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Journal articles on the topic 'MATHEMATICS / Algebra / Elementary'

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Published: 29 July 2024

Last updated: 30 July 2024

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1

Parajuli, Krishna Kanta. "Elementary Algebra on Vedic Mathematics." Mathematics Education Forum Chitwan 6, no. 6 (December 31, 2021): 82–94. http://dx.doi.org/10.3126/mefc.v6i6.42410.

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The South Asian region has a long history of discovering new ideas, ideologies, and technologies. Since the Vedic period, the land has been known as a fertile place for innovative discoveries. The Vedic technique used by Bharati Krishna Tirthaji is unique among South Asian studies. The focus of this study was mostly on algebraic topics, which are typically taught in our school level. The study also looked at how Vedic Mathematics solves issues of elementary algebra using Vedic techniques such as Paravartya Yojayet, Sunyam Samyasamuccaye, Anurupye Sunyamanyat, Antyayoreva and Lopanasthapanabhyam. The comparison and discussion of the Vedic with the conventional techniques indicate that the Vedic Mathematics and its five unique formulas are more beneficial and realistic to those learners who are experiencing problems with elementary level algebra utilizing conventional methods.

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2

Mathieu, Martin. "Elementary operators on prime C*-algebras II." Glasgow Mathematical Journal 30, no. 3 (September 1988): 275–84. http://dx.doi.org/10.1017/s0017089500007369.

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Compact elementary operators acting on the algebra ℒ(H) of all bounded operators on some Hilbert space H were characterised by Fong and Sourour in [9]. Akemann and Wright investigated compact and weakly compact derivations on C*-algebras [1], and also studied compactness properties of the sum and the product of the right and the left regular representation of an element in a C*-algebra [2]. They used the concept of a compact Banach algebra element due to Vala [17]: an element a in a Banach algebra A is called compact if the mapping x → axa is compact on A. This notion has been further investigated by Ylinen [18, 19, 20], who showed in particular that a is a compact element of the C*-algebra A if x ↦ axa is weakly compact on A [19].

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3

Timoney, Richard M. "AN INTERNAL CHARACTERIZATION OF COMPLETE POSITIVITY FOR ELEMENTARY OPERATORS." Proceedings of the Edinburgh Mathematical Society 45, no. 2 (June 2002): 285–300. http://dx.doi.org/10.1017/s0013091500000687.

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AbstractComplete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

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4

Tasic, Milan, Predrag Stanimirovic, Ivan Stanimirovic, Marko Petkovic, and Nebojsa Stojkovic. "Some useful MATHEMATICA teaching examples." Facta universitatis - series: Electronics and Energetics 18, no. 2 (2005): 329–44. http://dx.doi.org/10.2298/fuee0502329t.

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We show how a computer algebra system in MATHEMATICA can be used in several elementary courses in mathematics for students. We have also developed an application in programming language DELPHI for testing students in MATHEMATICA.

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5

Chrastina, Jan. "From elementary algebra to Bäcklund transformations." Czechoslovak Mathematical Journal 40, no. 2 (1990): 239–57. http://dx.doi.org/10.21136/cmj.1990.102378.

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6

Bernau, S. J., and C. B. Huijsmans. "Almost f-algebras and d-algebras." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 2 (March 1990): 287–308. http://dx.doi.org/10.1017/s0305004100068560.

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In this paper we study the classes of almost f-algebras and d-algebras. Apart from a survey of known properties we present some ‘new’ properties of almost f-algebras and d-algebras and we consider their connection with f-algebras. To be more precise, we show, among other things, in an elementary intrinsic manner, that every Archimedean almost f-algebra is commutative. This result is a considerable improvement upon the fact that every Archimedean f-algebra is commutative. Furthermore, we give a description of the set of nilpotent elements in both an Archimedean d-algebra and an Archimedean almost f-algebra.

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7

Magajna, Bojan. "Uniform approximation by elementary operators." Proceedings of the Edinburgh Mathematical Society 52, no. 3 (September 23, 2009): 731–49. http://dx.doi.org/10.1017/s0013091507001290.

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AbstractOn a separable C*-algebra A every (completely) bounded map which preserves closed two-sided ideals can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at ∞ of locally trivial C*-bundles of finite type.

