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In the paper James et al. (1995), the authors established a compact framework for general n × n system of chromatography (1.1) by using the kinetic formulation coupled with the compensated compactness method. However, how to construct suitable approximated solutions { u i l } of system (1.1) and then to prove the compactness of η ( u i l ) t + q ( u i l ) x in H l o c − 1 , for the entropy–entropy flux pairs ( η , q ) constructed by the kinetic formulation, with respect to the sequence { u i l } , is an open problem. In this paper, we construct the approximated solutions { u i ε } by using the parabolic viscosity method. By carefully calculating the Riemann invariants of system (1.1) , we obtained all necessary estimates in the compact framework of James et al. (1995), and gave a complete proof of the global existence of weak solutions for the Cauchy problem (1.1) with the bounded, nonnegative initial data (1.2) . As a direct by-product, when the total variation of the initial data is bounded, we obtained a simple proof of the existence of global weak solutions by applying the Div-Curl lemma in the compensated compactness theorem to some pairs of functions ( c , f ( u i ) ) , where c is a constant. [ABSTRACT FROM AUTHOR]

In a paper published in 1939, Ernest Nagel described the role that projective duality had played in the reformulation of mathematical understanding through the turn of the nineteenth century, claiming that the discovery of the principle of duality had freed mathematicians from the belief that their task was to describe intuitive elements. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon. In this paper I will argue that a reassessment is overdue. Something beyond doubt is that category theory has an enormous amount to say on the subject, for example, in terms of arrow reversal, dualising objects and adjunctions. These developments have coincided with changes in our understanding of identity and structure within mathematics. While it transpires that physicists have employed the term ‘duality’ in ways which do not always coincide with those of mathematicians, analysis of the latter should still prove very useful to philosophers of physics. Consequently, category theory presents itself as an extremely important language for the philosophy of physics. [ABSTRACT FROM AUTHOR]

In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying and analysing structural properties of Brauer configurations and Brauer configuration algebras, we show that a Brauer configuration algebra is multiserial; that is, its Jacobson radical is a sum of uniserial modules whose pairwise intersection is either zero or a simple module. The paper ends with a detailed study of the relationship between radical cubed zero Brauer configuration algebras, symmetric matrices with non-negative integer entries, finite graphs and associated symmetric radical cubed zero algebras. [ABSTRACT FROM AUTHOR]

Let S be a set of at least two vertices in a graph G. A subtree T of G is an S -Steiner tree if S ⊆ V (T). Two S -Steiner trees T 1 and T 2 are internally disjoint (resp. edge-disjoint) if E (T 1) ∩ E (T 2) = 0̸ and V (T 1) ∩ V (T 2) = S (resp. E (T 1) ∩ E (T 2) = 0̸). Let κ G (S) (resp. λ G (S)) be the maximum number of internally disjoint (resp. edge-disjoint) S -Steiner trees in G , and let κ k (G) (resp. λ k (G)) be the minimum κ G (S) (resp. λ G (S)) for S ranges over all k -subsets of V (G). It is well-known that the connectivity of the line graph of a graph G is closely related to the edge-connectivity of G. Chartrand et al., Li et al., and Li et al. showed that κ k (L (G)) ≥ λ k (G) for k = 2 , 3 , 4 , respectively. In [Discrete Math. 341(2018), 1945–1951.], Li et al. also proposed the following conjecture: κ k (L (G)) ≥ λ k (G) for integer k ≥ 2 and a graph G with at least k vertices. In this paper, we show that κ k (L (G)) ≥ λ k (G) − m a d (G) 2 2 for general k , where m a d (G) is the maximum average degree of a graph G. We also show that κ k (L (G)) ≥ λ k (G) − r 2 2 for any integers k and r such that k ≤ r 2 . Finally, we show that the conjecture holds for k = 5 , and the conjecture holds for the wheel graph and the complete bipartite graph K 3 , n. [ABSTRACT FROM AUTHOR]

This paper discusses the theory of oscillatory normal modes, and its extension to general multi-body mechanical systems. We first review the efforts to generalize modal analysis to the non-linear case. This body of knowledge is vast and spread across several subfields of mathematics, physics, and engineering. We concisely summarize these results, and connect them together using a language familiar to the control theorist. We then propose a direct way of extending nonlinear modes to mechanical systems with a non Euclidean metrics (e.g. robots). [ABSTRACT FROM AUTHOR]

