Vector semispace

Till now, all MQSM rely on overlap integrals between electron densities. However, this is not the only possibility. Based on the development of the theory of vector semispaces, Carbo et al. have shown that one can extend the QSM theory to include different kinds of operators [18,19]. In general, a MQSM can be obtained via... 

Definition 4, in appendix A, succinctly presents the structural details of first order eDF, and connects their form with the ASA eDF. The ASA eDF coefficients vector, w = w, must be optimised, in order to obtain an approximate fimction, completely adapted to ab initio eDF. The w vector component values shall be restricted to lie within the boundaries of some Vector Semispace (VSS) s Appendix A, and his element sum,, w, shall be... 

A Vector Semispace (VSS) over the PD real field R, is a Vector Space (VS) with the vector sum part provided by a structure of abelian semigroup. ... 

The work mentioned so far relies on quantum similarity indices that in turn rely on the overlap measure of electron densities. Simple overlap between electron density functions is, however, not the only possible choice. Based on the developed theory of vector semispaces and tagged sets, Carbo et al. have... 

It is immediately clear that the resulting matrix S is no longer a symmetrical matrix. On the other hand, the sum of all elements in a column of S belong to a unit shell vector semispace (see below), meaning that Vf (s ) 1. It is clear that the columns of the stochastic matrix become a new descriptor set for the... 

A continuing effort has been maintained by the scientific community to provide a firm theoretical and mathematical basis for molecular quantum similarity. According to Carbo-Dorca et al., the basis of molecular quantum similarity and of quantum QSAR (as described later) is fovmded in the concepts of tagged sets and vector semispaces. To make quantum similarity understandable to the novice, the following explanatory paragraphs provide first the required mathematical basis and second an extensive discussion of some useful aspects of vector semispaces. 

After the seminal structure building of the QS formalism, several additional studies appeared over time, which developed new theoretical details. Especially noteworthy is the concept of vector semispace (VSS). This mathematical construction will be shown to be the main platform on which several QS ideas are built, related in turn, to probability distributions and hence to quantum mechanical probability density functions. Such quantum mechanical density distributions form a characteristic functional set, which can be easily connected to VSS properties. Construction of the so-called quantum objects (QO) and their collections the QO sets (QOS) (see, for example, Carbo-Dorca ), easily permit the interpretation of the nature of quantum similarity measures for relationships between such quantum mechanically originated elements. Within quantum similarity context, QOS appear as a particular kind of tagged sets, where objects are submicroscopic systems and their density functions become tags. 

Mezey, Eds., JAI Press, London, 1998, pp. 43-72. Fuzzy Sets and Boolean Tagged Sets Vector Semispaces and Convex Sets Quantum Similarity Measures and ASA Density Functions Diagonal Vector Spaces and Quantum Chemistry. 

One must be aware that nowadays the mathematical concept of distance has evolved into an intricate labyrinth of alternative definitions and variants however, one can safely rely on the classic Euclidean concepts for practical QS purposes. From the QS point of view, any DF can be studied as a function belonging to a vector semispace. " Furthermore, DF can be seen as vectors belonging to infinite-dimensional Hilbert semispaces and thus can be also subject to comparative measures of distances and angles between the two of them. A pair of DF may, in this way, be considered as vectors subtending an angle a and situated in a plane... 

A vector semispace is a vector space without reciprocal elements and iJius defined over the positive real field. 

A vector semispace is considered here as a vector space defined over the positive real numbers. In this way the additive group of the semispace is a semigroup, a group without reciprocal elements. DF sets like the quantum object tags already discussed in MQOS definition are subsets of some Hilbert semispace. 

Obviously enough, for higher order QS integrals, this kind of reduced CSI is difficult to visualize. However, some hint permitting the same interpretation as in the lower order CSI can be found, generalizing the two and three DF cases. In any / -tuple DF case, the considered set of p DF, if associated to p different molecules, they must be linearly independent. Thus, this DF set generates a / -dimensional vector semispace. 

In no way there will appear any problem with negative signs in the cubic root of Eq. (26), because of the already commented vector semispace nature of all the involved tags, which bear everywhere positive definite elements. 

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