In this paper we study static spaces introduced in Hawking and Ellis (1975) [1], Fischer and Marsden (1975) [3] and Riemannian manifolds possessing solutions to the critical point equation introduced in Besse (1987) [11], Hwang (2000) [12]. In both cases, on the manifolds there is a function satisfying a particular Ricci-Hessian type equation (1.6). With an idea similar to that used in Cao et al. (2012) [15,16], we have made progress in solving the problem raised in Fischer and Marsden (1975) [3] of classifying vacuum static spaces and in proving the conjecture proposed in Besse (1987) [11] concerning manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3. © 2013 Elsevier B.V.

Communication complexity studies the amount of communication necessary to compute a function whose value depends on information distributed among several entities. Yao [Yao79] initiated the study of communication complexity more than 40 years ago, and it has since become a central eld in theoretical computer science with many applications in diverse areas such as data structures, streaming algorithms, property testing, approximation algorithms, coding theory, and machine learning. The textbooks [KN06,RY20] provide excellent overviews of the theory and its applications.

The Belle II detector will provide a major step forward in precision heavy flavor physics, quarkonium and exotic states, searches for dark sectors, and many other areas. The sensitivity to a large number of key observables can be improved by about an order of magnitude compared to the current measurements, and up to two orders in very clean search measurements. This increase in statistical precision arises not only due to the increased luminosity, but also from improved detector efficiency and precision for many channels. Many of the most interesting observables tend to have very small theoretical uncertainties that will therefore not limit the physics reach. This book has presented many new ideas for measurements, both to elucidate the nature of current anomalies seen in flavor, and to search for new phenomena in a plethora of observables that will become accessible with the Belle II dataset. The simulation used for the studiesinthis book was state ofthe artat the time, though weare learning a lot more about the experiment during the commissioning period. The detector is in operation, and working spectacularly well.