Abstract:
We present sampling theorems for reproducing kernel Banach spaces on Lie groups. Recent approaches to this problem rely on integrability of the kernel and its local oscillations. In this paper we replace the integrability conditions by requirements on the derivatives of the reproducing kernel. The results are then used to obtain frames and atomic decompositions for Banach spaces of distributions stemming from a cyclic representation, and it is shown that this is particularly easy, when the cyclic vector is a G{\aa}rding vector for a square integrable representation.

Abstract:
A fully irreducible outer automorphism phi of the free group F_n of rank n has an expansion factor which often differs from the expansion factor of the inverse of phi. Nevertheless, we prove that the ratio between the logarithms of the expansion factors of phi and its inverse is bounded above by a constant depending only on the rank n. We also prove a more general theorem applying to an arbitrary outer automorphism of F_n and its inverse, and their entire spectrum of expansion factors.

Abstract:
In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators.

Abstract:
We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse (2),. We show the equivalent relationship between the existence and limiting expression of (2), and some limiting processes of matrices and deduce the new determinantal representations of (2),, based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equation =,？()？,()？.

Abstract:
We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.

Abstract:
The group, Drazin and Koliha-Drazin inverses are particular classes of commuting outer inverses. In this note, we use the inverse along an element to study some spectral conditions related to these inverses in the case of bounded linear operators on a Banach space.

Abstract:
The study of solving inverse singular value problems for nonnegative matrices has been around for decades. It is clear that an inverse singular problem is trivial if the desirable matrix is not restricted to a certain structure. Provided with singular values and diagonal entries, this paper presents a numerical procedure, based on the successive projection process, to solve inverse singular value problems for nonnegative matrices subject to given diagonal entries. Even if we focus on the specific type of inverse singular value problems with prescribed diagonal entries, this entire procedure can be carried over with little effort to other types of structure. Numerical examples are used to demonstrate the capacity and efficiency of our method.

Abstract:
Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.

Abstract:
We investigate the relative perturbation bound of the group inverse and also consider the perturbation bound of the generalized Schur complement in a Banach algebra.