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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]

To reconcile the standard possible worlds model of knowledge with the intuition that ordinary agents fall far short of logical omniscience, a Stalnakerian strategy appeals to two components. The first is the idea that mathematical and logical knowledge is at bottom metalinguistic knowledge. The second is the idea that non-ideal minds are often fragmented. In this paper, we investigate this Stalnakerian reconciliation strategy and argue, ultimately, that it fails. We are not the first to complain about the Stalnakerian strategy. But in contrast to existing complaints, we want to cause trouble for the strategy directly on its home turf. That is, we will advance our objection while granting both the plausibility of the fragmentation component—save for an extreme version of it—and that of the metalinguistic component. Once our central objection to the Stalnakerian strategy is in place, we will show how it negatively affects Adam Elga and Augustín Rayo's recent attempt to apply the Stalnakerian strategy in the context of Bayesian decision theory. [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]

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]

We prove a theorem that generalizes Lemma 4 in the paper of A. S. Mamontov [Sib. Math. J., 54, No. 1, 114-118 (2013)] concerning the validity of the Baer–Suzuki theorem in groups of period 12. The results of the present work can be used in studying groups with a given set of element orders, also called a spectrum. [ABSTRACT FROM AUTHOR]

It has recently been argued that foundationalists, those who take grounding to be well-founded, should not understand the well-foundedness of grounding as the condition that every grounding chain terminates in the downward direction, because this interpretation of well-foundedness fails to correctly classify certain complex grounding structures. Some structures that plausibly would be acceptable to the foundationalist are classified as not well-founded and others that plausibly would not be acceptable to the foundationalist are classified as well-founded. In this paper I provide a characterisation of well-foundedness in terms of termination that correctly characterises all these difficult cases. This acts as a response to these recent arguments. Furthermore, it allows us to better evaluate their importance: these arguments have not shown that foundationalists are wrong to harbour the intuition that grounding chains must terminate in the downward direction, rather, they have shown that foundationalists need to be more clear about this intuition and how it is born out in more complex grounding structures. [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]

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]

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]

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]

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]

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]

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]

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]

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]

This paper introduces two central questions (1) what are available methodologies for making claims about mathematical practice and (2) when and how do claims about mathematical practice influence mathematics instruction. To motivate these questions, we critically analyze the relationship between research into mathematicians' practice and the design of mathematics instruction in three ways. First, we describe three influential research programs in mathematics education and illustrate how each research program uses claims about mathematical practice to inform their instructional goals. Second, by examining these important works, we highlight intrinsic difficulties in investigating mathematical practice. Our conclusion is that every research methodology for investigating mathematical practice is fundamentally limited and we require triangulation from multiple methods and theoretical lenses to fully understand mathematical practice. Third, we highlight reasons for why mathematical practice sometimes should not inform mathematics instruction. We conclude this paper by discussing how the articles in this special issue address our two central questions. [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 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]

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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]

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]

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]

In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure. [ABSTRACT FROM AUTHOR]

For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form x ¨ + λ x l = ϵ f (ω t , x , x ˙) , x ∈ R , where λ is a constant, 0 < ϵ ≪ 1 is a small parameter, l > 1 is an integer, ω ∈ R d is a frequency vector, and f : T d × R 2 → R 1 is real analytic and non-degenerate in x up to a given order p ≥ 0 , i.e., [ f (· , 0 , 0) ] = [ ∂ f (· , 0 , 0) ∂ x ] = [ ∂ 2 f (· , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p - 1 f (· , 0 , 0) ∂ x p - 1 ] = 0 and [ ∂ p f (· , 0 , 0) ∂ x p ] ≠ 0 , where [ ] denotes the average value of a continuous function on T d . In the case that λ = 0 and f is independent of x ˙ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when p = 1 . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that p > 1 is odd. In the case λ ≠ 0 , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when p = 0 . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case λ ≠ 0 by allowing p > 0 . Under the conditions that 0 ≤ p < l / 2 and λ [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 when l - p is even, we obtain a general result which particularly implies the following: (1) If either l is odd and λ < 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 , then as ϵ sufficiently small response solutions exist for each ω satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and λ > 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] < 0 , then there exists an ϵ ∗ > 0 sufficiently small and a Cantor set E ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ E and ω satisfying a Diophantine condition. Similar results are also obtained in the case λ = ± ϵ which particularly concern the existence of large amplitude response solutions. [ABSTRACT FROM AUTHOR]

