Search

The International Sirius Mathematics Center, located on the Black Sea coast, was founded in 2019 by the Talent and Success Foundation for organizing mathematical conferences and other events promoting the development of mathematics in Russia. In 2024, the Center has acquired its own Russian-language journal for publication of original research papers by participants of conferences held at the Center. The authors are invited to submit papers to the journal at https://ef.msp.org/submit/sirius [ABSTRACT FROM AUTHOR]

Very recently, P. Saha et al. [A global optimality result in probabilistic spaces using control function. Optimization. 2021;70:2387–2400] studied the existence of best proximity points for Banach type ϕ-proximal contractions in the framework of Menger probabilistic metric spaces. In this paper, we present a simple proof of their main existence result which is based on a fixed point theorem for ϕ-contractive self-mappings in complete Menger PM-spaces due to B.S. Choudhury and K. Das [A new contraction principle in Menger spaces. Acta Math Sin. 2008;24:13791386]. [ABSTRACT FROM AUTHOR]

Tablet‐based technology has become a conduit for the administration of curriculum‐based measurement (CBM) tools. Using tablets to administer CBM probes has many advantages; however, little is known about how students perform on CBMs when administered via a tablet. The current investigation compared digits correct per minute obtained from 44 third‐grade students on CBM addition fluency probes administered via three different modalities. Students completed the probes using traditional paper and pencil, a tablet using their fingers to write the answer, or a tablet application using a keyboard to type the answer. A within‐subjects group design showed students performed significantly better on the tablet when using their finger to write the answer, even though most students (58%) indicated this was their least preferred modality. The discussion focuses on the implications for educators and the use of different CBM modalities for interindividual and intraindividual comparisons. Practitioner points: Third‐grade students responded to math facts faster when writing with their fingers on an iPad, even compared to paper and pencil.The results are inconsistent with other studies that found students were faster when using paper and pencil.Finally, most third‐grade students preferred answering addition facts with a pencil on paper. [ABSTRACT FROM AUTHOR]

Computer-based curriculum-based measurement (CBM) is a relatively common practice, but surprisingly few studies have examined the reliability of computer-based CBM. This study sought to examine the reliability of CBM administered via paper/pencil versus the computer. Twenty-one of 25 students in two third-grade classes (N = 21) participated in two generalizability studies. The primary facet of interest, format of assessment, was examined for two measures, a fact operation (fact family measure) and a more complex operation (multidigit addition). Researchers administered four alternate forms of each measure under both computer and paper/pencil conditions. Results indicated ideal reliability of measurement within all conditions and results were consistent across assessment formats for the fact operation. However, assessment format explained a large proportion of variance for the more complex skill (multidigit addition). Results indicate a need to consider the reliability of transitioning assessment from paper/pencil to computer-based response on a measure-by-measure basis. [ABSTRACT FROM AUTHOR]

Example-generation tasks have been suggested as an effective way to both promote students’ learning of mathematics and assess students’ understanding of concepts. E-assessment offers the potential to use example-generation tasks with large groups of students, but there has been little research on this approach so far. Across two studies, we investigate students’ responses to a particular sequence of example-generation tasks, posed either on paper or through e-assessment. We identify a striking difference in students’ example-generation strategies and success, for e-assessment and paper-based versions of the same tasks. This suggests the use of example-generation tasks in e-assessment may not be straightforward, and we conclude by discussing the implications for teaching and research. [ABSTRACT FROM AUTHOR]

Prospective authors are requested to submit new, unpublished manuscripts for inclusion in the upcoming event described in this call for papers. [ABSTRACT FROM AUTHOR]

This paper focuses on the Southern African Association for Research in Mathematics, Science and Technology Education's (SAARMSTE's) role as a platform supporting research dissemination and connecting researchers in early grades mathematics (EGM) in the Southern African region. A review of the Long Papers in SAARMSTE over the last 20 years supports the finding of the other review papers in this Special Issue: that there has been substantial growth of attention to EGM since 2013. However, two distinctions are marked when looking at conference papers rather than journal papers. Firstly, there is a particularly large expansion of work in the last 5 years, with a broadening base of participation in this work. Second, looking across all the formats of conference presentations indicates SAARMSTE's role in supporting and building EGM as a key focus of research attention, and bringing together regional and international groups with interests in this area. Given that conference proceedings usually offer a broader picture of emerging interests than journal papers, we reflect on the range of foci of attention within EGM in the SAARMSTE Proceedings, and trends within this. These trends also help us to point to areas that are likely to be of key interest in the next decade. [ABSTRACT FROM AUTHOR]

