26. CMB 2008 (vol 51 pp. 249)
 Mangoubi, Dan

On the Inner Radius of a Nodal Domain
Let $M$ be a closed Riemannian manifold.
We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$.
We give upper and lower bounds on the inner radius of the type
$C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on
a local behavior of eigenfunctions discovered by Donnelly and
Fefferman and a Poincar\'{e} type inequality proved by Maz'ya.
Sharp lower bounds are known
only in dimension two. We give an account of this case too.
Categories:58J50, 35P15, 35P20 

27. CMB 2008 (vol 51 pp. 100)
 Petkov, Vesselin

Dynamical Zeta Function for Several Strictly Convex Obstacles
The behavior of the dynamical zeta function $Z_D(s)$ related to
several strictly convex disjoint obstacles is similar to that of the
inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function
$\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing
only over primitive periodic rays. In this paper we examine the
analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re
s = s_2$, where $s_2$ is the abscissa of absolute convergence of the
series obtained by the second iterations of the primitive periodic
rays. We show that at least one of the functions $Z_D(s), \Pi(s)$
has a singularity at $s = s_2$.
Keywords:dynamical zeta function, periodic rays Categories:11M36, 58J50 

28. CMB 2007 (vol 50 pp. 447)
 Śniatycki, Jędrzej

Generalizations of Frobenius' Theorem on Manifolds and Subcartesian Spaces
Let $\mathcal{F}$ be a family of vector fields on a manifold or a
subcartesian space spanning a distribution $D$. We prove that an orbit $O$
of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$
and it has constant rank on $O$. This result implies Frobenius' theorem, and
its various generalizations, on manifolds as well as on subcartesian spaces.
Keywords:differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theorem Categories:58A30, 58A40 

29. CMB 2007 (vol 50 pp. 113)
 Li, ZhenYang; Zhang, Xi

Hermitian Harmonic Maps into Convex Balls
In this paper, we consider Hermitian harmonic maps from
Hermitian manifolds into convex balls. We prove that there exist
no nontrivial Hermitian harmonic maps from closed Hermitian
manifolds into convex balls, and we use the heat flow method to
solve the Dirichlet problem for Hermitian harmonic maps when the
domain is a compact Hermitian manifold with nonempty boundary.
Keywords:Hermitian harmonic map, Hermitian manifold, convex ball Categories:58E15, 53C07 

30. CMB 2006 (vol 49 pp. 337)
 Berlanga, R.

Homotopy Equivalence and Groups of MeasurePreserving Homeomorphisms
It is shown that the group of compactly
supported, measurepreserving homeomorphisms of a
connected, second countable manifold is locally contractible in the direct limit topology.
Furthermore, this group is weakly homotopically equivalent to the more general group of
compactly supported homeomorphisms.
Categories:57S05, 58F11 

31. CMB 2006 (vol 49 pp. 226)
 Engman, Martin

The Spectrum and Isometric Embeddings of Surfaces of Revolution
A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.
Categories:58J50, 58J53, 53C20, 35P15 

32. CMB 2006 (vol 49 pp. 36)
33. CMB 2005 (vol 48 pp. 340)
 Andruchow, Esteban

Short Geodesics of Unitaries in the $L^2$ Metric
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$,
and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group
$U_\M$ as a subset of $\l2$ and characterize the shortest smooth
curves joining two fixed unitaries in the $L^2$ metric. As a
consequence of this we obtain that $U_\M$, though a complete (metric)
topological group, is not an embedded riemannian submanifold of $\l2$
Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds. Categories:46L51, 58B10, 58B25 

34. CMB 2005 (vol 48 pp. 283)
 Thibault, Lionel; Zagrodny, Dariusz

Enlarged Inclusion of Subdifferentials
This paper studies the integration of inclusion of subdifferentials. Under
various verifiable conditions, we obtain that if two proper lower
semicontinuous functions $f$ and $g$ have the subdifferential of $f$
included in the $\gamma$enlargement of the subdifferential of $g$, then
the difference of those functions is $ \gamma$Lipschitz over their
effective domain.
Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable function Categories:49J52, 46N10, 58C20 

