Enumerating several aspects of non-decreasing Dyck paths

A Dyck path is non-decreasing if the $y$-coordinates of its valleys form a non-decreasing sequence. In this paper we give enumerative results and some statistics of several aspects of non-decreasing Dyck paths. We give the number of pyramids at a fixed level that the paths of a given length have, count the number of primitive paths, count how many of the non-primitive paths can be expressed as a product of primitive paths, and count the number of paths of a given height and a given length. We present and prove our results using combinatorial arguments, generating functions (using the symbolic method) and parameterize the results studied here using the Riordan arrays. We use known bijections to connect direct column-convex polyominoes, Elena trees, and non-decreasing Dyck paths.     

Nonlinear Instability of the Two-Dimensional Striation Model About Smooth Steady States

The two-dimensional striation model consists of a nonlinear system of PDE's which arises in the modeling of the ionospheric plasma. The local-in-time existence of strong solutions is first proved using Banach's fixed point theorem. Then, under physically relevant assumptions, the system is shown to be nonlinearly unstable as soon as it is linearly unstable. Moreover, the instability occurs before the possible blow-up time of the solution. The proof relies on an earlier work of Hwang and Guo (2003 Hwang , H. J. , Guo , Y. ( 2003 ). On the dynamical Rayleigh–Taylor instability . Arch. of Rat. Mech. Anal. 167 : 235 – 253 .[Crossref], [Web of Science ®] , [Google Scholar]). The first step of the proof is to investigate under which conditions the linearized system is unstable and to prove that its spectrum is bounded, by means of a variational formulation. The second one consists in constructing a family of solutions depending on the parameter δ measuring the smallness of the perturbation to the steady-state. Thanks to the boundedness of the linearized spectrum, this family of solutions is shown to be unstable by means of a power series expansion in δ.     

Modeling,Analysis, and Simulation of Biofilm Formation and Decay

In this article, we give a brief overview of our recent work on continuum mechanical modelling and simulation of microbial films. This comprises some classical tasks of applied mathematics such as computational fluid dynamics, analysis of partial differential equations, and mathematical biology.     

η-Alkynyl and vinylidene transition metal complexes: 10. Reaction of the metal-acetylide [(η-C5H5)(CO)(NO)W---CC---C(CH3)3]Li with 1,2-diiodoethane as electrophile and with various nucleophiles

Treatment of the η¹-acetylide complex [(η⁵-C₅H₅)(CO)(NO)W---CC---C(CH₃)₃]Li (4) with 1,2-diiodoethane in THF at −78 °C, followed by the addition of Li---CC---R [R=C(CH₃)₃, C₆H₅, Si(CH₃)₃, 6a–6c] or n-C₄H₉Li and protonation with H₂O, afforded the corresponding oxametallacyclopentadienyl complexes (η⁵-C₅H₅)W(I)(NO)[η²-O=C(CC---R)CH=CC(CH₃)₃] (7a–7c), 8c and (η⁵-C₅H₅)W(I)(NO)[η²-O=C(n-C₄H₉)CH=CC(CH₃)₃] (9). The formation of these metallafuran derivatives is rationalized by the electrophilic attack of 1,2-diiodoethane onto the metal center of 4 to form first the neutral complex [(η⁵-C₅H₅)(I)(CO)(NO)W---CC---C(CH₃)₃] (5). Subsequent nucleophilic addition of Li---CC---R 6a–6c or n-C₄H₉Li and a reductive elimination step followed by protonation leads to the products 7a–7c and 9. One reaction intermediate could be trapped with CF₃SO₃CH₃ and characterized by a crystal structure analysis. The identity of another intermediate was established by infrared spectroscopic data. The oxametallacyclopentadienyl complex 10 forms in the presence of excess 1,2-diiodoethane through an alternative pathway and crystallizes as a clathrate containing iodine.