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This paper considers a specific autonomous Schwarzian differential equation given by Sf=c(f−σ1)(f−σ2)/[(f−τ1)(f−τ2)]$$\begin{equation*} \hspace*{35pt} S_f=c(f-\sigma _1)(f-\sigma _2)/[(f-\tau _1)(f-\tau _2)] \end{equation*}$$and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Choquard equation with weight (−Δ)su=1|x|N−α∗h(x)|u|ph(x)|u|p−2uinℝN,$$ {\left(-\Delta \right)}&#x0005E;su&#x0003D;\left(\frac{1}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{N-\alpha }}\ast h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;p\right)h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}&#x0005E;N, $$where 02s,p>2,α>0$$ 02s,p>2,\alpha >0 $$ and h$$ h $$ is a positive weight function satisfying h(x)≥C|x|a$$ h(x)\ge C{\left&#x0007C;x\right&#x0007C;}&#x0005E;a $$ at infinity, for some a≥0$$ a\ge 0 $$. We establish, in this paper, a Liouville type theorem saying that if maxN−4s−2a,0<α

We show that if G$G$ is a finitely generated group such that its profinite completion Ĝ$\widehat{G}$ is "far from being projective" (i.e., the kernel of the universal Frattini cover of Ĝ$\widehat{G}$ is not a small profinite group), then the class of existentially closed G$G$‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is "far from being projective," the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper. [ABSTRACT FROM AUTHOR]

Background: Despite its potentially significant functional and emotional impact, acalculia is still too rarely assessed and managed by speech and language therapists. Research on the rehabilitation of numerical transcoding remains scarce in the literature and, despite positive results, presents a low level of evidence. Aims: The present study aims to evaluate the effectiveness of a targeted rehabilitation of numerical transcoding in two patients suffering from a chronic secondary acalculia. Methods & Procedures: Two post‐brain injury females with secondary acalculia took part in a single‐case experimental design with multiple baseline across subjects according to a three‐phase experimental protocol: baseline involving global cognitive rehabilitation (5–7 measurements with randomized sequential introduction); targeted intervention (10 measurements); follow‐up (2 immediate measurements and 1 month after the end of the intervention). Repeated outcome measures consisted of six lists composed of numbers of equivalent difficulty that were used alternately to assess numerical transcoding. We used a reverse digit span as a control measure to assess the specificity of the intervention. Rehabilitation lasted 5 weeks and consisted of errorless learning with colour cues, tables and number–words cards. Outcomes & Results: During baseline period involving global cognitive rehabilitation, transcoding scores remained unchanged. In contrast, there was a significant improvement in scores for both patients during the intervention phase targeting transcoding and maintenance of benefits 1‐month post‐intervention. Conclusions & Implications: This study demonstrates that a specific rehabilitation targeting numerical transcoding following chronic secondary acalculia can be effective in improving transcoding skills. WHAT THIS PAPER ADDS: What is already known on the subject: Transcoding difficulties in patients with acalculia can cause a significant disability in everyday life activities. In secondary acalculia, rehabilitation of cognitive functions associated with number processing (attention, working memory, language) is not sufficient for improvement of transcoding. What this paper adds to existing knowledge: An intervention specifically targeting numerical transcoding significantly and durably improves the skills of patients with chronic secondary acalculia. What are the potential or actual clinical implications of this work?: Procedural error‐free intervention using colour cueing, tables, cards with number–words, copy and repetition seems effective to improve transcoding skills in chronic acalculia. [ABSTRACT FROM AUTHOR]

