On the Shape Dependence of Entanglement Entropy

We study the shape dependence of entanglement entropy (EE) by deforming symmetric entangling surfaces. We show that entangling surfaces with a rotational or translational symmetry extremize (locally) the EE with respect to shape deformations that break some of the symmetry (i.e. the 1st order correction vanishes). This result applies to EE and Renyi entropy for any QFT in any dimension. Using Solodukhin's formula in $4d$ and holography in any $d$, we calculate the 2nd order correction to the universal EE for CFTs and simple symmetric entangling surfaces. In all cases we find that the 2nd order correction is positive, and thus the corresponding symmetric entangling surface is a local minimum. Some of the results are extended to free massive fields and to 4d Renyi entropy.Comment: 24 pages, 2 figures, minor changes, added reference

Wider Thin-Very Tall Superatomic Boolean Algebras

For each regular cardinal k > w we show the consistent existence of a thin very tall superatomic Boolean algebra of width k.Comment: There is a gap in claim 4.

Non-linear Waves in Plasmas and Disk-like Galaxies

Nonlinear waves in plasmas and disk-like galaxie

Presidential Leadership in the Space Age

Mankind’s quest to reach the moon consisted of many people in leadership positions. In the US, however, many of the decisions behind the space race, especially funding for it, were made by four men: Dwight D. Eisenhower, John F. Kennedy, Lyndon Baines Johnson, and Richard Nixon. While some presidents (namely Kennedy) receive more credit than others for their work on bringing man to the moon, each of them passed influential policy that was vital in the development of Apollo: Eisenhower founded NASA and began research on the Saturn V, Kennedy gave vision and urgency to the program, Johnson gave massive funding to it, and Nixon oversaw the program and decided what would come next