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Mathematical competitions feature in most developed countries as a part of the secondary school experience, but to a lesser extent in developing countries. In this paper we investigate how widespread these competitions are in Africa, both historically as well as currently, focussing both on national Olympiads and on the participation of African countries in competitions beyond their borders, such as the continent-wide Pan African Mathematical Olympiad and the global International Mathematical Olympiad. Since we could not find a consolidated source of information on African competitions, we include many details and links, hoping that this paper will serve as a valuable starting point for future investigators. Next, we present some survey findings on the attitudes towards and experiences of mathematical competitions, both from the perspective of teachers in secondary schools as well as from secondary school students, and whether these experiences have served to improve or degrade their perceptions of mathematics in general. Among other results, we find that teachers view their students' participation in mathematics competitions as being important, with the biggest barrier to participation being that there are not enough competitions at appropriate levels available, and that students enjoy mathematics competitions and are more likely to pursue a STEM career because of them. Finally, we present some suggestions on how this research may be taken further, and on how the state of mathematical competition culture in Africa may be improved. [ABSTRACT FROM AUTHOR]

In this paper, we consider mathematical competitions for pre-university students, such as the International Mathematical Olympiad (IMO) and many national and regional Olympiads following a similar model. The problems proposed in these contests must be solvable by 'elementary' methods (i.e., without using calculus) and belong traditionally to four areas, namely, algebra, combinatorics, geometry, and number theory. Usually there is no explicit curriculum prescribing what parts of these subjects may be involved in the problems, however there is a hidden curriculum (HC) that may be deduced from the problems proposed and from what is considered a complete solution. In this paper, we try to characterize this HC showing typical problems, subjects, solutions, and strategies for each of the main areas. We also make a comparison of the HC with the traditional secondary school curriculum (SSC), with the following two goals: (1) to identify the main points that should be addressed in the preparation of students aspiring to participate in mathematical competitions, and (2) to suggest possible changes to it and diminish the gap between school and contemporary mathematics. [ABSTRACT FROM AUTHOR]

For the Helmholtz equation Δu + k2u = 0 in 2D domain S, there exists a unique solution if k2 is not exactly equal to an eigenvalue λ of the Laplace eigenvalue problem Δu + λu = 0 in S. One important criterion for numerical methods is that there must exist no spurious (i.e., superfluous) eigenvalues so that the above unique solution can be obtained correctly. For exterior problems, the method of fundamental solutions (MFS) using Hankel functions suffers from spurious eigenvalues. New modified Hankel functions have been proposed in Zhang et al. (Appl. Numer. Math. 145, 236–260, 2019) to eliminate all spurious eigenvalues. In this paper, we study bounded multiply connected domains. The MFS algorithms without spurious eigenvalues and their strict analysis are our goals. First, we study bounded simply connected domains by the MFS. The algorithms using Hankel functions are free from spurious eigenvalues. A brief error analysis is provided. Next, we focus on bounded multiply connected domains and choose an annular domain for analysis. The Hankel functions and the modified Hankel functions in [47] are chosen as the exterior fundamental solutions (FS) and the interior FS, respectively. Such combined FS eliminate all spurious eigenvalues, and new error and stability analyses are explored. Bounds of errors involve degeneracy, and those of the condition number involve a gap to eigenvalues. Numerical experiments are carried out to support the analysis made, and better pseudo-boundaries of source nodes are also investigated numerically. The new error and stability analyses in this paper are new and essential, thus providing a solid theoretical basis of the MFS for the Helmholtz equation in 2D bounded simply connected and multiply connected domains. [ABSTRACT FROM AUTHOR]

This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment. [ABSTRACT FROM AUTHOR]