• A cross-sectional study on the relation between math performance and, cheating is proposed. • Data comes from an experiment performed in 10 primary school located in the outskirts of Goma (Congo, DR). • 157 children were administered a questionnaire including a Dice Rolling Task and a Dictator Game. • Children's Total score performance (i.e. the sum of all individual subjects' scores) is positively associated with cheating levels. • Math-skilled children cheat more (on average) when gains from cheating are larger. This paper provides a novel contribution on the relation between school performance and cheating behavior by analyzing a sample of 157 pupils in 10 primary schools located in the outskirts of Goma (Congo, DR). In two subsequent school years, children were administered a questionnaire that included a modified Dice Rolling Task (DRT) and a Dictator Game (DG), while information on their school performance was obtained through the collection of school reports. We analyze whether cheating (measured through DRT) could be explained by school performance (measured by Math and Total scores) when controlling for individual (such as age, sex, altruism and previously recorded cheating attitudes) and background (such as class, school, interviewer) characteristics. Our results show that cheating is positively and significantly correlated with better school performance. Our main contribution shows that children better at math, in contrast to generally high-performing students, are more likely to lie when rewards from lying (i.e. the difference between two dice in a given roll) are higher. [ABSTRACT FROM AUTHOR]

• Pre-service mathematics teachers' narrated failure, identity and resilience are explored. • Mindset framework fruitfully brings together the constructs of failure, identity and resilience. • Self-expectations and goals are a key component of one's subjective understanding of failure. • Rethinking and redefining self-expectations and goals is a strategy that leads to resilience. • The constructs of failure and failure resilience need a subjective approach to investigation. In educational research, failure has often been touched upon as a by-product, but rarely has failure been investigated without attempting to find ways around it instead of attempting to simply understand it better. This paper provides insight into a pilot study on understanding math failure through narrative. I analyse two pre-service mathematics teachers' narratives about failure and identity in order to illustrate their personal understandings of their own failures. The analysed stories provide insight into the subjects and their failure resilience, and initiate a discussion on subjective understanding of resilience. These stories also illustrate the need to further research regarding the subjective understandings of failure in general and in math specifically. [ABSTRACT FROM AUTHOR]

In this note, we correct an inaccuracy of Eq. (13) in the proof of Theorem 1 in the paper (Higham and Kloeden, 2007). We also give a complete proof of the theorem by applying the Burkholder–Davis–Gundy Inequality. [ABSTRACT FROM AUTHOR]

Let κ (G) , μ n − 1 (G) , λ 2 (G) and q 2 (G) denote the vertex-connectivity, the algebraic connectivity, the second largest adjacency eigenvalue, and the second largest signless Laplacian eigenvalue of G , respectively. In this paper, we prove that for an integer k > 0 and any simple graph G of order n with maximum degree Δ and minimum degree δ ≥ k , the vertex-connectivity κ (G) ≥ k if μ n − 1 (G) > H 2 (Δ , δ , k) or λ 2 (G) < δ − H 2 (Δ , δ , k) or q 2 (G) < 2 δ − H 2 (Δ , δ , k) , where H 2 (Δ , δ , k) = (k − 1) n Δ (n − k + 1) (k − 1) + 4 (δ − k + 2) (n − δ − 1) , which improves the result in [Appl. Math. Comput. 344–345 (2019) 141–149] and the result in [Electron. J. Linear Algebra 34 (2018) 428–443]. Analogue results involving μ n − 1 (G) , λ 2 (G) and q 2 (G) to characterize vertex-connectivity of regular graphs, triangle-free graphs and graphs with fixed girth are also presented. [ABSTRACT FROM AUTHOR]

