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376. CMB 2003 (vol 46 pp. 265)
Reducing Spheres and Klein Bottles after Dehn Fillings Let $M$ be a compact, connected, orientable, irreducible 3manifold with a
torus boundary. It is known that if two Dehn fillings on $M$ along the
boundary produce a reducible manifold and a manifold containing a Klein
bottle, then the distance between the filling slopes is at most three. This
paper gives a remarkably short proof of this result.
Keywords:Dehn filling, reducible, Klein bottle Category:57M50 
377. CMB 2003 (vol 46 pp. 95)
Cercles de remplissage for the Riemann Zeta Function The celebrated theorem of Picard asserts that each nonconstant entire
function assumes every value infinitely often, with at most one
exception. The Riemann zeta function has this Picard behaviour in a
sequence of discs lying in the critical band and whose diameters tend
to zero. According to the Riemann hypothesis, the value zero would be
this (unique) exceptional value.
Keywords:cercles de remplissage, Riemann zeta function Category:30 
378. CMB 2003 (vol 46 pp. 130)
On Frankel's Theorem In this paper we show that two minimal hypersurfaces in a manifold with
positive Ricci curvature must intersect. This is then generalized to show
that in manifolds with positive Ricci curvature in the integral sense two
minimal hypersurfaces must be close to each other. We also show
what happens if a manifold with nonnegative Ricci curvature admits
two nonintersecting minimal hypersurfaces.
Keywords:Frankel's Theorem Category:53C20 
379. CMB 2003 (vol 46 pp. 122)
On Certain Finitely Generated Subgroups of Groups Which Split Define a group $G$ to be in the class $\mathcal{S}$ if for any
finitely generated subgroup $K$ of $G$ having the property that
there is a positive integer $n$ such that $g^n \in K$ for all
$g\in G$, $K$ has finite index in $G$. We show that a free
product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong
to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of
$C$ is finitely generated.
Keywords:free product with amalgamation, $\HNN$ group, graph of groups, fundamental group Categories:20E06, 20E08, 57M07 
380. CMB 2002 (vol 45 pp. 483)
Diffraction of Weighted Lattice Subsets A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.
Keywords:diffraction, Dirac combs, lattice subsets, homometric sets Categories:52C07, 43A25, 52C23, 43A05 
381. CMB 2002 (vol 45 pp. 428)
Criteria for Simultaneous Solutions of $X^2  DY^2 = c$ and $x^2  Dy^2 = c$ The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2  DY^2 =
c$ and $x^2  Dy^2 = c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}\cite{alfnme}.
Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions Categories:11A55, 11R11, 11D09 
382. CMB 2002 (vol 45 pp. 337)
Surjectivity of $\mod\ell$ Representations Attached to Elliptic Curves and Congruence Primes For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.
Keywords:torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, nonsplit Cartan Categories:11G05, 11F80 
383. CMB 2002 (vol 45 pp. 378)
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 
384. CMB 2002 (vol 45 pp. 161)
Sur les singularitÃ©s de la fonction croissance d'une variÃ©tÃ© non simplement connexe Si $M$ est une vari\'et\'e de dimension $n$, compacte non simplement
connexe, on caract\'erise les m\'etriques riemanniennes sur $M$ dont
la fonction croissance a exactement deux singularit\'es.
Keywords:fonction croissance, singularitÃ©s, fonction de Morse, Cutlocus Category:53B20 
385. CMB 2002 (vol 45 pp. 272)
The Transfer in the Invariant Theory of Modular Permutation Representations II In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.
Keywords:polynomial invariants of finite groups, permutation representation, transfer Category:13A50 
386. CMB 2002 (vol 45 pp. 265)
On the Smirnov Class Defined by the Maximal Function H.~O.~Kim has shown that contrary to the case of
$H^p$space, the Smirnov class $M$ defined by the radial maximal
function is essentially smaller than the classical Smirnov class
of the disk. In the paper we show that these two classes have the
same corresponding locally convex structure, {\it i.e.} they have the
same dual spaces and the same Fr\'echet envelopes. We describe a
general form of a continuous linear functional on $M$ and
multiplier from $M$ into $H^p$, $0 < p \leq \infty$.
Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelope Categories:46E10, 30A78, 30A76 
387. CMB 2002 (vol 45 pp. 213)
Griffiths Groups of Supersingular Abelian Varieties The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over
an algebraically closed field is defined to be the group of homologically
trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of
algebraically trivial algebraic cycles. The main result of this paper is
that the Griffiths group $\Gr^2 (A_{\bar{k}})$ of a supersingular
abelian variety $A_{\bar{k}}$ over the algebraic closure of a finite
field of characteristic $p$ is at most a $p$primary torsion group.
As a corollary the same conclusion holds for supersingular Fermat
threefolds. In contrast, using methods of C.~Schoen it is also
shown that if the Tate conjecture is valid for all smooth
projective surfaces and all finite extensions of the finite ground
field $k$ of characteristic $p>2$, then the Griffiths group of any ordinary
abelian threefold $A_{\bar{k}}$ over the algebraic closure of $k$ is
nontrivial; in fact, for all but a finite number of primes $\ell\ne p$ it
is the case that $\Gr^2 (A_{\bar{k}}) \otimes \Z_\ell \neq 0$.
