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Abstract Basic numerical skills provide an important foundation for the learning of mathematics. Thus, it is critical that researchers and educators have access to valid and reliable ways of assessing young children's numerical skills. The purpose of this study was to evaluate the concurrent, predictive, and incremental validity of a two-minute paper-and-pencil measure of children's symbolic (Arabic numerals) and non-symbolic (dot arrays) comparison skills. A sample of kindergarten children (M age = 5.86, N = 439) were assessed on the measure along with a number line estimation task, a measure of arithmetic, and several control measures. Results indicated that performance on the symbolic comparison task explained unique variance in children's arithmetic performance in kindergarten. Longitudinal analyses demonstrated that both symbolic comparison and number line estimation in kindergarten were independent predictors of 1st grade mathematics achievement. However, only symbolic comparison remained a unique predictor once language skills and processing speed were taken into account. These results suggest that a two-minute paper-and-pencil measure of children's symbolic number comparison is a reliable predictor of children's early mathematics performance. Highlights • We tested the validity and reliability of a recently developed Numeracy Screener. • Performance on the symbolic comparison task predicted kindergarten children's arithmetic performance. • Performance on the symbolic comparison task in kindergarten predicted school mathematics achievement one year later. • The Numeracy Screener may be a useful tool for early mathematics researchers and educators alike. [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]

I agree wholeheartedly with the authors of "At Least One Black Sheep: Pragmatics and Mathematical Language" on their main thesis that the language of mathematics is permeated by phenomena that belong to pragmatics. I also agree with what they take to be the opposite view, which is presented in Mohan Ganesalingam's monograph The Language of Mathematics (2013). I think that the conflict is to a large extent terminological and has to do with what is meant by pragmatics. [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]

Abstract: In this paper, we focus on the proposed algorithms to solve a linear programming problem with the convex combination of the max–min and the max–average composition and the max–star composition, respectively. They have been proposed by Ghodousian and Khorram [A. Ghodousian, E. Khorram, Solving a linear programming problem with the convex combination of the max–min and the max–average fuzzy relation equations, Appl. Math. Comput. 180 (2006) 411–418] and Khorram et al. [E. Khorram, A. Ghodousian, A. Abbasi Molai, Solving linear optimization problems with max–star composition equation constraints, Appl. Math. Comput. 179 (2006) 654–661], respectively. Firstly, we show that the “Tabular method algorithm” in the first paper and the “First procedure” in the second paper may not lead to the optimal solutions of the two models in some cases. Secondly, we generalize the proposed algorithm by Abbasi Molai and Khorram [A. Abbasi Molai, E. Khorram, A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian, Appl. Math. Comput. 190 (2007) 1161–1167] to solve the two models. In fact, it modifies the presented algorithms in the two papers. Finally, some numerical examples are given to illustrate the purposes. [Copyright &y& Elsevier]

Abstract: It is shown that the main theorem of Arslan’s paper (Theorem 2, 2011), as stated, is incorrect. Under additional conditions, we present a short proof of the corrected version of the theorem. We also give a proof of a theorem of Rao and Shanbhag (1991) , employed by Arslan, without the use of the Kolmogorov Consistency Theorem. [Copyright &y& Elsevier]

In mathematics, for professionals, the most compelling way to back a statement is to prove it. This could even be true, as recent research indicates, for 9-year-olds (or for even younger learners), given appropriate conditions in the early grades. In this article, based on data drawn from the three preceding papers, I explore some ways in which the building of an argument can emerge in step with the development of understanding. [Copyright &y& Elsevier]

Abstract: We prove a theorem for the existence of solutions to a variational problem, under assumptions that do not require the convexity of the integrand. [Copyright &y& Elsevier]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

Abstract: In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. [Copyright &y& Elsevier]

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]

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]

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]

Abstract: This paper considers systems that have a measurement that is computed from the post-processing of a short duration image. The measurement can be regarded as the integral of a linear function of the state variables of the system. The input to the system is assumed to be generated with a zero-order hold whose sampling frequency is the same as that of the measurement. The paper presents a discrete-time finite dimensional state variable model for such systems. [Copyright &y& Elsevier]

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]

