In this paper, k-harmonic submanifolds with parallel mean curvature vector in pseudo-Riemannian space forms are studied. We first investigate the fundamental distinction between biharmonic and k-harmonic submanifolds, proving that any nonminimal biharmonic submanifold with non light-like mean curvature vector in a nonflat pseudo-Riemannian space form can be not k-harmonic ( k≥ 3 ). Then, we give a partial classification of these pseudo-umbilical submanifolds.
In this paper, we prove that s-perspective, a-perspective and ad-perspective conditions are equivalent conditions, for finite dimensional left R-modules, where R is an K-algebra. We construct more examples of s-perspective modules and hence a-perspective modules which do not have essentially unique direct complements. We introduce a stronger variation of s-perspective modules named as ss-perspective. Let R be any ring and let M be any left R-module. We will say that M is ss-perspective if for any isomorphic two direct summands A and B of M, the equalities M=A⊕ X = B⊕ Y imply that A=B or X=Y . We prove that the class of ss-perspective modules is contained properly in the class of s-perspective modules. We also prove that for commutative rings, s-perspectivity and ss-perspectivity coincide. Finally, we prove that for a left R-module M with S=End_R(M) , if S_S is ss-perspective then M is ss-perspective, while we prove that if _SS is ss-perspective and M is weak duo, then M is ss-perspective.
We study a general class of nonlinear complementarity problems including the subclass of polynomial ones. As a solution method, we propose an algorithm based on branch-reduce-and-bound paradigm. In contrast to most existing solution methods, our algorithm works under very general conditions. Computational experiments on a number of complementarity problems show the practicability of our solution approach.
The aim of this paper is to investigate type-II E_∞ -Benson properly efficient solutions with respect to the recession cone for the vector optimization problem (in short, VOP) in real locally convex Hausdorff topological linear spaces. We establish both linear and nonlinear scalarization theorems for VOP in the sense of type-II E_∞ -Benson properly efficient solutions, employing the Gerstewitz functional, the Hiriart-Urruty functional, and augmented dual cones. The results presented in this paper generalize and improve several known results in the literature.
Let (R, 𝔪) be a Noetherian local ring, I an ideal of R and M a finitely generated R-module. In this paper, we define and study E-depth and Ef-depth of M in I. We prove that E-depth (resp. Ef-depth under mild conditions) of M in I is the common length of all maximal sequential sequences (resp. maximal sequential f-sequences) of M in I. We show that E-depth and Ef-depth do not decrease under small perturbations. We describe the non sequentially Cohen–Macaulay locus nSCM(M) of M in terms of support and non Cohen–Macaulay locus of deficiency modules of M. Using this description, we prove that the dimension of non-sequentially Cohen–Macaulay locus with respect to a sequential f-sequence does not increase under small pertubations.
