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The Riemann hypothesis (RH) is one of the major unsolved problems in pure mathematics. In the present paper, a parameterized family of non-cooperative games is constructed with the property that, if RH is true, then any game in the family admits a unique Nash equilibrium. We argue that this result is not degenerate. Indeed, neither is the conclusion a tautology, nor is RH used to define the family of games. • The Riemann Hypothesis (RH) is a famous open problem in mathematics. • This paper constructs generalized difference-form contests related to RH. • These contests admit a unique Nash equilibrium if RH holds true. • We argue that the result is not degenerate in any obvious way. [ABSTRACT FROM AUTHOR]

This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov–Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12–26) and extends the convergence results from O (10 7) degrees of freedom (DOFs) to O (10 9) DOFs using a new scalable parallel MPI/OpenMP implementation. Novel contributions of this paper include an alternative definition of coarse-grid systems based on restriction of fine-grid operators, yielding superior convergence results. In the uniform refinement setting, a detailed convergence study is provided, demonstrating h and p robust convergence and linear scaling with respect to the wave frequency. The paper concludes with numerical results on hp -adaptive simulations including a large-scale seismic modeling benchmark problem with high material contrast. [ABSTRACT FROM AUTHOR]

Leibniz's famous proposition that God has created the best of all possible worlds holds a significant place in his philosophical system. However, the precise manner in which God determines which world is the best remains somewhat ambiguous. Leibniz suggests that a form of "Divine mathematics" is employed to construct and evaluate possible worlds. In this paper, I uncover the underlying mechanics of Divine mathematics by formally reconstructing it. I argue that Divine mathematics is a one-player combinatorial game, in which God's goal is to find the best combination among many possibilities. Drawing on the combinatorial theory, I provide new solutions to some puzzles of compossibility. [ABSTRACT FROM AUTHOR]

Assessments constitute a fundamental and inevitable component of any educational journey. Manual effort required for the evaluation of these assessments is very high. Automation of the evaluation process and grading helps in making the review process more efficient, objective, and scalable, thereby reducing the workload of human reviewers. Automating the grading process for multiple-choice and short-answer assessments is relatively straightforward, but it poses significant challenges when applied to the evaluation of formal languages, particularly in the context of mathematical assessments. In this paper a model that automatically evaluates and grades the Quadratic Equation problems is presented. The study is conducted using a manually curated dataset comprising 1200 solutions to various quadratic equation problems. Embeddings of the quadratic solutions are generated using Google's T5 Model. These embeddings are then used to train different traditional and ensembled machine learning models along with complex Deep learning models like LSTM and Bi LSTM. An in-depth analysis of the fine-tuned T5 model's performance, evaluating its effectiveness in comparison with the pretrained T5 model in automatic grading of quadratic equation problems has been explored. Fine-tuning significantly contributes to the reduction of error by 70% and a noticeable increase in the R2 value to 97%. [ABSTRACT FROM AUTHOR]

Consider a polynomial f (z) = a n z n +... + a 1 z + a 0 of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the p th Hadamard power of f : f [ p ] (z) : = a n p z n +... + a 1 p z + a 0 p is stable if p is a positive integer number. However, it turns out that f [ p ] does not need to be stable for all real p > 1. A counterexample is known for n = 8 and p = 1.139. On the other hand, f [ p ] is stable for n = 1 , 2 , 3 , 4 , and every p > 1. In this paper we fill the gap by showing that f [ p ] is stable for n = 5 and constructing counterexamples for n ≥ 6. Moreover, by means of Rouché's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results. [ABSTRACT FROM AUTHOR]

Over the past few decades, Piaget's forms of abstraction have proved productive for developing explanatory models of student and teacher knowledge, yet the broader applicability of his abstraction forms to mathematics education remains an open question. In this paper, we adopt the Piagetian forms of abstraction to accomplish two interrelated goals. Firstly, we analyze instructional tasks to develop hypothetical accounts of the abstractions that might occur during students' engagement with them. Secondly, we draw on middle- and secondary-grades classroom data to discuss the abstractions that occurred during the implementation of those instructional tasks. Because this paper represents an initial attempt at extending the applicability of Piagetian forms of abstraction, we close with potential implications of such use and possible avenues for future research. Most notably, we highlight the complexities involved in supporting abstraction through curriculum and instruction. • We examine task materials and implementation. • We differentiate between forms of abstraction. • We identify abstraction's viability for such analysis. • We differentiate between student learning outcomes. • We suggest avenues for future research. [ABSTRACT FROM AUTHOR]

