The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \ldots, u_n \in \mathbb R^d$ that are $\epsilon$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, \ldots, v_n \in \mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1, \ldots, u_n$, the squared distance (to the set of exact solutions) is defined as $\inf_{v} \sum_{i=1}^n \| u_i - v_i \|_2^2$ where the infimum is over the set of exact solutions. Previous results show that the squared distance of any $\epsilon$-nearly solution is at most $O({\rm{poly}}(d,n,\epsilon))$ and there are $\epsilon$-nearly solutions with squared distance at least $\Omega(d\epsilon)$. The fundamental open question is whether the squared distance can be independent of the number of vectors $n$. We answer this question affirmatively by proving that the squared distance of any $\epsilon$-nearly solution is $O(d^{13/2} \epsilon)$. Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any $\epsilon$-nearly solution is $O(d^2 n \epsilon)$. Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an $\epsilon$-nearly solution is $O(d^{5/2} \epsilon)$ when $n$ is large enough and $\epsilon$ is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor changes in various place

Ancient channels of the Susquehanna River beneath Chesapeake Bay and the Delmarva Peninsula

Three generations of the ancestral Susquehanna River system have been mapped beneath Chesapeake Bay and the southern Delmarva Peninsula. Closely spaced seismic reflection profiles in the bay and boreholes in the bay and on the southern Delmarva Peninsula allow detailed reconstruction of each paleochannel system. The channel systems were formed during glacial low sea-level stands, and each contains a channel-fill sequence that records the subsequent transgression. The trunk channels of each system are 2 to 4 km wide and are incised 30 to SO m into underlying strata; they have irregular longitudinal profiles and very low gradients within the Chesapeake Bay area

Television Revenue and the Structure of Athletic Contests: The Case of the National Basketball Association

This study points out that changes in the format of the playoffs in the National Basketball Association have had the effect of lengthening the championship series and reducing the variance in the length of the series. These are valuable goals for the organization because they provide higher television ratings and revenues for the teams involved. Empirical evidence presented here suggests that viewership among 1.2 million additional households is at stake.Basketball; Television

The plug-based nanovolume Microcapillary Protein Crystallization System (MPCS)

The Microcapillary Protein Crystallization System (MPCS) is a new protein-crystallization technology used to generate nanolitre-sized crystallization experiments for crystal screening and optimization. Using the MPCS, diffraction-ready crystals were grown in the plastic MPCS CrystalCard and were used to solve the structure of methionine-R-sulfoxide reductase

The plug-based nanovolume Microcapillary Protein Crystallization System (MPCS)

This is the published version. Copyright International Union of CrystallographyThe Microcapillary Protein Crystallization System (MPCS) embodies a new semi-automated plug-based crystallization technology which enables nanolitre-volume screening of crystallization conditions in a plasticware format that allows crystals to be easily removed for traditional cryoprotection and X-ray diffraction data collection. Protein crystals grown in these plastic devices can be directly subjected to in situ X-ray diffraction studies. The MPCS integrates the formulation of crystallization cocktails with the preparation of the crystallization experiments. Within microfluidic Teflon tubing or the microfluidic circuitry of a plastic CrystalCard, ~10-20 nl volume droplets are generated, each representing a microbatch-style crystallization experiment with a different chemical composition. The entire protein sample is utilized in crystallization experiments. Sparse-matrix screening and chemical gradient screening can be combined in one com­prehensive `hybrid' crystallization trial. The technology lends itself well to optimization by high-granularity gradient screening using optimization reagents such as precipitation agents, ligands or cryoprotectants

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

The Protein Maker: an automated system for high-throughput parallel purification

The Protein Maker instrument addresses a critical bottleneck in structural genomics by allowing automated purification and buffer testing of multiple protein targets in parallel with a single instrument. Here, the use of this instrument to (i) purify multiple influenza-virus proteins in parallel for crystallization trials and (ii) identify optimal lysis-buffer conditions prior to large-scale protein purification is described