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The Borodin–Kostochka conjecture says that for a graph G , if Δ (G) ≥ 9 , then χ (G) ≤ max { Δ (G) − 1 , ω (G) }. Cranston and Rabern in [SIAM J. Discrete. Math. 27 (2013) 534–549] proved the conjecture holding for K 1 , 3 -free graphs. In this paper, we prove that the conjecture holds for K 1 , 3 ¯ -free graphs, where K 1 , 3 ¯ denotes the complement of K 1 , 3. [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]

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]

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]

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]

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]

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]

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]

For a graph H , let δ (H) be the minimum degree and let c (H) be the length of a longest cycle in H. A 2-factor of H is a spanning subgraph of H in which every component is a cycle. In [Discrete Math. 313 (2013) 1934-1943], Čada and Chiba asked if H is a 3-connected claw-free graph of order n with δ (H) sufficiently large, does H have a 2-factor F such that c (F) ≥ min ⁡ { 9 δ (H) − 3 , n } ? In this paper, to answer their question, we show that if H is a 3-connected claw-free graph of order n with δ (H) ≥ 15 , then H has a 2-factor F such that c (F) ≥ min ⁡ { 9 δ (H) − 3 , n }. [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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Anyone involved in mathematics education will at some point encounter a classroom situation in which a student's statement provides an opportunity for the teacher to enhance students' mathematical understanding. This paper aims to characterize such classroom situations, known as critical events, as noticed by pre-service mathematics teachers during classroom observations and teaching. Analysis of 80 critical events reported by 20 pre-service mathematics teachers during a field-based teaching preparation program yielded a model that enables teacher-educators and researchers to characterize critical events while reflecting on the wide variations within these events. The model comprises three axes: the participants in the event, the focus of the event, and the dimensions in learning and teaching mathematics (i.e., cognitive, affective and social). The paper discusses the theoretical and empirical construction of the model and demonstrates its potential contribution using examples taken from our research. • Pre-service teachers identified critical events during classroom observations. • A model for characterizing critical events was constructed theoretically and empirically. • The model affords eliciting the multidimensional nature and complexity of critical events. • The model's potential contribution is presented using examples from our research. [ABSTRACT FROM AUTHOR]

This paper uses a new Kudryashov approach to construct various novel exact optical solutions of the time-fractional Fokas system with conformable fractional derivatives. The acquired optical solutions are distributed into several categories of bell shape, singular, exponential, dark, bright, dark–bright, and periodic soliton solutions. The graphical simulations of several graphs are given for a better understanding of the effect of the fractional order derivative and the dynamical behavior of the present optical solutions with various values of time parameter. In order to highlight the significance of the Fokas system, a class of the acquired solutions are given using suitable parameter values on three-dimensional (3D), two-dimensional (2D), and contour graphs. As a result, this technique is productive and practically effective for understanding nonlinear problems in engineering, physics, and mathematics. [ABSTRACT FROM AUTHOR]

In this paper I tackle the question What is the mathematics in mathematics education? By providing three different frames for the word mathematics. 1. Frame 1: Mathematics as an abstract body of knowledge/ideas, the organization of that into systems and structures, and a set of methods for reaching conclusions. 2. Frame 2: Mathematics as contextual, ever present, as a lens or language to make sense of the world. 3. Frame 3: Mathematics as a verb (not a noun), a human activity, part of one's identity. After introducing the frames and examining their distinction and their overlap, I discuss their implication with respect to student-centered classroom, context, and culture. • Explicating the definition of mathematics is essential for discussions among educators. • Different definitions of mathematics lead to different implications on education. • Different definitions of mathematics lead to different versions of student-centered mathematics instruction. [ABSTRACT FROM AUTHOR]

The chromatic edge-stability number e s χ (G) of a graph G is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of G. A graph G is called (3 , 2) -critical if χ (G) = 3 , e s χ (G) = 2 and for any edge e ∈ E (G) , e s χ (G − e) < e s χ (G). In this paper, we characterize (3 , 2) -critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, Discrete Math. 343 (2020) 111845]. [ABSTRACT FROM AUTHOR]

