Boolean hypercubes and the structure of vector spaces

Ramon Carbó-dorca

doi:10.33187/jmsm.413116

Research Article

Year 2018, Volume: 1 Issue: 1, 1 - 14, 27.05.2018

Ramon Carbó-dorca

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

Cited By: 8

Abstract

References

[1] S. K. Berberian, Introduction to Hilbert Space, Oxford University Press, 1961.
[2] F. Ayres Jr., Matrices, Schaum Pub. Co. (Mc Graw Hill), 1962.
[3] G. E. Shilov, Linear Algebra, Dover Publications Inc. (New York), 1971.
[4] H. Schneider, G. Ph. Barker, Matrices and Linear Algebra, Dover Publications Inc. (New York), 1989.
[5] P. O. Löwdin, Linear Algebra for Quantum Theory, J. Wiley & Sons, 1998.
[6] G. A. Korn, Th. M. Korn, Mathematical Handbook for Scientists and Engineers, Dover Publications Inc. (New York), 2000.
[7] L. Hogben (Editor), Handbook of Linear Algebra, Chapman & Hall/CRC Taylor & Francis Group (Boca Raton), 2007.
[8] T. Gowers (Editor), The Princeton Companion to Mathematics, Princeton Univ. Press (Princeton), 2008.
[9] R. Carbó-Dorca, On the statistical interpretation of Density Functions: ASA, Convex Sets, discrete Quantum Chemical molecular representations, Diagonal Vector Spaces and related problems, J. Math. Chem. 23 (1998), 365-375.
[10] R. Carbó-Dorca, Shell partition and metric semispaces: Minkowski norms, root scalar products, distances and cosines of arbitrary order, J. Math. Chem. 32 (2002), 201-223.
[11] P. Bultinck and R. Carbó-Dorca, A mathematical discussion on density and shape functions, vector semispaces and related questions, J. Math. Chem. 36 (2004), 191-200.
[12] R. Carbó-Dorca and S. Van Damme, Riemann spaces, molecular density function semispaces, quantum similarity measures and quantum quantitative structure-properties relationships (QQSPR), An Homage to Dr. J. J. Bonet, Afinidad 64 (2007), 147-153.
[13] R. Carbó-Dorca, Molecular Quantum Similarity Measures in Minkowski Metric Vector Semispaces, J. Math. Chem. 40 (2008), 628-636.
[14] R. Carbó-Dorca and E. Besalu, Shells, point cloud huts, generalized scalar products, cosines and similarity tensor representations in vector Semispaces, J. Math. Chem. 50 (2012), 210-219.
[15] R. Carbó-Dorca, Natural Vector Spaces, (Inward Power and Minkowski Norm of a Natural Vector, Natural Boolean Hypercubes) and Fermat’s Last Theorem, J. Math. Chem. 55 (2017), 914-940.
[16] R. Carbó-Dorca, C. Munoz-Caro, A. Nino and S. Reyes, Refinement of a Generalized Fermat’s Last Theorem Conjecture in Natural Vector Spaces, J. Math. Chem. 55 (2017) 1869-1877.
[17] R. Carbó-Dorca, Multimolecular Polyhedra and QSPR, J. Math. Chem. 52 (2014), 1848-1856.
[18] R. Carbó-Dorca, Quantum Polyhedra, Definitions, Statistics and the Construction of a Collective Quantum Similarity Index, J. Math. Chem. Vol. 53, pp 171-182, 2015.
