On the construction of copulas and quasi-copulas with given diagonal sections

We study a method, which we call a copula (or quasi-copula) diagonal splice, for creating new functions by joining portions of two copulas (or quasi-copulas) with a common diagonal section. The diagonal splice of two quasi-copulas is always a quasi-copula, and we find a necessary and sufficient condition for the diagonal splice of two copulas to be a copula. Applications of this method include the construction of absolutely continuous asymmetric copulas with a prescribed diagonal section, and determining the best-possible upper bound on the set of copulas with a particular type of diagonal section. Several examples illustrate our results.     

Multivariate copulas, quasi-copulas and lattices

We investigate some properties of the partially ordered sets of multivariate copulas and quasi-copulas. Whereas the set of bivariate quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind-MacNeille completion of the set of bivariate copulas, we show that this is not the case in higher dimensions.     

Extreme value attractors for star unimodal copulasAttracteurs de valeurs extrêmes pour les copules 2-unimodales

We determine maximum attractors for copulas star (or 2-) unimodal (about a point $\text{(a,b)∈}\text{R}{}^2$). If (a,b)≠(1,1) these attractors form a two-parameter family of copulas extending that of Cuadras–Augé, whereas if (a,b)=(1,1) they cover all maximum value copulas. We also examine the relationship between unimodality and Archimax copulas. To cite this article: I. Cuculescu, R. Theodorescu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 689–692.     

A Characterization of Quasi-copulas

The notion of quasi-copula was introduced by C. Alsina, R. B. Nelsen, and B. Schweizer (Statist. Probab. Lett.(1993), 85–89) and was used by these authors and others to characterize operations on distribution functions that can or cannot be derived from operations on random variables. In this paper, the concept of quasi-copula is characterized in simpler operational terms and the result is used to show that absolutely continuous quasi-copulas are not necessarily copulas, thereby answering in the negative an open question of the above mentioned authors.     

Archimax copulas and invariance under transformations

Copulas which are invariant under transformations by means of increasing bijections on the unit interval are investigated, and the relationship to maximum attractors and Archimax copulas is discussed. To cite this article: E.P. Klement et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Intervals of 1-Lipschitz aggregation operators,quasi-copulas,and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections. A significant part of this work was done during a visit of the second author at the Johannes Kepler University, Linz (Austria). The second author was supported by the grant VEGA 1/3012/06 and by the Science and Technology Assistance Agency (Contract No. APVT-20-003204). Both authors would like to thank the anonymous referee whose comments (including the two copulas C ₁ and C ₂ given in the Conclusion) not only solved the originally stated open problem in the negative, but also allowed them to formulate two more interesting open problems.     

Best-possible bounds on the set of copulas with given degree of non-exchangeability

We establish best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability. These bounds are shown to be best-possible bounds as well for the set of copulas with given degree of non-exchangeability, and, consequently, also on the set of bivariate distribution functions of continuous random variables with given margins and given degree of non-exchangeability. Non-exchangeability of a (quasi-)copula is measured in the sense of Nelsen, i.e.?proportional to the maximal absolute difference between this (quasi-)copula and its transpose.     

Intervals of 1-Lipschitz aggregation operators, quasi-copulas, and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections.     

Copules gaussiennes dans les chaînes triplet partiellement de Markov

Hidden Markov chains, which are widely used in different data restoration problems, have recently been generalised to pairwise partially Markov chains, in which the hidden chain is no longer necessarily Markovian and the distribution of the observed chain, conditional on the hidden one, is of any form. First, we show the applicability of the models in the Gaussian case, with a particular attention to long range correlation noises. Second, we show that the use of copulas allows one to take into account any other form of marginal distributions of the observed chain, conditionally to the hidden one. We end by extending the latter model to a triplet partially Markov chain case. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Flipping and cyclic shifting of binary aggregation functions