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8

Bains, R. "Elementary linear algebra." Engineering Analysis with Boundary Elements 9, no. 4 (January 1992): 368. http://dx.doi.org/10.1016/0955-7997(92)90033-4.

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9

Earnest, Darrell, and Aadina A. Balti. "Instructional Strategies for Teaching Algebra in Elementary School: Findings from a Research-Practice Collaboration." Teaching Children Mathematics 14, no. 9 (May 2008): 518–22. http://dx.doi.org/10.5951/tcm.14.9.0518.

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Incorporating algebra into the elementary grades has become a focus for teachers, principals, and administrators around the country. Algebra is commonly regarded as a gateway to future opportunity (e.g., Moses and Cobb 2001), and elementary mathematics standards at both the state and national levels now reflect this effort to provide students with opportunities to learn critical concepts before middle and high school (California Department of Education 1997; NCTM 2000). However, implementing algebra standards at the elementary level is challenging—how do mathematics educators effectively and meaningfully incorporate algebraic ideas into K–5 curriculum? When elementary teachers are unfamiliar with early algebra, lessons designed and labeled as algebraic may become arithmetic exercises; the algebra then remains hidden from both the teacher and students in the implementation. The result is that the algebra standard is only superficially addressed.

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10

LYAKHOVSKY, V. D., and M. E. SAMSONOV. "ELEMENTARY PARABOLIC TWIST." Journal of Algebra and Its Applications 01, no. 04 (December 2002): 413–24. http://dx.doi.org/10.1142/s0219498802000306.

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The twist deformations for simple Lie algebras [Formula: see text] whose twisting elements ℱ are known explicitly are usually defined on the carrier subspace injected in the Borel subalgebra [Formula: see text]. We consider the case where the carrier of the twist intersects nontrivially with both [Formula: see text] and [Formula: see text]. The main element of the new deformation is the parabolic twist ℱ℘ whose carrier is the minimal parabolic subalgebra of simple Lie algebra [Formula: see text]. It has the structure of the algebra of two-dimensional motions, contains [Formula: see text] and intersects nontrivially with [Formula: see text]. The twist ℱ℘ is constructed as a composition of the extended jordanian twist [Formula: see text] and the factor [Formula: see text]. The latter can be considered as a special deformed version of the jordanian twist. The twisted costructure is found for [Formula: see text] and the corresponding universal ℛ-matrix is presented. The parabolic twist can be composed with certain types of chains of extended jordanian twists for algebras A2(n-1). The chains enlarged by the parabolic factor ℱ℘ perform the explicit quantization of the new set of classical r-matrices.

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11

Diniz, Diogo, Claudemir Fidelis, and Plamen Koshlukov. "Graded identities for algebras with elementary gradings over an infinite field." Proceedings of the Edinburgh Mathematical Society 65, no. 1 (January 10, 2022): 146–66. http://dx.doi.org/10.1017/s0013091521000857.

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AbstractLet $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

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Bay-Williams, Jennifer M. "Principles and Standards: What is Algebra in Elementary School." Teaching Children Mathematics 8, no. 4 (December 2001): 196–200. http://dx.doi.org/10.5951/tcm.8.4.0196.

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Patterns have long been part of early mathematics experiences. The K–4 Patterns and Relationships Standard in Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) was replaced in Principles and Standards for School Mathematics (NCTM 2000) with a K–12 Algebra Standard. This Standard encompasses patterns, functions, and some topics that are beyond what traditionally was considered to be algebra. However, the word algebra, often associated with content covered in a traditional middle school or high school course, can evoke feelings of anxiety and raise questions of appropriateness when discussed in relation to elementary school children. What is algebra in elementary school if it is more than identifying and extending patterns in the early grades yet is not the abstract content of an algebra course?

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Pham My, Hanh, and Giau Tran Thi Ngoc. "Vận dụng phương pháp mô hình hóa trong giảng dạy học phần đại số sơ cấp ngành Sư phạm Toán." Dong Thap University Journal of Science 10, no. 1 (February 15, 2021): 26–32. http://dx.doi.org/10.52714/dthu.10.1.2021.841.

