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The reviewing process is simple anonymous, that is the reviewers are kept anonymous while the authors are not. 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.

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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.

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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.

Current Volume:

The current volume is the volume of 2022.

• In this paper, we study the Cauchy problem of Boltzmann equation with soft potential in modulation spaces. Our aim is to obtain the global existence of solution to the space-homogeneous Boltzmann equation. To realize this goal, the boundedness of Boltzmann operator in modulation space is established. In addition, Banach fixed-point theorem is applied with careful estimate of time integral for the contraction mapping.
• In this article, we develop the geometry of canonical stratifications of the spaces $\overline{\mathcal{M}}_{0,n}$ and prepare ground for studying the action of the Galois group $Gal ({\overline{\mathbb{Q}}} /\mathbb{Q})$ upon strata. We define and introduce a version of a gravity operad constructed for a class of moduli spaces $\overline{\mathcal{M}}_{0,n}$, equipped with a hidden holomorphic involution. This additional symmetry is associated to a split quaternionic structure. We introduce a categorical framework to present this object. Interaction between the geometry, physics and the arithmetics are discussed. An important feature is that 0-divisors of the split quaternion algebra imply additional singular points, and lead to investigations concerning the geometry and mixed Hodge structures.
• Building on a result by Tao, we show that a certain type of simple closed curve in the plane given by the union of the graphs of two $1$-Lipschitz functions inscribes a square whose sidelength is bounded from below by a universal constant times the maximum of the difference of the two functions.
• The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: \[X_{n}=a_n+\sum_{j=1}^m b_j X_{\lfloor\frac{n}{m_j}\rfloor},\] where the $m_i$'s are integers with $m_i\ge 2$. The main novelty of this work is there is no assumption of regularity or monotonicity for $(a_n)$. Then, this result can be applied to various sequences of random variables $(a_n)_{n\ge 0}$, for example such that $\sup_{n\ge 1}\mathbb{E}(|a_n|)<+\infty$.
• We are interested in a sufficient condition given in an article by P. Lefèvre, É. Matheron and A. Primot to obtain the Blum-Hanson property and we then partially answer two questions asked in this same article on other possible conditions to have this property for a separable Banach space.
• We give alternative computations of the Schur multiplier of $Sp(2g,\mathbb{Z}/D\mathbb{Z})$, when $D$ is divisible by 4 and $g\geq 4$: a first one using $K$-theory arguments based on the work of Barge and Lannes and a second one based on the Weil representations of symplectic groups arising in abelian Chern-Simons theory. We can also retrieve this way Deligne's non-residual finiteness of the universal central extension $\widetilde{Sp(2g,\mathbb{Z})}$. We prove then that the image of the second homology into finite quotients of symplectic groups over a Dedekind domain of arithmetic type are torsion groups of uniformly bounded size. In contrast, quantum representations produce for every prime $p$, finite quotients of the mapping class group of genus $g\geq 3$ whose second homology image has $p$-torsion. We further derive that all central extensions of the mapping class group are residually finite and deduce that mapping class groups have Serre's property $A_2$ for trivial modules, contrary to symplectic groups. Eventually we compute the module of coinvariants $H_2(\mathfrak{sp}_{2g}(2))_{Sp(2g,\mathbb{Z}/2^k\mathbb{Z})}=\mathbb{Z}/2\mathbb{Z}$.
• We discuss here a classical Crank-Nicolson numerical scheme to approximate the solutions of a nonlinear equation Schrödinger that reads \begin{equation*} u_t-i\partial_x\left( \nu(x)\partial_x u\right)-iu\log |u|^2=0. \end{equation*}

Acknowledgements

This journal owes much to the following persons: Denis Bitouzé for LaTex support, Nadine Demarelle for printing service, Mohammed Khabzaoui for computer engineering issues and Omar Aouadi for secretarial tasks.

Supports

The Paul Painlevé Laboratory and the Department of Mathematics of Lille provide 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.