Expand all Collapse all | Results 1 - 25 of 27 |
1. CJM 2014 (vol 67 pp. 152)
On Homotopy Invariants of Combings of Three-manifolds Combings of compact, oriented $3$-dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$-structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$-valued invariant $\theta_G$ of torsion combings
on closed $3$-manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$-structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$-invariant, which is the simplest configuration
space integral invariant of rational homology $3$-balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the Casson-Walker invariant.
The article also includes a self-contained presentation of combings
for $3$-manifolds.
Keywords:Spin$^c$-structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integrals Categories:57M27, 57R20, 57N10 |
2. CJM Online first
Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory We provide some new local obstructions to
approximating
tropical curves in
smooth tropical surfaces. These obstructions are based on
a
relation between tropical and complex intersection theories which is
also established here. We give
two applications of the methods developed in this paper.
First we classify all locally irreducible approximable 3-valent fan tropical
curves in a
fan tropical plane.
Secondly, we prove that a generic non-singular
tropical surface
in tropical projective 3-space contains finitely
many approximable tropical lines
if
it is of degree 3, and contains no approximable tropical lines if
it is of degree 4 or more.
Keywords:tropical geometry, amoebas, approximation of tropical varieties, intersection theory Categories:14T05, 14M25 |
3. CJM Online first
Toric Degenerations, Tropical Curve, and Gromov-Witten Invariants of Fano Manifolds In this paper, we give a tropical method for computing Gromov-Witten
type invariants
of Fano manifolds of special type.
This method applies to those Fano manifolds which admit toric
degenerations
to toric Fano varieties with singularities allowing small resolutions.
Examples include (generalized) flag manifolds of type A, and
some moduli space
of rank two bundles on a genus two curve.
Keywords:Fano varieties, Gromov-Witten invariants, tropical curves Category:14J45 |
4. CJM 2013 (vol 66 pp. 566)
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms |
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of
MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.
Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence Categories:22E50, 11F70, 22E55, 22E35 |
5. CJM 2013 (vol 65 pp. 1217)
Beltrami Equation with Coefficient in Sobolev and Besov Spaces Our goal in this work is to present some function spaces on the
complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of
the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f
(z)$, have first derivatives locally in $X(\mathbb C)$, provided that the
Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.
Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³n-Zygmund operators Categories:30C62, 35J99, 42B20 |
6. CJM 2012 (vol 65 pp. 120)
Universal Families of Rational Tropical Curves We introduce the notion of families of $n$-marked
smooth rational tropical curves over smooth tropical varieties and
establish a one-to-one correspondence between (equivalence classes of)
these families and morphisms
from smooth tropical varieties into the moduli space of $n$-marked
abstract rational tropical curves $\mathcal{M}_{n}$.
Keywords:tropical geometry, universal family, rational curves, moduli space Categories:14T05, 14D22 |
7. CJM 2012 (vol 64 pp. 254)
Corrigendum to ``On $\mathbb{Z}$-modules of Algebraic Integers'' We fix a mistake in the proof of Theorem 1.6 in the paper in the title.
Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theorem Categories:11R04, 11R06 |
8. CJM 2011 (vol 64 pp. 573)
Fundamental Group of Simple $C^*$-algebras with Unique Trace III We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$-algebras.
We show that there exist separable stably projectionless $C^*$-algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.
Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function Categories:46L05, 46L08, 46L35 |
9. CJM 2011 (vol 64 pp. 345)
Salem Numbers and Pisot Numbers via Interlacing We present a general construction of Salem numbers via rational
functions whose zeros and poles mostly lie on the unit circle and
satisfy an interlacing condition. This extends and unifies earlier
work. We then consider the ``obvious'' limit points of the set of Salem
numbers produced by our theorems and show that these are all Pisot
numbers, in support of a conjecture of Boyd. We then show that all
Pisot numbers arise in this way. Combining this with a theorem of
Boyd, we produce all Salem numbers via an interlacing construction.
Keywords:Salem numbers, Pisot numbers Category:11R06 |
10. CJM 2011 (vol 64 pp. 409)
Lifting Quasianalytic Mappings over Invariants Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 |
11. CJM 2011 (vol 63 pp. 460)
Monotonically Controlled Mappings We study classes of mappings between finite and infinite dimensional
Banach spaces that are monotone and mappings which are differences
of monotone mappings (DM). We prove a RadÃ³-Reichelderfer estimate
for monotone mappings in finite dimensional spaces that remains
valid for DM mappings. This provides an alternative proof of the
FrÃ©chet differentiability a.e. of DM mappings. We establish a
Morrey-type estimate for the distributional derivative of monotone
mappings. We prove that a locally DM mapping between finite
dimensional spaces is also globally DM. We introduce and study a new
class of the so-called UDM mappings between Banach spaces, which
generalizes the concept of curves of finite variation.
Keywords: monotone mapping, DM mapping, RadÃ³-Reichelderfer property, UDM mapping, differentiability Categories:26B05, 46G05 |
12. CJM 2009 (vol 61 pp. 641)
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions |
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.
