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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]

In 1975, Symons [[5]] introduced a subsemigroup of HT ht , defined as HT ht , for a nonempty subset I Y i of I X i , determining all its automorphisms. The first paper [[21]] introduces a construction of a new graph associated with a semihypergroup, using the fundamental relation HT ht . In particular, it is shown that the Cayley graphs generated by transposition trees on the set HT ht are HT ht -extendable and their extendability number is HT ht for any integer HT ht . Connections between algebraic structure theory and graph theory have been established in order to solve open problems in one theory with the help of the tools existing in the other, emphasizing the remarkable properties of one theory with techniques involving the second. [Extracted from the article]

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]

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 set of polynomial mappings of n-dimensional complex space into itself the Jacobian matrix of which has a constant nonzero determinant is known to be very vast in any dimension that exceeds one. The well-known Jacobian conjecture states that any such mapping is polynomially invertible. Even though the computation of the determinant of the Jacobian matrix is very well supported in modern computer algebra systems, the algorithmic inversion of a polynomial mapping is still a problem of considerable computational complexity. In this paper, we present a Mathematica package JC that can be used for construction and inversion of polynomial mappings and more general analytic mappings with the unit determinant of the Jacobian matrix. The package includes functions that allow one to algorithmically construct these mappings for a given dimension of the space of variables and a given degree of mapping components. The package, together with a library of datasets for testing it and results of computational experiments, is available for free public use at https://www.researchgate.net/publication/358409332_JC_Package_and_Datasets. [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]

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]

Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks. [ABSTRACT FROM AUTHOR]

Advanced mathematics is seen as an integral component of secondary teacher preparation, and thus most secondary teacher preparation programs require their students to complete an array of advanced mathematics courses. In recent years, though, researchers have questioned the utility of proposed connections between advanced and secondary mathematics. It is simply not clear in many cases—to researchers, teacher educators, and teachers themselves—exactly how advanced mathematics content is related to secondary content. In this paper, we propose using a conceptual analysis—a form of theory in which one explicitly describes ways of reasoning about a particular mathematical idea—to address this issue. Specifically, we use conceptual analyses for the foundational notions of equivalence and inverse to illustrate how the ways of reasoning needed to support productive engagement with tasks in advanced mathematics can mirror and reinforce those that are similarly productive in school mathematics. To do so, we propose conceptual analyses for the key concepts of equivalence and inverse and show how researchers can use these conceptual analyses to identify connections to school mathematics in advanced mathematical tasks that might otherwise be obscured and overlooked. We conclude by suggesting ways in which conceptual analyses might be productively used by both teacher educators and future teachers. [ABSTRACT FROM AUTHOR]

Copyright of Ñawi: Arte, Diseño y Comunicación is the property of Escuela Superior Politecnica del Litoral 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.)

Francis Sowerby Macaulay began his career working on Brill and Noether's theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay's work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected. [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]

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]

Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra A. The characterization of normal block Toeplitz matrices with entries from a commutative algebra A of normal matrices is also obtained. [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]

Mathematics and early state formation, or the Janus face of early Mesopotamian mathematics: bureaucratic tool and expression of scribal professional autonomy: revised contribution to the symposium Mathematics and the State. I think linear algebra is a representative test case, being one of the core courses of undergraduate mathematics and a mathematical discipline with a millennia-long historical tradition. All that debate is about secondary school mathematics teaching, but the tide of changes is already reaching university mathematics as well. He emphasized the role of elementary matrices, introduced the I LU i -factorization of matrices, and suggested solving systems of simultaneous linear equations HT ht for a nondegenerate square matrix I A i not by Gaussian elimination, but by inversion of I A i : HT ht . [Extracted from the article]

In the present paper we give a matrix form of local derivations of the complex finite dimensional simple (non-Lie) Malcev algebra M, and a direct proof of the statement that every 2-local derivation of M is a derivation. We have some description of local and 2-local derivations of complex finite-dimensional semisimple binary Lie algebras. [ABSTRACT FROM AUTHOR]

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings. [ABSTRACT FROM AUTHOR]

This paper is concerned with basic concepts and some results on regular Γ-nearness semigroup and ideals of a Γ-nearness semigroup. Also, it is given some properties about ideals of a regular Γ-nearness semigroup and an example about the subject. Furthermore, we study relations among ideals of a Γ-nearness semigroup. [ABSTRACT FROM AUTHOR]

Sequence knitting comes from a straightforward directive; determine a short basic sequence of stitches and create a piece by continuous repetition of that sequence. This paper determines the number of sequence knitting patterns. We restrict our attention to sequences containing only knit and purl stitches and to flat knitting. We further assume that a sequence is restarted at the beginning of each row of knitting, but we do not assume that the number of stitches cast on is a multiple of the sequence length. We consider the case that a pattern and its left/right mirror image are the same, and the case that a pattern and its left/right mirror image are not necessarily the same. [ABSTRACT FROM AUTHOR]

