Institut de Mathématiques de Marseille

Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.

Let B \mathfrak {B} be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial S ( α ; N ) = ∑ b ∈ B , b ≤ N e ( b α ) S(\alpha ;N) = \sum _{b\in \mathfrak {B},b\le N}e(b\alpha ) satisfies S ( α ; N ) N / log ⁡ N ≪ A , A ′ 1 φ ( q ) + q N ( log ⁡ N ) 7 + 1 ( log ⁡ N ) A \begin{equation*} \frac {S(\alpha ; N)}{N/\sqrt {\log N}}\ll _{A,A’} \frac {1}{\varphi (q)} + \sqrt {\frac {q}{N}}(\log N)^{7} +\frac {1}{(\log N)^A} \end{equation*} for any A , A ′ ≥ 0 A,A’\ge 0 and when ( a , q ) = 1 (a,q)=1 and | q α − a | ≤ ( log ⁡ N ) A ′ / N |q\alpha -a|\le (\log N)^{A’}/N . We use this estimate together with a variant of the circle method influenced by Green and Tao’s Transference Principle to obtain the number of representations of a large enough odd integer N N as a sum b + b 1 + b 2 b+b_1+b_2 , where b ∈ B b\in \mathfrak {B} while b 1 b_1 (resp. b 2 b_2 ) belongs to a general subset B 1 \mathfrak {B}_1 (resp. B 2 \mathfrak {B}_2 ) of B \mathfrak {B} of relative positive density. We further show that the above bound is effective when 0 ≤ A > 1 / 2 0\le A>1/2 .

We study two positional games played on hypergraphs, whose edges may be interpreted as winning sets. Two players take turns picking a previously unpicked vertex of the hypergraph. We say a player fills an edge if that player has picked all the vertices of that edge. In the Maker-Maker convention, whoever first fills an edge wins, or we get a draw if no edge is filled. In the Maker-Breaker convention, the first player aims at filling an edge while the second player aims at preventing the first player from filling an edge. Our main result is that, for both games, deciding whether the first player has a winning strategy is a PSPACE-complete problem even when restricted to 4-uniform hypergraphs (of bounded maximum degree). For the Maker-Maker convention, this improves on the known PSPACE-completeness result for hypergraphs of rank 4. For the Maker-Breaker convention, this improves on the known PSPACE-completeness result for 5-uniform hypergraphs, and closes the complexity gap since the problem for hypergraphs of rank 3 is known to be solvable in polynomial time. As a corollary of our construction, we actually get a stronger result: deciding whether the first player has a winning strategy for the vertex-$C_4$-game played on arbitrary graphs, where the winning sets are the vertex sets of 4-cycles, is a PSPACE-complete problem for both conventions.

Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is algebraically closed in $F$. We give general results on the descent over the fields $k= \F_{p^t}$ for $t$ dividing $n$. Then, we completely handle the bi-cyclic case of the descent over the fields $k_1=\F_{p}$ and $k_2= \F_{p^2}$ of all the sub-extensions of $F$ defined over $\F_{p^4}$. We give explicit examples with small prime numbers $p$.

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal $\mathrm{L}^q$-regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.

Résumé Étant donnée une suite $A = (a_n)_{n\geqslant 0}$ d’entiers naturels tous au moins égaux à 2, on pose $q_0 = 1$ et, pour tout entier naturel n , $q_{n+1} = a_n q_n$ . Tout nombre entier naturel $n\geqslant 1$ admet une unique représentation dans la base A , dite de Cantor, de la forme $$ \begin{align*} m = \sum_{j\geqslant 0}\varepsilon_j(m)q_j\quad \text{avec} \quad 0\leqslant \varepsilon_j(m) < a_j = \frac{q_{j+1}}{q_j}. \end{align*} $$ Dans cet article, nous évaluons la somme $$ \begin{align*} S = \sum_{n \leqslant x}\Lambda(n) f(n) \end{align*} $$ où $\Lambda $ est la fonction de von Mangoldt et f une fonction fortement multiplicative en base A . L’estimation des sommes de type I et II associées repose sur le bon contrôle de transformées de Fourier discrètes de fonctions construites à partir de f par décalage dans la numération en base A . Cette approche pouvant échouer si la suite $(a_n)_{n\geqslant 0}$ est trop irrégulière, nous introduisons la notion de base de Cantor tempérée et obtenons dans ce cadre une majoration générale de la somme S . Nous étudions plusieurs exemples dans la base $A = (j+2)_{j\geqslant 0}$ , dite factorielle. En particulier, si $s_A$ désigne la fonction somme de chiffres dans cette base et p parcourt la suite des nombres premiers, nous montrons que la suite $(s_A(p))_{p\in \mathcal {P}}$ est bien répartie dans les progressions arithmétiques, et que la suite $(\alpha s_A(p))_{p\in \mathcal {P}}$ est équirépartie modulo $1$ pour tout nombre irrationnel $\alpha $ .