Making is a playful exploration of tools and materials to design personally meaningful artifacts, providing a particularly impactful entry point for traditionally underrepresented youth in science, technology, engineering, and mathematics (STEM) fields. However, it remains unclear how these constructionist explorations translate to eventual professional and educational STEM opportunities, especially for women. This paper tracks an exemplary case in a makerspace to theorize, describe, and analyze the behavioral patterns of young women as they engage in making and move toward expertise in STEM. Building on a material-based and constructionist notion of making, we use mediated discourse analysis to examine how recognition (materialized in artifacts as displaying, legitimizing, and circulating emergent STEM expertise) leads to transformational development over time. We introduce the notion of tinkering with development , which conceptualizes playful project design, spatial project placements, and emergent online project sharing as drivers of human developmental trajectories. Implications of this work include a set of design principles to support makerspaces and other constructionist learning environments to foster participation in STEM. Further, implications for constructionist theory and STEM gender representation are discussed. Image 1 • Recognition plays a vital role in STEM learning and pathway development. • Honoring women's STEM expertise can counteract their turning away from STEM. • Constructionism has underspecified how sharing can cultivate recognition. • Displaying, legitimizing, and circulating expertise can foster recognition. • Personally meaningful artifacts produce and display growing expertise. [ABSTRACT FROM AUTHOR]

Due to the lack of phase-change energy storage modules in the TRNSYS software, this paper applies the numerical simulation method to develop a TRNSYS module. Research has been conducted on the characteristics of the shell-and-tube phase-change energy storage system in order to provide a reasonable basis for its application in practical engineering. According to the principles of energy conservation, a numerical model has been proposed to calculate the temperature field and fluid temperature field of the phase-change unit, and a mathematical model is applied to guide the editing of the TRNSYS module. A Fluent model was established to compare the TRNSYS simulation results and verify the physical validity of the module. In terms of the thermal outlet temperature of the HTF, the maximum relative error between the two in the melting phase is noted to be 1.17%, and the maximum relative error at the solidification stage is 1.83%.Different flow parameters of the system have been studied. The results of the study indicate that differences in inlet temperatures have a greater influence on the average temperature and outlet temperature of the PCM, whereas differing inlet flows have a negligible influence on the phase-change unit. • A numerical simulation method for shell-and-tube phase change energy storage unit was proposed. • The phase change energy storage module in TRNSYS was established. • Using CFD simulation to compare and analyze the accuracy of the module in TRNSYS. • The established TRNSYS module is used to analyze the influence of different parameters on the phase transition process. [ABSTRACT FROM AUTHOR]

• Closely examining the mathematical details of classroom interaction is essential. • Engaging in details of others' mathematical ideas relates to achievement. • Productive teacher support depends on following up on invitations to students. • Interaction varies across classroom participation structures. This paper traces the development of a program of work that seeks to understand the teacher practices that promote productive dialogue and learning in mathematics classrooms. This work originated in cooperative/collaborative learning research, with a focus on understanding group dynamics effective for student learning. Over time, it layered on increasing attention to the teacher's role, and branched out to consider the multiple participation structures in the classroom (whole-class discussion, small-group collaborative work, private student-student conversations). Each phase of this work brought in a heightened focus on the details of classroom interaction around mathematics—both in the thinking that students share and in the practices teachers use to support student participation—to better understand the development of students' mathematics learning. [ABSTRACT FROM AUTHOR]

Ultrasound shear wave elastography (SWE) is an imaging modality used for noninvasive, quantitative evaluation of tissue mechanical properties. SWE uses an acoustic radiation force to produce laterally propagating shear waves that can be tracked in spatial and temporal domains in order to obtain the wave velocity. One of the ways to study the viscoelasticity is through examining the shear wave velocity dispersion curves. In this paper, we present an alternative method to two-dimensional Fourier transform (2D-FT). Our unique approach (2P-CWT) considers shear wave propagation measured in two lateral locations only and uses wavelet transformation analysis. We used the complex Morlet wavelet function as the mother wavelet to filter two shear waves at different locations. We examined how the first signal position and the distance between the two locations affect the shear wave velocity dispersion estimation in 2P-CWT. We tested this new method on a digital phantom data created using the local interaction simulation approach (LISA) in viscoelastic media with and without added white Gaussian noise to the wave motion. Moreover, we tested data acquired from custom made tissue mimicking viscoelastic phantom experiments and ex vivo porcine liver measurements. We compared results from 2P-CWT with the 2D-FT technique. 2P-CWT provided dispersion curves estimation with lower errors over a wider frequency band in comparison to 2D-FT. Tests conducted showed that the two-point technique gives results with better accuracy in simulation results and can be used to measure phase velocity of viscoelastic materials. [ABSTRACT FROM AUTHOR]

