Hydromagnetic stability of the magnetosphere boundary

Hydromagnetic stability of magnetosphere-solar wind interfac

On The Hereditary Discrepancy of Homogeneous Arithmetic Progressions

We show that the hereditary discrepancy of homogeneous arithmetic progressions is lower bounded by $n^{1/O(\log \log n)}$. This bound is tight up to the constant in the exponent. Our lower bound goes via proving an exponential lower bound on the discrepancy of set systems of subcubes of the boolean cube $\{0, 1\}^d$.Comment: To appear in the Proceedings of the American Mathematical Societ

Approximating Hereditary Discrepancy via Small Width Ellipsoids

The Discrepancy of a hypergraph is the minimum attainable value, over two-colorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related the natural extension of this quantity to matrices to rounding algorithms for linear programs, and gave a determinant based lower bound on the hereditary discrepancy. Matousek (2011) showed that this bound is tight up to a polylogarithmic factor, leaving open the question of actually computing this bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time $\tilde{O}(\log^3 n)$-approximation to hereditary discrepancy, as a by-product of their work in differential privacy. In this paper, we give a direct simple $O(\log^{3/2} n)$-approximation algorithm for this problem. We show that up to this approximation factor, the hereditary discrepancy of a matrix $A$ is characterized by the optimal value of simple geometric convex program that seeks to minimize the largest $\ell_{\infty}$ norm of any point in a ellipsoid containing the columns of $A$. This characterization promises to be a useful tool in discrepancy theory

Two-stream instability in gravitating plasmas with magnetic field and rotation

Gas stream instability investigated using moment equations in gravitating plasma clouds with magnetic field and uniform rotation - plasma physic

The Geometry of Differential Privacy: the Sparse and Approximate Cases

In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work. For a set of $d$ linear queries over a database $x \in \R^N$, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, an $O(\log^2 d)$ approximation to the optimal mechanism is known. Our first contribution is to give an $O(\log^2 d)$ approximation guarantee for the case of (\eps,\delta)-differential privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex geometry. We next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when $d > n \triangleq \|x\|_1$. It is known that better mechanisms exist in this setting. Our second main contribution is to give an (\eps,\delta)-differentially private mechanism which is optimal up to a \polylog(d,N) factor for any given query set $A$ and any given upper bound $n$ on $\|x\|_1$. This approximation is achieved by coupling the Gaussian noise addition approach with a linear regression step. We give an analogous result for the \eps-differential privacy setting. We also improve on the mean squared error upper bound for answering counting queries on a database of size $n$ by Blum, Ligett, and Roth, and match the lower bound implied by the work of Dinur and Nissim up to logarithmic factors. The connection between hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix $A$