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In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra structures. The Clifford-Weyl superalgebras can be realized as tensor product of the algebra of alternating and symmetric tensors respectively, on the even and odd parts of their underlying superspace. For Physical applications in elementary particles, we add star structures to these algebras and investigate the basic relations. Ortho-symplectic Lie algebras are naturally present in these algebras and their representations on these algebras can be described easily. [ABSTRACT FROM AUTHOR]

An Egyptian fraction is a finite sum of distinct rational numbers of the form 1 m, where m is a nonzero integer. It is well known that every rational number can be expressed as an Egyptian fraction. The purpose of this paper is to explore natural analogs of this concept for commutative integral domains. [ABSTRACT FROM AUTHOR]

Algebraic topology is a branch of mathematics which use the concepts of abstract algebra to study topological spaces in which to find algebraic invariants that classify topological spaces up to homeomorphism. In this paper, the basic properties of some concepts on algebraic topology such as the topological ring, the topological module, and the topological free module were recalled, which were helped to define new concepts and proof some of there properties as a new results. Then, some results related to the relation between the free topological module and homomorphism topology. As in any application will be introduced, the tensor concept will be chosen to explain the new results related to the topological module. [ABSTRACT FROM AUTHOR]

The retention of foundational knowledge is crucial in learning and teaching mathematics. However, a significant part of university students do not achieve long-term knowledge and problem-solving skills. A possible tool to increase further retention is testing, the strategic use of retrieval to enhance memory. In this study, the effect of a special kind of testing versus worked examples was investigated in an authentic educational setting, in an algebra course for pre-service mathematics teachers. The potential benefits of using tests versus showing students worked examples at the end of each practice session during a semester were examined. According to the results, there was no difference between the effectiveness of the two methods in the medium term—on the midterm that students took on the 6th week and the final that students took on the 13th week of the semester, the testing group performed the same as the worked example group. However, testing was more beneficial regarding long-term retention in studying and solving problems in abstract mathematics. Analyzing the results of the post-test that students took five months after their final test, the authors found that the improvement of those students who learned the material with testing was significantly larger than that of the worked example group. These findings suggest that testing can have a meaningful effect on abstract algebra knowledge and a long-lasting impact on solving complex, abstract mathematical problems. [ABSTRACT FROM AUTHOR]

This paper aims to expand soft int-group theory by analyzing its many aspects and structural properties regarding soft cosets and soft quotient groups, which are crucial concepts of the theory. All the characteristics of soft cosets are given in accordance with the properties of classical cosets in abstract algebra, and many interesting analogous results are obtained. It is proved that if an element is in the e-set, then its soft left and right cosets are the same and equal to the soft set itself. The main and remarkable contribution of this paper to the theory is that the relation between the e-set and the normality of the soft int-group is obtained, and it is proved that if the e-set has an element other than the identity of the group, then the soft int-group is normal. Based on this significant fact, it is revealed that if the soft set is not normal, then there do not exist any equal soft left (right) cosets. These relations are quite striking for the theory, since based on these facts, we show that the normality condition on the soft int-group is unnecessary in many definitions, propositions, and theorems given before. Furthermore, we come up with a fascinating result, unlike classical algebra that to construct a soft quotient group and to hold the fundamental homomorphism theorem, the soft int-group needs not to be normal. It is also demonstrated that the soft int-group is an abelian (normal) int-group if and only if the soft quotient group of G relative to the soft group is abelian. Finally, the torsion soft-int group and p-soft int-group are introduced, and we show that soft int-group is a torsion soft-int group (p-soft int-group) if and only if the soft quotient group G/fG is a torsion (p-group), respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of σ-symmetric amenability of Banach algebras and investigate some hereditary properties of them. We also apply our results to several abstract Segal algebras and group algebras. [ABSTRACT FROM AUTHOR]

