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

The North-Western European Journal of Mathematics adheres to the principles of Fair Open Access, and is a member of the Free Journal Network. 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

Articles submission: At the 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.

LaTeX details:

• LaTeX class: We strongly encourage the authors to prepare their article to appear in the NWEJM journal with the dedicated nwejmart' LaTeX class (currently, documented only in French), available with up to date MiKTeX 2.9 and TeX Live 2016 distributions.
• Incompatibilities:>
• Our production system is not compatible with the xypic' bundle. For commutative diagramms, please use instead the user-friendly and modern tikz-cd' package.
• We much more prefer graphics not created with PSTricks' and derived. Consider using PGF'/TikZ' or derived packages. In case you really need one of these packages, please contact us.
• Bibliography: The bibliography must be provided as a .bib' file in the format of the biblatex' package.
• Figures: The figures must be provided in PDF or EPS formats.

Here you can find an example of a LaTEX and a biblatex file. The template and the documentation (for the time being only in French). 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 2018.
• We study in this paper the functional equation $\alpha \mathbf{u}(t)+\mathcal{C}\star(\chi \mathbf{u})(t)=\mathbf{f}(t)$ where $\alpha\in\mathbb{C}^{d\times d}$, $\mathbf{u},\mathbf{f}:\mathbb{R}\rightarrow\mathbb{C}^d$, $\mathbf{u}$ being unknown. The term $\mathcal{C}\star(\chi \mathbf{u})(t)$ denotes the discrete convolution of an almost zero matricial mapping $\mathcal{C}$ with discrete support together with the product of $\mathbf{u}$ and the characteristic function $\chi$ of a fixed segment. This equation combines some aspects of recurrence equations and/or delayed functional equations, so that we may construct a matricial based framework to solve it. We investigate existence, unicity and determination of the solution to this equation. In order to do this, we use some new results about linear independence of monomial words in matrix algebras.
• By a recent work of Gran-Kadjo-Vercruysse, the category of cocommutative Hopf algebras over a field of characteristic zero is semi-abelian. In this paper, we explore some properties of this category, in particular we show that its abelian core is the category of commutative and cocommutative Hopf algebras.
• We investigate the inverse problem of simultaneously estimating the state and the spatial diffusion coefficient for an age-structured population model. The time evolution of the population is supposed to be known on a subdomain in space and age. We generalize to the infinite dimensional setting an adaptive observer originally proposed for finite dimensional systems.
• We discuss comparison principles, the asymptotic behaviour, and the occurrence of blow up phenomena for nonlinear parabolic problems involving the $p$--Laplacian operator of the form $\left\{\begin{array}{ll} \partial_t u=\Delta_p u+f(t,x,u)&\mbox{in}\ \Omega\ \mbox{ for }\ t>0,\\ \sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0&\mbox{on}\ \partial\Omega\ \mbox{ for }\ t>0,\\ u(0,\cdot)=u_0 &\mbox{in}\ \overline{\Omega},\\ \end{array}\right.$ where $\Omega$ is a bounded domain of ${\mathbb R}^N$ with Lipschitz boundary, and where $\Delta_p u:={\mathrm div}\, \left(|\nabla u|^{p-2}\nabla u\right)$ is the $p$--{\itshape Laplacian} operator for $p>1$. As for the {\itshape dynamical} time lateral boundary condition $\sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0$ the coefficient $\sigma$ is assumed to be a nonnegative constant. In particular, the asymptotic behaviour in the large for the parameter dependent nonlinearity $f(\cdot,\cdot,u)=\lambda|u|^{q-2}u$ will be investigated by means of the evolution of associated norms.
• Let $X$ be a hyperkähler variety, and assume $X$ has a non-symplectic automorphism $\sigma$ of order $>\frac{1}{2}\dim X$. Bloch's conjecture predicts that the quotient $X/<\sigma>$ should have trivial Chow group of $0$--cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non--symplectic automorphism.

## 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.