Mathematics is related to apheresis medicine. The safety of donor and patient in taking blood components is utterly important. Knowledge of total blood and plasma volume is needed and have to be calculated. By increasing quality, the safety of not only the donor and patient, but also the operator will increase, as well as the efficiency of operating an apheresis collection facility. Various concepts, formulas and calculations methods and their significance in apheresis are given in this paper. [ABSTRACT FROM AUTHOR]

This paper investigates data activities in an afterschool setting, offering a deeper understanding of the social nature of students' informal inferences by investigating how informal inferences are negotiated in group interactions, influenced by social norms, and how statistical concepts come into play in learners' informal inferential reasoning (IIR). Analyses take up a multi-sited orientation to investigate how youth used quantitative and contextual resources during a research activity to make meaning of data and negotiate emergent social tensions. Findings show how data activities that are part of informal inferential reasoning, such as collection, interpretation, generalization, inference, and representation unfolded as social, political, and personal. Implications call for designs for learning that better support working with data and understanding real-world phenomena and sociopolitical issues in ways that leverage youths' experiences, enabling them to take part in social action as critical community actors. • We illustrate the consequential nature of youths' negotiations of 'truths' about particular social phenomena (here, gender identity and bathroom restriction laws). • A multi-sited orientation illuminates how statistical concepts such as frequency counts and quantitative reasoning emerged as co-constructed aspects of meaning-making. • Analysis traces how youth used data activities to bridge their truths-under-negotiation with broader sociopolitical understandings of transgender people's experiences. • We show data activities, as part of informal inferential reasoning, such as collection, interpretation, and representation unfolded as social, political, and personal. • Implications focus on informal inferential reasoning as a crucial aspect of citizenship to be leveraged across and outside of disciplinary boundaries in designs for learning. [ABSTRACT FROM AUTHOR]

'Procedural' fluency in mathematics is often judged solely on numerical representations. 'Mathematical' fluency incorporates explaining and justifying as well as producing correct numerical solutions. To observe mathematical fluency, representations additional to a student's numerical work should be considered. This paper presents analysis of students' oral responses. Findings suggested oral responses are important vantage points from which to view fluency – particularly characteristics harder to notice through numerical work such as reasoning. Students' oral responses were particularly important when students' written (language) responses were absent/inconsistent. Findings also revealed the importance of everyday language alongside technical terms for observing reasoning as a fluency characteristic. Students used high modality verbs and language features, such as connectives, to explain concepts and justify their thinking. The results of this study purport that to gain a fuller picture of students' fluency, specifically their explanations or reasoning, students' oral responses should be analyzed, not simply numerical work. • Oral representations are useful for observing mathematical fluency. • Students' oral responses explain actions, concepts and reasons to show mathematical fluency. • Students use high modality words as evidence of well-developed levels of fluency. • Everyday verbs and connectives are important for observing reasoning as a fluency characteristic. • When students' oral and written representations are not aligned, observing mathematical fluency can be difficult. [ABSTRACT FROM AUTHOR]

In this paper, we present the first meta-analysis of the efficacy of game-based interventions on reducing students' levels of maths anxiety. After searching for randomised studies describing game-based interventions to reduce maths anxiety, 22 effect sizes with 913 participants described in 15 peer-review articles met the selection criteria. A random effects meta-analysis indicated a reduction of maths anxiety with a small effect size (mean effect size ES = −0.24, CI = [ − 0.47, −0.01]), marginally significant at 0.05 level but not robust to a leave-one-out sensitivity analysis. Several factors moderated the results: non-digital games were more effective, while digital games had a negligible mean effect size of ES = −0.10, CI = [ − 0.24, 0.03]. The effect size was also moderated by the total duration of the intervention, to the advantage of longer interventions, and by the type of gameplay: games had a greater effect on maths anxiety reduction when they promoted collaborative and social interactions. Such features were mainly present in non-digital games, while all bar one of the digital interventions used single-player games. The results obtained, which were particularly weak for digital games, indicated the need to develop and test games explicitly designed for maths-anxious students to increase the impact of game-based interventions. This will require investigation into the relationship between game features and maths anxiety through analysis of the behaviour of anxious and non-anxious students at play. Among the features that an anxiety-aware game could employ, we suggest collaborative gameplay, social interactions, adaptability, features promoting intrinsic motivation and embedding real-time measurements of maths anxiety in the game. • There is weak and non-robust evidence that games can reduce math anxiety levels. • The effect is weaker for digital games, that showed a negligible effect, and stronger for non-digital games. • Collaborative games and longer interventions showed a stronger effect. • The results indicate the need to design games for math anxious students and analyse the game experience of players. [ABSTRACT FROM AUTHOR]

