In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$. Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$. In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$.

The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$.

Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$.

We study pencils of curves on a germ of complex reduced surface $(S,0)$. These are families of curves parametrized by $ \mathbb{P}^1 $ having 0 as the unique common point. We prove that for $w\in \mathbb{P}^1$, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$, where f and g are generators of the pencil.

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least

\begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*}

$\exp\!(n\log n-\Theta (n))$

$\ell$

$(k-\ell )\mid n$

$(k-\ell )\mid k$

$(k-\ell )\nmid k$

$\ell \lt k/2$

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

All vital functions of living cells rely on the production of various functional molecules through gene expression. The production periods are burst-like and stochastic due to the discrete nature of biochemical reactions. In certain contexts, the concentrations of RNA or protein require regulation to maintain a fine internal balance within the cell. Here we consider a motif of two types of RNA molecules – mRNA and an antagonistic microRNA – which are encoded by a shared coding sequence and form a feed forward loop (FFL). This control mechanism is shown to be perfectly adapting in the deterministic context. We demonstrate that the adaptation (of the mean value) becomes imperfect if production occurs in random bursts. The FFL nevertheless outperforms the benchmark feedback loop in terms of counterbalancing variations in the signal. Methodologically, we adapt a hybrid stochastic model, which has widely been used to model a single regulatory molecule, to the current case of a motif involving two species; the use of the Laplace transform thereby circumvents the problem of moment closure that arises owing to the mRNA–microRNA interaction. We expect that the approach can be applicable to other systems with nonlinear kinetics.

A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.

The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$.

We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$. For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$, apart from four cases, which are all isomorphic to Onsager’s algebra.

In this note, we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.

Dans cet article, nous étudions la cohomologie de de Rham du premier revêtement de la tour de Drinfel’d. En particulier, nous obtenons une preuve purement locale du fait que la partie supercuspidale réalise la correspondance de Jacquet-Langlands locale pour $\mathrm {GL}_n$ en la comparant à la cohomologie rigide de certaines variétés de Deligne-Lusztig. Les représentations obtenues sont analogues à celles qui apparaissent dans la cohomologie $\ell $-adique lorsqu’on oublie l’action du groupe de Weil. La preuve repose sur une généralisation d’un résultat d’excision de Grosse-Klönne et de la description explicite du premier revêtement en tant que revêtement cyclique obtenu par l’auteur dans un travail précédent.

We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.