The High‐Pressure Phase of MgB2C2

A high‐pressure modification of MgB₂C₂ was synthesized and structurally characterized. The compound crystallizes in the orthorhombic space group Pnnm, with the lattice parameters a = 7.19633(3) Å, b = 4.61791(13) Å and c = 2.77714(8) Å. The compound contains heterographene B–C nets, isoelectronic to graphite, just like the ambient pressure modification. The layers are intercalated by magnesium atoms, which are arranged in a chain‐like manner. According to Density Functional Theory (DFT) calculations, the high‐pressure form of MgB₂C₂ is a semiconductor with a band gap of 1.02 eV. The compound does not undergo a superconducting transition down to 2 K.     

Constructing the Cubus simus and the Dodecaedron simum via paper folding

The archimedean solids Cubus simus (snub cube) and Dodecaedron simum (snub dodecahedron) cannot be constructed by ruler and compass. We explain that for general reasons their vertices can be constructed via paper folding on the faces of a cube, respectively dodecahedron, and we present explicit folding constructions. The construction of the Cubus simus is particularly elegant. We also review and prove the construction rules of the other Archimedean solids.     

A fully discrete Galerkin method for Abel-type integral equations

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in fractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.     

Piers′ Borane-Induced Tetramerization of Arylacetylenes

We herein report that the reaction of Piers′ borane, i. e. HB(C₆F₅)₂, with an excess of arylacetylenes at room temperature leads to tetramerization of the acetylene and the diastereoselective formation of boryl-substituted tetra-aryl-tetrahydropentalenes. The reaction mechanism was investigated by isotope labeling experiments and DFT computations. These investigations indicate that a series of 1,2-carboboration reactions form an octatetraene that undergoes an electrocyclization. Two skeletal rearrangements then presumably lead to the formation of the tetrahydropentalene core. Overall, this intricate and unprecedented transformation comprises five carbon-carbon bond formations in a single reaction.     

The Structure of the Copper(II) Complex with 4,5-Dichlorophthalic Acid: The 1:1 Complex ‘Adduct’ Salt <Emphasis Type="Italic">trans</Emphasis>-Tetraaqua(2-carboxy-4,5-dichlorobenzoato)(4,5-dichlorobenzene-1,2-dicarboxylic acid)copper(II) 2-Carboxy-4,5-dichlorobenzoate

Abstract  

The unusual (1:1) complex ‘adduct’ salt of copper(II) with 4,5-dichlorophthalic acid (H₂DCPA), having formula [Cu(H₂O)₄(C₈H₃Cl₂O₄) (C₈H₄Cl₂O₄)]·(C₈H₃Cl₂O₄) has been synthesized and characterized using single-crystal X-ray diffraction. Crystals are monoclinic, space group P2₁/c, with Z = 4 in a cell with dimensions a = 20.1376(7), b = 12.8408(4) c = 12.1910(4) Å, β = 105.509(4)°. The complex is based on discrete tetragonally distorted octahedral [CuO₆] coordination centres with the four water ligands occupying the square planar sites [Cu–O, 1.962(4)–1.987(4) Å] and the monodentate carboxyl-O donors of two DCPA ligand species in the axial sites. The first of these bonds [Cu–O, 2.341(4) Å] is with an oxygen of a HDCPA monoanion, the second with an oxygen of a H₂DCPA acid species [Cu–O, 2.418(4) Å]. The un-coordinated ‘adduct’ molecule is a HDCPA counter anion which is strongly hydrogen-bonded to the coordinated H₂DCPA ligand [O···O, 2.503(6) Å] while a number of peripheral intra- and intermolecular hydrogen-bonding interactions give a two-dimensional network structure.     

Asymmetric Synthesis of α‐Tocopherol

α‐Tocopherol was synthesized from a chiral intermediate α‐hydroxy ester by means of two ring‐closing methods to yield the chromanol in 94 % diastereomeric excess.     