[19] R. Carbó-Dorca and D. Barrag´an, Communications on Quantum Similarity (4): Collective distances computed by means of Similarity Matrices, as generators of intrinsic ordering among Quantum Multimolecular Polyhedra, WIREs Comput. Mol. Sci. 5, (2015) 380-404.
[20] R. Carbó-Dorca, An Isometric Representation Problem Related with Quantum Multimolecular Polyhedra and Similarity, J. Math. Chem. 53 (2015), 1750-1758.
[21] R. Carbó-Dorca, An Isometric Representation Problem in Quantum Multimolecular Polyhedra and Similarity: (2) Synisometry, J. Math. Chem. 53 (2015), 1867-1884.
[22] R. Carbó-Dorca, Quantum Molecular Polyhedra in LCAO-MO Theory, Mol. Phys. 114 (2016), 1236-1249.
[23] R. Carbó-Dorca and S. Gonz´alez, Notes in QSPR (4): Quantum Multimolecular Polyhedra, Collective Vectors, Quantum Similarity and Quantum QSPR fundamental equation, Management Studies 4 (2016), 33-47.
[24] R. Carbó-Dorca, A Study on the Centroid Vector of a Polyhedron, J. Math. Chem 54 (2016), 51-71.
[25] R. Carbó-Dorca, Time Vectors and Particle Swarms Defined as Polyhedra in Spherically Enfolded Spaces, J. Math. Chem. 54 (2016), 1751–1757.
[26] R. Carbó-Dorca, A Theorem on the Gram matrix of a Polyhedron, J. Math. Chem. 55 (2017), 79-97.
[27] R. Carbó-Dorca, Fuzzy sets and Boolean Tagged sets, J. Math. Chem. 22 (1997), 143-147.
[28] R. Carbó-Dorca; Fuzzy sets and Boolean Tagged sets, vector semispaces and convex sets, QSM and ASA density functions, diagonal vector spaces and quantum Chemistry, Adv. Molec. Simil., JAI Press, 2 (1998), 43-72.
[29] R. Carbó-Dorca, About Erd¨os Discrepancy Conjecture, J. Math. Chem. 54 (2016), 657-660.
[30] R. Carbó-Dorca, N-dimensional Boolean Hypercubes and the Goldbach Conjecture, J. Math. Chem. 54 (2016), 1213-1220.
[31] R. Carbó-Dorca, A study on Goldbach Conjecture, J. Math. Chem. 54 (2016), 1798-1809.
[32] R. Carbó-Dorca, Boolean Hypercubes as Time Representation Holders, J. Math. Chem. 56 (2018), 1349–1352.
[33] R. Carbó-Dorca, About some questions relative to the arbitrariness of signs: Their possible consequences in matrix signatures definition and quantum chemical applications, J. Math. Chem. 33 (2003), 227-244.
[34] R. Carbó-Dorca, Inward Matrix Products: Extensions and Applications to Quantum Mechanical Foundations of QSAR, J. Mol. Struct. (Teochem) 537 (2001), 41-54.
[35] R. Carbó-Dorca, Inward Matrix product algebra and calculus as tools to construct space-time frames of arbitrary dimensions, J. Math. Chem. 30 (2001), 227-245
[36] R. Carbó-Dorca, Applications of Inward Matrix Products and Matrix Wave Functions to H¨uckel MO Theory, Slater Extended Wave Functions, Spin Extended Functions and Hartree Method, Int. J. Quantum Chem. 91 (2003), 607-617.

