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Dyslexia is a type of Specific Learning Disability (SLD) that affects the educational development of a student, as it is the reason of difficulties not only in writing or reading but also in other disciplines like Mathematics. Research has proven that Information and Communication Technologies (ICT) can have a positive effect to the educational development of a dyslexic student. In this paper, twenty 6 th -grade primary-school students were observed while using computers during exams - ten with Dyslexia and ten with no SLDs. All students were called to solve a digital test and a paper test that were including some basic mathematic operations. The researcher recorded and examined the differences between the results from the two tests. Descriptive and deductive statistical analysis showed that students had better results when examined via a computer compared to a traditional written examination. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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

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]

Abstract: The authors presented in their paper a theoretical route describing the formation of myrcene (2) in dependence of the reaction temperature during the production from β-pinene (1) [M.B. Kolicheski, L.C. Cocco, D.A. Mitchell, M. Kaminski, J. Anal. Appl. Pyrol. 80 (2007) 92]. Pyrolysis experiments carried out seem to correlate with the mathematical model. Initial reaction schemes for the formation of 2 as well as for decomposition reactions were presented. However, reconsideration of the data reviled, serious interpretation flaws regarding content and reaction mechanisms were found. [Copyright &y& Elsevier]

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]

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]

In mathematics, a dissection of a square (or rectangle) into non-congruent rectangles is a Mondrian partition. If all the rectangles have the same area, it is called a perfect Mondrian partition. In this paper, we present a computational result by which we can affirm that there is no perfect Mondrian partition of a length n square for n ≤ 1000. Using the same algorithm we have been able to establish that there is no perfect Mondrian partition of a n × m rectangle for n , m ≤ 400. [ABSTRACT FROM AUTHOR]

Let G be a connected simple graph with vertex set V (G) and edge set E (G). For any subset η of V (G) with | η | ≥ 2 , let π G (η) denote the maximum number t of paths P 1 , P 2 , ... , P t in G such that P i contains η , V (P i) ∩ V (P j) = η and E (P i) ∩ E (P j) = 0̸ for any distinct i , j ∈ { 1 , 2 , ... , t }. For an integer k with 2 ≤ k ≤ | V (G) | , the k -path connectivity π k (G) of G , which can more accurately assess the reliability of networks, is defined as min { π G (η) | η ⊆ V (G) and | η | = k }. Since deciding whether π G (η) ≥ ℓ for a general graph is NP-complete in [Graphs Combin. 37(2021)2521–2533], there are few results about k -path connectivity even for k = 3. In this paper, we obtain the exact value of the 3-path connectivity of the exchanged hypercube E H (s , t) and show that π 3 (E H (s , t)) = 3 × min { s , t } + 2 4 which improves the known result about the 3-tree connectivity [Appl. Math. Comput. 347(2019)342–353]. As a corollary, the 3-path connectivity of the n -dimensional dual cube D n is obtained directly. [ABSTRACT FROM AUTHOR]

Diagnosability is a fundamental consideration when designing an interconnected network. The PMC and MM ∗ fault diagnosis models are the two most commonly used models. Both the g -good-neighbour diagnosability and g -extra diagnosability of an interconnection network have been two of the hot topics in the intersectional research areas of Graph theory and Computer Science, which become increasingly attractive for new solutions to real-world problems. However, there are still some problems in the transformation from the concepts of Computer Science to that of mathematics. In this paper, we systematically study such problems and give a strict proof from concepts to mathematical conclusions. In the terms of results, we not only give the relationship between g -good-neighbour diagnosabilities of the network under PMC model and MM ∗ model, but also between g -extra diagnosabilities of the network under PMC and MM ∗ models. To apply our results, we give an application on the enhanced hypercube in the end and derive a lemma explaining whether these are 3-cycles in enhanced hypercubes and how many common neighbours for two vertices of enhanced hypercubes under different values of k in the meantime. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