In this paper, we present a qualitative study on what values students perceive in their abstract algebra course. We interviewed six undergraduates early in their abstract algebra course and then again after their course was completed about what motivated them to learn abstract algebra and what value they saw in the subject. The key finding from the analysis was that participants found intrinsic value (i.e., their enjoyment of the subject) to be essential to learning abstract algebra. While participants desired utility value in the form of mathematical applications, they ultimately did not find this necessary to learn abstract algebra. Finally, some participants had different motivations for learning abstract algebra than for learning other branches of advanced mathematics, such as real analysis, suggesting that motivation research in mathematics education should not treat mathematics as a unitary construct. We offer analysis about how the nature of advanced theoretical proof-oriented mathematics may have contributed to these findings. [ABSTRACT FROM AUTHOR]

AbstractInteractive data visualization has become a staple of modern data presentation. Yet, despite its growing popularity, we still lack a general framework for turning raw data into summary statistics that can be displayed by interactive graphics. This gap may stem from a subtle yet profound issue: while we would often like to treat graphics, statistics, and interaction in our plots as independent, they are in fact deeply connected. This article examines this interdependence in light of two fundamental concepts from category theory: groups and monoids. We argue that the knowledge of these algebraic structures can help us design sensible interactive graphics. Specifically, if we want our graphics to support interactive features which split our data into parts and then combine these parts back together (such as linked selection), then the statistics underlying our plots need to possess certain properties. By grounding our thinking in these algebraic concepts, we may be able to build more flexible and expressive interactive data visualization systems. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Proof comprehension is an important skill for students to develop in their proof-based courses, yet students are rarely afforded opportunities to develop this skill. In this paper, we describe two implementations of an activity structure that was developed to give students the opportunity to engage with complex proofs and to develop their proof comprehension skills. We share one implementation in an introduction to proof course and one implementation in an abstract algebra course. In particular, we aim to elucidate the details of facilitating these days in class and offer suggestions on how other instructors can adapt this proof reading activity structure in their own classes. [ABSTRACT FROM AUTHOR]

In this paper, we apply the theory of modules on hoops and introduce two concepts of modules on hoops and provide special examples and interesting results. Both concepts are correct and logical. The first concept is very close to the definition of module in abstract algebra. In this case, we investigate some important results in modules such as sub-modules and quotient structures. But if we want to investigate the relationship between hoop-modules and other modules on logical algebraic structures such as B C K -modules and M V -modules, we need to define the second definition of hoop-modules. In this case, we can get that a B C K -modules and an M V -module from any hoop-module. [ABSTRACT FROM AUTHOR]

Understanding and constructing mathematical proofs is fundamental for students in abstract algebra courses. The computational thinking approach can aid the process of compiling mathematical proofs. This study examined the impact of integrating computational thinking components in constructing mathematical proofs. The researcher employed a sequential explanatory approach to ascertain the enhancement of algebraic proof capability based on computational thinking through the t- test. A total of 32 prospective teachers in mathematics education programs were provided with worksheets for seven meetings, which were combined with computational thinking components. Quantitative data were collected from initial and subsequent test instruments. Moreover, three prospective teachers were examined through case studies to investigate their mathematical proof capability using computational thinking components, including decomposition, abstraction, pattern recognition, and algorithmic thinking. The study's findings indicated that CT intervention enhanced students' logical reasoning, proof-writing abilities, and overall engagement with abstract algebra concepts. The findings illustrate that integrating computational thinking into learning strategies can provide a framework for developing higher-order thinking skills, especially in proving, which are essential for studies in mathematics education programs.

Introduction. The widespread mass dissemination of digital technologies in all spheres of human activity places high demands on the quality of university students’ training in the field of IT. However, the quality of such training, and especially its fundamental nature, in many universities lags behind the requirements of the time. Additionally, a significant gap has emerged in higher education regarding the development of basic education curriculum for IT training. Aim. The present study aims to explore the potential of establishing the methodological foundation for the fundamental nature of education by incorporating mathematical principles into various specialised disciplines. This involves integrating discrete and continuous modelling principles and algorithmisation to create synergy. Results and scientific novelty. The synergy between discretion and continuity in mathematics, physics, and information processes is analysed. This analysis characterises the role of discrete mathematics in achieving a synergetic effect in teaching mathematics and computer science. It reveals the fundamental importance of mathematics in teaching formal modelling and algorithmisation languages. The significance of abstract algebra in the introductory teaching of formal languages at both school and university levels is justified. The significance of structures and algorithms, which are prevalent in discrete mathematics for training highly skilled programmers, is emphasised. Practical significance. The research findings will be of interest to both educational theorists and teachers, who provide IT training for students in various fields.