Science teaching relies heavily on visual illustrations and visually organized assignments as methods of teaching and proving concepts like the Pythagorean theorem. As Visually Impaired Persons (VIPs) cannot see, they derive no benefit from the use of such illustrations alone. Often, material objects are used to help VIPs understand mathematical concepts by means of tactile and haptic perception. In this paper, I will show the practices employed in the pursuit of transforming a visually available material object into something of specific scientific relevance for tactile and haptic exploration. This article thus emphasizes the matter – in the original sense of materiality – of math. Based on video-recorded data, this paper provides an in-depth interactional analysis of a single case in which a VIP engages in a mathematics learning situation with an assistant. This research is based on and contributes to ethnomethodology and conversation analysis by critically examining how objects are used to teach VIPs in math classes. The research has implications for a broadened understanding of how learning mathematics is not just a matter of logics, verbal descriptions or the visual analysis of charts and diagrams, but can also inherently involve the tangible materiality of mathematical representations. [ABSTRACT FROM AUTHOR]

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order n with size m > ⌊ n 2 4 ⌋ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order n. Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size m. In this paper, we prove Q -spectral versions of Mantel's theorem. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) and edge set E (G) , and let d (u , w) denote the length of a u − w geodesic in G. For any vertex v ∈ V (G) and any edge e = x y ∈ E (G) , let d (e , v) = min { d (x , v) , d (y , v) }. For any distinct edges e 1 , e 2 ∈ E (G) , let R { e 1 , e 2 } = { z ∈ V (G) : d (z , e 1) ≠ d (z , e 2) }. Kelenc, Tratnik and Yero [Discrete Appl. Math. 251 (2018) 204-220] introduced the notion of an edge resolving set and the edge dimension of a graph: A vertex subset S ⊆ V (G) is an edge resolving set of G if | S ∩ R { e 1 , e 2 } | ≥ 1 for any distinct edges e 1 , e 2 ∈ E (G) , and the edge dimension , edim (G) , of G is the minimum cardinality among all edge resolving sets of G. For a function g defined on V (G) and for U ⊆ V (G) , let g (U) = ∑ s ∈ U g (s). A real-valued function g : V (G) → [ 0 , 1 ] is an edge resolving function of G if g (R { e 1 , e 2 }) ≥ 1 for any distinct edges e 1 , e 2 ∈ E (G). The fractional edge dimension , edim f (G) , of G is min { g (V (G)) : g is an edge resolving function of G }. Note that edim f (G) reduces to edim (G) if the codomain of edge resolving functions is restricted to { 0 , 1 }. In this paper, we introduce and study the fractional edge dimension of graphs, and we obtain some general results on the edge dimension of graphs. We show that there exist two non-isomorphic graphs on the same vertex set with the same edge metric coordinates. We construct two graphs G and H such that H ⊂ G and both edim (H) − edim (G) and edim f (H) − edim f (G) can be arbitrarily large. We show that a graph G with edim (G) = 2 cannot have K 5 or K 3 , 3 as a subgraph, and we construct a non-planar graph H satisfying edim (H) = 2. It is easy to see that, for any connected graph G of order at least three, 1 ≤ edim f (G) ≤ | V (G) | 2 ; we characterize graphs G satisfying edim f (G) = 1 and examine some graph classes satisfying edim f (G) = | V (G) | 2 . We also determine the fractional edge dimension for some classes of graphs. [ABSTRACT FROM AUTHOR]

In this paper we show how simple text-driven variations of given statements in mathematics can lead to interesting new problems and push forward a whole theory around simple initial questions. We exemplify this in two cases. Case 1 deals with problem-posing activities suitable for pupils and case 2 is a rational reconstruction of the organisation of mathematical knowledge within problems of graph colorings. Mathematicians learn to systematically look for subsequent problems around a given problem. We argue that this toy-model captures a nontrivial part of professional mathematical research within the pure fields and conjecture that it even grasps high level developments in mathematics. By doing this, we implicitly encourage a very simplistic view on criteria, so to speak a "cowpath" approach to progress in mathematics. The term "cowpath" is borrowed from architecture and software design, where it is commonly used. While we can contemplate which pathways are ideal, we may also just plant grass and see where people choose to walk. Those pathways are also self-enforcing, since we are less hesitant to walk on those rather than criss-cross the landscape. [ABSTRACT FROM AUTHOR]

This paper is concerned with an indirect chemotaxis-consumption model with sensitivity function χ (v). Frassu and Viglialoro (Appl. Math. Lett.; 2022; 132; 108108) showed global existence and boundedness in the case that χ is a constant. However, there is a room for improvement of the main claim. This paper improves the result and extends it in the case that χ depends on v. [ABSTRACT FROM AUTHOR]

A (d , h) -decomposition of a graph G is an ordered pair (D , H) such that H is a subgraph of G of maximum degree at most h and D is an acyclic orientation of G − E (H) with maximum out-degree at most d. In this paper, we prove that for l ∈ { 5 , 6 , 7 , 8 , 9 } , every planar graph without 4- and l -cycles is (2 , 1) -decomposable. As a consequence, for every planar graph G without 4- and l -cycles, there exists a matching M , such that G − M is 3-DP-colorable and has Alon-Tarsi number at most 3. In particular, G is 1-defective 3-DP-colorable, 1-defective 3-paintable and 1-defective 3-choosable. These strengthen the results in [Discrete Appl. Math. 157 (2) (2009) 433–436] and [Discrete Math. 343 (2020) 111797]. [ABSTRACT FROM AUTHOR]

