Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

A recurring theme in attempts to break the curse of dimensionality in the numerical approximation of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class \(\Sigma _n\) of functions, which can be written as a sum of rank-one tensors using a total of at most \(n\) parameters, and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side \(f\) of the elliptic PDE can be approximated with a certain rate \(\mathcal {O}(n^{-r})\) in the norm of \({\mathrm H}^{-1}\) by elements of \(\Sigma _n\), then the solution \(u\) can be approximated in \({\mathrm H}^1\) from \(\Sigma _n\) to accuracy \(\mathcal {O}(n^{-r'})\) for any \(r'\in (0,r)\). Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second “basis-free” model of tensor-sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent to which such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.     

Decarboxylative [4+2] Cycloaddition by Synergistic Palladium and Organocatalysis

A novel reaction based on synergistic catalysis, combining palladium‐ and organocatalysis has been developed. The palladium catalyst activates vinyl benzoxazinanones via a decarboxylation to undergo a [4+2] cycloaddition with iminium‐ion activated α,β‐unsaturated aldehydes. The reaction is demonstrated to proceed for a number of combinations of vinyl benzoxazinanones reacting with α,β‐unsaturated aldehydes, providing highly substituted vinyl tetrahydroquinolines in good to high yields, and excellent enantio‐ and diastereoselectivities (>98 % ee and >20:1 d.r.). The palladium catalyst used in the synergistic catalysis can be re‐used in a one‐pot sequential coupling reaction with an aromatic boronic acid forming the coupling product in 95 % yield, >20:1 d.r. and 99 % ee.     

Autocatalytic water dissociation on Cu(110) at near ambient conditions

Autocatalytic dissociation of water on the Cu(110) metal surface is demonstrated on the basis of X-ray photoelectron spectroscopy studies carried out in situ under near ambient conditions of water vapor pressure (1 Torr) and temperature (275-520 K). The autocatalytic reaction is explained as the result of the strong hydrogen-bond in the H2O-OH complex of the dissociated final state, which lowers the water dissociation barrier according to the Br?nsted-Evans-Polanyi relations. A simple chemical bonding picture is presented which predicts autocatalytic water dissociation to be a general phenomenon on metal surfaces.     

Synthesis of Bifunctional Lipoxin-Derived Enzyme-Triggered CO-Releasing Molecules (LipET-CORMs)

In an attempt to develop new anti-inflammatory agents which act by co-release of carbon monoxide (CO) and a specialized pro-resolving mediator, we designed conjugates of a lipoxin A₄ analogue and an acyloxycyclohexadiene-Fe(CO)₃ complex as an esterase-triggered CO-releasing molecule (ET-CORM). After adjustment of the protecting group strategy, two of such compounds were successfully prepared by total synthesis (12 steps; 4–5 % overall yield) starting from deoxy-d -ribose and exploiting a Wittig olefination and an intermolecular Heck reaction as key C−C bond-forming steps. A crucial late reduction of an aryl-ketone moiety in the presence of a highly sensitive dienol ester functionality was achieved with BH₃-SMe₂ in the presence of catalytic amounts of NaBH₄. Both target compounds were dose-dependently toxic towards cultured human umbilical vein endothelial cells (HUVEC), with LipET-CORM 1-A being slightly more toxic. While induction of heme oxygenase 1 (HO-1) in HUVEC was observed for both compounds, they did not inhibit TNF-α-mediated VCAM-1 expression in these cells. In M2 polarized macrophages HO-1 expression was more pronounced as compared to M1 polarized macrophages. In both types of macrophages HO-1 expression was downregulated by lipopolysaccharide, but only in M2 macrophages HO-1 expression was rescued by LipET-CORM. 15-Lipoxygenase (15-LO) was only expressed in M2 macrophages and was not influenced by LipET-CORM. Collectively our data demonstrate that LipET-CORMs induce HO-1 expression in endothelial cells and M2 polarized macrophages. The role of the intra-cellular released lipoxin A₄ in resolution of inflammation, however, remains to be assessed.     

Surface immobilization of biomolecules by click sulfonamide reaction

Alkyne-modified biomolecules can be immobilized site- and chemoselectively on sulfonylazide slides under very mild conditions by means of the click sulfonamide reaction.     

Influence of an electron-deficient bridging o-carborane on the electronic properties of an [FeFe] hydrogenase active site model

The IR carbonyl stretching frequencies of [Fe2(SRS)(CO)6] complexes correlate well with their first reduction potential; an [FeFe] hydrogenase model with a very mild reduction potential has been realized by using a strongly electron deficient carborane-dithiolate bridge.     

Catalytic chain-breaking pyridinol antioxidants

When assayed for their capacity to inhibit azo-initiated peroxidation of linoleic acid in a water/chlorobenzene two-phase system, tellurium-containing 3-pyridinols were readily regenerable by N-acetylcysteine contained in the aqueous phase. The best inhibitors quenched peroxyl radicals more efficiently than alpha-tocopherol, and the duration of inhibition was limited only by the availability of the thiol reducing agent. The compounds were also found to catalyze reduction of hydrogen peroxide in the presence of thiol reducing agent.     

A test complex for Gorensteinness

Let be a commutative noetherian ring with a dualizing complex. By recent work of Iyengar and Krause (2006), the difference between the category of acyclic complexes and its subcategory of totally acyclic complexes measures how far is from being Gorenstein. In particular, is Gorenstein if and only if every acyclic complex is totally acyclic.

In this note we exhibit a specific acyclic complex with the property that it is totally acyclic if and only if is Gorenstein.