The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) A = (A α β x α β) , where A α β is a 2 × 2 matrix over a field F and x α β is an indeterminate for α = 1 , 2 , ⋯ , μ and β = 1 , 2 , ⋯ , ν . This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (SIAM J Matrix Anal Appl 16(3):719–734, 1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos et al. (Comput Complex 27:561–593, 2018) and Garg et al. (Found. Comput. Math. 20:223–290, 2020), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial O ((μ ν) 2 min { μ , ν }) -time algorithm for computing the symbolic rank of a (2 × 2) -type generic partitioned matrix of size 2 μ × 2 ν . Our algorithm is inspired by the Wong sequence algorithm by Ivanyos et al. for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of A for an arbitrary field F . [ABSTRACT FROM AUTHOR]

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. [ABSTRACT FROM AUTHOR]

Identity research in mathematics education has become increasingly prominent over the past two decades. In the last few years, there have been several reviews of identity literature in the field of mathematics education generally, or specifically focused on mathematics learner identities or mathematics teacher identity. We begin our paper by summarizing the key findings of these reviews, pointing to various categorizations proposed by their authors, and the key critiques raised therein. We then report on our more recent review of identity literature in mathematics education published in the top twenty mathematics education journals over the past 5 years (47 articles in total). This review enables us firstly to speak descriptively about the extent to which the field is evolving (or not) in relation to earlier findings (dominant regions, focus of research, methods and perspectives used). Secondly, following deeper analysis of the papers, we speak to the way in which the field is addressing (or not) key critiques raised by reviews. Our analysis points to several theoretical, methodological and empirical absences and challenges requiring further engagement. We further highlight key absences in the research, particularly a lack of cross-national comparative studies and studies that connect learner and teacher identities. We then provide a summary discussion and analysis of the contributions of the present special issue papers and engage with how these take the field forward. Finally, we engage with the implications of our analysis and recommendations for future directions for research. [ABSTRACT FROM AUTHOR]

In the paper "Why Cryptosystems Fail", Ross Anderson ponders the question about why cryptosystems really fail. Obviously, there may be weak crypto-algorithms, too short key lengths and flawed crypto-protocols. However, these were not the main reason why cryptosystems failed. Anderson discovered that the problem had more to do with misplaced trust and misconceptions of the threats the systems faced. Now, more than 25 years later, it seems prudent to revisit the question of why cryptosystems fail. We investigate the original paper, and evaluate to what extent the situation is similar today. [ABSTRACT FROM AUTHOR]

Until the late 90's, conventional bibliometric indices such as, publication count, citation count, and number of co-authors have extensively been employed by the scientific community to rank the researchers. In 2005, inception of a renowned ranking measure h-index has grabbed the substantial importance; the community considered h-index as a quintessential ranking measure. Subsequently, different variants and extensions of h-index have also been proposed. To date, plethora of the studies exists that harnesses h-index, its variants and extensions for researchers' ranking. Nonetheless, the community does not agree upon a single measure that can be deemed as an optimal ranking measure. This is due to the fact that most of the contemporary studies have evaluated them either by employing a small amount of data or presumed their significance on the basis of hypothetical or imaginary case scenarios. We argue that comprehensive empirical investigation of these measures must be performed in order to tackle their real behavior. This study evaluates the h-index and its extensions by employing a comprehensive data set of authors from Mathematics discipline. The first experimental step involves the computation of correlation among the obtained values of the extensions to determine the similarity and divergence among them. Afterwards, we considered the data of international award winners from four prestigious Mathematics societies as benchmark to validate the potential of these measures by analyzing the dependence of societies on them. The outcomes revealed that overall 45% of the authors have appeared at the top occurrences of the ranked list. Out of all extensions, fraction count on paper has outperformed by bringing 55% of the awardees at top 10% of its ranked list. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the Cauchy problem of inhomogeneous convolution equations of a fairly general nature. To solve the problems posed here, we apply the operator method proposed in some earlier papers by the author. The solutions of the problems under consideration are found using an effective method in the form of well-convergent vector-valued power series. The proposed method ensures the continuity of the obtained solutions with respect to the initial data and the inhomogeneous term of the equation. [ABSTRACT FROM AUTHOR]

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used. [ABSTRACT FROM AUTHOR]

