Rational choice based on vague preferences

This paper studies vague preferences. In contrast to the concept of fuzzy preferences, the more general notion of vague preferences does not entail the assumption that the different degrees of preference are completely ordered. It is shown that the more general concept of a vague preference order arises very naturally in the context of aggregating a set of (exact) weak orders. Furthermore, necessary and sufficient conditions for the rationalizability of an exact choice function by means of a vague preference order are given. It turns out that in the general case these conditions are weaker than in the case where the degrees of preference are completely ordered. Consequently, there exist choice functions which are rationalizable by a general vague preference order but which are not rationalizable by a fuzzy preference order, or more generally by any vague preference order which induces a complete ranking of the degrees of preference. Examples of such choice functions are provided.This is a revised version of a discussion paper which was written at the University of Karlsruhe. I am grateful to Georg Bol and an anonymous referee for valuable comments. The research has been made possible by a fellowship of the DFG (Deutsche Forschungsgemeinschaft) which is also gratefully acknowledged.     

Kinetic theory of granular shear flow: Constitutive relations for the hard-disk model

A kinetic theory for the constitutive Theological relations of rapid granular shear flow of hard circular disks, characterized by a coefficient of restitutione and a surface roughness coefficient, is formulated. From a set of general constitutive equations for single-particle dynamical variables, the approximate expressions for the limit of small and large dimensionless dissipative parameterR _t are obtained. HereR _t is defined as the ratio /, where is the fluctuation of translational velocity from the mean flow velocity, is the diameter of a disk, and is the shear rate. At smallR _t the theoretical predictions can be compared with exact computer simulation results of granular dynamics that are also reported. The agreement between theory and simulation is better than expected; the present theory is accurate up to high packing density in this region ofR _t .     

The theoretical basis of column chromatography in multicomponent separations. Part 2: Peak capacity. Column systems with good selectivity

The selectivity of a column system, S, defined by equation 28, includes a variation coefficient (β) of plate number with capacity ratio which has a large influence on the peak capacity, as shown in Fig. 18. Some typical chromatograms are given. In order to predict S for a column system from Kovat's Index, equations 40 and 41 are given to calculate the constant of the carbon number rule for squalane at different temperatures or for different stationary phases. The specific retention value of heptane on squalane at different temperatures can be calculated from equation 42. The nonpolarity index, defined by equation 43, was used to calculate the retention value of heptane on various stationary phases. In liquid chromatography, the order of elution may be reversed by changing the composition of the eluent on the same chemically bonded silica (manufactured in China). The linear relations between the log retention values of different kinds of solutes or of a single solute on the silicas of different surface areas when using the same eluent are given.     

The systems Zr(Nb,Ti)(R)O2−δ, R=Yb, Ca—optimization of mixed conductivity and comparison with results of other systems (R=Y and Gd)

In this work we address the optimization of mixed conductivity in fluorite compounds based on zirconia. Phase relations of the new systems YbO_1.5-NbO_2.5-ZrO₂, and CaO-NbO_2.5-ZrO₂ are presented. The limit of the cubic defect fluorite phase in YbO_1.5-NbO_2.5-ZrO₂ closely resembles that of the system YO_1.5-NbO_2.5-ZrO₂, whilst in CaO-NbO_2.5-ZrO₂ is narrow extending to include composition Ca_0.255Nb_0.15Zr_0.595O_1.82 at 1500°C. The influence of dopant ion size, charge and composition on ionic conduction is assessed and parallels are drawn with the systems YO_1.5-NbO_2.5-ZrO₂ and YO_1.5-TiO₂-ZrO₂. Comparison of these results with published data on the Ti containing systems CaO-TiO₂-ZrO₂, GdO_1.5-TiO₂-ZrO₂ shows that the highest mixed conducting compositions can only be offered in the system YO_1.5-TiO₂-ZrO₂ out of all the systems here studied.     

The Ternary System Li2O-Al2O3-B2O3: Compounds and Phase Relations

The ternary system Li₂O-Al₂O₃-B₂O₃ is reinvestigated with solid-state reaction and X-ray powder diffraction technique to clarify some long-standing uncertainties. The phase relations are constructed based on the phase identifications of 51 ternary samples. Six ternary compounds, Li₂AlB₅O₁₀, LiAlB₂O₅, Li₃AlB₂O₆, Li₂AlBO₄, LiAl₇B₄O₁₇ and a compound with a composition close to 0.66Li₂O·0.06Al₂O₃·0.28B₂O₃, are observed or confirmed in this system, and the thermal stability of these ternary compounds is also discussed on the basis of DTA experimental results.     

Kinetic analysis of thermogravimetric data

Thermal decomposition of 6 complexes of the type AH[Cr(NCS)₄ (am)₂]· nH₂O is studied with derivatograph. The formation of Cr(NCS)₃ as a labile intermediate is presumed. For some decomposition stages kinetic parameters are derived. The kinetic compensation effect is discussed.     

