Expand all Collapse all | Results 376 - 400 of 408 |
376. CMB 2000 (vol 43 pp. 330)
Maximal Operators and Cantor Sets We consider maximal operators in the plane, defined by Cantor sets of
directions, and show such operators are not bounded on $L^2$ if the
Cantor set has positive Hausdorff dimension.
Keywords:maximal functions, Cantor set, lacunary set Categories:42B25, 43A46 |
377. CMB 2000 (vol 43 pp. 268)
Cockcroft Properties of Thompson's Group In a study of the word problem for groups, R.~J.~Thompson
considered a certain group $F$ of self-homeomorphisms of the Cantor
set and showed, among other things, that $F$ is finitely presented.
Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed
that $F$ is the fundamental group of a finite two-complex $Z^2$
having Euler characteristic one and which is {\em Cockcroft}, in
the sense that each map of the two-sphere into $Z^2$ is
homologically trivial. We show that no proper covering complex of
$Z^2$ is Cockcroft. A general result on Cockcroft properties
implies that no proper regular covering complex of any finite
two-complex with fundamental group $F$ is Cockcroft.
Keywords:two-complex, covering space, Cockcroft two-complex, Thompson's group Categories:57M20, 20F38, 57M10, 20F34 |
378. CMB 2000 (vol 43 pp. 21)
The Commutant of an Abstract Backward Shift A bounded linear operator $T$ on a Banach space $X$ is an abstract
backward shift if the nullspace of $T$ is one dimensional, and the
union of the null spaces of $T^k$ for all $k \geq 1$ is dense in
$X$. In this paper it is shown that the commutant of an abstract
backward shift is an integral domain. This result is used to
derive properties of operators in the commutant.
Keywords:backward shift, commutant Category:47A99 |
379. CMB 2000 (vol 43 pp. 60)
Trivial Units in Group Rings Let $G$ be an arbitrary group and let $U$ be a subgroup of the
normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$
as a subgroup of finite index, then $U = G$. This result can be used
to give an alternative proof of a recent result of Marciniak and
Sehgal on units in the integral group ring of a crystallographic group.
Keywords:units, trace, finite conjugate subgroup Categories:16S34, 16U60 |
380. CMB 2000 (vol 43 pp. 25)
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
extended-real-valued 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 |
381. CMB 2000 (vol 43 pp. 3)
Resolutions of Associative and Lie Algebras Certain canonical resolutions are described for free associative and
free Lie algebras in the category of non-associative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaf-labeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 |
382. CMB 1999 (vol 42 pp. 478)
A Remark On the Moser-Aubin 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 Moser-Aubin inequality.
Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functions Categories:26D15, 58G30 |
383. CMB 1999 (vol 42 pp. 427)
Ramanujan and the Modular $j$-Invariant A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$-invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 |
384. CMB 1999 (vol 42 pp. 335)
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups We derive a necessary and sufficient condition for HNN-extensions
of cyclic subgroup separable groups with cyclic associated
subgroups to be cyclic subgroup separable. Applying this, we
explicitly characterize the residual finiteness and the cyclic
subgroup separability of HNN-extensions of abelian groups with
cyclic associated subgroups. We also consider these residual
properties of HNN-extensions of nilpotent groups with cyclic
associated subgroups.
Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite Categories:20E26, 20E06, 20F10 |
385. CMB 1999 (vol 42 pp. 321)
Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces We shall study some connection between averaging operators and
martingale inequalities in rearrangement invariant function spaces.
In Section~2 the equivalence between Shimogaki's theorem and some
martingale inequalities will be established, and in Section~3 the
equivalence between Boyd's theorem and martingale inequalities with
change of probability measure will be established.
Keywords:martingale inequalities, rearrangement invariant function spaces Categories:60G44, 60G46, 46E30 |
386. CMB 1999 (vol 42 pp. 285)
On Kloosterman Sums with Oscillating Coefficients In this paper we prove: for any positive integers $a$ and $q$ with
$(a,q) =1$, we have uniformly
$$
\sum_{\substack{n \leq N \\ (n,q) = 1, \,n\on \equiv 1 (\mod q)}}
\mu (n) e \left( \frac{a\on}{q} \right) \ll Nd (q) \left\{
\frac{\log^{\frac52} N}{q^{\frac12}} + \frac{q^{\frac15}
\log^{\frac{13}5} N}{N^{\frac15}} \right\}.
$$
This improves the previous bound obtained by D.~Hajela,
A.~Pollington and B.~Smith~\cite{5}.
