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.
Suppose that $\g \in C^2\bigl([0,\infty)\bigr)$ is a real-valued function
such that $\g(0)=\g'(0)=0$, and $\g''(t)\approx t^{m-2}$, for some integer
$m\geq 2$. Let $\Gamma (t)=\bigl(t,\g(t)\bigr)$, $t>0$, be a curve in the
plane, and let $d \lambda =dt$ be a measure on this curve. For a
function $f$
on $\bR^2$, let
$$
Tf(x)=(\lambda *f)(x)=\int_0^{\infty} f\bigl(x-\Gamma(t)\bigr)\,dt,
\quad x\in\bR^2 .
$$
An elementary proof is given for the optimal $L^p$-$L^q$ mapping
properties of $T$.
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.
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.
Let $G$ be a real Lie group and $X$ a real analytic manifold.
Suppose that $G$ acts analytically on $X$ with finitely many
orbits. Then the orbits are subanalytic in $X$. As a consequence
we show that the micro-support of a $G$-equivariant sheaf on $X$ is
contained in the conormal variety of the $G$-action.
The growth properties at infinity for eigenfunctions corresponding to
embedded eigenvalues of the Neumann Laplacian on horn-like 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.
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.
We prove that the $L^2$ restriction theorem, and $L^p \to L^{p'}$,
$\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages
imply certain geometric restrictions on the underlying
hypersurface. We deduce that these bounds imply that a certain
number of principal curvatures do not vanish.
We shall provide a characterization of all isoparametric hypersurfaces
$M$'s in a real space form $\tilde{M}(c)$ by observing the extrinsic
Wshape of geodesics of $M$ in the ambient manifold $\tilde{M}(c)$.
An explicit classification is given of blocks of cyclic defect of
cyclotomic Schur algebras and of cyclotomic Hecke algebras, over
discrete valuation rings.
Let $G$ be a hermitian quaternionic group. We determine complementary
series for representations of $G$ induced from super-cuspidal
representations of a Levi factor of the Siegel maximal parabolic
subgroup of $G$.
A counterexample is given to a conjecture of Ikeda by finding a class of
Gorenstein rings of embedding dimension $3$ with larger Dilworth number than
Sperner number. The Dilworth number of $A[Z/pZ\oplus Z/pZ]$ is computed
when $A$ is an unramified principal Artin local ring.
We observe that any set of uniqueness for the Dirichlet space $\cD$
is a set of uniqueness for the class $S$ of normalized univalent
holomorphic functions.
The Four Color Theorem says that the faces (or vertices) of a plane
graph may be colored with four colors. Vizing's Theorem says that the
edges of a graph with maximum degree $\Delta$ may be colored with
$\Delta+1$ colors. In 1972, Kronk and Mitchem conjectured that the
vertices, edges, and faces of a plane graph may be simultaneously
colored with $\Delta+4$ colors. In this article, we give a simple
proof that the conjecture is true if $\Delta \geq 6$.
An infinite family of perfect, non-extremal Riemann surfaces
is constructed, the first examples of this type of surfaces.
The examples are based on normal subgroups of the modular group
$\PSL(2,{\sf Z})$ of level $6$. They provide non-Euclidean
analogues to the existence of perfect, non-extremal positive
definite quadratic forms. The analogy uses the function {\it syst\/}
which associates to every Riemann surface $M$ the length of a systole,
which is a shortest closed geodesic of $M$.
Soient $A$ un anneau noeth\'erien, $B$ un anneau r\'egulier
essentiellement de type fini sur $A$. Si la cohomologie
d'Andr\'e-Quillen $H^q (A,B,B) = 0$ pour tout $q \geq 2$ alors $A$
est un anneau r\'egulier.
Let $E$ be a stable rank 2 vector bundle on a smooth projective
curve $X$ and $V(E)$ be the set of all rank~1 subbundles of $E$
with maximal degree. Here we study the varieties (non-emptyness,
irreducibility and dimension) of all rank~2 stable vector bundles,
$E$, on $X$ with fixed $\deg(E)$ and $\deg(L)$, $L \in V(E)$ and
such that $\card \bigl( V(E) \bigr) \geq 2$ (resp. $\card \bigl(
V(E) \bigr) = 2$).
We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet
spaces $E$ and $F$ that we have the topological identity
$$
\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H}
\Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega
\biggr).
$$
Analogous results for the compact open topology have long been
established. We also give an example to show that the (Qno)
hypothesis on both $E$ and $F$ is necessary.
In this paper, we show that the absolute value of the signature of
the $2$-parallel version of a link is less than or equal to the
nullity of it and show that the signature, nullity, and Minkowski
units of the $2$-parallel version of a certain class of links are
always equal to $0$, $2$, and $1$ respectively.
We prove that Ornstein transformations are almost surely totally
ergodic provided only that the cutting parameter is not bounded.
