A Meyers type regularity result for approximations of second order elliptic operators by P1 finite elements

We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by P1 finite elements. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.     

Spectroscopic Studies of Mononuclear Molybdenum Enzyme Centers

A concise review is provided of the contributions that various spectroscopic methods have made to our understanding of the physical and electronic structures of mononuclear molybdenum enzymes. Contributions to our understanding of the structure and function of each of the major families of these enzymes is considered, providing a perspective on how spectroscopy has impacted the field.     

Synthesis of Conformationally Restricted Carbocyclic Nucleosides: The Role of the O(4′)-Atom in the Key Hydration Step of Adenosine Deaminase

Conformationally restricted carbocyclic nucleosides with either a northern(N)-type conformation, i.e., N-type 2′-deoxy-methanocarba-adenosine 8 ((N)MCdAdo), or a southern(S)-type conformation, i.e. S-type 2′-deoxy-methanocarba-adenosine 9 , ((S)MCdAdo), were used as substrates for adenosine deaminase (ADA) to assess the enzyme's preference for a fixed conformation relative to the flexible conformation represented by the carbocyclic nucleoside aristeromycin ( 10 ). Further comparison between the rates of deamination of these compounds with those of the two natural substrates adenosine (Ado; 1 ) and 2′-deoxyadenosine (dAdo; 2 ), as well as with that of the conformationally locked nucleoside LNA-Ado ( 11 ), which, like the natural substrates, has a furanose O(4′) atom, helped differentiate between the roles of the O(4′) anomeric effect and sugar conformation in controlling the rates of deamination by ADA. Differences in rates of deamination as large as 10000 can be attributed to the combined effect of the O(4′) atom and the enzyme's preference for an N-type conformation. The hypothesis proposed is that ADA's preference for N-type substrates is not arbitrary; it is rather the direct consequence of the conformationally dependent O(4′) anomeric effect, which is more efficient in N-type conformers in promoting the formation of a covalent hydrate at the active site of the enzyme. The formation of a covalent hydrate at the active site of ADA precedes deamination. A new and efficient synthesis of the important carbobicyclic template 14a , a useful intermediate for the synthesis of (N)MCdAdo ( 8 ) and other conformationally restricted nucleosides, is also reported.     

A limiting case for the divergence equation

We consider the equation $\text{ div}\,\mathbb{Y }=f$ , with $f$ a zero average function on the torus $\mathbb{T }^d$ . In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution $\mathbb{Y }\in W^{1,d}\cap L^\infty $ for a datum $f\in L^d$ . We extend their result to the critical Sobolev spaces $W^{s,p}$ with $(s+1)p=d$ and $p\ge 2$ . More generally, we prove a similar result in the scale of Triebel–Lizorkin spaces. We also consider the equation $\text{ div} \,\mathbb{Y }=f$ in a bounded domain $\varOmega $ subject to zero Dirichlet boundary condition.     

H1–L1 Boundedness of Riesz Transforms on Riemannian Manifolds and on Graphs

We prove that the Riesz transforms are bounded from H ¹ to L ¹ on complete Riemannian manifolds and on graphs with the doubling property and the Poincaré inequality.     

Selective Phenol Alkylation: An Improved Preparation of Efaproxiral

An improved, two‐step synthesis of efaproxiral, used in breast cancer therapy, is described, utilizing inexpensive commodity chemicals for starting materials. Selective amide formation and O‐alkylation in the presence of multireactive functional groups is demonstrated, thus avoiding protection/deprotection steps.     

Fractional Poincaré inequalities for general measures

We prove a fractional version of Poincaré inequalities in the context of ${\mathbb{R}}^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein–Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.