A hybrid structure is an arrangement that makes use of many hierarchical reporting structures and is applied to algebraic structures such as groups and rings. In the discipline of abstract algebra, an ideal of a near-ring is a unique subset of its elements in ring theory. Ideals generalize specific subsets of integers, such as even numbers or multiples of three. Researchers have been using mathematical theories of fuzzy sets in ring theory to explain the uncertainties that emerge in various domains such as art and science, engineering, medical science, and in environment. By fusing soft sets and fuzzy sets, a new mathematical tool that has significant advantages in dealing with uncertain information is provided. Consequently, there is always some discrepancy between reality's haziness and its mathematical model's precision. Hence ring theory has been widely used in many researches but there is some uncertainty in converting the fuzzy sets to a hybrid structure of any algebraic structure. Many approaches were done in groups. Therefore, the Hybrid structure of fuzzy sets in near rings is introduced, in which the fuzzy ideals are converted to hybrid ideals and fuzzy maximal ideals are converted to hybrid maximal ideals. For hybridization, firstly the hybrid structure is established and then sub-near rings and near rings are also determined. Then the hybrid structure of sub-near rings and ideals is introduced. This converts the fuzzy ideals and fuzzy maximal ideals to hybrid ideals and hybrid maximal ideals. The result obtained by the proposed model efficiently solved the uncertainty problems and the effectiveness of the proposed approach shows the best class, mean, worst class, and time complexity. [ABSTRACT FROM AUTHOR]

This essay utilises Suzanne Bachelard's essay 'Forme et Contenu' as an entry point to sketch out the possibility (or impossibility) of a dialectic of mathemata following the work of Jean Cavaillès and Jean Hyppolite. Firstly, I outline what Bachelard sees as a 'dialectic' of form and content in the history of abstract algebra, highlighting a parallel between Bachelard's reading of algebra and Hyppolite reading of the relation of the human subject to logos. Secondly, I move towards Jean Hyppolite's discussion of Lautman in 'Mathematical Thought' which allows for a revealing of convergences between the work of Hyppolite and Cavaillès. I conclude by noting that the similarities between Cavaillès and Hyppolite – despite possible protestations by Hyppolite – go against Hegel's understanding of mathematics, leading to the possibility of opening up a 'Hegelian mathematics against Hegel'. [ABSTRACT FROM AUTHOR]

In this paper, we examine the notions of mj-clean ring and strongly mj-clean ring. And we will provide some of its basic properties. We examine the relationship of mj-clean ring with m-clean ring and j-clean ring. We prove that R is strongly mj-clean ring if and only if Mn(R) is strongly mj-clean ring. We prove that mj-clean ring is Dedekind-finite; i.e., ab = 1 implies that ba = 1. [ABSTRACT FROM AUTHOR]

The primary motivation for this work is to develop the concept of Marshall's quotient applicable to non-commutative multi-rings endowed with involution, expanding upon the main ideas of the classical case—commutative and without involution—presented in Marshall's seminal paper. We define two multiplicative properties to address the involutive case and characterize their Marshall quotient. Moreover, this article presents various cases demonstrating that the "multi" version of rings with involution offers many examples, applications, and relatives in (multi)algebraic structures. Therefore, we established the first steps toward the development of an expansion of real algebra and real algebraic geometry to a non-commutative and involutive setting. [ABSTRACT FROM AUTHOR]

Fourier analysis is an indispensable technology, but so is mathematics. In this article, we review the history of modular forms and give an overview of the relationship among representation theory, Fourier analysis, and modular forms. The explanation of difficult terms is confined to footnotes, and we focus on the relationship between the concepts. Finally, we discuss the remaining difficulties in the modern theory of modular forms, challenges, and the author's research. [ABSTRACT FROM AUTHOR]

