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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 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]

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]

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 aims to investigate the following question: How do students use mathematics in modeling activities? With this purpose, the paper reports on mathematization and use of mathematics, and deals with empirical data with focus on modeling activities performed by students in the first year and in the fourth year of a degree in mathematics. After the description of a theoretical framework of modelling, what can be seen by means of a qualitative analysis is that the perception of messy world situations leads to idealization, and the idealized situation acts as the basis for mathematization in each activity. The mathematization in turn leads to different mathematics concepts, tools and procedures. The use of mathematics that students perform is anchored in their previous experiences, be it in their experiences with the concepts and tools of mathematics, or in their experiences with mathematical modeling practices. Besides that, after this process students have advanced and have expanded their knowledge and made significant progress by means of a balance between teacher guidance and students’ independence. Moreover, beyond mathematical knowledge, the research indicated that success in performing modeling activities also requires knowledge-based mathematical modeling anticipation. [ABSTRACT FROM AUTHOR]

The lace expansion for the Ising two-point function was successfully derived in (Sakai in Commun Math Phys 272:283–344, 2007, Proposition 1.1). It is an identity that involves an alternating series of lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster x-space decay (as the two-point function cubed) above the critical dimension d c ( = 4 for finite-variance models) if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of (Sakai in Commun Math Phys 272:283–344, 2007, Lemma 4.2), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic (Sakai 2007, Lemma 4.2), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in (Sakai 2007, Proposition 4.1) but nonetheless obey the same fast decay above the critical dimension d c . Consequently, the lace-expansion results for the Ising and φ 4 models in the literature are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients. [ABSTRACT FROM AUTHOR]

Properties of arithmetic operations are important in mathematics and play a significant role in elementary school students' learning of mathematics. At the same time, it is quite challenging to help students learn arithmetic operational properties beyond memorization, so that they know when and how to use a specific arithmetic property. This paper reports a study containing two parts to explore students' learning of the Commutative Property of Addition (CPA) as related to the content presentation in textbooks. Part one is to assess selected students' performance on CPA at the end of Grade two, and part two is to explore possible contributions of textbooks to students' performance in Shanghai. Firstly, we evaluated the performance of 41 second-graders on CPA by collecting data through a paper-and-pencil test, which was followed up with individual interviews. Through analyzing how textbooks present and organize this topic, we sought to better understand from a curriculum standpoint how students' performance on CPA is impacted by textbooks. Results revealed that Shanghai textbooks make a positive contribution to students' learning performance. The findings from this study help to shed light on how those textbooks aid student learning, and provide an example for further reflection on student learning of arithmetic operational properties in different curricula. [ABSTRACT FROM AUTHOR]

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form x · π (y) ⊕ h (y) [Maiorana–McFarland (M )] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the M class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in 4 p k variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results. [ABSTRACT FROM AUTHOR]

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo (Commun Contemp Math 6: 765–780, 2004) is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial α -curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial α -curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory (J. Differ. Geom. 109: 223–256, 2018) for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combinatorial α -curvature problem, which generalize the results obtained in Gu-Luo-Sun-Wu (J. Differ. Geom. 109: 223–256, 2018), Xu (Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I. arXiv:1806.04516v2) on prescribing combinatorial curvatures on surfaces. Gu-Luo-Sun-Wu (J. Differ. Geom. 109: 223–256, 2018) conjectured that one can prove Kazdan-Warner's theorems in Kazdan (Ann Math 99: 14–47, 1974), Kazdan (Ann Math 101: 317-331, 1975) via approximating smooth surfaces by polyhedral surfaces. This paper takes the first step in this direction. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up phenomena connected to the following porous-medium problem with gradient terms under Robin boundary conditions: { u t = Δ u m + k 1 u p − k 2 | ∇ u | q in Ω × (0 , t ∗) , ∂ u ∂ ν + γ u = 0 on ∂ Ω × (0 , t ∗) , u (x , 0) = u 0 (x) ≥ 0 in Ω ‾ , where Ω ⊂ R n (n ≥ 3 ) is a bounded and convex domain with smooth boundary ∂Ω. The constants p, q, m are positive, and p > q > m > 1 , q > 2 . By making use of the Sobolev inequality and the differential inequality technique, we obtain a lower bound for the blow-up time of the solution. In addition, an example is given as an application of the abstract results obtained in this paper. Our results can be regarded as an answer to the open question raised by Li et al. in (Z. Angew. Math. Phys. 70:1–18, 2019). [ABSTRACT FROM AUTHOR]

