We address the problem of estimating the ratio q p where p is a density function and q is another density, or, more generally an arbitrary function. Knowing or approximating this ratio is needed in various problems of inference and integration often referred to as importance sampling in statistical inference. It is also closely related to the problem of covariate shift in transfer learning. Our...

In this paper, a link between the Hyers–Ulam stability and the Moore–Penrose inverse is established, that is, a closed operator has the Hyers–Ulam stability if and only if it has a bounded Moore–Penrose inverse. Meanwhile, the stability constant can be determined in terms of the Moore– Penrose inverse. Based on this result, some conditions for the perturbed operators having the Hyers– Ulam stab...

We investigate the properties of the operator ∆(σ∆·) : H0(Ω) → H−2(Ω), where σ is a given parameter whose sign can change on the bounded domain Ω. Here, H0(Ω) denotes the subspace of H2(Ω) made of the functions v such that v = ν · ∇v = 0 on ∂Ω. The study of this problem arises when one is interested in some configurations of the Interior Transmission Eigenvalue Problem. We prove that ∆(σ∆·) : H...

Chern insulators are periodic band insulators with the property that their projector onto the occupied bands have non-zero Chern number. Chern insulator with a homogeneous boundary display continuum spectrum that fills the entire insulating gap. The local density of states corresponding to this part of the spectrum is localized near the boundary, hence the terminology edge spectrum. An interest...

Let A and B be bounded linear operators on a Banach space such that ABA = A2 and BAB = B2. Then A and B have some spectral properties in common. This situation is studied in the present paper. 1. Terminology and motivation Throughout this paper X denotes a complex Banach space and L(X) the Banach algebra of all bounded linear operators on X. For A ∈ L(X), let N(A) denote the null space of A, an...

Let Ω be a nonempty open subset of R. A consequence of (i) is that there is no Fredholm operator between L(Ω) and L(Ω), 1 < p < ∞. A consequence of (ii) is that there is no Fredholm operator between L(Ω) and L∞(Ω). Proof. Put X0 = kerT and Y1 = T (X). Then dimX0 < ∞, codimY1 < ∞, and Y1 is closed in Y . Since X0 is finite dimensional, it has an algebraic topological complement X1. Then T |X1 : ...

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn : n ∈ N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn + Dn+2 ⊆ 2Dn+1 for every n ∈ N. The second part ...

exist and have no zero eigenvalue. A typical example for A(t) is the div-grad-curl operator on a 3-manifold twisted by a connection which depends on t. Atiyah et al proved that the Fredholm index of such an operator DA is equal to minus the “spectral flow” of the family {A(t)}t∈R. This spectral flow represents the net change in the number of negative eigenvalues of A(t) as t runs from −∞ to ∞. ...