Journal of Mathematics

In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators and , both of which are positive and have spectra within the interval , denoted as and . In addition, let us introduce two monotone continuous functions, namely, and , defined on the interval . Let be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers and , we have . Under these specified conditions, we establish the following inequality: for all , this outcome highlights the intricate relationship between the numerical range of the expression when transformed by the convex function and the norm of . Importantly, this inequality holds true for a broad range of values of . Furthermore, we provide supportive examples to validate these results.

Let be a graph with vertices and is an -clique of . A vertex is said to resolve a pair of cliques in if where is the distance function of . For a pair of cliques , the resolving neighbourhood of and , denoted by , is the collection of all vertices which resolve the pair . A subset of is called an -clique metric generator for if for each pair of distinct -cliques and of . The -clique metric dimension of , denoted by , is defined as is an -clique metric generator of . In this paper, the -clique metric dimension of corona and edge corona of two graphs are computed. In addition, an integer linear programming model is presented for the -clique metric basis for a given graph and its -cliques.

We consider an ordered vector space . We define the net to be unbounded order convergent to (denoted as ). This means that for every , there exists a net (potentially over a different index set) such that , and for every , there exists such that whenever . The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every -convergent net implies -Cauchy, and vice versa. Let be an order dense subspace of the directed ordered vector space . If is a -band in , then we establish that is a -band in .

In this study, we defined some new sequence spaces using regular Tribonacci matrix. We examined some properties of these spaces such as completeness, Schauder basis. We have identified , and duals of the newly created spaces.

In the present paper, we first show that the existence of the solutions of the operator equation is related to the similarity of operators of class , and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of for some singular inner functions . As an application, we prove that every -quasinormal operator and -centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.

The small finitistic dimension of a ring is determined as the supremum projective dimensions among modules with finite projective resolutions. This paper seeks to establish that, for a coherent ring with a finite weak (resp. Gorenstein) global dimension, the small finitistic dimension of is equal to its weak (resp. Gorenstein) global dimension. Consequently, we conclude some new characterizations for (Gorenstein) von Neumann and semihereditary rings.