Child Maltreatment Fatalities (CMFs) have been alleged to exert great pressures on the delivery of Child Protective Services. Anecdotal accounts implicate CMFs as a cause of "vicious cycles" defined as sequences of scandal, outrage, and reform that destabilize and re-stabilize agency operations and personnel. In this paper we test the CMF cycle hypothesis using 22 years of data from the National Child Abuse and Neglect Data System (NCANDS). We employ Fourier time series and panel regression and find some evidence to support a cycle of CMF case accumulations that repeat at an 8 to 10 year interval. We show that previous levels of CMFs are much better predictors of subsequent CMFs than risk assessment factors used in much of the child welfare literature. We discuss the reasons why predictive analytics using traditional risk factors will continue to perform unsatisfactorily, and why dynamic models of CMFs will always perform better. Finally we consider how our research on the relative efficacy of traditional risk-as-hazard modelling vis a vis dynamic modelling can be used by academics and researchers to help practitioners operate more effective child welfare agencies. • Child Maltreatment Fatalities (CMFs) follow a periodic cycle. • Dynamic models outperform traditional models in predicting subsequent CMFs. • Complete elimination of CMFs is an unrealistic goal in child welfare. [ABSTRACT FROM AUTHOR]

A number of recent studies found evidence for shared structural representations across different cognitive domains such as mathematics, music, and language. For instance, Scheepers et al. (2011) showed that English speakers' choices of relative clause (RC) attachments in partial sentences like The tourist guide mentioned the bells of the church that … can be influenced by the structure of previously solved prime equations such as 80-(9 + 1) × 5 (making high RC-attachments more likely) versus 80-9 + 1 × 5 (making low RC-attachments more likely). Using the same sentence completion task, Experiment 1 of the present paper fully replicated this cross-domain structural priming effect in Russian, a morphologically rich language. More interestingly, Experiment 2 extended this finding to more complex three-site attachment configurations and showed that, relative to a structurally neutral baseline prime condition, N1-, N2-, and N3-attachments of RCs in Russian were equally susceptible to structural priming from mathematical equations such as 18+(7+(3 + 11)) × 2, 18 + 7+(3 + 11) × 2, and 18 + 7 + 3 + 11 × 2, respectively. The latter suggests that cross-domain structural priming from mathematics to language must rely on detailed, domain-general representations of hierarchical structure. [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]

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]

This paper presents an empirical study examining the professional development needs of out-of-field mathematics teachers in the Irish context (n = 202). An online questionnaire gathered data on cohort demographics and confidence with regard to teaching mathematics. A paper-and-pencil test evaluated teachers' cognitive and conceptual proficiency with curriculum-aligned mathematical content. Low achievement levels and high occurrence of conceptual errors indicate inadequate subject content knowledge and difficulties with the content of the curriculum. Additionally, these teachers described themselves as either somewhat or very confident in teaching all content. The discrepancies which exist between confidence levels and content proficiency highlight the importance of designing appropriate professional development programs that are situated and practice based. [ABSTRACT FROM AUTHOR]

This paper provides new results on the application of semidefinite optimization to satisfiability by studying the connection between semidefinite optimization and minimal unsatisfiability. We use a semidefinite least squares problem to assign weights to the clauses of a propositional formula in conjunctive normal form. We then show that these weights are a measure of the necessity of each clause in rendering the formula unsatisfiable, the weight of a necessary clause is strictly greater than the weight of any unnecessary clause. In particular, we show the following results: first, if a formula is minimal unsatisfiable, then all of its clauses have the same weight; second, if a clause does not belong to any minimal unsatisfiable subformula, then its weight is zero. An additional contribution of this paper is a demonstration of how the infeasibility of a semidefinite optimization problem can be tested using a semidefinite least squares problem by extending an earlier result for linear optimization. The connection between the semidefinite least squares problem and Farkas’ Lemma for semidefinite optimization is also discussed. [ABSTRACT FROM AUTHOR]

While a wealth of resources is available for teaching research writing of traditional IMRD research papers, instructors have little to draw on when working with graduate students in mathematics. The present study offers insight into recent mathematical research articles through an exploration guided by phrase-frames, recurring multiword units with a variable slot or blank. The paper introduces the concept of key phrase-frames. In doing so, the potential of phrase-frames for understanding subregisters and for teaching is shown. The starting point of this exploration is a 2,289,670-word corpus of 128 recent mathematics research articles collected from eight scholarly mathematics journals. Five- and six-gram phrase-frames were generated in KfNgram. Key phrase frames are identified as those with an occurrence of at least 20 per million words, occurring across 75% of journals and with a sMAPE of 1.95 cut off or higher when compared to the academic section of the Corpus of Contemporary American English as a reference. The 180 resulting frames were coded for open slot position and common pattern before being consolidated and functionally grouped. Core functions related to the aboutness, coherence and moves of the text. The discussion centers on how these groups reveal different aspects of mathematical texts. [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 paper, nonlinear equations of three-dimensional motion are established for straight fluid-conveying pipes with general boundary conditions. Firstly ten springs at the two ends of the pipes are used to represent the general boundary conditions, and the displacements are expressed as the superposition of a Fourier cosine series and four supplementary functions to satisfy the boundary conditions. Then in the Lagrangian framework, three coupled equations of motion in compact matrix form are derived by using the extended Hamilton's principle, and the effect of the extra linear springs and lumped masses at arbitrary positions along the pipe are also considered in these equations. The natural frequencies of pipes with several different boundary conditions are computed by the linearized dynamic equations in order to validate the method through comparisons with results from other reliable sources. Nonlinear dynamic behavior of the pipes with different boundary conditions is analyzed by solving the nonlinear equations using the Runge–Kutta scheme at last. It is found that the method presented in the paper could conveniently and effectively solve the nonlinear vibration problem of fluid-conveying pipes with general boundary conditions and additional springs and masses. [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]

• CSRP prekindergarten program positively impacted basic reading knowledge. • CSRP prekindergarten program positively impacted early mathematics knowledge. • No difference in program effects for Latinx children. • Positive smaller program effects for Black children. • Program effects suggestively different for children from low-income families. With a large volume of literature pointing to the beneficial impact of pre-kindergarten, researchers, educators, and policymakers are now raising questions about what works for whom, under what conditions, as well as conducting cost-benefit analyses of such endeavors. However, few studies exist on the differential and contextual effects of these programs, creating opportunities for new research in this area. This paper addresses this issue by reporting on the short-term effects of a state-funded pre-kindergarten program, including the potential differential effects among students of color and students from lower income families. Using data from the Connecticut School Readiness Program (CSRP), we use a regression discontinuity design to examine the causal impact of attending the CSRP on average, and by race and income across four different outcomes (reading, oral language, vocabulary, and mathematics). We find that, in the aggregate, attending the CSRP provided the strongest and most consistent outcomes in reading and mathematics. However, heterogeneity analyses indicate variation in impact. The magnitude and significance of reading and mathematics effects were positive but smaller for Black children. Null effects were detected for Latinx students. There was suggestive evidence that mathematics impacts may be smaller for students from low-income families, while reading impacts are uniformly positive irrespective of family income. Possible explanations for differential impact findings are explored. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a new approach for truss optimization against buckling that incorporates both the structural design and the imposed initial imperfections. Euler buckling of slender members, global buckling and stability of sequences of bars are all considered by optimizing the geometric nonlinear response, instead of by imposing a large number of constraints. The proposed optimization problem formulation consists of two alternating steps: (1) Shape optimization that aims to find the worst-case imperfection shape of the joints' positions; (2) Sizing-topology optimization that aims to find the minimum volume of a buckling-resistant structure with the given optimized imperfection. Sensitivity analysis for the two optimization phases follows the adjoint method and solutions are obtained using first-order sequential approximate methods. It is shown that optimizing the imperfection shape stabilizes the convergence as acute instantaneous buckling is avoided. Several numerical examples demonstrate the capability of the proposed formulation to generate buckling-resistant designs with robustness against imperfections. [ABSTRACT FROM AUTHOR]

