Lusin-type theorems for Cheeger derivatives on metric measure spaces

A theorem of Lusin states that every Borel function on $R$ is equal almost everywhere to the derivative of a continuous function. This result was later generalized to $R^n$ in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincar\'e inequalities, which admit a form of differentiation by a famous theorem of Cheeger.Comment: 16 pages. Comments welcom

Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Banach space. In contrast, we also show that if an Ahlfors regular Lipschitz differentiability space has charts of non-maximal dimension, then these charts are strongly unrectifiable in the sense of Ambrosio-Kirchheim.Comment: 22 page

Bi-Lipschitz Pieces between Manifolds

A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors $s$-regular, topological $d$-manifolds. In general, these manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff convergence of manifolds and a topological theorem of Bonk and Kleiner. This also yields a new proof of the uniform rectifiability of some metric manifolds.Comment: 38 page

Double swivel toggle release

A pyrotechnic actuated structural release device is disclosed which is mechanically two fault tolerant for release. The device comprises a fastener plate and fastener body each attachable to one of a pair of structures to be joined. The fastener plate and the fastener body are fastened by a dual swivel toggle member. The toggle member is supported at one end on the fastener plate and mounted for universal pivotal movement thereon. Its other end is received in a central opening in the fastener body, and has a universally mounted retainer ring member. The toggle member is restrained by three retractable latching pins symmetrically disposed in equiangular spacing about the axis of the toggle member and positionable in latching engagement with the retainer ring member on the toggle member. Each pin is retractable by a pyrotechnic charge, the expanding gases of which are applied to a pressure receiving face on the latch pins to effect retraction from the ring member. While retraction of all three pins releases the ring member, the fastener is mechanically two fault tolerant since the failure of any single one or pair of the latch pins to retract results in an asymmetrical loading on the ring member and its dual pivotal movement ensures a release

nPI Resummation in 3D SU(N) Higgs Theory

We test the utility of the nPI formalism for solving nonperturbative dynamics of gauge theories by applying it to study the phase diagram of SU(N) Higgs theory in 3 Euclidean spacetime dimensions. Solutions reveal standard signatures of a first order phase transition with a critical endpoint leading to a crossover regime, in qualitative agreement with lattice studies. The location of the critical endpoint, x sim 0.14 for SU(2) with a fundamental Higgs, is in rough but not tight quantitative agreement with the lattice. We end by commenting on the overall effectiveness and limitations of an nPI effective action based study. In particular, we have been unable to find an nPI gauge-fixing procedure which can simultaneously display the right phase structure and correctly handle the large-VEV Higgs region. We explain why doing so appears to be a serious challenge.Comment: 24 pages plus appendices, 8 figure