105,714 research outputs found
A Minimization Approach to Conservation Laws With Random Initial Conditions and Non-smooth, Non-strictly Convex Flux
We obtain solutions to conservation laws under any random initial conditions
that are described by Gaussian stochastic processes (in some cases
discretized). We analyze the generalization of Burgers' equation for a smooth
flux function for
under random initial data. We then consider a piecewise linear, non-smooth and
non-convex flux function paired with general discretized Gaussian stochastic
process initial data. By partitioning the real line into a finite number of
points, we obtain an exact expression for the solution of this problem. From
this we can also find exact and approximate formulae for the density of shocks
in the solution profile at a given time and spatial coordinate . We
discuss the simplification of these results in specific cases, including
Brownian motion and Brownian bridge, for which the inverse covariance matrix
and corresponding eigenvalue spectrum have some special properties. We
calculate the transition probabilities between various cases and examine the
variance of the solution in both and . We also
describe how results may be obtained for a non-discretized version of a
Gaussian stochastic process by taking the continuum limit as the partition
becomes more fine.Comment: 36 pages, 5 figures, small update from published versio
A non-partitionable Cohen-Macaulay simplicial complex
A long-standing conjecture of Stanley states that every Cohen-Macaulay
simplicial complex is partitionable. We disprove the conjecture by constructing
an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our
construction also disproves the conjecture that the Stanley depth of a monomial
ideal is always at least its depth.Comment: Final version. 13 pages, 2 figure
Extremal Lipschitz functions in the deviation inequalities from the mean
We obtain an optimal deviation from the mean upper bound \begin{equation}
D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\
x\in\R\label{abstr} \end{equation} where \F is the class of the integrable,
Lipschitz functions on probability metric (product) spaces. As corollaries we
get exact solutions of \eqref{abstr} for Euclidean unit sphere with
a geodesic distance and a normalized Haar measure, for equipped with a
Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond
graph equipped with uniform measure and Hamming distance. We also prove that in
general probability metric spaces the in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page
Oka properties of complements of holomorphically convex sets
Our main theorem states that the complement of a compact holomorphically
convex set in a Stein manifold with the density property is an Oka manifold.
This gives a positive answer to the well-known long-standing problem in Oka
theory whether the complement of a polynomially convex set in
is Oka. Furthermore, we obtain new examples of nonelliptic Oka
manifolds which negatively answer Gromov's question. The relative version of
the main theorem is also proved. As an application, we show that the complement
of a totally real affine subspace is
Oka if and .Comment: 15 page
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