We study the projective curvature tensor on multiply warped product manifolds. We investigate that the factor manifolds of a projectively flat multiply warped product manifold necessarily have constant sectional curvature. Moreover, we establish necessary and sufficient conditions under which such manifolds are projectively flat, symmetric, and admit a divergence-free curvature tensor. These results extend and generalize several earlier works in related geometric settings and highlight the deep interplay between curvature properties and the warping functions, which play a fundamental role in determining the geometric structure of the manifold.
In this paper, we investigate the existence of infinitely many positive solutions for a class of Kirchhoff-type problems involving a & ell;(& centerdot;)-Laplacian-like operator. Using a direct variational method and the framework of variable exponent Sobolev spaces, we prove the existence of infinitely many distinct positive solutions. Moreover, under suitable assumptions on the nonlinearity, the W 1 ,& ell; ( & centerdot; ) (D)-norms and L infinity-norms of these solutions are shown to tend to zero.
The concept of mutual-visibility introduced by Di Stefano in 2022 has already been extended in several directions. In k-distance mutual-visibility, a set S of vertices in a graph G admits the defining condition if for any two vertices in S there exists a geodesic between them of length at most k whose internal vertices are not in S. Recently, the mutual-visibility chromatic number of a graph was introduced, and in this paper we combine both ideas as follows. For any positive integer k, the k-distance mutual-visibility coloring of G is a partition of V (G) into subsets each of which is a k-distance mutual-visibility set. The minimum cardinality of such a partition is called the k-distance mutual-visibility chromatic number of G and is denoted by chi(& micro;k) (G). The special cases k = 1 and k >= diam(G) lead to the clique cover number theta(G) and the mutual-visibility chromatic number chi(& micro;)(G), respectively. So, our attention is given to 1 < k < diam(G) with k = 2 producing the most interesting results. We prove several bounds on chi(& micro;2) related to the order and (total) domination, and provide bounds and exact values in various graph products.
Let F be the free metabelian Leibniz algebra over with free generators x 1 and x 2; here metabelian means that the derived algebra [F, F] is abelian. We also consider the free metabelian bicommutative algebra M over with free generators x 1 and x 2. In the present study, we obtain a free generating set of [F, F]D2n as a module, where D 2n is the dihedral group of order 2n. Next, we show that the algebra MD 2n of D 2n-invariants is three generated as a -module.
Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called weak Moore-Penrose inverse in R. The weak Moore- Penrose inverse generalizes the notion of weak dual generalized inverse, introduced by Li and Wang for dual matrices in 2023, to the case of & lowast;-ring R. An element a is an element of R is weak Moore-Penrose invertible if there exists some x is an element of R such that a & lowast; axaa & lowast; = a & lowast; aa & lowast;, xax = x, (ax)& lowast; = ax and (xa)& lowast;= xa. If such an x exists, then it is unique (denoted by a w dagger). We prove that a w dagger exists if and only if (a & lowast; aa & lowast;)dagger exists, in which case a w dagger = a & lowast;(a & lowast; aa & lowast;)dagger a & lowast;. Then, we consider the relationship between weak Moore- Penrose inverse and other generalized inverses. Finally, the equivalent conditions under which a w dagger is normal, Hermitian and idempotent are investigated, respectively.
