Trace inequalities involving Hermitian matrices

Some trace inequalities for Hermitian matrices and matrix products involving Hermitian matrices are presented.     

X-ray structure analysis of 3β-hydroxy-4-(1,4-oxazin-4-yl)-androstane (C31H54NO2)

The crystal structure of 3β-hydroxy-4-(1,4-oxazin-4-yl)-androstane (C₃₁H₅₄NO₂) has been determined by X-ray crystallographic techniques. The compound crystallizes in the orthorhombic space group P2₁2₁2 with the following unit-cell parameters: a = 7.124(1) Å, b = 10.127(1) Å, c = 40.660(1) Å, V = 2933.4(1) ų, and Z = 4. The structure has been solved by direct methods and refined to an R factor of 0.067. Three six-membered rings, A, C, and E, exist in the chair conformation, while the ring B adopts a distorted half-chair conformation. The five-membered ring, D, has a distorted envelope conformation. The crystal structure is stabilized by strong intermolecular O-H...O hydrogen bonds.     

Synthesis and structure determination of 7a-Aza-B-homostigmast-5-eno [7a, 7-d] tetrazole-3β-yl chloride (C29H47N4Cl)

7a-Aza-B-homostigmast-5-eno [7a, 7-d] tetrazole-3β-yl chloride (C₂₉H₄₇N₄Cl) was synthesized for its crystallographic analysis and to investigate the role of intra- and intermolecular interactions in steroids. It crystallizes in the monoclinic space group C2 with unit cell parameters: a = 38.481(2), b = 6.661(3), c = 11.111 (6) ?, β = 94.49 (4)^o; λ( MoKα) = 0.71069 ?, V = 2839(2) ?³, and Z = 4. The structure has been solved by direct methods using X-ray diffraction techniques. The final reliability index for the computed structure is 0.0597 for 1252 observed reflections. Except the five-membered ring, all other rings of the steroid nucleus exist in non-planar conformations. The structure is stabilized by C–H···N intermolecular interaction.Supplementary material CCDC-267926 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge at www.ccdc.cam.an.uk/uk/conts/retrieving.html or from the Cambridge Crystallographic Data Centre (CCDC), 12 Union Road, Cambridge CB2 1EZ, United Kingdom; Fax: $+$44(0) 1223-336033; e-mail: deposit@ccdc.cam.ac.uk.     

Structure of a liquid crystal of 4-cyano-4′-n-octyloxybiphenyl

The structure of a liquid crystal of 4-cyano-4′-n-octyloxybiphenyl (C₂₁H₂₅NO) is determined by X-ray diffraction analysis. The compound crystallizes in the triclinic crystal system with unit cell parameters a = 7.322(1) Å, b = 12.693(3) Å, c = 20.393(2) Å, α = 92.45(1)°, β = 99.96(1)°, γ = 99.35(2)°, and space group $P\bar 1$ . The structure is solved by the direct method and refined to R = 0.057. Two independent molecules are located in the asymmetric unit. No short intermolecular contacts are observed in the crystal packing.     

Crystallographic Analysis of Acetyl β ‐boswellic acid

The crystal structure of the title compound (3 α ‐acetoxy‐urs‐12‐en‐24‐oic acid, C₃₂H₅₀O₄) has been determined by X‐ray crystallographic techniques. The compound crystallizes into orthorhombic space group P2₁2₁2₁ with unit cell parameters : a = 12.773(2), b=16.381(4), c=27.929(7)Å. The structure has been solved by direct methods and refined to R = 0.054 for 4930 observed reflections. The structure contains two crystallographically independent molecules in the asymmetric unit which are almost identical in geometry. Rings A, B, D and E have chair conformations while ring C assumes a sofa conformation in both the molecules. The molecules in the structure are linked together by intra‐ and intermolecular O‐H…= and C‐H…O hydrogen bonds.     

3 β ‐Acetoxy‐5 α ‐cholestan‐6‐one: A Steroid

The crystal and molecular structure of 3 β ‐Acetoxy‐5 α ‐cholestan‐6‐one has been determined by X‐ray crystallographic techniques. The compound crystallizes in the space group P2₁ with cell parameters : a = 13.060(3), b=6.299(2), c=17.152(6)Å; β =96.47(3)o, V = 1402.02ų, Z = 2, R = 0.072 for 1921 observed reflections. The six‐membered rings (A, B and C) exist in the chair conformations while the five‐membered ring‐d assumes half‐chair. All rings of the steroid skeleton are trans connected.     

Crystallographic Investigations of 3β‐acetoxy‐5α‐cholestan‐6‐one‐semicarbazone ‐ A Steroid

The crystal structure of 3b‐acetoxy‐5a‐cholestan‐6‐one‐semicarbazone (C₃₀H₅₁O₃N₃) has been determined by X‐ray diffraction methods. It crystallizes in the orthorhombic space group P2₁2₁2₁ with cell parameters a = 11.641(1), b = 16.552(1) c = 31.181(4) Å and Z = 8. The structure has been refined to an R‐value of 0.050 for 4407 observed reflections. Two molecules in the asymmetric unit have been observed. In both the crystallographically independent molecules, all the three six‐membered rings (A, B and C ) of steroid nucleus exist in chair conformation, while the five‐ membered ring D exists in 13β distorted‐envelope in molecule‐I and 13β, 14α half‐chair conformation in molecule‐II. Three intermolecular N‐H … O hydrogen bonds have been observed.     

An organocatalytic direct Mannich-cyclization cascade as [3+2] annulation: asymmetric synthesis of 2,3-substituted pyrrolidines

A new method for asymmetric synthesis of 2,3-substituted pyrrolidines from N-PMP aldimines and succinaldehyde via formal [3+2] cycloaddition is reported. This reaction involves proline catalyzed direct Mannich reaction and acid catalyzed reductive cyclization with high yields (up to 78%) and excellent enantioselectivities (up to >99%).