Real Kasner and related complex “windmill” vacuum spacetime metrics

The Kasner family of vacuum solutions of Einstein's field equations admits a simply-transitiveH ₄, a four-parameter local homothetic group of motions which has an AbelianG ₃ subgroup. It is shown that a complex transformation of coordinates and constants exists which maps this family from the normal Kasner form into a form of vacuum metrics whose Weyl tensors are each Petrov type I and which were published in 1932 by Lewis. These metrics also admit a similarH ₄; however for one particular metric (for one parameter value) theH ₄ becomes aG ₄ and the resultant metric is one which was rediscovered by Petrov in 1962. These Lewis metrics are thus shown to be Kasner metrics over complex fields. Here they are calledwindmill metrics because of the rotating relationship between the coordinates and the Killing vector fields admitted. The principal null directions of thereal Kasner and the windmill metrics are discussed; the two families then provide illustrations of two degenerate classes of spacetime metrics whose Weyl tensors are of Petrov type I, as discussed elsewhere by Arianrhod and McIntosh. An extension of the windmill-type generation of metrics to some other families of metrics is also discussed.     

Azapropellanes as phase-transfer catalysts—II: A CMR and PMR study of the three-dimensional structure of 1-azoniapropellanes containing 5- and 6-membered rings

An improved synthesis of 1-azoniatricyclo[4.4.3.0^1,6]tridecane (2) and the preparation of the remaining member of the series, 1-azoniatricyclo[4.3.3.0^1,6]dodecane salts (3) have been achieved. Using a combination of ¹H (at 400 MHz) and ¹³C NMR spectra has allowed the assignment of the signals in the well-resolved ¹H spectra. The results indicate that 1 exists in the all-chair form which undergoes racemization by ring inversion with a rate constant of 0.7 sec^? and further that the counterion in these salts is restricted to associating with only one face of the tetrahedral N atom.     

Symmetry adapted functions belonging to the dirac groups

An earlier analysis of the canonical form of a pair of invertible operators obeying the exchange rule is extended to cover a set of operators, between each pair of which a relation of this type exists; and for which a power of each operator is the unit matrix. Such relations define a system which may be regarded as a generalization of the Dirac matrices of relativistic quantum mechanics. We concentrate upon the group theoretic aspects of such a system and its matrix representations. Applications arise from the fact that all projective representations of finite abelian groups take the form of a Dirac Group. In particular, the representations of the magnetic space groups, which are projective representations of the lattice groups, arise in this manner.     

Performance Analysis of Cyclical Simulated Annealing Algorithms

Generalized hill climbing (GHC) algorithms provide a framework for modeling local search algorithms for addressing intractable discrete optimization problems. Measures for assessing the finite-time performance of GHC algorithms have been developed using this framework, including the expected number of iterations to visit a predetermined objective function value level. This paper analyzes how the expected number of iterations to visit a predetermined objective function value level can be estimated for cyclical simulated annealing. Cyclical simulated annealing uses a cooling schedule that cycles through a set of temperature values. Computational results with traveling salesman problem instances taken from TSPLIB show how the expected number of iterations to visit solutions with predetermined objective function levels can be estimated for cyclical simulated annealing.AMS 2000 Subject Classification 90-08 Computational Methods: Local Search, 90C59 Heuristics: Simulated Annealing     

Spatial photorelease of oligonucleotides, using a safety-catch photolabile linker

[reaction: see text] We report the development of a safety-catch photolabile linker that allows the light-directed synthesis and spatially selective photorelease of oligonucleotides from microarrays. The linker remains stable to light during DNA synthesis, and is activated for photorelease after acidic hydrolysis. We demonstrate that the photoreleased oligonucleotides can be amplified by PCR to produce double stranded DNA. The advantages offered by this linker could aid the development of an automated gene synthesis platform.     

New mock theta conjectures Part I

In their paper “A survey of classical mock theta functions”, Gordon and McIntosh observed that the classical mock \(\theta \)-functions, including those found by Ramanujan, can be expressed in terms of two ‘universal’ mock \(\theta \)-functions denoted by \(g_{2}\) and \(g_{3}\). These identities are known as mock \(\theta \)-conjectures. The fifth- and seventh-order mock \(\theta \)-conjectures were proved by Dean Hickerson. In the survey paper the authors gave mock \(\theta \)-conjectures for the other mock \(\theta \)-functions and referred the proofs to a future paper with this title, listed in their references as [GM4]. The purpose of this paper is to prove these identities for the functions of orders 2 and 3.