Fast Cross-Polytope Locality-Sensitive Hashing

We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys $\mathcal{O}(d \ln d )$ hash computation time. Building on a recent result (by Andoni, Indyk, Laarhoven, Razenshteyn, Schmidt, 2015), we show that optimal asymptotic sensitivity for cross-polytope LSH is retained even when the dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform followed by discrete pseudo-rotation, reducing the hash computation time from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \ln d )$. Moreover, our scheme achieves the optimal rate of convergence for sensitivity. By incorporating a low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to require only $\mathcal{O}(\ln^9(d))$ random bitsComment: 14 pages, 6 figure

Assessing Devolution in the Canadian North: A Case Study of the Yukon Territory

Despite a rich literature on the political and constitutional development of the Canadian territorial North, few scholars have examined the post-devolution environment in Yukon. This lacuna is surprising since devolution is frequently cited as being crucial to the well-being of Northerners, leading both the Government of Nunavut and the Government of the Northwest Territories to lobby the federal government to devolve lands and resources to them. This paper provides an updated historical account of devolution in Yukon and assesses its impact on the territory since 2003. Relying mainly on written resources and 16 interviews with Aboriginal, government, and industry officials in the territory, it highlights some broad effects of devolution and specifically analyzes the processes of obtaining permits for land use and mining. Our findings suggest that devolution has generally had a positive effect on the territory, and in particular has led to more efficient and responsive land use and mining permit processes

Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups

The little Grothendieck problem consists of maximizing $\sum_{ij}C_{ij}x_ix_j$ over binary variables $x_i\in\{\pm1\}$, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C a dn x dn positive semidefinite matrix, the objective is to maximize $\sum_{ij}Tr (C_{ij}^TO_iO_j^T)$ restricting $O_i$ to take values in the group of orthogonal matrices, where $C_{ij}$ denotes the (ij)-th d x d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve this problem and show a constant approximation ratio. Our method is based on semidefinite programming. For a given $d\geq 1$, we show a constant approximation ratio of $\alpha_{R}(d)^2$, where $\alpha_{R}(d)$ is the expected average singular value of a d x d matrix with random Gaussian $N(0,1/d)$ i.i.d. entries. For d=1 we recover the known $\alpha_{R}(1)^2=2/\pi$ approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous problem over the Unitary Group. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of~$\frac1{2\sqrt{2}}$. The little Grothendieck problem falls under the class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and it provides a more efficient algorithm with better approximation ratios and matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints.Comment: Updates in version 2: extension to the complex valued (unitary group) case, sharper lower bounds on the approximation ratios, matching integrality gap, and a generalized rank constrained version of the problem. Updates in version 3: Improvement on the expositio

How Hot Is Radiation?

A self-consistent approach to nonequilibrium radiation temperature is introduced using the distribution of the energy over states. We begin rigorously with ensembles of Hilbert spaces and end with practical examples based mainly on the far from equilibrium radiation of lasers. We show that very high, but not infinite, laser radiation temperatures depend on intensity and frequency. Heuristic "temperatures" derived from a misapplication of equilibrium arguments are shown to be incorrect. More general conditions for the validity of nonequilibrium temperatures are also established.Comment: 26 pages, revised, LaTeX, 3 encapsulated PostScript figure

Christopher Kennedy folio

A folio of English-language poetry by Christopher Kennedy

A Bochner Formula on Path Space for the Ricci Flow

We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer-Naber \cite{HN18a}. Our results are parabolic counterparts of the recent results in the elliptic setting from \cite{HN18b}

N-heterocyclic germylenes: structural characterisation of some heavy analogues of the ubiquitous N-heterocyclic carbenes

The X-ray crystal structures of three N-heterocyclic germylenes (NHGes) have been elucidated including the previously unknown 1,3-bis(2,6-dimethylphenyl)diazagermol-2-ylidene (1). In addition, the X-ray crystal structures of the previously synthesised 1,3-bis(2,4,6-trimethylphenyl)diazagermol-2-ylidene (2) and 1,3-bis(2,6-diisopropylphenyl)diazagermol-2-ylidene (3) are also reported. The discrete molecular structures of compounds 1 to 3 are comparable, with Ge-N bond lengths in the range 1.835-1.875 Å, while the N-Ge-N bond angles range between 83.6 and 85.2°. Compound 2 was compared to the analogous N-heterocyclic carbene species, 1,3-bis(2,4,6-trimethylphenyl)imidazol-2-ylidene (IMes). The major geometrical difference observed, as expected, was the bond angle around the divalent group 14 atom. The N-Ge-N bond angle was 83.6° for compound 2 versus the N-C-N bond angle of 101.4° for IMes. The Sn equivalent of (1), 1,3-bis(2,6-dimethylphenyl)diazastannol-2-ylidene (4), has also been synthesised and its crystal structure is reported here. In order to test their suitability as ligands, compounds 1 to 3 were reacted with a wide range of transition metal complexes. No NHGes containing metal complexes were observed. In all cases the NHGe either degraded or gave no reaction