Keywords:Kloosterman sums, oscillating coefficients, estimate Category:10G10 |
387. CMB 1999 (vol 42 pp. 274)
The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite,
real rank zero $C^\ast$-algebras $E$ and $E'$ for which
there is an isomorphism of ordered groups
$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to
\bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible
with all the coefficient transformations. The $C^\ast$-algebras
$E$ and $E'$ are not isomorphic since there is no $\Theta$
as above which is also compatible with the Bockstein operations.
By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair
of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras
with similar properties.
Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classification Categories:46L35, 46L80, 19K14 |
388. CMB 1999 (vol 42 pp. 190)
Topological Quantum Field Theory and Strong Shift Equivalence Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of
a closed $(d+1)$-dimensional manifold $M$, we define an invariant
taking values in a strong shift equivalence class of matrices. The
notion of strong shift equivalence originated in R.~Williams' work
in symbolic dynamics. The Turaev-Viro module associated to a TQFT
and an infinite cyclic covering is then given by the Jordan form of
this matrix away from zero. This invariant is also defined if the
boundary of $M$ has an $S^1$ factor and the infinite cyclic cover
of the boundary is standard. We define a variant of a TQFT
associated to a finite group $G$ which has been studied by Quinn.
In this way, we recover a link invariant due to D.~Silver and
S.~Williams. We also obtain a variation on the Silver-Williams
invariant, by using the TQFT associated to $G$ in its unmodified form.
Keywords:knot, link, TQFT, symbolic dynamics, shift equivalence Categories:57R99, 57M99, 54H20 |
389. CMB 1999 (vol 42 pp. 139)
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions |
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions Every weakly compact composition operator between weighted Banach
spaces $H_v^{\infty}$ of analytic functions with weighted sup-norms is
compact. Lower and upper estimates of the essential norm of
continuous composition operators are obtained. The norms of the point
evaluation functionals on the Banach space $H_v^{\infty}$ are also
estimated, thus permitting to get new characterizations of compact
composition operators between these spaces.
Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operator Categories:47B38, 30D55, 46E15 |
390. CMB 1999 (vol 42 pp. 198)
Commutators and Analytic Dependence of Fourier-Bessel Series on $(0,\infty)$ In this paper we study the boundedness of the commutators $[b,
S_n]$ where $b$ is a $\BMO$ function and $S_n$ denotes the $n$-th
partial sum of the Fourier-Bessel series on $(0,\infty)$.
Perturbing the measure by $\exp(2b)$ we obtain that certain
operators related to $S_n$ depend analytically on the functional
parameter $b$.
Keywords:Fourier-Bessel series, commutators, BMO, $A_p$ weights Category:42C10 |
391. CMB 1999 (vol 42 pp. 104)
InstabilitÃ© de vecteurs propres d'opÃ©rateurs linÃ©aires We consider some geometric properties of eigenvectors of linear
operators on infinite dimensional Hilbert space. It is proved that
the property of a family of vectors $(x_n)$ to be eigenvectors
$Tx_n= \lambda_n x_n$ ($\lambda_n \noteq \lambda_k$ for $n\noteq k$)
of a bounded operator $T$ (admissibility property) is very instable
with respect to additive and linear perturbations. For instance,
(1)~for the sequence $(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$ to
be admissible for every admissible $(x_n)$ and for a suitable
choice of small numbers $\epsilon_n\noteq 0$ it is necessary and
sufficient that the perturbation sequence be eventually scalar:
there exist $\gamma_n\in \C$ such that $v_n= \gamma_n v_{k}$ for
$n\geq k$ (Theorem~2); (2)~for a bounded operator $A$ to transform
admissible families $(x_n)$ into admissible families $(Ax_n)$ it is
necessary and sufficient that $A$ be left invertible (Theorem~4).
