Basic Algebra Research Articles

Type [formula omitted] Temperley-Lieb algebra quotients and Catalan combinatorics

We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type C˜, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.

PENGEMBANGAN BAHAN AJAR BERBANTUAN SOFTWERE MAPEL UNTUK MENINGKATKAN PEMAHAMAN KONSEP MATEMATIS MAHASISWA

This research is a type of research and development which focuses on developing teaching materials for Basic Algebra courses which aim to improve understanding of mathematical concepts. Broadly speaking, the development model used in this study consists of 3 stages, namely the preliminary, development and evaluation stages. In the preliminary stage, the researcher conducted a literature study and analysis of course descriptions and identified teaching materials for Basic Algebra courses. The teaching materials developed are tested for validity, practicality and effectiveness. From the results of the module validation analysis, it was obtained 84.25 with good categories. The small scale trial obtained 38.7 with good category. The results of the analysis of the practicality of teaching materials during four meetings obtained an average score of 25.25 from the ideal score of 30 or 84.17 in the very good category. The result of the analysis of the effectiveness of the average score of the teaching profile of the lecturers using basic algebra teaching materials based on the maple-assisted scientific approach is 50.25 from the ideal score of 56 or 89.73% with the very good category. The average score of the learning activity profile of students who use basic algebra teaching materials based on the maple-assisted scientific approach is 44.5 from the ideal score of 52 or 85.58% in the very good category. From the analysis of the results of the test for understanding mathematical concepts, the total average is 0.58 with the moderate category, which can be concluded that the teaching materials for basic algebra courses with a scientific approach

Research of Teaching Model in Math and Algebra

Teaching mode based on the intelligent inquiry focuses on guiding students to experience empirical thinking accumulation and developing students’ intellectual ability. Various steps form wisdom learning while concerning the differences in students’ subjects, the combination of the steps will change correspondingly, thereby contributes explicitly to all kinds of the subjects. Therefore, the pursuing goals reflected by the disciplinary knowledge motivate the variation of the teaching model. This article uses technical means as the supportive environment, focusing on the teaching of Math and Algebra in junior high school, and analyzed the application processes of the intelligent inquiry teaching model variants.

Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids

In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-algebroids and split Lie 2-bialgebroids. We further introduce the notion of a weak Dirac structure of a CLWX 2-algebroid and show that the graph of a Maurer–Cartan element of the homotopy Poisson algebra of degree 3 associated to a split Lie 2-bialgebroid is a weak Dirac structure. Various examples including the string Lie 2-algebra, split Lie 2-algebroids constructed from integrable distributions and left-symmetric algebroids are given.

Axiomatizing Maximal Progress and Discrete Time

Milner's complete proof system for observational congruence is crucially based on the possibility to equate τ divergent expressions to non-divergent ones by means of the axiom recX.(τ.X + E) = recX.τ.E. In the presence of a notion of priority, where, e.g., actions of type δ have a lower than silent τ actions, this axiom is no longer sound. Such a form of is, however, common in timed process algebra, where, due to the interpretation of δ as a time delay, it naturally arises from the maximal progress assumption. We here present our solution, based on introducing an auxiliary operator pri(E) defining a priority scope, to the long time open problem of axiomatizing using standard observational congruence: we provide a complete axiomatization for a basic process algebra with and (unguarded) recursion. We also show that, when the setting is extended by considering static operators of a discrete time calculus, an axiomatization that is complete over (a characterization of) finite-state terms can be developed by re-using techniques devised in the context of a cooperation with Prof. Jos Baeten. Research partly supported by the H2020-MSCA-RISE project ID 778233 Behavioural Application Program Interfaces (BEHAPI).

Analytical modeling of matrix–vector multiplication on multicore processors

The efficiency of matrix–vector multiplication is of considerable importance. No current approaches can optimize this sufficiently well under severe time constraints. All major existing methods are based on either manual‐tuning or auto‐tuning and can therefore be time‐consuming. We introduce an alternative model‐driven approach, which is used to map the implementation of matrix–vector multiplication to a target architecture and analytically obtain its parameters. The approach yields the performance that is competitive with optimized Basic Linear Algebra Subprograms (BLAS)‐like dense linear algebra libraries without the need for manual‐tuning or auto‐tuning. Our method provides competitive performance across hardware architectures and can be utilized to obtain single‐threaded and multi‐threaded implementations on multicore processors. We expect that this approach allows the community to progress from valuable engineering solutions to techniques with a broader application.