Let N be the set of all positive integers. For a set A of positive integers, let A ∼ N denote that A contains all but finitely many positive integers. For an integer h ⩾ 2 , define h A = { a 1 + ⋯ + a h : a 1 , ⋯ , a h ∈ A } and h × A = { a 1 + ⋯ + a h : a 1 , ⋯ , a h ∈ A , a i ≠ a j for i ≠ j }. In 2023, Chen and Yu [Discrete Math. 346 (2023), Paper No. 113388.] proved that, there exists a set B of positive integers such that: lim x → ∞ ⁡ B (x) / x = 1 / 2 , B ⋃ (2 B) ∼ N , B ⋃ (2 × B) ≁ N , and B ⋃ (2 × B) ⋃ (3 × B) ∼ N. In this paper, we construct a somewhat dense set B satisfying the above properties. That is, there exists a set B of positive integers such that: lim inf x → ∞ B (x) / x = 1 / 2 , lim sup x → ∞ B (x) / x = 1 , B ⋃ (2 B) ∼ N , B ⋃ (2 × B) ≁ N , and B ⋃ (2 × B) ⋃ (3 × B) ∼ N. [ABSTRACT FROM AUTHOR]

We correct an error occurred in Lemma 10, and a couple of typos in the following paper P. Dulio, S.M.C. Pagani A rounding theorem for unique binary tomographic reconstruction , Discrete Appl. Math. 268 (2019) 54–69. The corrections do not alter anyone of the obtained results. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

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]

• 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]

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: A integro-differential equation (IDE) model of a cyclic competitive system is analyzed. Linear stability analysis are performed about the spatial constant solutions, and conditions are derived under which they are stable or unstable. Spatial patterns along with the instability of the nontrivial equilibrium are also observed. [Copyright &y& Elsevier]

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

This paper examines mathematics teaching and learning, specifically of Geometry, in Greek girls' schools in the 19th century. It explores how educational laws and school practice defined its teaching. Research has proved that female students received only the basics in Geometry, substantially less than what was offered to male students in all-boys' schools. Also, the Geometry textbooks designed for girls are discussed. The problems considered in the article are at the intersection of economic, political and ideological issues. The study draws on a wide range of primary sources such as school archives and records as well as government gazettes. In particular, the school archives of the Philekpedeutiki Etaireia provide this research with a rich source of information regarding female schooling in 19th century. [ABSTRACT FROM AUTHOR]

Track-to-track association (T2TA) aims at unifying batch numbers of tracks, reducing redundancy, and clarifying the situation. It is the precondition and foundation of track fusion, situation awareness, and traffic control. Existing methods based on statistical reasoning or fuzzy math bring some problems that are difficult to solve simultaneously, such as unreasonable assumptions, unsuitable models, uncertain thresholds, and long association times. In the light of the above problems, in this paper, we proposed a T2TA method via Track Fusion and Track Segmentation (TF-TS). The track fusion module fuses and extracts track features from several tracks to reduce the dependency on prior assumptions, motion models, and thresholds. The track segmentation mapping module transforms track tensors into association matrices directly to improve association efficiency. With four kinds of constraints, the association matrices inferred by TF-TS are close to the real one. TF-TS can reduce the dependence on the assumptions, motion models, and thresholds. It can also reduce the traversal calculation of tracks and increase the association efficiency. The global AIS tracks are used to train and test our model, and association results demonstrate that the proposed method can associate tracks in complex scenarios. Moreover, the efficiency is improved and the demand for the real-time association is satisfied. [ABSTRACT FROM AUTHOR]

• Collective engagement structures describe group's engagement type. • Case study illustrates one group's collective engagement. • Engagement structures and positioning theory ground the framework. This paper proposes a new social construct and conceptual-analytic framework for the study of group engagement in mathematics, which we call collective engagement structures. This new framework is offered to address research and practice-identified needs to better understand how individual student dispositions and preferences might interact with group processes to inform student engagement and learning. We demonstrate the collective engagement structures framework using a case study analysis of three middle school students working in a group on a complex mathematics problem. Our analysis details how three students' individual engagement structures and positional influence interact to form two collective engagement structures: Follow The Leader and Let's Figure This Out. Our findings indicate that students' collective engagement may present itself in identifiable and patterned ways, providing fresh theoretical and practical insight into how to better structure group-centered instruction. [ABSTRACT FROM AUTHOR]

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

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]