H-, C-NMR and ethylene polymerization studies of zirconocene/MAO catalysts: effect of the ligand structure on the formation of active intermediates and polymerization kinetics

Using ¹H- and ¹³C-NMR spectroscopies, cationic intermediates formed by activation of L₂ZrCl₂ with methylaluminoxane (MAO) in toluene were monitored at Al/Zr ratios from 50 to 1000 (L₂ are various cyclopentadienyl (Cp), indenyl (Ind) and fluorenyl (Flu) ligands). The following catalysts were studied: (Cp-R)₂ZrCl₂ (R=Me, 1,2-Me₂, 1,2,3-Me₃, 1,2,4-Me₃, Me₄, Me₅, n-Bu, t-Bu), rac-ethanediyl(Ind)₂ZrCl₂, rac-Me₂Si(Ind)₂ZrCl₂, rac-Me₂Si(1-Ind-2-Me)₂ZrCl₂, rac-ethanediyl(1-Ind-4,5,6,7-H₄)₂ZrCl₂, (Ind-2-Me)₂ZrCl₂, Me₂C(Cp)(Flu)ZrCl₂, Me₂C(Cp-3-Me)(Flu)ZrCl₂ and Me₂Si(Flu)₂ZrCl₂. Correlations between spectroscopic and ethene polymerization data for catalysts (Cp-R)₂ZrCl₂/MAO (R=H, Me, 1,2-Me₂, 1,2,3-Me₃, 1,2,4-Me₃, Me₄, Me₅, n-Bu, t-Bu) and rac-Me₂Si(Ind)₂ZrCl₂ were established. The catalysts (Cp-R)₂ZrCl₂/AlMe₃/CPh₃⁺B(C₆F₅)₄⁻ (R=Me, 1,2-Me₂, 1,2,3-Me₃, 1,2,4-Me₃, Me₄, n-Bu, t-Bu) were also studied for comparison of spectroscopic and polymerization data with MAO-based systems. Complexes of type (Cp-R)₂ZrMe⁺←Me⁻-Al≡MAO (IV) with different [Me-MAO]⁻ counteranions have been identified in the (Cp-R)₂ZrCl₂/MAO (R=n-Bu, t-Bu) systems at low Al/Zr ratios (50-200). At Al/Zr ratios of 500-1000, the complex [L₂Zr(μ-Me)₂AlMe₂]⁺[Me-MAO]⁻ (III) dominates in all MAO-based reaction systems studied. Ethene polymerization activity strongly depends on the Al/Zr ratio (Al/Zr=200-1000) for the systems (Cp-R)₂ZrCl₂/MAO (R=H, Me, n-Bu, t-Bu), while it is virtually constant in the same range of Al/Zr ratios for the catalytic systems (Cp-R)₂ZrCl₂/MAO (R=1,2-Me₂, 1,2,3-Me₃, 1,2,4-Me₃, Me₄) and rac-Me₂Si(Ind)₂ZrCl₂/MAO. The data obtained are interpreted on assumption that complex III is the main precursor of the active centers of polymerization in MAO-based systems.     

Subsolidus phase relations and crystal structures of R-Ca-Cu-O (R=Nd, Sm, Gd, Tm) systems

The subsolidus phase relations of R₂O₃-CaO-CuO ternary systems (R=Nd, Sm, Gd, Tm) have been investigated by X-ray powder diffraction. All samples were synthesized at about 950° in air. There exists a ternary compound Ca_14−xR_xCu₂₄O₄₁ (x = 4 for R=Nd, Gd and x = 5 for R = Sm) and a ternary solid solution Ca_2+xR_2−xCu₅O₁₀ (R=Nd, Sm, Gd, Tm) with a wide composition range Δx of about 0.6. The compound Ca_14−xR_xCu₂₄O₄₁ possesses a layered orthorhombic structure and is isostructural to Sr_14−xCa_xCu₂₄O₄₁. The lattice parameters a and c of the compound are basically independent of the ionic radius of R, while the lattice parameter b and unit-cell volume V decrease substantially with the decrease of the ionic radii of R. The Ca_2+xR_2−xCu₅O₁₀ solid solution is isostructural to Ca_2+xY_2−xCu₅O₁₀, the structure of which is based on an orthorhombic “NaCuO₂-type” subcell containing infinite one-dimensional chains of edge-shared square planar cuprate groups crosslinked by the layered cations Ca and R that locate in the inter-chain tunnels.     

Subsolidus phase relations and crystal structure of compounds in the PrOx-CaO-CuO system

The subsolidus phase relations of the PrO_x-CaO-CuO pseudo-ternary system sintered at 950-1000°C have been investigated by X-ray powder diffraction. In this system, there exist one compound Ca₁₀Pr₄Cu₂₄O₄₁, one Ca₂Pr₂Cu₅O₁₀-based solid solution, seven three-phase regions and two two-phase regions. The crystal structures of Ca₁₀Pr₄Cu₂₄O₄₁ and Ca₂Pr₂Cu₅O₁₀-based solid solution have been determined. Compound Ca₁₀Pr₄Cu₂₄O₄₁ crystallizes in an orthorhombic cell with space group D_2h²⁰−Cccm, Z=4. Its lattice parameters are a=11.278(2) Å, b=12.448(3) Å and c=27.486(8) Å. The crystal structure of Ca₂Pr₂Cu₅O₁₀-based solid solution is an incommensurate phase based on the orthorhombic NaCuO₂ type subcell. The lattice parameters of the subcell of the Ca_2.4Pr_1.6Cu₅O₁₀ are a₀=2.8246(7) Å, b₀=6.3693(5) Å, c₀=10.679(1) Å, and those of the orthorhombic superstructure are with a=5a₀, b=b₀, c=5c₀. The Ca_2.4Pr_1.6Cu₅O₁₀ structure can also be determined by using a monoclinic supercell with space group C_2h⁵−P2₁/c, Z=4, a=5a₀, b=b₀, and β=104.79(1)° or 136.60(1)°, V=5a₀b₀c₀.