51. CMB 2001 (vol 44 pp. 376)
52. CMB 2001 (vol 44 pp. 323)
 Schuman, Bertrand

Une classe d'hamiltoniens polynomiaux isochrones
Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan
$\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones
qui sont des perturbations polynomiales de $H_0$. On obtient alors
une condition n\'ecessaire d'isochronisme, et un crit\`ere de choix
pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant
une g\'en\'eralisation du caract\`ere isochrone des perturbations
hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S].
Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the
plane $\Rset^2$. We obtain a necessary condition for a system to be
isochronous. We can think of this result as a generalization of the
isochronous behaviour of the homogeneous polynomial perturbation of
the Hamiltonian $H_0$ considered in [L], [P], [S].
Keywords:Hamiltonian system, normal forms, resonance, linearization Categories:34C20, 58F05, 58F22, 58F30 

53. CMB 2001 (vol 44 pp. 210)
 Leung, Man Chun

Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n2}}=0$ in $\R^n$
We construct unbounded positive $C^2$solutions of the equation
$\Delta u + K u^{(n + 2)/(n  2)} = 0$ in $\R^n$ (equipped
with Euclidean metric $g_o$) such that $K$ is bounded between two
positive numbers in $\R^n$, the conformal metric $g=u^{4/(n2)}g_o$
is complete, and the volume growth of $g$ can be arbitrarily fast
or reasonably slow according to the constructions. By imposing natural
conditions on $u$, we obtain growth estimate on the $L^{2n/(n2)}$norm
of the solution and show that it has slow decay.
Keywords:positive solution, conformal scalar curvature equation, growth estimate Categories:35J60, 58G03 

54. CMB 2001 (vol 44 pp. 160)
55. CMB 2001 (vol 44 pp. 140)
 Gotay, Mark J.; Grabowski, Janusz

On Quantizing Nilpotent and Solvable Basic Algebras
We prove an algebraic ``nogo theorem'' to the effect that a
nontrivial \pa\ cannot be realized as an associative algebra with the
commu\ta\tor bracket. Using it, we show that there is an
obstruction to quantizing the \pa\ of polynomials generated by a
nilpotent \ba\ on a \sm. This result generalizes \gr 's famous
theorem on the impossibility of quantizing the Poisson algebra of
polynomials on $\r^{2n}$. Finally, we explicitly construct a
polynomial quantization of a \sm\ with a solvable \ba, thereby showing
that the obstruction in the nilpotent case does not extend to the
solvable case.
Categories:81S99, 58F06 

56. CMB 2001 (vol 44 pp. 129)
 CurrásBosch, Carlos

LinÃ©arisation symplectique en dimension 2
In this paper the germ of neighborhood of a compact leaf in a
Lagrangian foliation is symplectically classified when the compact
leaf is $\bT^2$, the affine structure induced by the Lagrangian
foliation on the leaf is complete, and the holonomy of $\bT^2$ in
the foliation linearizes. The germ of neighborhood is classified by a
function, depending on one transverse coordinate, this function is
related to the affine structure of the nearly compact leaves.
Keywords:symplectic manifold, Lagrangian foliation, affine connection Categories:53C12, 58F05 

57. CMB 2001 (vol 44 pp. 105)
58. CMB 2000 (vol 43 pp. 427)
 Ivey, Thomas A.

Helices, Hasimoto Surfaces and BÃ¤cklund Transformations
Travelling wave solutions to the vortex filament flow generated by
elastica produce surfaces in $\R^3$ that carry mutually orthogonal
foliations by geodesics and by helices. These surfaces are classified
in the special cases where the helices are all congruent or are all
generated by a single screw motion. The first case yields a new
characterization for the B\"acklund transformation for constant
torsion curves in $\R^3$, previously derived from the wellknown
transformation for pseudospherical surfaces. A similar investigation
for surfaces in $H^3$ or $S^3$ leads to a new transformation for
constant torsion curves in those spaces that is also derived from
pseudospherical surfaces.
Keywords:surfaces, filament flow, BÃ¤cklund transformations Categories:53A05, 58F37, 52C42, 58A15 

59. CMB 2000 (vol 43 pp. 183)
60. CMB 2000 (vol 43 pp. 51)
 Edward, Julian

Eigenfunction Decay For the Neumann Laplacian on HornLike Domains
The growth properties at infinity for eigenfunctions corresponding to
embedded eigenvalues of the Neumann Laplacian on hornlike domains
are studied. For domains that pinch at polynomial rate, it is shown
that the eigenfunctions vanish at infinity faster than the reciprocal
of any polynomial. For a class of domains that pinch at an exponential
rate, weaker, $L^2$ bounds are proven. A corollary is that eigenvalues
can accumulate only at zero or infinity.
Keywords:Neumann Laplacian, hornlike domain, spectrum Categories:35P25, 58G25 

61. CMB 2000 (vol 43 pp. 25)
 Bounkhel, M.; Thibault, L.

Subdifferential Regularity of Directionally Lipschitzian Functions
Formulas for the Clarke subdifferential are always expressed in the
form of inclusion. The equality form in these formulas generally
requires the functions to be directionally regular. This paper
studies the directional regularity of the general class of
extendedrealvalued functions that are directionally Lipschitzian.
Connections with the concept of subdifferential regularity are also
established.
Keywords:subdifferential regularity, directional regularity, directionally Lipschitzian functions Categories:49J52, 58C20, 49J50, 90C26 

62. CMB 1999 (vol 42 pp. 478)
 Pruss, Alexander R.

A Remark On the MoserAubin Inequality For Axially Symmetric Functions On the Sphere
Let $\scr S_r$ be the collection of all axially symmetric functions
$f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2}
x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$.
We prove that
$$
\inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} \nabla f^2 \, d\omega
+ 2\int_{\Sph^2} f \, d\omega \log \int_{\Sph^2} e^{2f} \, d\omega > \oo,
$$
and that this infimum is attained. This complements recent work of
Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang
concerning the MoserAubin inequality.
Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functions Categories:26D15, 58G30 