In the classical bargaining problem, we propose a very mild axiom of individual rationality, which we call possibility of utility gain. This requires that for at least one bargaining problem, there exists at least one player who reaches a higher utility level than their disagreement utility. This paper shows that the Nash solution (Nash in Econometrica 18(2):155–162, 1950) is characterized by possibility of utility gain and continuity with respect to feasible sets together with Nash's axioms except weak Pareto optimality. We also show that in Nash's theorem, weak Pareto optimality can be replaced by conflict-freeness (introduced by Rachmilevitch in Math Soc Sci 76(C):107–109, 2015). This demands that when the agreement most preferred by all players is feasible, this should be chosen. Furthermore, we provide alternative and unified proofs for other efficiency-free characterizations of the Nash solution. This clarifies the role of each axiom in the related results. [ABSTRACT FROM AUTHOR]

This note corrects an error in our paper Anosov flows, growth rates on covers and group extensions of subshifts, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section. [ABSTRACT FROM AUTHOR]

In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings. [ABSTRACT FROM AUTHOR]

In this paper, for d ≥ 1 and s ∈ (0 , d 2) , we study the Bianchi–Egnell quotient Q (f) = inf f ∈ H ˙ s (R d) \ B ‖ (- Δ) s / 2 f ‖ L 2 (R d) 2 - S d , s ‖ f ‖ L 2 d d - 2 s (R d) 2 dist H ˙ s (R d) (f , B) 2 , f ∈ H ˙ s (R d) \ B , where S d , s is the best Sobolev constant and B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d = 1 , there is a neighborhood of B on which the quotient Q (f) is larger than the lowest value attainable by sequences converging to B . This behavior is surprising because it is contrary to the situation in dimension d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d = 1 , Q (f) has no minimizer on H ˙ s (R d) \ B , which again would be contrary to the situation in d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d ≥ 1 . For d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d = 1 we make some numerical observations which support the conjecture stated above. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space (Xia and Wang in Math Ann 385(1–2):863–879, 2023), and Szegö–Weinberger inequalities in rank-1 symmetric spaces obtained by Aithal and Santhanam (Trans Am Math Soc 348(10):3955–3965, 1996). [ABSTRACT FROM AUTHOR]

In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed by Hwang and Singer (Trans Am Math Soc 354(6):2285–2325, 2002) on C n + 1 and on O (- k) . For the metrics in both the families, we prove the existence of an asymptotic expansion for their ϵ -functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on C n + 1 are not projectively induced, and that the Burns–Simanca metric is characterized among the scalar flat metrics on O (- k) to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the ϵ -function vanishes. [ABSTRACT FROM AUTHOR]

The Journal of Mathematics Teacher Education (JMTE) has been publishing research on mathematics teacher education since 1998. The current issue explores the diversity of research in this field, specifically focusing on the objects of study. The concept of relation, proposed within the Anthropological Theory of the Didactic (ATD), serves as a modeling tool for analyzing the various aspects of mathematics teacher education. The nine papers in this issue examine the relation between institutional positions (such as teachers or teacher educators) and objects (such as mathematical concepts or students' thinking) within different educational institutions. The concept of relation allows for a comprehensive understanding of the objects of study in the field of mathematics teacher education. [Extracted from the article]

