Ergodic Actions and Spectral Triples

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C*-algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the "algebraic" existence of ergodic action and the "analytic" finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples - including noncommutative tori and quantum Heisenberg manifolds.Comment: 18 page

What is the Epistemic Significance of Disagreement?

Over the past decade, attention to epistemically significant disagreement has centered on the question of whose disagreement qualifies as significant, but ignored another fundamental question: what is the epistemic significance of disagreement? While epistemologists have assumed that disagreement is only significant when it indicates a determinate likelihood that one’s own belief is false, and therefore that only disagreements with epistemic peers are significant at all, they have ignored a more subtle and more basic significance that belongs to all disagreements, regardless of who they are with—that the opposing party is wrong. It is important to recognize the basic significance of disagreement since it is what explains all manners of rational responses to disagreement, including assessing possible epistemic peers and arguing against opponents regardless of their epistemic fitness

The role of titanium in electromigrated tunnel junctions

A standard route for fabrication of nanoscopic tunnel junctions is via electromigration of lithographically prepared gold nanowires. In the lithography process, a thin adhesion layer, typically titanium, is used to promote the adhesion of the gold nanowires to the substrate. Here, we demonstrate that such an adhesion layer plays a vital role in the electrical transport behavior of electromigrated tunnel junctions. We show that junctions fabricated from gold deposited on top of a titanium adhesion layer are electrically stable at ambient conditions, in contrast to gold junctions without a titanium adhesion layer. We furthermore find that electromigrated junctions fabricated from pure titanium are electrically exceptionally stable. Based on our transport data, we provide evidence that the barrier in gold-on-titanium tunnel devices is formed by the native oxide of titanium

Young measures supported on invertible matrices

Motivated by variational problems in nonlinear elasticity depending on the deformation gradient and its inverse, we completely and explicitly describe Young measures generated by matrix-valued mappings \{Y_k\}_{k\in\N} \subset L^p(\O;\R^{n\times n}), \O\subset\R^n, such that \{Y_k^{-1}\}_{k\in\N} \subset L^p(\O;\R^{n\times n}) is bounded, too. Moreover, the constraint $\det Y_k>0$ can be easily included and is reflected in a condition on the support of the measure. This condition typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if $Y:=\nabla y$ for y\in W^{1,p}(\O;\R^n). Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in W^{1,\infty}(\O;\R^n) where the inverted gradients are also bounded in L^\infty(\O;\R^{n\times n}). This extends the original results due to D. Kinderlehrer and P. Pedregal