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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.

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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.

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