Solving multivariate functional equations

This paper presents a new method to solve functional equations of multivariate generating functions, such as $$F(r,s)=e(r,s)+xf(r,s)F(1,1)+xg(r,s)F(qr,1)+xh(r,s)F(qr,qs),$$ giving a formula for $F(r,s)$ in terms of a sum over finite sequences. We use this method to show how one would calculate the coefficients of the generating function for parallelogram polyominoes, which is impractical using other methods. We also apply this method to answer a question from fully commutative affine permutations.Comment: 11 pages, 1 figure. v3: Main theorems and writing style revised for greater clarity. Updated to final version, to appear in Discrete Mathematic

The relation between discs and young companions - Observational studies

The direct imaging technique brings advantages with respect to other, indirect methods of detecting planets. It is sensitive to larger separations, it can detect companions on a variety of orbital configurations, and it allows to simultaneously image both a companion and the circumstellar disc it resides in, thus being the perfect tool to study companion-disc interactions. Direct observations of Hα emission from young planetary and low-mass stellar companions can also shed light on the early gas accretion phase of planet formation. In this Thesis I use the direct imaging technique to study various aspects of planet-disc interaction and planet formation and evolution. I present the detection of a previously unknown low-mass stellar companion around HD 193571, observed as part of the NaCo Imaging Survey for Planets around Young Stars (ISPY). The companion appears to reside within the gap between the host star and its surrounding disc, making this the third low-mass stellar companion discovered within a debris disc. This system is thus the perfect laboratory where to study the relative importance between self- and companion-stirring models in discs. I also present the detection of Hα emission from the known substellar companion around the young star PZ Tel. The derived Hα luminosity, combined with age and disc information, indicates that the emission is likely due to chromospheric activity of the companion. This detection further proves the capability of using high-contrast imaging instruments and techniques to detect Hα signatures from companions around young stars. On a larger scale, I present the L’ band Imaging Survey to find Exoplanets in the North (LIStEN), which targeted ∼30 nearby stars with known and well characterised circumstellar discs. LIStEN focuses on characterising the population of wide-orbit giant planets around disc-hosting stars, as well as studying the intricacies of companion-disc interactions. I present the survey’s scientific goals, data selection and observational strategy, as well as the data reduction and analysis. No new planetary companions were detected, and the mass detection limits derived from the observations are combined with information on the disc size and morphology to constrain the presence of unseen planetary and low-mass stellar companion around these disc-hosting stars

Some linear recurrences and their combinatorial interpretation by means of regular languages

AbstractIn this paper we apply ECO method and the concept of numeration systems to give a combinatorial interpretation to linear recurrences of the kind an=kan−1+han−2, where k>|h|⩾0. In particular, we define a language L such that the words of L having length n satisfy the recurrence, and then we describe a recursive construction for this language, according to the ECO method, and the corresponding finite succession rule