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Although women's representation in STEM fields and occupations has increased, science and math continue to be stereotyped as male domains. This paper links psychological and sociological explanations for gendered disparities in STEM by examining the relationship between the local "micro-situational" female learning environment and the gender gap in academic self-concept in math and science. We applied hybrid models to TIMSS 2015 data comprised of a pseudo-panel of repeated measures for individual student and peer achievement, academic self-concept, utility value, and interest-enjoyment value in math/science (at age 14). We analyzed data from three countries, including a subsample of students who were taught by the same teacher in both math and science, thus eliminating unobserved teacher heterogeneity. Results indicate that female peer climate in the classroom is important for understanding how girls' self-concept in math/science is formed, even though it was unrelated to the gender gap. [ABSTRACT FROM AUTHOR]

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds (M^{n},g) with boundary and with dimension n<8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static (M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n < 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field. [ABSTRACT FROM AUTHOR]

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

Nowadays, typing is considered as one of the highly important aspects of accessibility that the visually impaired and blinds struggle with the most as existing virtual keyboards are complex and slow. This paper proposes a new text entry method named SwingBoard for visually impaired and blind smartphone users to solve their accessibility problem. It supports a–z, 0–9, 7 punctuations, 12 symbols, and eight keyboard functionalities that are arranged in 8 zones (specific range of angles), four segments, two modes, and different gestures. The proposed keyboard is suitable for the either single-handed or both-handed operation that tracks swipe angle and length to trigger any of the 66 key events. The key triggering process is based on only swiping the finger at different angles with different lengths. Typing speed of SwingBoard is increased by including some effective features such as the quick alphabet and number mode shifting, haptic feedback feature, talkback on swipe to learn the map quickly, and customizable swipe length feature. At the end of 150 one-minute tests, seven blind participants reached an average of 19.89 words per minute (WPM) with an 88% accuracy rate which is one of the fastest-ever recorded average typing speeds for the blind. Almost all users found SwingBoard effective, easy to learn and want to keep using it. SwingBoard is a handy virtual keyboard for visually impaired people with amazing typing speed and accuracy. Rehabilitation would be easier for a visually impaired/disabled person who can adapt the current technology-oriented world and use communication tools easily. As visually impaired and blind people faced a lot of problems in their regular life because of their inability to see things, coping with the evolving world's demands from a person to do a certain task using smart devices would be easier for them with the proposed virtual keyboard topology. As the number of visually impaired and blind users of smartphones rises, faster typing becomes a vital aspect of the smartphone experience. Deaf-blind communities are expanding but solutions for them are not increasing at the same pace because of the limitation of implication. Research on a virtual keyboard with the proposed eyes-free swipe-based typing operation and ears-free reliability on haptic feedback would enable others to create new solutions. [ABSTRACT FROM AUTHOR]

Purpose: This commentary discusses the paper by Reardon et al. (2024; this issue) entitled, "Overcoming implementation challenges through using a train-the-trainer approach to teach numeracy in a special school setting." This commentary outlines the necessary contribution this paper adds to the substantive area of research it is couched within while also identifying potential areas of future research to expand the understanding of this phenomenon and its impact upon practice. Design/methodology/approach: What is fidelity of implementation, how do we measure it and how does it impact our interpretations of experimental findings? This commentary focuses specifically on the nebulous construct of fidelity in experimental studies and how this impacts experimental findings' internal and external validity. Findings: Although fidelity is frequently referenced as an important aspect to consider, the measurement of the construct has been critiqued in experimental studies. To understand if an intervention was "implemented as intended," the core dimensions of the intervention must be considered in the measurement process, as well as potential confounding variables. Originality/value: With an increased need for experimental work to inform what works, for whom and under what conditions, there becomes a need to better investigate the implementation of the intervention in these contexts – thus, fidelity must be reconceptualized. This commentary provides an overview of this dilemma with potential ideas to investigate moving forward. [ABSTRACT FROM AUTHOR]

