A Comprehensive Survey on the Metric Dimension Problem of Graphs and Its Types
Issue:
Volume 9, Issue 1, February 2023
Pages:
1-5
Received:
20 June 2023
Accepted:
5 July 2023
Published:
13 July 2023
Abstract: Consider a robot that is navigating a graph-based environment and trying to figure out where it is at the moment. It can send a signal to determine how far away it is from every set of fixed landmarks. We address the problem of finding exactly the minimum number of landmarks required and their perfect placement to make sure the robot can always locate itself. The graph's metric dimension is the quantity of landmarks, and the graph's metric basis is the set of nodes on which they are distributed. The metric dimension of a graph is the smallest set of nodes needed to uniquely identify every other node using the shortest path distances. Optimization, network theory, navigation, pattern recognition, image processing, locating the origin of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces are a few examples of applications for metric dimension. Also, Due to its many and varied applications in fields like social sciences, communications networks, algorithmic designs, and others, the study of dominance is the kind of metric dimension that is developing at the fastest rate. This survey provides a self-contained introduction to the metric dimension and an overview of several metric dimension results and applications. We also present algorithms for computing the metric dimension of families of graphs.
Abstract: Consider a robot that is navigating a graph-based environment and trying to figure out where it is at the moment. It can send a signal to determine how far away it is from every set of fixed landmarks. We address the problem of finding exactly the minimum number of landmarks required and their perfect placement to make sure the robot can always loc...
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Ring Theory Application to Musical Tones Compositions
Temitayo Emmanuel Olaosebikan,
Adesoji Adewumi Obayomi,
Friday Ogoigbe Egbon
Issue:
Volume 9, Issue 1, February 2023
Pages:
6-9
Received:
26 May 2023
Accepted:
27 June 2023
Published:
6 September 2023
Abstract: This paper critically analyzed the idea of using musical tones and mathematical techniques to composed a beautiful, listenable musical sound. observed the tone behavior and connection between mathematical concepts (such as rings) and musical notes. The twelve, (12), musical notes were used for the composition, which formed a ring with addition, +, and multiplication, *, operations, such that additive semigroup and multiplicative group forms a commutative group and semigroup respectively. This was done as a specific technique for the composition of music in which, with the aid of mathematical matrix developed, an equal importance is given to all twelve musical tones in the chromatic scale. The ring theorem is a branch of mathematics that is applicable to music composition, contrary to what people might think, as shown by a practical demonstration of mathematical techniques applied to the twelve musical tones that produced a good sound when played on the keyboard, one of the fundamental musical instruments used for translation of keys to sound. There were some claims that had supporting evidence as a prove that, mathematics is one of the tools for music composition. The research is a prove that, ring theory which is an aspect of mathematics that can be used to formulate music using the twelve musical notes, the behaviors and sound produced attested to the fact that mathematics is musical friendly.
Abstract: This paper critically analyzed the idea of using musical tones and mathematical techniques to composed a beautiful, listenable musical sound. observed the tone behavior and connection between mathematical concepts (such as rings) and musical notes. The twelve, (12), musical notes were used for the composition, which formed a ring with addition, +, ...
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