Expand all Collapse all | Results 1 - 15 of 15 |
1. CJM 2012 (vol 65 pp. 757)
Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved The squared distance curvature is a kind of two-point curvature the
sign of which turned out crucial for the smoothness of optimal
transportation maps on Riemannian manifolds. Positivity properties of
that new curvature have been established recently for all the simply
connected compact rank one symmetric spaces, except the Cayley
plane. Direct proofs were given for the sphere, an indirect one
via the Hopf fibrations) for the complex and quaternionic
projective spaces. Here, we present a direct proof of a property
implying all the preceding ones, valid on every positively curved
Riemannian locally symmetric space.
Keywords:symmetric spaces, rank one, positive curvature, almost-positive $c$-curvature Categories:53C35, 53C21, 53C26, 49N60 |
2. CJM 2011 (vol 64 pp. 151)
Moments of the Rank of Elliptic Curves Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$-function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of Heath-Brown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of low-lying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.
Keywords:elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rank Categories:11G05, 11G40 |
3. CJM 2011 (vol 63 pp. 1284)
Non-Existence of Ramanujan Congruences in Modular Forms of Level Four Ramanujan famously found congruences like $p(5n+4)\equiv 0
\operatorname{mod} 5$ for the partition
function. We provide a method to find all simple
congruences of this type in the coefficients of the inverse of a
modular form on $\Gamma_{1}(4)$ that is non-vanishing on the upper
half plane. This is applied to answer open questions about the
(non)-existence of congruences in the generating functions for
overpartitions, crank differences, and 2-colored $F$-partitions.
Keywords:modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crank Categories:11F33, 11P83 |
4. CJM 2011 (vol 63 pp. 689)
Higher Rank Wavelets A theory of higher rank multiresolution analysis is given in the
setting of abelian multiscalings. This theory enables the
construction, from a higher rank MRA, of finite wavelet sets
whose multidilations have translates forming an orthonormal basis
in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide
simple examples we construct many nonseparable higher rank
wavelets. In particular we construct \emph{Latin square
wavelets} as rank~$2$ variants of Haar wavelets. Also we construct
nonseparable scaling functions for rank $2$ variants of Meyer
wavelet scaling functions, and we construct the associated
nonseparable wavelets with compactly supported Fourier transforms.
On the other hand we show that compactly supported scaling
functions for biscaled MRAs are necessarily separable.
Keywords: wavelet, multi-scaling, higher rank, multiresolution, Latin squares Categories:42C40, 42A65, 42A16, 43A65 |
5. CJM 2009 (vol 62 pp. 646)
Reducibility in A_{R}(K), C_{R}(K), and A(K) Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let
$A_{\mathbb R}(K)$ denote the real Banach algebra of all real
symmetric continuous functions on $K$ that are analytic in the
interior $K^\circ$ of $K$, endowed with the supremum norm. We
characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$
which are reducible.
In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a
new proof (not relying on Banach algebra theory or elementary stable
rank techniques) of the fact that the Bass stable rank of $A(K)$ is
$1$.
Finally, we also characterize all compact real symmetric sets $K$ such
that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass
stable rank $1$.
Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibility Categories:46J15, 19B10, 30H05, 93D15 |
6. CJM 2009 (vol 62 pp. 109)
Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.
Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues Categories:15A42, 15A57 |
7. CJM 2009 (vol 61 pp. 1239)
Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition
for higher rank graphs.
We present a detailed analysis of when this occurs
in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof
of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$.
The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown
that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq
\mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$
where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra.
Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectation Categories:47L55, 47L30, 47L75, 46L05 |
8. CJM 2009 (vol 61 pp. 740)
On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if
$\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We
prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0)
realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup
of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our
result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits
buildings.
Keywords:Coxeter group, flat rank, $\cat0$ space, building Categories:20F55, 51F15, 53C23, 20E42, 51E24 |
9. CJM 2009 (vol 61 pp. 264)
On $\BbZ$-Modules of Algebraic Integers Let $q$ be an algebraic integer of degree $d \geq 2$.
Consider the rank of the multiplicative subgroup of $\BbC^*$ generated
by the conjugates of $q$.
