1,168 research outputs found
Convergence of HX Preconditioner for Maxwell's Equations with Jump Coefficients (ii): The Main Results
This paper is the second one of two serial articles, whose goal is to prove
convergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) for
Maxwell's equations with jump coefficients. In this paper, based on the
auxiliary results developed in the first paper (Hu, 2017), we establish a new
regular Helmholtz decomposition for edge finite element functions in three
dimensions, which is nearly stable with respect to a weight function. By using
this Helmholtz decomposition, we give an analysis of the convergence of the HX
preconditioner for the case with strongly discontinuous coefficients. We show
that the HX preconditioner possesses fast convergence, which not only is nearly
optimal with respect to the finite element mesh size but also is independent of
the jumps in the coefficients across the interface between two neighboring
subdomains.Comment: with 25 pages, 2 figure
Efficient Algorithms for the Consensus Decision Problem
We address the problem of determining if a discrete time switched consensus
system converges for any switching sequence and that of determining if it
converges for at least one switching sequence. For these two problems, we
provide necessary and sufficient conditions that can be checked in singly
exponential time. As a side result, we prove the existence of a polynomial time
algorithm for the first problem when the system switches between only two
subsystems whose corresponding graphs are undirected, a problem that had been
suggested to be NP-hard by Blondel and Olshevsky.Comment: Small modifications after comments from reviewer
Substitute Valuations: Generation and Structure
Substitute valuations (in some contexts called gross substitute valuations)
are prominent in combinatorial auction theory. An algorithm is given in this
paper for generating a substitute valuation through Monte Carlo simulation. In
addition, the geometry of the set of all substitute valuations for a fixed
number of goods K is investigated. The set consists of a union of polyhedrons,
and the maximal polyhedrons are identified for K=4. It is shown that the
maximum dimension of the maximal polyhedrons increases with K nearly as fast as
two to the power K. Consequently, under broad conditions, if a combinatorial
algorithm can present an arbitrary substitute valuation given a list of input
numbers, the list must grow nearly as fast as two to the power K.Comment: Revision includes more background and explanation
Geometric entropy, area, and strong subadditivity
The trace over the degrees of freedom located in a subset of the space
transforms the vacuum state into a density matrix with non zero entropy. This
geometric entropy is believed to be deeply related to the entropy of black
holes. Indeed, previous calculations in the context of quantum field theory,
where the result is actually ultraviolet divergent, have shown that the
geometric entropy is proportional to the area for a very special type of
subsets. In this work we show that the area law follows in general from simple
considerations based on quantum mechanics and relativity. An essential
ingredient of our approach is the strong subadditive property of the quantum
mechanical entropy.Comment: Published versio
On the vertices of the k-additive core
The core of a game v on N, which is the set of additive games φ dominating v such that φ(N)=v(N), is a central notion in cooperative game theory, decision making and in combinatorics, where it is related to submodular functions, matroids and the greedy algorithm. In many cases however, the core is empty, and alternative solutions have to be found. We define the k-additive core by replacing additive games by k-additive games in the definition of the core, where k-additive games are those games whose Möbius transform vanishes for subsets of more than k elements. For a sufficiently high value of k, the k-additive core is nonempty, and is a convex closed polyhedron. Our aim is to establish results similar to the classical results of Shapley and Ichiishi on the core of convex games (corresponds to Edmonds' theorem for the greedy algorithm), which characterize the vertices of the core.Cooperative games; Core; k-additive games; Vertices
- …