可加性与正弦可加性之间的相互依存关系

IF 0.9 Q2 MATHEMATICS
Bruce Ebanks
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引用次数: 0

摘要

让 S 是一个半群,K 是一个域。一个函数(f:S)是可加的,如果对于所有在S中的(x,y),(f(xy) = f(x) + f(y))是可加的,并且函数(g,h:S (rightarrow K\) 形成了一对正弦,如果它们满足正弦加法法则的话。将这两个等式相加,我们就得到函数等式 (*) \(f(xy) + g(xy) = f(x) + f(y) + g(x)h(y) + h(x)g(y)\).关于可加性和正弦可加性的异化问题问的是:(*) 是否意味着 f 是可加的,(g, h) 是一对正弦。为了完全回答这个问题,我们要找到未知函数 \(f,g,h:S \rightarrow {\mathbb {C}}\) 的 (*) 的一般解。这个解说明了可加性和正弦可加性之间的大量相互依存关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Interdependence of additivity and sine additivity

Let S be a semigroup and K a field. A function \(f:S \rightarrow K\) is additive if \(f(xy) = f(x) + f(y)\) for all \(x,y \in S\), and functions \(g,h:S \rightarrow K\) form a sine pair if they satisfy the sine addition law \(g(xy) = g(x)h(y) + h(x)g(y)\) for all \(x,y \in S\). Adding these two equations we arrive at the functional equation (*) \(f(xy) + g(xy) = f(x) + f(y) + g(x)h(y) + h(x)g(y)\). The alienation question for additivity and sine additivity asks whether (*) implies that f is additive and (gh) is a sine pair. To fully answer this question we find the general solution of (*) for unknown functions \(f,g,h:S \rightarrow {\mathbb {C}}\). The solution illustrates a significant amount of interdependence between additivity and sine additivity.

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来源期刊
Afrika Matematika
Afrika Matematika MATHEMATICS-
CiteScore
2.00
自引率
9.10%
发文量
96
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