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The structure of every musical instrument is related to the applied topics of mathematics like logarithms, golden ratio, etc., and this paper practically explores into this topic to know how music is related to the mathematical concepts. It can be realized that our sensitivity to sound is linked to the logic of our brains. Every musical instrument has distinct sounds while playing the same frequency. Many people know that music is continuous in one frequency. But in the quality of the sound, the discontinuity occurs in some musical instruments. We can see how discontinuity occurs in such behavior of the musical waveform in one frequency. [ABSTRACT FROM AUTHOR]

The notion of occasionally weakly compatible mappings, and Semi compatible mappings in fuzzy metric space is discussed in this research. The paper investigates the Common fixed-point theorems in fuzzy metric spaces. By taking the property of commutativity of pair mappings, the pairwise semi-compatible mappings and occasionally weakly compatible mappings, satisfying constructive type condition for six self-mappings the common fixed-point theorems in fuzzy metric spaces were proved. The result improves and generalizes other similar results in the literature. [ABSTRACT FROM AUTHOR]

Group theory and Combinatorics play important role in design security protocols, algorithms and techniques for various security applications. Secure group-oriented communication is crucial to a wide range of applications in Internet of Things (IoT). Key management is the mechanism which is used to work out the problem of creation, establishment, to distribute, periodic refresh and the maintenance of the cryptographic keys. It is not enough to care only about primitives that satisfy a stated security objective in the domain of the IoT. The design of group key establishment techniques for securing group communications between resource-constrained IoT devices is presented in this work. Furthermore, the paper assesses possible ways for tailoring current security protocols to the peculiarities of IoT devices and networks. [ABSTRACT FROM AUTHOR]