The theory of complex dynamics is an area of pure mathematics. Even though the theory of dynamical systems belongs to analysis, it is studied in relation to various fields of mathematics, including algebra and geometry. I study it using the theory of non-archimedean numbers from number theory, which seems distant from dynamics. In this article, we discuss how attractive these theories are. [ABSTRACT FROM AUTHOR]

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. [ABSTRACT FROM AUTHOR]

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes. [ABSTRACT FROM AUTHOR]

We are used to thinking of formal principles as either good or bad for gender expression. But might they be both, in some cases? This essay examines the recent work of queer formalism in narrative through the lens of a poetic verse form: the thirty-nine-line scheme of the sestina. Through this scheme, I show that formal principles contained in a single literary form can be conducive to the a priori antithetical notions of antinarrativity and queer relationality of form. On the one hand, the sestina exhibits a narrative-like metonymic extension through the repetition of end-words, a gesture that has been made queer through contemporary poems. On the other hand, the sestina inverts the linear temporality of narrative in its unique possibilities of anti-teleology through cyclic and non-linear stanzaic traversal, stemming from mathematical properties within abstract algebra. The duality produced in this single poetic form adds additional dimensions to queer formalist debates by splitting the problem of antinarrativity from the problem of formal constraint. I also show how Sara Ahmed’s Queer Phenomenology: Orientations, Objects, Others (2006) offers a reading of the orientations invoked by the sestina’s changing configurations of end-words. [ABSTRACT FROM AUTHOR]

This article presents an inquiry-oriented lesson for teaching Lagrange's theorem in abstract algebra. This lesson was developed and refined as part of a larger grant project focused on how to "Orchestrate Discussions Around Proof" (ODAP, the name of the project). The lesson components were developed and refined with attention to how well they supported active and broad student participation. By guiding student exploration of a few key example groups and structuring the exploration to identify recurrent structure, students formulate key lemmas that they can combine to prove Lagrange's theorem. [ABSTRACT FROM AUTHOR]

The algebraic group $ \operatorname {Spin}_{3\times 3} $ Spin 3 × 3 arises from spinning collections of the numbers 1–9 on a $ 3\times 3 $ 3 × 3 game board. The authors have been using this group, as well as a corresponding online application, to introduce undergraduate students to core concepts in group theory. We discuss the benefits of using this deceptively simple, toy-like puzzle in terms of student learning and engagement. Practical exercises as well as use cases outside the abstract algebra classroom are provided at the end. [ABSTRACT FROM AUTHOR]

Introduction/purpose: This research introduces the concept of a δ⋇- algebra, a unique structure in the field of abstract algebra. The study aims to explore the defining features and distinct properties of δ⋇- algebras, distinguishing them from other algebraic systems and examining their interrelations with other types of algebras. Methods: The methodology includes the formal definition and characterization of δ⋇-algebras, a comparative analysis with the existing algebraic structures, and an exploration of their interconnections. An algorithm is developed to verify whether a given structure meets the conditions of a δ⋇-algebra. Results: The results reveal that δ⋇-algebras possess unique properties not found in other algebraic systems. The comparative study clarifies their distinctive place within the algebraic landscape and highlights significant interrelations with other structures. The verification algorithm proves effective in identifying δ⋇-algebras, providing a systematic approach for further study. Conclusions: In conclusion, δ⋇-algebras represent a significant addition to abstract algebra, offering new theoretical insights and potential for future research. The study’s findings enhance the understanding of algebraic systems and their interconnections, opening new avenues for exploration in the field. [ABSTRACT FROM AUTHOR]

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c ∈ (0 , 1 ] such that I = (− x , y) ∈ R 2 | c x ≤ y ≤ x c and x , y ≥ 0 . This shows that its shape has to be symmetric with the graph y = − x . [ABSTRACT FROM AUTHOR]