This paper conceptualizes children's mathematical thinking from a materialist perspective on language and mathematics. This perspective considers human and non-human bodies as ontologically equivalent; that is, as both being agentive, vibrant, and animated, thus resisting static representations. This conceptualization is an alternative to the interactionist and language-based models that have dominated research in language and mathematics. This conceptual approach informs the paper's non-hierarchical analysis of a bilingual classroom studying the concept of equivalent fractions. The non-hierarchical nature of this analysis permits a dynamic shift of focus from learners to materials to mathematical concepts. This non-traditional analysis informs the paper's final discussion regarding the nature of children's mathematical thinking. The discussion highlights how working outside language-based interactionist models provides an alternative view of children and artefacts as equal partners working within and as part of the vibrant boundaries between matter and meaning. [ABSTRACT FROM AUTHOR]

Teacher noticing has become increasingly acknowledged as a fundamental aspect of teacher professional competence. Teacher education scholars have examined how the development of noticing might be supported both in initial teacher education and in professional development. In mathematics teacher education, several studies have explored the use of video as a supporting tool for teacher noticing. It remains unclear how this body of work builds on the various theoretical perspectives of noticing prevalent in the literature, thus broadening our understanding of noticing. Furthermore, the field has not examined systematically the extent to which research has leveraged the affordances of digital video technologies, and whether scholars have employed different research methods to answer questions that are critical to teacher educators. This survey paper reviews studies published in the last two decades on programs centered on mathematics teacher noticing that used video as a supporting tool for teacher learning. Thirty-five peer-reviewed papers written in English were identified and coded along three dimensions: (1) theoretical perspectives; (2) use of video technologies; and (3) research questions and methods. This review summarizes important findings and highlights several directions for future research. Most studies involved pre-service teachers, and only a few centered on in-service teachers. Developers of the large majority of programs took a cognitive psychological perspective and focused on the attending/perceiving and interpreting/reasoning facets of noticing. Few studies used video-based software and few studies used grouping, and even fewer used randomized grouping. Evidence of program effects on responding and decision making, and on instructional practice, is limited and should be extended in the future. [ABSTRACT FROM AUTHOR]

In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill's equation based on L p -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill's equations of potential a (t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS. [ABSTRACT FROM AUTHOR]

We consider a class of sparsity-inducing optimization problems whose constraint set is regularizer-compatible, in the sense that, the constraint set becomes easy-to-project-onto after a coordinate transformation induced by the sparsity-inducing regularizer. Our model is general enough to cover, as special cases, the ordered LASSO model in Tibshirani and Suo (Technometrics 58:415–423, 2016) and its variants with some commonly used nonconvex sparsity-inducing regularizers. The presence of both the sparsity-inducing regularizer and the constraint set poses challenges on the design of efficient algorithms. In this paper, by exploiting absolute-value symmetry and other properties in the sparsity-inducing regularizer, we propose a new algorithm, called the doubly majorized algorithm (DMA), for this class of problems. The DMA makes use of projections onto the constraint set after the coordinate transformation in each iteration, and hence can be performed efficiently. Without invoking any commonly used constraint qualification conditions such as those based on horizon subdifferentials, we show that any accumulation point of the sequence generated by DMA is a so-called ψ opt -stationary point, a new notion of stationarity we define as inspired by the notion of L-stationarity in Beck and Eldar (SIAM J Optim 23:1480–1509, 2013) and Beck and Hallak (Math Oper Res 41:196–223, 2016). We also show that any global minimizer of our model has to be a ψ opt -stationary point, again without imposing any constraint qualification conditions. Finally, we illustrate numerically the performance of DMA on solving variants of ordered LASSO with nonconvex regularizers. [ABSTRACT FROM AUTHOR]

