Abstract
Collective distances in quantum multimolecular polyhedra (QMP), which can be set as a scalar indices associated to the QMP variance vector, enhance the role of the pair density similarity matrix. This paper describes a simplified efficient algorithm to compute triple, quadruple or higher order density similarity hypermatrices via an approximate isometry: a synisometric decomposition of the pair similarity matrix. Synisometries pretend to avoid the use of Minkowski metrics in QMP description problems, where the double density similarity matrix possesses negative eigenvalues. The synisometric decomposition of the similarity matrix opens the way to the general use of higher order approximate similarity elements in quantum QSAR and in the construction of scalar condensed vector statistical-like indices, for instance skewness and kurtosis. This might lead the way to describe, without excessive complication and within a real field computational framework, the collective structure of quantum multimolecular polyhedra.
Similar content being viewed by others
References
R. Carbó-Dorca, An isometric representation problem in quantum multimolecular polyhedra and similarity. J. Math. Chem. doi: 10.1007/s10910-015-0516-4
R. Carbó-Dorca, E. Besalú, Shells, point cloud huts, generalized scalar products, cosines and similarity tensor representations in vector semispaces. J. Math. Chem. 50, 210–219 (2012)
R. Carbó-Dorca, Collective Euclidian distances and quantum similarity. J. Math. Chem. 51, 338–353 (2013)
R. Carbó-Dorca, Multimolecular polyhedra and QSPR. J. Math. Chem. 52, 1848–1856 (2014)
R. Carbó-Dorca, Quantum polyhedra, definitions, statistics and the construction of a collective quantum similarity index. J. Math. Chem. 53, 171–182 (2015)
R. Carbó-Dorca, D. Barragán, Communications on quantum similarity (4): collective distances computed by means of similarity matrices, as generators of intrinsic ordering among quantum multimolecular polyhedra. IQCC Technical Report TC-2015-1. WIREs (in press)
R. Carbó-Dorca, D. Barragán, Collective vectors, condensed indices and quantum similarity. IQCC Technical Report TC-2015-2. Front. Comput. Chem. (in press)
R. Carbó-Dorca, S. Rodríguez, Quantum multimolecular polyhedra, collective vectors, quantum similarity and quantum QSPR fundamental equation. IQCC Technical Report TC-2015-4. Management Editions (submitted for publication)
R. Carbó-Dorca, Aromaticity, quantum molecular polyhedra and quantum QSPR. IQCC Technical Report TC-2015-8. J. Comp. Chem. (submitted for publication)
R. Carbó-Dorca, Inward matrix products: extensions and applications to quantum mechanical foundations of QSAR. J. Mol. Struct. (Teochem) 537, 41–54 (2001)
R. Carbó, B. Calabuig, E. Besalú, A. Martínez, Triple density molecular quantum similarity measures: a general connection between theoretical calculations and experimental results. Mol. Eng. 2, 43–64 (1992)
E. Besalú, R. Carbó, J. Mestres, M. Solà, Foundations and recent developments of quantum molecular similarity. Top. Curr. Chem. Mol. Similarity I 173, 31–62 (1995). (Springer, Berlin)
D. Robert, R. Carbó-Dorca, Analyzing the triple density molecular quantum similarity measures with the INDSCAL model. J. Chem. Inf. Comp. Sci. 38, 620–623 (1998)
R. Carbó-Dorca, Triple density quantum similarity measures and the tensorial representation of quantum object sets, in Quantum Chemistry: Theory and Practice, ed. by T. Chakraborty (Apple Academic Press and Taylor and Francis Group, London, 2012)
J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University Press, Oxford, 1965)
R. Carbó-Dorca, A quantum similarity matrix Aufbau procedure. J. Math. Chem. 44, 228–234 (2008)
R. Carbó-Dorca, E. Besalú, L.D. Mercado, Communications on quantum similarity (3): a geometric-quantum similarity molecular superposition (GQSMS) algorithm. J. Comput. Chem. 32, 582–599 (2011)
R. Carbó-Dorca, Shell Partition and metric semispaces, Minkowski norms, root scalar products, distances and cosines of arbitrary order. J. Math. Chem. 32, 201–223 (2002)
P. Bultinck, R. Carbó-Dorca, A mathematical discussion on density and shape functions, vector semispaces and related questions. J. Math. Chem. 36, 191–200 (2004)
R. Carbó-Dorca, Inward matrix products: extensions and applications to quantum mechanical foundations of QSAR. J. Mol. Struct. Teochem 375, 41–54 (2001)
R. Carbó, E. Besalú, Definition, mathematical examples and quantum chemical applications of nested summation symbols and logical Kronecker deltas. Comput. Chem. 18, 117–126 (1994)
L.D. Mercado, R. Carbó-Dorca, Quantum similarity and discrete representation of molecular sets. J. Math. Chem. 49, 1558–1572 (2011)
R. Carbó-Dorca, Quantum similarity matrices column set as holograms of DF molecular point clouds. J. Math. Chem. 50, 2339–2341 (2012)
Acknowledgments
The author wishes to specify that this work has been developed under a Prometeo fellowship, and acknowledges the SENESCYT, Government of Ecuador, for granting it.
Conflict of interest
The author confirms that this paper contents have no conflict of interest.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carbó-Dorca, R. An isometric representation problem in quantum multimolecular polyhedra and similarity: (2) synisometry. J Math Chem 53, 1876–1884 (2015). https://doi.org/10.1007/s10910-015-0525-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-015-0525-3