Mathematics can be used as a tool to question and critique society and, in doing so, give us more information about the world around us and how it operates. This however, is not a common perspective that is conveyed to students during their undergraduate mathematics coursework. This paper contributes to the understanding of how undergraduate mathematics students question and critique society via mathematical modeling tasks. In two courses at two universities, 27 mathematics majors and secondary preservice teachers engaged in the modeling process situated in authentic contexts to learn specific concepts and make mathematical connections across domains and disciplines. Both courses culminated in a final project in which students created and investigated solutions to their own modeling tasks. In this paper, we describe how our courses (1) centered justice as pedagogy, (2) were environments for student agency and exploration, and (3) explicitly demonstrated how mathematics and social justice are intertwined. Drawing on frameworks of mathematical modeling for social justice, we present and analyze student-created tasks to showcase how they utilized mathematics as a tool to question and critique the world around them. [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]

Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629-655, 2017]. Following the pure algebraic approach by Keckic and Lazovic, in the author’s paper [S. Ivkovic, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*- algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semiWeyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivkovic, Banach J. Math. Anal., 17:51, 2023]. [ABSTRACT FROM AUTHOR]

English Teaching Quality Assessment is a crucial aspect of evaluating and enhancing the effectiveness of English language education programs. To achieve a comprehensive understanding of teaching quality, a Fuzzy Comprehensive Evaluation Model is employed. This model combines qualitative and quantitative assessments, considering various factors such as teaching methods, student engagement, curriculum design, and instructor proficiency. Based on the multi-level model of Fuzzy mathematics, this paper designs a set of teaching Quality Assessment training suitable for English Teaching for Medical Students and applies it to teaching courses to improve their understanding of comprehensive functions and provide reference for future online teaching. In terms of curriculum structure, this paper uses Fuzzy analysis method and literature analysis method. Based on the literature, this paper discusses the training module, training theme, training content, training time and training mode. Summarize and organize the consultant's opinions, and finally formulate th e training program. According to the teaching syllabus of comprehensive Quality Assessment student, teaching courseware was prepared, teachers conducted teaching demonstrations and recorded them as high-definition videos, and then cloud classroom teaching platform was built to teach community English. In terms of teaching practice, this subject took the tertiary medical institutions as the object, and investigated and evaluated the qualified primary care staff. Before training, the Fuzzy mathematics multi-level knowledge questionnaire, English Teaching Quality Assessment knowledge questionnaire, English Teaching Quality Assessment attitude, behavioral intention and self-efficacy scale were used to investigate the English Teaching. After completion, the methods were used to investigate English Teaching, and the different degrees of student effects were compared. Finally, a self-designed questionnaire was used to investigate English Teaching to understand the comprehensive evaluation of the satisfaction degree of student courses. [ABSTRACT FROM AUTHOR]

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

The majority of children in mainstream (no fee, poorly resourced, African language dominant) public schools in South Africa fall significantly behind curricular norms in mathematics by Grade 3; the early learning gap grows across time. The provision of quality instructional materials is a well-recognised component of effective strategies to improve early grade mathematics in low- and middle-income contexts. However, there has been little explication of what constitutes quality instructional materials, nor long-term design work to contribute theory to this instructional design challenge. Across 15 years, the author has been working in an education design hub with teacher educators and teachers in the rural Eastern Cape, with the goal of improving early grade mathematics. The gains in mathematics performance have been among the most significant reported in research literature. This paper presents the design principles emerging from the work. Engaging the theoretical layers discussed by diSessa and Cobb, the paper contributes to design theory by emphasising the relationship between explicit instructional assumptions and design principles, integrating four ontological categories (motive, pedagogy, language use and mathematical meaning-making). Since 2011, the Department of Basic Education has provided learner workbooks to support teaching and learning in foundation phase mathematics at system scale. It is hoped that this paper will contribute to the ongoing development of this vital national resource. The paper makes explicit instructional assumptions and design principles (available for critique, adaptation, and refinement), and provides a more nuanced framework through which to assess and improve materials across time. [ABSTRACT FROM AUTHOR]

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. [ABSTRACT FROM AUTHOR]