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14

Gogić, Ilja. "On derivations and elementary operators on C*-algebras." Proceedings of the Edinburgh Mathematical Society 56, no. 2 (March 21, 2013): 515–34. http://dx.doi.org/10.1017/s0013091512000302.

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AbstractLet A be a unital C*-algebra with the canonical (H) C*-bundle $\mathfrak{A}$ over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of $\mathfrak{A}$ is a prime C*-algebra. We also consider separable C*-algebras A for which $\mathfrak{A}$ is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of $\mathfrak{A}$ have uniformly finite dimensions, and each restriction bundle of $\mathfrak{A}$ over a set where its fibres are of constant dimension is of finite type as a vector bundle.

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15

Berman, Barbara, and Fredda Friederwitzer. "Algebra Can Be Elementary … When It's Concrete." Arithmetic Teacher 36, no. 8 (April 1989): 21–24. http://dx.doi.org/10.5951/at.36.8.0021.

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Effective mathematics requires more than the use of symbols or numerals. All students, and even many adults, need concrete models to understand mathematical concepts. Elementary and middle school children can understand and master basic concepts of mathematics, including algebra, when concrete materials are used to model these concepts. This article presents a rationale for these statements and describes a model for teaching early algebraic concepts using manipulative materials.

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Strand, Krista, and Briana Mills. "Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Algebra." Mathematics Enthusiast 11, no. 2 (August 1, 2014): 385–432. http://dx.doi.org/10.54870/1551-3440.1307.

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17

Asfaw, Ayanaw Yigletu, Kassa Michael Weldeyesus, and Mulugeta Atnafu Ayele. "The effect of assessment for learning on prospective teachers’ learning of algebra through a professional development program." International Electronic Journal of Mathematics Education 17, no. 4 (October 28, 2022): em0718. http://dx.doi.org/10.29333/iejme/12587.

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Three mathematics teacher educators who were assigned to teach fundamental concepts of algebra course in a teacher education college, as well as their prospective elementary mathematics teachers who took the course, took part in a job-embedded, context-specific, and content-based comprehensive professional development (PD) program on assessment for learning (AfL). Three teacher educators and 129 prospective elementary mathematics teachers from three teacher education colleges took part in the study. The findings demonstrated that prospective elementary mathematics teachers in the intervention group significantly outperformed prospective teachers in both of the comparative groups in their post-test scores in algebra. Furthermore, it was found out that after the intervention, there was no statistically significant mean difference among achiever levels in the treatment group on post-test scores, despite a statistically significant mean difference among achiever levels in their pre-test scores. This research adds to our knowledge of the impact of comprehensive, job-embedded, context-specific, and content-based PD on prospective teachers’ achievements in algebra in elementary school mathematics teacher education. Implications of implementing AfL as well as recommendations for further research are highlighted.

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18

Sangwin, Christopher J. "Assessing elementary algebra with STACK." International Journal of Mathematical Education in Science and Technology 38, no. 8 (November 9, 2007): 987–1002. http://dx.doi.org/10.1080/00207390601002906.

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19

Hudson, Richard. "Mathematics in Language." Cognitive Semantics 6, no. 2 (August 12, 2020): 243–78. http://dx.doi.org/10.1163/23526416-bja10005.

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Abstract Elementary mathematics is deeply rooted in ordinary language, which in some respects anticipates and supports the learning of mathematics, but which in other respects hinders this learning. This paper explores a number of areas of arithmetic and other elementary areas of mathematics, considering for each area whether it helps or hinders the young learner: counting and larger numbers, sets and brackets, algebra and variables, zero and negation, approximation, scales and relationships, and probability. The conclusion is that ordinary language anticipates the mathematics of counting, arithmetic, algebra, variables and brackets, zero and probability; but that negation, approximation and probability are particularly problematic because mathematics demands a different way of thinking, and different mental capacity, compared with ordinary language. School teachers should be aware of the mathematics already built into language so as to build on it; and they should also be able to offer special help in the conflict zones.

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20

Suh, Jennifer M. "Investigations: Developing Algebra-'Rithmetic in the Elementary Grades." Teaching Children Mathematics 14, no. 4 (November 2007): 246–53. http://dx.doi.org/10.5951/tcm.14.4.0246.