Abstract Given a finite group G , we say that G has property P n if for every prime integer p , G has at most n − 1 irreducible characters whose degrees are multiples of p. In this paper, we classify all finite groups that have property P 4. We show that the groups satisfying property P 4 are exactly the finite groups with at most three nonlinear irreducible characters, one solvable group of order 168, SL 2 (3) , A 5 , S 5 , PSL 2 (7) and A 6. [ABSTRACT FROM AUTHOR]

Abstract There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called (m , n) -cupping degree and show that there exists an isolated (2 , ω) -cupping degree, and there exists a proper (2 , 1) -cupping degree. It helps understanding various degree structures in the Ershov Hierarchy. [ABSTRACT FROM AUTHOR]

Over the last two decades structural realism has been given progressively more elaborated formulations. Steven French has been at the forefront of the development of the most conceptually sophisticated and historically sensitive version of the view. In his book, The Structure of the World (French (2014)), French shows how structural realism, the view according to which structure is all there is (ontic structural realism), is able to illuminate central issues in the philosophy of science: underdetermination, scientific representation, dispositions, natural modality, and laws of nature. The discussion consistently sheds novel light on the problems under consideration while developing insightful and provocative views. In this paper, I focus on the status of mathematics within French's ontic structural realism, and I raise some concerns about its proper understanding vis-à-vis the realist components of the view. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new shock model called Marshall–Olkin run shock model is defined and studied. According to the model, two components are subject to shocks that may arrive from three different sources, and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3, i = 1 , 2. Reliability and mean residual life functions of such components are studied when the times between shocks follow phase-type distribution. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the n -width problem for the Sobolev space of periodic functions, H per r (0 , 2 π). Building on a theorem of Pinkus we show that it admits optimal even-dimensional spline spaces of all degrees ≥ r − 1. Then, by using a theorem of Karlovitz, we show that it does not admit any optimal spline space of odd dimension > 1. [ABSTRACT FROM AUTHOR]

Abstract The sensitivity conjecture proposed by Nisan and Szegedy in 1994, which asserts that for any Boolean function, its sensitivity complexity is polynomially related to the block sensitivity complexity, is one of the most important and challenging problems in the study of decision tree complexity. Despite a lot of efforts, the best known upper bound of block sensitivity, as well as the certificate complexity, is still exponential in terms of sensitivity. In this paper, we give a better upper bound for certificate complexity and block sensitivity, b s (f) ≤ C (f) ≤ (8 9 + o (1)) s (f) 2 s (f) − 1 , where b s (f) , C (f) and s (f) are the block sensitivity, certificate complexity and sensitivity, respectively. The proof is based on a deep investigation on the structure of the sensitivity graph. We also provide a tighter relationship between the 0-certificate complexity C 0 (f) and 0-sensitivity s 0 (f) for functions with small 1-sensitivity s 1 (f). [ABSTRACT FROM AUTHOR]

• We examine the relationship between teacher meanings and conventions. • Teachers often do not dissociate conventions from logico-mathematical operations. • We provide implications for teaching in relation to conventions of graphing. In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals' meanings for graphs and related topics. Specifically, we draw on data from pre-service (PST) and in-service (IST) teachers to characterize such relationships. We use PSTs' responses during clinical interviews to illustrate three themes: (a) some PSTs' responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs' meanings; (b) some PSTs' responses implied they understood certain practices in U.S. school mathematics as customary choices not necessary to represent particular mathematical ideas; and (c) some PSTs' responses exhibited what we or they perceived to be contradictory actions and claims. We then compare our PST findings to data collected with ISTs. [ABSTRACT FROM AUTHOR]

• Mathematicians may value pedagogical content knowledge but lack confidence in teaching for its development. • Mathematicians and mathematics educators may value pedagogical content knowledge for similar reasons. • Expectancy-value theory may explain a lack of focus on developing teachers' pedagogical content knowledge. Recent studies suggest that change is needed in undergraduate mathematics capstone courses for prospective secondary teachers. One promising but infrequently used strategy for improvement is to incorporate tasks that explicitly focus on pedagogical content knowledge (PCK). Expectancy-value theory provides an account for why instruction of these courses does not more regularly employ this strategy. To make this argument, this paper uses an interview-based study of 9 mathematicians that investigated the process of prioritizing tasks and goals for these courses. As the study found, these mathematicians valued developing teachers' PCK. However, they were unconfident of their ability to teach with tasks and goals focused on developing teachers' PCK relative to more purely mathematical tasks and goals. The central implication is that interventions in mathematicians' teaching must take into account the possibility that it may be just as important to improve confidence and resources as it is to change values. [ABSTRACT FROM AUTHOR]