A flaw in the main definition of the original paper has been corrected. Other typos have been fixed. An error in a lemma proof has been amended. With these changes, all the results in the aforementioned paper are now valid. [ABSTRACT FROM AUTHOR]

Touchscreen devices have become prominent in a variety of educational contexts due to their use of hands-on and interactive learning styles, and particularly for their direct engagement, rapid orientation, and multi-touch affordances, which textbooks and oral instruction may not always provide. As such, over the past decade, both the number of touchscreen applications for mathematics learning and the body of research examining their efficacy have increased. Yet, there are still few existing systematic reviews of how these touchscreen-based mathematics applications promote K-12 student learning. In response, this paper provides an in-depth examination of extant studies which adopt practices that utilize touchscreen technology for K-12 mathematics teaching and learning. The Web of Science database was searched for research published between 2010 and 2023, and 34 papers were chosen to review the empirical bases of touchscreen-based mathematics learning. These articles were analyzed based on the educational contexts found, touchscreen actions prompted, and the mathematical meanings elicited from these actions. Most significantly, we propose a touchscreen action matrix that characterizes the actions performed on the touchscreen platform and presents the mathematical meanings derived from these action categories. We hope that this review and characterization provide guidance for researchers, designers, and educators in eliciting mathematical knowing with action-based design in touchscreen medium. • 34 studies of touchscreen-based mathematics applications are systematically reviewed. • Most apps target number concepts via gamified environments at the early primary level. • We proposed a 2 × 2 touchscreen action matrix to characterize actions on touchscreens. • Number of Input, Temporality, Spatiality & Speed are the four dimensions of interest. • We present findings of how mathematical meanings are elicited from touch taxonomies. [ABSTRACT FROM AUTHOR]

Despite much research exploring affect within a mathematics context, poor student engagement in mathematics persists internationally. There is limited research examining mathematics classroom practice and students' affect, with the potential to inform teachers' practice. This scoping review analysed this limited literature. Of 250 papers analysed, only 26 papers adequately described classroom practices, considered students' affective state in response to the practice, and considered mathematics learning outcomes. Collectively, these explore very few of the practices used in mathematics classrooms, highlighting the need for more and improved practice-focussed research to better inform efforts to improve student engagement in mathematics. • Despite extensive mathematics affect research the problem of poor student engagement in mathematics persists internationally. • This scoping review demonstrates there is limited research examining classroom practices, affect and mathematics outcomes. • Improved focus on classroom practice in research examining mathematics and affect is required. • The explanatory model presented describes the interaction of students' affect, classroom practices, and mathematical learning. [ABSTRACT FROM AUTHOR]

The concept of closure operator is key in several branches of mathematics. In this paper, closure operators are extended to relational structures, more specifically to fuzzy relations in the framework of complete fuzzy lattices. The core of the work is the search for a suitable definition of (strong) fuzzy closure relation, that is, a fuzzy relation whose relation with fuzzy closure systems is one-to-one. The study of the properties of fuzzy closure systems and fuzzy relations helps narrow down this exploration until an appropriate definition is settled. [ABSTRACT FROM AUTHOR]

In recent years, systems of nonlinear equations have attracted widespread attention and have been extensively studied. The recent designed Fletcher-Reeves (FR) type methods of Papp and Rapajić [Appl. Math. Comput. 269 (2015) 816–823] [27] are efficient in solving large-scale monotone nonlinear equations due to the simple iterative form. In this paper, we propose an accelerated variant of these FR-type methods for approximating the solutions of nonlinear equations involving monotone and Lipschitz continuous mappings. Under suitable assumptions, we prove that the sequence generated by the new algorithm converges globally. Some numerical results are reported to illustrate the computational performance of the new methods. [ABSTRACT FROM AUTHOR]

Stature estimation is a core component to the biological profile in forensic anthropology casework. Here we provide mathematical equations for estimating stature for contemporary American Indians (AI), which currently are lacking in forensic anthropology. Drawing on postmortem computed tomography data from the New Mexico Decedent Image Database we regressed cadaveric length on four long bone length measures of the tibia, femur, and humerus to produce 11 combinations of models. Separate regression models were calculated for the entire pooled sample, by sex, broad AI language groups, and age + sex subsamples and compared. Sex-specific models were statistically better than general models, which were more accurate than language group and age + sex models. Equations were created for general and sex-specific models. Application to an independent test sample demonstrates the equations are accurate for stature estimation with overestimates of less than 1 cm. The equations provide similar levels of precision to stature estimation programs like the FORDISC 3.0 module and other stature equations in the literature. We provide recommendations for equation use in casework based on our results. These equations are the first for estimating stature in contemporary AI. This paper demonstrates the appropriateness of these newly created stature equations for use in New Mexico and the surrounding region. • This research provides the first stature estimation equations based on contemporary American Indians. • Stature equations derive from postmortem CT scans of maximum humeral, tibial, and femoral length and femoral bicondylar length. • Sex-specific models perform better than general models. • An independent test sample indicates the newly created equations are accurate and precise. • We present updated stature estimation equations for Southwest American Indians for use in forensic anthropology casework. [ABSTRACT FROM AUTHOR]