Another significant research area relevant to teacher awareness and its development is teacher reflection on teaching. In this process the role of the researcher/mathematics teacher educator involves development of awareness in mathematics, teaching, teacher education and research. [Extracted from the article]

In this paper, we report an enquiry into elementary preservice teachers' learning, as they engage in doing mathematics for themselves. As a group of researchers working in elementary Initial Teacher Education in English universities, we co-planned and taught sessions on growing pattern generalisation. Following the sessions, interviews of fifteen preservice teachers at two universities focused on their expressed awareness of their approach to the mathematical activity. Preservice teachers' prospective planning and post-teaching evaluations of similar activities in their classrooms were also examined. We draw on aspects of enactivism and the notion of reflective "spection" in the context of teacher learning, tracing threads between preservice teachers' retro-spection of learning and pro-spection of teaching. Our analysis indicates that increasing sensitivity to their own embodied processes of generalisation offers opportunities for novice teachers to respond deliberately, rather than to react impulsively, to different pedagogical possibilities. The paper contributes a new dimension to the discussion about the focus of novice elementary school teachers' retrospective reflection by examining how deliberate retrospective analysis of doing mathematics, and not only of teaching actions, can develop awarenesses that underlie the growth of expertise in mathematics teaching. We argue that engaging preservice teachers in mathematics to support deliberate retrospective analysis of their mathematics learning and prospective consideration of the implications for teaching can enable more critical pedagogical choices. [ABSTRACT FROM AUTHOR]

The complete forcing number of a graph G is the cardinality of a minimum subset of E(G) to which the restriction of every perfect matching M is a forcing set of M. In a previous paper (He et al., J Math Chem 59:1767–1784, 2021), we presented a complete forcing set of a hexagonal system in terms of elementary edge-cut cover, and a lower bound for the complete forcing number of a normal hexagonal system by matching numbers of some subgraphs of its inner dual graph. In this paper, we show that the complete forcing number of a normal hexagonal system without 2 × 3 subsystems attains the above lower bound. As an example, we present an expression of the complete forcing numbers of pyrene systems. Besides, for a parallelogram hexagonal system, we obtain that its complete forcing number is larger than the lower bound by at most 1. [ABSTRACT FROM AUTHOR]

This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” (IBE) as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths. [ABSTRACT FROM AUTHOR]

In this paper, we use two approaches to define the concept of fuzzy nominal sets: classic and universal algebraic. We see that the fuzzy nominal sets obtained using the universal algebraic approach (so-called fuzzy nominal sets) are within finitely supported mathematics, whereas the fuzzy nominal sets derived using the classical approach (so-called fuzzy nominal ν supp -sets) are within ordinary mathematics and each fuzzy nominal set can be considered as a fuzzy nominal ν supp -set. We also go over the presheaf representation of fuzzy nominal sets and some other properties of these various types of fuzzy nominal sets. [ABSTRACT FROM AUTHOR]

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph K m , n , extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr (K 10 , n) ≥ 4.87057 n 2 - 10 n , cr (K 11 , n) ≥ 5.99939 n 2 - 12.5 n , cr (K 12 , n) ≥ 7.25579 n 2 - 15 n , cr (K 13 , n) ≥ 8.65675 n 2 - 18 n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite. [ABSTRACT FROM AUTHOR]

In this paper, we explore the version of Hairer's regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct and stochastically estimate the renormalized model postulated in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039), avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up. [ABSTRACT FROM AUTHOR]

Automatic generation of math word problems (MWPs) is a challenging task in Natural Language Processing (NLP), particularly connecting it to real-life problems because it can benefit students in developing a higher level of mathematical thinking. However, most of the MWPs are presented within a scholastic setting in a decontextualized way. This paper describes a prototype system that generates authentic math word problems (i.e., real-life problems) using authentic contextual recognition with controlled difficulty levels. Our innovative approach includes the acquisition of authentic contextual information through recognition technology, an instructional-based prompt generator for three different difficulty levels, and question generation through the Generative Pre-trained Transformer (GPT) model. We evaluated the performance of the system in terms of the quality of generated questions using automatic evaluation and human evaluation. Further, we assessed the usability of the system using heuristic evaluation. The automatic evaluation showed our generated MWPs were relevant for geometry topics and varied in sentence generation. The human evaluation found our system generated more realistic problems with satisfactory language quality and mathematical validity. Our system also produced more questions with higher difficulty levels based on human evaluation. Heuristic evaluation captured the usability of the system and highlighted the potential to be applied in a pedagogical context for mathematics learning. [ABSTRACT FROM AUTHOR]

