We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the one-step (abelian) and the two-step ones, there are only six (here f(m, p) ...

It is of fundamental interest to study the geometric and analytic properties of compact Einstein manifolds and their moduli. In dimension 2 these problems are well understood. A 2-dimensional Einstein manifold, (M, g), has constant curvature, which after normalization, can be taken to be −1, 0 or 1. Thus, (M, g) is the quotient of a space form and the metric, g, is completely determined by the ...

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski–Demianski metrics one obtains a family of local toric Kähler–Einstein metrics. These can be used to construct local Sasaki–Einstein metrics in five dimensions, which are generalisations of the Y p,q metrics. In fact, we find that the local metrics are diffeomorphic to those recently found by Cvetic, Lu, Page...

A Riemannian metric is said to be Einstein if the Ricci curvature is a constant multiple of the metric. Given a manifold M , one can ask whether M carries an Einstein metric, and if so, how many. This fundamental question in Riemannian geometry is for the most part unsolved (cf. [Bes]). As a global PDE or a variational problem, the question is intractible. It becomes more manageable in the homo...

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski–Demianski metrics one obtains a family of local toric Kähler–Einstein metrics. These can be used to construct local Sasaki–Einstein metrics in five dimensions which are generalisations of the Y p,q manifolds. In fact, we find that these metrics are diffeomorphic to those recently found by Cvetic, Lu, Page an...

Burris, Stanley. ‘‘George Boole.’’ In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta. 2010. Available from http://plato.stanford.edu/archives/sum2010/entries/boole/ Krantz, Steven G. Discrete Mathematics Demystified. New York: McGraw-Hill, 2009. Nahin, Paul J. The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age. Princeton, NJ: Princ...

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a la...

We propose that periodically in spaceand time-modulated potentials (dynamic lattices) can efficiently remove the excited (the high-energy and large momentum) components of the trapped Bose-Einstein condensates (BECs) and, consequently, can result in efficient cleaning of the BECs. We prove the idea by numerically solving the mean-field models (the Schrödinger equation for noninteracting condens...

The study of the so called “genuine” Bose-Einstein correlations (BEC) of three identical charged pions requires the subtraction of the 2nd order BEC of the pion pairs. The OPAL subtraction method used in the BEC study of the π±π±π± system, which utilised mixed charged pions triplets of the data themselves, is investigated in detail. The derivation of the subtraction formula is explained.

In this paper, we analyze phase separation of multi-component Bose-Einstein Condensates (BECs) in the presence of a strong optical lattice. This paper is in threefold. We first prove that when the inter-component scattering lengths go to infinity, phase separation of a multi-component BEC occurs. Furthermore, particles repel each other and form segregated nodal domains. Secondly, we show that t...