Language is a source of power that preserves the status quo in its relationships with learners and mathematics. However, there is a dearth of research that examines how multiple systems of power (e.g., racism, cisheteropatriarchy, colonialism) shape variation in structural inequality and experiences of oppression for language learners. This conceptual paper argues that intersectionality is a promising analytical lens to advance interdisciplinary research on language in mathematics education that nuances knowledge about linguistic justice for racially minoritized and Indigenous learners. To develop this argument, we present insights from a scoping review of research in language education and mathematics education. Our review shows how intersectional analyses can interrogate inequities of language in mathematics education contexts, which are entrenched in ideologies of neutrality that leave exclusionary practices of teaching and learning unchecked. The review also depicts how uptake of intersectionality extends sociopolitical research on language in mathematics education that left interplay between racism, colonialism, linguicism, and other power systems unexamined for nuanced insights on linguistic justice. Following the review, we look across three cases of research studies from our review that vary in terms of their designs and methodologies to illustrate the analytical possibilities of intersectionality in future research on language in mathematics education. We conclude with a discussion of how adopting the lens of intersectionality is necessary to advance interdisciplinary scholarship for linguistic justice. [ABSTRACT FROM AUTHOR]

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in Ai W et al (Math Progr 128(1):253–283, 2011), Huang Y, Zhang S (Math Oper Res 32(3):758–768, 2007), Sturm JF, Zhang S (Math Oper Res 28(2):246–267 2003). The enhanced property can be used to drive some improved results in joint numerical range, S -Procedure and quadratically constrained quadratic programming (QCQP) in the quaternion domain, demonstrating the capability of our new decomposition technique. [ABSTRACT FROM AUTHOR]

Decision-making under uncertainty involves three fundamental components: acts, states of nature, and consequences, as first introduced by Savage (The foundations of statistics, Wiley, New York, 1954). An act is uniquely determined by a number of random consequences that are associated with different states of nature. If the consequences are identical across all states of nature, then the act is state-independent. Prior research on distortion risk measures (DRMs) has primarily focused on state-independent acts. In this paper, we extend the research to state-dependent acts by introducing a state-dependent DRM (SDRM) under the Anscombe–Aumann's framework (Anscombe and Aumannin in Ann Math Stat 34(1):199–205, 1963). The proposed SDRM is the weighted average of DRMs at each state, where the weights are determined by the decision maker's (DM's) subjective probabilities of the states. In situations where there is incomplete information about the DM's true distortion function and/or the true subjective probabilities of the states, we introduce a preference robust SDRM (PRSDRM) for acts. The PRSDRM is based on the worst-case state-dependent distortion function and the worst-case subjective probabilities over a dependent joint ambiguity set constructed with partially available information. To compute the PRSDRM numerically, we show that when the distortion functions are concave, it can be formulated as a biconvex program and further as a convex program by changing some variables. As a motivation and application, we use the PRSDRM for decision-making problems and propose an alternating iterative algorithm for solving it. Finally, we conduct numerical experiments to assess the performance of our proposed model and computational scheme. [ABSTRACT FROM AUTHOR]

The paper deals with the code length distributions for which the proportion of the codewords of a given length to all words of this length is a suitable power of 1/2. We derive a lower bound and an upper bound for the number of prefix codes with the above property. We also investigate the ratio of prefix codes to all uniquely decodable codes, which relates to the more general results of Woryna (Discret Math 340(2):51–57, 2017; Discret Appl Math 244:205–213, 2018). [ABSTRACT FROM AUTHOR]

In this paper we characterize random variables which are stable but not strictly stable in the sense of generalized convolution. We generalize the results obtained in Jarczyk and Misiewicz (J Theoret Probab 22:482-505, 2009), Misiewicz and Mazurkiewicz (J Theoret Probab 18:837-852, 2005), Oleszkiewicz (in Milman VD and Schechtman Lecture Notes in Math. 1807, Geometric Aspects of Functional Analysis (2003), Israel Seminar 2001–2002, Springer-Verlag, Berlin). The main problem was to find the solution of the following functional equation for symmetric generalized characteristic functions φ , ψ : ∀ a , b ≥ 0 ∃ c (a , b) ≥ 0 ∃ d (a , b) ≥ 0 ∀ t ≥ 0 φ (a t) φ (b t) = φ (c (a , b) t) ψ (d (a , b) t) , (A) where both functions c and d are continuous, symmetric, homogeneous but unknown. We give the solution of equation (A) assuming that for fixed ψ , c , d there exist at least two different solutions of (A). To solve (A) we also determine the functions that satisfy the equation (f (t (x + y)) - f (t x)) (f (x + y) - f (y)) = (f (t (x + y)) - f (t y)) (f (x + y) - f (x)) , (B) x , y , t > 0 , for a function f : (0 , ∞) → R . As an additional result we infer that each Lebesgue measurable or Baire measurable function f satisfying equation (B) is infinitely differentiable. [ABSTRACT FROM AUTHOR]