If (S, ◦,≤) is an ordered hypersemigroup, an equivalence relation ρ on S is called congruence if (a, b) ∈ ρ implies (a ◦ x, b ◦ x) ∈ ρ and (x ◦ a, x ◦ b) ∈ ρ for every x ∈ S ; in the sense that for every u ∈ a ◦ x there exists v ∈ b ◦ x such that (u, v) ∈ ρ and for every u ∈ x◦a there exists v ∈ x◦b such that (u, v) ∈ ρ. It has been proved in Turk J Math 2021(5) [On the paper “A study on (strong) order-congruences in ordered semihypergroups”] that if S is an ordered hypersemigroup, then there exists a congruence ρ on S such that S/ρ is an ordered hypersemigroup. This result, is the main result for an involution ordered hypersemigroup by Xinyang Feng, Jian Tang and Yanfeng Luo in U.P.B. Sci. Bull, Series A, 2018, but its proof is wrong; the correct proof is given in the present paper. Examples illustrate the results. [ABSTRACT FROM AUTHOR]

Curriculum-based measurement (CBM) tools are increasingly administered through technology-based modalities such as computers and tablets. Two studies were conducted to examine whether students perform similarly on paper-based and tablet-based math fact probes. Ten students completed 1-min addition or multiplication math probes using a single-case multielement design. Students completed the probes using traditional paper and pencil, a tablet using a stylus to write the answer, or a tablet application using a keyboard to type the answer. Visual analysis of time series graphs showed that the majority of students performed better on the paper-based probes than on either the keyboard or stylus probes. Nonoverlap of all pairs (NAP) effect size measures indicated medium to large differences between paper and stylus probes and paper and keyboard probes but weak effects between stylus and keyboard probes. Discussion focuses on the implications for educators and the use of different CBM modalities for interindividual and intraindividual comparisons. [ABSTRACT FROM AUTHOR]

The work focuses on the comparison of basic strategies in the teaching of mathematics. The first option is to use a calculator and ICT in order to solve more complex tasks and not waste time on algorithmically simple tasks. The second option is to solve all tasks with a paper-and-pencil strategy, while the number of tasks will be lower due to the fact that we need time for numerical calculation. Pupils of the 7th grade of elementary school were compared with pupils up to two grades higher in order to find out the differences between their performances. Pupils of the 7th grade do not use calculators during the math classes and thus have to cope with each task without help. Pupils of higher grades have been using calculators for numerically more challenging tasks for the last two years. Results of the nonmetric multidimensional scaling showed that the 7th grade pupils have the different position on the one-dimensional scale than the pupils of higher grades.43T [ABSTRACT FROM AUTHOR]

In this short note we prove that the Kurepa (K) function is different from the Smarandache-Kurepa (SK) function, therefore, these functions are not the same, as Mijajlović has unfoundedly accused the prestigious Encyclopedia of Mathematics and this author. This note is an answer to Mijajlović's paper (Žarko Mijajlović, Fifty years of Kurepa's !n hypothesis, Bulletin T.CLIV de l'Académie serbe des sciences et des arts - 2021 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No. 46, 169-181 (2021). [ABSTRACT FROM AUTHOR]

Many complex physical systems exhibit a rich variety of discrete behavioural modes. Often, the system complexity limits the applicability of standard modelling tools. Hence, understanding the underlying physics of different behaviours and distinguishing between them is challenging. Although traditional machine learning techniques could predict and classify behaviour well, typically they do not provide any meaningful insight into the underlying physics of the system. In this paper we present a novel method for extracting physically meaningful clusters of discrete behaviour from limited experimental observations. This method obtains a set of physically plausible functions that both facilitate behavioural clustering and aid in system understanding. We demonstrate the approach on the V-shaped falling paper system, a new falling paper type system that exhibits four distinct behavioural modes depending on a few morphological parameters. Using just 49 experimental observations, the method discovered a set of candidate functions that distinguish behaviours with an error of 2.04%, while also aiding insight into the physical phenomena driving each behaviour. [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]

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function f : R → R which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points (x k) k ∈ N converging to a point x ∗ where f is discontinuous, lower semicontinuous and whose objective function value f (x ∗) is strictly less than lim k → ∞ f (x k) . Moreover the dDSM generates trial points in only one of the continuity sets of f near x ∗ . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample. [ABSTRACT FROM AUTHOR]

