Symmetry analysis of the interplay between local moment anisotropies and propagation vectors in the borocarbides

Representational analysis is used to examine the interplay between Fermi surface nesting and local moment effects in the formation of the magnetically ordered states of the rare-earth nickel borocarbides, RNi₂B₂C. We derive compatibility tables for the propagation wavevector k, the local moment anisotropy and the ordered moment direction μ _k for materials with this tetragonal crystal structure. The magnetic structures observed in the rare-earth nickel borocarbides are discussed in this context.     

CyClick Chemistry for the Synthesis of Cyclic Peptides

Here, we report a novel “CyClick” strategy for the macrocyclization of peptides that works in an exclusively intramolecular fashion thereby precluding the formation of dimers and oligomers via intermolecular reactions. The CyClick chemistry is highly chemoselective for the N‐terminus of the peptide with a C‐terminal aldehyde. In this protocol, the peptide conformation internally directs activation of the backbone amide bond and thereby facilitates formation of a stable 4‐imidazolidinone‐fused cyclic peptide with high diastereoselectivity (>99 %). This method is tolerant to a variety of peptide aldehydes and has been applied for the synthesis of 12‐ to 23‐membered rings with varying amino acid compositions in one pot under mild reaction conditions. The reaction generated peptide macrocycles featuring a 4‐imidazolidinone in their scaffolds, which acts as an endocyclic control element that promotes intramolecular hydrogen bonding and leads to macrocycles with conformationally rigid turn structures.     

An alternative route to tethered Ru(II) transfer hydrogenation catalysts

A new route towards a series of tethered η⁶-arene/Ru(II) catalysts for use in the transfer and pressure hydrogenation of ketones and aldehydes to alcohols is reported. The route proceeds through the formation of an amide from the diamine precursor, followed by reduction, rather than the direct alkylation of the diamine. This has the advantage that dialkylation of the amine is avoided during the synthesis. Through this new route, both racemic and enantiomerically-pure η⁶-arene/Ru(II) tethered catalysts can be prepared in high yield.     

The use of a [4 + 2] cycloaddition reaction for the preparation of a series of 'tethered' Ru(II)-diamine and aminoalcohol complexes

A series of catalysts have been prepared for use in the asymmetric transfer hydrogenation of ketones. The complexes were prepared using a [4 + 2] cycloaddition reaction at a key step in the reaction sequence. This provides a means for the synthesis of catalysts with modifications at specific sites.     

Alkali metal complexes of a phosphine-borane-stabilised carbanion: influence of co-ligands on structure

The adducts [[(Me(3)Si)(2){Me(2)P(BH(3))}C]K(L)(n)](m) [L = THF, n = 0.5, m = infinity (2a); L = tmeda (2b), pmdeta (2c), n = 1, m = 2] may be synthesised by treatment of solvent-free [[(Me(3)Si)(2){Me(2)P(BH(3))}C]K](infinity) (2) with the corresponding Lewis base (tmeda = N,N,N',N'-tetramethylethylenediamine; pmdeta = N,N,N',N',N'-pentamethyldiethylenetriamine). X-Ray crystallography reveals that, whereas 2 crystallises with a complex 2-dimensional sheet structure, 2a crystallises as a ribbon-type one-dimensional polymer and both 2b and 2c crystallise as dimers. The corresponding complex with 12-crown-4, [K(12-crown-4)(2)][(Me(3)Si)(2){Me(2)P(BH(3))}C] (2d) crystallises as a separated ion pair. The complexes [[(Me(3)Si)(2){Me(2)P(BH(3))}C]M(pmdeta)](n) [M = Na, n = 1 (6); M = Rb, n = 2 (7)] may be synthesised by treatment of [(Me(3)Si)(2){Me(2)P(BH(3))}C]M with pmdeta. Whereas crystallises as a discrete monomer, compound 7 crystallises as a dimer. Compounds 2, 2a-2d, 6, 7 and the corresponding caesium derivative [[(Me(3)Si)(2){Me(2)P(BH(3))}C]Cs(pmdeta)](2) () provide an opportunity to consider the influence of the ionic radius of the metal and the nature of the co-ligands on the structures of alkali metal complexes of a phosphine-borane-stabilised carbanion.     

Notes on the Roots of Ehrhart Polynomials

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n², where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n². For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of Ehrhart polyomials in the three-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.     

A Blichfeldt-type inequality for the surface area

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that #(K?\Bbb Zⁿ) ￡ n! vol(K)+n\#(K\cap{\Bbb Z}^n)\le n! {\rm vol}(K)+n , whenever K ì \Bbb RⁿK\subset{\Bbb R}^n is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely #(K?\Bbb Zⁿ) < vol(K) + ((?n+1)/2) (n-1)! F(K)\#(K\cap{\Bbb Z}^n) < {\rm vol}(K) + ((\sqrt{n}+1)/2) (n-1)! {\rm F}(K) . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where t ? \Bbb Rⁿ\\Bbb Zⁿt\in{\Bbb R}^n\setminus{\Bbb Z}^n and P is an n-dimensional polytope with integral vertices. Then we have #((t+P)?\Bbb Zⁿ) ￡ n! vol(P)\#((t+P)\cap{\Bbb Z}^n)\le n! {\rm vol}(P) . Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K?\Bbb Zⁿ) < vol(K) + 2 F(K)\#(K\cap{\Bbb Z}^n)< {\rm vol}(K) + 2 {\rm F}(K) .     

Decoupling of Higgs from supersymmetric ϱ parameter

The one loop, large Higgs mass corrections to the ? parameter are calculated in a manifestly supersymmetric way, in a model with soft breaking in which Yukawa couplings are neglected. We find that the Higgs decouples from ? and, in a phenomenologically viable class of models, which are a subset of the above model, the ? parameter is independent of Higgs mass.