51. CJM 2000 (vol 52 pp. 1057)
 Urakawa, Hajime

The Spectrum of an Infinite Graph
In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.
Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrum Categories:05C50, 58G25 

52. CJM 2000 (vol 52 pp. 897)
 Christiansen, T. J.; Joshi, M. S.

Higher Order Scattering on Asymptotically Euclidean Manifolds
We develop a scattering theory for perturbations of powers of the
Laplacian on asymptotically Euclidean manifolds. The (absolute)
scattering matrix is shown to be a Fourier integral operator
associated to the geodesic flow at time $\pi$ on the boundary.
Furthermore, it is shown that on $\Real^n$ the asymptotics of certain
shortrange perturbations of $\Delta^k$ can be recovered from the
scattering matrix at a finite number of energies.
Keywords:scattering theory, conormal, Lagrangian Category:58G15 

53. CJM 2000 (vol 52 pp. 757)
 Hanani, Abdellah

Le problÃ¨me de Neumann pour certaines Ã©quations du type de MongeAmpÃ¨re sur une variÃ©tÃ© riemannienne
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 MongeAmp\`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 nondecreasing function in $u$.
Keywords:connexion de LeviCivita, Ã©quations de MongeAmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 

54. CJM 2000 (vol 52 pp. 849)
 Sukochev, F. A.

Operator Estimates for Fredholm Modules
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 selfadjoint 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 noncommutative $L_p$space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{1/2}$ belongs to the
noncommutative 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.
Categories:46L50, 46E30, 46L87, 47A55, 58B15 

55. CJM 2000 (vol 52 pp. 695)
 Carey, A.; Farber, M.; Mathai, V.

Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion
Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and
represents a volume element there. Here we utilise the theory of
determinant lines of Hilbertian modules over finite von~Neumann
algebras as developed in \cite{CFM}. This specialises to the
RaySingerQuillen holomorphic torsion in the finite dimensional case.
We compute a metric variation formula for the holomorphic $L^2$
torsion, which shows that it is {\it not\/} in general independent of
the choice of Hermitian metrics on the complex manifold and on the
holomorphic Hilbertian bundle, which are needed to define it. We
therefore initiate the theory of correspondences of determinant lines,
that enables us to define a relative holomorphic $L^2$ torsion for a
pair of flat Hilbertian bundles, which we prove is independent of the
choice of Hermitian metrics on the complex manifold and on the flat
Hilbertian bundles.
Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost KÃ¤hler manifolds, von~Neumann algebras, determinant lines Categories:58J52, 58J35, 58J20 

56. CJM 2000 (vol 52 pp. 582)
 Jeffrey, Lisa C.; Weitsman, Jonathan

Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems
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 $d$.
Category:58F05 

57. CJM 2000 (vol 52 pp. 119)
58. CJM 1999 (vol 51 pp. 952)
 Deitmar, Anton; Hoffmann, Werner

On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rankone 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
squareintegrable $\Gamma$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 DeGeorgeWallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 

59. CJM 1999 (vol 51 pp. 816)
 Hall, Brian C.

A New Form of the SegalBargmann Transform for Lie Groups of Compact Type
I consider a twoparameter 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 infinitedimensional analysis in joint work with
B.~Driver, but are treated here by finitedimensional means. These
transforms interpolate between two previously known transforms, and
all should be thought of as generalizations of the classical
SegalBargmann transform. I consider also the limiting cases $s
\rightarrow \infty$ and $s \rightarrow t/2$.
Categories:22E30, 81S30, 58G11 

60. CJM 1999 (vol 51 pp. 585)
 Mansfield, R.; MovahediLankarani, H.; Wells, R.

Smooth Finite Dimensional Embeddings
We give necessary and sufficient conditions for a normcompact 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.
Keywords:tangent space, diffeomorphism, manifold, spherically compact, paratingent, quasibundle, embedding Categories:57R99, 58A20 

