Abstract:
We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice on the average height of planted plane trees, and results by Fulmek and Katori et al. on the expected value, respectively the higher moments, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information we prove a reciprocity relation satisfied by the derivatives of one of Jacobi's theta functions, which generalises the well known reciprocity law for Jacobi's theta functions.

Abstract:
We revisit the product convolution semigroup of probability densities e_c(t),c>0 on the positive half-line with moments (n!)^c and determine the asymptotic behaviour of e_c(t) for large and small t>0. This shows that (n!)^c is indeterminate as Stieltjes moment sequence if and only if c>2

Abstract:
We determine the asymptotic behaviour of the number of the Eulerian circuits in undirected simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). We also prove some new properties of the Laplacian matrix.

Abstract:
System of partial differential equations with a convolution terms and non-local nonlinearity describing oscillations of plate due to Berger approach and with accounting for thermal regime in terms of Coleman-Gurtin and Gurtin-Pipkin law and fading memory of material is considered. The equation is transformed into a dynamical system in a suitable Hilbert space which asymptotic behaviour is analysed. Existence of the compact global attractor in this dynamical system and some of its properties are proved in this article. Main tool in analysis of asymptotic behaviour is stabilizability inequality.

Abstract:
The asymptotic behaviour at late times of inhomogeneous axion-dilaton cosmologies is investigated. The space-times considered here admit two abelian space-like Killing vectors. These space-times evolve towards an anisotropic universe containing gravitational radiation. Furthermore, a peeling-off behaviour of the Weyl tensor and the antisymmetric tensor field strength is found. The relation to the pre-big-bang scenario is briefly discussed.

Abstract:
This paper is concerned with the oscillation and asymptotic behaviour of nonoscillatory solutions of nonlinear difference equation of a particular form.

Abstract:
We investigate the asympotic behaviour of the moduli space of morphisms from the rational curve to a given variety when the degree becomes large. One of the crucial tools is the homogeneous coordinate ring of the variey. First we explain in details what happens in the toric case. Then we examine the general case.

Abstract:
We consider compact Hankel operators realized in $ \ell^2(\mathbb Z_+)$ as infinite matrices $\Gamma$ with matrix elements $h(j+k)$. Roughly speaking, we show that if $h(j)\sim (b_{1}+ (-1)^j b_{-1}) j^{-1}(\log j)^{-\alpha}$ as $j\to \infty$ for some $\alpha>0$, then the eigenvalues of $\Gamma$ satisfy $\lambda_{n}^{\pm} (\Gamma)\sim c^{\pm} n^{-\alpha}$ as $n\to \infty$. The asymptotic coefficients $c^{\pm}$ are explicitly expressed in terms of the asymptotic coefficients $b_{1} $ and $b_{-1}$. Similar results are obtained for Hankel operators $\mathbf \Gamma$ realized in $ L^2(\mathbb R_+)$ as integral operators with kernels $\mathbf h(t+s)$. In this case the asymptotics of eigenvalues $\lambda_{n}^{\pm} (\mathbf \Gamma)$ are determined by the behaviour of $\mathbf h(t)$ as $t\to 0$ and as $t\to \infty$.

Abstract:
Let k be a field of characteristic 0, R = k[x_1, ..., x_d] be a polynomial ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H^0_m(R/I^n) and n^d. We show that this quantity always has a limit as n goes to infinity. We also give an example for which the limit is irrational; in particular, this proves that the length of H^0_m(R/I^n) is not asymptotically a polynomial in n.