Juan B. Gil

Geometry and Spectra of Closed Extensions of Elliptic Cone Operators

We study the geometry of the set of closed extensions of index 0 of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

On Rays of Minimal Growth for Elliptic Cone Operators

We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays.

Trace expansions for elliptic cone operators with stationary domains

We analyze the behavior of the trace of the resolvent of an ellip- tic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal exten- sion of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first compo- nent and expand the component of finite rank in the case where the domain is stationary. The results make use, and develop further, our previous investi- gations on the analytic and geometric structure of the resolvent. The analysis of nonstationary domains, considerably more intricate, is pursued elsewhere.

Aspects of boundary problems in analysis and geometry

I Rarefied Gases.- 1. Macroscopic limits of the Boltzmann equation: a review.- 2. Moment equations for charged particles: global existence results.- 3. Monte-Carlo methods for the Boltzmann equation.- 4. Accurate numerical methods for the Boltzmann equation.- 5. Finite-difference methods for the Boltzmann equation for binary gas mixtures.- II Applications.- 6. Plasma kinetic models: the Fokker-Planck-Landau equation.- 7. On multipole approximations of the Fokker-Planck-Landau operator.- 8. Traffic flow: models and numerics.- 9. Modelling and numerical methods for granular gases.- 10. Quantum kinetic theory: modelling and numerics for Bose-Einstein condensation.- 11. On coalescence equations and related models.

Arithmetic in the Ring of Formal Power Series with Integer Coefficients

of polynomials R[x] over R, namely the ring ^[[x]] of formal powers series in one variable over R, is hardly ever mentioned in such a course. In most cases, it is relegated to the homework problems (or to the exercises in the textbooks), and one learns that, like R[x], R[[x]] is an integral domain provided that R is an integral domain. More surprising is to learn that, in contrast to the situation of polynomials, in R[[x]] there are many invertible elements: while the only units in R[x] are the units of R, a necessary and sufficient condition for a power series to be invertible is that its constant term be invertible in R. This fact makes the study of arithmetic in /?[[*]] simple when R is a field: the only prime element is the variable x. As might be expected, the study of prime factorization in Z[[x]] is much more interesting (and complicated), but to the best of our knowledge it is not treated in detail in the available literature. After some basic considerations, it is apparent that the question of deciding whether or not an integral power series is prime is a difficult one, and it seems worthwhile to develop criteria to determine irreducibility in Z[[x]] similar to Eisensteins criterion for polynomials. In this note we propose an easy argument that provides us with an infinite class of irreducible power series over Z. As in the case of Eisensteins criterion in 7L\x\, our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials. It is important to note that irreducibility in Z[x] and in Z[[x]] are, in general, un related. For instance, 6 + x + x2 is irreducible in Z[x] but can be factored in Z[[x]], while 2 + Ix + 3x2 is irreducible in Z[[x]] but equals (2 + x)(l +3x) as a polyno mial (observe that this is not a proper factorization in Z[[x]] since 1 + 3x is invert

Colored partitions of a convex polygon by noncrossing diagonals

For any positive integers a and b , we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to b modulo a . For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions.

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ

We consider polynomials with integer coefficients and discuss their factorization properties in ℤ[[x]], the ring of formal power series over ℤ. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over ℤ[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

A family of Bell transformations

We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full range of applications remains unexplored, in this paper we show a glimpse of the versatility of Bell transformations by discussing the enumeration of several combinatorial configurations, including rational Dyck paths, rooted planar maps, and certain classes of permutations.

On rational Dyck paths and the enumeration of factor-free Dyck words

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

On factor-free Dyck words with half-integer slope

We study a class of rational Dyck paths with slope (2m+1)/2 corresponding to factor-free Dyck words, as introduced by P. Duchon. We show that, for the slopes considered in this paper, the language of factor-free Dyck words is generated by an auxiliary language that we examine from the algebraic and combinatorial points of view. We provide a lattice path description of this language, and give an explicit enumeration formula in terms of partial Bell polynomials. As a corollary, we obtain new formulas for the number of associated factor-free generalized Dyck words.