Abstract: Although most modern processors have hardware support for double precision or double-extended precision floating-point multiplication, this support is inadequate for many scientific computations. This paper presents the architecture of a quadruple precision floating-point multiplier that also supports two parallel double precision multiplications. Since hardware support for quadruple precision arithmetic is expensive, a new technique is presented that requires much less hardware than a fully parallel quadruple precision multiplier. With this architecture, quadruple precision multiplication has a latency of three cycles and two parallel double precision multiplications have latencies of only two cycles. The multiplier is pipelined so that two double precision multiplications can begin every cycle or a quadruple precision multiplication can begin every other cycle. The technique used for the dual-mode quadruple precision multiplier is also applied to the design of a dual-mode double precision floating-point multiplier that performs a double precision multiplication or two single precision multiplications in parallel. Synthesis results show that the dual-mode double precision multiplier requires 43% less area than a conventional double precision multiplier. The correctness of all the multipliers presented in this paper is tested and verified through extensive simulation. [Copyright &y& Elsevier]

A quandle is a set with a self-distributive binary operation satisfying a certain condition. Here we construct a monoid (a semi-group with the identity) associated with a quandle. This monoid has a structure of a quandle, which contains the original quandle as a sub-quandle. We call it the enveloping monoidal quandle. The purpose of this paper is to introduce the notion of the enveloping monoidal quandle, and to investigate it. [Copyright &y& Elsevier]

Abstract: The concept of labelled stratified graph (LSG) was introduced in Ţăndăreanu (Knowledge Inform. Syst. 2(4) (2000) 438) in connection with that of knowledge base with output (KBO). The aim of this paper is to present a distinct facet of this concept. We prove several algebraic properties for LSGs and we conclude that a LSG can be used independently of a KBO. In order to realize this aim we define a partial order on the set of all LSGs over a labelled graph G, an equivalence relation on and a partial order on the factor set. The set endowed with becomes a join semilattice with greatest element. Each equivalence class contains an unique LSG, which is named distinguished representative of C. This is the least element of. Particularly we obtain the distinguished representative for the supremum of two classes (DRS) and the greatest distinguished LSG (the least LSG of the greatest element of, denoted GD). Two applications are presented, one for DRS and one for GD. Several opens problems are briefly exposed in the last section. [Copyright &y& Elsevier]

• Corrected transfer functions of Al-Alaoui and Tustin based discretizations of the Laplacian operator are discussed. • The updated MAPLE code that generates the discretized transfer functions is provided. • The corrected bode plots of the discretized transfer functions are illustrated. This paper points out a number of mathematical inaccuracies in the recent paper "FPGA Realization of Fractional Order Neuron" by S.A. Malik and A.H. Mir. The comments in this paper pertain mainly with the errors presented in Section 3 of that paper referencing the discretization of the fractional operator. Proposed corrections to the formulas and figures are presented along with a sample Maple code to verify the results. [ABSTRACT FROM AUTHOR]

Structuralists typically appeal to some variant of the widely popular 'mapping' account of mathematical representation to suggest that mathematics is applied in modern science to represent the world's physical structure. However, in this paper, I argue that this realist interpretation of the 'mapping' account presupposes that physical systems possess an 'assumed structure' that is at odds with modern physical theory. Through two detailed case studies concerning the use of the differential and variational calculus in modern dynamics, I show that the formal structure that we need to assume in order to apply the mapping account is inconsistent with the way in which mathematics is applied in modern physics. The problem is that a realist interpretation of the 'mapping' account imposes too severe of a constraint on the conformity that must exist between mathematics and nature in order for mathematics to represent the structure of a physical system. • This paper presents a critique of the 'mapping' account of mathematical representation. • The 'mapping' account requires that physical systems possess an assumed structure. • This structure is at odds with modern physical theory. • I argue that a realist interpretation of the account imposes a severe constraint on the conformity between math and nature. • The argument is defended through two case studies concerning the differential and variational calculus. [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]

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]

