Boolean hypercubes and the structure of vector spaces

Year 2018, , 1 - 14, 27.05.2018

Ramon Carbó-dorca

https://doi.org/10.33187/jmsm.413116

Cited By: 8

Abstract

The present study pretends to describe an alternative way to look at Vector Spaces as a scaffold to produce a meaningful new theoretical structure to be used in both classical and quantum QSPR. To reach this goal it starts from the fact that N-Dimensional Boolean Hypercubes contain as vertices the whole information maximally expressible by means of strings of N bits. One can use this essential property to construct the structure of $N$-Dimensional Vector Spaces, considering vector classes within a kind of Space Wireframe related to a Boolean Hypercube. This way of deconstruct-reconstruct Vector Spaces starts with some newly coined nomenclature, because, through the present paper, any vector set is named as a Vector Polyhedron, or a polyhedron for short if the context allows it. Also, definition of an Inward Vector Product allows to easily build up polyhedral vector structures, made of inward powers of a unique vector, which in turn one might use as Vector Space basis sets. Moreover, one can construct statistical-like vectors of a given Vector Polyhedron as an extended polyhedral sequence of vector inward powers. Furthermore, the Complete Sum of a vector is defined simply as the sum of all its elements. Once defined, one can use it to compute, by means of inward products, generalized scalar products, generalized vector norms and statistical-like indices attached to a Vector Polyhedron.

Keywords

Vector spaces, vector semispaces, vector polyhedra, boolean hypercubes, vector space wireframe, perfect, whole and hollow vectors, complete sum of a vector, inward vector product, inward vector power, power basis sets, generalized scalar products, generalized vector norms, statisticallike vectors and indices

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