We thus obtain a larger class of Ornstein transformations with the
mixing property.
The purpose of this note is to give a simple combinatorial
construction of the map from the canonically compactified moduli
spaces of punctured complex projective lines to the moduli spaces
$\P_r$ of polygons with fixed side lengths in the Euclidean space
$\E^3$. The advantage of this construction is that one can obtain a
complete set of linear relations among the cycles that generate
homology of $\P_r$. We also classify moduli spaces of pentagons.
We show that the use of orbifold bundles enables some questions to
be reduced to the case of flat bundles. The identification of
moduli spaces of certain parabolic bundles over elliptic surfaces
is achieved using this method.
If $A$ is a prime C$^*$-algebra, $a \in A$ and $\lambda$ is in the
numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there
exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^*
(a-\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of
$W(a)$, the same conclusion holds without the assumption that $A$ is
prime. Given any element $a$ in a von Neumann algebra (or in a
general C$^*$-algebra) $A$, all normal elements in the weak* closure
(the norm closure, respectively) of the C$^*$-convex hull of $a$ are
characterized.
We show a result slightly more general than the following. Let $K$
be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a
lower semi-continuous metric on $K$. Then each continuous function
$f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on
$K$ which is Lipschitz in $d$. The extension has the same supremum
norm and the same Lipschitz constant.
As a corollary we get that a Banach space $X$ is reflexive if and only
if each bounded, weakly continuous and norm Lipschitz function
defined on a weakly closed subset of $X$ admits a weakly continuous,
norm Lipschitz extension defined on the entire space $X$.
The results herein continue observations on norm form equations and
continued fractions begun and continued in the works
\cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}.
Using spaces introduced by Anick, we construct a decomposition into
indecomposable factors of the double loop spaces of odd primary Moore
spaces when the powers of the primes are greater than the first power.
If $n$ is greater than $1$, this implies that the odd primary part
of all the homotopy groups of the $2n+1$ dimensional sphere lifts
to a $\mod p^r$ Moore space.
We prove that $\{(n^p-n)/p\}_p \in \prod_p \mathbf{F}_p$, with $p$
ranging over all primes, is independent of $1$ over the integers,
assuming a conjecture in elementary number theory generalizing
the infinitude of Mersenne primes. This answers a question of
Buium. We also prove a generalization.
For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
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.
We examine which representations of the absolute Galois group of a
field of finite characteristic with image over a finite field of the
same characteristic may be constructed by the Galois group's action on
the division points of an appropriate Drinfeld module.
Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos
\frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place
au-dessus de $p$. Lorsque l'image de $(2 - 2\cos \frac{2\pi}{r})$
dans $\F$ n'est pas un carr\'e, nous donnons une construction
g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de
Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement
de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3,
sec.~6.3], et son existence est pr\'edite par le crit\`ere de
rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction
s'obtient en tordant la representation galoisienne associ\'ee aux
points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a
multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et
Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un
corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien.
Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que
l'on retrouve quand $r = 2$ et $r = 3$.
Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos
\frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place
above $p$. When the image of $(2 - 2\cos \frac{2\pi}{r})$ in $\F$ is
not a square, we describe a geometric construction of a regular
extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension
corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$''
in the sense of [3, sec.~6.3], and its existence is predicted by the
rigidity criterion of Belyi, Fried, Thompson and Matzat. Its
construction is obtained by twisting the mod $p$ galois representation
attached to a family of abelian varieties with real multiplications by
$K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian
varieties are defined in general over a quadratic field, and are
isogenous to their galois conjugate. Our construction generalises a
method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.
If $n$ and $m$ are positive integers, necessary and sufficient
conditions are given for the existence of a finite commutative ring $R$
with exactly $n$ elements and exactly $m$ prime ideals. Next,
assuming the Axiom of Choice, it is proved that if $R$ is a
commutative ring and $T$ is a commutative $R$-algebra which is
generated by a set $I$, then each chain of prime ideals of $T$ lying
over the same prime ideal of $R$ has at most $2^{|I|}$ elements. A
polynomial ring example shows that the preceding result is
best-possible.
We consider \cst-algebras which are inductive limits of finite
direct sums of copies of $ C([0,1]) \otimes \Otwo$. For such
algebras, the lattice of closed two-sided ideals is proved to be
a complete invariant.
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.
Let $\T$ denote the unit circle in the complex plane, and let $X$ be a
Banach space that satisfies\break Burkholder's UMD condition. Fix a natural
number, $N \in \N$. Let $\od$ denote the reverse lexicographical order
on $\Z^N$. For each $f \in L^1 (\T^N,X)$, there exists a strongly
measurable function $\wt{f}$ such that formally, for all $\bfn \in
\Z^N$, $\Dual{{\wt{f}}} (\bfn) = -i \sgn_\od (\bfn)
\Dual{f} (\bfn)$. In this paper, we present a summation method for
this conjugate function directly analogous to the martingale methods
developed by Asmar and Montgomery-Smith for scalar-valued functions.