COVID-19 pandemic has raised concerns for a number of strategical social interactions. From quarantines and social distancing to vaccination and geopolitics, the cooperation or even the coordination of efforts has been challenging, as governments, economic sector, institutions, and individuals, often have conflicting interests. Game theory, which is a mathematical tool extensively used in order to determine the outcomes of strategical interactions, has been applied to a number of issues arising from the pandemic. In this paper, 60 publications are analyzed to identify the key contributions of game theory to addressing COVID-19. First, 10 keywords-topic are chosen and each publication is examined to consider if it deals with these topics. Then R package "igraph" and cluster Louvain are used in order to group the papers to 3 clusters. The results indicate that there is a temporal variation in the topics examined. In the beginning the main concern has been coupling the modelling of the transmission of the virus with social distancing responses from the public and quarantining policy. Then the economic implications of the pandemic drew the attention and finally the application of the vaccination campaigns and the cooperation of the individuals with vaccination guidelines. The paper targets on bridging the gap between game theorists and epidemiologists, illustrating how modelling agents interactions can improve responses. The clusters created can help researchers identify groups of problems that can be addressed jointly, using similar approaches. The analysis indicates that game theory can reveal social aspects that can improve the responses to future emergencies. [ABSTRACT FROM AUTHOR]

COVID-19 pandemic has raised concerns for a number of strategical social interactions. From quarantines and social distancing to vaccination and geopolitics, the cooperation or even the coordination of efforts has been challenging, as governments, economic sector, institutions, and individuals, often have conflicting interests. Game theory, which is a mathematical tool extensively used in order to determine the outcomes of strategical interactions, has been applied to a number of issues arising from the pandemic. In this paper, 60 publications are analyzed to identify the key contributions of game theory to addressing COVID-19. First, 10 keywords-topic are chosen and each publication is examined to consider if it deals with these topics. Then R package "igraph" and cluster Louvain are used in order to group the papers to 3 clusters. The results indicate that there is a temporal variation in the topics examined. In the beginning the main concern has been coupling the modelling of the transmission of the virus with social distancing responses from the public and quarantining policy. Then the economic implications of the pandemic drew the attention and finally the application of the vaccination campaigns and the cooperation of the individuals with vaccination guidelines. The paper targets on bridging the gap between game theorists and epidemiologists, illustrating how modelling agents interactions can improve responses. The clusters created can help researchers identify groups of problems that can be addressed jointly, using similar approaches. The analysis indicates that game theory can reveal social aspects that can improve the responses to future emergencies. [ABSTRACT FROM AUTHOR]

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as "ten-roots" (mūla), and we discover that these arrangements form a regular numerical sequence, called "oblong numbers". Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics. [ABSTRACT FROM AUTHOR]

In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's 'Why a diagram is (sometimes) worth 10,000 words', between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's 'characteristic visual shape' in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry. [ABSTRACT FROM AUTHOR]

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process–object transitions and externalist approaches that stress mathematical artifacts—such as physical manipulatives and formulas—as constituting mathematical objects. We search for a view on process–object duality that overcomes the dualism of mind and body. One such approach is commognition that describes mathematical objects as discursive entities. This paper expands the nature of mathematical objects beyond discourse and highlights the role of learners' interaction with the environment by adopting ecological onto-epistemology. We develop a functional dynamic systems perspective on process–object duality in mathematics learning emphasizing embodied actions and the re-invention of artifacts' affordances. As a main result, we reconsider process–object duality as a reification of repetitive actions into a cultural artifact that consists of two steps: (1) forming a new sensory-motor coordination that brings new perception to the fore and (2) crystallizing a new artifact in a mathematical environment that captures this new perception. An empirical example from research on embodied action-based design for trigonometry illustrates our theoretical ideas. [ABSTRACT FROM AUTHOR]

This note is devoted to developing Morrey regularity theory for the following system of Rivière \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f belongs to some Morrey space. Our results extend the L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24]. [ABSTRACT FROM AUTHOR]

The aim of this paper is to transfer the restrictedness theory to Hom-Lie algebras. The concept of restricted Hom-Lie algebras, which is introduced in [S. Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11 2023, 24, Paper No. 4955], will be used in this paper. First, the existence and uniqueness of p-structures on a Hom-Lie algebra is studied. Then the definition of a restrictable Hom-Lie algebra is given and the equivalence relation between restrictable Hom-Lie algebras and restricted Hom-Lie algebras is constructed. Finally, the p-envelopes of a Hom-Lie algebra are defined and studied. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the new class of left and right B-Weyl operators, which naturally extends the conventional concepts of left and right Weyl operators. Our contributions encompass demonstrating the stability of the left (and right) B-Weyl operators under small perturbations. We further characterize the left (and right) B-Weyl operators as the direct sum of a closed left (and right) Drazin invertible operator and a finite rank operator. Additionally, we present some characterizations of the left and right B-Weyl spectra, utilizing the left and right Drazin spectra as essential components. Furthermore, our obtained results play a pivotal role in exploring the interrelations between the left and right B-Weyl spectra and other spectra integral to the realm of B-Fredholm theory. This paper seeks to enhance and extend the recent research explored in [F. Abdmouleh and T. Ben Lakhal, Left and right B-Fredholm operators, Ukrainian Math. J. 74 2023, 10, 1479–1489] to a larger class in the unbounded B-Fredholm operators theory. [ABSTRACT FROM AUTHOR]