Autism spectrum disorder (ASD) is a pervasive brain development disease. Recently, the incidence rate of ASD has increased year by year and posed a great threat to the lives and families of individuals with ASD. Therefore, the study of ASD has become very important. A suitable feature representation that preserves the data intrinsic information and also reduces data complexity is very vital to the performance of established models. Topological data analysis (TDA) is an emerging and powerful mathematical tool for characterizing shapes and describing intrinsic information in complex data. In TDA, persistence barcodes or diagrams are usually regarded as visual representations of topological features of data. In this paper, the Regional Homogeneity (ReHo) data of subjects obtained from Autism Brain Imaging Data Exchange (ABIDE) database were used to extract features by using TDA. The average accuracy of cross validation on ABIDE I database was 95.6% that was higher than any other existing methods (the highest accuracy among existing methods was 93.59%). The average accuracy for sampling with the same resolutions with the ABIDE I on the ABIDE II database was 96.5% that was also higher than any other existing methods (the highest accuracy among existing methods was 75.17%). [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Avalanche Principle in [Ann. of. Math. (2), 2001, 154(1): 155–203]. [ABSTRACT FROM AUTHOR]

Over the last decades, digital technologies (DTs) have become ubiquitous in mathematics education. Still, their integration into classroom teaching and learning varies enormously. In this narrative overview, we focus on the different purposes for which DTs are used in mathematics education in order to study how the effectiveness of DTs depends on researchers' underlying goals and perspectives. We set up an experience- and literature-based framework including five different purposes. Applying this framework gave rise to the following results: (1) there is evidence for the benefit of using DTs for mathematics learning; (2) research on DTs leads to new theoretical developments and (3) to new design paradigms; (4) issues of equity with respect to access to and use of DTs are important but under-researched; and (5) DTs challenge curricula and teaching and assessment practices. While early research on the use of DTs focused on questions such as "does it work?" or "does it work better?", the maturing of the field has shifted to more nuanced questions. As a future research agenda, we recommend further study of how the use of DTs in mathematics education impacts the time required for learning as well as the temporality of teaching and learning, how it changes the nature of doing mathematics and the relation to basic skills and higher-order skills in particular, how curricula, teaching practices, and assessment might change due to the availability of sophisticated mathematical tools, how DTs and other resources might be combined in teaching and learning, and how they may help to address equity issues in education. These questions will prompt the development of new theoretical constructs and approaches. [ABSTRACT FROM AUTHOR]

Background: Large scale international assessments depend on invariance of measurement across countries. An important consideration when observing cross-national differential item functioning (DIF) is whether the DIF actually reflects a source of bias, or might instead be a methodological artifact reflecting item response theory (IRT) model misspecification. Determining the validity of the source of DIF has implications for how it is handled in practice. Method: We demonstrate a form of sensitivity analysis that can point to model misspecification induced by item complexity as a possible cause of DIF, and show how such a cause of DIF might be accommodated through attempts to generalize the IRT model for the studied item(s) in psychometrically and psychologically plausible ways. Results: In both simulated illustrations and empirical data from TIMSS 2011 and TIMSS 2019 4th and 8th Grade Math and Science, we have found that using a form of proposed IRT model generalization can substantially reduce DIF when IRT model misspecification is at least a partial cause of the observed DIF. Conclusions: By demonstrating item complexity as a possible valid source of DIF and showing the effectiveness of the proposed approach, we recommend additional attention toward model generalizations as a means of addressing and/or understanding DIF. [ABSTRACT FROM AUTHOR]

Kovaleva and Spieksma (SIAM J Discrete Math 20(3):48–768, 2006) considered the problem of stabbing a given set of horizontal line segments with the smallest number of horizontal and vertical lines. The standard LP relaxation for this problem is easily shown to have an integrality gap of at most 2 by treating the horizontal and vertical lines separately. However, Kovaleva and Spieksma observed that threshold rounding can be used to obtain an integrality gap of e / (e - 1) ≈ 1.58 which is also shown to be tight. This is one of the rare known examples where the obvious upper bound of 2 on the integrality gap of the standard LP relaxation can be improved. Our goal in this paper is to extend their proof to two other problems where the goal is to stab a set R of objects with horizontal and vertical lines: in the first problem R is a set of horizontal and vertical line segments, and in the second problem R is a set of unit sized squares. The proof of Kovaleva and Spieksma essentially shows the existence of an appropriate threshold which yields the improved approximation factor. We begin by showing that a random threshold picked from an appropriate distribution works. This reduces the problem to finding an appropriate distribution for a desired approximation ratio. In the first problem, we show that the required distribution can be found by solving a linear program. In the second problem, while it seems harder to find the optimal distribution, we show that using the uniform distribution an improved approximation factor can still be obtained by solving a number of linear programs. [ABSTRACT FROM AUTHOR]