This paper extends some results of Mattila (J. Fractal Geom. 66 (2021) 389–401 and Ann. Acad. Sci. Fenn. A Math. 42 (2017) 611–620), in particular, removing assumptions of positive lower density. We give conditions on a general family Pλ:Rn→Rm,λ∈Λ$P_{\lambda }:\mathbb {R}^n\rightarrow \mathbb {R}^m, \lambda \in \Lambda$, of orthogonal projections which guarantee that the Hausdorff dimension formula dimA∩Pλ−1{u}=s−m$\dim A\cap P_{\lambda }^{-1}\lbrace u\rbrace =s-m$ holds generically for measurable sets A⊂Rn$A\subset \mathbb {R}^{n}$ with positive and finite s$s$‐dimensional Hausdorff measure, s>m$s>m$. As an application we prove for Borel sets A,B⊂Rn$A,B\subset \mathbb {R}^{n}$ with positive s$s$‐ and t-dimensional$t{\text{-dimensional}}$ measures that if s+(n−1)t/n>n$s + (n-1)t/n > n$, then dimA∩(g(B)+z)⩾s+t−n$\dim A\cap (g(B)+z) \geqslant s+t - n$ for almost all rotations g$g$ and for positively many z∈Rn$z\in \mathbb {R}^{n}$. We shall also give an application to the estimates of the dimension of the set of exceptional rotations. [ABSTRACT FROM AUTHOR]

We establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics. [ABSTRACT FROM AUTHOR]

Background: Game‐based learning can frame problem‐solving as a sense‐making experience with domain‐specific tasks for school students. However, multiple challenges arise when trying to support learners in such a complex, problem‐oriented learning environment. Objectives and Methods: With an architecture‐themed mathematics learning game, we conducted two mixed‐method studies to explore the impact and design of game‐based mathematical experience on the math problem‐solving performance of middle school students. Results and Conclusions: The study findings suggested a positive impact of game‐based math experience on math problem‐solving for middle school students. Problematization‐oriented game‐based math tasks with structuring features enhanced students' reasoning with problems and channelled it to doing mathematics. Takeaways: The current research findings support the initiative to frame learning as a sense‐making experience with domain‐specific tasks and inform the design of game‐based mathematical experience and learning support. Lay Description: What is already known about this topic: Realistic mathematical practices frame learning as a sense‐making experience.Challenges arise when involving learners in realistic mathematical experiences driven by domain‐specific tasks. What this paper adds: Game‐based simulation of math tasks framed learning as sense‐making experience.Game‐based math experiencing promoted math contextual problem‐solving performance.Structuring features enhanced problematized experiencing with math tasks. Implications for practice and/or policy: Teachers can integrate and facilitate game‐based math experiencing in class.Efforts should be made to make game‐based math tasks meaningful for students. [ABSTRACT FROM AUTHOR]

Background: Even though monitoring and control enactment are key aspects of self‐regulated learning (SRL), Adaptive learning technologies (ALTs) may reduce the need for learners to monitor and control their learning. Personalized dashboards are effective in supporting learners' monitoring and can potentially support control behaviour. Allowing learners to enact control over their learning process, seems to hold potential for increasing their motivation. Objectives: Therefore, this study's aim was to investigate the relation between control enactment and motivation. We examined how learners enacted control while learning with an ALT with personalized dashboards and examined the relation between learners' enactment of control and their motivation. Methods: Seventy‐eight primary school learners (Grade 5) participated. During the lesson, learners worked on mathematics in the ALT and concurrently were shown personalized visualizations that supported monitoring and enacting control over their learning process. Learners could enact control to change problems' difficulty to easy, medium, or hard. Motivation was measured before and after learning. Results: The SEM analyses showed that how learners enacted control was related to their motivation. Choosing difficult problems was related to more enjoyment and competence while choosing easy problems related to more pressure and tension. Learners who complied with the suggested difficulty level experienced less choice, but also less pressure/tension and more enjoyment and competence. Conclusions: These results provide avenues to account for broader learner characteristics like SRL and motivation to optimize learning. This way, hybrid systems in which control enactment is a shared responsibility of the system and learner, can be improved to support SRL development. Lay Description: What is currently known: Monitor and control enactment are key aspects of self‐regulated learning (SRL).Adaptive learning technologies may reduce the need for learners to monitor and control their learning.Personalized dashboards support learners' monitoring behaviour and can potentially also support control enactment.Enactment of control could positively affect learners' motivation. What does this paper add: There is a large variation in how learners enact control and whether they comply with the scaffold.Learners who chose more difficult problems experienced more enjoyment, while learners choosing easy problems, experienced more pressure.Learners complied with the scaffold, experienced less choice, but also less pressure and more enjoyment.This study provides avenues to account for broader learner characteristics like SRL and motivation to optimize learning. Practical implications: Support control enactment by actively involve learners in choice during the learning process.Offer monitor and enactment in conjunction, as learners can only make well‐founded choices if they can monitor their learning process.Intrinsically and extrinsically motivated learners showed different choice patterns and learning experience.It is important to have an understanding and monitoring of learners' motivation. [ABSTRACT FROM AUTHOR]

