We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on . We also construct a solution of the equation in that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
Given a quadratic irrational , we are interested in how some numerical schemes applied to a convenient function provide subsequences of convergents to . We investigate three numerical schemes: secant-like methods and formal generalizations, which lead to linear recurring subsequences; the false position method, which leads to arithmetical subsequences of convergents and gives some interesting series expansions; Newton’s method, for which we complete a result of Edward Burger [1] about the existence...
Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give...
Generalizing a result of Pourchet, we show that, if are power sums over satisfying suitable necessary assumptions, the length of the continued fraction for tends to infinity as . This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers , .
Let d be a positive integer and α a real algebraic number of degree d + 1. Set . It is well-known that , where ||·|| denotes the distance to the nearest integer. Furthermore, for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that for any integer n ≥ 1.
We investigate and refine a device which we introduced in [3] for the study of continued fractions. This allows us to more easily compute the period lengths of certain continued fractions and it can be used to suggest some aspects of the cycle structure (see [1]) within the period of certain continued fractions related to underlying real quadratic fields.