This paper is about the immanence between architecture and mathematics. At the peak of the use of digital design tools, the aim of this paper is to explain the definition of “parametric architect”, which is thought to be known by everyone. In this study, where the definition of parametric architecture is sought in the relationship that architecture establishes with mathematics, firstly, a historical projection is cast to explain the derivation of the concept of “parametric architect” through different concepts, from the platonic dimensions of mathematics to architecture's search for perfection; the need for precision and accuracy; and the transformation of geometric and graphic intuition into representation. The virtuosity of “parametric architects” is then discussed through the representations of virtuoso architects such as Zaha Hadid and Antoni Gaudí. This is followed by a discussion of the mathematical transformation required of the “parametric architect” working to meet the materialist demands of architectural manipulations as a field of computational thinking. Instead of compressing it into power-producing positions such as the starchitect and God's architect, the discussion of virtuosity, which this paper seeks to underline, gives place to the recreation of the “parametric architect” through Patrik Schumacher and Mark Burry. As a result, the insights gained from this research define “parametric architecture”, which seems to be a way to overcome the architectural practice trapped in the limits of descriptive geometrical thinking, as a new and evolving virtual ground shift/joining through the creativity-oriented relationship established with mathematics. Thus, it offers implications for architectural practice and theory. [ABSTRACT FROM AUTHOR]

This paper concerns the bubbling phenomena for the L2$L^2$‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non‐linear Schrödinger equations (NLS). However, unlike the single‐bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders |t|2+$|t|^{2+}$ and |t|3−$|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure. [ABSTRACT FROM AUTHOR]

This paper presents a modular approach to motion planning with provable stability guarantees for robots that move through changing environments via periodic locomotion behaviors. We focus on dynamic walkers as a paradigm for such systems, although the tools developed in this paper can be used to support general compositional approaches to robot motion planning with dynamic movement primitives (DMPs). By formulating the planning process as a switching system with multiple equilibria (SSME), we prove that the system's evolution remains within explicitly characterized trapping regions in the state space under suitable constraints on the frequency of switching among the DMPs. These conditions encapsulate the low-level stability limitations in a form that can be easily communicated to the planner. Furthermore, we show how the available primitives can be safely composed online in a receding horizon manner to enable the robot to react to moving obstacles. The proposed framework can be applied in a wide class of 3D bipedal walking models, and offers a modular approach for integrating readily available low-level locomotion control and high-level planning methods. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish new Maximum Principles for parabolic equations in the framework of mixed local/nonlocal operators. In particular, these results apply to the case of mixed local/nonlocal Neumann boundary conditions, as introduced by Dipierro, Proietti Lippi, and Valdinoci [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), pp. 1093–1166]. Moreover, they play an important role in the analysis of population dynamics involving the so-called Allee effect, which is performed by Dipierro, Proietti Lippi, and Valdinoci [J. Math. Biol. 89 (2024), Paper No. 19]. This is particularly relevant when studying biological populations, since the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. [ABSTRACT FROM AUTHOR]

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most $ 2n $ 2 n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of Abramovich Y.A., Lipecki Z. [On ideals and sublattices in linear lattices and F-lattices. Math Proc Cambridge Philos Soc. 1990;108(1):79–87.; On lattices and algebras of simple functions. Comment Math Univ Carolin. 1990;31(4):627–635.], as well as Kakutani's characterization of closed sublattices of $ C(K) $ C (K) spaces. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

Giving the proper citations, it is shown that, except of Lemma 2.4 and Theorem 2.6, almost all the results of the paper in the title have been previously published for ordered hypersemigroups in Eur. J. Pure Appl. Math. and they are not new. There are also two results obtained from ordered semigroups just putting a "o" instead of "o" (that isn't a correct way to work), without reference to ordered semigroups on which the results on ordered hypersemigroups are based. One of them can be obtained as corollary to a theorem in Eur. J. Pure Appl. Math. as well, and it is not new. [ABSTRACT FROM AUTHOR]

The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences. [ABSTRACT FROM AUTHOR]