Galois connections seem to be ubiquitous in mathematics. They have been used to model solutions for both pure and application-oriented problems. Throughout the paper, the general framework is a complete fuzzy lattice over a complete residuated lattice. The existence of three fuzzy Galois connections (two antitone and one isotone) between three specific ordered sets is proved in this paper. The most interesting part is that fuzzy closure systems, fuzzy closure operators and strong fuzzy closure relations are formal concepts of these fuzzy Galois connections. [ABSTRACT FROM AUTHOR]

Recently, graph-based deep learning models have exhibited remarkable performance in generating solution expressions for the math word problem (MWP). However, most of these models have not taken into account the limitations and errors in constructing prior knowledge graphs, which may affect their accuracy and reliability in practical applications. In addition, during graph learning, they focus on extracting information from each given graph, while neglecting the adaptability and unification of graph representation learning. In this paper, we propose a novel multi-view graph learning-to-tree model with dual-strategy (MVG-DS-T), in which it performs adaptive and consistent multi-view representation learning through two benchmark graphs. Specifically, we construct benchmark graphs via semantic dependency parsing of MWP text, considering both semantic and quantitative aspects, i.e., semantic graph and quantitative graph. Then, the reconstruction strategy is employed to reconstruct the structure of the benchmark graphs to capture the adaptive representation information suitable for downstream tasks, while the alignment strategy is utilized to overcome the limitation of independent view representations by unifying the semantic and quantity embedding information through graph structure. Also, an adaptive length normalized loss balancing term for the tree-based decoder is introduced to control the model focus on label length during training, resulting in better equation generation. Extensive experiments demonstrate the effectiveness of the proposed approach on the MWP task. The empirical results show that MVG-DS-T achieves performance comparable to that of the state-of-the-art graph-based models in the existing literature. [ABSTRACT FROM AUTHOR]

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal). This article has been retracted at the request of the Editors-in-Chief after receiving a complaint about citation issues. The editors solicited further independent reviews which indicated that the statements in this paper substantially overlap with the PhD thesis of Z. Furdzik [O własnościach pewnych rozkładów przestrzeni topologicznych na iloczyny kartezjańskie, 1968 (unpublished, in Polish), Institute of Mathematics of the Polish Academy of Sciences, Warsaw], without proper citation. [ABSTRACT FROM AUTHOR]

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal). This article has been retracted at the request of the Editors-in-Chief after receiving a complaint about citation issues. The editors solicited further independent reviews which indicated that the statements in this paper substantially overlap with the PhD thesis of Z. Furdzik [O własnościach pewnych rozkładów przestrzeni topologicznych na iloczyny kartezjańskie, 1968 (unpublished, in Polish), Institute of Mathematics of the Polish Academy of Sciences, Warsaw], without proper citation. [ABSTRACT FROM AUTHOR]

Future experts, decision-makers, stakeholders and future citizens – they all go to school, which could be a place to prepare students to facing post-normal problems in their future roles. While the literature on mathematics and science education shows increased interest in post-normal science to critically understand contemporary problems, mathematics is still generally taught as a neutral subject with either correct or wrong answers. In this paper, I lean on post-normal science literature and the slowly growing literature on ethics and mathematics education to develop a framework for identifying how mathematics education can prepare students for ethical aspects of post-normal problems. Three areas of ethics associated with post-normal problems are explored through concepts from ethics and mathematics education: ethical perspectives of the problem, ethical perspectives embedded in knowledge, and ethics in interpersonal care in discussions and care for students' self-esteem in participatory processes. This framework is used to analyse and discuss three classroom projects to provide examples of what these areas may contain. The projects partly align with post-normal science, and are about clothing and sustainability, traffic safety, and argumentation related to open an offshore area to oil exploitation. The ethical areas are discussed in terms of ethical awareness and democratic practices. • Three teaching and learning situations with 13–15-year-olds are analysed. • Ecological, societal and individual stakes of problems were addressed. • Ethics embedded in knowledge production could be more explicit. • Care for participants and their self-appreciation is key is extended peer-reviews. • Mathematics class is suitable for addressing ethics of post-normal problems. [ABSTRACT FROM AUTHOR]

Laurie in [Math. Comp., 65(1996), pp. 739–747] introduced anti-Gaussian quadrature rule, that gives an error equal in magnitude but of opposite sign to that of the corresponding Gaussian quadrature rule. Guided by that idea, in this paper we consider a set of anti-Gaussian quadrature rules for the optimal set of quadrature rules in Borges' sense (see [Numer. Math., 67(1994), pp. 271–288]), as well as the corresponding class of multiple orthogonal polynomials. The main properties of such quadrature rules and multiple orthogonal polynomials are proved and numerical methods for their constructions are presented. Some numerical examples are also included. [ABSTRACT FROM AUTHOR]