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The “Investigations” department features children's hands-on and minds-on explorations in mathematics and presents teachers with open-ended investigations to enhance mathematics instruction. These tasks invoke problem solving and reasoning, require communication skills, and connect various mathematical concepts and principles. The ideas presented here have been tested in classroom settings.

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21

Bates, Teresa. "Applications of the Gauge-invariant uniqueness theorem for graph algebras." Bulletin of the Australian Mathematical Society 66, no. 1 (August 2002): 57–67. http://dx.doi.org/10.1017/s0004972700020670.

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We give applications of the gauge-invariant uniqueness theorem, which states that the Cuntz-Krieger algebras of directed graphs are characterised by the existence of a canonical action of . We classify the C*-algebras of higher order graphs, identify the C*-algebras of cartesian product graphs with a certain fixed point algebra and investigate a relation called elementary shift equivalence on graphs and its effect on the associated graph C*-algebras.

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22

McGlaughlin, Sean M., Andrew J. Knoop, and Gregory A. Holliday. "Differentiating Students with Mathematics Difficulty in College: Mathematics Disabilities vs. No Diagnosis." Learning Disability Quarterly 28, no. 3 (August 2005): 223–32. http://dx.doi.org/10.2307/1593660.

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Difficulties with college algebra can be the gatekeeper for earning a degree. Students struggle with algebra for many reasons. The focus of study was to examine students struggling with entry-level algebra courses and differentiate between those who were identified as having a mathematics disability and those who were not. Variables related to working memory, math fluency, nonverbal/visual reasoning, attention, and reading were analyzed using a MANOVA and separate ANOVAs. Significant differences were found on all but attention, supporting the findings of research on students in elementary and secondary education. Implications include a focus on techniques that help to remediate these specific deficits.

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23

Li, Fang. "Characterization of left Artinian algebras through pseudo path algebras." Journal of the Australian Mathematical Society 83, no. 3 (December 2007): 385–416. http://dx.doi.org/10.1017/s144678870003799x.

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AbstractIn this paper, using pseudo path algebras, we generalize Gabriel's Theorem on elementary algebras to left Artinian algebras over a field k when the quotient algebra can be lifted by a radical. Our particular interest is when the dimension of the quotient algebra determined by the nth Hochschild cohomology is less than 2 (for example, when k is finite or char k = 0). Using generalized path algebras, a generalization of Gabriel's Theorem is given for finite dimensional algebras with 2-nilpotent radicals which is splitting over its radical. As a tool, the so-called pseudo path algebra is introduced as a new generalization of path algebras, whose quotient by ken is a generalized path algebra (see Fact 2.6).The main result is that(i) for a left Artinian k–algebra A and r = r(A) the radical of A. if the quotient algebra A/r can be lifted then A ≅ PSEk (Δ, , ρ) with Js ⊂ (ρ) ⊂ J for some s (Theorem 3.2);(ii) If A is a finite dimensional k–algebra with 2-nilpotent radical and the quotient by radical can be lifted, then A ≅ k(Δ, , ρ) with 2 ⊂ (ρ) ⊂ 2 + ∩ ker (Theorem 4.2),where Δ is the quiver of A and ρ is a set of relations.For all the cases we discuss in this paper, we prove the uniqueness of such quivers Δ and the generalized path algebras/pseudo path algebras satisfying the isomorphisms when the ideals generated by the relations are admissible (see Theorem 3.5 and 4.4).

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Behncke, Horst, and George A. Elliott. "Lexicographic Direct Sums of Elementary C*-Algebras." Canadian Journal of Mathematics 39, no. 2 (April 1, 1987): 257–96. http://dx.doi.org/10.4153/cjm-1987-012-3.

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Besides the simple ones, there are several other kinds of C*-algebras which it has proved interesting to try to classify. For instance, a large body of results relates to the extensions of one given C*-algebra, possibly simple, by another. Extrapolating in this direction, we have considered the class of C*-algebras which can be decomposed in the strongest possible nontrivial sense in terms of their simple subquotients, and such that these simple subquotients in turn are as uncomplicated as possible.We have found that the classification of these C*-algebras, namely, the lexicographic direct sums of elementary C*-algebras, is to a large degree tractable, and yet involves an interesting new invariant in the antiliminary case, which is the case of no minimal ideals. Even the postliminary case, which is the case that the ordered set of simple subquotients satisfies the decreasing chain condition, is not without interest as an extension of the case of finitely many simple subquotients, analysed in the earlier papers [1] and [2].