Open science is receiving widespread attention globally, and preprinting offers an important way to implement open science practices in scholarly publishing. To develop a systematic understanding of researchers' adoption of and attitudes toward preprinting, we conducted a survey of authors of research papers published in 2021 and early 2022. Our survey results show that the United States and Europe led the way in the adoption of preprinting. The United States and European respondents reported a higher familiarity with and a stronger commitment to preprinting than their colleagues elsewhere in the world. The adoption of preprinting is much stronger in physics and astronomy as well as mathematics and computer science than in other research areas. Respondents identified free accessibility of preprints and acceleration of research communication as the most important benefits of preprinting. Low reliability and credibility of preprints, sharing results before peer review and premature media coverage are the most significant concerns about preprinting, emphasized in particular by respondents in the life and health sciences. According to respondents, the most crucial strategies to encourage preprinting are integrating preprinting into journal submission workflows and providing recognition for posting preprints. [ABSTRACT FROM AUTHOR]

Background Study: The effects of the flipped classroom (FC) on students' academic performance remain ambiguous, and the use of pre‐class videos may be one of the main factors hindering students' progress. A software package called GeoGebra has been proposed as a substitute for pre‐class videos to aid students' learning. Objective: This study investigates the effect of the FC using GeoGebra (FCG) on mathematical performance and memory among senior secondary students. Methods: One hundred and sixty students of a senior high school in China were selected. A quasi‐experimental design was implemented in which the classes were split into the intervention group (FCG), control group A (FCV) and control group B (TL) by random assignment. Pretest, Post‐test 1 and Post‐test 2 were introduced. Results and Conclusion: The results indicate that both the FCG and the FC with pre‐class videos (FCV) were more effective than traditional lectures in enhancing students' academic test scores, and that the effect of the FCG was comparable to its video counterpart, F(2, 136) = 3.752, ρ = 0.026, η2 = 0.052. Although their effects were similar, the qualitative results revealed some hidden benefits associated with the replacement of pre‐class video with GeoGebra. Furthermore, although they were equally effective in promoting students' performance in the short‐term, the long‐term effects on academic performance of both forms of FC were non‐significant when compared with traditional lectures. Lay Description: What is already known about this topic: Flipped classroom is an effective strategy in teaching mathematics.Among those challenges, video may be one of the main factors hindering the effect of flipped classroom. What this paper adds: GeoGebra could be an effective alternative to video.Effect between the flipped classroom using GeoGebra and the flipped classroom using video were similar.Both the flipped classroom with GeoGebra and the flipped classroom with videos were more effective than traditional lectures. Implications for practice and/or policy: Students' short‐term performance could be enhanced by visualisation provided by flipped classroom.To enhance students' long‐term performance, an appropriate amount of homework is suggested. [ABSTRACT FROM AUTHOR]