In this paper, we deal with extremal eigenvalues of the adjacency matrices of complete signed graphs. The complete signed graphs with maximal index (i.e. the largest eigenvalue) with n vertices and m ≤ ⌊ n 2 / 4 ⌋ negative edges have been already determined. We address the remaining case by characterizing those with m > ⌊ n 2 / 4 ⌋ negative edges. We also identify the unique signed graph with maximal index among complete signed graphs whose negative edges induce a tree of diameter at least d for any given d. This extends a recent result by Li, Lin, and Meng [Discrete Math. 346 (2023), 113250] who established the same result for d = 2. Finally, we prove that the smallest minimum eigenvalue of complete signed graphs with n vertices whose negative edges induce a tree is − n 2 − 1 − 1 + O (1 n). [ABSTRACT FROM AUTHOR]

Factors affecting academic achievement were widely analyzed and compared. This paper studies the effect of some context variables, including Gender, Age and School Resources (namely, General School Resources index ?GSR? and Resources for Mathematics Instruction ?RMI?) on the performance in Mathematics of the Moroccan?s fourth-grade students that participated on the TIMSS 2019 survey. The data collection method uses secondary data from this survey. Two statistical software packages are used (R Core Team, 2017 and SPSS 20) to treat data and apply multilevel modeling. The results of this modeling show that the variance explained at the school level is 39.41 [ABSTRACT FROM AUTHOR]

Decomposition integrals provide a framework for non-linear integrals that include Choquet, Shilkret, the PAN, and the concave integrals. All of these integrals found their applications in mathematics, notably in decision-making and economy. An important class of decomposition integrals is the class of integrals extending Lebesgue integral in the sense that the decomposition integral with respect to classical measures coincides with Lebesgue integral. In this paper, we consider finite spaces X only and discuss some necessary and sufficient conditions for this property. Also, some construction methods are given and exemplified. [ABSTRACT FROM AUTHOR]

Given a real number α ∈ (0 , 1) , we define the Webster sequence of density α to be W α = (⌈ (n − 1 / 2) / α ⌉) n ∈ N , where ⌈ x ⌉ is the ceiling function. It is known that if α and β are irrational with α + β = 1 , then W α and W β partition N. On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of N into sequences W α , W β , W γ with α , β , γ irrational. In this paper, we consider the question of how close one can come to a three-part partition of N into Webster sequences with prescribed densities α , β , γ. We show that if α , β , γ are irrational with α + β + γ = 1 , there exists a partition of N into sequences of densities α , β , γ , in which one of the sequences is a Webster sequence and the other two are "almost" Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the n th element of each sequence in O (1) time and the second algorithm gives the assignment of the m th positive integer to a sequence in O (1) time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems. [ABSTRACT FROM AUTHOR]

The paper develops an approximate semi-analytical solution for the computation of the third statistical cross-moments of modal responses in a stochastic dynamic analysis. These moments would require heavy twofold numerical integration in a general context but are drastically simplified in the proposed formulation by taking advantage of the assumed distinctness between the low characteristic frequency of the loading and the natural frequencies of the structure. This condition is typically respected and acknowledged in wind engineering where the buffeting analysis of large structures hinges on the Background/Resonant decomposition. As such, the proposed formulation extends to third statistical order the existing developments for the estimation of the modal variances and covariances. It allows the third order spectral analysis of large structures to be conducted within a reasonable amount of time. It also reveals the existence of three main components to the response: background, bi-resonant and tri-resonant. The latter one is specific to this very own problem and is shown to be important when the sum of two natural frequencies is equal to a third one, although the structural behavior is linear. Mathematics highlight this and other findings which are then illustrated on a minimum working example, easily reproducible by readers. Overall, it clearly demonstrates the benefits of the proposed decomposition in terms of both behavioral comprehension and computational consumption. • Third order cross-correlation moments of modal responses are determined. • Simplified expressions are given by means of the multiple timescale spectral analysis. • The paper reports the discovery of a tri-resonant component in these moments. • Tri-resonance takes place when one natural frequency is the sum of two others. [ABSTRACT FROM AUTHOR]

Using multiple datasets on learning outcomes of over 2 million children in the age group 8–16 years from rural India, the paper examines the prevalence and persistence of gender gap in performance in mathematics. We observe significant and persistent gender gap in mathematics, which is not found in reading outcomes. The northern states lag behind and the southern ones show a 'reverse gender gap'. Pre-existing household practices and societal gender norms are found to explain this difference significantly. The findings call for the need to understand these gaps granularly and inform gender-specific educational interventions for mathematics. • The paper examines gender gap in mathematics learning outcomes among rural Indian children. • We find significant gender gap, which is also persistent across time. • Substantial variation across states is observed with 'reverse gender gap' in some. • Patriarchal household practices and gender norms explain this gap significantly. • Gender linked education interventions are needed in states showing persistent gap. [ABSTRACT FROM AUTHOR]

