Multilinear algebra

WhiD74 White, D. E. Linear and multilinear aspects of isomorph rejection. Linear and multilinear algebra (1974) 211-226. 

WhiD75a White, D. E. Multilinear enumerative techniques. Linear and Multilinear Algebra 4 (1975) 341-352. 

WhiD75b White, D. E. Redfield s theorems and multilinear algebra. Canad. J. Math. 27 (1975) 704-714. 

WilS73a Williamson, S. G. Tensor compositions and lists of combinatorial structures. Linear and Multilinear Algebra 1... 

M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973,1975)... 

Much of numerical computation in chemistry revolves about numerical multilinear algebra. By this term I denote the manipulation of arrays whose dimension may exceed two. Mindful of Richard and Ledgard s admonition (16) that language design should never be overly ambitious, I propose only data structures, operations, and syntax for programming multilinear algebra. Multilin exceeds Bohlender s Pascal extension in two ways its provision for more than two dimensions, and its explicit declaration of data representations. 

De Lathauwer L, Signal processing based on multilinear algebra, Ph.D. thesis, K.U. Leuven, 1997. 

However, useful methods exist that treat simultaneous equations without resorting to formalized methods of multilinear algebra. We shall discuss two of these methods because of their utility and frequent occurrence in practical problems ... 

In other words, Eq. (6.18) is satisfied by an appropriate function /(/I) = F( 2 - 11). Obviously, 2 and 2 obey the above relations. The reason for symmetry (6.18) will be explained in Appendix A in terms of a duality transformation well-known in the multilinear algebra literature. In Appendix A one can also find a possible generalization of indices N and Neff. Various examples of Nes and related indices will be given throughout the rest of this chapter. 

In this Appendix we clarify the cause for postulating symmetry relation (6.18). For this aim we introduce a formal operation which can be named the duality transformation and which is well known in multilinear algebra as the Hodge star operation, or Hodge dual [132]. In the RDM theory an equivalent transformation was applied in [19, 133], without recognizing it as a Hodge dual. The following simple example helps to explain this notion in the more familiar terms of many-electron state vectors. 

Cvetkovid, D. Doob, M. Development in the Theory of Graph Spectra. Linear Multilinear Algebra 1985, 18, 153-181. 

Cvetkovid, D. Rowlinson, P. The Largest Eigenvalue of a Graph A Survey. Linear Multilinear Algebra 1990, 28, 3. 

Algebraic expressions for terms M and C were derived using Dewar s PMO method (for C in a version similar to the co-technique [57] in order to calculate carbocation stabilization energies). The size factor S is simply a cubic function of the number of carbon atoms [97], The three independent variables of the model were assumed to be linearly related to the experimental Iball indices (vide supra). By multilinear regression analysis (sample size = 26) an equation was derived for calculating Iball indices from the three theoretical parameters. The correlation coefficient for the linear relation between calculated and experimental Iball indices is r = 0.961. 

Chapter 2 (Statistical Space for Multivariate Correlations) Aims to prepare the conceptual-computational ground for correlating chemical structure with biological activity by the celebrated quantitative stractuie-activity relationships (QSARs). Additionally, the fundamental statistical advanced frameworks are detailed to best understand the classical multilinear regression analysis generalized by an algebraic (in quantum Hilbert space) reformulation in terms of data vectors and orthogonal conditions (explained in see Chapter 3). 

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