This paper explores how the twin processes of neoliberalism and neoconservatism work together on, and through, curricula and their associated pedagogies. It bridges the gap between policy and classroom practice, focusing on the particular example of the school subject of mathematics and the notion of mastery, operationalised in the English education system as Teaching for Mastery (TfM). From this context, it develops a theoretical argument using Dean's analytics of government as part of a broader Foucauldian frame, to analyse how TfM is constructed as a particular policy truth. It then shifts the analysis from a wide, social one to the individual classroom level using a psychological argument to critique TfM in its own terms, examining the onto‐epistemological nature of mathematics as a subject. In doing so, it explores ways in which mastery might be problematic in classrooms, even whilst appearing to offer a solution at policy level to long‐standing problems in English schooling. The aim is not to suggest that TfM has nothing to offer, but to point to ways in which it draws on the psychology of teaching and learning in a very particular manner, inscribing pupils with very specific mathematical subjectivities. By providing this insight into how neoliberal policy positions play out at practitioner level via curricula and pedagogies, the paper raises questions which are philosophical, political, and ethical, regarding the potential effect of TfM on teachers' and pupils' experiences of mathematics in schools, including implications for equity of this experience amongst the latter. [ABSTRACT FROM AUTHOR]

The safety of chemicals, drugs, novel foods and genetically modified crops is often tested using repeat-dose sub-acute toxicity tests in rats or mice. It is important to avoid misinterpretations of the results as these tests are used to help determine safe exposure levels in humans. Treated and control groups are compared for a range of haematological, biochemical and other biomarkers which may indicate tissue damage or other adverse effects. However, the statistical analysis and presentation of such data poses problems due to the large number of statistical tests which are involved. Often, it is not clear whether a “statistically significant” effect is real or a false positive (type I error) due to sampling variation. The author's conclusions appear to be reached somewhat subjectively by the pattern of statistical significances, discounting those which they judge to be type I errors and ignoring any biomarker where the p-value is greater than p = 0.05. However, by using standardised effect sizes (SESs) a range of graphical methods and an over-all assessment of the mean absolute response can be made. The approach is an extension, not a replacement of existing methods. It is intended to assist toxicologists and regulators in the interpretation of the results. Here, the SES analysis has been applied to data from nine published sub-acute toxicity tests in order to compare the findings with those of the author's. Line plots, box plots and bar plots show the pattern of response. Dose-response relationships are easily seen. A “bootstrap” test compares the mean absolute differences across dose groups. In four out of seven papers where the no observed adverse effect level (NOAEL) was estimated by the authors, it was set too high according to the bootstrap test, suggesting that possible toxicity is under-estimated. [ABSTRACT FROM AUTHOR]

In an iterated non-cooperative game, if all the players act to maximize their individual accumulated payoff, the system as a whole usually converges to a Nash equilibrium that poorly benefits any player. Here we show that such an undesirable destiny is avoidable in an iterated Rock-Paper-Scissors (RPS) game involving two rational players, X and Y. Player X has the option of proactively adopting a cooperation-trap strategy, which enforces complete cooperation from the rational player Y and leads to a highly beneficial and maximally fair situation to both players. That maximal degree of cooperation is achievable in such a competitive system with cyclic dominance of actions may stimulate further theoretical and empirical studies on how to resolve conflicts and enhance cooperation in human societies. [ABSTRACT FROM AUTHOR]

