The Abelian Groups of Large Order: Perspective from (Fuzzy) Subgroups of Finite p-Groups
Sunday Adesina Adebisi,
Mike Ogiugo,
Michael Enioluwafe
Issue:
Volume 6, Issue 3, May 2021
Pages:
45-48
Received:
8 May 2021
Accepted:
29 May 2021
Published:
7 June 2021
Abstract: In the recent past, results have shown that Nilpotent groups such as p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian product (otherwise known as the product set) plays vital roles in the course of synthesizing the abstract groups. Previous studies have determined the number of distinct fuzzy subgroups of various finite p-groups including those of square-free order. However, not much work has been done on the fuzzy subgroup classification for the nilpotent groups formed from the Cartesian products of p-groups through their computations. Here, part of our intention is therefore trying to make some designs so as to classify the nilpotent groups formed from the Cartesian products of p-groups through their computations. The Cartesian products of p-groups were taken to obtain nilpotent groups. Results up to two dimensions are now obtainable. In this paper, the fuzzy subgroups of the nilpotent product of two abelian subgroups of orders 2n and 128. The integers n ≥ 7 have been successfully considered and the derivation for the explicit formulae for its number distinct fuzzy subgroups were calculated. Some methods were once being used in counting the chains of fuzzy subgroups of an arbitrary finite p-group G. Here, the adoption of the famous Inclusion-Exclusion principle is very necessary and imperative so as to obtain a reasonable, and as much as possible accurate.
Abstract: In the recent past, results have shown that Nilpotent groups such as p-groups, have normal series of finite length. Any finite p-group has many normal subgroups and consequently, the phenomenon of large number of non-isomorphic subgroups of a given order. This makes it an ideal object for combinatorial and cohomological investigations. Cartesian pr...
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Enhancing Parallel Scheduling of Grid Jobs in a Multicored Environment
Goodhead Tomvie Abraham,
Evans Fiebibiseighe Osaisai,
Abalaba Ineyekineye
Issue:
Volume 6, Issue 3, May 2021
Pages:
49-58
Received:
17 May 2021
Accepted:
9 June 2021
Published:
21 June 2021
Abstract: The computing Grid has emerged as a platform to solve the complex and ever-increasing processing need of man and advances in computing technology have birthed the multicore era aimed for high throughput and efficient parallel computing. However, most systems still rely on the underlying hardware for parallelism despite the hard evidence that sequential algorithms do not optimally exploit parallel systems. This research seeks to harness the benefits of multicore systems using job and machine grouping methods to enhance parallelism in the scheduling of Grid jobs. The paper presents the result of two separate experiments on a method that parallelize scheduling algorithm on two multicore platforms. An arbitrary method was employed to group machines; a summation of the total processing power of machines in each group was made. To ensure load balancing, jobs were allocated to machine groups based on the ratio of the total processing power of the machines in each group. The MinMin Grid scheduling algorithm was implemented independently within the groups using a range of threads varied in powers of two. Also, the numbers of groups were varied between 2, 4, and 8. The same experiment was executed on a single processor computer; a duocore machine and a quadcore machine. A performance improvement of 16% to 85% was recorded by the group method against the best ordinary MinMin results and an improvement of 50% to 84% was recorded by the group method against the ordinary MinMin on corresponding machines. We prove that an increase in the number of groups results in improved performance on corresponding machines (approximately 2 times using 2 groups, approximately 3 times using four groups, and approximately 6 times using 8 groups). And most importantly, we established that as the number of processors increases, the grouping method makes more significant improvements over the ordinary MinMin scheduling algorithm executed on the multicore systems.
Abstract: The computing Grid has emerged as a platform to solve the complex and ever-increasing processing need of man and advances in computing technology have birthed the multicore era aimed for high throughput and efficient parallel computing. However, most systems still rely on the underlying hardware for parallelism despite the hard evidence that sequen...
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