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A Complete, Integrated Time Crystal Model of a Human Brain
Published in Anirban Bandyopadhyay, Nanobrain, 2020
Distributed parallel computing in parts of cortex shows tensor algebra like signal processing (Anastasio and Robinson, 1990). If tensors are there, theories of a holographic brain projecting Schrödinger’s wave could not remain far (Nobili, 1985). Clocked cell cycles clock (Edmunds and Adams, 1981) to process three distinct classes of tensors simultaneously. All large-scale models of a human brain (Eliasmith et al., 2012) are linear, 4D, 8D, and 12D network of time crystals is the singular undefined model of the human brain. Octonion algebra could estimate the phase elements of the tensors interacting with each other (Furey, 2018). However, dodecanion algebra would loop the interactive elements, once? Twice? In reality, looping isolated elements mean triggering the PPM. Four-dimensional unified brain theory (Friston, 2010) is not transformed into a 12-dimensional brain here, rather, a variable dimension brain that explores the topology of dimensional transformation would be the right explanation of the time crystal brain model discussed here. Imagine some thoughts are 2-dimensional, some are 8-dimensional, and some events are 10-dimensional (Figure 7.17), the truth lies in the pattern of imaginary worlds active among 4, 8, and 12 choices, it is non-computable (Stulf et al., 2015).
Nonassociative Algebras
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Murray R. Bremner, Lucia I. Murakami, Ivan P. Shestakov
Evaluation of the monomials gives the matrix in Table 69.1; its reduced row echelon form appears in Table 69.2. The nullspace contains the identities satisfied by the octonion algebra: the span of the rows of the matrix in Table 69.3. These rows represent the linearizations of the right alternative identity (row 1), the left alternative identity (row 2), and the flexible identity (row 5), together with the assocyclic identities (x, y, z) = (y, z, x) and (x, y, z) = (z, x, y) (rows 3 and 4).
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.