Prahalad Chunnilal Vaidya was an eminent general relativist, gravitation theorist, physicist, mathematician, educationist and a follower of Gandhian philosophy in post-independence India. Apart from his scientific career, Vaidya is renowned for his Vaidya metric. Though he has contributed significantly to the scientific community, very few people are aware about him or his contributions. As 23 May 2019 was the 101st birth anniversary of Vaidya, here we present his contributions and their impact through a bibliometric study. This study presents an analysis of published research works of Vaidya. During his active career, he had contributed 102 publications, including 90 journal articles, 5 conference papers, 4 book chapters, 2 review articles and 1 report. His publications have received 2004 citations, h-index of 18 and i10-index of 29 from 1979 to 2018. We analysed Vaidya's publications by type, country, journals, publishers, citations, year-wise growth, collaborated authors, top cited papers, etc. This bibliometric study is an effort to draw the attention of the younger generation of scientists to Vaidya and his contributions. [ABSTRACT FROM AUTHOR]

Cooperation among kin is common in animal societies. Kin groups may form by individuals directly discriminating relatives based on kin recognition cues, or form passively through natal philopatry and limited dispersal. We describe the genetic landscape for a primitively eusocial wasp, Polistes dominula, and ask whether individuals choose cooperative partners that are nearby and/or that are genetic relatives. Firstly, we genotyped an entire sub-population of 1361 wasps and found genetic structuring on an extremely fine scale: the probability of finding genetic relatives decreases exponentially within just a few meters of an individual’s nest. At the same time, however, we found a lack of genetic structuring between natural nest aggregations within the population. Secondly, in a separate dataset where ~2000 wasps were genotyped, we show that wasps forced experimentally to make a new nest choice tended to choose new nests near to their original nests, and that these nests tended to contain some full sisters. However, a significant fraction of wasps chose nests that did not contain sisters, despite sisters being present in nearby nests. Although we cannot rule out a role for direct kin recognition or natal nest-mate recognition, our data suggest that kin groups may form via a philopatric rule-of-thumb, whereby wasps simply select groups and nesting sites that are nearby. The result is that most subordinate helpers obtain indirect fitness benefits by breeding cooperatively. [ABSTRACT FROM AUTHOR]

The Semantic Link Network is a general semantic model for modeling the structure and the evolution of complex systems. Various semantic links play different roles in rendering the semantics of complex system. One of the basic semantic links represents cause-effect relation, which plays an important role in representation and understanding. This paper verifies the role of the Semantic Link Network in representing the core of text by investigating the contribution of cause-effect link to representing the core of scientific papers. Research carries out with the following steps: (1) Two propositions on the contribution of cause-effect link in rendering the core of paper are proposed and verified through a statistical survey, which shows that the sentences on cause-effect links cover about 65% of key words within each paper on average. (2) An algorithm based on syntactic patterns is designed for automatically extracting cause-effect link from scientific papers, which recalls about 70% of manually annotated cause-effect links on average, indicating that the result adapts to the scale of data sets. (3) The effects of cause-effect link on four schemes of incorporating cause-effect link into the existing instances of the Semantic Link Network for enhancing the summarization of scientific papers are investigated. The experiments show that the quality of the summaries is significantly improved, which verifies the role of semantic links. The significance of this research lies in two aspects: (1) it verifies that the Semantic Link Network connects the important concepts to render the core of text; and, (2) it provides an evidence for realizing content services such as summarization, recommendation and question answering based on the Semantic Link Network, and it can inspire relevant research on content computing. [ABSTRACT FROM AUTHOR]

In this study, we address the question whether (and to what extent, respectively) altmetrics are related to the scientific quality of papers (as measured by peer assessments). Only a few studies have previously investigated the relationship between altmetrics and assessments by peers. In the first step, we analyse the underlying dimensions of measurement for traditional metrics (citation counts) and altmetrics–by using principal component analysis (PCA) and factor analysis (FA). In the second step, we test the relationship between the dimensions and quality of papers (as measured by the post-publication peer-review system of F1000Prime assessments)–using regression analysis. The results of the PCA and FA show that altmetrics operate along different dimensions, whereas Mendeley counts are related to citation counts, and tweets form a separate dimension. The results of the regression analysis indicate that citation-based metrics and readership counts are significantly more related to quality, than tweets. This result on the one hand questions the use of Twitter counts for research evaluation purposes and on the other hand indicates potential use of Mendeley reader counts. [ABSTRACT FROM AUTHOR]

