Associated to a finite graph $X$ is its quantum automorphism group $G(X)$. We prove a formula of type $G(X*Y)=G(X)*_{\mathrm{w}}G(Y)$, where $*_{\mathrm{w}}$ is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula $\mu(G*_{\mathrm{w}}H)=\mu(G)\boxtimes\mu(H)$, where $\mu$ is the associated spectral measure. This is verified in two situations: one using free probability techniques, the other one using planar algebras.

We prove that a (bounded, linear) operator acting on an infinite-dimensional, separable, complex Hilbert space can be written as a product of two quasi-nilpotent operators if and only if it is not a semi-Fredholm operator. This solves the problem posed by Fong and Sourour in 1984. We also consider some closely related questions. In particular, we show that an operator can be expressed as a product of two nilpotent operators if and only if its kernel and co-kernel are both infinite dimensional. This answers the question implicitly posed by Wu in 1989.

We prove the existence of multiple bound states of the nonlinear Schrödinger equation −Δu + V(x)u = f(u). Here the linear potential V is continuous and bounded from below, and the nonlinearity f is of asymptotically linear type. We show that, under certain assumptions on the spectrum of the Schrödinger operator −Δ + V and the asymptotic behaviour of f(u)/u, the above equation has at least four non-trivial solutions, two of them sign changing.

We consider elliptic systems with discontinuous coefficients and prove that if the known term belongs to the Morrey space Lp,λ, then the highest-order derivatives of the local solution belong to the same space. We also obtain local Hölder continuity for lower-order derivatives.

In this paper we consider the Navier–Stokes equations in Rn, n ≥ 3. We prove the asymptotic stability for weak solutions in the marginal class u ∈ L2(0, ∞; BMO), where ‘BMO’ denotes the bounded mean oscillation function, with arbitrary initial and external perturbations.

We investigate geometrically exact generalized continua of micromorphic type in the sense of Eringen. The two-field problem for the macrodeformation φ and the affine microdeformation P̄ ∈ GL+(3, R) in the quasistatic, conservative load case is investigated in a variational form. Depending on material constants, two existence theorems in Sobolev spaces are given for the resulting nonlinear boundary-value problems. These results comprise existence results for the micro-incompressible case P̄ ∈ SL(3, R) and the Cosserat micropolar case P̄ ∈ SO(3, R). In order to treat external loads, a new condition, called bounded external work, has to be included, which overcomes the conditional coercivity of the formulation. The possible lack of coercivity is related to fracture of the micromorphic solid. The mathematical analysis uses an extended Korn first inequality. The methods of choice are the direct methods of the calculus of variations.

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

The long-time behaviour of solutions to a semilinear damped wave equation in a three-dimensional bounded domain with the nonlinearity rapidly oscillating in time (f = f(ε, u, t/ε)) is studied. It is proved that (under natural assumptions) the behaviour of solutions whose initial energy is not very large can be described in terms of global (uniform) attractors Aε of the corresponding dynamical processes and that, as ε → 0, these attractors tend to the global attractor A0 of the corresponding averaged system. We also give the detailed description of these attractors in the case where the limit attractor A0 is regular.

Moreover, we give explicit examples of semilinear hyperbolic equations where the uniform attractor Âε (for the initial data belonging to the whole energy phase space) contains the irregular resonant part, which tends to infinity as ε → 0, and formulate the additional restrictions on the nonlinearity f which guarantee that this part is absent.

We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

This paper is concerned with multi-dimensional non-isentropic Euler–Poisson equations for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyse the quasi-neutral limit for Cauchy problems with prepared initial data. It is shown that the small-parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Moreover, the formal limit is justified.

We consider the existence of stationary or pinned waves of reaction–diffusion equations in heterogeneous media. By combining averaging, homogenization and dynamical-systems techniques we prove under mild non-degeneracy conditions that if the heterogeneity is periodic with period ε, pinned solutions persist at most for intervals in parameter space whose length is O(e−c/√ε).

We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the result was previously only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partly ordered measure spaces.

We prove the existence of radial solutions ofconcentrating on a sphere for potentials which might be zero and might decay to zero at infinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov–Schmidt reduction.

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

Wir entwickeln eine Strategie zum Beweis der Positivität der Koeffizienten von Kazhdan-Lusztig-Polynomen für beliebige Coxeter-Gruppen.

We develop a strategy to prove the positivity of coefficients of Kazhdan–Lusztig polynomials for arbitrary Coxeter groups.

Dans ce papier nous étudions une correspondance de Jacquet–Langlands locale pour toutes les représentations lisses irréductibles. La correspondance est caractérisée par le fait qu’elle respecte la correspondance de Jacquet–Langlands classique et commute avec le foncteur d’induction parabolique. Elle est compatible dans un sens à préciser au foncteur de Jacquet et à l’involution d’Aubert–Schneider–Stuhler. Nous utilisons cette correspondance pour montrer qu’une certaine classe de représentations d’une forme intérieure de $\mathrm{GL}_n$ sur un corps $p$-adique sont unitarisables. C’est le premier pas dans la preuve de la conjecture U1 de Tadić.

We study a local Jacquet–Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet–Langlands correspondence and it commutes with the parabolic induction functor. It has good behavior with respect to the Jacquet’s functor and the involution of Aubert–Schneider–Stuhler. Using this correspondence, we prove some particular cases of the global Jacquet–Langlands correspondence and we deduce that a certain class of representations of an inner form of $\mathrm{GL}_n$ over a $p$-adic field are unitarizable. This is the first step towards the proof of Conjecture U1 of Tadić.

In this paper we study the category of non-degenerate modules over the Harish-Chandra Schwartz algebra of a p-adic connected reductive group. We construct functors of parabolic induction and restriction and show that they are exact and both ways adjoint to each other.

We say that $n$-vertex graphs $G_1,G_2,\ldots,G_k$ pack if there exist injective mappings of their vertex sets onto $[n] = \{1, \ldots,n \}$ such that the images of the edge sets do not intersect. The notion of packing allows one to make some problems on graphs more natural or more general. Clearly, two $n$-vertex graphs $G_1$ and $G_2$ pack if and only if $G_1$ is a subgraph of the complement $\overline{G}_2$ of $G_2$.

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.