Boolean hypercubes and the structure of vector spaces

Year 2018, Volume: 1 Issue: 1, 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

References

[1] S. K. Berberian, Introduction to Hilbert Space, Oxford University Press, 1961.
[2] F. Ayres Jr., Matrices, Schaum Pub. Co. (Mc Graw Hill), 1962.
[3] G. E. Shilov, Linear Algebra, Dover Publications Inc. (New York), 1971.
[4] H. Schneider, G. Ph. Barker, Matrices and Linear Algebra, Dover Publications Inc. (New York), 1989.
[5] P. O. Löwdin, Linear Algebra for Quantum Theory, J. Wiley & Sons, 1998.
[6] G. A. Korn, Th. M. Korn, Mathematical Handbook for Scientists and Engineers, Dover Publications Inc. (New York), 2000.
[7] L. Hogben (Editor), Handbook of Linear Algebra, Chapman & Hall/CRC Taylor & Francis Group (Boca Raton), 2007.
[8] T. Gowers (Editor), The Princeton Companion to Mathematics, Princeton Univ. Press (Princeton), 2008.
[9] R. Carbó-Dorca, On the statistical interpretation of Density Functions: ASA, Convex Sets, discrete Quantum Chemical molecular representations, Diagonal Vector Spaces and related problems, J. Math. Chem. 23 (1998), 365-375.
[10] R. Carbó-Dorca, Shell partition and metric semispaces: Minkowski norms, root scalar products, distances and cosines of arbitrary order, J. Math. Chem. 32 (2002), 201-223.
[11] P. Bultinck and R. Carbó-Dorca, A mathematical discussion on density and shape functions, vector semispaces and related questions, J. Math. Chem. 36 (2004), 191-200.
[12] R. Carbó-Dorca and S. Van Damme, Riemann spaces, molecular density function semispaces, quantum similarity measures and quantum quantitative structure-properties relationships (QQSPR), An Homage to Dr. J. J. Bonet, Afinidad 64 (2007), 147-153.
[13] R. Carbó-Dorca, Molecular Quantum Similarity Measures in Minkowski Metric Vector Semispaces, J. Math. Chem. 40 (2008), 628-636.
[14] R. Carbó-Dorca and E. Besalu, Shells, point cloud huts, generalized scalar products, cosines and similarity tensor representations in vector Semispaces, J. Math. Chem. 50 (2012), 210-219.
[15] R. Carbó-Dorca, Natural Vector Spaces, (Inward Power and Minkowski Norm of a Natural Vector, Natural Boolean Hypercubes) and Fermat’s Last Theorem, J. Math. Chem. 55 (2017), 914-940.
[16] R. Carbó-Dorca, C. Munoz-Caro, A. Nino and S. Reyes, Refinement of a Generalized Fermat’s Last Theorem Conjecture in Natural Vector Spaces, J. Math. Chem. 55 (2017) 1869-1877.
[17] R. Carbó-Dorca, Multimolecular Polyhedra and QSPR, J. Math. Chem. 52 (2014), 1848-1856.
[18] R. Carbó-Dorca, Quantum Polyhedra, Definitions, Statistics and the Construction of a Collective Quantum Similarity Index, J. Math. Chem. Vol. 53, pp 171-182, 2015.
[19] R. Carbó-Dorca and D. Barrag´an, Communications on Quantum Similarity (4): Collective distances computed by means of Similarity Matrices, as generators of intrinsic ordering among Quantum Multimolecular Polyhedra, WIREs Comput. Mol. Sci. 5, (2015) 380-404.
[20] R. Carbó-Dorca, An Isometric Representation Problem Related with Quantum Multimolecular Polyhedra and Similarity, J. Math. Chem. 53 (2015), 1750-1758.
[21] R. Carbó-Dorca, An Isometric Representation Problem in Quantum Multimolecular Polyhedra and Similarity: (2) Synisometry, J. Math. Chem. 53 (2015), 1867-1884.
[22] R. Carbó-Dorca, Quantum Molecular Polyhedra in LCAO-MO Theory, Mol. Phys. 114 (2016), 1236-1249.
[23] R. Carbó-Dorca and S. Gonz´alez, Notes in QSPR (4): Quantum Multimolecular Polyhedra, Collective Vectors, Quantum Similarity and Quantum QSPR fundamental equation, Management Studies 4 (2016), 33-47.
[24] R. Carbó-Dorca, A Study on the Centroid Vector of a Polyhedron, J. Math. Chem 54 (2016), 51-71.
[25] R. Carbó-Dorca, Time Vectors and Particle Swarms Defined as Polyhedra in Spherically Enfolded Spaces, J. Math. Chem. 54 (2016), 1751–1757.
[26] R. Carbó-Dorca, A Theorem on the Gram matrix of a Polyhedron, J. Math. Chem. 55 (2017), 79-97.
[27] R. Carbó-Dorca, Fuzzy sets and Boolean Tagged sets, J. Math. Chem. 22 (1997), 143-147.
[28] R. Carbó-Dorca; Fuzzy sets and Boolean Tagged sets, vector semispaces and convex sets, QSM and ASA density functions, diagonal vector spaces and quantum Chemistry, Adv. Molec. Simil., JAI Press, 2 (1998), 43-72.
[29] R. Carbó-Dorca, About Erd¨os Discrepancy Conjecture, J. Math. Chem. 54 (2016), 657-660.
[30] R. Carbó-Dorca, N-dimensional Boolean Hypercubes and the Goldbach Conjecture, J. Math. Chem. 54 (2016), 1213-1220.
[31] R. Carbó-Dorca, A study on Goldbach Conjecture, J. Math. Chem. 54 (2016), 1798-1809.
[32] R. Carbó-Dorca, Boolean Hypercubes as Time Representation Holders, J. Math. Chem. 56 (2018), 1349–1352.
[33] R. Carbó-Dorca, About some questions relative to the arbitrariness of signs: Their possible consequences in matrix signatures definition and quantum chemical applications, J. Math. Chem. 33 (2003), 227-244.
[34] R. Carbó-Dorca, Inward Matrix Products: Extensions and Applications to Quantum Mechanical Foundations of QSAR, J. Mol. Struct. (Teochem) 537 (2001), 41-54.
[35] R. Carbó-Dorca, Inward Matrix product algebra and calculus as tools to construct space-time frames of arbitrary dimensions, J. Math. Chem. 30 (2001), 227-245
[36] R. Carbó-Dorca, Applications of Inward Matrix Products and Matrix Wave Functions to H¨uckel MO Theory, Slater Extended Wave Functions, Spin Extended Functions and Hartree Method, Int. J. Quantum Chem. 91 (2003), 607-617.

