For any simple complex Lie group, we classify irreducible finite-dimensional representations rho for which the longest element w_0 of the Weyl group acts non-trivially on the zero-weight space. Among irreducible representations that have zero among their weights, w_0 acts by Id or -Id if and only if the highest weight of rho is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.

Given an integer $k >0$, our main result states that the sequence of orders of the groups $SL_k (\mathbb{Z}_n)$ (respectively, of the groups $GL_k (\mathbb{Z}_n)$ is Cesàro equivalent as $n\to\infty$ to the sequence $C_1(k)n^{k^2−1}$ (respectively, $C_2(k)n^{k^2}$, where the coefficients $C_1(k)$ and $C_2(k)$ depend only on $k$; we give explicit formulas for $C_1(k)$ and $C_2(k)$. This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function $\varphi(n)$ is Cesàro equivalent to $n\frac{6}{\pi^2}$. We present some experimental facts related to the main result.

W. Thurston constructed a combinatorial model of the Mandelbrot set M2M2such that there is a continuous and monotone projection of M2M2to this model. We propose the following related model for the space MD3MD3of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3(P,c1,c2)∈MD3, then every point *z* in the Julia set of the polynomial *P * defines a unique maximal finite set AzAzof angles on the circle corresponding to the rays, whose impressions form a continuum containing *z *. Let G(z)G(z)denote the convex hull of AzAz. The convex sets G(z)G(z)partition the closed unit disk. For (P,c1,c2)∈MD3(P,c1,c2)∈MD3let <img height="16" border="0" style="vertical-align:bottom" width="14" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si6.gif">c1⁎be the *co-critical point of *c1c1. We tag the marked dendritic polynomial (P,c1,c2)(P,c1,c2)with the set <img height="18" border="0" style="vertical-align:bottom" width="159" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si14.gif">G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combtheir collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3MD3to <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combso that <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combserves as a model for MD3MD3.

We obtain explicit formulae, in the form of regularized multiplicative functionals related to certain Blaschke products, of the Radon-Nikodym derivatives between reduced Palm measures of all orders for determinantal point processes associated with a large class of weighted Bergman spaces on the disk. Our method also applies to determinantal point processes associated with weighted Fock spaces. © 2015 Académie des sciences.

We obtain explicit formulae, in the form of regularized multiplicative functionals related to certain Blaschke products, of the Radon–Nikodym derivatives between reduced Palm measures of all orders for determinantal point processes associated with a large class of weighted Bergman spaces on the disk. Our method also applies to determinantal point processes associated with weighted Fock spaces.

The paper deals with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces of either measure-valued functions or vector measures. Bilinear finite element discretizations are constructed and their stability and error analysis is accomplished.

We unify several Bellman function problems into one setting. For that purpose we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in R^2). We show how the unit ball in the BMO space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions.

We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation ψ of the maximal stable state μ ≡ 3 is obtained by adding extra grains at several points. It appears that the result ψ◦ of the relaxation of ψ coincides with μ almost everywhere; the set where ψ◦ ̸= μ is called the deviation locus. The scaling limit of the deviation locus turns out to be a distinguished tropical curve passing through the perturbation points.