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation ( ) ( ) where are whole numbers. Results of the present paper show that the exponential Diophantine equation ( ) ( ) where are whole numbers, has no solution in the whole number. [ABSTRACT FROM AUTHOR]

This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz König, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of Lévy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels. [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]

Multiplace functions, which are also called functions of many variables, and their algebras called Menger algebras have been studied in various fields of mathematics. Based on the theory of many-sorted algebras, the primary aim of this paper is to present the ideas of Menger systems and Menger systems of full multiplace functions which are natural generalizations of Menger algebras and Menger algebras of n -ary operations, respectively. Two specific types of n -ary operations, which are called idempotent cyclic and weak near-unanimity generated by cyclic and weak near-unanimity terms, are provided. The Menger algebras under consideration have a two-element universe, the elements of which are two specific n -ary operations. Additionally, we provide necessary and sufficient conditions in which the abstract Menger algebra and the Menger algebras of these two n -ary operations are isomorphic. An abstract characterization of unitary Menger systems via systems of idempotent cyclic and weak near-unanimity multiplace functions is generally investigated. A strong connection between clone of terms and Menger systems of full multiplace functions is also investigated. [ABSTRACT FROM AUTHOR]

This paper investigates the solutions (if any) of the Diophantine equation 3x + 35y = Z², where x, y, and are whole numbers. Diophantine equations are drawing the attention of researchers in diversified fields over the years. These are equations that have more unknowns than a number of equations. Diophantine equations are found in cryptography, chemistry, trigonometry, astronomy, and abstract algebra. The absence of any generalized method by which each Diophantine equation can be solved is a challenge for researchers. In the present communication, it is found with the help of congruence theory and Catalan's conjecture that the Diophantine equation 3x + 35y = Z² has only two solutions of (x, y, z) as (1, 0, 2) and (0, 1, 6) in non-negative integers. [ABSTRACT FROM AUTHOR]

Most studies on the switching game Lights Out and its variants focus on the solvability of given games or the number of solvable games, but when the game is viewed in a coding-theoretical perspective, more interesting questions with special symbolizations in coding theory will naturally pop up, such as finding the minimal number of lit lights among all solvable games apart from the solved game, or finding the minimal number of lit lights that the player can achieve from a given unsolvable game, etc. However, these problems are usually hard to solve in general from the perspective of algorithmic complexity. This study considers a Lights Out variant called two-state Alien Tiles, which toggles all the lights in the same row and those in the same column of the clicked light. We investigate its properties, discuss several coding-theoretical problems about this game, and explore this game as an error-correcting code and investigate its optimality. The purpose of this paper is to propose ways of playing switching games in a think-outside-the-box manner, which benefits the recreational mathematics community. [ABSTRACT FROM AUTHOR]

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings $\mathbb {Z}_{p^{n}}$ (where $p$ is a prime) and $\mathbb {Z}_{p^{m}q^{n}}$ (where $p < q$ are distinct primes and $m\leq n$). Our ultimate objective is to characterize such codes’ minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper offers a wide variety of minimal codes over (commutative) rings for the first time. Novel perspectives and developments in this direction are expected in the future. [ABSTRACT FROM AUTHOR]

In this paper, we aim to extend the studies [M. R. Alimoradi, R. Rezaei and M. Rahimi, Some notes on ideals in soft rings, Journal Australian Journal of Basic and Applied Sciences6(3) (2012) 717–721; F. Koyuncu and B. Tanay, Some soft algebraic structures, Journal of New Results Science10(2016) 38–51] as regards maximal, prime and principal soft ideal of soft rings, characterize soft rings with these soft ideals and also provide some more relations between maximal, prime and principal soft ideals of soft rings. The notions of maximality and primeness points of soft ideals of a soft rings are defined, maximal and prime idealistic soft rings as well as maximal, prime and principal soft ideal of soft rings and their basic properties are more investigated under certain conditions, especially by means of homomorphism and epimorphism of rings. We apply some of the basic results about maximal ideals and prime ideals in classical abstract algebra to maximal, prime and principal idealistic soft rings and we give some of their interrelations between each others. [ABSTRACT FROM AUTHOR]

Windmill images and shapes have a long history in geometry and can be found in problems in different mathematical contexts. In this paper, we share and discuss various problems involving windmill shapes and solutions from geometry, algebra, to elementary number theory. These problems can be used, separately or together, for students to explore rotational symmetry and connect seemingly unrelated problems. [ABSTRACT FROM AUTHOR]

Connecting and comparing across student strategies has been shown to be productive for students in elementary and secondary classrooms. We have recently been working on a project converting such practices from the K–12 level to the undergraduate classroom. In this paper, we share a particular instantiation of this practice in an abstract algebra setting. Students compare across two common proof approaches to showing that the Abelian property is preserved by isomorphism. We share a complete sample lesson where students make sense of the theorem and the two proof approaches, then leverage the differences between them in order to modify both proofs and mathematical statements. We conclude with the students' reflections on the activities, and share our learnings from adapting best practices from K–12 to this new setting. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space Bmin in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of Bmin with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that Bmin is a symmetric Segal subalgebra of L1(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of Bmin on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which Amin with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product. [ABSTRACT FROM AUTHOR]

