Modified Jacobi-Perron Algorithm Research Articles

Convergence of β-Modified Jacobi-Perron Algorithm over the Field of Formal Power Series

The aim of this paper is to study multidimentional $\beta$-continued fraction algorithm over the field of formal power series. In the case of the Modified Jacobi-Perron algorithm, we prove that it converges.

True orbit simulation of piecewise linear and linear fractional maps of arbitrary dimension using algebraic numbers.

We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems-that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system-we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.

A new multidimensional continued fraction algorithm

It has been believed that the continued fraction expansion of(α,β)(\alpha ,\beta )(1,α,β(1,\alpha ,\betais aQ{\mathbb Q}-basis of a real cubic field))obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of(⟨33⟩,⟨93⟩)(\langle \sqrt [3]{3}\rangle , \langle \sqrt [3]{9}\rangle )(⟨x⟩\langle x\rangledenoting the fractional part ofxx). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of(α,β)(\alpha ,\beta )with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.

On optimal simultaneous rational approximation to [formula omitted] with [formula omitted] being some kind of cubic algebraic function

It is shown that each rational approximant to ( ω , ω 2 ) τ given by the Jacobi–Perron algorithm (JPA) or modified Jacobi–Perron algorithm (MJPA) is optimal, where ω is an algebraic function (a formal Laurent series over a finite field) satisfying ω 3 + k ω - 1 = 0 or ω 3 + kd ω - d = 0 . A result similar to the main result of Ito et al. [On simultaneous approximation to ( α , α 2 ) with α 3 + k α - 1 = 0 , J. Number Theory 99 (2003) 255–283] is obtained.

On simultaneous approximation to ( α, α2) with α3+ kα−1=0

We show that the modified Jacobi–Perron algorithm gives the best simultaneous approximation to ( α, α 2) with α 3+ kα−1=0. We claim the following facts: (1) the limit set of {( q n (q nα−p n), q n (q nα 2−r n) | n=1,2,…} become an ellipse, where ( p n , q n , r n ) is the nth convergent ( p n / q n , r n / q n ) of ( α, α 2) by the modified Jacobi–Perron algorithm, (2) the limit set of {( q (qα−p), q (qα 2−r) | q∈ Z,q>0} belongs to outside of the ellipse in (1).

Continued Fractions and the d-Dimensional Gauss Transformation

In this paper we study a multidimensional continued fraction algorithm which is related to the Modified Jacobi–Perron algorithm considered by Podsypanin and Schweiger. We demonstrate that this algorithm has many important properties which are natural generalisations of properties of one-dimensional continued fractions. For this reason, we call the transformation associated to the algorithm the d-dimensional Gauss transformation. We construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure. We also discuss the ergodic properties of this invariant measure.

A simple proof of the exponential convergence of the modified Jacobi–Perron algorithm

The modified Jacobi–Perron algorithm was shown by Ito etal in 1993 to be strongly convergent (in the sense of Brentjes 1981) almost everywhere with exponential rate. Their proof relies on very complicated computations. We will show that the original article by Podsypanin in 1977 on the modified Jacobi–Perron algorithm almost contains a proof of this convergence with exponential rate. The only ingredients missing in that article are some ergodic-theoretical facts about the transformation generating the approximations. This leads to a very simple proof of the before mentioned exponential convergence in the modified Jacobi–Perron algorithm.

On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof

The following theorem was published in [2].Theorem. There exists a constant δ > 0 such that for Lebesgue almost every (α, β) ∈ X = [0, 1] × [0, 1], there exists no = no(α, β) such that for any n > nowhere the integers pn, qn, rn are provided by the modified Jacobi-Perron algorithm.

The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials

We study the operator moment problem for a ( p + 2)-diagonal nonsymmetric matrix operator and give an explicit form of the Favard theorem for vector orthogonal polynomials. As a main tool we use the development of a system of resolvent functions of the operator in a vector continued fraction by a modified Jacobi-Perron algorithm.

Almost everywhere exponential convergence of the modified Jacobi—Perron algorithm

AbstractWe prove that there exists a constant δ > 0 such that for almost every pair of numbers α and β there exists n0= n0(α,β) such that for any n ≥ n0where the integers pn,qn, rn are provided by the modified Jacobi-Perron algorithm.

Errata: “Rational approximations of algebraic irrationals by means of a modified Jacobi-Perron algorithm,” vol. 32 (1965) pp. 161–176

Errata: “Rational approximations of algebraic irrationals by means of a modified Jacobi-Perron algorithm,” vol. 32 (1965) pp. 161–176