Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

On homotopy rigidity of the functor ΣΩ on co-H-spaces

In this paper we study the homotopy rigidity property of the functors ΣΩ and Ω. Our main result is that both functors are homotopy rigid on simply-connected p-local finite co-H-spaces. The result is obtain by a subtle interplay of homotopy decomposition techniques, modular representation theory and the counting principle.     

UCT-Kirchberg algebras have nuclear dimension one

We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial _K0$K_0$ and finite _K1$K_1$ have nuclear dimension 1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras. We then prove that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras to obtain our main theorem.     

Rate of convergence in the law of large numbers for supercritical general multi-type branching processes

We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman’s martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the processes in focus.     

Two Aromatic Ester Organoimido Derivatives of Hexa­molybdate: Syntheses,Crystal Structures,and Spectroscopic Studies

Two alkylimido derivatives of hexamolybdate, (Bu₄N)₂[Mo₆O₁₈(≡N‐o‐COOCH₃C₆H₄)] ( 1 ) and (Bu₄N)₂[Mo₆O₁₈(≡N‐o‐COOCH₂CH₃C₆H₄)] ( 2 ), were synthesized in high purity and good yields by the reaction of [(C₄H₉)₄N]₄[α‐Mo₈O₂₆] and methyl anthranilate or ethyl‐o‐aminobenzoate hydrochloride with N,N′‐dicyclohexylcarbodiimide (DCC) as a dehydrating agent in dry acetonitrile solution, which were characterized by elemental analyses, IR, UV/Vis, and ¹H NMR spectroscopy as well as ESI‐MS, and single‐crystal X‐ray diffraction study. Compound 1 crystallizes in the monoclinic space group P2₁/n with one‐dimensional chain structure via intramolecular hydrogen bond. Compound 2 also crystallizes in the monoclinic space group P2₁/n with dimer structure by intramolecular hydrogen bonds and π–π interactions between the pairs of cluster anions.     

On the Solvent‐free Structure of a Jäger‐type Cobalt(II) N2O2‐Chelate Complex

A combined synchrotron X‐ray and density functional theory (DFT) study on the structure of a Jäger‐type N₂O₂ chelate complex was carried out. The ethoxy‐substituted bis(3‐oxo‐enaminato)cobalt(II) complex ( 1 ) was an original sample from the laboratory of the late Professor Ernst‐G. Jäger (University of Jena, Germany). Single‐crystal X‐ray analysis revealed essentially flat molecules of 1 , which are unsolvated and coordinatively unsaturated. The DFT calculations on the isolated molecule predict a planar structure for the non‐hydrogen atoms, which is a local minimum on the energy surface. The crystal packing is achieved through off‐set stacking (staircase arrangement), resulting in a herringbone pattern in the space group P2₁2₁2₁. The structure of 1 is compared to known structures of related bis(3‐oxo‐enaminato)cobalt(II) complexes ( 2 – 4 ). Original bulk material of 1 was investigated by scanning electron microscopy (SEM), powder X‐ray diffraction (PXRD), melting point determination, and infrared (IR) spectroscopy.     

Synthesis and conformational analysis of new troponyl aromatic amino acid

Synthetic peptides are in huge demand in expansion of potential peptide mimics, which may have improved or comparable function as natural one. With these concerns, phenyl bearing aromatic amino acids and peptides has extensively explored, because phenyl residue has high probability in forming stable secondary structure, owing to the presence of an extra stabilizing factor as π–π non-covalent interactions. Apart from phenyl bearing benzenoid aromatic amino acids, a few non-benzenoid aromatic derivatives such as tropolone and related compounds are also occurred in nature, but troponyl containing amino acids and peptides are very poorly understood. Tropolonyl derivatives also contain carbonyl functional group, which may play an important role to provide stable conformation in peptide. Herein we report the synthesis, and conformational analysis of rationally designed new unnatural δ-amino acid, troponyl aminoethylglycine (Tr-aeg), which contains troponyl residue as side chain in flexible aminoethylglycine (aeg) amino acid backbone. We also demonstrate the role of troponyl carbonyl of Tr-aeg residue in hydrogen bonding with adjacent amide NH of their hybrid di/tri-peptides with NMR methods and DFT calculations. In future, Tr-aeg amino acid would be a potential building block in development of promisable peptide mimics.     

Catalyzation of 1,4-additions of arylboronic acids to α,β-unsaturated substrates using nickel(I) complexes

Nickel(I) complexes were generated in situ from Ni (PPh₃)₂Cl₂ using activated iron and the complexes combined with N,N′-bis(4-fluorobenzylidene) ethane-1,2-diamine (BFBED) were then used as a catalyst for the 1,4-addition reaction of arylboronic acids to α,β-unsaturated substrates. The reaction proceeded to completion and did not require the addition of a base but the addition of potassium iodide is crucial to this cross-coupling reaction. Moreover, experimental observations suggested a possible Ni(I)–Ni(III) catalytic cycle mechanism.     

BF3·OEt2-catalyzed synthesis of 1-(tetrahydropyran-3-yl)-1,3-dihydroisobenzofuran and trans-fused hexahydropyrano[3,2-c]chromene derivatives

A novel intramolecular Prins cyclization of (E)-5-(2-(hydroxymethyl)phenyl)pent-4-en-1-ol with aldehydes has been achieved using 10 mol % BF₃·Et₂O to produce 1-(tetrahydropyran-3-yl)-1,3-dihydroisobenzofuran derivatives in good to excellent yields with high selectivity. Similar type of coupling with salicylaldehydes provides the trans-fused hexahydropyrano[3,2-c]chromene derivatives in excellent yields.     

An efficient and regioselective one-pot multi-component synthesis of pyrimido[4,5-d]pyrimidine derivatives in water

Pyrimido[4,5-d]pyrimidine derivatives 4 have been prepared in an efficient and regioselective manner in water via multi-component reaction of isothiocyanate 1, aromatic aldehyde 2, N,N-dimethyl-6-amino uracil 3 in the presence of p-toluenesulfonic acid (p-TSA) as a Lewis acid catalyst.