For each nonzero h ∈ F[E]?, where F is a field, let Ah be the unital associative algebra generated by elements x, y, satisfying the relation yx - xy = h. This gives a parametric family of subalgebras of the Weyl algebra A1, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the Hochschild cohomology HH●.Ah/over a field of an arbitrary characteristic. In case F has a positive characteristic, the center Z.Ah/of Ah is nontrivial and we describe HH●.Ah/as a module over Z.Ah/. The most interesting results occur when F has a characteristic 0. In this case, we describe HH●.Ah/as a module over the Lie algebra HH1.Ah/and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when HH●.Ah/is a semisimple HH1.Ah/-module. [ABSTRACT FROM AUTHOR]

Although cryptography is monumentally important in keeping our information safe and secure, this interdisciplinary field of study can also be used as a powerful teaching tool by putting mathematics in a dramatic and realistic setting. Cryptography also allows for a natural way to introduce topics such as modular arithmetic, matrix operations, and elementary group theory. This paper reports the results of a preliminary qualitative study of students' thinking and understanding of cryptography using both the SOLO taxonomy and open coding as they participated in task-based interviews. Students interviewed were assessed on how they made connections to various areas of mathematics through solving cryptography problems. Analyzing these interviews showed that students have a strong foundation in number-theoretic concepts such as divisibility and modular arithmetic. Also, students were able to use probability intuitively to make sense of and solve problems. Finally, participants' SOLO levels ranged from uni-structural to extended abstract, with the multi-structural level being the most common (three participants). This gives evidence to suggest that students should be given the opportunity to make mathematical connections early and often in their academic careers. Results indicate that although cryptography need not be a required course for mathematics majors, concepts from this field could be introduced in courses such as linear algebra, abstract algebra, number theory, and discrete mathematics. [ABSTRACT FROM AUTHOR]

We report on a group theory activity in which learners explored dihedral symmetries through a tangible geometric model. This approach has historical roots in the work of Felix Klein's Erlangen Program and his Elementary Mathematics from an Advanced Standpoint. We situate our study with respect to this history as well as current educational research in spatial visual reasoning, abstract algebra, and teacher knowledge. Our findings highlight opportunities that tangible geometric models can provide for fostering structural and interconnected understanding characteristic of teachers' knowledge at the mathematical horizon. [ABSTRACT FROM AUTHOR]

In this paper, we define the conceps of complex fuzzy Lie subalgebras and complex fuzzy ideals of Lie algebras with respect to t-norms and investigate some of characteristics and relationship between them. Next, we introduce the concepos of quotient subalgebras, intersection, sum and direct product of them and prove some results about them. Finally, we introduce and study the image and the inverse image of them under Lie algebra homomorphisms. [ABSTRACT FROM AUTHOR]

In our earlier paper, we gave an abstract formulation of a theorem of Sierpinski in uncountable commutative groups. In this paper, we prove a result which generalizes the earlier formulation. [ABSTRACT FROM AUTHOR]

In this paper, an (implicative) ideal and a fuzzy ideal of Sheffer stroke BG-algebra are defined and some properties are presented. Then a fuzzy implicative and a sub-implicative ideals of a Sheffer stroke BG-algebra are described. Morever, an implicative Sheffer stroke BG-algebra and a medial Sheffer stroke BG-algebra are defined, and it is expressed that every medial Sheffer stroke BG-algebra is an implicative Sheffer stroke BG-algebra. Also, a fuzzy (completely) closed ideal and a fuzzy p-ideal are determined. Finally, the relationships between these structures are shown. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q and E be a set in F q d. The distance set of E is defined by Δ (E) : = { ‖ x − y ‖ : x , y ∈ E } , where ‖ α ‖ = α 1 2 + ... + α d 2. Iosevich, Koh and Parshall (2018) proved that if d ≥ 2 is even and | E | ≥ 9 q d / 2 , then F q = Δ (E) Δ (E) = { a b : a ∈ Δ (E) , b ∈ Δ (E) ∖ { 0 } }. In other words, for each r ∈ F q ⁎ there exist (x , y) ∈ E 2 and (x ′ , y ′) ∈ E 2 such that ‖ x − y ‖ ≠ 0 and ‖ x ′ − y ′ ‖ = r ‖ x − y ‖. Geometrically, this means that if the size of E is large, then for any given r ∈ F q ⁎ we can find a pair of edges in the complete graph K | E | with vertex set E such that one of them is dilated by r ∈ F q ⁎ with respect to the other. A natural question arises whether it is possible to generalize this result to arbitrary subgraphs of K | E | with vertex set E and this is the goal of this paper. In this paper, we solve this problem for k -paths (k ≥ 2) , simplexes and 4-cycles. We are using a mix of tools from different areas such as enumerative combinatorics, group actions and Turán type theorems. [ABSTRACT FROM AUTHOR]