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Fernandez, Maria L., and Cynthia O. Anhalt. "Transition Toward Algebra." Mathematics Teaching in the Middle School 7, no. 4 (December 2001): 236–41. http://dx.doi.org/10.5951/mtms.7.4.0236.

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IN PREPARING STUDENTS FOR ALGEBRA, EDucators must consider the range of mathematics and the continued development of mathematics as a science of pattern and order. Informal explorations involving physical models, data, graphs, and symbols are vital to a solid foundation for students' transition into algebra. Also important is developing a deep understanding of such concepts as decimals, fractions, ratio and proportion, measurement, integers, functional relationships, and variables. As students in elementary school and middle school work with number and operation and other mathematics strands, the focus should be on informal discussion and investigations that lead students to build, describe, represent patterns, develop and apply relationships, make and verify rules or generalizations, and explore mathematical properties (NCTM 1989, 2000).

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Carducci, Olivia M. "FOUR ELEMENTARY LINEAR ALGEBRA PROJECTS." PRIMUS 3, no. 4 (January 1993): 337–44. http://dx.doi.org/10.1080/10511979308965714.

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27

Arı, Ayşe Arzu, and Barış Demir. "An Innovative Approach to Linear Algebra Course: Using Manga Comics." Shanlax International Journal of Education 11, S1-Oct (October 1, 2023): 29–38. http://dx.doi.org/10.34293/education.v11is1-oct.6371.

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The aim of this study is to investigate the effect of using manga comics in Linear Algebra course on elementary mathematics teacher candidates’ attitudes and achievement towards the course. Pre-test-post-test control group experimental design was used in the study. The study was conducted with 48 elementary mathematics teacher candidates’ taking Linear Algebra 1 course. In the study, the experimental group received lessons that were taught under the guidance of the researcher using the Linear Algebra Manga Guidebook. On the other hand, the control group received lessons taught by the researcher using the traditional teaching method. In this 8-week study, Attitude Scale towards Linear Algebra Course and Linear Algebra Course Achievement exam prepared by the researchers were applied as pre-test and post-test. At the end of the study, a significant difference was obtained between the attitude towards linear algebra course and achievement scores of the experimental group and the control group in favour of the experimental group.

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Bay-Williams, Jennifer M. "Teaching Mathematics Then and Now: Still Hazy after All These Years? A Reaction Paper to Algebra in the Elementary Schools." Teaching Children Mathematics 13, no. 5 (January 2007): 248–51. http://dx.doi.org/10.5951/tcm.13.5.0248.

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In his song “Still Crazy after All These Years,” Paul Simon (1974) writes, “I seem to lean on old familiar ways.” Shipley's article “Algebra in the Elementary Schools,” written in 1912, offers the opportunity for us to reflect on what we have accomplished, what we have learned, and what remains hazy even 100 years later with respect to teaching and learning mathematics. Do we still lean on old familiar ways of teaching algebra? This article has many messages that are relevant to us today in thinking about algebra and about teaching and learning mathematics in general. In addition, it raises the following questions: How does Shipley's conception of algebra compare with our conception of it today? What broader issues does he address that are still prevalent today? How far have we come in our implementation of his ideas? What aspects of teaching and learning mathematics in elementary school remain hazy after all these years?

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Barksdale, James B. "Elementary Affine Transformation Models." Mathematics Teacher 81, no. 2 (February 1988): 127–30. http://dx.doi.org/10.5951/mt.81.2.0127.

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Elementary algebra is often the earliest mathematics course for college students. As the reader is well aware, one of the first topics in such a course is the notion of an affine function, f(x) = mx + k, and its straight-line graph. Of course, many reasons account for the early introduction of this topic, including conceptual simplicity, fundamental character, and broad applicability

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Cargal, James M. "Some notes on elementary Boolean algebra." International Journal of Mathematical Education in Science and Technology 19, no. 2 (March 1988): 231–41. http://dx.doi.org/10.1080/0020739880190203.

