New biorthogonal potential--density basis functions

We use the weighted integral form of spherical Bessel functions, and introduce a new analytical set of complete and biorthogonal potential--density basis functions. The potential and density functions of the new set have finite central values and they fall off, respectively, similar to $r^{-(1+l)}$ and $r^{-(4+l)}$ at large radii where $l$ is the latitudinal quantum number of spherical harmonics. The lowest order term associated with $l=0$ is the perfect sphere of de Zeeuw. Our basis functions are intrinsically suitable for the modeling of three dimensional, soft-centred stellar systems and they complement the basis sets of Clutton-Brock, Hernquist & Ostriker and Zhao. We test the performance of our functions by expanding the density and potential profiles of some spherical and oblate galaxy models.Comment: 8 pages, 6 figures, Accepted for publication in Monthly Notices of the Royal Astronomical Societ

On the potential functions for a link diagram

For an oriented diagram of a link $L$ in the 3-sphere, Cho and Murakami defined the potential function whose critical point, slightly different from the usual sense, corresponds to a boundary parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of $\pi_1(S^3 \setminus L)$. They also showed that the volume and Chern-Simons invariant of such a representation can be computed from the potential function with its partial derivatives. In this paper, we extend the potential function to a $\mathrm{PSL}(2,\mathbb{C})$-representation that is not necessarily boundary parabolic. Under a mild assumption, it leads us to a combinatorial formula for computing the volume and Chern-Simons invariant of a $\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold.Comment: 22 page

An embedding potential definition of channel functions

We show that the imaginary part of the embedding potential, a generalised logarithmic derivative, defined over the interface between an electrical lead and some conductor, has orthogonal eigenfunctions which define conduction channels into and out of the lead. In the case of an infinitely extended interface we establish the relationship between these eigenfunctions and the Bloch states evaluated over the interface. Using the new channel functions, a well-known result for the total transmission through the conductor system is simply derived.Comment: 14 pages, 2 figure

Zernike functions, rigged Hilbert spaces, and potential applications

Producción CientíficaWe revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1, 1) ⊕ su(1, 1). We show how they induce discrete as well as continuous bases that coexist in the framework of rigged Hilbert spaces. We also discuss some other interesting properties of Zernike polynomials and Zernike functions. One of the areas of interest of Zernike functions has been their applications in optics. Here, we suggest that operators on the spaces of Zernike functions may play a role in optical image processing

Representation of multivariate functions via the potential theory

In this paper, by the use of Potential Theory, some representation results for multivariate functions from the Sobolev spaces in terms of the double layer potential and the fundamental solution of Laplace's equation are pointed out. Applications for multivariate inequalities of Ostrowski type are also provided