A value on ′AN

We prove here the existence of a value (of norm 1) on the spaces N A and even A N, the closure in the variation distance of the linear space spanned by all games f, where is a non-atomic, non-negative finitely additive measure of mass 1 and f a real-valued function on [0,1] which satisfies a much weaker continuity at zero and one. 1991 Mathematics Subject Classification. 90A08, 90A07This research was in part supported by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Prime Ministers Science Policy Office.     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Some rings are hereditary rings

LetR be a bounded Noetherian Prime ring. The Asano-Michler theorem shows thatR is a bounded Dedekind ring if every prime ideal ofR is invertible. We provide a simple proof of the Asano-Michler theorem, and we suggest some possible generalizations. We also prove that if the proper residue rings ofR areQF-rings thenR is a bounded Dedekind ring, and generalize this result toLD-rings.     

Hydrothermal syntheses, characterizations and crystal structures of three new cadmium (II) amino-diphosphonates: effects of substitute groups on the structures of metal phosphonates

Hydrothermal reactions of cadmium(II) chloride with three amino-diphosphonic acids, C₆H₅CH₂N(CH₂PO₃H₂)₂ (H₄L¹), C₆H₅CH₂CH₂N(CH₂PO₃H₂)₂ (H₄L²) and 4-CH₃-C₆H₄CH₂N (CH₂PO₃H₂)₂) (H₄L³) resulted in three new metal amino-diphosphonates, namely, Cd(H₃L¹)₂, 1 Cd(H₃L²)₂·2H₂O 2 and Cd(H₃L³)₂3. In all three complexes, the Cd(II) ion is octahedrally coordinated by six phosphonate oxygen atoms from six ligands. Complexes 1 and 3 have a similar structure in which the CdO₆ octahedra are cross-linked by bridging ligands into a double chain along the c-axis, such double chains are further interlinked via hydrogen bonds between non-coordinated phosphonate oxygen atoms to form 〈100〉 and 〈200〉 layers with the phenyl groups of the ligands orientated toward the interlayer space. The structure of complex 2 features a 〈100〉 cadmium(II) diphosphonate layer. The effects of the substitute groups attached to the amine groups on the structures of the metal phosphonates are also discussed.     

Hydrothermal synthesis, structure and thermal stability of diamine templated layered uranyl-vanadates

Six new layered uranyl vanadates (NH₄)₂[(UO₂)₂V₂O₈] (1), (H₂EN)[(UO₂)₂V₂O₈] (2), (H₂DAP)[(UO₂)₂V₂O₈] (3), (H₂PIP)[(UO₂)₂(VO₄)₂].0,8H₂O (4), (H₂DMPIP)[(UO₂)₂V₂O₈] (5), (H₂DABCO)[(UO₂)₂(VO₄)₂] (6) were prepared from mild-hydrothermal reactions using 1,2-ethylenediamine (EN); 1,3-diaminopropane (DAP); piperazine (PIP); 1-methylpiperazine (MPIP); 1,4-diazabicyclo[2,2,2]octane (DABCO). The structures of 1, 4, 5 and 6 were solved using single-crystal X-ray diffraction data while the structural models of 2 and 3 were established from powder X-ray diffraction data. In compounds 1, 2, 3 and 5, the uranyl-vanadate layers are built from dimers of edge-shared UO₇ pentagonal bipyramids and dimers of edge-shared VO₅ square pyramids further connected through edge-sharing. In 1 and 3, the layers are identical to that occurring in the carnotite group of uranyl-vanadates. In 2 and 5, the V₂O₈ dimers differ in orientation leading to a new type of layer. The layers of compound 4 and 6 are built from chains of edge-shared UO₇ pentagonal bipyramids connected by VO₄ tetrahedra and are of uranophane-type anion topology. For the six compounds, the ammonium or organoammonium cation resides in the space between the inorganic layers. Crystallographic data: 1 monoclinic, space group P2₁/c with a=6.894(2), b=8.384(3), c=10.473(4) Å and β=106.066(5)°, 2 monoclinic, space group P2₁/a with a=13.9816(6), b=8.6165(3), c=10.4237(3) Å and γ=93.125(3)°, 3 orthorhombic, space group Pmcn with a=14.7363(8), b=8.6379(4) and c=10.4385(4) Å, 4 monoclinic, space group C2/m with a=15.619(2), b=7.1802(8), c=6.9157(8) Å and β=101.500(2)°, 5 monoclinic, space group P2₁/b with a=9.315(2), b=8.617(2), c=10.5246(2) Å and γ=114.776(2)°, 6 monoclinic, space group C2/m with a=17.440(2), b=7.1904(9), c=6.8990(8) Å and β=98.196(2)°.     

