Xavier Mary

Generalized inverses modulo ℋ in semigroups and rings

The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.

The inverse along a lower triangular matrix

Abstract In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions.

Splines with non positive kernels

Non parametric regression methods can be presented in two main clusters. The one of smoothing splines methods requiring positive kernels and the other one known as Nonparametric Kernel Regression allowing the use of non positive kernels such as the Epanechnikov kernel. We propose a generalization of the smoothing spline method to include kernels which are still symmetric but not positive semi denite (they are called indenite). The general relationship between smoothing splines, Reproducing Kernel Hilbert Spaces (RKHS) and positive kernels no longer exists with indenite kernels. Instead the splines are associated with functional spaces called Reproducing Kernel Krein Spaces (RKKS) endowed with an indenite inner product and thus not directly associated with a norm. Smoothing splines in RKKS have many of the interesting properties of splines in RKHS, such as orthogonality, projection and representer theorem. We show that smoothing splines can be dened in RKKS as the regularized solution of the interpolation problem. Since no norm is available in an RKKS, Tikhonov regularization cannot be dened. Instead, we propose the use of conjugate gradient type iterative methods, with early stopping as a regularization mechanism. Several iterative algorithms are collected which can be used to solve the optimization problems associated with learning in indenite spaces. Some preliminary experiments with indenite kernels for spline smoothing reveal the computational eciency of this approach.

Centralizer's applications to the inverse along an element

In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements. Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution.

Reverse Order Law for the Group Inverse in Semigroups and Rings

In this paper, we provide equivalent conditions for the two-sided reverse order law for the group inverse (ab)# = b # a # and (ba)# = a # b #, in semigroups and rings. Moreover, we prove that, under finiteness conditions, these conditions are also equivalent with the one-sided reverse order law (ab)# = b # a #.

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

Theory of subdualities

We present a new theory of dual systems of vector spaces that extends the existing notions of reproducing kernel Hilbert spaces and Hilbert subspaces. In this theory, kernels (understood as operators rather than kernel functions) need not be positive or self-adjoint. These dual systems, called subdualities, enjoy many properties similar to those of Hilbert subspaces and include the notions of Hilbert subspaces or Kreîn subspaces as particular cases. Some applications to Green operators or invariant subspaces are given.

Targeted Learning Using Adaptive Survey Sampling

Consider the following situation: we wish to build a confidence interval (CI) for a real-valued pathwise differentiable parameter Ψ evaluated at a law P0, ψ0 ≡ Ψ(P0), from a data set O1, …, O N of independent random variables drawn from P0 but, as is often the case nowadays, N is so large that we will not be able to use all data. To overcome this computational hurdle, we decide (a) to select n among N observations randomly with unequal inclusion probabilities and (b) to adapt TMLE from the smaller data set that results from the selection.

Weak inverses of products — Cline’s formula meets Jacobson lemma

We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommuting weak inverses, in semigroups and general rings. In particular, we provide various isomorphisms between the (one-sided, commuting and bicommuting) weak inverses of ab and those of ba (or ab and cd when ab and cd satisfy additional identities).