Using appropriate tools strategically to aid in problem solving is a crucial skill identified in K-12 mathematics curriculum standards. As more assessments transition from paper-and-pencil to digital formats, a variety of interactive tools have been made available to test takers in digital testing platforms. Using onscreen calculators as an example, this study illustrates how process data obtained from student interactions with a digitally-based large-scale assessment can be leveraged to explore how and how well test takers use interactive tools and unveil their mathematical problem-solving processes and strategies. Specifically, sequence mining techniques using the longest common subsequence were applied on process data collected from a nationally representative sample who took the National Assessment of Educational Progress (NAEP) mathematics assessment to examine patterns of eighth-grade students' calculator-use behaviors and the content of calculator input across a series of items. Sequences of keystrokes executed on the onscreen calculator by test takers were compared to reference sequences identified by content experts as proficient and efficient use to infer how well and how consistently the calculator was used. Results indicated that calculator-use behaviors and content differed by item characteristics. Students were more likely to use calculators on calculation-demanding items that involve intensive and complex computations than on items that involve simple or no computation. Using the calculator on more calculation-demanding items and using it in a manner that is more efficient and more similar to reference sequences on these items were related to higher mathematical proficiency. Findings have implications for assessment design and can be used in educational practices to provide educators with actionable process-related information on tool use and problem solving. • We study digital tool use and problem solving in the NAEP mathematics assessment. • Calculator-use behaviors and content differ by item characteristics. • Calculator use on calculation demanding items is related to mathematical achievement. • Patterns of tool use have implications for assessment design and educational practice. [ABSTRACT FROM AUTHOR]

In their recent paper in this journal, Delgrande, Renne and Sack study qualitative probability and discuss its role in Artificial Intelligence. Building on related work by de Finetti, Scott, Segerberg and others, the authors provide general axioms for qualitative probability. In this paper we investigate Koopman's conditional qualitative probability from the computational viewpoint. An important part of his work, published in the Annals of Mathematics in 1940-1941, deals with finite conjunctions K of statements of the form "the probability of a given h does not exceed the probability of b given k ", with a , b , h , k elements of a boolean algebra. Upon coding these elements by boolean formulas, we provide a decision procedure to check the consistency of any such K. As an immediate consequence, also inferences in Koopman's probability theory are shown to be computable. These problems of qualitative probability theory significantly generalize Boole's (typically quantitative) problem of estimating the possible probabilities of a new event given the probabilities of other events. Boole's classical problem today is known as the optimization version of the probabilistic satisfiability problem PSAT. In 1986 Nilsson published an influential paper on this subject in this journal. The scope of our results is much larger than that of PSAT, because Koopman's conjunctions K also formalize the key notion of independence. Some familiarity with boolean logic and finite boolean algebras is the only prerequisite for this paper. [ABSTRACT FROM AUTHOR]

