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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Copyright of Military Technical Courier / Vojnotehnicki Glasnik is the property of Military Technical Courier / Vojnotehnicki Glasnik and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

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]

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]

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]

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]

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]

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 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]

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 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]

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]

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]

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]

In this paper, we introduce the complex (anti complex) fuzzy topological space (X, T) with complex (anti complex) gradation of openness under T-norm (C-conorm), which X is itself a T-complex (C-anti complex) fuzzy subset of a nonempty set M. We show that the set of all T-complex gradations of openness on X is a semicomplete lattice. Some example such as T-complex fuzzy subspace of ᴧRm, the exterior algebra on Rm are given. [ABSTRACT FROM AUTHOR]

In this paper, we study smooth Lie algebra bundles of finite Type. We discuss tangent bundle and Lie algebra bundle induced by Lie group bundle of finite type. Finite type property of top space bundles and RMS bundle for are also examined. [ABSTRACT FROM AUTHOR]

Algebraic hyperstructures are the classical generalizations of algebraic structures which has several applications in uncertainity theory [6], rough set theory [7], lattice based probability theories, analysis etc. Davvaz et.al.[5] extensively studied the chemical and biological applications of hyperstructures by exploring several inheritance examples of algebraic hyperstructures. This paper focusses on the occurences of twin zeros and triple zeros in Hyperlattices with respect to hyperideals. A lattice is a partially ordered set in which every pair of elements has a least upper bound (supremum or join) and a greatest lower bound (infimum or meet). Multilattice is a generalization of a lattice introduced by Benado [3]. They extended the concept of supremum and infimum to"multi" versions, allowing for the consideration of suprema and infima over multiple elements instead of just pairs. This provides a more flexible framework for dealing with larger collections of elements. A lattice can also be viewed as an algebraic structure with two binary operations: join (supremum) and meet (infimum). These operations are used to define the least upper bound and greatest lower bound of elements in the lattice, respectively. Konstantinidou [12], further generalized lattices by replacing the binary operations of join and meet with hyperoperations. However, with these generalizations some properties are not retained. Later, Konstantinidou [11] discussed the concept of distributivity of hyperlattices, particularly of P-hyperlattices. Rasouli and Davvaz [17] considered special relations on hyperlattices, called regular relations and showed that the quotient structure with respect to regular relations form a lattice. Rasouli and Davvaz [16] defined a topology on the spectrum of join hyperlattices and showed that it forms a T0-space. Ameri [2] and others have explored the distributivity and dual distributivity of elements in a hyperlattice. [ABSTRACT FROM AUTHOR]

We introduce learning modules in cryptography that can be crafted to motivate many abstract mathematical ideas, and we illustrate with a sample module. These modules can be used in a variety of ways, such as the core for a cryptography course or as motivating topics in other courses such as abstract and linear algebra or number theory. [ABSTRACT FROM AUTHOR]

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included. [ABSTRACT FROM AUTHOR]

We present a detailed description of a Cryptography and Computer Security course that has been offered at Knox College for the last 15 years. While the course is roughly divided into two sections, Cryptology and Computer Security, our emphasis here is on the Cryptology section. The course puts the cryptologic material into its historical context and also includes reference to ciphers used and discussed in literature. Through lectures and assignments on cryptanalysis, the course motivates students to learn a variety of topics in advanced discrete mathematics, including discrete probability, elementary statistics, number theory, and abstract algebra. The students have found this approach motivating and we believe our experiences are relevant to instructors teaching this material in a variety of different courses and institutions. [ABSTRACT FROM AUTHOR]

We introduce the notion of lifting Baer modules, as a generalization of both Baer and lifting modules and give some of their properties. A module M is called lifting Baer if right annihilator of a left ideal of End(M) lies above a direct summand of M. Also, we define the concepts of r-supplemented and amply r-supplemented modules. It is shown that an amply r-supplemened module M that every supplement submodule, is a direct summand of M, is lifting Baer. The relationships between Baer modules and lifting Baer modules are investigated. Morever, we prove that the endomorphism ring of any lifting Baer module is lifting Baer ring. [ABSTRACT FROM AUTHOR]