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg–Witten functional that in particular includes the Kapustin–Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais–Smale compactness. [ABSTRACT FROM AUTHOR]

In this study, we explore the valuation challenge posed by American options subject to regime switching, utilizing a model defined by a complex system of parabolic variational inequalities within an infinite domain. The initial pricing model is transformed into a linear complementarity problem (LCP) in a bounded rectangular domain, achieved through the application of a priori estimations and the introduction of an appropriate artificial boundary condition. To discretize the LCP, we employ a finite difference method (FDM), and address the resulting discretized system using a primal-dual active set (PDAS) strategy. The PDAS approach is particularly advantageous for its ability to concurrently determine the option's price and the optimal exercise boundary. This paper conducts an extensive convergence analysis, evaluating both the truncation error associated with the FDM and the iteration error of the PDAS. Comprehensive numerical simulations are performed to validate the method's accuracy and efficiency, underscoring its significant potential for application in the field of financial mathematics. [ABSTRACT FROM AUTHOR]

Picard's theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard's theorem and entire solutions of (i) difference equation Δ η f + a (z) P (f) = 0 in C and (ii) differential equation D k f + h (z) g (f (z)) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the boundedness of solutions of the Ginzburg–Landau system where and is the subelliptic operator In the stationary case, where the solutions are independent of the time variable, our result can be seen as an extension of some results in [A. Farina, B. Sciunzi, and N. Soave, Commun. Contemp. Math. 22 (5), Article no. 1950044 (2020)] from the Laplace operator to the subelliptic operator . [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads ∂ t u − F (u) (− Δ) s / 2 u + (− Δ) s / 2 (u F (u)) = 0 , x ∈ T d , with s ∈ (0 , 1 ] . We are interested in solutions stemming from periodic positive bounded initial data. The given function F ∈ C ∞ (R +) must satisfy F ′ > 0 a.e. on (0 , + ∞) . For instance, all the functions F (u) = u n with n ∈ N ∗ are admissible non-linearities. The local theory can also be developed on the whole space, however the most complete well-posedness result requires the periodic setting. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in L ∞ . We show that any weak solution is instantaneously regularized into C ∞ . We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations, in particular (Ann. Fac. Sci. Toulouse, Math. 25(4):723–758, 2016; Ann. Fac. Sci. Toulouse, Math. 27(4):667–677, 2018). [ABSTRACT FROM AUTHOR]

In this paper, we introduce some versions of relative connectedness of subspaces of a topological space and we give some facts and relations among them. We prove that these relative versions satisfy some of the classical properties of connectedness. Additionally, we apply our results to the theory of hyperspaces, aiming to address a general problem posed by Arhangel'skii (Comment Math Univ Carolin 36:305–325, 1995, Problem 3). [ABSTRACT FROM AUTHOR]

Nearly 30 years after the end of apartheid in South Africa, the education system has yet to provide learners with an adequate school experience. With school closures during the pandemic, the urgency to provide a quality education in the townships became even more dire. In collaboration with the leadership of Charles Duna Primary School in Gqeberha, South Africa, Partnerships for Educational Excellence and Research (PEER) International reestablished a partnership to design a flexible implementation of methods and curricula to build numeracy repertoires. Professional development efforts leverage evidence-based best teaching and learning strategies included in the Morningside Model of Generative Instruction (Johnson et al., 2020). Weekly videoconferences target the development of basic computation skill acquisition and math problem solving skills. The dramatically different environment and available resources in township schools is a prevailing consideration in this transfer of instructional technology. Our project started early in 2022 and continues with optimism and determination. The purpose of this paper is to describe our constructional approach (Goldiamond, 1974/2002 Behavior and Social Issues, 11(2), 108-197) when invited to improve classroom conditions for teachers and learners. [ABSTRACT FROM AUTHOR]

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein's approach had roots in Poncelet's principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann's theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker's work had a strongly qualitative character that imbued Klein's early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet's earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). [ABSTRACT FROM AUTHOR]

