Quantization of gauge fields through fibre bundle multiconnectivity

A geometric model for the quantum nature of interaction fields is proposed. We utilize a trivial fibre bundle whose typical fibre has a multiconnectivity characterized by a discrete group Γ. By seeing Γ as a gauge group with global action on each fibre, we show that the corresponding field strength is non-zero only on the future part of the light cone whose vertex is at the interaction point. When the interaction is submitted to the symmetries of a Lie group G, we consider the gauge group G x Γ. The field strength of the gauge having this group includes a term expressing the quantization of the interaction field described by G. This geometric interpretation of quantization makes use of topological arguments similar to those applied to explain the Aharonov-Bohm effect. Two examples show how this interpretation applies to the cases of electromagnetic and gravitational fields.      

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by [De Vylder and Goovaerts, 1999] and [De Vylder and Goovaerts, 2000]. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

Metric and isoperimetric problems in symplectic geometry

Our first result is a reduction inequality for the displacement energy. We apply it to establish some new results relating symplectic capacities and the volume of a Lagrangian submanifold in a number of different settings. In particular, we prove that a Lagrange submanifold always bounds a holomorphic disc of area less than , where is some universal constant. We also explain how the Alexandroff-Bakelman-Pucci inequality is a special case of the above inequalities. Our inequality on displacement of reductions is also applied to yield a relation between length of billiard trajectories and volume of the domain. Two simple results concerning isoperimetric inequalities for convex domains and the closure of the symplectic group for the norm are included.
     

Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions

We survey the properties of two parameters introduced by C. Ding and the author for quantifying the balancedness of vectorial functions and of their derivatives. We give new results on the distribution of the values of the first parameter when applied to F + L, where F is a fixed function and L ranges over the set of linear functions: we show an upper bound on the nonlinearity of F by means of these values, we determine then the mean of these values and we show that their maximum is a nonlinearity parameter as well, we prove that the variance of these values is directly related to the second parameter. We briefly recall the known constructions of bent vectorial functions and introduce two new classes obtained with Gregor Leander. We show that bent functions can be used to build APN functions by concatenating the outputs of a bent (n, n/2)-function and of some other (n, n/2)-function. We obtain this way a general infinite class of quadratic APN functions. We show that this class contains the APN trinomials and hexanomials introduced in 2008 by L. Budaghyan and the author, and a class of APN functions introduced, in 2008 also, by Bracken et al.; this gives an explanation of the APNness of these functions and allows generalizing them. We also obtain this way the recently found Edel?CPott cubic function. We exhibit a large number of other sub-classes of APN functions. We eventually design with this same method classes of quadratic and non-quadratic differentially 4-uniform functions.     

New variants of bundle methods

In this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization, convex—concave games and variational inequalities. We outline the ideas underlying these methods and present rate-of-convergence estimates.Corresponding author.     

Etude de l'algorithme de Rémès en l'absence de condition de Haar

Summary Using a geometric interpretation, the unicity of a best approximation and the convergence of the Remes-algorithm without the Haar-condition are investigated (in the case of the Tchebycheff-approximation of a continuous function by a linear combination of two continuous functions). Replacing the Haar-condition by a weaker one which is generally satisfied, one obtains a convergence theorem that is weaker than the classical convergence theorem but is sufficient for the applicatons. A generalization of these results is indicated.     

Internal Nucleophilic Termination in Acid‐Mediated Polyene Cyclizations. Part 3

Treatment of the unsaturated allenic alcohols (E)‐ 7 , (Z)‐ 7, 10, 13 , and 19 with an excess of FSO₃H in 2‐nitropropane at ?90° to ?30° afforded, in 68–85% yield, diastereoisomer mixtures of racemic tricyclic ethers 14a – d and 20a – d , respectively (Schemes 3 and 5), with high stereoselectivity (see Table and Scheme 6). These stereospecific transformations represent the first reported examples of an acid‐mediated polyene cyclization, in which an alkene is the initiating group and an allenic alcohol serves as the internal terminator. In close analogy to our earlier work, a nonsynchronous process is postulated, whereby the stereochemical course of cyclization is directed by the conformational structure of an intermediate cyclohexyl cation (see Schemes 3 and 6). In addition, the organoleptic properties of 14c and 20c , racemic didehydro and methyl didehydro analogues, respectively, of the known odorant Ambrox^® ((?)‐ 4f ), are briefly discussed.