61. CJM 1999 (vol 51 pp. 250)
 Combari, C.; Poliquin, R.; Thibault, L.

Convergence of Subdifferentials of Convexly Composite Functions
In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'eKuratowski
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 secondorder derivability of
convexly composite functions.
Keywords:epiconvergence, Mosco convergence, PainlevÃ©Kuratowski convergence, primallowernice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 

62. CJM 1999 (vol 51 pp. 266)
 Deitmar, Anton; Hoffman, Werner

Spectral Estimates for Towers of Noncompact Quotients
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
cutoff Laplacian. The proof generalizes the heatkernel method which
has been applied by Donnelly in the case of a fixed lattice~$\Ga$.
Categories:11F72, 58G25, 22E40 

63. CJM 1999 (vol 51 pp. 26)
 Fabian, Marián; Mordukhovich, Boris S.

Separable Reduction and Supporting Properties of FrÃ©chetLike Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chetlike
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 BishopPhelps density theorem.
Moreover, in this way we establish new characterizations of Asplund
spaces in terms of $\epsilon$normals.
Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chetlike normals and subdifferentials, supporting properties, Asplund spaces Categories:49J52, 58C20, 46B20 

64. CJM 1998 (vol 50 pp. 497)
65. CJM 1998 (vol 50 pp. 134)
 Médan, Christine

On critical level sets of some two degrees of freedom integrable Hamiltonian systems
We prove that all Liouville's tori generic bifurcations of a
large class of two degrees of freedom integrable Hamiltonian
systems (the so called JacobiMoserMumford 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'fandDikii's system). We also check the Bott property
for two interesting systems: the Lagrange top and the geodesic
flow on an ellipsoid.
Categories:70H05, 70H10, 58F14, 58F07 

66. CJM 1997 (vol 49 pp. 820)
 Robart, Thierry

Sur l'intÃ©grabilitÃ© des sousalgÃ¨bres de Lie en dimension infinie
Une des questions fondamentales de la th\'eorie des groupes de
Lie de dimension infinie concerne l'int\'egrabilit\'e des
sousalg\`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 sousalg\`ebres de Lie
ferm\'ees $\cal H$ de $\cal G$ nonint\'egrables en un
sousgroupe 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
physiquemath\'ematique et les groupes formels de
diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.
Categories:22E65, 58h05, 17B65 

67. CJM 1997 (vol 49 pp. 855)
 Smith, Samuel Bruce

Rational Classification of simple function space components for flag manifolds.
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.
Keywords:Rational homotopy theory, SullivanHaefliger model. Categories:55P62, 55P15, 58D99. 

68. CJM 1997 (vol 49 pp. 583)
 Pal, Janos; Schlomiuk, Dana

Summing up the dynamics of quadratic Hamiltonian systems with a center
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
algebrogeometric 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.
Categories:34C, 58F 

69. CJM 1997 (vol 49 pp. 212)
 Coll, B.; Gasull, A.; Prohens, R.

Differential equations defined by the sum of two quasihomogeneous vector fields
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
quasihomogeneous 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 nonlinearities.
Categories:34C05, 58F21 

70. CJM 1997 (vol 49 pp. 359)
 Sawyer, P.

Estimates for the heat kernel on $\SL (n,{\bf R})/\SO (n)$
In \cite{Anker}, JeanPhilippe 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.
Categories:58G30, 53C35, 58G11 

71. CJM 1997 (vol 49 pp. 338)
 Rousseau, C.; Toni, B.

Local bifurcations of critical periods in the reduced Kukles system
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
ChristopherLloyd 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.
Categories:34C25, 58F14 

72. CJM 1997 (vol 49 pp. 232)
 Edward, Julian

Spectral theory for the Neumann Laplacian on planar domains with hornlike ends
The spectral theory for the Neumann Laplacian on planar domains with
symmetric, hornlike 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.
Categories:35P25, 58G25 