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

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

Abstract: In this note we show that a multivalued bounded quasimonotone operator of type between a reflexive Banach space and its dual is pseudomonotone. Hence we deduce that a result of Su Ke and He Zhen [Su Ke, He Zhen, Solutions of nonlinear evolution inclusions, J. Math. Anal. Appl. 313 (2006) 84–97] follows from a theorem of Bian and Webb [W. Bian, J.R.L. Webb, Solutions of nonlinear evolution inclusions, Nonlinear Anal. 37 (1999) 915–932]. [Copyright &y& Elsevier]

Abstract: The aim of this short note is to show on a numerical example that one of the two optimization algorithms proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] may not lead to the optimal solution in some cases. Besides it will be pointed out that the other algorithm (correct, but unfortunately ineffective for larger problems) proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] can be extended to a wider class of non-convex optimization problems. [Copyright &y& Elsevier]

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

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

A nonoscillation criterion for a second-order linear difference equation is established correcting a result in [1]. [Copyright &y& Elsevier]

• The first paper to address the bid construction problem with heterogeneous fleet. • An efficient MIP formulation with symmetry-breaking constraints. • Exact and heuristic approaches based on adaptive large neighborhood search. • The heuristic provides optimal or near-optimal solutions to benchmark instances. • Instances with up to 500 contracts are solved with a gap lower than 2.53%. This paper solves the bid construction problem (BCP) for heterogeneous truckload operations. Exact and heuristic solution methods are proposed to solve the BCP. The heuristic framework is based on a destroy-repair principle coupled with local search procedures. A heuristic variant integrating a post-optimization method considering solutions obtained during the heuristic search is also proposed. The exact approach uses CPLEX to solve an adapted arc-based mathematical formulation with our best heuristic solution as an initial solution. Computational results show that the proposed methods perform very well in terms of CPU time and solution quality for existing and new sets of instances. [ABSTRACT FROM AUTHOR]

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

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

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

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]

Abstract Currently, wind data measurement using lidars or other remote sensing instruments are increasing. Lidar measures the wind data over the full wind turbine, but it is not available for a complete period. This paper investigates the application of measure-correlate-predict technique to short-term lidar measurements in order to extrapolate wind shear index. The wind shear exponent from lidar measurements was compared from mast measurements because lidar and mast have a different measurement methodology. The short-term measurement campaign was conducted at Dhanushkodi to analyze the wind data. This paper discussed the application of measure-correlate-predict method to evaluate the performance of the measure-correlate-predict to extrapolate the short term wind shear exponent. A quantitative analysis of the measure-correlate-predict method has been made. The new approach of measuring wind shear exponent using lidar and measure-correlate-predict was used to test the measured data. The six metrics used to evaluate the MCP predictions in the present analysis. The five parameters used to compare the predicted distribution with the actual long-term wind data. The shear measured by lidar over a height range from 10 m to 220 m. Subsequently, the mast data and the MCP method are used to extrapolate the lidar measurement to get the long-term shear exponent. The reference wind data is used by the wind shear exponent from 10 m to 100 m from the met mast. Due to the reduction of error in the wind shear exponent measurement, the uncertainty can sufficiently reduce in the estimation of wind shear exponent. Comparison of the two correlations signifies that the LR method shows in linear relations that have higher intercepts and lower slopes. A difference is found for shear exponent in the range of 0.13–0.28. Highlights • The wind shear exponent from lidar was compared using mast measurement. • The observation from the mast and lidar at Dhanushkodi are used for the comparison. • The measure-correlate-predict technique is applied to provide longer wind statics than the sample period. • The LR method shows the higher intercepts with a difference is found for shear exponent in the range of 0.13–0.28. • The observations show that the LR method result in the best estimates of the normalized biased. [ABSTRACT FROM AUTHOR]

Abstract In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ϕ such that P (G n ⊨ ϕ) does not converge as n → ∞ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices { 1 , ... , n }). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a mathematical proof of the existence of a global minimum of Self-Optim (Self-Optimizing Inverse Analysis Method) cost functional is presented based upon weak-solution theory of partial differential equations. The Self-Optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem. Furthermore, discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified. This paper provides a rigorous mathematical foundation for applications of the Self-Optim method to various inverse problems in mechanics. [ABSTRACT FROM AUTHOR]