Using a stochastic integral representation and an application of
Garling's characterization of UMD spaces, we prove that the associated
maximal operator satisfies a weak-type $(1,1)$ inequality with a
constant independent of the dimension~$N$.
In this paper, we determine some sufficient conditions for an $A +
XB[X]$ domain to be an HFD. As a consequence we give new examples
of HFDs of the type $A + XB[X]$.
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about
equivalence of $L_q$-norms of logconcave random variables
(Kahane-Khinchin's inequality) to the quasi-convex case. We
construct examples of quasi-convex bodies $K_n \subset \R$ which
demonstrate that this equivalence fails for uniformly distributed
vector on $K_n$ (recall that the uniformly distributed vector on a
convex body is logconcave). Our examples also show the lack of the
exponential decay of the ``tail" volume (for convex bodies such
decay was proved by M.~Gromov and V.~Milman).
It is shown that given a local $L$-function defined by Langlands-Shahidi
method, there exists a highly ramified character of the group which when
is twisted with the original representation leads to a trivial
$L$-function.
Let $D$ be a family of $k$-subsets (called blocks) of a $v$-set
$X(v)$. Then $D$ is a $(v,k,t)$ covering design or covering if every
$t$-subset of $X(v)$ is contained in at least one block of $D$. The
number of blocks is the size of the covering, and the minimum size of
the covering is called the covering number. In this paper we consider
the case $t=2$, and find several infinite classes of covering numbers.
We also give upper bounds on other classes of covering numbers.
A binary structure $S$ has the pigeonhole property ($\mathcal{P}$) if
every finite partition of $S$ induces a block isomorphic to $S$. We
classify all countable tournaments with ($\mathcal{P}$); the class of
orders with ($\mathcal{P}$) is completely classified.
The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$,
where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved
that, subject to $n^c a_n$ being eventually monotonic for each
constant $c$ and to the existence of $\alpha := \lim
\frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only
if $\alpha < p$.
We construct a countably infinite family of pairwise non-isomorphic
maximal ${\mathbb Q}[X]$-orders such that the full $2$ by $2$
matrix rings over these orders are all isomorphic.
Let $\cZ$ be the unital simple nuclear infinite dimensional
$C^*$-algebra which has the same Elliott invariant as $\bbC$,
introduced in \cite{JS}. A $C^*$-algebra is called $\cZ$-stable
if $A \cong A \otimes \cZ$. In this note we give some necessary
conditions for a unital simple $C^*$-algebra to be $\cZ$-stable.
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 well-known
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.
Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a
subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider
the {\it generalized matrix function} on $m\times m$ matrices $B =
(b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m
b_{j\sigma(j)}$ and the {\it generalized numerical range} of an
$n\times n$ complex matrix $A$ associated with $d^H$ defined by
$$
\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that }
X^*X = I_m\}.
$$
It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$.
We show that there exist normal matrices $A$ for which $\wmp(A)$ is
not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove
that a normal matrix $A $ with eigenvalues lying on a straight line
has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some
nonzero $\nu \in \IC$. These results extend those of Hu, Hurley
and Tam, who studied the special case when $2 \le m \le 3 \le n$
and $H = S_m$.
We generalize to a field of characteristic zero certain properties of
the invariant functions of the coadjoint representation of solvable
Lie algebras with abelian nilradicals, previously obtained over the
base field $\bbC$ of complex numbers. In particular we determine
their number and the restricted type of variables on which they
depend. We also determine an upper bound on the maximal number of
functionally independent invariants for certain families of solvable
Lie algebras with arbitrary nilradicals.
Many classes of singularly perturbed reaction-diffusion equations
possess localized solutions where the gradient of the solution is
large only in the vicinity of certain points or interfaces in the
domain. The problems of this type that are considered are an
interface propagation model from materials science and an
activator-inhibitor model of morphogenesis. These two models are
formulated as nonlocal partial differential equations. Results
concerning the existence of equilibria, their stability, and the
dynamical behavior of localized structures in the interior and on
the boundary of the domain are surveyed for these two models. By
examining the spectrum associated with the linearization of these
problems around certain canonical solutions, it is shown that the
nonlocal term can lead to the existence of an exponentially small
principal eigenvalue for the linearized problem. This eigenvalue
is then responsible for an exponentially slow, or metastable,
motion of the localized structure.
We extend Maxwell's representation of harmonic polynomials to $h$-harmonics
associated to a reflection invariant weight function $h_k$. Let $\CD_i$,
$1\le i \le d$, be Dunkl's operators associated with a reflection group.
For any homogeneous polynomial $P$ of degree $n$, we prove the
polynomial $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ is
a $h$-harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and
$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for
$h$-harmonics. We also discuss self-adjoint operators acting on the
space of $h$-harmonics.