Learning beginning algebra in a computer-intensive environment

We present a design research on learning beginning algebra in an environment where spreadsheets were available at all times but the decision about using them or not, and how, in any particular situation was left to the students. Students’ activity is analyzed in Kieran’s framework of generational, transformational and global/meta-level activity, and compared to the designers’ intentions. We do this by focusing on the activity of one student in four sessions spread over several months and discussing the activity of 51 additional students in view of the analysis of the focus student. We show that the environment enables a number of different entries into algebra and as such supports students in becoming autonomous learners of algebra, and in making the shift from arithmetic to algebra via generational and global/meta-level activity before dealing with the more technical transformational activities.     

Learning mathematics in a computer algebra environment: obstacles are opportunities

Using computer algebra is not as easy it may seem. Students often encounter obstacles while working in a computer algebra environment. In this paper, global and local obstacles are distinguished, and obstacles from both categories are identified. The theory of instrumentation provides a framework for interpreting an obstacle as an unbalance of the conceptual and technical aspects of an instrumentation scheme. It is argued that making the obstacles explicit and trying to overcome them leads to conceptual development. Therefore, obstacles are opportunities for learning.     

Generating numerical algorithms using a computer algebra system

An useful application of computer algebra systems is the generation of algorithms for numerical computations. We have shown in Gander and Gruntz (SIAM Rev., 1999) how computer algebra can be used in teaching to derive numerical methods. In this paper we extend this work, using essentially the capability of computer algebra system to construct and manipulate the interpolating polynomial and to compute a series expansion of a function. We will automatically generate formulas for integration and differentiation with error terms and also generate multistep methods for integrating differential equations. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D25, 65D30, 65D32, 65L06     

Ergodic sequences in the Fourier-Stieltjes algebra and measure algebra of a locally compact group

Let be a locally compact group. Blum and Eisenberg proved that if is abelian, then a sequence of probability measures on is strongly ergodic if and only if the sequence converges weakly to the Haar measure on the Bohr compactification of In this paper, we shall prove an extension of Blum and Eisenberg's Theorem for ergodic sequences in the Fourier-Stieltjes algebra of We shall also give an improvement to Milnes and Paterson's more recent generalization of Blum and Eisenberg's result to general locally compact groups, and we answer a question of theirs on the existence of strongly (or weakly) ergodic sequences of measures on

     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space

There are two extensions of Virasoro algebra with particular importance in superstring theory: the Ramond algebra and the Neveu-Schwarz algebra, which are Z₂-graded extensions of the Virasoro algebra. In this paper, we show that the support of a simple weight module over the Ramond algebra with an infinite-dimensional weight space coincides with the weight lattice and that all intersections of non-trivial weight spaces and odd part or even part of the module are infinite-dimensional. This result together with the one that we have obtained over the Neveu-Schwarz algebra generalizes the result for the Virasoro algebra to the super-Virasoro algebras.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

The subconstituent algebra of strongly regular graphs associated with a Latin square

It is well-known that one may construct several strongly regular graphs on the positions of a Latin square, where adjacency corresponds to any subset of the relations on distinct positions of being in the same row, being in the same column, having the same entry, or none of these. We describe the local spectrum and subconstituent (Terwilliger) algebras of such strongly regular graphs.      

Lie subalgebras in a certain operator Lie algebra with involution

We show in a certain Lie*-algebra, the connections between the Lie subalgebra G ⁺:= G + G* + [G, G*], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgebras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion: Let A ⊂ B(X) be a space of self-adjoint operators and ℒ:= A ⊕ iA the corresponding complex Lie*-algebra. G ⁺ = G + G* + [G, G*] and G are two LM-decomposable Lie subalgebras of ℒ with the decomposition G ⁺ = R(G ⁺) + S, G = R _G + S _G , and R _G ⊆ R(G ⁺). Then G ⁺ is ideally finite iff R _G ⁺:= R _G + R* _G ^* + [R _G , R _G ^*] is a quasisolvable Lie subalgebra, S _G ⁺:= S _G + S _G ^* + [S _G , S _G ^*] is an ideally finite semisimple Lie subalgebra, and [R _G , S _G ] = [R _G ^*, S _G ] = {0}.     

