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Specific Adjacency in Bonding
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Mihai V. Putz, Marina A. Tudoran
The SAIB method presented by Tudoran & Putz (2015) was, is a topo-reactivity method based on replacing the 1 and 0 values (existence and respectively non-existence of a bond between atoms) in the adjacency matrix whit the reactivity indices obtained from each type of bond and was, until now, successfully applied only to polycyclic aromatic hydrocarbons (finite systems) with the maximum of 30 carbon atoms. As a result of studying 1D polymers at the beginning and later multidimensional systems (Cataldo et al., 2010), the extension of topological methods (TM) based on distance matrix invariants to infinite systems appears as a good possibility due to their behavior of the polynomial type. It appears that TM simulations are able to isolate the regions stabilized by topology connections at the lattice specific critical size (Putz et al., 2015). In this context, the SAIB method can be further improved if the adjacency matrix can be replaced with the distance matrix, i.e., if one can determine a recurrence formula which can determine the type of bond only by knowing the distance between the neighbors for a reference vertex/atom from the lattice.
Two Decades of Multidimensional Systems Research and Future Trends
Published in Krzysztof Gałkowski, Jeff David Wood, Multidimensional Signals, Circuits and Systems, 2001
• Algebraic aspects of multidimensional convolutional codes, which can be studied in the setting of a ring /"„[¿i, z?, • • ■, г„], where Тя is a finite field with q elements has attracted some attention recently because of the potentials of n-D convolutional codes in the encoding and decoding of multidimensional data (Fornasini & Valcher 1994). (Fornasini & Valcher 1997). The resources of multidimensional systems theory including the different notions of primeness, were summarized in (Wood et al. 1998) for the ring /C[zi,¿2, ■ • •, г„] (and the Laurent polynomial ring where the indctcrmi-natcs occur with their respective inverses), when the field of coefficients is arbitrary
Multidimensional realisation theory and polynomial system solving
Published in International Journal of Control, 2018
Philippe Dreesen, Kim Batselier, Bart De Moor
Recent years have witnessed a surge in research on multidimensional systems theory, identification and control (Batselier & Wong, 2016; Bose, 2007; Hanzon & Hazewinkel, 2006a; Ramos & Mercère, 2016; Rogers et al., 2015; Zerz, 2000, 2008). There is a broad scientific interest regarding multidimensional systems, as they offer an extension to the well-known class of one-dimensional linear systems, in which the system trajectories depend on a single variable (such as time or frequency), to a dependence on several independent variables (such as a two-dimensional position, spatio-temporal systems, parameter varying systems, etc.). However, the analysis of multidimensional systems is known to be more complicated than that of one-dimensional systems.