Leibniz algebras are non-antisymmetric generalizations of Lie algebras. In this paper, we investigate the properties of complete Leibniz algebras under certain conditions on their extensions. Additionally, we explore the properties of derivations and direct sums of Leibniz algebras, proving several results analogous to those in Lie algebras. [ABSTRACT FROM AUTHOR]

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

The article is a review of two books by John Stillwell: "Algebraic Number Theory for Beginners: Following a Path From Euclid to Noether" and "The Story of Proof: Logic and the History of Mathematics." The reviewer acknowledges that Stillwell's works may not be directly relevant to modeling and applied mathematics, but praises his writing skills. The review discusses the content and structure of "Algebraic Number Theory for Beginners," noting that it may not be suitable for beginners due to unclear prerequisites and a lack of depth in certain topics. The reviewer also briefly mentions the overlap between algebraic number theory and elementary number theory. The review concludes with a brief mention of "The Story of Proof," describing it as a senior-level book on logic and the history of mathematics. [Extracted from the article]

A module M is called ef-extending if every closed submodule which contains essentially a finitely generated submodule is a direct summand of M. In this paper, we prove some properties of rings via ef-extending modules and essentially finite injective modules. It is shown that a module M is an ef-extending module and whenever M = H ⊕K with H essentially finite, then H is essentially finite K-injective if and only if for essentially finite submodules N1,N2 of M with N1 ∩N2 = 0, there exist submodules M1,M2 of M such that Ni is essential in Mi (i = 1, 2) and M1 ⊕M2 is a direct summand of M. A ring R is right co-Harada if and only if R is right (or left) perfect with ACC on right annihilators and R(N)R is ef-extending as a right R-module, iff R is right (or left) perfect and R(N) R is an ef-extending module. Some properties of ef-extending modules over excellent extension rings are considered. [ABSTRACT FROM AUTHOR]

This work presents the concept of interval-valued neutrosophic INK-subalgebras, also known as IV N INKsubalg's, which are the level and strong level neutrosophic INK-subalgebras. Next, we establish and validate a few theorems that establish the connection between these concepts and neutrosophic INK-subalgebras. We define the images and inverse images of IV N INK-subalgebras and study the transformations of the homomorphic images and inverse images of interval valud neutrosophic (briefly, IV N) INK-subalgebra into IV N INK-subalgebras. [ABSTRACT FROM AUTHOR]

Efim Zelmanov is a Russian-American mathematician who has made significant contributions to the field of algebra. He received his PhD from Novosibirsk State University and has held positions at various universities in the United States, Korea, and China. Zelmanov specializes in combinatorial problems of nonassociative algebra and group theory and was awarded the Fields Medal in 1994 for his solution of the restricted Burnside problem. He has been a member of prestigious scientific academies and has had influential mentors throughout his career. Zelmanov has also been involved in outreach activities to promote mathematics and is currently based in China. [Extracted from the article]

Though posters have been used as a pedagogical tool in a variety of fields, the literature suggests that the use of posters as an educational assessment tool by the mathematics community has been limited. This is unfortunate given the numerous potential benefits of a poster, including the development of oral communication skills and the ability to examine a mathematical topic at a deeper level. We describe the use of a poster project in an introductory abstract algebra course with specific implementation guidance including a schedule of assignments and assessment criteria. We also include reflections on the success of this project over several semesters as well as advice for adapting the project to a remote course. Samples of student posters are available upon request to the authors. [ABSTRACT FROM AUTHOR]

We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor's use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor's simultaneous use of various forms of embodiment primed students for the formalism and symbolism, highlighted and disambiguated students' referents, amplified students' contributions to develop conceptual fluency, and linked students' body form catchments to motivate the FHT. Our results offer practical implications for teaching by illustrating examples of how embodiment can assist in making abstract concepts concrete. [ABSTRACT FROM AUTHOR]

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle's kinetic energy is less than 0 − m 0 c 2 . Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero. [ABSTRACT FROM AUTHOR]