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Johanning, Debra, William B. Weber, Christine Heidt, Marian Pearce, and Karen Horner. "Links to Literature: The Polar Express to Early Algebraic Thinking." Teaching Children Mathematics 16, no. 5 (December 2009): 300–307. http://dx.doi.org/10.5951/tcm.16.5.0300.

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Algebra in the early grades (pre-K-2) is a relatively new focus area in the mathematics education community. As students, many of today's elementary school teachers did not study algebra until they were in high school.

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Maida, Paula. "Using Algebra without Realizing It." Mathematics Teaching in the Middle School 9, no. 9 (May 2004): 484–88. http://dx.doi.org/10.5951/mtms.9.9.0484.

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The phrase “Algebra for All” has been suggested as a standard for mathematics education. “By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more-sophisticated work in algebra in the middle grades and high school” (NCTM 2000, p. 37) Although current mathematics teachers are working toward achieving this goal, it is crucial to prepare future elementary and middle school teachers to integrate algebraic concepts into their lessons.

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Lin, Cheng-Yao, Yi-Yin Ko, and Yu-Chun Kuo. "Changes in Pre-Service Teachers’ Algebraic Misconceptions by Using Computer-Assisted Instruction." International Journal for Technology in Mathematics Education 21, no. 3 (September 1, 2014): 89–101. http://dx.doi.org/10.1564/174427114838794361.

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In order to carry out current reforms regarding algebra and technology in elementary school mathematics successfully, pre-service elementary mathematics teachers must be equipped with adequate understandings of algebraic concepts and self-confidence in using computers for their future teaching. This paper examines the differences in preservice teachers’ misconceptions of algebra and their attitudes towards using computers in teaching mathematics between computer-assisted instruction and lecture instruction. The results of the study suggest that pre-service teachers in both instructions have similar algebraic misconceptions and attitudes about teaching mathematics with computers. Even though there was no significant difference in algebraic misconceptions and beliefs about integrating computers into teaching mathematics between the participants in computer-assisted instruction and lecturebased instruction, both instructions were effective to increase pre-service teachers’ knowledge of parentheses usage, symbolism, and equality and attitudes of using computer in teaching mathematics in terms of confident and liking. It is recommended that more attention should be paid to teaching and learning algebra with computer-supported instruction in mathematics teacher education programs so that pre-service teachers gain experience in integrating technology into classrooms.

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Gogić, Ilja. "Elementary operators and subhomogeneous C*-algebras." Proceedings of the Edinburgh Mathematical Society 54, no. 1 (October 28, 2010): 99–111. http://dx.doi.org/10.1017/s0013091509001114.

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AbstractLet A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

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Martinez, Joseph G. R. "Building Conceptual Bridges from Arithmetic to Algebra." Mathematics Teaching in the Middle School 7, no. 6 (February 2002): 326–31. http://dx.doi.org/10.5951/mtms.7.6.0326.

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How can we help students make the transition from elementary school mathematics to middle school algebra? Although algebraic thinking and some important concepts are now routinely taught in the early grades, some students still approach algebra as an alien domain with an incomprehensible language.

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Meel, David E. "JOURNAL WRITING: ENLIVENING ELEMENTARY LINEAR ALGEBRA." PRIMUS 9, no. 3 (January 1999): 205–25. http://dx.doi.org/10.1080/10511979908965928.

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GOLDBRING, ISAAC, and THOMAS SINCLAIR. "ON THE AXIOMATIZABILITY OF C*-ALGEBRAS AS OPERATOR SYSTEMS." Glasgow Mathematical Journal 61, no. 03 (September 19, 2018): 629–35. http://dx.doi.org/10.1017/s001708951800040x.

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AbstractWe show that the class of unital C*-algebras is an elementary class in the language of operator systems and that the algebra multiplication is a definable function in this language. Moreover, we prove a general model theoretic fact which implies that the aforementioned class is ∀∃∀-axiomatizable. We conclude by showing that this class is, however, neither ∀∃-axiomatizable nor ∃∀-axiomatizable.

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38

BROWN, JONATHAN, and JONATHAN BRUNDAN. "ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES." Journal of the Australian Mathematical Society 86, no. 1 (February 2009): 1–15. http://dx.doi.org/10.1017/s1446788708000608.

