Nonexistence of Generalized Apparent Horizons in Minkowski Space

We establish a Positive Mass Theorem for initial data sets of the Einstein equations having generalized trapped surface boundary. In particular we answer a question posed by R. Wald concerning the existence of generalized apparent horizons in Minkowski space

Wald Statistics in high-dimensional PCA

In this note we consider PCA for Gaussian observations $X_1,\dots, X_n$ with covariance $\Sigma=\sum_i \lambda_i P_i$ in the 'effective rank' setting with model complexity governed by $\mathbf{r}(\Sigma):=\text{tr}(\Sigma)/\| \Sigma \|$. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector $\hat P_r$. This can be used to construct non-asymptotic confidence ellipsoids and tests for spectral projectors $P_r$. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(\Sigma) \gg \sqrt{n}$.Comment: 11 page

Entropy for gravitational Chern-Simons terms by squashed cone method

In this paper we investigate the entropy of gravitational Chern-Simons terms for the horizon with non-vanishing extrinsic curvatures, or the holographic entanglement entropy for arbitrary entangling surface. In 3D we find no anomaly of entropy appears. But the squashed cone method can not be used directly to get the correct result. For higher dimensions the anomaly of entropy would appear, still, we can not use the squashed cone method directly. That is becasuse the Chern-Simons action is not gauge invariant. To get a reasonable result we suggest two methods. One is by adding a boundary term to recover the gauge invariance. This boundary term can be derived from the variation of the Chern-Simons action. The other one is by using the Chern-Simons relation $d\bm{\Omega_{4n-1}}=tr(\bm{R}^{2n})$. We notice that the entropy of $tr(\bm{R}^{2n})$ is a total derivative locally, i.e. $S=d s_{CS}$. We propose to identify $s_{CS}$ with the entropy of gravitational Chern-Simons terms $\Omega_{4n-1}$. In the first method we could get the correct result for Wald entropy in arbitrary dimension. In the second approach, in addition to Wald entropy, we can also obtain the anomaly of entropy with non-zero extrinsic curvatures. Our results imply that the entropy of a topological invariant, such as the Pontryagin term $tr(\bm{R}^{2n})$ and the Euler density, is a topological invariant on the entangling surface.Comment: 19 pag

Physical Process Version of the First Law of Thermodynamics for Black Holes in Higher Dimensional Gravity

The problem of physical process version of the first law of black hole thermodynamics for charged rotating black hole in n-dimensional gravity is elaborated. The formulae for the first order variations of mass, angular momentum and canonical energy in Einstein (n-2)-gauge form field theory are derived. These variations are expressed by means of the perturbed matter energy momentum tensor and matter current density.Comment: 6 pages, REVTEX, to be published in Phys.Rev.D1

Collaborative Piano Student Recital, November 17, 1993

This is the concert program of the Collaborative Piano Student Recital on Wednesday, November 17, 1993 at 8:00 p.m., at the Concert Hall, 855 Commonwealth Avenue. Works performed were Sonata for Piano and Violin, Op. 30 No. 3 by Ludwig van Beethoven, Chanson triste by Henri Duparc, Extase by H. Duparc, L'invitation au voyage by H. Duparc, Allerseelen by Richard Strauss, Nacht by R. Strauss, Zueignung by R. Strauss, Von ewiger Liebe by Johannes Brahms, O kühler Wald by J. Brahms, and "Song to the Moon" from "Rusalka" by Antonin Dvorák. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund