We study the approximate range searching for three variants of the clustering problem with a set P of n points in d-dimensional Euclidean space and axis-parallel rectangular range queries: the k-median, k-means, and k-center range-clustering query problems. We present data structures and query algorithms that compute (1 + ε)-approximations to the optimal clusterings of P ∩Q efficiently for a qu...

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1 = (c+ c )/2 and b2 = (c− c)/2i. Also write B for the spectral scale of {b1, b2} relative to τ . In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information ...

In this paper, we introduce the notions of C-numerical range and C-spectrum of matrix polynomials. Some algebraic and geometrical properties are investigated. We also study the relationship between the C-numerical range of a matrix polynomial and the joint C-numerical range of its coefficients.