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AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

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Hersberger, Jim, and Bill Frederick. "Flower Beds and Landscape Consultants: Making Connections in Middle School Mathematics." Mathematics Teaching in the Middle School 1, no. 5 (April 1995): 364–67. http://dx.doi.org/10.5951/mtms.1.5.0364.

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40

Aldhafeeri, S., and M. Bani-Ata. "On root-involutions and root-subgroups of E6(K) for fields K of characteristic two." Journal of Algebra and Its Applications 18, no. 01 (January 2019): 1950017. http://dx.doi.org/10.1142/s0219498819500178.

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The purpose of this paper is to investigate the root-involutions and root-subgroups of the Chevalley group [Formula: see text] for fields [Formula: see text] of characteristic two. The approach we follow is elementary and self-contained depends on the notion of [Formula: see text]-sets which we have introduced in [Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beit. Algebra Geom. 58 (2017) 529–534]. The approach is elementary on the account that it consists of little more than naive linear algebra. It is remarkable to mention that Chevalley groups over fields of characteristic two have not much been researched. This work may contribute in this regard. This paper is divided into three main sections: the first section is a combinatorial section, the second section is on relations among [Formula: see text]-sets, the last one is on Lie algebra.

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Lendert, Rola M., S. N. Aulele, and Y. A. Lesnussa. "ANALISIS PENGARUH DAERAH ASAL SMA TERHADAP NILAI UJIAN MAHASISWA DENGAN MENGGUNAKAN UJI WALD-WOLFOWITZ." VARIANCE : Journal of Statistics and Its Applications 1, no. 1 (August 8, 2019): 11–15. http://dx.doi.org/10.30598/variancevol1iss1page11-15.

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Origin is a factor that influences one's behavior about the value of the Mathematics Student's exam that comes from the Ambon city area and outside the city of Ambon. In this research, the method used to look at two independent samples is the Wald-Wolfowitz Test in the academic year 2016/2017. The purpose of this study to determine the influence of the origin of high school to the value of the middle exam of Elementary Linear Algebra in Department Mathematics FMIPA Unpatti. The result shows that there is no difference in the middle exam of Elementary Linear Algebra value between students who are in Ambon city and students who are outside Ambon city.

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42

Herman, Allen. "A Constructive Brauer-Witt Theorem for Certain Solvable Groups." Canadian Journal of Mathematics 48, no. 6 (December 1, 1996): 1196–209. http://dx.doi.org/10.4153/cjm-1996-063-1.

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AbstractDivision algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

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Oliveira, Hélia, Irene Polo Blanco, and Ana Henriques. "EXPLORING PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS’ KNOWLEDGE: A FOCUS ON FUNCTIONAL THINKING." Journal on Mathematics Education 12, no. 2 (April 12, 2021): 257–78. http://dx.doi.org/10.22342/jme.12.2.13745.257-278.

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The importance of students being acquainted with algebraic ideas before secondary education has been revealed in the research literature. It is therefore essential that prospective elementary teachers (PTs) be prepared to instill an early algebra perspective in their teaching. However, PTs often show difficulties in algebra content knowledge, which need to be diagnosed aiming to assist them in developing the required knowledge to teach according to that perspective. This study aims to understand what aspects of functional thinking Spanish and Portuguese elementary PTs exhibit at the beginning of their teacher education program. The findings show that although PTs from both countries use different strategies to generalize functional relationships, the occurrence of successful strategies is low. Also, most participants provide local approaches in their interpretation of relationships between variables and reveal difficulties in understanding and connecting different representations of functions. These difficulties show that PTs lack important knowledge about functional thinking. By providing a framework concerning the functional thinking required for PTs to teach within an early algebra perspective, we shed light on a necessary step for teacher education programs to diagnose PTs’ functional thinking and to assist them in developing the needed mathematical knowledge to teach accordingly.

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Lim, Lek-Heng. "Tensors in computations." Acta Numerica 30 (May 2021): 555–764. http://dx.doi.org/10.1017/s0962492921000076.

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The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.

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Schiller, Diane. "An Infinite Variety of Names." Arithmetic Teacher 35, no. 3 (November 1987): 12–13. http://dx.doi.org/10.5951/at.35.3.0012.

