12,078 research outputs found
Numerical techniques for conformal mapping onto a rectangle
This paper is concerned with the problem of determining approximations to the function F which maps conformally a simply-connected domain onto a rectangle R, so that four specified points on are mapped Ω∂respectively onto the four vertices of R. In particular, we study the following two classes of methods for the mapping of domains of the form . (i) Methods which approximate where f is an approximation to the conformal map of Q onto the unit disc, and S is a simple Schwarz-Christoffel transformation. (ii) Methods based on approximating the conformal map of a certain symmetric doubly-connected domain onto a circular annulus.
Keywords: Conforma
Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions
We consider the class of biorthogonal polynomials that are used to solve the
inverse spectral problem associated to elementary co-adjoint orbits of the
Borel group of upper triangular matrices; these orbits are the phase space of
generalized integrable lattices of Toda type. Such polynomials naturally
interpolate between the theory of orthogonal polynomials on the line and
orthogonal polynomials on the unit circle and tie together the theory of Toda,
relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish
corresponding Christoffel-Darboux formulae . For all these classes of
polynomials a 2x2 system of Differential-Difference-Deformation equations is
analyzed in the most general setting of pseudo measures with arbitrary rational
logarithmic derivative. They provide particular classes of isomonodromic
deformations of rational connections on the Riemann sphere. The corresponding
isomonodromic tau function is explicitly related to the shifted Toeplitz
determinants of the moments of the pseudo-measure. In particular the results
imply that any (shifted) Toeplitz (Hankel) determinant of a symbol (measure)
with arbitrary rational logarithmic derivative is an isomonodromic tau
function.Comment: 35 pages, 1 figur
On Christoffel and standard words and their derivatives
We introduce and study natural derivatives for Christoffel and finite
standard words, as well as for characteristic Sturmian words. These
derivatives, which are realized as inverse images under suitable morphisms,
preserve the aforementioned classes of words. In the case of Christoffel words,
the morphisms involved map to (resp.,~) and to
(resp.,~) for a suitable . As long as derivatives are
longer than one letter, higher-order derivatives are naturally obtained. We
define the depth of a Christoffel or standard word as the smallest order for
which the derivative is a single letter. We give several combinatorial and
arithmetic descriptions of the depth, and (tight) lower and upper bounds for
it.Comment: 28 pages. Final version, to appear in TC
On the Structure of Bispecial Sturmian Words
A balanced word is one in which any two factors of the same length contain
the same number of each letter of the alphabet up to one. Finite binary
balanced words are called Sturmian words. A Sturmian word is bispecial if it
can be extended to the left and to the right with both letters remaining a
Sturmian word. There is a deep relation between bispecial Sturmian words and
Christoffel words, that are the digital approximations of Euclidean segments in
the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic}
bispecial Sturmian words are precisely the maximal internal factors of
\emph{primitive} Christoffel words. We extend this result by showing that
bispecial Sturmian words are precisely the maximal internal factors of
\emph{all} Christoffel words. Our characterization allows us to give an
enumerative formula for bispecial Sturmian words. We also investigate the
minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167
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