Optimality of a Standard Adaptive Finite Element Method

In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance ε > 0 in energy norm by a continuous piecewise linear function on some partition with O(ε^-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.     

The algebraic structure of cohomological field theory

The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.     

Fermions coupled to autonomous (λΦ 4)3+1 theory

Recent Gaussian-effective-potential results for “autonomous”λΦ ⁴ theory, an apparently nontrivial, positive-λ, 3+1 dimensional theory, are extended to include a Yukawa coupling to fermions. The formerly massless bosons of the symmetric phase acquire a mass.     

Resolution of the renormalisation-scheme ambiguity in perturbative QCD

Next-to-leading-order QCD predictions depend on the arbitrary choice of renormalisation scheme. I resolve this ambiguity and show how to find the unique, optimum prediction, given the result in some arbitrary, initial scheme.     

Synthesis of tetrahydrofuran lignans, (±)-galbelgin and (±)-grandisin

A stereoselective synthesis procedure for lignans of the all-trans αα'-diaryl-ββ'-dimethyltetrahydrofurans is described. The lignans, galbelgin (4) and grandisin (37) were synthesized by routine reduction of the tetrahydrofuran dicarboxylic esters (24 and 34), obtained by mild acid treatment of the readily available diaryl dilactones (23 and 33).     

Miscibility gaps in fully pressure-ionized binary alloys

Binary alloys in which all the electrons form a uniform, degenerate, non-relativistic gas are predicted to exhibit miscibility gaps. Approximate analytical formulae are given for the critical temperatures and compositions. Astrophysical implications are mentioned.