Existence of families of spacetimes with a Newtonian limit

Jürgen Ehlers developed frame theory to better understand the relationship between general relativity and Newtonian gravity. Frame theory contains a parameter λ, which can be thought of as 1/c ², where c is the speed of light. By construction, frame theory is equivalent to general relativity for λ > 0, and reduces to Newtonian gravity for λ = 0. Moreover, by setting , frame theory provides a framework to study the Newtonian limit . A number of ideas relating to frame theory that were introduced by Jürgen have subsequently found important applications to the rigorous study of both the Newtonian limit and post-Newtonian expansions. In this article, we review frame theory and discuss, in a non-technical fashion, some of the rigorous results on the Newtonian limit and post-Newtonian expansions that have followed from Jürgen’s work.     

Cosmological Post-Newtonian Expansions to Arbitrary Order

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter e = v_T/c{\epsilon=v_T/c} (0 < e < e₀){(0< \epsilon < \epsilon_0)}, where c is the speed of light, and v _T is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M @ [0,T)×\mathbb T³{M\cong [0,T)\times \mathbb {T}^3}, and converge as e\searrow 0{\epsilon \searrow 0} to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter e{\epsilon} to any specified order with expansion coefficients that satisfy e{\epsilon}-independent (nonlocal) symmetric hyperbolic equations.     

The Groups of <Emphasis Type="Italic">Zc</Emphasis>-Automaton Transformations

Under study are the ZC-automata and the transformation groups determined by them. We establish relationships between the group of ZC-automaton transformations and the group of infinite unitriangular integer matrices. We describe the derived series of the group of ZC-automaton transformations and present conditions for the representability of residually solvable groups by ZC-automaton transformations. We construct a continual family of ZC-automata with two states, each of which generates a free semigroup.     

Biosynthesis of the angiogenesis inhibitor borrelidin by Streptomyces parvulus Tü4055: cluster analysis and assignment of functions

The biosynthetic gene cluster for the angiogenesis inhibitor borrelidin has been cloned from Streptomyces parvulus Tü4055. Sequence analysis indicates that the macrolide ring of borrelidin is formed by a modular polyketide synthase (PKS) (borA1-A6), a result that was confirmed by disruption of borA3. The borrelidin PKS is striking because only seven rather than the nine modules expected for a nonaketide product are encoded by borA1-A6. The starter unit of the PKS has been verified as trans-cyclopentane-1,2-dicarboxylic acid (trans-1,2-CPDA), and the genes involved in its biosynthesis identified. Other genes responsible for biosynthesis of the nitrile moiety, regulation, and self-resistance were also identified.     

ChemInform Abstract: Rare‐Earth Manganese Germanides RE2+xMnGe2+y (RE: La,Ce) Built from Four‐Membered Rings and Stellae quadrangulae of Mn‐Centred Tetrahedra.

La_2+xMnGe_2+y (I) and Ce_2+xMnGe_2+y (II) are prepared by arc‐melting of the elements in molar ratios of 2:1:2 followed by annealing (sealed silica tubes, 800 °C, 2 weeks).     

Faithful Group Actions on Rooted Trees Induced by Actions of Quotients

We develop some new techniques of constructing (finite state) actions on rooted homogeneous trees and apply them to various groups. In particular we show that there is a faithful action of each amalgameted free product of the form ??_?? on a rooted homogeneous tree of finite degree, described by finite state automorphisms.     

Post-Newtonian Expansions for Perfect Fluids

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations that have a first post-Newtonian expansion. The results here are based on the elliptic-hyperbolic formulation of the Einstein-Euler equations used in [15], which contains a singular parameter , where v _T is a characteristic velocity associated with the fluid and c is the speed of light. As in [15], energy estimates on weighted Sobolev spaces are used to analyze the behavior of solutions to the Einstein-Euler equations in the limit , and to demonstrate the validity of the first post-Newtonian expansion as an approximation.