Background: Sponsors that seek to commercialize new drugs apply to the Food and Drug Administration (FDA) which independently analyzes the raw data and reports the results on its website. Objectives: This study sought to determine if there are differences between the FDA assessments and journal reports on biologic agents developed for the treatment of rheumatoid arthritis. Methods: Available data on FDA-approved drugs were extracted from the website, and a systematic literature search was conducted to identify matching studies in peer-reviewed medical journals. Outcome measures were the American College of Rheumatology response criteria ACR20 (efficacy) and withdrawal due to adverse events (safety). As effect size odds ratios were estimated for each active trial arm vs. control arm (i.e. for both sources: FDA and journal report), followed by calculation of the ratios of the FDA and journal report odds ratios. A ratio of odds ratios not equal to 1 was categorized as a discrepancy. Results: FDA reports were available for 8 of 9 FDA-approved biologic agents for rheumatoid arthritis; all identified trials (34) except one were published in peer-reviewed journals. Overall, discrepancies were noted for 20 of the 33 evaluated trials. Differences in the apparent benefit reporting were found in 39% (24/61) pairwise comparisons and in 11 cases these were statistically significant; the FDA report showed greater benefit than the journal publication in 15 comparisons and lesser benefit in 9. Differences in the reported harms were found in 51% (28/55) pairwise comparisons and were statistically significant in 5. The “signal” in FDA reports showed a less harmful effect than the journal publication in 17 comparisons whereas a more harmful effect in 11. The differences were attributed to differences in analytic approach, patient inclusion, rounding effect, and counting discrepancies. However, no differences were categorized as critical. Conclusion: There was no empirical evidence to suggest biased estimates between the two sources. Increased and detailed transparency in publications would improve the understanding and credibility of published results. Further, the FDA report was found to be a useful source when data are missing in the published report (i.e. reporting bias). [ABSTRACT FROM AUTHOR]

An $(r, \delta)$ -locally repairable code ($(r, \delta)$ -LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al.. An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. , and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of , , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in , , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method. [ABSTRACT FROM AUTHOR]

We correct the proof of Theorem 2 in the mentioned paper concerning finite-difference equilibrium solutions. [ABSTRACT FROM AUTHOR]

We give a corrected proof of the last theorem (Theorem 4.11) of Regularized Riesz energies of submanifolds, published in Math. Nachr. 291 (2018), no. 8–9, 1356–1373. Namely, we prove that the Riesz energy EΩ(z) of a compact body Ω is Möbius invariant if and only if the dimension of Ω is even and z=−2dimΩ. [ABSTRACT FROM AUTHOR]

The article looks at the challenges and issues associated with mathematics of paper folding such as Japanese paper craft origami. Topics discussed include geometric connections between mathematics and origami folding such as symmetries, use of five identical modular pieces for making origami star, and mathematical calculation for making origami.

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

Faithfully flat descent of pseudo-coherent complexes in rigid geometry was proved by Mathew. In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic varieties, which have been introduced by Clausen and Scholze by means of condensed mathematics. [ABSTRACT FROM AUTHOR]

This article describes the outcomes of a faculty partnership between a math professor and a composition professor to improve the final paper in a research-focused senior capstone math course. Much attention has been paid in recent years to the importance of undergraduate research experiences and how to best create and support such endeavors. However, little space has been dedicated to helping math faculty design and scaffold writing assignments to support their students' development as scholars. We describe how our partnership took place, the changes we made to existing assignments and the creation of new ones, and an assessment of our efforts completed by an interdisciplinary group of faculty and students, which showed marked improvement in the areas of information literacy, critical thinking, and quality of writing. [ABSTRACT FROM AUTHOR]

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some questions instructors can ask themselves to design novel and engaging activities when constrained to teaching from a particular curriculum. [ABSTRACT FROM AUTHOR]

Among the problems identified at Technical and Vocational Education and Training (TVET) colleges, low achievement in mathematical subjects is the most prominent one. This paper documents a qualitative case study undertaken at TVET College in Gauteng with the purpose of exploring the National Certificate Vocational (NC(V)) Level 4 students' errors and associated misconceptions when answering optimisation questions in differential calculus. The participants were 60 students who were registered for a course in mathematics. Data were generated from the written student responses to two non-routine test items followed by interviews. Using the Newman error hierarchical model to analyse the data, it was discovered that students errors were conceptual and procedural as well as systematic and non-systematic. The literature also alluded to instructional approaches as some of the causes of students' misconceptions and the errors and that interventions should target students as well as the instructors. [ABSTRACT FROM AUTHOR]

In this paper, we consider a random one-dimensional Schrödinger operator H ω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize H ω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that H ω is actually self-adjoint with probability one. When the white noise in H ω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of H ω . We shall further prove that the negative part of the spectrum of H ω is discrete, with no accumulation points other than 0. [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]

