Crystal and Molecular Structure of 3-Methoxy-estra-1,3,5(10)-trien-17β-ol

The crystal structure determination of the title compound was undertaken to determine the conformational flexibility of the steroid skeleton. 3-Methoxy-estra-1,3,5(10)-trien-17β-ol (C₁₉H₂₆O₂) crystallizes in the space group P2₁2₁2₁ with Z = 12, a = 12.589(2), b = 16.274(5), and c = 23.535(6) Å. The structure was solved by direct methods and refined to R = 0.056. Three symmetry independent molecules having different molecular conformations were observed in the crystal. The most flexible region of the molecules is the B ring which adopts following conformations: 7α,8β-half-chair, 8β-sofa, and an intermediate form between half-chair and sofa.     

A Representation of Quantum Measurement in Order-Unit Spaces

A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lüders-von Neumann quantum measurement as a probability conditionalization rule. A major result shows that the operator algebras must be replaced by order-unit spaces with some specific properties in the generalized approach, and it is analyzed under which conditions these order-unit spaces become Jordan algebras. An application of this result provides a characterization of the projection lattices in operator algebras.     

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.     

A metric on the manifold of immersions and its Riemannian curvature

E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n–1) dimensional manifold into ⁿ , and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.     

Cut sharing for multistage stochastic linear programs with interstage dependency

Multistage stochastic programs with interstage independent random parameters have recourse functions that do not depend on the state of the system. Decomposition-based algorithms can exploit this structure by sharing cuts (outer-linearizations of the recourse function) among different scenario subproblems at the same stage. The ability to share cuts is necessary in practical implementations of algorithms that incorporate Monte Carlo sampling within the decomposition scheme. In this paper, we provide methodology for sharing cuts in decomposition algorithms for stochastic programs that satisfy certain interstage dependency models. These techniques enable sampling-based algorithms to handle a richer class of multistage problems, and may also be used to accelerate the convergence of exact decomposition algorithms. Research leading to this work was partially supported by the Department of Energy Contract DE-FG03-92ER25116-A002; the Office of Naval Research Contract N00014-89-J-1659; the National Science Foundation Grants ECS-8906260, DMS-8913089; and the Electric Power Research Institute Contract RP 8010-09, CSA-4O05335. This author's work was supported in part by the National Research Council under a Research Associateship at the Naval Postgraduate School, Monterey, California.     

Syntheses and Crystal Structures of Potassium–Lanthanoid(III) Complexes with the 1,2‐Benzenedisulfonate Ligand

The complexes [K(H₂O)₂LnL₂] (Ln = La or Nd; L = 1,2‐benzenedisulfonate) and [K(H₂O)Yb(H₂O)₄L₂] were initially isolated fortuitously from attempts to prepare the corresponding Ln₂L₃ complexes from Ln₂O₃ and H₂L in water. Indeed the bulk products from these reactions have the composition Ln₂L₃. Subsequently, deliberate syntheses by reacting equimolar amounts of Ln₂L₃ with K₂L in water gave the complexes in good yield. X‐ray crystal structures of [K(H₂O)₂LnL₂] (Ln = La or Nd) showed the complexes to be isostructural with a two dimensional polymeric network structure in which LnL₂ units are linked into chains crosslinked by potassium ions. Each Ln is nine coordinate with solely sulfonate oxygen donor atoms. Between adjacent lanthanoid ions there are three different types of sulfonate bridges and two examples of each. Most noteworthy is highly unsymmetrical bridging through μ‐η²‐sulfonate oxygen atoms. Consequently, one Ln–O bond is ca. 0.5 Å longer than the other eight. Potassium is nine‐coordinate with seven sulfonate oxygen atoms and two aqua ligands, and surprisingly <K–O(sulfonate)> is much longer than <K–O(H₂O)>. Pairs of potassium ions are linked by two μ‐η²‐sulfonate oxygen atoms, which are unsymmetrically bridging. The structure of [K(H₂O)Yb(H₂O)₄L₂] comprises discrete tetranuclear units containing two independent ytterbium ions, each coordinated by four water molecules and two chelating (via seven membered rings) disulfonate ligands, and two potassium ions, each coordinated by six sulfonate oxygen atoms and a water molecule. For each potassium, four of the coordinated sulfonate oxygen atoms are from sulfonate ligands bonded to one ytterbium atom and two from sulfonate ligands attached to the other ytterbium atom. In contrast to the Nd and La complexes, <K–O(sulfonate)> is shorter than <K–O(H₂O)>.     

Distinct dynamic behaviors of water molecules in hydrated pores

Water molecules confined inside narrow pores are of great importance in understanding the structure, stability, and function of water channels. Here we report that besides the H-bonding water that structures the pore, the permanent presence of a significant, fast-moving fraction of incompletely H-bonded water molecules inside the pore should control the free entry and exit of water. This is achieved by means of complementary DSC and solid-state NMR studies. We also present compelling evidence from X-ray diffraction data that the cluster formed by six water molecules in the most stable cage-like structure is sufficiently hydrophobic to be stably adsorbed in a nonpolar environment.     

Heavy Alkali Metal Manganate Complexes: Synthesis,Structures and Solvent-Induced Dissociation Effects

Rare examples of heavier alkali metal manganates [{(AM)Mn(CH₂SiMe₃)(N^‘Ar)₂}_∞] (AM=K, Rb, or Cs) [N^‘Ar=N(SiMe₃)(Dipp), where Dipp=2,6-iPr₂-C₆H₃] have been synthesised with the Rb and Cs examples crystallographically characterised. These heaviest manganates crystallise as polymeric zig-zag chains propagated by AM⋅⋅⋅π-arene interactions. Key to their preparation is to avoid Lewis base donor solvents. In contrast, using multidentate nitrogen donors encourages ligand scrambling leading to redistribution of these bimetallic manganate compounds into their corresponding homometallic species as witnessed for the complete Li - Cs series. Adding to the few known crystallographically characterised unsolvated and solvated rubidium and caesium s-block metal amides, six new derivatives ([{AM(N^‘Ar)}_∞], [{AM(N^‘Ar)⋅TMEDA}_∞], and [{AM(N^‘Ar)⋅PMDETA}_∞] where AM=Rb or Cs) have been structurally authenticated. Utilising monodentate diethyl ether as a donor, it was also possible to isolate and crystallographically characterise sodium manganate [(Et₂O)₂Na(ⁿBu)Mn[(N^‘Ar)₂], a monomeric, dinuclear structure prevented from aggregating by two blocking ether ligands bound to sodium.