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On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem

Received: 3 November 2021     Accepted: 30 November 2021     Published: 11 December 2021
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Abstract

The main idea of this work is based on the question: how can we control the electric circuits between a number of electric bulbs and a number of electric sources. This generates the correspondences between two discrete sets. The correspondence is based on the notion of discrete function and repetitive arrangements. The normal construction and notions of the work are introduced gradually and are detailed at every stage. Our constant endevour has been to ensure that every sentence in the work has a logical position. Here appears many questions: how to construct all discrete fuctions, which is the total number of these funtions, which is the relation between the number of bulbs and the number of sources, can we constract and control only a partial number of electric circuits (by direct access method) etc. The work answers all these questions by specialised algorithms: the construction algorithm and the decomposition algorithm. The algorithms use the rule from left to right to construct all possille discrete functions and, hence, all electric circuits. The decomposition algorithm supplies an access direct method. So we can control any part of the whole set of circuits. A lot of notions and specific notations are used to develop and illustrate the work. For combinations we have to show the constructon elements. A lot of examples explain this important notion. The work contains a lot of numerical examples and applications. The last section of the work deals with the bijective (and invertible) functions. Specialized notions and notations are used. Numerical examples and geometric designs illustrate the theory.

Published in Mathematics Letters (Volume 7, Issue 4)
DOI 10.11648/j.ml.20210704.11
Page(s) 45-53
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Arrangements, Repetitive Arrangements, Discrete Functions, Decomposition Algorithm, Rule Left Right

References
[1] Oscar Levin (2015). Discrete Mathematics. An Open Introduction. American Institute of Mathematics. Open Textbook Initiative.
[2] Susanna S. Epp (2010-08-04). Discrete Mathematics with Applications. Thomson Books Code. ISBN 978-0 201-72634-3.
[3] Tom Jekyns, Ben Stephenson (2010). Fundamentals of Discrete Mathematical Functions. Computer Science, A problem solving primer. Springer.
[4] Keneth H. Rosen (2007). Discrete Mathematics and its Applications. McGraw-Hill College. ISBN 978-0-07-288008-3.
[5] Popa V. Mircea (2005). Matematica Aplicata. Universitatea Lucian Blaga Sibiu.
[6] Popa V. Mircea (2005). Aranjamente generalizate. Educatia Matematica. Universitatea Lucuan Blaga, Sibiu. Vol. 1, Nr. 2, pag. 49-58.
[7] Andrew Simpson (2002). Discrete Mathematics by Examples. McGraw-Hill Incorporated. ISBN 978-0-07-709840-7.
[8] Popoviciu Nicolae (1998). Transformata Fourier Rapidă şi Teoria Numerelor. Editura Academiei Tehnice Militare, Bucuresti, 1998, 206 de pagini. (Fast Fourier Transform and Number’s Theory. Convolution Theory).
[9] Kenneth H. Rosen (1991). Discrete Mathematics and its Applications. ISBN 978 1260 09.
[10] Norman L. Biggs (1990). Discrete Mathematics (revised edition).
[11] Popa V. Mircea (1986). Asupra numerotarii bijectiilor intre doua multimi multiple. Gazeta Matematica, Perfectionare Metodica si Metodologica. Vol. VII, Nr. 2, Bucuresti, pag. 78-81.
[12] Popa V. Mircea (1980). Unele generalizari in Combinatorica. Buletinul Stiintific al Institutului de Invatamant Superior, Sibiu. Vol. III, pag. 33-30.
[13] Vermani Lekh R. A Cours in Discrete Mathematical Structures. Imperial College Press.
[14] Balakrishan V. K. Introductory Discrete Mathematics. Edition Dover Publications.
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  • APA Style

    Nicolae Popoviciu. (2021). On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem. Mathematics Letters, 7(4), 45-53. https://doi.org/10.11648/j.ml.20210704.11

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    ACS Style

    Nicolae Popoviciu. On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem. Math. Lett. 2021, 7(4), 45-53. doi: 10.11648/j.ml.20210704.11

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    AMA Style

    Nicolae Popoviciu. On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem. Math Lett. 2021;7(4):45-53. doi: 10.11648/j.ml.20210704.11

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  • @article{10.11648/j.ml.20210704.11,
      author = {Nicolae Popoviciu},
      title = {On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem},
      journal = {Mathematics Letters},
      volume = {7},
      number = {4},
      pages = {45-53},
      doi = {10.11648/j.ml.20210704.11},
      url = {https://doi.org/10.11648/j.ml.20210704.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210704.11},
      abstract = {The main idea of this work is based on the question: how can we control the electric circuits between a number of electric bulbs and a number of electric sources. This generates the correspondences between two discrete sets. The correspondence is based on the notion of discrete function and repetitive arrangements. The normal construction and notions of the work are introduced gradually and are detailed at every stage. Our constant endevour has been to ensure that every sentence in the work has a logical position. Here appears many questions: how to construct all discrete fuctions, which is the total number of these funtions, which is the relation between the number of bulbs and the number of sources, can we constract and control only a partial number of electric circuits (by direct access method) etc. The work answers all these questions by specialised algorithms: the construction algorithm and the decomposition algorithm. The algorithms use the rule from left to right to construct all possille discrete functions and, hence, all electric circuits. The decomposition algorithm supplies an access direct method. So we can control any part of the whole set of circuits. A lot of notions and specific notations are used to develop and illustrate the work. For combinations we have to show the constructon elements. A lot of examples explain this important notion. The work contains a lot of numerical examples and applications. The last section of the work deals with the bijective (and invertible) functions. Specialized notions and notations are used. Numerical examples and geometric designs illustrate the theory.},
     year = {2021}
    }
    

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    T1  - On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem
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    Y1  - 2021/12/11
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    N1  - https://doi.org/10.11648/j.ml.20210704.11
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    JO  - Mathematics Letters
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    AB  - The main idea of this work is based on the question: how can we control the electric circuits between a number of electric bulbs and a number of electric sources. This generates the correspondences between two discrete sets. The correspondence is based on the notion of discrete function and repetitive arrangements. The normal construction and notions of the work are introduced gradually and are detailed at every stage. Our constant endevour has been to ensure that every sentence in the work has a logical position. Here appears many questions: how to construct all discrete fuctions, which is the total number of these funtions, which is the relation between the number of bulbs and the number of sources, can we constract and control only a partial number of electric circuits (by direct access method) etc. The work answers all these questions by specialised algorithms: the construction algorithm and the decomposition algorithm. The algorithms use the rule from left to right to construct all possille discrete functions and, hence, all electric circuits. The decomposition algorithm supplies an access direct method. So we can control any part of the whole set of circuits. A lot of notions and specific notations are used to develop and illustrate the work. For combinations we have to show the constructon elements. A lot of examples explain this important notion. The work contains a lot of numerical examples and applications. The last section of the work deals with the bijective (and invertible) functions. Specialized notions and notations are used. Numerical examples and geometric designs illustrate the theory.
    VL  - 7
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Author Information
  • Department of Informatics, Faculty of Informatics, Hyperion University, Bucharest, Romania

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