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 $H\left( p\right) =\left\vert p\right\vert ^{j}$ for $j\geq2$ 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 $t$ and spatial coordinate $x$. 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 $w\left(x,t\right)$ in both $x$ and $t$. 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 $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ 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 $\sup$ 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 $\mathbb{C}^{n}$ $(n>1)$ 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 $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)\neq(2,1),(2,2),(3,3)$.Comment: 15 page