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An elementary school mathematics program can be structured to allow students to practice computational skills and also become familiar with concepts that need to be mastered in algebra. A basic notion needed for successful work in algebra is the idea that any number has an infinite variety of expressions. This same idea occurs in elementary school mathematics. For example, 10 = 9 + I = 8 + 2 = 7 + 3 = 6 + 4 = 5 + 5 = 100/10 = 2 × 5 = 90/9 = … = 20/2. The isolated knowledge that 3 + 7 = 10 is far less important than the chain of relationships in the previous sentence. An elementary school teacher can give students practice in computation (finding the sum to 10) while beginning to develop this idea of multiple names for any number. The activities described in this article begin with students' life experiences, continue through concrete and pictorial experiences, and conclude with abstract activities and problem solving.

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Shipley, James H. "Teaching Mathematics Then and Now: Algebra in the Elementary Schools." Teaching Children Mathematics 13, no. 5 (January 2007): 244–47. http://dx.doi.org/10.5951/tcm.13.5.0244.

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I have never yet heard of a high school teacher's being satisfied with the first-year pupil's knowledge of common fractions, and the other day a 1. A. [a specific grade level] girl frankly admitted that she could not multiply 12 2/3 by 15 3/4 because she had skipped one grade, had then had algebra and geometry, and the review didn't touch on mixed numbers. Of course, this was not the fault of any teacher, but the fault of a system which is trying to crowd too many things into too small a space. The real efficiency of the schools does not depend upon their being housed in million-dollar buildings, under a well-organized administrative force, excellent equipment, etc.; it depends on two things,—what is taught, and how it is taught; and more especially upon the latter; for every teacher of mathematics knows that a pupil can derive as much permanent good from the study of a few topics or theorems, so presented as to be pleasing to the pupil, or at least interesting, and at the same time make him think, as he can from ten times the amount of material “rammed home” with the sole object of being reproduced at examinations. The policy of standardizing everything by examinations is doing our expensive school system an untold injury; the report of the city superintendent compares the schools according to the number of their pupils who pass the examinations, and the principal warns the teacher that he is rated according to the number of his pupils that pass, and this pressure is passed on to the pupil. Until some method of close class-room observation and supervision is introduced with a view to allowing free rein to a teacher's individuality and originality even at the expense of his pupil's failing the conventional examinations, it is hardly worth while suggesting other changes.

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Macdonald, Alan. "An elementary construction of the geometric algebra." Advances in Applied Clifford Algebras 12, no. 1 (June 2002): 1–6. http://dx.doi.org/10.1007/bf03161249.

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COYETTE, CÉCILE. "MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS." Journal of Algebra and Its Applications 11, no. 03 (May 24, 2012): 1250052. http://dx.doi.org/10.1142/s0219498811005804.

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The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].

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DI VINCENZO, ONOFRIO M., and VINCENZO NARDOZZA. "GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS." Journal of Algebra and Its Applications 06, no. 03 (June 2007): 385–401. http://dx.doi.org/10.1142/s0219498807002193.

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Let F be a field and let E be the Grassmann algebra of an infinite dimensional F-vector space. For any p,q ∈ ℕ, the algebra Mp,q(E) can be turned into a ℤp+q × ℤ2-algebra by combining an elementary ℤp+q-grading with the natural ℤ2-grading on E. The tensor product Mp,q(E) ⊗ Mr,s(E) can be turned into a ℤ(p+q)(r+s) × ℤ2-algebra in a similar way. In this paper, we assume that F has characteristic zero and describe a system of generators for the graded polynomial identities of the algebras Mp,q(E) and Mp,q(E) ⊗ Mr,s(E) with respect to these new gradings. We show that this tensor product is graded PI-equivalent to Mpr+qs,ps+qr(E). This provides a new proof of the well known Kemer's PI-equivalence between these algebras. Then we classify all the graded algebras Mp,q(E) having no non-trivial monomial identities, and finally calculate how many non-isomorphic gradings of this new type are available for Mp,q(E).

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Lukierski, Jerzy. "Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras." Symmetry 13, no. 8 (July 21, 2021): 1309. http://dx.doi.org/10.3390/sym13081309.

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We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

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