The need to complete multiple tasks concurrently is a common occurrence both daily life and in occupational activities, which can often include simultaneous cognitive and physical demands. As one example, there is increasing availability of head-worn display technologies that can be employed when a user is mobile (e.g., while walking). This new method of information presentation may, however, introduce risks of adverse outcomes such as a decrement to gait performance. The goal of this study was thus to quantify the effects of a head-worn display (i.e., smart glasses) on motor variability during gait and to compare these effects with those of other common information displays (i.e., smartphone and paper-based system). Twenty participants completed four walking conditions, as a single task and in three dual-task conditions (three information displays). In the dual-task conditions, the information display was used to present several cognitive tasks. Three different measures were used to quantify variability in gait parameters for each walking condition (using the cycle-to-cycle standard deviation, sample entropy, and the “goal-equivalent manifold” approach). Our results indicated that participants used less adaptable gait strategies in dual-task walking using the paper-based system and smartphone conditions compared with single-task walking. Gait performance, however, was less affected during dual-task walking with the smart glasses. We conclude that the risk of an adverse gait event (e.g., a fall) in head-down walking conditions (i.e., the paper-based system and smartphone conditions) were higher than in single-task walking, and that head-worn displays might help reduce the risk of such events during dual-task gait conditions. [ABSTRACT FROM AUTHOR]

Quantifying the impact of a scholarly paper is of great significance, yet the effect of geographical distance of cited papers has not been explored. In this paper, we examine 30,596 papers published in Physical Review C, and identify the relationship between citations and geographical distances between author affiliations. Subsequently, a relative citation weight is applied to assess the impact of a scholarly paper. A higher-order weighted quantum PageRank algorithm is also developed to address the behavior of multiple step citation flow. Capturing the citation dynamics with higher-order dependencies reveals the actual impact of papers, including necessary self-citations that are sometimes excluded in prior studies. Quantum PageRank is utilized in this paper to help differentiating nodes whose PageRank values are identical. [ABSTRACT FROM AUTHOR]

In Kim et al. (Nagoya Math J 178: 37-53, 2005), Lee and Lee (J Geometry Phys 56(10): 2011-2023, 2006), the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to n-valued maps in Deconinck and Dekimpe (J Fixed Point Theory Appl 25(4): Paper No. 84, 29, 2023). In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any n-valued affine map on an infra-nilmanifold. [ABSTRACT FROM AUTHOR]

We attempt to identify the 25 most-cited statistical papers, providing some brief commentary on each paper on our list. This list consists, to a great extent, of papers that are on non-parametric methods, have applications in the life sciences, or deal with the multiple comparisons problem. We also list the most-cited papers published in 1993 or later. In contrast to the overall most-cited papers, these are predominately papers on Bayesian methods and wavelets. We briefly discuss some of the issues involved in the use of citation counts. [ABSTRACT FROM AUTHOR]

Research on fostering learning about percentages within intelligent tutoring systems (ITSs) is limited. Additionally, there is a lack of data‐driven approaches for improving the design of ITS to facilitate learning about percentages. To address these gaps, we first investigated whether students' understanding of basic mathematical skills (eg, arithmetic, measurement units and geometry) and fractions within an ITS predicts their understanding of percentages. We then applied a psychological network analysis to evaluate interdependencies within the data on 44 subtopics of basic mathematical concepts, fractions and percentages. We leveraged a large‐scale dataset consisting of 2798 students using the ITS bettermarks and working on approximately 4.1 million mathematical problems. We found that advanced arithmetic, measurement units, geometry and fraction understanding significantly predicted percentage understanding. Closer inspection indicated that percentage understanding was best predicted by problems sharing similar features, such as fraction word problems and fraction/natural number multiplication/division problems. Our findings suggest that practitioners and software developers may consider revising specific subtopics which share features with percentage problems for students struggling with percentages. More broadly, our study demonstrates how evaluating interdependencies between subtopics covered within an ITS as a data‐driven approach can provide practical insights for improving the design of ITSs. Practitioner notes What is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics. What is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics. [ABSTRACT FROM AUTHOR]

A well-known theorem of Philip Hall states that if a group G has a nilpotent normal subgroup N such that $G/N'$ is nilpotent, then G itself is nilpotent. We say that a group class is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that $G/N'$ belongs to. Hall classes have been considered by several authors, such as Plotkin ['Some properties of automorphisms of nilpotent groups', Soviet Math. Dokl. 2 (1961), 471–474] and Robinson ['A property of the lower central series of a group', Math. Z. 107 (1968), 225–231]. A further detailed study of Hall classes is performed by us in another paper ['Hall classes of groups', to appear] and we also investigate the behaviour of the class of finite-by- groups for a given Hall class ['Hall classes in linear groups', to appear]. The aim of this paper is to prove that for most natural choices of the Hall class , also the classes $(\mathbf{L}\mathfrak{F})\mathfrak{Y}$ and are Hall classes, where L is the class of locally finite groups and is the class of locally finite groups of finite exponent. [ABSTRACT FROM AUTHOR]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