We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work (Raphaël and Rodnianski in Publ Math Inst Hautes Études Sci 115:1–122, 2012) of Raphaël and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time with the blow-up rate λ (t) = (T - t) e - | log (T - t) | + O (1) . In this paper, we show that this e O (1) -factor in fact converges to the universal constant 2 e - 1 , and hence these solutions contract at the universal rate λ (t) = 2 e - 1 (T - t) e - | log (T - t) | (1 + o t → T (1)) . Our proof is inspired by recent works on type-II blow-up dynamics for parabolic equations. The key improvement is in the construction of an explicit invariant subspace decomposition for the linearized operator perturbed by the scaling generator in the dispersive case, from which we obtain a more precise ODE system determining λ (t) . [ABSTRACT FROM AUTHOR]

Some of the key challenges for STEM education in Turkey include the problems encountered by teachers when integrating diverse disciplines and technologies into STEM activities, the need for teachers to acquire the skills necessary to prepare and implement appropriate STEM activities for the educational level of their students and their school context, the difficulties faced by teachers when following the Engineering Design (ED) problem-solving process in the implementation of STEM activities, and teachers' heavy workloads, which impede the collaboration required to develop and implement STEM activities. A field study was carried out in Samsun, Turkey between 2017 and 2018 to develop a Design Thinking (DT) approach customized to support STEM teachers. The field study adopted an action research methodology and involved two main studies with fifth-grade teachers and students, facilitated by a researcher-designer. Qualitative methods including co-design workshops, focus groups, interviews, and observation were employed to gather data. The paper presents a seven-step DT approach, customized to support pre-service and in-service secondary school teachers in collaboratively designing and implementing classroom STEM activities. The results indicate that structuring the STEM activity design as a specialized DT process provides a complementary framework. Collaboration among teachers in a structured STEM activity design process supports the integration of disciplines and the development of STEM activities that correspond to the students' level of knowledge and ability. Empathy building, which facilitates familiarization with other teachers and students, enhances collaboration among teachers and aids the development of STEM activities relevant to the students' level. [ABSTRACT FROM AUTHOR]

This paper finds new quasi-polynomials over Z for the number p k (n) of partitions of n with parts at most k. Methods throughout are elementary. We derive a small number of polynomials (e.g., one for k = 3 , two for k = 4 or 5, six for k = 6 ) that, after addition of appropriate constant terms, take the value p k (n) . For example, for 0 ≤ r < 6 and for all q ≥ 0 , p 3 (6 q + r) = p 3 (r) + π 0 (q , r) , a polynomial of total degree 2 in q and r. In general there are M ⌊ k / 2 ⌋ = lcm { 1 , 2 , ... , ⌊ k / 2 ⌋ } such polynomials. In two variables q and s, they take the form ∑ a i , j q i s j with a i , j ∈ Z , which we call the proper form for an integer-valued polynomial. They constitute a quasi-polynomial of period M ⌊ k / 2 ⌋ for the sequence (p k (n) - p k (r)) with n ≡ r (mod M k) . For each k the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on k. A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over Q for p k (n) that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At k = 6 we find serious errors in Sylvester's calculation of his "waves." Sylvester JJ (Q J Pure Appl Math 1:141–152, 1855). The results are generalized to the (not very different) problem called "making change," where significant improvements to existing approaches are found. We find an infinitude of new congruences for p k (n) for k = 3 , 4 , and one new one for k = 5 . Reduced modulo m the periodic sequence (p k (n)) is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case. [ABSTRACT FROM AUTHOR]

In this paper we develop a Nash-Moser iteration type reducibility approach to prove the (inverse) localization for some d-dimensional discrete almost-periodic operators with power-law long-range hopping. We also provide a quantitative lower bound on the regularity of the hopping. As an application, some results of Sarnak (Comm Math Phys 84(3):377–401, 1982), Pöschel (Comm Math Phys 88(4):447–463, 1983), Craig (Comm Math Phys 88(1):113–131, 1983) and Bellissard et al. (Comm Math Phys 88(4):465–477, 1983) are generalized to the power-law hopping case. [ABSTRACT FROM AUTHOR]

The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu (in Commun Math Phys 350(2):699–747, 2017) via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect. [ABSTRACT FROM AUTHOR]

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Γ -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Γ -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004). [ABSTRACT FROM AUTHOR]

In the educational context, one of the main goals is to reduce the disparities among students, generally at the national level, to allow all individuals to achieve a similar cultural background. Using data from a large-scale standardised test administered by INVALSI (National Institute for the Evaluation of the Educational System), this paper offers a first longitudinal analysis of the performance in the maths test of a cohort of students enrolled in 2013/2014 at grade 8 and observed up to grade 13. The aim is to identify those obstacles that undermine students' learning to help adopt informed educational actions. Specific features of these data are their hierarchical structure and the presence of not vertically scaled scores. Two approaches have been followed for their analysis: growth models and growth percentiles. Coherently with the literature, our results suggest the presence of a gender gap, a significant impact of the type of school, and of social-cultural background. Differently from previous research on the INVALSI data, we evaluate these time-invariant covariates' effects on students' performance over different school cycles. [ABSTRACT FROM AUTHOR]

