A bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.

中文翻译：

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重新审视非奇异双线性映射

双线性映射$\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$是非奇异的，如果$\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$暗示$\overrightarrow {a}=\overrightarrow {0}$要么$\overrightarrow {b}=\overrightarrow {0}$. 这些地图对拓扑学家很感兴趣，并且有助于研究真实射影空间上的向量丛。本文的主要目的是在范围内制作此类地图的示例$24\leqslant r\leqslant 32,\24\leqslant s\leqslant 32,$使用八元数（也称为凯莱数）的算术作为一种有效的工具。虽然先前在低维情况下的构造使用临时技术，但我们的构造遵循系统程序并将这些技术纳入统一的视角。