We introduce two types of transformations of random variables, called flipping and cyclic shifting. As these transformations preserve monotonicity at the level of univariate cumulative distribution functions, they can be used to develop corresponding coordinate-wise transformations of binary aggregation functions. We lay bare the admissibility of these transformations, i.e. the necessary and sufficient conditions under which they result in a binary aggregation function. We investigate which additional properties, such as the 1-Lipschitz property and 2-increasingness, entail these admissibility conditions. Moreover, we point out which of these properties are preserved under flipping and/or cyclic shifting. Interestingly, quasi-copulas remain quasi-copulas under flipping, while copulas remain copulas under flipping as well as under cyclic shifting.     

Baire category and zero sets

Quasi-all continuous functions have a zero set which is a perfect Kronecker set with Hausdorff dimension zero. To cite this article: T. Körner, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Ordinal sums and idempotents of copulas

We prove that the ordinal sum of n-copulas is always an n-copula and show that every copula may be represented as an ordinal sum, once the set of its idempotents is known. In particular, it will be shown that every copula can be expressed as the ordinal sum of copulas having only trivial idempotents. As a by-product, we also characterize all associative copulas whose n-ary forms are n-copulas for all n.     

Exponential rates of convergence by an iteration technique

The goal of this Note is to introduce a technique leading to a convergence of exponential type for the solution of problems set in cylinders becoming unbounded in some directions. To cite this article: M. Chipot, K. Yeressian, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Affability of Euclidean tilings

We prove that every minimal equivalence relation on a Cantor set arising from the continuous hull of an aperiodic and repetitive Euclidean tiling is affable. To cite this article: F. Alcalde Cuesta et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Représentation du cône polaire des fonctions convexes et applicationsRepresentation of the polar cone of convex functions and applications

Following Y. Brenier, we give a representation of the polar cone of the set K of the gradient of convex functions, implying the set of measure-preserving maps. This can also be formulated in terms of doubly stochastic measures, and has a geometrical characterization.We deduce an Euler–Lagrange equation and regularity results for some minimization problems in the set K. To cite this article: G. Carlier, T. Lachand-Robert, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 571–576.     

Un théorème de représentation des solutions de viscosité d'une équation d'Hamilton–Jacobi–Bellman ergodique dégénérée sur le tore

We consider an ergodic Hamilton–Jacobi–Bellman equation coming from a stochastic control problem in which there are exactly k points where the dynamics vanishes and the Lagrangian is minimal. Under a stabilizability assumption, we state that the solutions of the ergodic equation are uniquely determined by their value on these k points, and that the set of solutions is sup-norm isometric to a non-empty closed convex set whose dimension is less or equal to k. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the asymptotic behaviour of elliptic problems with periodic data

We study the asymptotic behaviour of the solution of elliptic problems with periodic data when the size of the domain on which the problem is set becomes unbounded. To cite this article: M. Chipot, Y. Xie, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A density result for the variation of a material with respect to small inclusions

We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ?, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ? in $?=0$. In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials. To cite this article: J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

An optimization problem related to the modeling of atmospheric inorganic aerosols

A mathematical model for the computation of chemical equilibrium of atmospheric inorganic aerosols is proposed. The equilibrium is given by the minimum of the Gibbs free energy for a system involving an aqueous phase, a gas phase and solid salts. A primal-dual method solving the Karush–Kuhn–Tucker conditions is detailed. An active set/Newton method permits the computation of the minimum and track solid salts at the equilibrium. To cite this article: N.R. Amundson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the summability of the formal solutions for some PDEs with irregular singularity

In this Note, we consider some classes of nonlinear partial differential equations with regular singularity with respect to t=0 and irregular one with respect to x=0. Our purpose is to establish a result which is similar to the k-summability property, known in the case of singular ordinary differential equations. We can prove that, except at most a countable set, the formal solution is Borel summable or k-summable with respect to x in all other directions. To cite this article: Z. Luo et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).