For a lifeguard at a crowded city pool, rotating from station to station, those periodic fifteen-minute breaks between stations are precious commodities. Moreover, since these breaks provide some rare quality time with the other break guards, a lifeguard's question is this: given a certain rotation consisting of stations and breaks, how many of one's breaks are shared with each coworker? Abstract algebra comes to the rescue: we show how the answer, for all coworkers art once, can be packaged in a generating function, computed by an easy calculation in the group ring of the cyclic group. [ABSTRACT FROM AUTHOR]

We provide a concrete example of two constructible polygons in the plane whose common area bisecting line is not constructible. [ABSTRACT FROM AUTHOR]

In this article, we study modules that satisfy the double infinite chain condition on uncountably generated submodules, briey called u:c:g: DICC modules. We show that if a quotient finite dimensional module M satisfies the double infinite chain condition on uncountably generated submodules, then it has Krull dimension. We study submodules N of a module M such that whenever M/N satisfies the double infinite chain condition so does M. Moreover, we observe that an α-atomic module, where α > ω1 is an ordinal number, satisfies the previous chain condition if and only if it satisfies the descending chain condition on uncountably generated submodules. [ABSTRACT FROM AUTHOR]

Cell average decomposition (CAD) plays a critical role in constructing boundpreserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant--Friedrichs--Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss--Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D P2 and P3 spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional P2 and P3 spaces. Yet, it remained unknown for a decade what CAD is optimal for higher-degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carath\'eodory's theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D Pk spaces and general 2D Qk spaces of an arbitrary k\geq 1. For the widely used 2D Pk spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high-order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results. [ABSTRACT FROM AUTHOR]

In this study, we introduce the concepts of MBJ-Neutrosophic WI-ideal and MBJ-Neutrosophic lattice ideal of lattice Wajsberg algebras. We demonstrate that every MBJ-Neutrosophic WI-ideal of lattice Wajsberg algebra is an MBJ-Neutrosophic lattice ideal of lattice Wajsberg algebra. Additionally, we talk about its opposite. Furthermore, we discover that in lattice H-Wajsberg algebra, every MBJ-Neutrosophic lattice ideal is an MBJ-Neutrosophic WI-ideal. [ABSTRACT FROM AUTHOR]

In this paper, we consider the q-analog of Kostant’s Partition Function of Lie algebras sl4(C) and sp6(C) and present a closed formula for the values of these functions. [ABSTRACT FROM AUTHOR]

In tertiary mathematics courses, students often have difficulties acquiring an understanding of the mathematical concepts covered. One approach to address this problem is to implement so-called Concept-Tests. These are multiple-choice questions whose distractors represent common problems and misconceptions related to the concepts. While there exist lots of such questions for calculus, Concept-Test questions focusing on basic concepts of abstract algebra are still rare, although previous research has shown that students have many problems with these. We therefore developed such questions for important concepts of basic group and ring theory in the years 2020–2022. In this paper, we first want to present the questions and the developmental process. Furthermore, we want to present an empirical study investigating to what extent the questions helped students in a proof-oriented abstract algebra course to acquire an understanding of the concepts covered. This study especially indicates that the developed Concept-Test questions provided good starting points for conceptual changes. [ABSTRACT FROM AUTHOR]

Counting transcripts of mRNA are a key method of observation in modern biology. With advances in counting transcripts in single cells (single-cell RNA sequencing or scRNA-seq), these data are routinely used to identify cells by their transcriptional profile, and to identify genes with differential cellular expression. Because the total number of transcripts counted per cell can vary for technical reasons, the first step of many commonly used scRNA-seq workflows is to normalize by sequencing depth, transforming counts into proportional abundances. The primary objective of this step is to reshape the data such that cells with similar biological proportions of transcripts end up with similar transformed measurements. But there is growing concern that normalization and other transformations result in unintended distortions that hinder both analyses and the interpretation of results. This has led to an intense focus on optimizing methods for normalization and transformation of scRNA-seq data. Here, we take an alternative approach, by avoiding normalization and transformation altogether. We abandon the use of distances to compare cells, and instead use a restricted algebra, motivated by measurement theory and abstract algebra, that preserves the count nature of the data. We demonstrate that this restricted algebra is sufficient to draw meaningful and practical comparisons of gene expression through the use of the dot product and other elementary operations. This approach sidesteps many of the problems with common transformations, and has the added benefit of being simpler and more intuitive. We implement our approach in the package countland, available in python and R. [ABSTRACT FROM AUTHOR]