In the paper, we investigate the uniqueness problem of a meromorphic function and its difference operator to the most general setting via two shared set problems and thus improve a recent result of Chen–Chen (Bull Malays Math Sci Soc 35(3): 765-774, 2012). [ABSTRACT FROM AUTHOR]

For $k\geq 2$ and a nonzero integer n , a generalised Diophantine m -tuple with property $D_k(n)$ is a set of m positive integers $S = \{a_1,a_2,\ldots , a_m\}$ such that $a_ia_j + n$ is a k th power for $1\leq i. Define $M_k(n):= \text {sup}\{|S| : S$ having property $D_k(n)\}$. Dixit et al. ['Generalised Diophantine m -tuples', Proc. Amer. Math. Soc. 150 (4) (2022), 1455–1465] proved that $M_k(n)=O(\log n)$ , for a fixed k , as n varies. In this paper, we obtain effective upper bounds on $M_k(n)$. In particular, we show that for $k\geq 2$ , $M_k(n) \leq 3\,\phi (k) \log n$ if n is sufficiently large compared to k. [ABSTRACT FROM AUTHOR]

If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

With constantly changing political landscapes affecting the ability of college students with minoritized identities of sexuality and/or gender (MIoSG; Vaccaro et al., 2015) to thrive on campus, higher educators need to understand student reported stressors to design more inclusive learning environments. Building from minority stress theory and using data from a grounded theory study with 56 collegiate STEM students with MIoSG, this paper documents stressors that students reported as contributing to diminished wellbeing. We used constant comparative grounded theory analysis to identify stressors shared by all participants, which included lack of belonging and invisibility in competitive STEM cultures as well as exclusionary interactions with STEM peers. Recommendations include the design and delivery of holistic education and support services on campus. [ABSTRACT FROM AUTHOR]

For K / F a finite Galois extension of number fields, the relative Pólya group Po (K / F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K / F. The notion of Ostrowski quotient Ost (K / F) , as the cokernel of the capitulation map into Po (K / F) , has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define Po (K / F) and Ost (K / F) which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For E an elliptic curve defined over F , we define the Ostrowski quotient Ost (E , K / F) and the coarse Ostrowski quotient Ost c (E , K / F) of E relative to K / F , for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group Ost c (E , K / F) and we analyze this theorem, in some details, for the class of curves E over quadratic extensions K / F. [ABSTRACT FROM AUTHOR]

In this paper, we prove the convexity of trace functionals (A , B , C) ↦ Tr | B p A C q | s , for parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems. [ABSTRACT FROM AUTHOR]

In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed L 2 -inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted L 2 -inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form [B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904]. [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]

Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

The paper addresses the potential relationships between shared authority in mathematics classrooms and students' mathematical reasoning. Even though tensions and challenges related to shared authority are explicated in the literature, there are few examples of how these issues play out in mathematics teaching. We investigate the case of a mathematics teacher attempting to share authority as well as applying several moves recognized as supporting meaningful student learning. Data has been collected in a fourth-grade Norwegian classroom and is analyzed by means of open coding, inspired by literature. We identify the moves used by the teacher, and we rank these moves along two dimensions: (1) their potential to support mathematical reasoning and (2) their potential for sharing authority. From this, we uncover how a teacher's work of orchestrating mathematical discussions involves moves in all four quadrants, and we discuss how the interplay of moves affects the authority structures and the collaborative reasoning in the classroom. [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]

Some global error bounds with undetermined parameters, which are not always valid, for the extended vertical linear complementarity problems (LCP) of CKV-type matrices and CKV-type B -matrices, are presented by Yan and Wang (Jpn. J. Ind. Appl. Math. 41:129–150, 2024). In this paper, new global error bounds for the extended vertical LCP of CKV-type matrices and CKV-type B -matrices are given, which depend only on the entries of the involved matrices. Numerical examples show that the new bounds are better than those provided in Zhang et al. (Comput. Optim. Appl. 42(3):335–352, 2009) and Wang et al. (Comput. Appl. Math. 40:148, 2021) in some cases. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]