In this conceptual paper, we explore a framework of reasoning practices for decision-making in context. We extend prior work related to socioscientific issues (SSI) (Sadler et al., 2007) to consider the applicability of this framework to sociomathematical issues, specifically using the context of fairness in political (re)districting. We illustrate the usefulness of the SSI framework for sociomathematical issues drawing on student work and reflections from two undergraduate courses. We conclude by suggesting adjustments to reasoning practices of the SSI framework that might uniquely reflect the nature of sociomathematical reasoning. We discuss implications of our findings for conceptualizing reasoning practices that include mathematical perspectives within social and political contexts and for mathematics education generally. • Reasoning about a societal issue consists of intertwining personal, practical, contextual, and mathematical meaning. • A framework based on Sadler, Barab & Scott's (2007) analysis of reasoning practices regarding socioscientific issues is useful for analyzing emergent student reasoning in the context of redistricting, a sociomathematical issue. • Students reason in multiple, sophisticated ways about the fairness of specific maps and redistricting processes. [ABSTRACT FROM AUTHOR]

Recognizing and describing children's mathematical thinking in humanizing ways, especially when students engage in confusion, productive struggle, and mistakes, is a complex and challenging process. This paper describes an exploratory, mixed-methods study about how elementary teacher candidates (TCs) describe children's thinking as a right to exercise and to value their humanity when learning mathematics. The study analyzed transcripts from 64 TCs' summative assessments, which consisted of mock parent-teacher conferences (MPTC). Findings suggest that TCs described children's confusion, productive struggle, and mistakes (RotL 1 and 2) as: a teacher's observation, an opportunity for students to correct or clarify their thinking, an opportunity for teachers to adjust instruction or provide support, and as a normal part of the learning process. More importantly, some TCs reassured children that learners have fundamental rights when learning mathematics, especially when feeling confused and claiming a mistake. Implications for research and teacher education are discussed. Keywords: Elementary, teacher education, mathematics, mixed methods, rehumanizing, Torres' rights of the learner • Topic modeling and sentiment analysis helps to quantify and describe valence that arises from large amounts of text. • 32 TCs identified children's confusion, generalized struggle, and mistakes (Torres' RotL 1 and 2) in their MPTC across 50 instances. • Some TCs described children's confusion and mistakes as rights to exercise, a potential rehumanizing stance. [ABSTRACT FROM AUTHOR]

Individual fairness requires that similar individuals are treated similarly. It is supposed to prevent the unfair treatment of individuals on the subgroup level and to overcome the problem that group fairness measures are susceptible to manipulation or gerrymandering. The goal of the present paper is to explore the extent to which individual fairness itself can be gerrymandered. It will be proved that individual fairness can be gerrymandered in the context of predicting scores. Then, it will be argued that individual fairness is a very weak notion of fairness for some choices of feature space and metric. Finally, it will be discussed which properties of (individual) fairness are desirable. [ABSTRACT FROM AUTHOR]

Most of the published results on solving operator equations are very restrictive i.e. they have been proved under certain additional assumptions and in fact we do not have general solvability conditions. There are many reasons why this is so, one of them being the fact that the usual methods employed when solving these equations involve the use of generalized inverses which exist and are bounded in the case of operators only under certain special conditions such as closedness of the range of operators. Using two previously considered systems of operator equations as examples, we will show that using a particular general approach we can give necessary and sufficient solvability conditions without any additional assumptions. We will consider the system A X C = C = C X A and generalize recent particular results from the paper C. Deng, et al. (Appl. Math. Letters 81, 86–92 (2018)), as well as B A X = B = X A B for which the particular results are given in the paper C. Deng, (J. Math. Anal. Appl. 398 (2013) 664–670). We intend to use our approach to initiate formulating various general solvability conditions for other systems of operator equations. [ABSTRACT FROM AUTHOR]

There has been significant growth in the literature base exploring questions of teaching in undergraduate mathematics. In this paper we synthesize the literature on the teaching of proof-based undergraduate mathematics, drawing on 104 published reports from a range of countries and research traditions. We primarily differentiate the papers into those which explore lecture-based pedagogy and student-centered pedagogy. For each type of instruction, we focus on three categories of findings from the literature: description of instruction, instructor beliefs and rationales, and the relationship between instruction and students (cognitive, participatory, affective, and equity oriented). Much is known about the enactment of lecture-based teaching, including instructors' cognitive and affective goals. The student-centered literature focuses on tensions and challenges implementing curricula with a greater focus on participatory goals. Overall, there are few studies that attempt to link instructors' classroom activity and students' learning. Similarly, attention to equity is relatively lacking in the extant research. • In the lecture setting, we have a rich understanding of what mathematicians do and why, but only limited work on its influence on student activity and outcomes. • In the student-centered setting, we have evidence of affective and participation benefits to students, but more work is needed on the teachers' role. • We propose a framework by which to conduct research on teaching in advanced mathematics in both lecture-based and student-centered formats. [ABSTRACT FROM AUTHOR]