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new (to the students) mathematics. In this way, students come to creatively establish new concepts that they may take ownership of. I provide insights from previous implementations and conclude by reflecting on what has (and has not) worked well in my experience with implementing the lesson. [ABSTRACT FROM AUTHOR]

We develop a novel, general framework for reduced-bias M -estimation from asymptotically unbiased estimating functions. The framework relies on an empirical approximation of the bias by a function of derivatives of estimating function contributions. Reduced-bias M -estimation operates either implicitly, solving empirically adjusted estimating equations, or explicitly, subtracting the estimated bias from the original M -estimates, and applies to partially or fully specified models with likelihoods or surrogate objectives. Automatic differentiation can abstract away the algebra required to implement reduced-bias M -estimation. As a result, the bias-reduction methods, we introduce have broader applicability, straightforward implementation, and less algebraic or computational effort than other established bias-reduction methods that require resampling or expectations of products of log-likelihood derivatives. If M -estimation is by maximising an objective, then there always exists a bias-reducing penalised objective. That penalised objective relates to information criteria for model selection and can be enhanced with plug-in penalties to deliver reduced-bias M -estimates with extra properties, like finiteness for categorical data models. Inferential procedures and model selection procedures for M -estimators apply unaltered with the reduced-bias M -estimates. We demonstrate and assess the properties of reduced-bias M -estimation in well-used, prominent modelling settings of varying complexity. [ABSTRACT FROM AUTHOR]

Algebraic graph theory is an important area of mathematics that looks into the complex relationships between different algebraic structures and the many features that graphs have. This interdisciplinary field integrates principles from abstract algebra, exploring structures such as rings, fields, and groups, with concepts from graph theory, committed to revealing the properties and topology of graphs. The study of graph theory focuses on elucidating graphs' features and topological aspects. In the context of this exploration, a graph denoted as G is categorized as a zero-divisor graph solely if the zero-divisors of the modular ring $\mathbb{Z}_n$ Z n form its vertex set. In the absence of this criterion, the graph does not attain the status of a zero-divisor graph. It is noteworthy that the modulo n operation plays a pivotal role in determining the adjacency of two vertices in this network, contingent on whether the product of those vertices yields zero. This study's scope includes a close examination of specific topological indices designed for various families of zero-divisor graphs. The focus is predominantly on indices such as the first, second, and second modified Zagrebs; the general and inverse general Randics; the third and fifth symmetric divisions; the harmonic and inverse sum indices; and other often overlooked topological indices. Furthermore, we broaden the analysis to encompass various entropies, including the first, second, and third redefined Zagrebs, across various families of zero-divisor graphs. The incorporation of numerical and graphical comparisons in this work aims to provide a more holistic understanding. These comparisons rely on topological indices computed across the previously expounded families of zero-divisor graphs. [ABSTRACT FROM AUTHOR]

Cuntz algebra O2 is the universal C*-algebra generated by two isometries s1, s2 satisfying s1s* 1 + s2s* 2 = 1. This is separable, simple, infinite C*-algebra containing a copy of any nuclear C*-algebra. The C*-algebra O2 plays a central role in the modern theory of C*-algebras and appears in many fundamental statements, including a formulation of the celebrated Uniform Coefficient Theorem (UCT). There are several extensions of this notion, including Cuntz algebra On, Cuntz-Krieger algebra OA for a matrix A, Cuntz-Pimsner algebra OX and its relaxation by Katsura for a C*-correspondence X, and Cuntz-Nica-Pimsner algebra NOX, for a product system X. We give an overview of the construction of these classes of C*-algebras with a focus on conditions ensuring their simplicity, which is needed in the Elliott Classification Program, as it stands now. The results we present are now part of the literature, but we hope to shed a light on recent developments in a fascinating area of modern operator algebras. [ABSTRACT FROM AUTHOR]