Low‐dimensional hydrogen‐bonded structures in the 1:1 and 1:2 proton‐transfer compounds of 4,5‐dichlorophthalic acid with the aliphatic Lewis bases triethylamine,diethylamine, n‐butylamine and piperidine

The structures of the proton‐transfer compounds of 4,5‐dichlorophthalic acid (DCPA) with the aliphatic Lewis bases triethylamine, diethylamine, n‐butylamine and piperidine, namely triethylaminium 2‐carboxy‐4,5‐dichlorobenzoate, C₆H₁₆N⁺·C₈H₃Cl₂O₄⁻, (I), diethylaminium 2‐carboxy‐4,5‐dichlorobenzoate, C₄H₁₂N⁺·C₈H₃Cl₂O₄⁻, (II), bis(butanaminium) 4,5‐dichlorobenzene‐1,2‐dicarboxylate monohydrate, 2C₄H₁₂N⁺·C₈H₂Cl₂O₄²⁻·H₂O, (III), and bis(piperidinium) 4,5‐dichlorobenzene‐1,2‐dicarboxylate monohydrate, 2C₅H₁₂N⁺·C₈H₂Cl₂O₄²⁻·H₂O, (IV), have been determined at 200 K. All compounds have hydrogen‐bonding associations, giving discrete cation–anion units in (I) and linear chains in (II), while (III) and (IV) both have two‐dimensional structures. In (I), a discrete cation–anion unit is formed through an asymmetric R₁²(4) N⁺—H...O₂ hydrogen‐bonding association, whereas in (II), chains are formed through linear N—H...O associations involving both aminium H‐atom donors. In compounds (III) and (IV), the primary N—H...O‐linked cation–anion units are extended into a two‐dimensional sheet structure via amide–carboxyl N—H...O and amide–carbonyl N—H...O interactions. In the 1:1 salts (I) and (II), the hydrogen 4,5‐dichlorophthalate anions are essentially planar with short intramolecular carboxyl–carboxyl O—H...O hydrogen bonds [O...O = 2.4223 (14) and 2.388 (2) Å, respectively]. This work provides a further example of the uncommon zero‐dimensional hydrogen‐bonded DCPA–Lewis base salt and the one‐dimensional chain structure type, while even with the hydrate structures of the 1:2 salts with the primary and secondary amines, the low dimensionality generally associated with 1:1 DCPA salts is also found.     

Unusual hydrate stabilization in the two‐dimensional layered structure of quinacrinium bis(2‐carboxy‐4,5‐dichlorobenzoate) tetrahydrate,a proton‐transfer compound of the drug quinacrine

The crystal structure of the hydrated proton‐transfer compound of the drug quinacrine [rac‐N′‐(6‐chloro‐2‐methoxyacridin‐9‐yl)‐N,N‐diethylpentane‐1,4‐diamine] with 4,5‐dichlorophthalic acid, C₂₃H₃₂ClN₃O²⁺·2C₈H₃Cl₂O₄⁻·4H₂O, has been determined at 200 K. The four labile water molecules of solvation in the structure form discrete ...O—H...O—H... hydrogen‐bonded chains parallel to the quinacrine side chain, the two N—H groups of which act as hydrogen‐bond donors for two of the water acceptor molecules. The other water molecules, as well as the acridinium H atom, also form hydrogen bonds with the two anion species and extend the structure into two‐dimensional sheets. Between these sheets there are also weak cation–anion and anion–anion π–π aromatic ring interactions. This structure represents the third example of a simple quinacrine derivative for which structural data are available but differs from the other two in that it is unstable in the X‐ray beam due to efflorescence, probably associated with the destruction of the unusual four‐membered water chain structures.     

Calculating the jet quenching parameter

Models of medium-induced radiative parton energy loss account for the strong suppression of high-p(T) hadron spectra in square root of (S)NN=200 GeV Au-Au collisions at BNL RHIC in terms of a single "jet quenching parameter" q. We observe that q can be given a model-independent, nonperturbative, quantum field theoretic definition in terms of the short-distance behavior of a particular lightlike Wilson loop. We then use the anti-de Sitter/conformal-field-theory correspondence to obtain a strong-coupling calculation of q in hot N=4 supersymmetric QCD, finding q(SYM)=26.69 square root of alpha(SYM)N(c)T(3) in the limit in which both N(c) and 4pialpha(SYM)N(c) are large. Thus, at strong coupling q is not proportional to the entropy density s, or to some "number density of scatterers" since, unlike the number of degrees of freedom, q does not grow like N(c)(2).     

Bi-Lipschitz parameterization of surfaces

We consider surfaces Z homeomorphic to the plane with complete, possibly singular Riemannian metrics. If we have _ZK⁺<2– for the positive and _ZK^–<C for the negative part of the integral curvature, then Z is L-bi-Lipschitz equivalent to ² with L depending only on >0 and C>0. This result implies a conjecture by J. Fu.Supported by NSF grant DMS-0200566.