Background: Students often show unfavourable attribution: they attribute poor school performance to stable factors such as lack of ability and good school performance to variable factors such as effort. However, attribution can be influenced by individualized digital re‐attributional feedback leading to positive motivational effects and higher learning outcomes. This is very promising, but it still is unclear, whether this digital re‐attributional feedback can also be successfully integrated in everyday classroom activities. Objectives: The present field experiment investigated how integrating digital re‐attributional feedback into classroom instruction affects student attribution, motivation and learning outcomes. Methods: In the experiment, 8th–10th grade high school students (N = 322) worked with a digital mathematics learning program which was integrated in a three‐week teaching unit. Half the students in each classroom received only standard feedback after each practice task (SF group); the other half received additionally an individual re‐attributional feedback (RF group) after every third task. Attribution, mathematics self‐concept, and self‐efficacy were measured by an online questionnaire twice a week; learning outcomes in mathematics were measured weekly. Results and Conclusion: Hierarchical analyses showed that re‐attributional feedback led to a more favourable attribution in case of success on stable factors. Especially low‐performing students benefited from this feedback. No effects on attribution in case of failure, self‐efficacy or learning outcomes could be found. Further research could investigate if certain adaptations to the digital re‐attributional feedback is more effective in a real classroom setting and has a broader impact on different students. Lay Description: What is already known about this topic: Working with computers in the classroom and at home is becoming more common.Re‐attributional feedback can increase student's motivation when given directly after solving a task and at a high density.Digital re‐attributive feedback can be given in a high density.Re‐attributional training sessions have been shown to be effective in individual training sessions or outside the students' classroom. What this paper adds: The study took place within a real teaching environment during mathematics class for 4 weeks.Digital re‐attributional feedback sessions were directly integrated into the students' online learning which was part of the teaching plan. The implications of study findings for practitioners: Computer‐based re‐attributional training show some limited effects in regular classroom instruction and could be well integrated in student online learning—when adapted accordingly.Low‐performing students show less favourable attribution than better‐performing students in case of success and benefit the most from re‐attributional training. [ABSTRACT FROM AUTHOR]

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

Background: Providing students with worked out problem solutions is a beneficial instructional technique in STEM disciplines, and studying examples that have been worked out incorrectly may be especially helpful for reducing misconceptions in students with low prior content knowledge. However, past results are inconclusive and the effects of incorrect worked examples alone or in combination with correct examples remains unclear. Objectives: We aim to address whether studying incorrect examples alone or in combination with correct examples can support the reduction of students' fraction misconceptions, operationalized as errors made with high confidence. Methods: After incorrectly solving a sampling problem, 130 students in 4th through 11th grade in the U.S. were randomly assigned to a condition in an online problem set focused on fraction equivalence. Students studied either single‐type worked examples (i.e., correct or incorrect; n = 49) or combination‐type worked examples (correct and incorrect; n = 41) or engaged in a problem‐solving control (n = 50). Results: Studying a combination of correct and incorrect worked examples was as effective as the problem‐solving control with feedback at improving fraction equivalence knowledge and reducing the rate of high‐confidence errors. Students in both the combination condition and the problem‐solving with feedback condition outperformed those who studied either correct or incorrect worked examples alone. Conclusions: Results support the inclusion of a combination of correct and incorrect worked examples when teaching students with low prior content knowledge. Studying a combination of example types within an online tutor helps to reduce misconceptions about fractions, a topic students commonly struggle with. A problem‐solving task with corrective feedback worked equally well. Lay Description: What is already known about this topic: Studying worked examples of mathematical solutions improves student learning.Incorrect examples, either alone or in combination with correct examples, are particularly effective for improving learning for students with low prior knowledge in the target content.Comparing correct and incorrect examples is also effective at improving problem‐solving but this effect may be specific to those with high prior knowledge.Still unknown are the impacts of these different combinations of examples on misconceptions in particular. What this paper adds: We show that studying a combination of correct and incorrect examples reduces math misconceptions.Problem‐solving with corrective feedback within an online tutor also has this effect. Implications for practice and/or policy: Using a combination of example types will help reduce students' misconceptions about fractions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the noncommutative analog of the normal subgroup theorem for certain groups. Inspired by Kalantar and Panagopoulos (arXiv:2108.02928, 2021, 16), we show that all Γ$\Gamma$‐invariant subalgebras of LΓ$L\Gamma$ and Cr∗(Γ)$C^*_r(\Gamma)$ are (Γ$\Gamma$‐)coamenable. The groups we work with satisfy a singularity phenomenon described by Bader et al. (Invent. Math. 229 (2022), 929–985). The setup of singularity allows us to obtain a description of Γ$\Gamma$‐invariant intermediate von Neumann subalgebras L∞(X,ξ)⊂M⊂L∞(X,ξ)⋊Γ$L^{\infty }(X,\xi)\subset \mathcal {M}\subset L^{\infty }(X,\xi)\rtimes \Gamma$ in terms of the normal subgroups of Γ$\Gamma$. [ABSTRACT FROM AUTHOR]

