If S is a given regular d-simplex of edge length a in the d-dimensional Euclidean space E, then the distances t 1 , · · ·, t d+1 of an arbitrary point in E to the vertices of S are related by the elegant relationThe purpose of this paper is to prove that this is essentially the only relation that exists among t 1 , · · · , t d+1 . The proof uses tools from analysis, algebra, and geometry.
Abstract. If S is a given regular n-simplex, n ≥ 2, of edge length a, then the distances a 1 , · · ·, a n+1 of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation φ n+1 (a, a 1 , · · · , a n+1 ) = 0, whereThe natural question whether this is essentially the only relation is answered positively by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled Distances from the vertices of a regular simplex. In that paper, the authors made use of the irreducibility of the polynomial φ in the case when n ≥ 2, t = n + 1, x = a = 0, and k = R, but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of φ without restrictions on t, x, a, and k. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley-Menger determinant that are more general than those established by C. D'Andrea and M. Sombra in Sib. J. Math. 46, 71-76.
In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras.
In this paper, we introduce the concept of fuzzy Hom-Lie subalgebras (ideals) of Hom-Lie algebras and we investigate some of their properties. We study the relationship between fuzzy Hom-Lie subalgebras (resp. ideals) and Hom-Lie subalgebras (resp. ideals). For a finite number of fuzzy Hom-Lie subalgebras, we construct a new fuzzy hom-Lie subalgebras on their direct sum. Finally, The properties of fuzzy Hom-Lie subalgebras and fuzzy Hom-Lie ideals under morphisms of Hom-Lie algebras are studied.
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