For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP = λP for some rank k orthogonal projection P . In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1 ≤ r < k it is shown that Λk(A) ⊆ Λk−r(A + F ) for any op...

Given P ⊂ R, a mixed integer set P I = P ∩ (Z × Rn−t), and a k-tuple of n-dimensional integral vectors (π1, . . . , πk) where the last n− t entries of each vector is zero, we consider the relaxation of P I obtained by taking the convex hull of points x in P for which π 1 x, . . . , π T k x are integral. We then define the k-dimensional lattice closure of P I to be the intersection of all such r...

In this paper, we show that injectivity with respect to the class $mathcal{D}$  of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category  well-behaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$-injectivity  well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {bf S-Act}, where $r...

A language L ⊆ A∗ is literally idempotent in case that uav ∈ L if and only if uav ∈ L for each u, v ∈ A∗, a ∈ A. Such classes result naturally by taking all literally idempotent languages in a classical (positive) variety or by considering a certain closure operator. We initiate their systematic study. Various classes of such languages can be characterized using syntactic methods. A starting ex...

Violator Spaces were introduced by J. Matoušek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by antiexchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator spaces ...

Anderson and Canary ([1]) have constructed examples of 3-manifolds M for which the space of convex co-compact representations of π1(M) in PSL2(C) is disconnected but has connected closure (in the topology of algebraic convergence). More precisely, they showed that for their examples the intersection of the closures of any two path components of the space of convex co-compact representations is ...