Optimal approximate matrix product in terms of stable rank

We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows. Here $\tilde{r}$ is the maximum stable rank, i.e. squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [MZ11, KVZ14], and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future. Our main theorem, via connections with spectral error matrix multiplication shown in prior work, implies quantitative improvements for approximate least squares regression and low rank approximation. Our main result has also already been applied to improve dimensionality reduction guarantees for $k$-means clustering [CEMMP14], and implies new results for nonparametric regression [YPW15]. We also separately point out that the proof of the "BSS" deterministic row-sampling result of [BSS12] can be modified to show that for any matrices $A, B$ of stable rank at most $\tilde{r}$, one can achieve the spectral norm guarantee for approximate matrix multiplication of $A^T B$ by deterministically sampling $O(\tilde{r}/\varepsilon^2)$ rows that can be found in polynomial time. The original result of [BSS12] was for rank instead of stable rank. Our observation leads to a stronger version of a main theorem of [KMST10].Comment: v3: minor edits; v2: fixed one step in proof of Theorem 9 which was wrong by a constant factor (see the new Lemma 5 and its use; final theorem unaffected

Lie algebras generated by extremal elements

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.Comment: 28 page

Maine’s Contested Waterfront: The Project to Remake Sebago Lake’s Lower Bay, 1906-1930

Throughout the nation’s history, few resources have been considered as ubiquitous as water. The issue of who controls the use of water, however, has seldom been straight forward. This was no less true in the Progressive Era, when many growing urban areas significantly altered their water infrastructure to meet increased demands. When debate arose over water use, these municipalities often relied on the relatively new authority of scientific knowledge, particularly in the area of public health and safety. In this article, the author describes how the Portland Water District was able to conserve Sebago Lake’s Lower Bay as a clean, reliable source of drinking water for Portland, Maine. A native of Portland, the author is a graduate of Brown University, where he earned his A.B. in history and geology-biology. He is currently a Ph.D. candidate and Irving and Rose Crown Fellow at Brandeis University, where he studies North American environmental history

Synthesis and Characterization of Carbon Monoxide Prodrugs

Carbon monoxide (CO) is an important endogenous signaling molecule that has pleiotropic effects through the regulation of a series of hemoprotein targets. It has been demonstrated repeatedly that there is a need for organic CO donors of various properties capable of the controlled release of CO since inhalation delivery has safety concerns and the reactivity and toxicity of metal-based donors are un-resolved issues. Our research group has previously described the synthesis and kinetic studies of organic CO prodrugs of various types with tunable release rates. In one series, the design makes use of an inverse-electron demand Diels-Alder cycloaddition, followed by a cheletropic reaction of the resulting norbornadienone intermediate to release CO. Herein, we describe the synthesis, characterization, and release kinetics of analogous CO prodrugs, with the aim of improved structural properties and/or release kinetics. These new CO prodrugs will add to the diverse set of CO donors available

Extrinsic models for the dielectric response of CaCu{3}Ti{4}O{12}

The large, temperature-independent, low-frequency dielectric constant recently observed in single-crystal CaCu{3}Ti{4}O{12} is most plausibly interpreted as arising from spatial inhomogenities of its local dielectric response. Probable sources of inhomogeneity are the various domain boundaries endemic in such materials: twin, Ca-ordering, and antiphase boundaries. The material in and neighboring such boundaries can be insulating or conducting. We construct a decision tree for the resulting six possible morphologies, and derive or present expressions for the dielectric constant for models of each morphology. We conclude that all six morphologies can yield dielectric behavior consistent with observations and suggest further experiments to distinguish among them.Comment: 9 pages, with 1 postscript figure embedded. Uses REVTEX and epsf macros. Also available at http://www.physics.rutgers.edu/~dhv/preprints/mc_ext/index.htm