• We disprove a conjecture proposed by Fiedorowicz, Sidorowicz, Sopena (Appl. Math. Comput. 410 (2021) 126253). • In addition, we find a result of the above paper is not correct. • Two sufficient conditions are given for a hamiltonian digraph D with w c → (D) = 2. Under an arc-coloring c of a digraph D , if for each pair of vertices (u , v) , there exists a directed walk from u to v satisfying that any two consecutive arcs of it have different colors, we say that D is properly-walk connected, and c is a proper-walk coloring of D. The proper-walk connection number w c → (D) of D is the least integer k such that D has a proper-walk coloring with k colors. Fiedorowicz, Sidorowicz, Sopena (Appl. Math. Comput. 410 (2021) 126253) conjectured that if D is a hamiltonian digraph with δ (D) ≥ 2 , then w c → (D) ≤ 2. In this paper, we disprove the conjecture by constructing two families of counterexamples. Also, we present some cases for a hamiltonian digraph D having w c → (D) = 2. In addition, we find that Observation 15 is not true in the same paper. [ABSTRACT FROM AUTHOR]

The generalized k -connectivity κ k (G) of a graph G , introduced by Hager in 1985, is a natural generalization of the concept of connectivity κ (G) , which is just for k = 2. Rongxia Hao et al. determined the generalized 4-connectivity of the total graphs of the complete equipartition bipartite graph and conjectured that κ 4 (T (K m , m + 1)) = 2 m − 1 and κ 4 (T (K m , n)) = 2 m for n > m + 1 and m ≥ 4 in [Appl. Math. Comput. 422 (2022)]. In this paper, we solved this conjecture. [ABSTRACT FROM AUTHOR]

• Following the untimely death of philosopher and mathematician William Kingdon Clifford in 1879, his wife, the novelist, Lucy Lane Clifford, survived him for fifty years, during which she refashioned Clifford's posthumous reputation to temper his philosophical radicalism. • Lucy Clifford's collaboration with her publisher and editor reveal her concern that William Clifford be remembered in alignment with gendered Victorian expectations of masculinity, as opposed to someone ruled by passion in his mathematical work. • Rather than hold William and Lucy Clifford apart as individuals, this article tracks their interrelationship and demonstrates how important their shared life as a married couple was to their embodied lived experience and to the work they did, drawing out connections between William as a creator of mathematics and Lucy as a creator of literature. The avowed atheist, evolutionary naturalist and mathematician William Kingdon Clifford is often remembered for his essay, "The Ethics of Belief," in which he opposed organized religion in any form. As a mathematician, Clifford was an early advocate of non-Euclidean geometry in England. Combining William Rowan Hamilton's work on quaternions with Hermann Grassmann's theory of linear extension, he invented an original system of geometric algebra. Breaking with conservative traditionalism in his philosophical and mathematical work, Clifford's marriage to the children's writer, novelist, and dramatist Lucy Lane was a relatively conventional, if brief, Victorian marriage. After his untimely death from consumption in 1879, Lucy outlived her husband by fifty years. Raising their two daughters and supporting herself after his passing, Lucy refashioned Clifford's posthumous reputation to temper his philosophical radicalism. Her collaboration with Clifford's publisher and editor reveal Lucy's concern that Clifford not be remembered as someone ruled by passion in his mathematical work. Her efforts to expunge writings suggestive of William's weakness, excitability, or inconstancy from the public record demonstrates her desire to craft an image of her husband in alignment with gendered expectations of masculinity. This paper argues that Lucy fashioning of William's memory conformed, rather than departed from, normative parameters of gender as defined by Victorian society. [ABSTRACT FROM AUTHOR]

In this paper we construct consistent examples of subgroups of 2 ω with Menger remainders which fail to have other stronger combinatorial covering properties. This answers several open questions asked by Bella, Tokgoz and Zdomskyy (Arch. Math. Logic 55 (2016), 767–784). [ABSTRACT FROM AUTHOR]

• The minimum over all maximum number of edge-disjoint paths to connect any 3-subset θ of V (G) is given. • The compact upper bound of 3-path connectivity π 3 (G) for a general graph G is gotten. • The structural proprieties of line graphs L (K m , n) and total graphs T (K m , n) of complete bipartite graphs K m , n are deeply explored. • The 3-path connectivity π 3 (L (K m , n)) and π 3 (T (K m , n)) are determined. Let θ be a subset of vertex set V (G) of a simply connected graph G , two θ -trees P 1 and P 2 of G are said to be internally disjoint if V (P 1) ∩ V (P 2) = θ and E (P 1) ∩ E (P 2) = ∅. For an integer k ≥ 2 , the k -path connectivity π k (G) (resp. k -tree connectivity κ k (G)) of a graph G is defined as min { π G (θ) | θ ⊆ V (G) and | θ | = k } (resp. min { κ G (θ) | θ ⊆ V (G) and | θ | = k }), where π G (θ) (resp. κ G (θ)) is the maximum number of pairwise internally disjoint θ -paths (resp. θ -trees) in G. The 3-tree connectivity of the line graph L (K m , n) and total graph T (K m , n) of the complete bipartite graph K m , n are gotten in [Appl. Math. Comput. 347(2019) 645-652]. In this paper, these results are improved from trees to paths. The exact values of the 3-path connectivity for L (K m , n) and T (K m , n) are gotten. That is, π 3 (L (K m , n)) = ⌊ 3 m + 2 n − 3 4 ⌋ − 1 for m = 3 and odd n , otherwise, π 3 (L (K m , n)) = ⌊ 3 m + 2 n − 3 4 ⌋ unless m = 1 and n = 1 , 2 ; π 3 (T (K m , n)) = m + ⌊ m 4 ⌋ for n ≥ m ≥ 1. In addition, the compact upper bound of π 3 (G) for a general graph G are gotten. [ABSTRACT FROM AUTHOR]