We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$
has norm $\pm 1$, or the rank is $d$.
In this paper we study some properties of $\BbZ[q]$ where $q$ is an
algebraic integer of full rank.
The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number
are also studied.
There are four main results.
\begin{compactenum}[\rm(1)]
\item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive
integer,
then there are only finitely many $m$ such that
$\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$.
\item If $q$ and $r$ are algebraic integers of degree $d$ of full rank
and $\BbZ[q^n] = \BbZ[r^n]$ for
infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$,
where
$r'$ is some conjugate of $r$ and $\omega$ is some root of unity.
\item Let $r$ be an algebraic integer of degree at most $3$.
Then there are at most $40$ Pisot numbers $q$ such that
$\BbZ[q] = \BbZ[r]$.
\item There are only finitely many Pisot-cyclotomic numbers of any fixed
order.
\end{compactenum}
Keywords:algebraic integers, Pisot numbers, full rank, discriminant Categories:11R04, 11R06 |
10. CJM 2008 (vol 60 pp. 1050)
Adjacency Preserving Maps on Hermitian Matrices Hua's fundamental theorem of the geometry of hermitian matrices
characterizes bijective maps on the space of all $n\times n$
hermitian matrices preserving adjacency in both directions.
The problem of possible improvements
has been open for a while. There are three natural problems here.
Do we need the bijectivity assumption? Can we replace the
assumption of preserving adjacency in both directions by the
weaker assumption of preserving adjacency in one direction only?
Can we obtain such a characterization for maps acting between the
spaces of hermitian matrices of different sizes? We answer all
three questions for the complex hermitian matrices, thus obtaining
the optimal structural result for adjacency preserving maps on
hermitian matrices over the complex field.
Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices Categories:15A03, 15A04, 15A57, 15A99 |
11. CJM 2005 (vol 57 pp. 82)
Jordan Structures of Totally Nonnegative Matrices An $n \times n$ matrix is said to be totally nonnegative if every
minor of $A$ is nonnegative. In this paper we completely
characterize all possible Jordan canonical forms of irreducible
totally nonnegative matrices. Our approach is mostly combinatorial
and is based on the study of weighted planar diagrams associated
with totally nonnegative matrices.
Keywords:totally nonnegative matrices, planar diagrams,, principal rank, Jordan canonical form Categories:15A21, 15A48, 05C38 |
12. CJM 2001 (vol 53 pp. 592)
Ideal Structure of Multiplier Algebras of Simple $C^*$-algebras With Real Rank Zero We give a description of the monoid of Murray-von Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.
Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid Categories:46L05, 46L80, 06F05 |
13. CJM 2000 (vol 52 pp. 123)
An Algorithm for Fat Points on $\mathbf{P}^2 Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 |
14. CJM 1998 (vol 50 pp. 719)
Indecomposable almost free modules---the local case Let $R$ be a countable, principal ideal domain which is not a field and
$A$ be a countable $R$-algebra which is free as an $R$-module. Then we
will construct an $\aleph_1$-free $R$-module $G$ of rank $\aleph_1$
with endomorphism algebra End$_RG = A$. Clearly the result does not
hold for fields. Recall that an $R$-module is $\aleph_1$-free if all
its countable submodules are free, a condition closely related to
Pontryagin's theorem. This result has many consequences, depending on
the algebra $A$ in use. For instance, if we choose $A = R$, then
clearly $G$ is an indecomposable `almost free' module. The existence of
such modules was unknown for rings with only finitely many primes like
$R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result
complements a classical realization theorem of Corner's showing that
any such algebra is an endomorphism algebra of some torsion-free,
reduced $R$-module $G$ of countable rank. Its proof is based on new
combinatorial-algebraic techniques related with what we call {\it rigid
tree-elements\/} coming from a module generated over a forest of trees.
Keywords:indecomposable modules of local rings, $\aleph_1$-free modules of rank $\aleph_1$, realizing rings as endomorphism rings Categories:20K20, 20K26, 20K30, 13C10 |
15. CJM 1997 (vol 49 pp. 963)
Homomorphisms from $C(X)$ into $C^*$-algebras Let $A$ be a simple $C^*$-algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHF-algebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$-algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AF-algebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 |