Multiplicative Maps With Specified (1, 1)-Entry

In this article, we will show that for many complex-valued maps f over a semigroup G of matrices, there exists a minimum k for the existence of a multiplicative map for which the (1, 1)-entry of ϕ( A ) is f ( A ). We obtain results on such multiplicative maps, and use them to classify all the multiplicative maps τon G such that f ( A ) = τ( g ( A )) for any A ε G , where f and g are given complex-valued maps.     

LFI代数的性质及其与剩余格的关系

讨论LFI代数的性质．也讨论具有可嵌入性的LFI代数的性质；还给出了定义在完备格上的LFI代数成为剩余格的充要条件。     

Bernoulli多项式与Euler—Maclaurin公式的推广

本文提出了广义Bernoulli多项式与广义Bernoulli数,并借此得到了一类含两端点连续阶导数值求积公式的误差渐近式和推广的Euler-Maclaurin求和公式.借助于计算机代数系统进行了公式的机械推导,并列出了部分推导结果.     

On a class of subalgebras of and the intersection of their free maximal ideals

Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

     

Powers of matrices over an extremal algebra with applications to periodic graphs

Consider the extremal algebra=({},min,+), using + and min instead of addition and multiplication. This extremal algebra has been successfully applied to a lot of scheduling problems. In this paper the behavior of the powers of a matrix over is studied. The main result is a representation of the complete sequence (A ^m ) _m which can be computed within polynomial time complexity. In the second part we apply this result to compute a minimum cost path in a 1-dimensional periodic graph.     

Asymptotic Ideals (Ideals in the Ring of Colombeau Generalized Constants with Continuous Parametrization)

We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.     

ON THE ZEROS OF POLYNOMIALS OVER QUATERNIONS

ABSTRACT

We review and expand some results on the number of zeros of polynomials over Hamilton's quaternions, with particular emphasis on those polynomials with coefficients in a degree two subfield.     

Cuspidal Modules as Summands of a Gel'fand–Graev Module

Abstract

Let G = GL _n (q) be the general linear group over a finite field 𝔽 _q with q elements. We call a Gel'fand–Graev module to be the module which affords the Gel'fand–Graev character defined in Definition I.1. It is known that every cuspidal module of G is isomorphic to a (unique) direct summand of a Gel'fand–Graev module. In this article, we investigate a certain endomorphism so that each irreducible cuspidal module is contained in a certain eigenspace corresponding to the cuspidal character. Furthermore, we determine the eigenvalue of that endomorphism by using character theory of finite general linear group.     

What “counts” as algebra in the eyes of preservice elementary teachers?

This study examined conceptions of algebra held by 30 preservice elementary teachers. In addition to exploring participants’ general “definitions” of algebra, this study examined, in particular, their analyses of tasks designed to engage students in relational thinking or a deep understanding of the equal sign as well as student work on these tasks. Findings from this study suggest that preservice elementary teachers’ conceptions of algebra as subject matter are rather narrow. Most preservice teachers equated algebra with the manipulation of symbols. Very few identified other forms of reasoning - in particular, relational thinking - with the algebra label. Several participants made comments implying that student strategies that demonstrate traditional symbol manipulation might be valued more than those that demonstrate relational thinking, suggesting that what is viewed as algebra is what will be valued in the classroom. This possibility, along with implications for mathematics teacher education, will be discussed.     

Absolute Valued Algebras Satisfying ( x 2, x 2, x 2) = 0

Abstract

Let A be an absolute valued algebra. We prove that if A satisfies the identity (x ², x ², x ²) = 0 for all x in A, and contains a central idempotent e, that is ex = xe for all x in A, then A is finite dimensional. This result enables us to prove that if A satisfies (x ², x ², x ²) = 0 and admits an involution then A is finite dimensional. To show that our assumptions on A are essential we recall that in El-Mallah [El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51:39–49] it was shown that the existence of a central idempotent in A is not a sufficient condition for A to be finite dimensional; and the example given in El-Mallah [El-Mallah, M. L. (2003). Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31(7):3135–3141] shows that there exist infinite dimensional semi-algebraic absolute valued algebras satisfying the identity (x ², x ², x ²) = 0.