A Software Library for Reliable Online-Arithmetic with Rational Numbers

An overview of a novel calculation framework for scientific computing in integrable spaces is introduced. This paper discusses some implementation issues adopted for a software library devoted to exact rational online-arithmetic operators for periodic rational operands codified in fractional positional notation

A Gauss-Kuzmin theorem for continued fractions associated with non-positive interger powers of an integer m≥2m \geq 2

We consider a family $\{\tau_m:m\geq 2\}$ of interval maps introduced by Hei-Chi Chan [5] as generalizations of the Gauss transformation. For the continued fraction expansion arising from $\tau_m$, we solve its Gauss-Kuzmin-type problem by applying the method of Rockett and Sz\"usz [18].Comment: 18 page

Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations

Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the $\beta$-subgraph centrality, which is based on the exponential of the matrix $\beta A$, where $A$ is the adjacency matrix of the graph and $\beta$ is a real parameter ("inverse temperature"). We prove that for algebraic $\beta$, two vertices with equal $\beta$-subgraph centrality are necessarily cospectral. We further show that two such vertices must have the same degree and eigenvector centralities. Our results settle a conjecture of Estrada and a generalization of it due to Kloster, Kr\'al and Sullivan. We also discuss possible extensions of our results.Comment: 8 pages, no figure

Robust Padé approximation via SVD

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity