46,216 research outputs found
How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability
Published versio
Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem
We investigate singular and degenerate behavior of solutions of the unstable
free boundary problem First, we construct a
solution that is not of class and whose free boundary consists of
four arcs meeting in a {\em cross}-shaped singularity. This solution is
completely unstable/repulsive from above and below which would make it hard to
get by the usual methods, and even numerics is non-trivial. We also show
existence of a degenerate solution. This answers two of the open questions in a
recent paper by R. Monneau-G.S. Weiss
Self-propagating High temperature Synthesis (SHS) in the high activation energy regime
We derive the precise limit of SHS in the high activation energy scaling
suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J.
Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit turns
out to be the Stefan problem for supercooled water with spatially inhomogeneous
coefficients. Although the present paper leaves open mathematical questions
concerning the convergence, our precise form of the limit problem suggest a
strikingly simple explanation for the numerically observed pulsating waves
Learning When Training Data are Costly: The Effect of Class Distribution on Tree Induction
For large, real-world inductive learning problems, the number of training
examples often must be limited due to the costs associated with procuring,
preparing, and storing the training examples and/or the computational costs
associated with learning from them. In such circumstances, one question of
practical importance is: if only n training examples can be selected, in what
proportion should the classes be represented? In this article we help to answer
this question by analyzing, for a fixed training-set size, the relationship
between the class distribution of the training data and the performance of
classification trees induced from these data. We study twenty-six data sets
and, for each, determine the best class distribution for learning. The
naturally occurring class distribution is shown to generally perform well when
classifier performance is evaluated using undifferentiated error rate (0/1
loss). However, when the area under the ROC curve is used to evaluate
classifier performance, a balanced distribution is shown to perform well. Since
neither of these choices for class distribution always generates the
best-performing classifier, we introduce a budget-sensitive progressive
sampling algorithm for selecting training examples based on the class
associated with each example. An empirical analysis of this algorithm shows
that the class distribution of the resulting training set yields classifiers
with good (nearly-optimal) classification performance
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