In this paper, we employ the techniques in [C. Cavaterra, S. Dipierro, Z. Gao and E. Valdinoci, Global gradient estimates for a general type of nonlinear parabolic equations, J. Geom. Anal. 32 2022, 2, Paper No. 65] and the approach in [H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc. 147 2019, 12, 5329–5338] to derive sharp gradient estimates for a positive solution to the heat equation u t = Δ ⁢ u + a ⁢ u ⁢ log ⁡ u in a complete noncompact Riemannian manifold (where a is a real constant). This is an extension of the gradient estimates of Dung and Dung. [ABSTRACT FROM AUTHOR]

This paper studies the inhomogeneous defocusing coupled Schrödinger system i ⁢ u ˙ j + Δ ⁢ u j = | x | - ρ ⁢ (∑ 1 ≤ k ≤ m a j ⁢ k ⁢ | u k | p ) ⁢ | u j | p - 2 ⁢ u j , ρ > 0 , j ∈ [ 1 , m ] . The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ⁢ (ℝ N) such that x ⁢ f ∈ L 2 ⁢ (ℝ N) . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x ⁢ u 0 ∈ L 2 enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 , where p 0 is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ⁢ ∥ u ⁢ (t) ∥ L r ⁢ (ℝ N) ≲ 1 for some α > 0 and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ ⁢ u 0 . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 , which restricts the space dimensions to N ≤ 3 . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 . [ABSTRACT FROM AUTHOR]

Developing mathematical connections for pupils continues to be at the forefront of educational policies worldwide. One strategy suggested to support the connections pupils make is the idea of developing "coherence". Research regarding coherence has increased within mathematics education, however, the term is used widely. It therefore becomes necessary for coherence to be further conceptualised. In this systematic scoping review, five databases were searched for literature on the topic of coherence in school-based mathematics education. A total of 11,587 pieces of literature were extracted. Following further screening, 35 papers were reviewed. The findings suggest that there is limited consensus regarding the conceptualisation of coherence, leading to fragmented perceptions across the literature. This paper presents a framework for coherence based on the TIMSS curriculum model to support international comparisons of coherence and provide clarity for mathematics education researchers and school colleagues when considering which aspect of coherence they are addressing. [ABSTRACT FROM AUTHOR]

Integrating socioscientific issues (SSI) into education is gaining widespread adoption in classrooms due to its positive impact on student's critical thinking, environmental awareness, holistic knowledge and/or the idea of combining science and mathematics. This paper explores the idea of using comics as appropriate media for elementary school students to engage with SSI content. Because of the difficulties in integrating SSI into classrooms, SSI-based learning is more commonly applied in high school and higher education settings, with limited implementation in earlier education. To answer this gap, we developed comics that have SSI content for elementary schools. Comics are chosen as reliable tools for visualising and simplifying complex concepts and making SSI content more accessible and engaging. This paper describes our comics on earthquake-related issues in Indonesia and the principles that guided its design. SSI inherently involves multiple perspectives, so the integration brings together science, and mathematics within the one comic. In integrating different disciplines of comic content, we utilised the rule-of-five framework, widely employed to merge five representational models (experiential, verbal, numerical, visual, and symbolic) commonly used in developing content combining two or more different academic disciplines. [ABSTRACT FROM AUTHOR]

