• Journal of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.241-250
• /
• 2003

Let (B, m$_{B}$, k) be a maximal commutative $textsc{k}$-subalgebra of M$_{m}$(k). Then, for some element z $\in$ Soc(B), a k-algebra R = B［X,Y］/I, where I ＝ (m$_{B}$X, m$_{B}$Y, X$^2$- z,Y$^2$- z, XY) will create an interesting maximal commutative $textsc{k}$-subalgebra of a matrix algebra which is neither a $C_1$-construction nor a $C_2$-construction. This construction will also be useful to embed a maximal commutative $textsc{k}$-subalgebra of matrix algebra to a maximal commutative $textsc{k}$-subalgebra of a larger size matrix algebra.gebra.a.

• Communications of the Korean Mathematical Society
• /
• v.23 no.2
• /
• pp.153-159
• /
• 2008

For $\mathbb{F}[e^{{\pm}x}]_{\{{\partial}\}}$, all the derivations of the evaluation algebra $\mathbb{F}[e^{{\pm}x}]_{\{{\partial}\}}$ is found in the paper (see [16]). For $M=\{{\partial}_1,\;{\partial}_1^2\},\;Der_{non}(\mathbb{F}[e^{{\pm}x}]_M))$ of the evaluation algebra $\mathbb{F}[e^{{\pm}x},\;e^{{\pm}y}]_M$ is found in the paper (see [2]). For $M=({\partial}_1^2,\;{\partial}_2^2)$, we find $Der_{non}(\mathbb{F}[e^{{\pm}x},\;e^{{\pm}y}]_M))$ of the evaluation algebra $\mathbb{F}[e^{{\pm}x},\;e^{{\pm}y}]_M$ in this paper.

• Journal for History of Mathematics
• /
• v.17 no.4
• /
• pp.37-44
• /
• 2004

Homological algebra has emerged and developed since 1950s. However, in 1890's Hilbert investigated the resolutions in his Syzygy Theorem which is a vital ingredient in homological algebra. In 1956 Serre has proved the finite global dimension of regular local rings. His result give a basic tool in homological algebra. This paper also deals with the depth formula that was raised by Auslander in 1961.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• v.11 no.5
• /
• pp.2346-2361
• /
• 2017

A major challenge in cloud infrastructure is the efficient allocation of virtual network elements on top of substrate network elements. Path algebra is a mathematical framework which allows the validation and convergence analysis of the mono-constraint or multi-constraint routing problems independently of the network topology or size. The present study proposes a new heuristic approach based on mathematical framework "paths algebra" to map virtual nodes and links to substrate nodes and paths in cloud. In this approach, we define a measure criterion to rank the substrate nodes, and map the virtual nodes to substrate nodes according to their ranks by using a greedy algorithm. In addition, considering multi-constraint routing in virtual link mapping stage, the used paths algebra framework allows a more flexible and extendable embedding. Obtained results of simulations show appropriate improvement in acceptance ratio of virtual networks and cost incurred by the infrastructure networks.

• Research in Mathematical Education
• /
• v.12 no.3
• /
• pp.215-233
• /
• 2008

Deep comprehension of basic mathematical notions and concepts is a basic condition of a successful teaching. Some elements of algebraic thinking belong to the elementary school mathematics. The question "What stays the same and what changes?" link arithmetic problems with algebraic conception of variable. We have studied beliefs and comprehensions of future elementary school mathematics teachers on early algebra. Pre-service teachers from three academic pedagogical colleges deal with mathematical problems from the pre-algebra point of view, with the emphasis on changes and invariants. The idea is that the intensive use of non-formal algebra may help learners to construct a better understanding of fundamental ideas of arithmetic on the strong basis of algebraic thinking. In this article the study concerning arithmetic series is described. Considerable number of pre-service teachers moved from formulas to deep comprehension of the subject. Additionally, there are indications of ability to apply the conception of change and invariance in other mathematical and didactical contexts.

• The Mathematical Education
• /
• v.40 no.2
• /
• pp.335-343
• /
• 2001

Algebra is important part of mathematics education. Recent days, many mathematics educators emphasize on real world situation. Form real situation, pupils make sense of concepts, and mathematize it by reflective thinking. After that they formalize the concepts in abstract. For example, operation in numbers develops these course. Operation in natural number is an arithmetic, but operation on real number is algebra. Transition from arithmetic to algebra has the cutting point in representing the concepts to mathematics sign system. In this note, we see the cognitive tendency of 10th grade about operation of real number, their cutting point of transition from arithmetic to algebra, and show some methods of helping pupils.

• The Pure and Applied Mathematics
• /
• v.15 no.4
• /
• pp.393-400
• /
• 2008

In this paper, we investigate the following additive functional inequality (0.1) ||f(x)+f(y)+f(z)+f(w)||${\leq}$||f(x+y)+f(z+w)|| in normed modules over a $C^*$-algebra. This is applied to understand homomor-phisms in $C^*$-algebra. Moreover, we prove the generalized Hyers-Ulam stability of the functional inequality (0.2) ||f(x)+f(y)+f(z)f(w)||${\leq}$||f(x+y+z+w)||+${\theta}||x||^p||y||^p||z||^p||w||^p$ in real Banach spaces, where ${\theta}$, p are positive real numbers with $4p{\neq}1$.

• Communications of Mathematical Education
• /
• v.23 no.4
• /
• pp.1131-1148
• /
• 2009

Algebra is one of the important subjects that not only mathematics but many science major students should know at least at the elementary level. Unfortunately abstract algebra, specially, is seen as an extremely difficult course to learn. One reason of difficulties is because of its very abstract nature, and the other is due to the lecture method that simply telling students about mathematical contents. In this paper we study about the teaching and learning abstract algebra in universities in corporation of a programming language such as ISETL. ISETL is a language whose syntax closely imitates that of mathematics. In asking students to read and write code in ISETL before they learn in class, we observe that students can much understand and construct formal statements that express a precise idea. We discuss about the classroom activities that may help students to construct and internalize mathematical ideas, and also discuss about some barriers we might overcome.

• The Pure and Applied Mathematics
• /
• v.20 no.4
• /
• pp.243-249
• /
• 2013

Let A be an algebra and D a derivation of A. Then D is called algebraic nil if for any $x{\in}A$ there is a positive integer n = n(x) such that $D^{n(x)}(P(x))=0$, for all $P{\in}\mathbb{C}[X]$ (by convention $D^{n(x)}({\alpha})=0$, for all ${\alpha}{\in}\mathbb{C}$). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra A maps into N(A), where N(A) denotes the set of all nilpotent elements of A. As an application, we deduce that any nilpotent derivation on a commutative complex algebra A maps into N(A), Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.5
• /
• pp.1209-1219
• /
• 2021

Let G be a topological group with a locally compact and Hausdorff topology. Let ω be a diagonally bounded weight on G. In this paper, (σ, σ)-derivation and (σ, 𝜏)-weak amenability of the Beurling algebra L¹_ω(G) are studied, where σ, 𝜏 are isometric automorphisms of L¹_ω(G). We prove that every continuous (σ, σ)-derivation from L¹_ω(G) into measure algebra M_ω(G) is (σ, σ)-inner and the Beurling algebra L¹_ω(G) is (σ, 𝜏)-weakly amenable.