S-expansions in Dimension Two
نویسندگان
چکیده
The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, can be used to transfer certain metrical and diophantine properties from one algorithm to the others. In particular, we are interested in the transferability of the density of the invariant measure. Finally, we use this method to construct an algorithm which improves approximation properties, as opposed to Brun’s algorithm.
منابع مشابه
Symmetric shift radix systems and finite expansions
Shift radix systems provide a unified notation to study several important types of number systems. However, the classification of such systems is already hard in two dimensions. In this paper, we consider a symmetric version of this concept which turns out to be easier: the set of such number systems with finite expansions can be completely classified in dimension two.
متن کاملEffective Stiffness Tensor of Composite Media-i. Exact Series Expansions
The problem of determining exact expressions for the effective stiffness tensor macroscopically anisotropic, two-phase composite media of arbitrary microstructure in arbitrary space dimension d is considered. We depart from previous treatments by introducing an integral equation for the “cavity” strain field. This leads to new, exact series expansions for the effective stiffness tensor of macro...
متن کاملMaster integrals with one massive propagator for the two - loop electroweak form factor
We compute the master integrals containing one massive propagator entering the two-loop electroweak form factor, i.e. the process f ¯ f → X, where f ¯ f is an on-shell massless fermion pair and X is a singlet particle under SU (2) L × U (1) Y , such as a virtual gluon or an hypothetical Z ′. The method used is that of the differential equation in the evolution variable x = −s/m 2 , where s is t...
متن کاملFrom Fourier to Gegenbauer: Dimension Walks on Spheres
In this article we provide a solution to Problem 2 in Gneiting, 2013. Specifically, we show that the evenresp. odd-dimensional Schoenberg coefficients in Gegenbauer expansions of isotropic positive definite functions on the sphere S d can be expressed as linear combinations of Fourier resp. Legendre coefficients, and we give closed form expressions for the coefficients involved in these expansi...
متن کاملInverse dimension expansions
Two toy Hamiltonians of anharmonic oscillators are considered. Both of them are found exactly solvable not only in the limit of infinite spatial dimension D but also in the next D ≫ 1 approximation. This makes all their bound states tractable via perturbation series in 1/ √ D ≪ 1.
متن کاملAsymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large
This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N = n+1 on two sets of variables, i.e., xu; p1×1 and xv; p2×1. These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p1 and p2 are fixed an...
متن کامل