Research Article
On Fuzzy Baire-Separated Spaces and Related Concepts
Ganesan Thangaraj*,
Natarajan Raji
Issue:
Volume 13, Issue 1, February 2024
Pages:
1-8
Received:
22 January 2024
Accepted:
27 February 2024
Published:
20 March 2024
DOI:
10.11648/j.pamj.20241301.11
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Abstract: In this paper, a new class of fuzzy topological spaces, namely fuzzy Baire-separated spaces is introduced in terms of fuzzy Baire sets. Several characterizations of fuzzy Baire-separated spaces are established. It is shown that fuzzy Baire sets lie between disjoint fuzzy P-sets and fuzzy Fσ- sets in a fuzzy Baire-separated space. Conditions under which fuzzy topological spaces become fuzzy Baire-separated spaces are established. Fuzzy nowhere dense sets are fuzzy closed sets in fuzzy nodec spaces and subsequently a question will arise. Which fuzzy topological spaces [other than fuzzy hyperconnected spaces, fuzzy globally disconnected spaces] have fuzzy closed sets with fuzzy nowhere denseness? For this, fuzzy topological spaces having fuzzy closed sets with fuzzy nowhere denseness are identified in this paper. It is verified that fuzzy ultraconnected spaces are non fuzzy Baire -separated spaces. The means, by which fuzzy weakly Baire space become fuzzy Baire -separated spaces and in turn fuzzy Baire - separated spaces become fuzzy seminormal spaces, are obtained. There are scope in this paper for exploring the inter-relations between fuzzy Baire spaces and Baire -separated spaces.
Abstract: In this paper, a new class of fuzzy topological spaces, namely fuzzy Baire-separated spaces is introduced in terms of fuzzy Baire sets. Several characterizations of fuzzy Baire-separated spaces are established. It is shown that fuzzy Baire sets lie between disjoint fuzzy P-sets and fuzzy Fσ- sets in a fuzzy Baire-separated space. Conditions under w...
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Research Article
Maximizing Efficiency in the Computation of Generalized Harmonic Numbers Through Recursive Binary Splitting
Oscar Luis Palacios-Vélez
,
Felipe José Antonio Pedraza-Oropeza*
Issue:
Volume 13, Issue 1, February 2024
Pages:
9-16
Received:
22 January 2024
Accepted:
2 April 2024
Published:
21 April 2024
Abstract: In this article, an efficient algorithm is implemented in Mathematica for the exact calculation of Generalized Harmonic Numbers (GHN). This is achieved through the combination of two methods. The first method is binary division, where terms formed by the powers of the reciprocals of odd and even numbers are summed separately. The second method is a recursive function that iterates the same sequence of operations until all calculations are completed. Within each cycle, the algorithm processes half of the remaining terms, a feature that significantly improves its efficiency. The computer code is notably concise, consisting of only 11 lines, depending on how they are counted. A remarkable event occurs when the argument is a power of two, as the code condenses into a single line. The most distinctive feature of this algorithm lies in the fact that to calculate the GHN for an argument ‘n’, it requires only the terms formed by the reciprocals of odd numbers. This provides a clear advantage over algorithms that use the complete numerical sequence of the reciprocals of all numbers from 1 to n. An intriguing aspect of this algorithm, is the unexpected discontinuity in the powers of two within the denominators of the common factors across each layer. Contrary to expected, these do not form a continuous sequence from 0 to number of layers − 1.
Abstract: In this article, an efficient algorithm is implemented in Mathematica for the exact calculation of Generalized Harmonic Numbers (GHN). This is achieved through the combination of two methods. The first method is binary division, where terms formed by the powers of the reciprocals of odd and even numbers are summed separately. The second method is a...
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