Teacher expectations not only relate positively to student achievement, but also to student beliefs such as their self-concept. Nevertheless, most studies focus on the relations with student achievement, followed by studies on beliefs. Beliefs are a significant determinant of academic success and can include student self-concept or emotions, such as anxiety. The extent to which anxiety can be influenced by teacher expectations has been investigated in very few studies. This paper examined how teacher expectations related to changes in student achievement, self-concept, and anxiety in mathematics within a school year. The data were from a longitudinal study "Outcomes of teacher education", funded by the Swiss National Science Foundation and based on 28 teachers and 509 primary school students. Teacher expectations were operationalized using the residual approach. Student mathematics achievement was assessed via a standardized mathematics test and their self-concept as well as their mathematics anxiety via a questionnaire. The multi-level structure was considered in the analyses, because the interclass-correlation of student mathematics achievement exceeded the critical value of 10%. The results showed that teacher expectations were positively related to student achievement as well as self-concept and negatively related to anxiety towards mathematics. The change in the explained variance was small (self-concept and anxiety) to large (achievement). The results extend findings on the expectation effect in the classroom, as they focus not only on student achievement but also on student beliefs and are discussed regarding their significance for academic success. [ABSTRACT FROM AUTHOR]

Proximal femur geometry is an important risk factor for diagnosing and predicting hip and femur injuries. Hence, the development of an automated approach for measuring these parameters could help physicians with the early identification of hip and femur ailments. This paper presents a technique that combines the active shape model (ASM) and deep learning methodologies. First, the femur boundary is extracted by a deep learning neural network. Then, the femur's anatomical landmarks are fitted to the extracted border using the ASM method. Finally, the geometric parameters of the proximal femur, including femur neck axis length (FNAL), femur head diameter (FHD), femur neck width (FNW), shaft width (SW), neck shaft angle (NSA), and alpha angle (AA), are calculated by measuring the distances and angles between the landmarks. The dataset of hip radiographic images consisted of 428 images, with 208 men and 220 women. These images were split into training and testing sets for analysis. The deep learning network and ASM were subsequently trained on the training dataset. In the testing dataset, the automatic measurement of FNAL, FHD, FNW, SW, NSA, and AA parameters resulted in mean errors of 1.19%, 1.46%, 2.28%, 2.43%, 1.95%, and 4.53%, respectively. [ABSTRACT FROM AUTHOR]

Today's adolescents are digital natives. Information and knowledge are abundant and easily accessible. Learning a subject perceived to be difficult such as mathematics may present a relatively high cost and emotional burden. This study investigated the relations among cost, academic emotion, and performance in mathematics. A longitudinal survey of two waves was conducted among 754 (52.12% males) seventh grade Taiwanese students at both Time 1 and Time 2 with an average age of 12.59 years (SD =0.53) at Time 1. Mathematics academic emotions, cost, and academic performance were significantly correlated. Cross-lagged analysis revealed that mathematics academic emotions (MAE) predicted students' mathematics performance but indirectly through perceived cost. Academic performance in mathematics was positively associated with subsequent mathematics academic emotions. A loop was detected between MAE and cost as well as cost and academic performance. This paper clarifies the interrelationships among these variables and reveals the crucial role of the perceived cost. Activities that aim to reduce students' perceived cost of engaging in mathematics tasks will improve their learning emotions and mathematics achievement. [ABSTRACT FROM AUTHOR]

