Neighborhoods and Covering Vertices by Cycles

G =(V,E) is a 2-connected graph, and X is a set of vertices of G such that for every pair x,x' in X, , and the minimum degree of the induced graph <X> is at least 3, then X is covered by one cycle. This result will be in fact generalised by considering tuples instead of pairs of vertices. Let be the minimum degree in the induced graph <X>. For any , . If , and , then X is covered by at most (p-1) cycles of G. If furthermore , (p-1) cycles are sufficient. So we deduce the following: Let p and t () be two integers. Let G be a 2-connected graph of order n, of minimum degree at least t. If , and , then V is covered by at most cycles, where k is the connectivity of G. If furthermore , (p-1) cycles are sufficient. In particular, if and , then G is hamiltonian. Received April 3, 1998     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000     

Sampling measures for Bergman spaces on the unit disk

We provide a characterization of the sampling measures for the Bergman spaces. These are the positive measures on the unit disk for which there exists a constant such that These are the continuous analogues of the sets of sampling characterized by K. Seip [13,14] and A. Schuster [12]. Our characterization is in terms of weak* limits of the Moebius transformations of the measure , and mimics the notion for sequences that sampling means being uniformly far from zero sets. Received: 26 October 1998 / in revised form: 25 Juni 1999     

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

Lifting chains of prime ideals to paravaluation rings

Résumé  SoitR ⊂T une extension des anneaux commutatifs et soit {P _α :α ∈I} une cha?ne croissante des idéaux premiers deR (I étant un ensemble totalement ordonné, peut-être infini). Alors il existe un anneau de paravaluationV deT et une cha?ne {Q _α} des idéaux premiers deV de sorte queR ⊂V etQ _α ∩R =P _α pour toutα ∈I. Tout d’abord, on établit le cas spécial dans lequelT est un corps; dans ce cas, on trouve en effet un tel anneau de valuationV deT. Ensuite, l’assertion ci-dessus pour le cas général découle comme conséquence. Dans le cas général, on peut aussi remplacer le mot “paravaluation” avec le mot “valuation” siR est un anneau de Marot etT est son anneau total de fractions.      

A solution to the problem of Lp-L^p-maximal regularity

We give a negative solution to the problem of the -maximal regularity on various classes of Banach spaces including -spaces with . Received June 11, 1999; in final form September 6, 1999 / Published online September 14, 2000     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.     

Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements

This paper axiomatizes classes of Abelian lattice-ordered groups with a finite upper bound on the number of pairwise disjoint positive elements; finds model-completions for these theories; derives corresponding Nullstellensätze; determines which model-completions eliminate quantifiers; and examines quantifier elimination in a different language and for positive formulas.