The mechanism of solvolysis of methyl perchlorate in hydroxylic solvents: calculation of the cavity terms

From standard Gibbs energies of transfer, corrected for cavity or nonelectrostatic effects, it is concluded that the solvolysis of methyl perchlorate takes place through a looser-than-usual S_N2 transition state with a charge separation of about 0.52 units.     

Determination of fluid--solid transitions in model protein solutions using the histogram reweighting method and expanded ensemble simulations

Protein crystallization conditions are usually identified by empirical screening methods because of the complexity of the process, such as the existence of nonequilibrium phases and the different crystal forms that may result from changes in solution conditions. Here the crystallization of a model protein is studied using computer simulation. The model consists of spheres that have both an isotropic interaction of short range and anisotropic interactions between patch-antipatch pairs. The free energy of a protein crystal is calculated using expanded ensemble simulations of the Einstein crystal, and NpT-Monte Carlo simulations with histogram reweighting are used to determine the fluid-solid coexistence. The histogram reweighting method is also used to trace out the complete coexistence curve, including multiple crystal phases, with varying reduced temperature, which corresponds to changing solution conditions. At a patch-antipatch interaction strength five times that of the isotropic interaction, the protein molecules form a stable simple cubic structure near room temperature, whereas an orientationally disordered face-centered-cubic structure is favored at higher temperatures. The anisotropic attractions also lead to a weak first-order transition between orientationally disordered and ordered face-centered-cubic structures at low temperature, although this transition is metastable. A complete phase diagram, including a fluid phase, three solid phases, and two triple points, is found for the six-patch protein model. A 12-patch protein model, consistent with the face-centered-cubic structure, leads to greater thermodynamic stability of the ordered phase. Metastable liquid-liquid phase equilibria for isotropic models with varying attraction tails are also predicted from Gibbs ensemble simulations.     

The formation of propylene fumarate oligomers for use in bioerodible bone cement composites

Poly(propylene fumarate) (PPF) oligomers were synthesized by step polymerization using bis(2-hydroxypropyl fumarate) or propylene bis(hydrogen maleate) as starting materials. Oligomers possessing identical degrees of polymerization (DP), but varying in their end group character (either hydroxyl or carboxyl) were first prepared and characterized, then used as part of a bone cement preparation consisting of oligomer, tricalcium phosphate, calcium carbonate, and methyl methacrylate. Compressive strength of the resulting composite appeared to be dependent on both the degree of polymerization of the PPF, and the nature of the oligomers' end groups.     

Structure Determination of a Complex Tubular Uranyl Phenylphosphonate, (UO(2))(3)(HO(3)PC(6)H(5))(2)(O(3)PC(6)H(5))(2).H(2)O, from Conventional X-ray Powder Diffraction Data

The three-dimensional structure of a complex tubular uranyl phosphonate, (UO(2))(3)(HO(3)PC(6)H(5))(2)(O(3)PC(6)H(5))(2).H(2)O, was determined ab initio from laboratory X-ray powder diffraction data and refined by the Rietveld method. The crystals belong to the space group P2(1)2(1)2(1), with a = 17.1966(2) ?, b = 7.2125(2) ?, c = 27.8282(4) ?, and Z = 4. The structure consists of three independent uranium atoms, among which two are seven-coordinated and the third is eight-coordinated. These metal atoms are connected by four different phosphonate groups to form a one-dimensional channel structure along the b axis. The phenyl groups are arranged on the outer periphery of the channels, and their stacking forces keep the channels intact in the lattice. The determination of this structure which contains 50 non-hydrogen atoms in the asymmetric unit, from conventional X-ray powder data, represents significant progress in the application of powder techniques to structure solution of complex inorganic compounds, including organometallic compounds.