In this paper, we extend to the setting of positive orthogonally bi-additive operators several results from Aliprantis and Burkinshaw (Math Z 185: 245-257, 1983), de Pagter (Math Anal Appl 472: 238-245, 2019), Pliev and Popov (Siberian Math J 57: 552-557, 2016), Pliev and Ramdane (Mediter J Math 15(2): 55, 2018). First, we show that a positive order bounded orthogonally bi-additive map T : I → W defined on a lateral ideal of a Cartesian product of vector lattices E and F and taking values in a Dedekind complete vector lattice W can be extended to the whole space E × F . Then we prove that a positive orthogonally bi-additive operator T : E × F → W is laterally-to-order continuous if and only if the kernel of each S with 0 ≤ S ≤ T is laterally closed. Finally, we calculate the laterally-to-order continuous part of a positive orthogonally bi-additive operator T : E × F → W . [ABSTRACT FROM AUTHOR]

We consider Nitsche's method for solving elliptic Dirichlet boundary control problems on curved domains with control constraints. By using Nitsche's method for the treatment of inhomogeneous Dirichlet boundary conditions, the L2 boundary control enters in the variational formulation in a natural sense. The idea was first used in Chang, et al. (Math. Anal. Appl.453, 529–557 2017) where the curved boundary was approximated by a broken line and a locally defined mapping was needed to obtain the numerical control on the curved boundary. In this paper, we develop a method defined on curved domains directly. We derive a priori estimates of quasi-optimal order for the control in the L2 norm, and quasi-optimal order for the state and adjoint state in energy norms. Numerical examples are provided to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies decision-making in the face of two stochastically independent sources of uncertainty. It characterizes axiomatically a Subjective Expected Utility representation of preferences where subjective beliefs consist of a product probability measure. The two key axioms in this characterization both involve some behavioral notions of stochastic independence. Our result can be understood as a purely subjective version of the Anscombe and Aumann (Ann Math Stat 34:199–205, 1963) theorem that avoids the controversial use of exogenous probabilities by appealing to stochastic independence. We also obtain an extension to Choquet Expected Utility representations. [ABSTRACT FROM AUTHOR]

For rational functions , where is a polynomial of degree at the most and , with we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions . In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807]. [ABSTRACT FROM AUTHOR]

We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional ∫ ∫ Ω (1 + | ∇ u (x , y) | 2 ) - 1 d x d y in the class of concave functions u : Ω → [0,M], where Ω ⊂ R 2 is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

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

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

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

In this paper, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics— CW -complexes—and argue that it is explanatory. With this, we see that there are cases of explanatory induction. [ABSTRACT FROM AUTHOR]

This paper is concerned with a quasilinear degenerate Keller–Segel system of parabolic–parabolic type. It was proved in Ishida and Yokota (J. Differ. Equ. 252:2469–2491, 2012) that if q > m + 2 n , then the system has a global weak solution under smallness conditions for initial data, where m describes the intensity of diffusion, q shows the nonlinearity, and n denotes the dimension. The smallness conditions were relaxed in Wang et al. (Z. Angew. Math. Phys. 70:18 pp., 2019) when q = 2 . The purpose of this paper is to obtain global existence and boundedness under more general conditions for initial data in the case m + 2 n < q < m + 4 n + 2 and to relax the conditions assumed in Ishida and Yokota (J. Differ. Equ. 252:2469–2491, 2012). [ABSTRACT FROM AUTHOR]