Linear transformations preserving algebraic elements of degree 2

ABSTRACT In the present paper, we will characterize linear maps of , the ring of all n × n matrices over an algebraically closed field of characteristic zero, which preserve the set of all algebraic elements of degree 2.

Brauer configuration algebras for multimedia based cryptography and security applications

The notion of visual cryptography was introduced without formalisms by Naor and Shamir in 1994. It provides a very powerful technique by which one secret can be distributed into two or more shares, when the shares on transparencies are superimposed exactly together, the original secret can be discovered without computer participation. In this paper, master or universal shares defined by some suitable Brauer configuration algebras are used to protect multimedia database systems and in particular digital biometric data (e.g., iris, fingerprints, handwritten signatures, DNA, faces, etc). Master shares allow to conceal effectively and efficiently multiple secrets between a number of trusted parties. In this proposal such master shares are encoded by sequences of integer numbers, thus saving storage space of the database servers involved in the protection process. According to the procedure, a master share is obtained via a digit sequencing algorithm (DSA), which define elements of a database as linear combinations of basic elements of a suitable Brauer configuration algebra (BCA).

Uniform, Integral, and Feasible Proofs for the Determinant Identities

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF (2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC 2 ; the latter is a first-order theory corresponding to the complexity class NC 2 consisting of problems solvable by uniform families of polynomial-size circuits and O (log 2 n )-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC 2 -circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).

Desain Didaktis Konteks Fabel Berbasis Pemahaman Matematis Siswa pada Materi Aljabar

This research is motivated by the importance of students 'mathematical understanding, but the fact is that students' mathematical understanding is still low in working on algebraic form questions. This problem brings out a learning obstacle, where students cannot interpret the algebraic form elements and cannot operate the algebraic form. This study aimed to design a didactic design phase using fables based on students' mathematical understanding. The research method is descriptive qualitative research. Design is Didactical Design Research (DDR) with 15 junior high school students as subjects. The data was collected using triangulation techniques, then analyzed descriptively. This study results in a didactic design in the form of a situation that can anticipate students' learning obstacles in algebraic form material and facilitate students' thinking process. Situations are created using exciting pictures and stories to motivate student learning. The results show that the situations that have been created make it easier for students to group similar and dissimilar terms and can operate the algebraic form easily and algorithmically. The students also can interpret the elements of the algebraic form along with addition and subtraction operations in the algebraic form based on the understanding of each student.

To tell a story, you need a protagonist: how dynamic interactive mediators can fulfill this role and foster explorative participation to mathematical discourse

This paper focuses on students’ mathematical discourse emerging from interactions in the digital environment GeoGebra, in which one can construct virtual objects that realize mathematical signifiers and then interact with them. These virtual object realizations can become dynamic interactive mediators (DIMs) that influence the development of the learners’ mathematical discourse. In this case study, I analyze in fine detail the discourse developed by two dyads of students in response to an unfamiliar interview question. One dyad came from a class in which GeoGebra was not part of classroom practice and included students who, according to the teacher’s evaluation, were standard-to-high achieving. The other dyad was from a generally demotivated and low-achieving class in which GeoGebra had become part of classroom practice. The analyses, focused especially on the low-achieving dyad, are guided by the question of how DIMs shaped these students’ discourse. According to the analysis, these students ended up succeeding where standard-to-high-achieving peers did not. Moreover, the detailed analysis of the ways in which the DIMs supported this dyad’s learning showed mechanisms that may be general rather than specific to this one case. This suggests that appropriate integration of DIMs into the teaching and learning of high school algebra can be beneficial for low-achieving students.