• Scepticism philosophical thought also questioned the status of mathematical certitude. • Christopher Clavius stressed the anti-sceptical function of mathematics, and Francisco Sánchez offered a refutation in a letter of 1589. • Artifacts and senses support the certainty of mathematics, according to Sánchez. • Conjectural empiricism could be applied to mathematical knowledge. • Clavius's Prolegomena are crucial to analyzing Sánchez's position on the certainty of mathematics. In this paper I analyse Francisco Sánchez's role in the Quaestio de certitudine mathematicarum debate. Despite some studies on the philosophical and medical scepticism of Sánchez and, his extant letter with Christopher Clavius, a participant in the debate, we have few analyses about Sánchez's position regarding the certainty of mathematics. Sánchez discussed some problems that Clavius analysed in his Prolegomena to propose an empirical basis for mathematics through a questioning of its certainty. I will trace the conceptual connections between Sánchez's 1589 letter to Clavius and the Quaestio debate, to introduce Sánchez's sceptical approach to analysing the certainty of mathematics. [ABSTRACT FROM AUTHOR]

This special issue focuses on perspectives on educational effectiveness in science and mathematics and addresses the role of non-cognitive measures in Trends in International Mathematics and Science Study (TIMSS). Moving beyond a mere ranking of countries based on their average achievement by making use of the large amount of collected non-cognitive measures, brings many challenges. This special issue aims to address some of these challenges by including contributions with a more conceptual or methodological focus that encompass several non-cognitive scales, international comparative studies or multiple data points over time. The conceptual contributions are clustered around aspects of engagement and motivation, the experience of school bullying and its effects, and quality and equality. The methodological contributions study topics such as measurement invariance, sampling and teacher-centric inferences, and trend scaling. We hope that the papers in this special issue may stimulate and guide future work in the area of TIMSS. • Introduction on a special issue regarding non-cognitive measures in TIMSS. • Exploring the role of non-cognitive measures in educational effectiveness research. • Elicits challenges regarding non-cognitive measures in TIMSS. • Includes contributions at a conceptual and methodological level. • Discusses implications for future research. [ABSTRACT FROM AUTHOR]

In a series of recently published studies purportedly on the "additive-area heuristic," Yousif & Keil (2019, 2020) argue for a systematic distortion in the perception of the cumulative area of an item array and further claim that previous findings of numerical cognition and magnitude perception in general are "at risk" (Yousif & Keil, 2021). This commentary describes serious stimulus design flaws present in all of Yousif and colleagues experiments that prevent from making such conclusions. Specifically, item arrays used in those studies demonstrate a skewed correlational structure between selected magnitude dimensions and exhibit unbalanced ranges across different magnitude dimensions of interest. Because the perception of magnitude dimensions interferes one another and because our perceptual system is sensitive to the statistical regularities of the sensory input, such a biased design makes it difficult, if not impossible, to interpret the choice behavior of an observer making magnitude judgments. By re-introducing the mathematical framework for a systematic construction of dot array stimuli (DeWind et al., 2015) and by re-analyzing the data from another recent study on area perception (Tomlinson et al., 2020), this paper explains-and introduces a MATLAB code for-an optimal method for designing and constructing dot arrays for magnitude perception studies. [ABSTRACT FROM AUTHOR]

South Africa exhibits one of the largest pro-girl gaps in education, yet the reasons behind this gap are poorly understood. This paper analyses South Africa's pro-girl gap in Grade 4 reading and Grade 5 mathematics achievement. I make use of Oaxaca-Blinder decomposition analysis to decompose the observed gender gaps into their explained and unexplained components, separately by school socio-economic quintile. This approach allows me to present new evidence that part of the country's pro-girl gaps in reading and mathematics are due to girls progressing through the early grades faster than boys. I also find that South African girls score higher on the constructs aimed at tapping student attitudes toward reading (reading self-efficacy and enjoyment, and engagement in reading lessons), and that these differences contribute significantly to the pro-girl gap in PIRLS reading achievement. Interestingly, despite outperforming boys in the TIMSS mathematics assessment, South African girls do not display more positive attitudes toward mathematics. The results suggest much more needs to be done to understand the reasons behind boys' disadvantage in the early grades, since much of the pro-girl achievement gap in Grade 4 reading and Grade 5 mathematics can be attributed the pro-girl advantage in grade completion in earlier grades. • South African girls outperform boys in both PIRLS and TIMSS. • This is partly due to girls progressing faster through the early grades. • Girls also display more positive attitudes toward reading than boys. [ABSTRACT FROM AUTHOR]