Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.     

L-fuzzy quantifiers of type determined by fuzzy measures

The aim of this paper is, first, to introduce two new types of fuzzy integrals, namely, -fuzzy integral and →-fuzzy integral. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Second, using these integrals, two classes of monadic L-fuzzy quantifiers of type 1 are defined. These L-fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like “all”, “some”, “many”, “none”, “at most half”, etc. Several semantic properties of these L-fuzzy quantifiers are studied.     

Computing the mean and the variance of the cedent’s share for largest claims reinsurance covers

We present mathematical results allowing one to evaluate the moments of order 1 and 2 of the cedent’s share in the framework of reinsurance treaties based on ordered claim sizes. These results consist of closed analytical formulas that do not involve any approximation procedure. This is illustrated by numerical examples when the claim number has the Poisson or the negative binomial distribution, and the claim cost has the exponential or the Pareto distribution.     

Valuation and hedging of participating life-insurance policies under management discretion

The valuation and hedging of participating life insurance policies, also known as with-profits policies, is considered. Such policies can be seen as European path-dependent contingent claims whose underlying security is the investment portfolio of the insurance company that sold the policy. The fair valuation of these policies is studied under the assumption that the insurance company has the right to modify the investment strategy of the underlying portfolio at any time. Furthermore, it is assumed that the issuer of the policy does not setup a separate portfolio to hedge the risk associated with the policy. Instead, the issuer will use its discretion about the investment strategy of the underlying portfolio to hedge shortfall risks. In that sense, the insurer’s investment portfolio serves simultaneously as the underlying security and as the hedge portfolio. This means that the hedging problem can not be separated from the valuation problem. We investigate the relationship between risk-neutral valuation and hedging of these policies in complete and incomplete financial markets.     

Forgetting the initial distribution for Hidden Markov Models

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the nonlinear state space model and the stochastic volatility model.     

(1)H and (13)C NMR spectral data for a tricyclic derivative of a Diels-Alder adduct

A complete NMR analysis with full assignment for (1)H and (13)C NMR spectral data for 5-(acetyloxy)-3-hydroxy-9,10-dimethoxy-6-oxo-11-oxatricyclo[6.2.1.0(2,7)]undec-2-yl acetate is described. This compound was prepared by rapid hydrogenation of the unstable Diels-Alder adduct obtained from the reaction between 3,4-dimethoxyfuran and 2,5-diacetoxy-1,4-benzoquinone. Full homonuclear hydrogen coupling constants measurements and molecular mechanics calculations were performed for the determination of the relative stereochemistry.     

Synthesis, crystal structure and charge distribution of Na7As11O31: An oxygen-deficient layered sodium arsenate

A new sodium arsenate with layer structure has been synthesized and its crystal structure solved and refined by single-crystal X-ray diffraction. The crystal is trigonal, space group , a=11.199(3) Å, c=5.411(2) Å, V=587.80(3) Å³, Z=1; the refinement converged to R=0.0282 and wR=0.0751 for 590 reflections with (I)>2sigma(I). The structural model gives the formula Na₇As₁₁O₃₂, which would be non-neutral; besides, the structural model is not validated by the charge distribution (CD) analysis, which gives an unsatisfactory agreement on the computed charges of the cations. The CD analysis suggest incomplete (5/6) occupation of the O5 site, which leads to the deficiency of an oxygen atom per unit cell and to formula Na₇As₁₁O₃₁: this new structural model corresponds to a neutral compound, is validated by the CD analysis, and results in better displacement parameters for O5 than its non neutral counterpart. The (001) anionic layers are built up from corner and edge sharing of As1 and As2 distorted octahedra and As3 distorted tetrahedra, the sodium cations playing the role of interlayer cations. The effects of the oxygen deficiency on the crystal structure are discussed.     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Moduli spaces and multiple polylogarithm motives

In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors A and B in the moduli spaces of n-pointed stable curves of genus 0, and prove that the cohomology of the pair is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple ζ-values. Then we apply our construction to the dilogarithm and calculate the period matrix which turns out to be same with the canonical one of Deligne.     

On Polynomially Normal Lattice Configurations

In this paper we extend Champernowne’s construction of a normal sequence in base b to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the constructed configuration with an arbitrary polynomial curve in the plane is a normal sequence in base b.