A complete asymptotic expansion of power means

The complete asymptotic expansion of power means in terms of Bell polynomials is obtained. Some results recently obtained by M. Bjelica are generalized.     

The shape invariant of triangles and trigonometry in two-point homogeneous spaces

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces P ⁿ and P ⁿ (n2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces H ⁿ and H ⁿ (n2).     

Link maps and the geometry of their invariants

The geometry of two types of link homotopy invariants of a link map f:S^pS^qS^m is discussed. The first one is the -invariant which greatly generalizes the classical notion of linking number. The second one, the -invariant, is closely related to the linking behaviour of f|s^p with only the double point set of f|S^q, and therefore measures (to some extend) the obstruction to embedding S^q. These invariants are related by a Hopf invariant homomorphism. In many cases link maps are classified up to link homotopy here, and a setting is provided e.g. for future injectivity results for . Also the image of is studied, yielding an interesting double filtration of stable homotopy groups of spheres.     

Decomposition of multiattribute expected-utility functions

This paper integrates and extends the theory of the decomposition of multiattribute expected-utility functions based on utility independence. In a preliminary section, the standard decision model of expected utility is briefly discussed, including the fact that the decision maker's preference forlotteries with two outcomes determines the utility function uniquely. The decomposition possibilities of a utility function are captured by the concept ofautonomous sets of attributes, an affine separability of some kind known as generalized utility independence.Overlapping autonomous sets lead to biaffine-associative, i.e.multiplicative oradditive decompositions. The multiplicative representation shows that autonomy has strongerclosure properties than utility independence, for instance with respect to set-theoretic difference. Autonomy is also a concept with a wider scope since it applies to the decomposition of Boolean functions, games and a number of other topics in combinatorial optimization. This relationship to the well-known theory ofsubstitution decomposition in discrete mathematics also reveals a kind of discrete core behind the decomposition of utility functions. The entirety of autonomous sets can be represented by a compact data structure, the so-calledcomposition tree, which frequently corresponds to a natural hierarchy of attributes. Multiplicative/additive ormulti-affine functions correspond to the hierarchy steps. The known representation of multi-affine functions is shown to be given by aMoebius inversion formula. The entire approach has the advantage that it allows the application of more sophisticated representation methods on a detailed level, whereas it employs onlyfinite set theory andarithmetic on the main levels in the hierarchy.     

Assignment of the 5,6-Double-Bond Configuration in Iloprost and Isoiloprost from 13C-NMR Shifts Determined by 2D Methods

The configurations of the 5,6-double bond in the carbacyclins iloprost ( 3 ; (E)) and isoiloprost ( 4 ; (Z)) are based on a complete assignment of the ¹³C and ¹H resonances determined by 1D and 2D ¹³C-NMR and ¹H-NMR methods.     

13C- and 1H-NMR. Spectra of ortho-benzoquinones. On the assignment problem in 13C spectra

¹³C- and ¹H-NMR. spectra of ortho-benzoquinone 1 and its methyl derivatives have been analysed. By means of heteronuclear double resonance experiments it is shown that assignments given in the literature for the olefinic carbon resonances of 1 and of a series of substituted ortho-benzoquinones have to be inverted. ¹³C-carbonyl frequencies of various six-membered cyclic ketones are presented.     

Filiation structurale des phases de´rive´es de BaFeF5

Au sein d'un re´seau de type BaFeF₅ des chai?nes d'octae`dres fluore´sa`sommets communs peuvent disparai?tre en donnant naissance a des canaux dont l'occupation partielle ou totale par des files de cations Sr²⁺ et de couples d'anions F^?, conduit respectivement aux phases Sr₃(FeF₆)₂ ou Sr₅T₃F₁₉ (T = Cr, Co, Ga). Une discussion cristallochimique permet d'expliquer l'existence d'un domaine de solutions solides au sein du diagramme ternaire SrF₂BaF₂FeF₃.