Gauge aspect of tetrad field in gravity

In general relativity, an inertial frame can only be established in a small region of spacetime, and a locally inertial frame is mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform a local Lorentz transformation in an inertial frame, and there exists freedom to choose the locally inertial frame at each spacetime. The local Lorentz transformations are known as non-Abelian gauge transformations for the tetrad field, and to fix the gauge freedom, corresponding to the Lorentz gauge $\partial^\mu\mathcal{A}_\mu=0$ and Coulomb gauge $\partial^i\mathcal{A}_i=0$ in electrodynamics, the Lorentz gauge and Coulomb gauge for the tetrad field are proposed in the present work. Moreover, properties of the Lorentz gauge and Coulomb gauge for tetrad field are discussed, which show the similarities to those in electromagnetic field.Comment: 4 pages, no figure, comments are welcome

Bilateral Random Projections

Low-rank structure have been profoundly studied in data mining and machine learning. In this paper, we show a dense matrix $X$'s low-rank approximation can be rapidly built from its left and right random projections $Y_1=XA_1$ and $Y_2=X^TA_2$, or bilateral random projection (BRP). We then show power scheme can further improve the precision. The deterministic, average and deviation bounds of the proposed method and its power scheme modification are proved theoretically. The effectiveness and the efficiency of BRP based low-rank approximation is empirically verified on both artificial and real datasets.Comment: 17 pages, 3 figures, technical repor

Stochastic collocation on unstructured multivariate meshes

Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification (UQ) community. Techniques for least-squares regularization, compressive sampling recovery, and interpolatory reconstruction are becoming standard tools used in a variety of applications. Selection of a collocation mesh is frequently a challenge, but methods that construct geometrically "unstructured" collocation meshes have shown great potential due to attractive theoretical properties and direct, simple generation and implementation. We investigate properties of these meshes, presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure