We study estimates of the type $$ \Vert \phi(D) - \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D - D_0 \Vert^{\alpha}, \quad \alpha = \frac12, 1 $$ where $\phi(t) = t(1 + t^2)^{-1/2}$, $D_0 = D_0^*$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $\calM$, $D - D_0$ is a bounded self-adjoint linear operator from $\calM$ and $(1 + D_0^2)^{-1/2} \in \emt$, where $\emt$ is a symmetric operator space associated with $\calM$. In particular, we prove that $\phi(D) - \phi(D_0)$ belongs to the non-commutative $L_p$-space for some $p \in (1,\infty)$, provided $(1 + D_0^2)^{-1/2}$ belongs to the non-commutative weak $L_r$-space for some $r \in [1,p)$. In the case $\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this result continues to hold under the weaker assumption $(1 + D_0^2)^{-1/2} \in \calC_p$. This may be regarded as an odd counterpart of A.~Connes' result for the case of even Fredholm modules.

Let $(M_n,g)$ be a strictly convex riemannian manifold with $C^{\infty}$ boundary. We prove the existence\break of classical solution for the nonlinear elliptic partial differential equation of Monge-Amp\`ere:\break $\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM \times \bbR)$ is an everywhere strictly positive function satisfying some assumptions, $\nu$ stands for the unit normal vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$ is a non-decreasing function in $u$.

This paper treats the moduli space ${\cal M}_{g,1}(\Lambda)$ of representations of the fundamental group of a Riemann surface of genus $g$ with one boundary component which send the loop around the boundary to an element conjugate to $\exp \Lambda$, where $\Lambda$ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over ${\cal M}_{g,1} (\Lambda)$ and exhibit natural homology cycles representing the Poincar\'e dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank $k$ and degree $d$.

Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997), 232--262) are corrected. A non-standard regularisation of the auxiliary operator $A$ appearing in Mourre theory is used.

Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.

I consider a two-parameter family $B_{s,t}$ of unitary transforms mapping an $L^{2}$-space over a Lie group of compact type onto a holomorphic $L^{2}$-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B.~Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases $s \rightarrow \infty$ and $s \rightarrow t/2$.

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a $C^1$ embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of $n$-dimensional points is contained in an $n$-dimensional $C^1$ submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of G.~Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.

In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlev\'e-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$.

We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

This paper deals with periodic solutions for the billiard problem in a bounded open set of $\hbox{\Bbbvii R}^N$ which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.

We prove that all Liouville's tori generic bifurcations of a large class of two degrees of freedom integrable Hamiltonian systems (the so called Jacobi-Moser-Mumford systems) are nondegenerate in the sense of Bott. Thus, for such systems, Fomenko's theory~\cite{fom} can be applied (we give the example of Gel'fand-Dikii's system). We also check the Bott property for two interesting systems: the Lagrange top and the geodesic flow on an ellipsoid.

Une des questions fondamentales de la th\'eorie des groupes de Lie de dimension infinie concerne l'int\'egrabilit\'e des sous-alg\`ebres de Lie topologiques $\cal H$ de l'alg\`ebre de Lie $\cal G$ d'un groupe de Lie $G$ de dimension infinie au sens de Milnor. Par contraste avec ce qui se passe en th\'eorie classique il peut exister des sous-alg\`ebres de Lie ferm\'ees $\cal H$ de $\cal G$ non-int\'egrables en un sous-groupe de Lie. C'est le cas des alg\`ebres de Lie de champs de vecteurs $C^{\infty}$ d'une vari\'et\'e compacte qui ne d\'efinissent pas un feuilletage de Stefan. Heureusement cette ``imperfection" de la th\'eorie n'est pas partag\'ee par tous les groupes de Lie int\'eressants. C'est ce que montre cet article en exhibant une tr\`es large classe de groupes de Lie de dimension infinie exempte de cette imperfection. Cela permet de traiter compl\`etement le second probl\`eme fondamental de Sophus Lie pour les groupes de jauge de la physique-math\'ematique et les groupes formels de diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.

Let $M(X,Y)$ denote the space of all continous functions between $X$ and $Y$ and $M_f(X,Y)$ the path component corresponding to a given map $f: X\rightarrow Y.$ When $X$ and $Y$ are classical flag manifolds, we prove the components of $M(X,Y)$ corresponding to ``simple'' maps $f$ are classified up to rational homotopy type by the dimension of the kernel of $f$ in degree two cohomology. In fact, these components are themselves all products of flag manifolds and odd spheres.

In this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection $I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$, $Q(x,y,z) = 0$ at a point $p$ of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

In this paper we prove, that under certain hypotheses, the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$, $\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar\'e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

In \cite{Anker}, Jean-Philippe Anker conjectures an upper bound for the heat kernel of a symmetric space of noncompact type. We show in this paper that his prediction is verified for the space of positive definite $n\times n$ real matrices.

In this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gr\"obner bases, theory of the resultant, Sturm's algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at $0$ or $\infty$. The proof uses Mourre theory.