Keywords:Griffiths group, Beauville conjecture, supersingular Abelian variety, Chow group Categories:14J20, 14C25 
388. CMB 2002 (vol 45 pp. 154)
On the Poisson Integral of Step Functions and Minimal Surfaces Applications of minimal surface methods are made to obtain information
about univalent harmonic mappings. In the case where the mapping arises
as the Poisson integral of a step function, lower bounds for the number
of zeros of the dilatation are obtained in terms of the geometry of the
image.
Keywords:harmonic mappings, dilatation, minimal surfaces Categories:30C62, 31A05, 31A20, 49Q05 
389. CMB 2002 (vol 45 pp. 138)
The Discriminant of a Dihedral Quintic Field Defined by a Trinomial $X^5 + aX + b$ Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be
a root of $X^5 + aX + b$. An explicit formula is given for the
discriminant of $Q(\theta)$.
Keywords:dihedral quintic field, trinomial, discriminant Categories:11R21, 11R29 
390. CMB 2002 (vol 45 pp. 109)
The Distribution of Totatives D.~H.~Lehmer initiated the study of the distribution of totatives, which
are numbers coprime with a given integer. This led to various problems
considered by P.~Erd\H os, who made a conjecture on such distributions.
We prove his conjecture by establishing a theorem on the ordering of
residues.
Keywords:Euler's function, totatives Categories:11A05, 11A07, 11A25 
391. CMB 2002 (vol 45 pp. 97)
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 
392. CMB 2001 (vol 44 pp. 398)
Exponents of Class Groups of Quadratic Function Fields over Finite Fields We find a lower bound on the number of imaginary quadratic extensions
of the function field $\F_q(T)$ whose class groups have an element of
a fixed order.
More precisely, let $q \geq 5$ be a power of an odd prime and let $g$
be a fixed positive integer $\geq 3$. There are $\gg q^{\ell
(\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D)
\leq \ell$ such that the class groups of the quadratic extensions
$\F_q(T,\sqrt{D})$ have an element of order~$g$.
Keywords:class number, quadratic function field Categories:11R58, 11R29 
393. CMB 2001 (vol 44 pp. 504)
Weak Amenability of a Class of Banach Algebras We show that, if a Banach algebra $\A$ is a left ideal in its second
dual algebra and has a left bounded approximate identity, then the
weak amenability of $\A$ implies the ($2m+1$)weak amenability of $\A$
for all $m\geq 1$.
Keywords:$n$weak amenability, left ideals, left bounded approximate identity Categories:46H20, 46H10, 46H25 
394. CMB 2001 (vol 44 pp. 323)
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 
395. CMB 2001 (vol 44 pp. 376)
A Note on $p$Harmonic $1$Forms on Complete Manifolds In this paper we prove that there is no nontrivial $L^{q}$integrably
$p$harmonic $1$form on a complete manifold with nonnegatively Ricci
curvature $(0

396. CMB 2001 (vol 44 pp. 337)
Spectral Transformations of the Laurent Biorthogonal Polynomials, II. Pastro Polynomials We continue to study the simplest closure conditions for chains of
spectral transformations of the Laurent biorthogonal polynomials
($\LBP$). It is shown that the 11periodic $q$closure condition
leads to the $\LBP$ introduced by Pastro. We introduce classes of
semiclassical and LaguerreHahn $\LBP$ associated to generic closure
conditions of the chain of spectral transformations.
Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformations Category:33D45 
397. CMB 2001 (vol 44 pp. 266)
Extension of Maps to Nilpotent Spaces We show that every compactum has cohomological dimension $1$ with respect
to a finitely generated nilpotent group $G$ whenever it has cohomological
dimension $1$ with respect to the abelianization of $G$. This is applied
to the extension theory to obtain a cohomological dimension theory condition
for a finitedimensional compactum $X$ for extendability of every map from
a closed subset of $X$ into a nilpotent $\CW$complex $M$ with finitely
generated homotopy groups over all of $X$.
Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space Categories:55M10, 55S36, 54C20, 54F45 
398. CMB 2001 (vol 44 pp. 210)
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 
399. CMB 2001 (vol 44 pp. 129)
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 
400. CMB 2001 (vol 44 pp. 126)
Each Copy of the Real Line in $\C^2$ is Removable Around 1995, Professors Lupacciolu, Chirka and Stout showed that a
closed subset of $\C^N$ ($N\geq 2$) is removable for holomorphic
functions, if its topological dimension is less than or equal to
$N2$. Besides, they asked whether closed subsets of $\C^2$
homeomorphic to the real line (the simplest 1dimensional sets) are
removable for holomorphic functions. In this paper we propose a
positive answer to that question.
Keywords:holomorphic function, removable set Category:32D20 