In the paper Targino (Outer Lipschitz Geometry of Complex Algebraic Plane Curves. International Mathematics Research Notices, rnac202. https://doi.org/10.1093/imrn/rnac202, 2022) Renato Targino shows that bi-Lipschitz type of a plane curve is determined by the local ambient topological properties of curves. Here we show that it is no longer true in higher dimensions. However we show that the bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane. [ABSTRACT FROM AUTHOR]

This paper examines how different approaches in mathematics education conceptualise the relationship between school mathematics and university mathematics. The approaches considered here include: (a) Klein's elementary mathematics from a higher standpoint; (b) Shulman's transformation of disciplinary subject matter into subject matter for teaching; and (c) Chevallard's didactic transposition of scholarly knowledge into knowledge to be taught. Similarities and contrasts between these three approaches are discussed in terms of how they frame the relationship between the academic discipline and the school subject, and to what extent they problematise the reliance and bias towards the academic discipline. The institutional position implicit in the three approaches is then examined in order to open up new ways of thinking about the relationship between school mathematics and university mathematics. [ABSTRACT FROM AUTHOR]

Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers—by which we mean teachers of students with ages, approximately, of 12–18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue. [ABSTRACT FROM AUTHOR]

Creating sustained, transformative change within and across organizations is challenging, particularly when those undertaking change act as individuals. COMmunities of Practice (CoPs) are organically created collaborations among like-minded participants, working toward a common set of goals (Lave & Wenger, 1991; Wenger-Trayner & Wenger-Trayner, 2014). CoPs offer an avenue for members to connect individuals across various boundaries. In this paper, we investigate the ways in which regional CoP leaders experience value participating in their community, using the Communities for Mathematics Inquiry in Teaching (COMMIT) Network as our unit of study. The COMMIT Network is a grant funded project aimed at engaging mathematics faculty at institutions of higher education in regional CoPs around teaching with inquiry. In this study we examine the experiences of CoP leaders nested within this network setting. We interviewed 19 leaders from eight United States regions to understand their perceptions of individual and collective value participating in the regional CoP and COMMIT Network structures. We framed our study on Wenger et al. (2011). Promoting and assessing value creation in communities and networks: A conceptual framework. Open University of the Netherlands.) Value Framework. Our findings show that leaders found Immediate Value as individuals participating in a collaborative, supportive CoP environment and they found Realized Value in terms of the impact their CoP could make on instructional practices, both in their region and the network. An unexpected finding examines how future opportunities for value creation may influence long-term sustainability and transformation of college mathematics instruction. We provide implications for the ways that regional CoPs, along with CoP networks, can provide value for members through such communities. [ABSTRACT FROM AUTHOR]

Despite the documented negative outcomes accompanying sexual harassment, the experience of sexual harassment among STEM faculty members remains underexamined. In this paper, we explore how two sexual harassment variables—gender harassment and sexual harassment climate—are linked to four facets of faculty well-being: job burnout, turnover intentions, psychological distress, and self-rated physical health. Using data from STEM faculty at a mid-sized university located in the upper Midwest (N = 117 faculty members), we find gender harassment is associated with lower self-rated physical health and higher turnover intentions among women STEM faculty. In contrast, gender harassment was not significant in predicting well-being outcomes among men STEM faculty. Instead, the sexual harassment climate features more prominently in the experiences of STEM men faculty, with the perception that sexual harassment charges will be treated seriously (the sexual harassment climate) being negatively related to men's job burnout and psychological distress and positively related to men's self-rated physical health. Taken together, our findings extend the existing literature by documenting that outcomes for STEM men faculty are more strongly shaped by the perceived institutional climate surrounding sexual harassment, whereas women's outcomes are more intricately linked to experiencing gender harassment. [ABSTRACT FROM AUTHOR]

