North-Western European Journal of Mathematics

ISSN=2496-5170

The Lille Main Square

## Aims and scope

The North-Western European Journal of Mathematics publishes research articles and surveys in all areas of pure and applied mathematics including history of mathematics. Articles can be submitted in Dutch, English, French or German.

## Policies

### Peer Review Process

The authors are asked to provide the names of one or two potential associate editors who will manage their article. Then, the editor in chief assigns the article to an associate editor who seeks reviewer(s). Once the reviews are available, the associate editor reports his decision to the whole editorial board which takes the final decision regarding the acceptance of the article.

### Open Access Policy

No charges are levied on authors. Articles are freely available to anyone as soon they are accepted. Hard copies of a volume are available at the cost of the paper and delivery.

Authors keep their copyrights. Thus they can reuse figures, article parts or any material they need for subsequent work. The contract license is the Creative Commons License CC-BY which is the most flexible contract maximising the readership on publication.

## Author Guidelines

At this initial stage, the authors have only to provide a file in the pdf format. Only when the article is accepted or subject to minor revisions are they asked to ensure it complies with the journal format.

Here is the LaTEX class we use : nwejm, please also note that the bibliography must be provided as a .bib file in the biblatex format.

For the time being, articles should be sent by mail to the managing editor (nicolas.wickerNOSPAMATuniv-lille1.fr) or you can submit them online.

## Current Volume:

The current volume is the volume of 2016, the first volume, of 2015, is being typeset using the new class, nwejmart.
• In this paper iteration stable (STIT) tessellations of the $d$-dimensional Euclidean space are considered. By a careful analysis of the capacity functional an alternative proof is given for the fact that STIT tessellations are mixing.
• The invariant $\Theta$ is the simplest $3$--manifold invariant defined by counting graph configurations. It is actually an invariant of rational homology $3$--spheres $M$ equipped with a combing $X$ over the complement of a point, where a combing is a homotopy class of nowhere vanishing vector fields. The invariant $\Theta(M,X)$ is the sum of $6 \lambda(M)$ and $\frac{p_1(X)}{4}$, where $\lambda$ denotes the Casson-Walker invariant, and $p_1$ is an invariant of combings, which is an extension of a first relative Pontrjagin class, and which is simply related to a Gompf invariant $\theta_G$. In Lescop (2015), we proved a combinatorial formula for the $\Theta$--invariant in terms of decorated Heegaard diagrams. In this article, we study the variations of the invariants $p_1$ or $\theta_G$ when the decorations of the Heegaard diagrams that define the combings change, independently. Then we prove that the formula of Lescop (2015) defines an invariant of combed once punctured rational homology $3$--spheres without referring to configuration spaces. Finally, we prove that this invariant is the sum of $6 \lambda(M)$ and $\frac{p_1(X)}{4}$ for integer homology $3$--spheres, by proving surgery formulae both for the combinatorial invariant and for $p_1$.
• We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.
• We study composition operators acting between $\mathcal{N}_p$-spaces in the unit ball in $\mathbb{C}^n$. We obtain characterizations of the boundedness and compactness of $C_{\varphi}:\mathcal{N}_p\longrightarrow\mathcal{N}_q$ for $p, q>0$.
• In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated the extent of asymptotic stability of the null solution for weakly coupled partially damped equations of the second order in time. The main point is that the damping operator acts only on the first component and, whenever it is bounded, the coupling is not strong enough to produce an exponential decay in the energy space associated to the conservative part of the system. As a consequence, for initial data in the energy space, the rate of decay is not exponential. Due to the nature of the result it seems at first sight impossible to obtain the asymptotic stability result by the classical Liapunov method. Surprisingly enough, this turns out to be possible and we exhibit, under some compatibility conditions on the operators, an explicit class of Liapunov functions which allows to do 3 different things: 1) When the problem is reduced to a stable finite dimensional space, we recover the exponential decay by a single differential inequality and we estimate the logarithmic decrement of the solutions with worst (slowest) decay. The estimate is optimal at least for some values of the parameters. 2) We explain the form of the stability result obtained by the previous authors when the coupling operator is a multiple of the identity, so that the decay is not exponential. 3) We obtain new exponential decay results when the coupling operator is strong enough (in particular unbounded). The estimate is again sharp for some solutions.

## Acknowledgements

This journal owes much to the following persons: Denis Bitouzé for LaTex support, Jean-Jacques Derycke for printing service, Sébastien Huart for computer engineering issues and Aurore Smets for secretarial tasks.

## Supports

The department of Mathematics of Lille 1 provides financial support which ensures that this journal stays open access. The French Mathematical Society (SMF), the Dutch Mathematical Society (KWG), the Luxembourg Mathematical Society (SML) and the Fields Institute support this project, of a mathematics journal by a non-profit publisher, making research available free of charge for readers and authors.

## Auxiliary Material

A slide to present the journal.

## Friends

We provide a tentative list of journal friends, i.e. diamond open access journals free of charge for authors and readers.