Highlights • Generating Pareto curves to show the optimal trade-off between Robustness and Performance. • Find the optimal PI and PID controller for First-Order and Integrating Plus Delay Processes. • Investigating the difference between ideal and serial PID, and the impact of filtering. • Comprehensive comparison between optimal-, SIMC- and "improved" SIMC-tuning. Abstract The first-order plus delay process model with parameters k (gain), τ (time constant) and θ (delay) is the most used representation of process dynamics. This paper has three objectives. First, we derive optimal PI- and PID-settings for this process. Optimality is here defined as the minimum Integrated Absolute Error (IAE) to disturbances for a given robustness level. The robustness level, which is here defined as the sensitivity peak (M s), may be regarded as a tuning parameter. Second, we compare the optimal IAE-performance with the simple SIMC-rules, where the SIMC tuning parameter τ c is adjusted to get a given robustness. The "original" SIMC-rules give a PI-controller for a first-order with delay process, and we find that this SIMC PI-controller is close to the optimal PI-controller for most values of the process parameters (k , τ , θ). The only exception is for delay-dominant processes where the SIMC-rule gives a pure integrating controller. The third objective of this paper is to propose and study a very simple modification to the original SIMC-rule, which is to add a derivative time τ d = θ /3 (for the serial PID-form). This gives performance close to the IAE-optimal PID also for delay-dominant processes. We call this the "improved" SIMC-rule, but we put "improved" in quotes, because this controller requires more input usage, so in practice the original SIMC-rule, which gives a PI-controller, may be preferred. [ABSTRACT FROM AUTHOR]

Highlights • The paper contributes set-membership regression (SMR), a new approach to nonlinear parameter estimation under measurement uncertainty. • Links between SMR and both guaranteed parameter estimation and classical statistical inference are investigated. • Several approaches to describing tight enclosures of the SMR regions using complete- search methods are developed. • Case studies are presented to illustrate various theoretical and computational aspects of SMR. Abstract This paper introduces set-membership nonlinear regression (SMR), a new approach to nonlinear regression under uncertainty. The problem is to determine the subregion in parameter space enclosing all (global) solutions to a nonlinear regression problem in the presence of bounded uncertainty on the observed variables. Our focus is on nonlinear algebraic models. We investigate the connections of SMR with (i) the classical statistical inference methods, and (ii) the usual set-membership estimation approach where the model predictions are constrained within bounded measurement errors. We also develop a computational framework to describe tight enclosures of the SMR regions using semi-infinite programming and complete-search methods, in the form of likelihood contour and polyhedral enclosures. The case study of a parameter estimation problem in microbial growth is presented to illustrate various theoretical and computational aspects of the SMR approach. [ABSTRACT FROM AUTHOR]

In this paper, we continue our study of 2-colorings in hypergraphs (see, Henning and Yeo, 2013). A hypergraph is 2-colorable if there is a 2-coloring of the vertices with no monochromatic hyperedge. It is known (see Thomassen, 1992) that every 4-uniform 4-regular hypergraph is 2-colorable. Our main result in this paper is a strengthening of this result. For this purpose, we define a vertex in a hypergraph H to be a free vertex in H if we can 2-color V ( H ) ∖ { v } such that every hyperedge in H contains vertices of both colors (where v has no color). We prove that every 4-uniform 4-regular hypergraph has a free vertex. This proves a conjecture in Henning and Yeo (2015). Our proofs use a new result on not-all-equal 3-SAT which is also proved in this paper and is of interest in its own right. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogeneous m -machine scheduling problem where unit-time jobs have to be scheduled within their time windows so that, for any subset of machines, the set of the time units at which at least one machine is busy, is an interval. For this problem, a time assignment of the jobs satisfying the time windows constraints is a schedule if it has a pyramidal profile and is a k -schedule if this profile property is satisfied up to level k only. We develop a generic algorithm to solve the problem and show it is correct using a proof that mainly relies on a necessary and sufficient condition for a schedule to exist proved in a previous paper. We finally show that the generic algorithm is polynomial. We conclude by giving the directions of ongoing works and by bringing open questions related to different variants of the basic non-idling m -machine scheduling problem. [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 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]

• 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]