In 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions u(·,t)$ \bm{u}(\cdot ,t)$ to the incompressible Navier–Stokes in Rn$ \mathbb {R}^{n}$: if the associated Stokes flows had their L2$L^{2}$ norms bounded by O(1+t)−α$ O(1 + t)^{-\;\!\alpha }$ for some 0<α⩽(n+2)/4$ 0 < \alpha \leqslant (n+2)/4$, then the same would be true of ∥u(·,t)∥L2(Rn)$ \Vert \hspace{0.56917pt} \bm{u}(\cdot ,t) \hspace{0.56917pt} \Vert _{L^{2}(\mathbb {R}^{n})}$. The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form (1+t)−α≲∥u(·,t)∥L2(Rn)≲(1+t)−α$(1+t)^{-\alpha }\lesssim \Vert \bm{u}(\cdot ,t)\Vert _{L^2(\mathbb {R}^n)} \lesssim (1+t)^{-\alpha }$. [ABSTRACT FROM AUTHOR]

Although the educational potential of social media has been widely acknowledged, it remains unclear whether social media use might present opportunities for the development of math identity. This study drew on the socialization framework of math identity to examine the relationships between social media use for math learning, bicultural identity integration, ethnic‐math beliefs and math identity among ethnic minority high school students in China. Structural equation modelling results showed that (a) social media use was positively related to math identity; (b) high bicultural identity integration was positively associated with math identity, while biased ethnic‐math beliefs were negatively related to math identity and (c) social media use suppressed the socialization process of math identity, weakening the maladaptive roles of low bicultural identity integration and biased ethnic‐math beliefs in shaping math identity. Our findings highlight the beneficial role of social media use in student math learning. Theoretical and practical implications are discussed. Practitioner notesWhat is already known about this topicSocial media use for learning is associated with students' academic motivation, engagement and academic achievement.Social media use may have affordances for identity development, especially for ethnic minorities.The formation of ethnic minority students' math identity is intertwined with their math and ethnic socialization.What this paper addsSocial media use for math learning was directly and positively related to ethnic minorities' math identity.This study quantitatively confirmed the associations of bicultural identity integration and ethnic‐math beliefs with math identity.Social media use suppressed the relations of bicultural identity integration and ethnic‐math beliefs with math identity.Implications for practice and/or policyPromoting social media use for math learning benefits ethnic minority students' math identity.Social media use for math learning could alleviate the maladaptive role of low bicultural identity integration and biased ethnic‐math beliefs in math identity development.Ethnic minority students' math identity socialization processes merit attention. [ABSTRACT FROM AUTHOR]

In this paper, we study the non‐Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky (Selecta Math. (N.S.) 4 (1998) 279–320). [ABSTRACT FROM AUTHOR]

There is an irreparable error in the proof of Theorem 3.26 in the above‐mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense. [ABSTRACT FROM AUTHOR]