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Shadows’ hypercube, vector spaces, and non-linear optimization of QSPR procedures

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Abstract

The role of the shadows’ hypercube is presented first to define a classification of the vectors in a vector space defined over the rational field. Up from this step, the inward product of vectors is presented as the basis for non-linear optimization of several vector scalar and matrix functions. The first use of the inward power of a vector shows that the simple least-squares fitting can be non-linearly optimized. Then the shadows’ hypercube is connected with the topological description of molecules. Further application of the developed theoretical background to the QSPR problem permits us to have some insight into the role of descriptor representation of molecular structures and the non-linear optimization of the involved equations.

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Notes

  1. While the original acrostic has been and still is applied in the literature: QSAR (quantitative structure activity relationships), in modern times one can employ the more general reference: QSPR (quantitative structure property relations). From now on, this last term will be adopted.

  2. Among the possible vector constructs, one can also refer, in general, to numeric multidimensional arrays, like matrices and hypermatrices, with conveniently reordered elements into one column or row.

  3. Here a column vector space has been chosen, but everything said can be applied to a dual row vector space. As in previous work, the bra-ket (row-column) notation of Dirac will be used.

  4. Here it is chosen the rational field which is more adequate to computational structure instead of the real field.

  5. In previous definitions a full vector is named as a perfect vector when all the vector elements are different and ordered in increasing canonical order, but here this restrictive ordered construction is not used.

  6. A semispace is a vector space with the vector addition defined with a semigroup structure. In a semispace there are no negative vectors, vectors bearing negative elements, nor negative scalars. Subtraction is not defined either.

  7. Here is called as inward product the product of two vectors which in the literature has been also named diagonal, or Hadamard, or Schur product.

  8. The Mersenne number \(\mu \left( N \right)\) associated to the dimension of the shadows’ hypercube and represented by the binary unity vector \(\left| {\mathbf{1}} \right\rangle\).

  9. Or nodes in case the topological matrix is the result of the transcription of some graph.

  10. Here, and in previous work, a set of vectors representing a set of molecular structures is named as a molecular, or as in the present case, descriptor polyhedron, not as a polytope, which is the name used to describe sets of points forming polyhedral structures of dimension large than three.

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  1. Institut de Química Computacional i Catàlisi, Universitat de Girona, 17003, Girona, Catalonia, Spain

    Ramon Carbó-Dorca

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The present writing is dedicated to Blanca Cercas, my wife. Without her love, heartful care and understanding, some past and most actual work could not have been done so smoothly.

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Carbó-Dorca, R. Shadows’ hypercube, vector spaces, and non-linear optimization of QSPR procedures. J Math Chem 60, 283–310 (2022). https://doi.org/10.1007/s10910-021-01301-y

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