Abstract This paper discusses the quasilinear Schrödinger equation − Δ u + V (x) u − Δ [ (1 + u 2) 1 2 ] u 2 (1 + u 2) 1 2 = K (x) f (u) , x ∈ R N , where N ⩾ 3. Under appropriate assumptions on the potentials V and K and local sublinear growth assumptions on the nonlinear term f , we get the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution. [ABSTRACT FROM AUTHOR]

Abstract Let I ⊆ k [ x 1 , ... , x n ] be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution F of k [ x 1 , ... , x n ] / I. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on F. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive. On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general. Finally, we show that it is enough to modify the last map of F to ensure that it admits the structure of a DG algebra. [ABSTRACT FROM AUTHOR]

Abstract Let g E H be the Eguchi–Hanson metric on the blow-up of ℂ 2 at the origin. In this paper we show that m g E H is not balanced for any positive integer m. [ABSTRACT FROM AUTHOR]

Abstract Science, Technology, Engineering and Mathematics (STEM) education is recognized as a top school education priority worldwide and Inquiry-based teaching and learning is identified as a promising approach. To effectively engage students in Inquiry tasks, appropriate guidance should be provided, usually by combining digital tools such online labs and modeling tools. This is a cumbersome task for teachers, since it involves manually assessing the type/level of tool-supported guidance to be provided and potentially refining these to meet guidance needs of individual students. In our research we target to investigate how to support this systematic reflection process with educational data analytics methods and tools from both the design and the delivery of Inquiry-based educational designs (IED). The contribution of this paper is to propose a novel "Teaching and Learning" Analytics method and research prototype tool, extending the scope of purely learning analytics methods, to analyze IED in terms of the tool-supported guidance they offer and relate these analyses to students' educational data that are already being collected by existing learning analytics systems, increasing teachers' awareness. A two-layer evaluation methodology positively assessed the capacity of our method to analyze IED and provided initial evidence that the insights generated offer statistically significant indicators that impact students' activity during the delivery of these IED. The insights of this work aim to contribute in the field of cognitive data analytics for teaching and learning, by investigating new ways to combine analyses of the educational design and students' activity, and inform teachers' reflection from a holistic perspective. Highlights • Tool-supported Guidance is essential for effective Inquiry-based education. • Teaching and Learning Analytics (TLA) can support teachers provide appropriate Guidance. • The TLA method and supporting tool provides analyses of the level of Guidance in Inquiry-based scenarios. • Analyses of the design can be investigated against customizable learners' data and profiles. • Insights from these combined analyses could help teachers improve their teaching designs. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we first characterize the structure of symmetries J such that a projection P is J -contractive. Then the minimal and maximal elements of the symmetries J with P ⁎ J P ⩽ J (or J P ⩾ 0) are given. Moreover, some formulas between P (2 I − P − P ⁎) + (P (2 I − P − P ⁎) − ) and P (P + P ⁎) − (P (P + P ⁎) + ) are established. [ABSTRACT FROM AUTHOR]

Abstract This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem, providing a linear algebra approach to the domination problem. We show that several Grundy domination type parameters are bounded above by minimum rank type parameters. We also give a method to calculate the L -Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the L -Grundy domination number. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide realizations of Herglotz functions in the quaternionic case where hyperholomorphic functions are meant in the sense of Fueter. We consider the counterparts of the Arveson space and of the Herglotz–Agler functions by introducing suitable positive kernels. Crucial tools are the so-called Fueter variables and the CK-product. Our realization results also show that positivity implies analyticity. [ABSTRACT FROM AUTHOR]

Abstract In the current paper, an error estimate has been proposed to find a second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. The convergence order of the proposed method is O (τ 2 + h 2). The numerical results show the efficiency of the new technique. [ABSTRACT FROM AUTHOR]