The Humble Administrator's Garden is a fine specimen of classical Chinese gardens (CCGs). Researching the Humble Administrator's Garden is helpful to understand the essence of the spatial design of CCGs. This paper puts forward a revised drawing principle for the axial map based on the theory of Space Syntax and compared the garden's spatial structure in the sixteenth and nineteenth centuries in Depthmap. It revealed that: (1) The spatial structure of the two different periods has a common "transportation system" and a series of connected spaces. (2) In the nineteenth century, groves and scenery spots disappeared, while corridors, platforms and courtyards appeared, deepening the spatial depth of the garden. The spatial topology changed from the chain-type to the ring-type. (3) The closed-ring space combination makes it easier for visitors to obtain the information of the global space through the local space, that is, it can make the space more convenient and efficient when visiting the garden. [ABSTRACT FROM AUTHOR]

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P- η -proximal mapping, we prove its equivalence with a class of generalized implicit Wiener–Hopf equations and employ the obtained equivalence relationship and Nadler's technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164–180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

Let (M, g) be a compact Riemann surface with unit area, h ∈ C ∞ (M) a function which is positive somewhere, ρ > 0 , p i ∈ M and α i ∈ (- 1 , + ∞) for i = 1 , ⋯ , ℓ , we consider the mean field equation Δ v + 4 π ∑ i = 1 ℓ α i 1 - δ p i = ρ 1 - h e v ∫ M h e v d μ , on M, where Δ and d μ are the Laplace–Beltrami operator and the area element of (M, g) respectively. Using variational method and blowup analysis, we prove some existence results in the critical case ρ = ρ ¯ : = 8 π (1 + min { 0 , min 1 ≤ i ≤ ℓ α i }) . These results can be seen as partial generalizations of works of Chen and Li (J Geom Anal 1:359–372, 1991), Ding et al. (Asian J Math 1:230–248, 1997), Mancini (J Geom Anal 26:1202–1230, 2016), Yang and Zhu (Proc Amer Math Soc 145:3953–3959, 2017), Sun and Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and h is positive, this is the most important contribution of our paper. [ABSTRACT FROM AUTHOR]

Deflationists hold that it does not seem to us, in experience, as though time robustly passes. There is some recent empirical evidence that appears to support this contention. Equally, empirical evidence suggests that we naïvely represent time as dynamical. Thus deflationists are faced with an explanatory burden. If, as they maintain, the world seems to us in experience as though it is non-dynamical, then why do we represent time as dynamical? This paper takes up the challenge of investigating, on the part of the deflationist, one candidate explanation. We hypothesise that people's belief that the future is what we call deliberatively open partly explains why they represent time as dynamical. In a series of two experiments we test this hypothesis. We find no evidence that beliefs about deliberative openness explain why people represent time as dynamical. Hence there remains an explanatory burden for the deflationist to discharge. [ABSTRACT FROM AUTHOR]

Over the past decade, teaching mathematics ambitiously has received increased attention. In this paper, we argue that to materialize this vision in contemporary classes we need to understand how practicing teachers experiment with certain aspects of this teaching and what challenges they encounter. Toward this end, we focus on an aspect of teaching ambitiously—designing and using enablers and extenders—and examine how four elementary schoolteachers experimented with it in their practice while also participating in a video-club setting. Drawing on a corpus of data including lesson plans, videotaped lessons, pre- and post-lesson interviews, end-of-program interviews, and videotaped video-club sessions, and looking across the four cases, we sketch how these teachers worked with enablers and extenders and the challenges they faced. Our analysis helped identify certain components entailed in working with enablers and extenders during the phases of lesson planning and enactment; it also yielded a classification of observed and reported challenges encountered as teachers engage with this work. This mapping of work associated with designing and using enablers and extenders, along with the classification of challenges generated, can inform professional learning development attempts aiming to support teachers enact ambitious teaching by identifying and naming separate components of practice that merit consideration and by providing insights into the types of scaffolds needed to support teachers in teaching ambitiously. [ABSTRACT FROM AUTHOR]