Inverse limits in representations of a restricted Lie algebra

Let(g,[p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 0.Then the inverse limits of "higher" reduced enveloping algebras {uχs(g)|s∈N} with χ running over g* make representations of g split into different "blocks".In this paper,we study such an infinitedimensional algebra Aχ(g):= ■Uχs(g) for a given χ∈g*.A module category equivalence is built between subcategories of U(g)-mod and Aχ(g)-mod.In the case of reductive Lie algebras,(quasi) generalized baby Verma modules and their properties are described.Furthermore,the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined,and a higher reciprocity in the case of regular nilpotent is obtained,generalizing the ordinary reciprocity.     

格蕴涵代数的LI-理想格及其素元刻画

运用格理论的原理和方法对格蕴涵代数L的LI-理想概念作进一步研究.首先,在L的全体LI-理想之集ф_(LI)(L)上定义了格运算■和■,蕴涵运算■以及伪补运算■,证明了(ф_(LI)(L),,■,■,■,{O},L)构成一个完备Heyting代数的结论.其次,利用运算固的性质给出了(ф_(LI)(L),,■,■,■,■,{O},L)成为Boolean代数的若干充要条件.最后,借助于L的素LI-理想之特性获得了格(ф_(LI)(L),,■,■,{O},L)中素元的若干等价刻画.     

Students' attitude towards computer algebra systems (CAS) and their choice of using CAS in problem-solving

We explore students choice of using computer algebra systems (CAS) in problem-solving relative to their self-reported attitude towards learning mathematics with CAS. Our research design is a case study of nine Norwegian upper-secondary mathematics students with a wide range of attitude towards CAS. Our findings on routine problems indicate that (1) students use CAS whenever students perceive the problem as time-consuming regardless of their attitude towards CAS, and (2) students attitude affects their use of CAS whenever students perceive the problem as non-time-consuming. Norway, among other countries, has implemented CAS as an essential digital resource towards learning mathematics in upper-secondary school. Our discussion focuses on the implications of our findings have on local mathematics educators and national policy-makers.     

Strong connectedness of the invertibles in a finite subdiagonal algebra

Suppose is a finite, subdiagonal subalgebra of a von Neumann algebra. We show that the invertible group of is strongly connected.

     

The centralizer of an element in a Lie algebra of type L

In this paper, we investigate the Lie algebra L(A, α, δ) of type L and obtain the respective sufficient conditions for L(A, α,δ) to be semisimple, and for Z(ω) = Fω as well, where 0 ≠ ω ∈ L(A, α, δ) and Z(ω) is the centralizer of ω.     

Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Let M be a type I von Neumann algebra with the center Z and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M) is inner. In particular, all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements of LS(M). The text was submitted by the authors in English.     

A seminorm with square property on a complex associative algebra is submultiplicative

The result stated in the title is proved as a consequence of an appropriate generalization replacing the square property of a seminorm with a similar weaker property which implies an equivalence to the supnorm of all continuous functions on a compact Hausdorff space also.

     

Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III

We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. The structural analysis of type III factors and the canonical extension of endomorphisms introduced by Izumi are our main technical tools.     

On k-regular functions with values in a universal Clifford algebra

In this paper, we mainly study properties of nullsolutions of the operator Dk (k∈N^∗=N?{0}), so-called k-regular functions. Firstly, we study the set of all homogeneous polynomials of degree p in x₁,…,xn which are k-regular in the whole Rn, clearly is a right module over C(Vn,n), we construct a basis for the right module . Secondly, we study the k-regular and analytic functions, and we give the Taylor expansions for these functions. At last, the corresponding Taylor expansions for k-regular functions are given since each k-regular function is a real analytic function.     

Creating discussions with classroom voting in linear algebra

We present a study of classroom voting in linear algebra, in which the instructors posed multiple-choice questions to the class and then allowed a few minutes for consideration and small-group discussion. After each student in the class voted on the correct answer using a classroom response system, a set of clickers, the instructor then guided a class-wide discussion of the results. We recorded the percentage of students voting for each option on each question used in 18 sections of linear algebra, taught by 10 instructors, at 8 institutions, over the course of 5 years, together recording the results of 781 votes on a collection of 311 questions. To find the questions most likely to provoke significant discussions, we identify the six questions for which votes were most broadly distributed. Here we present these questions, we discuss how we used them to advance student learning, and we discuss the common features of these questions, to identify why they were so good at stimulating discussions.