On Discrete Functions and Repetitive Arrangements with Algorithms to Construct All Discrete Functions and a Practical Problem
Issue:
Volume 7, Issue 4, December 2021
Pages:
45-53
Received:
3 November 2021
Accepted:
30 November 2021
Published:
11 December 2021
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.
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 notio...
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A New Proof of Young’s Inequality Using Multivariable Optimization
Issue:
Volume 7, Issue 4, December 2021
Pages:
54-58
Received:
18 August 2021
Accepted:
16 December 2021
Published:
24 January 2022
Abstract: In its standard form, Young's inequality for products is a mathematical inequality about the product of two numbers and it allows us to estimate a product of two terms by a sum of the same terms raised to a power and scaled. This inequality, though very simple, has attracted researchers working in many fields of mathematics due to its applications. Apart from the above standard form, there are numerous refinements and variants of Young’s inequality in the literature. Some of these variants are Young’s inequality for arbitrary products, Young’s inequality for increasing functions, Young’s inequality for convolutions, Young’s inequality for integrals, Young’s inequality for matrices, trace version of Young’s inequality, determinant version of Young’s inequality, and so on. The present study examines three variants of Young’s inequality, namely the standard Young’s inequality, Young’s inequality for increasing functions and Young’s inequality for arbitrary products. There are various proofs for these three variants in the literature. For example, just like several other classical important inequalities, these inequalities can be deduced from Jensen’s inequality. The objective of this article is to provide a new alternative proof for each of them. The significance of the article lies in its attempts to open a new direction of poof so that the same approach could be applied to other useful inequalities. The proofs to be presented are based on the methods of multivariable optimization theory.
Abstract: In its standard form, Young's inequality for products is a mathematical inequality about the product of two numbers and it allows us to estimate a product of two terms by a sum of the same terms raised to a power and scaled. This inequality, though very simple, has attracted researchers working in many fields of mathematics due to its applications....
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