35. CMB 2004 (vol 47 pp. 607)
 Plamenevskaya, Olga

A Residue Formula for $\SU(2)$Valued Moment Maps
Jeffrey and Kirwan gave expressions
for intersection pairings on the reduced space
$M_0=\mu^{1}(0)/G$ of a Hamiltonian $G$space $M$
in terms of multiple residues.
In this paper we prove a residue formula for
symplectic volumes of reduced spaces of a quasiHamiltonian
$\SU(2)$space. The definition of quasiHamiltonian
$G$spaces was recently introduced in .
Category:58F05 

36. CMB 2004 (vol 47 pp. 624)
 Zhang, Xi

A Compactness Theorem for YangMills Connections
In this paper, we consider YangMills connections
on a vector bundle $E$ over a compact Riemannian manifold $M$ of
dimension $m> 4$, and we show that any set of YangMills
connections with the uniformly bounded $L^{\frac{m}{2}}$norm of
curvature is compact in $C^{\infty}$ topology.
Keywords:YangMills connection, vector bundle, gauge transformation Categories:58E20, 53C21 

37. CMB 2004 (vol 47 pp. 515)
 Frigon, M.

Remarques sur l'enlacement en thÃ©orie des points critiques pour des fonctionnelles continues
Dans cet article, \`a partir de la notion d'enlacement introduite
dans ~\cite{F} entre des paires d'ensembles $(B,A)$ et $(Q,P)$,
nous \'etablissons l'existence d'un point critique d'une
fonctionnelle continue sur un espace m\'etrique lorsqu'une de ces
paires enlace l'autre. Des renseignements sur la localisation du
point critique sont aussi obtenus. Ces r\'esultats conduisent \`a
une g\'en\'eralisation du th\'eor\`eme des trois points critiques.
Finalement, des applications \`a des probl\`emes aux limites pour
une \'equation quasilin\'eaire elliptique sont pr\'esent\'ees.
Categories:58E05, 35J20 

38. CMB 2003 (vol 46 pp. 481)
 Bachir, M.; Lancien, G.

On the Composition of Differentiable Functions
We prove that a Banach space $X$ has the Schur property if and only if every
$X$valued weakly differentiable function is Fr\'echet differentiable. We
give a general result on the Fr\'echet differentiability of $f\circ T$, where
$f$ is a Lipschitz function and $T$ is a compact linear operator. Finally
we study, using in particular a smooth variational principle, the
differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces
of Lipschitz functions.
Categories:58C20, 46B20 

39. CMB 2002 (vol 45 pp. 378)
 FernándezLópez, Manuel; GarcíaRío, Eduardo; Kupeli, Demir N.

The Local MÃ¶bius Equation and Decomposition Theorems in Riemannian Geometry
A partial differential equation, the local M\"obius equation, is
introduced in Riemannian geometry which completely characterizes the
local twisted product structure of a Riemannian manifold. Also the
characterizations of warped product and product structures of
Riemannian manifolds are made by the local M\"obius equation and an
additional partial differential equation.
Keywords:submersion, MÃ¶bius equation, twisted product, warped product, product Riemannian manifolds Categories:53C12, 58J99 

40. CMB 2002 (vol 45 pp. 97)
 Haas, Andrew

Invariant Measures and Natural Extensions
We study ergodic properties of a family of interval maps that are
given as the fractional parts of certain real M\"obius
transformations. Included are the maps that are exactly
$n$to$1$, the classical Gauss map and the Renyi or backward
continued fraction map. A new approach is presented for deriving
explicit realizations of natural automorphic extensions and their
invariant measures.
Keywords:Continued fractions, interval maps, invariant measures Categories:11J70, 58F11, 58F03 

41. CMB 2002 (vol 45 pp. 3)
 Azagra, D.; Dobrowolski, T.

RealAnalytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
We prove that every infinitedimensional Banach space $X$ having a
(not necessarily equivalent) realanalytic norm is realanalytic
diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an
infinitedimensional Banach space and $F$ is a closed subspace of $X$
such that there is a realanalytic seminorm on $X$ whose set of zeros
is $F$, and $X/F$ is infinitedimensional, then $X$ and $X \setminus
F$ are realanalytic diffeomorphic. As an application we show the
existence of realanalytic free actions of the circle and the
$n$torus on certain Banach spaces.
Categories:46B20, 58B99 

42. CMB 2001 (vol 44 pp. 376)
43. 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 

44. 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 

45. 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 

46. CMB 2001 (vol 44 pp. 160)
47. 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 

48. CMB 2001 (vol 44 pp. 105)
49. 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 

50. CMB 2000 (vol 43 pp. 183)