathematics to careers, and building mathematics teacher capacity were all associated with the school's higher than expected mathematics performance. Rather than hindering the school's mathematics programme, its small size and rural context were used to enable practices that contributed to mathematics success. [ABSTRACT FROM AUTHOR]

This paper is the continuation of the previous one Jiao and Cui (Area-Minimizing Cones Over Grassmannian Manifolds. J. Geom. Anal. 32, 224 (2022). https://doi.org/10.1007/s12220-022-00963-7), where we re-proved the area-minimization of cones over Grassmannians of n-planes G (n , m ; F) (F = R , C , H) , Cayley plane O P 2 from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians G ~ (n , m ; R) by using Lawlor's Curvature Criterion Lawlor (Mem Amer Math Soc 91(446), 1991). Here, we make a further step on showing that the cones, of dimension no less than 8 , over minimal products of G (n , m ; F) are area-minimizing. Moreover, those cones are very similar to the classical cones over products of spheres, and for the critical situation—the cones of dimension 7 Lawlor (Mem Amer Math Soc 91(446), 1991), we gain more area-minimizing cones by carefully computing the Jacobian i n f v d e t (I - t H ij v) . Certain minimizing cones among them had been found from the perspective of R-spaces Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), or isoparametric theory Tang and Zhang (J Differ Geom 115(2):367–393, 2020) recently, and the generic ones in our results are completely new. We also prove that the cones over minimal product of general G ~ (n , m ; R) are area-minimizing, it can be seen as generalized results for some G ~ (2 , m ; R) shown in Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), Tang and Zhang (J Differ Geom 115(2):367–393, 2020). [ABSTRACT FROM AUTHOR]

Convexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-valued fuzzy sets in order to be able to cover some wider area of imprecision. We show some of its interesting properties, and study the preservation under the intersection and the cutworthy property. Finally, we applied convexity to decision-making problems. [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]

Due to the prior work by Köetter and Kschischang (IEEE Trans Inf Theory 54(8):3579–3591, 2008), subspace codes (especially constant-dimension codes, CDC for short) have attracted significant attention in recent years. The sub-code construction is currently one of the most popular constructions, which can be used to further improve the lower bounds for CDCs' construction. In this paper, we show how to improve the construction of subspace codes from two parallel versions of the sub-code construction. This construction allows us to obtain larger size codes for a given minimum distance, which exceeds the latest improvements on the sub-code construction (Cossidente et al. in Adv Math Commun, 2021, https://doi.org/10.3934/amc.2021007; Lao et al. in Cryptogr Commun, 2021) in the cases including A q (10 , 4 , 5) , A q (14 , 6 , 7) , and A q (16 , 8 , 8) , etc. [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]

This article provides a power series summability-based Korovkin type approximation theorem for any sequence of fuzzy positive linear operators. Using the notion of fuzzy modulus of smoothness, we also derive an associated approximation theorem concerning the fuzzy rate of convergence of these operators. Furthermore, through an example, we illustrate that our summability- based Korovin type theorem has an advantage over the fuzzy Korovkin type theorem proved in the seminal paper by Anastassiou (Stud. Univ. Babeş-Bolyai Math L(4):3–10, 2005) [ABSTRACT FROM AUTHOR]

Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students' self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students' self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (nStudents = 170, nTime-points = 596). Nine classes of Norwegian grade 6 (n = 77) and grade 10 students (n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students' self-efficacy across lesson sequences, (b) differences in the starting point and change of students' self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed. [ABSTRACT FROM AUTHOR]

The teaching of Mathematics – in particular, Geometry, through conventional methods has been a challenging task for tutors. Augmented Reality (AR) based applications available in commercial space, have not followed any structured pedagogical approach in the designing process, and also do not ensure that the learning time of students is spent prolifically. In this paper, we explore the use of AR in mathematics for geometry education, to aid visualization of multidimensional objects and long-term retention of concepts by the learners. For designing an appropriate AR application it is necessary to identify some principles which support better memory retention of the students. The application has been specifically designed on the basis of identified principles affecting memory retention. We further explain the development of an AR-based Geometry Learning Assistant (AR-GLA), using a structural approach to pedagogical-design for teaching 3-dimensional geometry to higher school students through improved visualization and enhance their memory retention for related concepts. A sample of 54 K-12 students and 2 teachers with expertise in mathematics were part of the experiment. The students were divided into two different groups; one of the groups was taught with AR-based content whereas the other group was given Interactive Simulation (IS) based learning. The results illustrated that AR-based learning provides better retention of memory as compared to IS-based learning were tested over a period of two months. [ABSTRACT FROM AUTHOR]

In this study, we focussed on designing and evaluating a learning tool for the research process in higher education. Mastering the research process seems to be a bottleneck within the academy. Therefore, there is a great need to offer students other ways to learn this skill in addition to books and lectures. The MethodViz tool supports ubiquitous aspects of the research process in their scientific works higher education students follow. Moreover, the tool facilitates and structures the process interactively. In this paper, we describe the creation process of the artefact and examine the characteristics and scope of MethodViz alongside the traits and ideas of design science research. The evaluation's results are encouraging and show that MethodViz has the potential to improve students' learning achievements. [ABSTRACT FROM AUTHOR]

The large rates of students' failure is a very frequent problem in undergraduate courses, being even more evident in exact sciences. Pointing out the reasons of such problem is a paramount research topic, though not an easy task. An alternative is to use Educational Data Mining techniques (EDM), which enables one to convert data from educational database into useful information, in order to understand and improve teaching and learning processes. In this way, the objective of this paper is to propose mathematical models based on EDM techniques to estimate the probability of a student in a mathematics degree course at IFSP (Federal Institute of São Paulo) to fail in exact sciences disciplines, and later on, indicate which aspects contribute significantly for the Students' failure rates in these branches. We present three logistic regression models that which were applied based on socioeconomic data and student performance over 4 years. For interpretation and evaluation of such models, odds ratio, ten-fold Cross Validation method and the metrics: accuracy, sensitivity, specificity and area under the ROC curve (AUC) were used. It was noted that through Cross Validation, the models achieved accuracy values accounting for over 70%, sensitivity over 70%, specificity over 60% and AUC over 0.75. Analyzing the predictive variables of these models, we identified that factors such as advantage age, rates of failure through the course and attendance in initial semesters can increase the probability of failure in exact science disciplines in the analyzed course. [ABSTRACT FROM AUTHOR]

In this paper we consider implications of the current world-wide inclusion of computational thinking in relation to children's development of algebraic thinking. Little is known about how newly developed visual programming environments such as Scratch could enhance early algebra learning. The study is based on examples of programming activities used by mathematics teachers in Sweden, teaching students aged 10–12 years during the first two years of implementing programming in the mathematics curriculum. Informed by Chevallard's praxeology in terms of praxis and logos, we describe, unpack, discuss and expand these activities. Core issues related to algebra found in the three activities are as follows: making implicit variables explicit; using a counter variable; and identifying parameters as a specific type of variable. Our findings show that, in addition to already identified uses of variables in early algebra, programming activities in the early years bring in new aspects and new ways of treating variables that could, potentially, enhance students' understanding of variables and generalization, provided that programming praxis is embedded in an appropriate algebra logos. [ABSTRACT FROM AUTHOR]