Litvak-Hinenzon et al. developed the phenomenological model to simulate the horizontal motion of particles in the atmosphere, and a series of their papers (see e.g., Litvak-Hinenzon et al. (Phys. D 164:213-250, 2002; Nonlinearity 15:1149-1177, 2002; SIAM J. Appl. Dyn. Syst. 3:525-573, 2004)) focused on studying the fate of parabolic resonance lower dimensional tori (as part of a quasi-periodic Hamiltonian pitchfork bifurcation (HPB) scenario in the unperturbed phenomenological model) under perturbations. However, the structural stability of quasi-periodic HPB involved therein has not been fully exposed theoretically (Litvak-Hinenzon et al. have only given some numerical explanations). Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), we consider a more general quasi-periodic HPB triggered by the Z 2 -invariant universal unfolding N b c k v = a y 2 2 − (λ + b I 1) x 2 2 + c x 4 4 with respect to Z 2 -equivariant BCKV-restricted morphisms of the planar singularity a 2 y 2 + c 4 x 4 (the coefficients a , b , c ≠ 0 , the I 1 is regarded as distinguished parameter with respect to the external parameter λ). We prove a KAM (Kolmogorov–Arnold–Moser) theorem concerning parabolic tori in such quasi-periodic HPB, by which Diophantine parabolic tori (and the whole corresponding Diophantine HPB scenario) survive non-integrable and Z 2 -invariant Hamiltonian perturbations, parametrized by pertinent large Cantor sets. In the context of Z 2 -symmetry, our results can be seen as rigorous proof of the structural stability problem of bifurcations of Floquet-tori triggered by the universal unfolding N b c k v with distinguished parameters which is proposed by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993). Ultimately, we similarly obtain the structural stability result of quasi-periodic HPB in the phenomenological model mentioned above, which can be utilized as a starting point for a deeper understanding of the various resonances and chaotic dynamics (in the gaps of the Cantor sets), just as Litvak-Hinenzon et al. did for normally parabolic tori undergoing a HPB. • We prove a KAM theorem in a quasi-periodic Hamiltonian pitchfork bifurcation. • We obtain the structural stability of a bifurcation in a phenomenological model. • We verify some of the previous numerical phenomena of Litvak-Hinenzon et al. [ABSTRACT FROM AUTHOR]

Empirical studies demonstrate that students have various difficulties in the mathematical modeling and therefore need teacher support. This research explains teacher support in mathematical modeling through the three characteristics of scaffolding (i.e., contingency, fading, and transfer of responsibility). Studies using this theoretical perspective involve open questions concerning which teacher support is suitable for effective scaffolding methods. This study aims to examine the scaffolding methods of teachers who used different participant structures during mathematical modeling activities. This paper argues that teachers' participation structures relate to their scaffolding methods. It also addresses issues that teachers may reflect on while supporting modeling activities. • Teachers' participant structures can relate to their support levels but not support areas. • The teacher who acted as a monitor provided higher support levels compared to the teacher who acted as a partner. • Guiding the understanding of the nature of a modelling task and assisting with modeling stages support modeling competencies. • Strategic and content-related strategic support can be recognized as effective scaffolding methods in modeling process. [ABSTRACT FROM AUTHOR]

Several studies in the mathematical education literature show the effect of students' high school skills in maths on their success at higher levels of education and work. In particular, the importance of maths course taking in US high schools is highlighted to be important for college enrollment and completion. The choice of taking additional maths courses or, as in Italy, of choosing a high-school curriculum with more maths, is not random: it depends on several substantial factors such as gender and socio-economic status. This selection bias implies that the differences in the academic outcomes might be traceable not only to mathematics ability and knowledge. In this paper, the aim is to estimate the treatment effect of attending a relatively new high school curriculum in Italy with more maths, with respect to the traditional track of the scientific "liceo", on two academic outcomes: university enrollment and first-year university performance. After having reduced the selection bias using a caliper multi-level propensity score matching procedure, a multi-state Markov model is used to study the treatment effect on the joint educational outcomes. • The influence of studying more mathematics in high school on academic outcomes. • Multi-level propensity score application on higher education data. • Application of multi-state Markov models. [ABSTRACT FROM AUTHOR]