Fermionic Topological Order on Generic Triangulations

Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if $${{\mathcal {P}}}$$ is any of its spectral projections, the Booleanization of the fundamental group $$\pi _1(M)$$ can be embedded inside the group of invertible elements of the corner algebra $${{\mathcal {P}}}\, \mathrm{CAR} \, {{\mathcal {P}}}$$ . As a consequence, $${{\mathcal {P}}}$$ decomposes in $$4^g$$ lower projections. Furthermore, a projective representation of $${{\mathbb {Z}}}_2^{4g}$$ is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group $$(2^X,\Delta )$$ , where X is the set of fermion sites and $$\Delta $$ is the symmetric difference of its sub-sets.

Roughness in basic algebras1

We introduce rough approximations into basic algebras. After investigating elementary properties of the upper (lower) approximations in basic algebras and discussing the convexity of these two approximations in linearly ordered basic algebras, we generalize related results for MV-algebras, lattice ordered effect algebras, and orthomodular lattices to basic algebras. We also study the relationship between upper (lower) rough ideals of basic algebras and upper (lower) approximations of their homomorphic images.

A Heart-Centered Stance: Receptivity to Algebra Teachers’ and Students’ Multidimensional Experiences

The algebra classroom in urban public high schools in the United States is a complex space, ripe with many challenges and opportunities. In this paper I introduce the notion of a heart-centered stance for the teacher and the educator, and a method of engaging in creative expression for reflection and introspection toward individual change in the rich context of the high school algebra classroom. My evolving relationships with two high school algebra teachers, observations of their classrooms, as well as my own self-study and professional growth, are incorporated into this paper as I introduce and exemplify two tenets of a heart-centered stance: multidimensionality of experience and receptivity to relatedness. This study suggests the possibility of using creative artistic expression and a self-study approach to support the transformation of educators’ perspectives toward research, creative activities, and outreach that are receptive to the mathematical experiences of teachers and students in our local communities.

Алгебры Ли дифференцирований линейных алгебр над полем

In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equa­tions. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of ap­plying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an exam­ple, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.

Numerical methods in Jost Bürgi's “Coss”

AbstractFrom 1579 to 1604 Jost Bürgi (1552–1632) worked at the court of the landgraves at Kassel both as constructor of scientific instruments and astronomical desk clocks and as astronomer and mathematician who analyzed the data gained by astronomical observations in order to design the technical devices that he constructed. Generally, little is known about the numerical methods which were used by craftsmen and engineers at that time. Bürgi, however, wrote up a manuscript on “Coss”, i. e., elementary algebra, in which he discusses several numerical methods to approximate zeros of polynomials.

Teachers’ perspective of algebra learning in junior high school

Teaching algebra in junior high school has its own difficulty. Algebra topic, such as using the variable, is introduced in the seventh grade of junior high school. Teachers must change students' ways of thinking from arithmetic thinking to algebraic thinking. The aim of this study is to know the teacher's perspective of algebra learning, especially about the number topic, in junior high school. The instrument was an open questionnaire, and it was given to 26 mathematics teachers at junior high school in Padang Pariaman region. The analysis used a descriptive approach. The result of the analysis shows that 18 teachers (69%) use media in learning mathematics, and most of them use media manipulative. There is no teacher using computer base media learning. Teachers find problems in algebra learning, such as students do not understand the type of number and the order of operations, which is happened because of misconception in mathematics.

Relations between infinitesimal non-commutative cumulants

Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici-Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of algebra of Grassmann numbers. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the algebra of Grassmann numbers defined over the base field. We also consider the infinitesimal analog of the Boolean Bercovici-Pata map.

On the structure of invertible elements in certain Fourier–Stieltjes algebras

For a locally compact abelian group $G$, J. L. Taylor (1971) gave a complete characterization of invertible elements in the measure algebra $M(G)$. Using the Fourier-Stieltjes transform, this characterization can be carried out in the context of Fourier-Stieltjes algebras $B(G)$. We obtain this latter characterization for the Fourier-Stieltjes algebra $B(G)$ of certain classes of locally compact groups, in particular, many totally minimal groups and the $ax+b$-group.

An elementary proof of Chollet’s permanent conjecture for 4 × 4 real matrices

Abstract A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.