Background: Mindfulness practices enhance executive function skills and academic achievement, spurring interest in integrating mindfulness interventions into education. Embedding mindfulness practice into a digital math game may provide a low‐cost, scalable way to induce mindfulness and boost game‐based learning, yet this approach remains unexplored. Objectives: We investigated the learning benefits of integrating mindfulness exercises in a digital math learning game and examined how students' trait mindfulness might moderate the outcomes. Methods: Two classroom studies were conducted with 404 5th and 6th grade students from six public schools in the U.S. (nStudy 1 = 227, nStudy 2 = 177). The two randomized controlled experiments assigned students to one of the three conditions: passive control (playing the digital learning game Decimal Point), story‐enriched active control, or mindfulness‐enriched condition. Trait mindfulness, learning gains, and in‐game problem‐solving (including problem‐solving duration, error count and correctness after reminder) were assessed. Study 2 included a manipulation check to better understand the effects of the mindfulness intervention. Results: Findings showed no significant differences in learning gains, problem‐solving duration or error count among the conditions. Students' trait mindfulness did not moderate these outcomes. Mindfulness reminders in the mindfulness‐enriched game led to more correct answers after errors than jokes in the story‐enriched game. Study 2 revealed that we failed to induce higher state mindfulness through the mindfulness inductions. Conclusions: Mindfulness prompts could be especially beneficial for students experiencing frustration during gameplay, warranting more exploration for digital game‐based instruction. We highlight barriers and future directions for fostering mindfulness through computer‐based instruction in classrooms. Lay Description: What is currently known about this topic?: Researchers focus on digital games as productive learning environments for math, with potential for higher learning gains compared with traditional methods. However, results are mixed, and not all games lead to improved math outcomes.Executive function (EF) skills are crucial for math learning, and mindfulness‐based interventions show promise in enhancing EF skills in school‐aged students.Embedding mindfulness practice into a digital math game may provide a low‐cost, scalable way to induce mindfulness and, in turn, boost EF skills and game‐based learning, yet this approach remains unexplored What does this paper add?: This paper presents the results of two randomized control trials investigating the feasibility and benefits of incorporating mindfulness exercises into a digital math game designed for middle school students. Benefits are compared to those of a story‐enriched and regular version of the same digital math learning game.The paper also explores variations in the effects of the mindfulness‐enriched game based on students' initial trait mindfulness levels.We observe that listening to mindfulness inductions at the beginning of game sessions do not induce mindfulness, and therefore does not benefit math learning.We find that mindfulness prompts received after recurrent errors can be beneficial for students' problem solving. Implications of study findings for practitioners: Our study provides important information on how digital learning game designers should attempt to induce mindfulness in a digital game to promote learning.Digital learning game designers should consider incorporating mindfulness exercises into their games strategically. Presenting mindfulness inductions in audio format at the beginning of game sessions may not induce mindfulness or benefit math learning. Instead, designers should focus on integrating mindfulness prompts at moments when students encounter frustration within the learning game.Beyond embedding audio mindfulness exercises in the game, learning designers should also consider alternative technological and game features to induce mindfulness within a learning game. [ABSTRACT FROM AUTHOR]

Background: With the growing integration of technology in education, the adoption of blended learning (b‐learning) has gained attention. B‐learning combines traditional classroom teaching with online components, holding potential to enhance student outcomes and educational efficiency. Yet, current research predominantly concentrates on higher education institutions in urban areas, creating a void in understanding its impact on K‐12 education, particularly in rural settings. Objectives: This qualitative study aims to propose recommendations for successful implementation of blended learning in rural K‐12 areas by addressing the key stakeholders influencing its adoption and identifying the main factors affecting its success. Methods: We performed a content analysis of grey literature documents detailing the implementation of b‐learning in K‐12 education. Additionally, interviews with crucial stakeholders such as teachers, principals, and experts in rural schools in South America provide insights into the challenges and prospects of b‐learning adoption in these contexts. Results and Conclusions: The study identifies pivotal stakeholders for effective b‐learning implementation, outlining their roles and addressing challenges inherent in rural settings. Recommendations for enhancing b‐learning's implementation in developing countries are also proposed. The research underscores the significance of involving diverse stakeholders such as governmental bodies, school leaders, educators, students, and families to ensure a holistic and efficient approach to blended learning. Lay Description: What is currently known about this topic: Blended Learning Definition and Benefits: Blended learning combines face‐to‐face and online instruction, enhancing student engagement and personalization by providing diverse learning pathways.Growth in K‐12 Implementation: Blended learning gained traction in K‐12 education, boosting academic outcomes, driven by recognition of learner diversity and potential in meeting students' needs.Contextual Challenges: Implementation challenges arise in low‐income households and rural areas, including limited access to technology and internet connectivity, impacting student engagement and success. What does this paper add: Focusing on Rural Dynamics: This study helps to fill a gap by investigating blended learning in rural K‐12 settings, spotlighting key players and success drivers in these unique contexts.Recommendations from Various Sources: By combining interviews and grey literature, this research generates comprehensive, practical recommendations grounded in both empirical insights and real‐world perspectives.Insights into Rural Hurdles: This paper extends the discussion on blended learning by addressing specific rural barriers, proposing family involvement, private sector partnerships for training, and tailored pedagogical strategies. Implications for practice or policy: Leadership and Collaboration: Effective school leadership and collaboration with local governments are pivotal for fruitful blended learning implementation, ensuring defined roles, responsibility, and accountability.Adaptable Curriculum Design: An adaptable, centralized curriculum aligns with recommendations, fostering efficient tracking of student progress and enabling customized learning.Engagement‐Centric Pedagogies: Recommendations spotlight interactive, student‐centered teaching methods, tailored to diverse student needs, nurturing active participation and elevated learning outcomes. [ABSTRACT FROM AUTHOR]