Sound, with all its manifestations, presents itself as a form of energy with properties and capabilities that remain partially shrouded in mystery. The art of composing music, the artful assemblage and fusion of sounds, still lingers in a realm of obscurity, fraught with complexities. The techniques wielded by composers often stem from an intrinsic gift that defies replication or sharing. However, akin to other art forms, music can be deciphered through the lenses of mathematics and geometry. Through the utilization of LINEAR ALGEBRA, a paramount mathematical discipline, a practical and efficient framework can be established for the comprehension and manipulation of musical compositions. Within this discourse, we delve into the essence of music and its fundamental principles, subsequently exploring the application of mathematical concepts; LINEAR ALGEBRA in particular; in the analysis of music. The findings put forth in this manuscript hold the potential to enlighten researchers and enthusiasts within the realm of music and mathematical sciences, enhance comprehension and knowledge, refine and enhance analytical methodologies and musical procedures across various facets of the music industry, and foster a deeper bond between the realms of mathematics and artistic expression.

In this paper, first we consider Ln as a semimodule over a complete bounded distributive lattice L. Then we define the basic concepts of module theory for Ln. After that, we proved many similar theorems in LINEAR ALGEBRA for the space Ln. An application of LINEAR ALGEBRA over lattices for solving LINEAR systems, was given.

In this article, a method based on LINEAR ALGEBRA approach is adopted to study the hybrid Sheffer– Appell polynomials. The recursive formulas and differential equation for these polynomials are derived by using the properties and relationships between the Pascal functional matrices and the Wronskian matrices. The corresponding results for some mixed type special polynomials are also obtained.

In this paper, a simple application of LINEAR ALGEBRA to the problem of ranking players in a tournament is given. We will see that in the case where every player plays exactly one match against each of the other players, one can use a certain eigenvector of the score matrix, known as Perron eigenvalue, to rank players.

Abstract:Let 𝑋 be a graph and G ≤ 𝐴𝑢𝑡(𝑋). the graph 𝑋 is called 𝐺 -symmetric if 𝐺 acts transitively on its arcs. The covering technique has long been known as a powerful tool in topology and graph theory. In this paper, by using the concepts of LINEAR ALGEBRA and covering techniques, we will classify the symmetric elementary abelian covers of the Nauru graph for one of its symmetric subgroups

It is known that even duals of a Banach ALGEBRA A with one of Arens products are Banach ALGEBRAs, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some ALGEBRAic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these ALGEBRAs. We also determine the spectrum and multiplier ALGEBRA for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of ALGEBRAs.

For a locally compact group G, let A(G) be the Fourier ALGEBRA and let A_M (G) be the completion of this ALGEBRA in its multiplier ALGEBRA. In this paper, we show that A(G) is an abstract Segal ALGEBRA in A_M (G). Also, a necessary and sufficient condition for equality of these two ALGEBRAs is given. Then we prove that A_M (G) is an ideal in its second dual if and only if G is discrete. We show that if G is a discrete group, then A_M (G) is a BSE ALGEBRA if and only if G is M-weakly amenable. As a corollary, it is proven that A_M (F_2 ) is a BSE ALGEBRA while A(F_2 ) is not. Finally, we examine our results for the Lebasque-Fourier ALGEBRA and also give a completely new proof for equality of the character space of A(G) and A_M (G).