Quantum Ergodicity and Averaging Operators on the Sphere

We prove quantum ergodicity for certain orthonormal bases of $L^2(\mathbb{S}^2)$, consisting of joint eigenfunctions of the Laplacian on $\mathbb{S}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.Comment: 27 page

Yield, Pests, and Water Use of Durum and Selected Crucifer Oilseeds in Two-Year Rotations

Cool-season oilseed crops are potential feedstock for biofuel production, but few studies have compared oilseed-durum (Triticum durum Desf.) rotations on yield, quality, water use, and pests associated with crops. We conducted an experiment under dryland conditions during 2007 to 2010 near Culbertson, MT, comparing crop productivity, water balance, and key weed and arthropod pests of 2-yr oilseed-durum rotations under zero tillage. Rotations included durum with three Brassicaceae sp., camelina [Camelina sativa (L.) Crantz], crambe (Crambe abyssinica Hochst. ex R.E. Fries), and canola-quality Brassica juncea L., and fallow. Over 4 yr, B. juncea had the highest seed and oil yields of crucifer entries. Water use was similar among oilseed crops, averaging 286 mm. Water use was similar for durum following oilseeds, averaging 282 mm, 72 mm less than for durum following fallow. Durum following fallow averaged 775 kg ha−1 greater grain yield than durum following oilseeds due to higher water availability and use. Camelina had greater weed biomass at harvest and lower densities of Plutella xylostella L. than other oilseeds. Durum in rotation with crambe had higher weed density and biomass at harvest than durum following B. juncea or fallow. Brassica juncea generally performed better than crambe or camelina, but each oilseed crop had several positive attributes. Oilseed-durum rotations can be used for biofuel feedstock and grain production, but long-term sustainability of 2-yr rotations on crop yields and pest management requires further study

Riemannian consensus for manifolds with bounded curvature

Consensus algorithms are popular distributed algorithms for computing aggregate quantities, such as averages, in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements lie in Euclidean space. In this work we propose Riemannian consensus, a natural extension of existing averaging consensus algorithms to the case of Riemannian manifolds. Unlike previous generalizations, our algorithm is intrinsic and, in principle, can be applied to any complete Riemannian manifold. We give sufficient convergence conditions on Riemannian manifolds with bounded curvature and we analyze the differences with respect to the Euclidean case. We test the proposed algorithms on synthetic data sampled from the space of rotations, the sphere and the Grassmann manifold.This work was supported by the grant NSF CNS-0834470. Recommended by Associate Editor L. Schenato. (CNS-0834470 - NSF

Rotation Coordinate Descent for Fast Globally Optimal Rotation Averaging

Under mild conditions on the noise level of the measurements, rotation averaging satisfies strong duality, which enables global solutions to be obtained via semidefinite programming (SDP) relaxation. However, generic solvers for SDP are rather slow in practice, even on rotation averaging instances of moderate size, thus developing specialised algorithms is vital. In this paper, we present a fast algorithm that achieves global optimality called rotation coordinate descent (RCD). Unlike block coordinate descent (BCD) which solves SDP by updating the semidefinite matrix in a row-by-row fashion, RCD directly maintains and updates all valid rotations throughout the iterations. This obviates the need to store a large dense semidefinite matrix. We mathematically prove the convergence of our algorithm and empirically show its superior efficiency over state-of-the-art global methods on a variety of problem configurations. Maintaining valid rotations also facilitates incorporating local optimisation routines for further speed-ups. Moreover, our algorithm is simple to implement; see supplementary material for a demonstration program.Comment: Accepted to CVPR 2021 as an oral presentatio