Following Krause [Kra99], we prove Krull-Schmidt theorems for thick subcategories of various triangulated categories: derived categories of rings, noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ...

We propose an independence criterion based on the eigenspectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernel-based independence criteria. Fir...

As classical information arises from probability correlation between two random variables, quantum information arises from entanglement [1, 2]. Motivated by the finding of an entangled state which does not violate Bell’s inequality, the problem of quantifying entanglement has received an increasing interest recently. Vedral et. al. [3] proposed three necessary conditions that any measure of ent...

Substituted pyrroles were prepared by a gold(I)-catalyzed acetylenic Schmidt reaction of homopropargyl azides. The reaction allows for regiospecific substitution at each position of the pyrrole ring under mild conditions. A mechanism in which azides serve as nucleophiles toward gold(I)-activated alkynes with subsequent gold(I)-aided expulsion of dinitrogen is proposed.