In this study, we describe a linear space AW,V a,p (Rd) of functions (Rd) whose fractional Fourier transforms belong to AW,V a,p(Rd) for. We show that (Rd) (Rd) becomes a Banach algebra with the sum norm ||f||AW,Va,p=||f||1,W and under (fractional convolution) convolution operation. Besides, we indicate that the space AW,Va,p(Rd) is an abstract Segal algebra, where is weight function of regular growth. Moreover, we find an approximate identity for AW,Va,p(Rd). We also discuss some other properties of AW,Va,p (Rd). Finally, we investigate some inclusions of this space. [ABSTRACT FROM AUTHOR]

The mathematics education research landscape possesses limited examples of large-scale research regarding students' experience with primary historical sources in the learning of mathematics at the undergraduate level. However, the field continues to investigate methods of determining how students might overcome obstacles to reach new outcomes. This article documents an exploration within the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project that combined transgressions theory and situated learning theory, thereby illuminating three components of the former--boundaries, transgressive actions, and outcomes. In this regard, we capture students' Primary Source Project (PSP) learning experiences in Abstract Algebra courses and bring attention to the challenges in delineating these elements due to drawing upon data not initially collected with transgressions or situated learning theory in mind. Despite this limitation, we underscore the potential of utilizing the transgressions framework in the analysis of students' unorthodox learning experiences, such as with PSPs, and provide insights for future studies in the field. [ABSTRACT FROM AUTHOR]

This article discusses a comprehensive book on the theory of multidimensional stationary time series, focusing on dimension reduction and prediction. The authors cover advanced topics in probability theory, linear algebra, and real and complex analysis. The book explores various tools such as harmonic analysis, abstract algebra, and state space methods. It also delves into the properties of multidimensional, weakly stationary time series and provides insights into dimension reduction and prediction in the time series and frequency domain. The book is aimed at graduate students and researchers studying time series analysis. [Extracted from the article]

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. [ABSTRACT FROM AUTHOR]

During the pandemic, students faced greater challenges in learning mathematics. Due to the outbreak of the pandemic, many changes were made to the learning environment to adapt to changing conditions brought about by the new learning methods. These include the addition of new learning tools and techniques, as well as the training of teachers and students on how to use them. Since class interaction is limited, e-learning materials were used by the students to understand mathematical concepts. This study aims to determine the usage and effects of e-learning materials in learning mathematics among Bachelor of Secondary Education majoring in Mathematics during the COVID-19 pandemic. This study utilized descriptive-correlational research. Data were analyzed using mean, standard deviation, Pearson's r, and coefficient of determination. The respondents were randomly selected third year students. The e-learning materials were classified as printed, audio-visual, and specialized applications while mathematics performance is measured in terms of their final grades in Abstract Algebra, Calculus 3 and Problem Solving, Mathematical Investigation, and Modelling. The result of the study shows that printed and audio-visual e-learning materials affect the respondents' mathematical performance while specialized applications only bear significance with Calculus 3. The study shows the importance of being resourceful and fostering individual learning among students. [ABSTRACT FROM AUTHOR]

We provide complete classifications of algebras of partial maps for a significant swathe of combinations of operations not previously classified. Our focus is the many subsidiary operations that arise in recent considerations of the 'override' and 'update' operations arising in specification languages. These other operations turn out to have an older pedigree: domain restriction, set subtraction and intersection. All signatures considered include domain restriction, at least as a term. Combinations of the operations are classified and given complete axiomatizations with and without the presence of functional composition. Each classification is achieved by way of providing a concrete representation of the corresponding abstract algebras as partial maps acting on special kinds of filters determined with respect to various induced orders. In contrast to many negative results in the broader area, all of the considered combinations lead to finite axiomatizations. [ABSTRACT FROM AUTHOR]