Expand all Collapse all  Results 351  375 of 419 
351. CMB 2003 (vol 46 pp. 373)
Potential Theory of the FarthestPoint Distance Function We study the farthestpoint distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 
352. CMB 2003 (vol 46 pp. 216)
Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range 
Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of
bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH
\to \HH$ leaving invariant various properties such as invertibility, positive
definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed
{\it a priori\/} continuous. It is shown that under an appropriate surjective
or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto
\xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, 1\}$,
where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis.
Examples are given to show that the surjective or injective assumption cannot
be relaxed. The results are extended to complex linear maps on the algebra of
bounded linear operators on $H$. Similar results are proved for the (real)
linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$
is a scalar and $K$ is compact.
Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical range Categories:47B15, 47B49 
353. CMB 2003 (vol 46 pp. 310)
Second Order Dehn Functions of Asynchronously Automatic Groups Upper bounds of second order Dehn functions of asynchronously
automatic groups are obtained.
Keywords:second order Dehn function, combing, asynchronously automatic group Categories:20E06, 20F05, 57M05 
354. CMB 2003 (vol 46 pp. 268)
Group Cohomology and $L^p$Cohomology of Finitely Generated Groups Let $G$ be a finitely generated, infinite group, let $p>1$, and let
$L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in
G} a_x ^p < \infty \}$. In this paper we will study the first
cohomology group of $G$ with coefficients in $L^p(G)$, and the first
reduced $L^p$cohomology space of $G$. Most of our results will be for a
class of groups that contains all finitely generated, infinite nilpotent
groups.
Keywords:group cohomology, $L^p$cohomology, central element of infinite order, harmonic function, continuous linear functional Categories:43A15, 20F65, 20F18 
355. 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 
356. 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 
357. 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 
358. 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 
359. 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 
360. 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 
361. 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 
362. 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 
363. 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 
364. 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 
365. 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 
366. 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 
367. 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 
368. 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 
369. 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 
370. 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 
371. 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 
372. 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 
373. 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 
374. 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

375. 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 