Keywords:eigenvectors, minimal families, reproducing kernels Categories:47A10, 46B15 |
392. CMB 1999 (vol 42 pp. 97)
On Analytic Functions of Bergman $\BMO$ in the Ball Let $B = B_n$ be the open unit ball of $\bbd C^n$ with
volume measure $\nu$, $U = B_1$ and ${\cal B}$ be the Bloch space on
$U$. ${\cal A}^{2, \alpha} (B)$, $1 \leq \alpha < \infty$, is defined
as the set of holomorphic $f\colon B \rightarrow \bbd C$ for which
$$
\int_B \vert f(z) \vert^2 \left( \frac 1{\vert z\vert}
\log \frac 1{1 - \vert z\vert } \right)^{-\alpha}
\frac {d\nu (z)}{1-\vert z\vert} < \infty
$$
if $0 < \alpha <\infty$ and ${\cal A}^{2, 1} (B) = H^2(B)$, the Hardy
space. Our objective of this note is to characterize, in terms of
the Bergman distance, those holomorphic $f\colon B \rightarrow U$ for
which the composition operator $C_f \colon {\cal B} \rightarrow
{\cal A}^{2, \alpha}(B)$ defined by $C_f (g) = g\circ f$,
$g \in {\cal B}$, is bounded. Our result has a corollary that
characterize the set of analytic functions of bounded mean
oscillation with respect to the Bergman metric.
Keywords:Bergman distance, \BMOA$, Hardy space, Bloch function Category:32A37 |
393. CMB 1999 (vol 42 pp. 13)
Dow's Principle and $Q$-Sets A $Q$-set is a set of reals every subset of which is a relative
$G_\delta$. We investigate the combinatorics of $Q$-sets and
discuss a question of Miller and Zhou on the size $\qq$ of the smallest
set of reals which is not a $Q$-set. We show in particular that various
natural lower bounds for $\qq$ are consistently strictly smaller than
$\qq$.
Keywords:$Q$-set, cardinal invariants of the continuum, pseudointersection number, $\MA$($\sigma$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcing Categories:03E05, 03E35, 54A35 |
394. CMB 1999 (vol 42 pp. 125)
Modular Vector Invariants of Cyclic Permutation Representations Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.
Keywords:polynomial invariants of finite groups Category:13A50 |
395. CMB 1999 (vol 42 pp. 118)
Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$ For a compact Hausdorff space with a dense set of isolated points, we
give a complete description of points of weak$^\ast$-norm continuity
in the dual unit ball of the space of Banach space valued functions
that are continuous when the range has the weak topology. As an
application we give a complete description of points of weak-norm
continuity of the unit ball of the space of vector measures when
the underlying Banach space has the Radon-Nikodym property.
Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-ideals Categories:46B20, 46E40 |
396. CMB 1998 (vol 41 pp. 497)
On the construction of HÃ¶lder and Proximal Subderivatives We construct Lipschitz functions such that for all $s>0$ they are
$s$-H\"older, and so proximally, subdifferentiable only on dyadic
rationals and nowhere else. As applications we construct Lipschitz
functions with prescribed H\"older and approximate subderivatives.
Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationals Categories:49J52, 26A16, 26A24 |
397. CMB 1998 (vol 41 pp. 348)
Characterizing continua by disconnection properties We study Hausdorff continua in which every set of certain
cardinality contains a subset which disconnects the space. We show
that such continua are rim-finite. We give characterizations of
this class among metric continua. As an application of our
methods, we show that continua in which each countably infinite set
disconnects are generalized graphs. This extends a result of
Nadler for metric continua.
Keywords:disconnection properties, rim-finite continua, graphs Categories:54D05, 54F20, 54F50 |
398. CMB 1998 (vol 41 pp. 267)
On the nonemptiness of the adjoint linear system of polarized manifold Let $(X,L)$ be a polarized manifold over the complex number field
with $\dim X=n$. In this paper, we consider a conjecture of
M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this
conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs
|L|\leq 0$ for any $n\geq 3$. Moreover we can generalize the
result of Sommese.
Keywords:Polarized manifold, adjoint bundle Categories:14C20, 14J99 |
399. CMB 1998 (vol 41 pp. 207)
An oscillation criterion for first order linear delay differential equations A new oscillation criterion is given for the delay differential
equation $x'(t)+p(t)x \left(t-\tau(t)\right)=0$, where $p$, $\tau
\in \C \left([0,\infty),[0,\infty)\right)$ and the function
$T$ defined by $T(t)=t-\tau(t)$, $t\ge 0$ is increasing and such
that $\lim_{t\to\infty}T(t)=\infty$. This criterion concerns the
case where $\liminf_{t\to\infty} \int_{T(t)}^{t}p(s)\,ds\le
\frac{1}{e}$.
Keywords:Delay differential equation, oscillation Category:34K15 |
400. CMB 1998 (vol 41 pp. 129)
Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the
weighted Bergman spaces of the unit ball in the $n$-dimensional complex
space is considered and two pluriharmonic symbols of commuting
Toeplitz type operators are completely characterized.
Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators. Categories:47B38, 32A37 |