There are 36 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Ramon Carbó-dorca 0000-0002-9219-0686
Publication Date	May 27, 2018
Submission Date	April 6, 2018
Acceptance Date	May 27, 2018
Published in Issue	Year 2018 Volume: 1 Issue: 1

Cite

APA	Carbó-dorca, R. (2018). Boolean hypercubes and the structure of vector spaces. Journal of Mathematical Sciences and Modelling, 1(1), 1-14. https://doi.org/10.33187/jmsm.413116
AMA	Carbó-dorca R. Boolean hypercubes and the structure of vector spaces. Journal of Mathematical Sciences and Modelling. May 2018;1(1):1-14. doi:10.33187/jmsm.413116
Chicago	Carbó-dorca, Ramon. “Boolean Hypercubes and the Structure of Vector Spaces”. Journal of Mathematical Sciences and Modelling 1, no. 1 (May 2018): 1-14. https://doi.org/10.33187/jmsm.413116.
EndNote	Carbó-dorca R (May 1, 2018) Boolean hypercubes and the structure of vector spaces. Journal of Mathematical Sciences and Modelling 1 1 1–14.
IEEE	R. Carbó-dorca, “Boolean hypercubes and the structure of vector spaces”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 1, pp. 1–14, 2018, doi: 10.33187/jmsm.413116.
ISNAD	Carbó-dorca, Ramon. “Boolean Hypercubes and the Structure of Vector Spaces”. Journal of Mathematical Sciences and Modelling 1/1 (May 2018), 1-14. https://doi.org/10.33187/jmsm.413116.
JAMA	Carbó-dorca R. Boolean hypercubes and the structure of vector spaces. Journal of Mathematical Sciences and Modelling. 2018;1:1–14.
MLA	Carbó-dorca, Ramon. “Boolean Hypercubes and the Structure of Vector Spaces”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 1, 2018, pp. 1-14, doi:10.33187/jmsm.413116.
Vancouver	Carbó-dorca R. Boolean hypercubes and the structure of vector spaces. Journal of Mathematical Sciences and Modelling. 2018;1(1):1-14.