We contribute to the ongoing discussion on mathematical structuralism by focusing on a question that has so far been neglected: when is a structure part of another structure? This paper is a first step towards answering the question. We will show that a certain conception of structures, abstractionism about structures, yields a natural definition of the parthood relation between structures. This answer has many interesting consequences; however, it conflicts with some standard mereological principles. We argue that the tension between abstractionism about structure and classical mereology is an interesting result and conclude that the mereology of abstract structures is a subject that deserves further exploration. We also point out some connections between our discussion of the mereology of structures and recent work on non-well-founded mereologies. [ABSTRACT FROM AUTHOR]

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

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

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

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

An introduction to the journal is presented in which the guest editors discuss various reports within the issue on topics including the recentadvances in neuroscience, cognitive and developmental and psychology and its contribution to science and mathematics.

The mechanism of cytoplasmic incompatibility (CI) is important in the study of Wolbachia invasion in wild mosquitoes. Su et al. (Bull Math Biol 84(9):95, 2022) proposed a delay differential equation model by relating the CI effect to maturation delay. In this paper, we investigate the dynamics of this model by allowing the same density-dependent death rate and distinct density-independent death rates. Through analyzing the existence and stability of equilibria, we obtain the parameter conditions for Wolbachia successful invasion if the maternal transmission is perfect. While if the maternal transmission is imperfect, we give the ranges of parameters to ensure failure invasion, successful invasion and partially suppressing, respectively. Meanwhile, numerical simulations indicate that the system may exhibit monostable and bistable dynamics when parameters vary. Particularly, in the bistable situation an unstable separatrix, like a line, exists when choosing constant functions as initial values; and the maturation delay affects this separatrix in an interesting way. [ABSTRACT FROM AUTHOR]

Successful early mathematical development is vital to children’s later education, employment, and wellbeing outcomes. However, established measurement tools are infrequently used to (i) assess children’s mathematical skills and (ii) identify children with or at-risk of mathematical learning difficulties. In response, this pre-registered systematic review aimed to provide an overview of measurement tools that have been evaluated for their psychometric properties for measuring the mathematical skills of children aged 0–8 years. The reliability and validity evidence reported for the identified measurement tools were then synthesised, including in relation to common acceptability thresholds. Overall, 41 mathematical assessments and 25 screeners were identified. Our study revealed five main findings. Firstly, most measurement tools were categorised as child-direct measures delivered individually with a trained assessor in a paper-based format. Secondly, the majority of the identified measurement tools have not been evaluated for aspects of reliability and validity most relevant to education measures, and only 15 measurement tools met the common acceptability thresholds for more than two areas of psychometric evidence. Thirdly, only four screeners demonstrated an acceptable ability to distinguish between typically developing children and those with or at-risk of mathematical learning difficulties. Fourthly, only one mathematical assessment and one screener met the common acceptability threshold for predictive validity. Finally, only 11 mathematical assessments and one screener were found to concurrently align with other validated measurement tools. Building on this current evidence and improving measurement quality is vital for raising methodological standards in mathematical learning and development research. [ABSTRACT FROM AUTHOR]

Though Bartlett has offered several constructive criticisms of the mathematical formulations that were included in my paper, Exponential Population Growth and Doubling Times, not one of them undermines either of the two basic points made in my paper—that exponential population growth is no longer being projected for any nation in the world and, this being so, writers and speakers addressing population issues no longer have reason to discuss either exponential growth or its frequent companion, "doubling times." In making my argument, included several paragraphs in which exponential growth, exponential curves, and doubling times were described. Bartlett has found these descriptions "distressing" and his paper explains his reasoning. So be it. I claim no scholarship in mathematics. In the writing of my paper believed—and still do—that my descriptions provide background material that is useful to readers not concerned with mathematical purity. [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]