Strong approximation in h-mass of rectifiable currents under homological constraint

Let {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that {h(0)=0} and {h(\theta)\geq\alpha|\theta|} for some {\alpha>0} . If {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with {\mathbb{M}_{h}(T)<\infty} and {\partial T} polyhedral, we prove that for every {\eta>0} , it decomposes as {T=P+\partial V} with P polyhedral, V rectifiable, {\mathbb{M}_{h}(V)<\eta} and {\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta} . In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint {\partial P=\partial T} . When {h^{\prime}(0^{+})} is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary. These results are motivated by the study of approximations of {\mathbb{M}_{h}} by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of {T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)} with respect to the topology of the flat norm.

On the structure of flat chains modulo p

In this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m-dimensional integral currents modulo p implies that the family of {(m-1)}-dimensional flat chains of the form pT, with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1.

Energetic materials: α-NTO crystallizes as a four-component triclinic twin

The prevalent polymorph of the energetic material 5-nitro-2,4-dihydro-1,2,4,-triazol-3-one, α-NTO, crystallizes as a four-component twin with triclinic symmetry (space group P\bar 1). All crystals under investigation were fourlings, i.e. they contained each of the four possible twin components. Complete data sets were collected for two crystals, one with a predominant amount of one individual component (55%) and one with approximately equal volumes of each component. In both cases the fourling components are related by the twofold axes inherent in the holohedral symmetry of a pseudo-orthorhombic superlattice with a o = a t , b o = b t and c o = a t + b t + 2c t . The triclinic unit cell contains four crystallographically independent planar molecules in the asymmetric unit, each of which forms a hydrogen-bonded flat chain parallel to a t . Pairs of chains are combined into planar ribbons by additional hydrogen bonds. Thus, two independent ribbons extend parallel to a t , creating a dihedral angle of ∼ 70°. The origin of the twinning is derived from consideration of the crystal packing and the hydrogen-bonding scheme.