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Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Every octonion is a real linear combination of the unit octonions {1, e1, e2, e3, e4, e5, e6, e7}. Their properties include: (a) e2 = −1; (b) eiej = −ejei when i 6= j; (c) the index doubling identity: eiej = ek =⇒ e2ie2j = e2k; and
Almost periodic oscillation of octonion-valued neural networks with delays on time scales
Published in International Journal of Systems Science, 2023
With the continuous deepening of the theoretical research of neural networks and the continuous expansion of their application scope, the signals they need to deal with become more and more complex, in recent years, neural networks with multi-dimensional domain values such as quaternion-valued neural networkd and Clifford-valued neural networks have attracted more and more researchers' attention (Cao & Li, 2022; Deng & Bao, 2019; Humphries et al., 2020a, 2020b; Li et al., 2021; Li & Li, 2022; Li et al., 2022; Popa, 2018b; Pratap et al., 2020; Song & Chen, 2018; Sriraman et al., 2020; Tu et al., 2019; Wang et al., 2021; Xia et al., 2022). Very recently, Popa (2016) proposed octonion-valued neural networks. Due to the potential application value of octonion-valued neural networks for dealing with multi-dimensional signals (Aimeur et al., 2020; Cariow & Cariowa, 2021; Saoud & Ghorbani, 2019; Wu et al., 2020), the study of the dynamics of octonion-valued neural networks has become a problem worthy of in-depth exploration. We know that octonions (Baez, 2002; Dickson, 1919) are non-associative generalisations of quaternions, so they are not part of Clifford algebra. Octonion algebra has many applications in fields such as physics, geometry, and signal processing (Dray & Manogue, 2015; Okubo, 1995; Sirley et al., 2020; Snopek, 2015). However, the non-associative and non-commutative properties of octonion algebra make it difficult to study various qualitative properties of octonion-valued neural networks. Therefore, the existing results about the dynamics of octonion-valued neural networks (Kandasamy & Rajan, 2020; Popa, 2018a, 2020) are vey few, and almost all of them are obtained by decomposing octonion-valued neural networks into real-valued ones. Therefore, it has important theoretical and practical value to study the long term qualitative behaviours of solutions of octonion-valued neural networks by direct approaches.