We note the condition 0 ≤ w(e) ≤ 1 is given since it is convenient for presenting our theoretical results, although we can define each problem without this condition. We say unweighted dense subgraph problem if w(e) = 1 for each edge. We define the density ∆ of the output subgraph H to be ∆ = 2w(F ) k(k−1) for the non-bipartite case and ∆ = w(F ) m′n′ for the bipartite case. In other words, den...

(a) P is a nonempty set; (b) ≤ is a partial ordering of P; i.e., ≤ is transitive, reflexive relation in P (and so p ≤ q ∧ q ≤ p ∧ p 6= q is not forbidden); (c) for all p ∈ P, p ≤ 1. (1 need not be the only such maximum element.) We shall often write P for 〈P,≤,1〉. Let P be a poset. If p is an element of P, then an extension of p is a q ∈ P such that q ≤ p. Two elements of P are compatible if th...

This paper describes a general-purpose system for computing registered stereoscopic video overlays of pre-operative imagery during minimally invasive surgery. There are three key elements to our approach. The first element is a real-time computer vision system that operates on stereoscopic video acquired during minimally invasive surgery to extract geometric information. We present two variatio...

The finite element implement of the generalized Lorenz gauged A formulation has been proposed for low-frequency modeling. However, the inverse of mass matrix of intermediate scalar in the finite element implement leads to additional computation cost and dense coefficient matrix. In this paper we propose to adopt a diagonal lumping mass matrix in the finite element discretization of the generali...

A hybrid finite element method/boundary element method (FEM/BEM) is a standard approach for calculating the magnetostatic potential within Micromagnetics [1]. This involves dealing with a dense N ×N -matrix Bij , N being the number of mesh surface nodes. In order to apply the method to ferromagnetic structures with a large surface one needs to apply matrix compression techniques on Bij . An eff...

We present a novel strategy for sparse direct factorizations that is geared towards the matrices that arise from hp-adaptive Finite Element Methods. In that context, a sequence of linear systems derived by successive local refinement of the problem domain needs to be solved. Thus, there is an opportunity for a factorization strategy that proceeds by updating (and possibly downdating) the factor...

We present a novel strategy for sparse direct factorizations that is geared towards the matrices that arise from hp-adaptive Finite Element Methods. In that context, a sequence of linear systems derived by successive local refinement of the problem domain needs to be solved. Thus, there is an opportunity for a factorization strategy that proceeds by updating (and possibly downdating) the factor...

We consider the following two selection principles for topological spaces: Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space; Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, su...

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...