We consider the sine-Gordon (SG) equation in 1 + 1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and ϕ 4 at the level of linear Bäcklund transformations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some sharp upper and lower bounds for the spectral radii of nonnegative k-uniform tensors (resp., nonnegative tensors) by using the i-th q-times-average slice sum which was introduced by C. Lü, L. H. You and Y. F. Huang [AIMS Math., 2020, 5(3): 1799–1819]. When k = 2, q =1, our results can deduce the main results of D. P. Huang and L. H. You [J. Appl. Math., 2016, 3: 1–7]. We also find that the upper bounds of nonnegative k-uniform tensors we obtained can not be compared with the upper bounds presented by [AIMS Math., 2020, 5(3): 1799–1819] by numerical examples. As applications, we obtain some sharp upper bounds for the α-spectral radii of k-uniform hypergraphs (resp., k-uniform directed hypergraphs). [ABSTRACT FROM AUTHOR]

The Weyl law of the Laplacian on the flat torus T n is concerning the number of eigenvalues ≤ λ 2 , which is equivalent to counting the lattice points inside the ball of radius λ in R n . The leading term in the Weyl law is c n λ n , while the sharp error term O (λ n - 2) is only known in dimension n ≥ 5 . Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators H V = - Δ g + V in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531)) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau's heat kernel estimates, Blair–Sire–Sogge's eigenfunction estimates, and Duhamel's principle for the wave equation. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve X were investigated and the sets of minimal generators were determined for all points in X (F q 2) and X (F q 6) \ X (F q 2) . This paper completes their work by settling the remaining cases, that is, for points in X (F ¯ q) \ X (F q 6) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in X (F q 7) \ X (F q) and we give a bound on the Feng–Rao minimum distance d ORD . For q = 3 we provide a table that also reports the exact values of d ORD . As a further application we construct quantum codes from F q 7 -rational points of the GK-curve. [ABSTRACT FROM AUTHOR]

Over the last three decades, educational researchers and policymakers have increasingly promoted instructional strategies that centralize group work in mathematics. One difficulty teachers face in implementing group-based instruction in mathematics involves facilitating meaningful group interaction amongst students. In this paper, we explore student noticing as a novel strategy for supporting college students to collaborate effectively during group work in mathematics. First, we construct a noticing framework named student noticing of collaborative practices which provides a lens for "seeing" how students notice their collaborative practices. Then, we use the framework to explore how 25 college students noticed their collaborative practices in mathematics. After working on a novel mathematics task in groups, the college students listened to audiorecordings of their group interactions and responded to reflection questions about the effectiveness of their collaboration. We identified themes regarding how and what students noticed related to their collaborative practices. The findings reveal that students attended to many aspects of their collaboration, including their talking turns and propensity to listen to others. Students demonstrated a desire to change their collaborative practices in the future. The findings imply that teachers and researchers might leverage student noticing as a tool for improvement in mathematics group-based classrooms. [ABSTRACT FROM AUTHOR]

In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics, 225, 2455–2463 (2010)]. That is to say: What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest ∣G∣? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices G of a given semimodular lattice L and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice G equal the length of L and the number of non-zero join-irreducible elements of L, respectively. Therefore, we solve the problem on the best cover-preserving embedding of a given semimodular lattice raised by Gábor Czédli and E. Tamás Schmidt. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for d ≥ 1 , there exist C 0 , C 1 > 0 , depending on ‖ m ‖ L 1 only, such that C 0 d ≤ supremum of total population ≤ C 1 d. However, when replaced by random diffusion, a conjecture, proposed by Wei-Ming Ni and justified in Bai et al. (Proc. Am. Math. Soc. 144:2161–2170, 2016), indicates that in the one-dimensional case, supremum of total population = 3 ‖ m ‖ L 1. This reflects serious discrepancies between models with local and nonlocal dispersal strategies. Furthermore, we provide an equivalent characterization about the combination of resource distribution and diffusion rate such that the corresponding total population could reach the optimal order d as d goes to infinity. [ABSTRACT FROM AUTHOR]

Mathematical education, both mass education, and university education of non-mathematicians, are in an abominable state, and rapidly degrading. We argue that the instruction of non-mathematicians should be dramatically reformed both as substance and style. With traditional approach, such a transformation would take decades, with unclear results. But we do not have this time. The advent of Computer Algebra Systems gives the mathematics community a chance to reverse the trend. We should make a serious attempt to seize this opportunity. In the present paper we describe one such project of reform implemented at the St Petersburg State University. [ABSTRACT FROM AUTHOR]

We consider the content and methods of teaching mathematics in primary school in the context of digitalization. The necessity of a significant updating of the curriculum is shown. In particular, this implies the use of digital tools as an object of study and a way of significantly improving the content and effectiveness of learning mathematics to prepare children for the future life and work. [ABSTRACT FROM AUTHOR]