Let $X=G/H$ be a reductive symmetric space and $K$ a maximal compact subgroup of $G$. We study Fourier transforms of compactly supported $K$-finite distributions on $X$ and characterize the image of the space of such distributions.

We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a non-commutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz–Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various authors. We prove that bivariant equivariant periodic cyclic homology is homotopy invariant, stable and satisfies excision in both variables. Moreover, we construct the exterior product which generalizes the obvious composition product. Finally, we prove a Green–Julg theorem in cyclic homology for compact groups and the dual result for discrete groups.

Nous présentons une étude des relations d’équivalence mesurées du point de vue spectral. En particulier nous montrons que pour une relation d’équivalence donnée, la propriété T de Kazhdan, l’ergodicité forte et la moyennabilité peuvent être caractérisées par la présence d’un trou dans le spectre de certaines marches aléatoires sur les orbites de cette relation (à coefficients dans des représentations hilbertiennes appropriées). Les démonstrations reposent de façon essentielle sur l’étude des relations d’équivalence moyennables (Connes–Feldman–Weiss), sur les caractérisations spectrales correspondantes, pour les groupes dénombrables, de la moyennabilité (théorème de Kesten) et de la propriété T de Kazhdan (Gromov, Ghys, etc.), ainsi que sur des résultats techniques sur les représentations unitaires de relations d’équivalence mesurées (en particulier au voisinage de la représentation triviale) que nous développons au cours de l’article. Enfin nous obtenons un analogue du “critère $\lambda_1>1/2$” pour les relations d’équivalence mesurées.

It is a well known theorem due to Kesten that amenability for discrete groups can be characterized in terms of the spectra of diffusion operators associated to random walks on the Cayley graph of these groups. In this paper we are interested in analogous results in the framework of discrete measured equivalence relations. Our main results concern characterizations of Kazhdan’s property T, amenability, and the non existence of amenable quotients (strong ergodicity), in terms of the spectra of diffusion operators associated to random walks and hilbertian representations of the underlying equivalence relation. Our arguments are based on the proof of Connes–Feldman–Weiss’s classification of amenable equivalence relations, the spectral characterizations, in the group case, of amenability (Kesten) and property T (Gromov, Ghys, etc.), as well as new results on the representation theory of measured equivalence relations (in particular in the neighbourhood of the trivial representation) and Kazhdan’s property T. As an application we show how Żuk’s ‘$\lambda_1>1/2$’ criterion for property T can be adapted to measured equivalence relations.

We study product theorems for matrix spaces. In particular, we prove the following theorems.

Theorem 1. For all $\varepsilon>0$, there is $\delta>0$ such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$ is a finite set, then either $A$ intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3|>|A|^{1+\delta}$.

Theorem 2. Let $A$ be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$ is contained in a virtually abelian subgroup, or $|A^3|>c|A|^{1+\delta}$ for some absolute constant $\delta>0$.

Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$ is the $3$-fold product set of $A$.

The continuing down-scaling trend of CMOS technology has brought serious deterioration in the accuracy of the SPICE (Simulation Program with Integrated Circuit Emphasis) device models used in the design of chip functions. This is due to in part to hot electron and quantum effects that occur in modern nano-scale MOSFET devices [13, 25, 28, 33, 34]. The focus of this paper is on modeling quantum confinement effects based on the Density-Gradient (DG) model [6, 9, 14], for application in SPICE. Analytic 1-D quantum mechanical (QM) effects correction formulae for the MOSFET inversion charge and electrostatic potential are derived from the DG model using matched asymptotic expansion techniques. Comparison of these new models with numerical data shows good results.

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.