Keywords:space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector Categories:53C40, 53C42 |
13. CJM 2009 (vol 61 pp. 566)
Convex Subordination Chains in Several Complex Variables In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.
Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordination Categories:32H02, 30C45 |
14. 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 |
15. CJM 2008 (vol 60 pp. 1067)
On Types for Unramified $p$-Adic Unitary Groups Let $F$ be a non-archimedean local field of residue characteristic
neither 2 nor 3 equipped with a galois involution with fixed field
$F_0$, and let $G$ be a symplectic group over $F$ or an unramified
unitary group over $F_0$. Following the methods of Bushnell--Kutzko for
$\GL(N,F)$, we define an analogue of a simple type attached to a
certain skew simple stratum, and realize a type in $G$. In
particular, we obtain an irreducible supercuspidal representation of
$G$ like $\GL(N,F)$.
Keywords:$p$-adic unitary group, type, supercuspidal representation, Hecke algebra Categories:22E50, 22D99 |
16. CJM 2008 (vol 60 pp. 721)
Uniform Linear Bound in Chevalley's Lemma We obtain a uniform linear bound for the Chevalley function at a point in
the source of an analytic mapping that is regular in the sense of
Gabrielov. There is a version of
Chevalley's lemma also along a fibre, or at a point of the image of a proper
analytic mapping. We get a uniform linear bound for the Chevalley
function of a closed Nash (or formally Nash) subanalytic set.
Keywords:Chevalley function, regular mapping, Nash subanalytic set Categories:13J07, 32B20, 13J10, 32S10 |
17. CJM 2007 (vol 59 pp. 696)
AlgÃ¨bres de Lie d'homotopie associÃ©es Ã une proto-bigÃ¨bre de Lie On associe \`a toute structure de proto-big\`ebre de Lie sur un espace
vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie
d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre
ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres,
tous les crochets $n$-aires sont nuls pour $n \geq 4$ du fait qu'ils
proviennent d'une structure de proto-big\`ebre de Lie. Plus
g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de
l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une
collection d'applications multilin\'eaires antisym\'etriques sur
l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les
identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres
de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour
le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de
$F$ et de~$F^*$.
To any proto-Lie algebra structure on a finite-dimensional vector
space~$F$, we associate homotopy Lie algebra structures defined on
the suspension of the exterior algebra of $F$ and that of its dual
$F^*$, respectively. In these algebras, all $n$-ary brackets for $n
\geq 4$ vanish because the brackets are defined by the proto-Lie
algebra structure. More generally, to any element of odd degree in
the exterior algebra of the direct sum of $F$ and $F^*$, we associate
a set of multilinear skew-symmetric mappings on the suspension of the
exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized
Jacobi identities, defining the homotopy Lie algebras, if the given
element is of square zero with respect to the big bracket of the
exterior algebra of the direct sum of $F$ and~$F^*$.
Keywords:algÃ¨bre de Lie d'homotopie, bigÃ¨bre de Lie, quasi-bigÃ¨bre de Lie, proto-bigÃ¨bre de Lie, crochet dÃ©rivÃ©, jacobiateur Categories:17B70, 17A30 |
18. CJM 2007 (vol 59 pp. 465)
Searching for Absolute $\mathcal{CR}$-Epic Spaces In previous papers, Barr and Raphael investigated the situation of a
topological space $Y$ and a subspace $X$ such that the induced map
$C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative
rings (with units). We call such an embedding a $\CR$-epic embedding
and we say that $X$ is absolute $\CR$-epic if every embedding of $X$
is $\CR$-epic. We continue this investigation. Our most notable
result shows that a Lindel\"of space $X$ is absolute $\CR$-epic if a
countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta
X$-neighbourhood of $X$. This condition is stable under countable
sums, the formation of closed subspaces, cozero-subspaces, and being
the domain or codomain of a perfect map. A strengthening of the
Lindel\"of property leads to a new class with the same closure
properties that is also closed under finite products. Moreover, all
\s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger
condition. We get some results in the non-Lindel\"of case that are
sufficient to show that the Dieudonn\'e plank and some closely related
spaces are absolute $\CR$-epic.
Keywords:absolute $\mathcal{CR}$-epics, countable neighbourhoo9d property, amply LindelÃ¶f, DiuedonnÃ© plank Categories:18A20, 54C45, 54B30 |
19. CJM 2005 (vol 57 pp. 1012)
Deformations of $G_2$ and $\Spin(7)$ Structures We consider some deformations of $G_2$-structures on $7$-manifolds. We
discover a canonical way to deform a $G_2$-structure by a vector field in
which the associated metric gets ``twisted'' in some way by the
vector cross product. We present a system of partial differential
equations for an unknown vector field $w$ whose solution would
yield a manifold with holonomy $G_2$. Similarly we consider analogous
constructions for $\Spin(7)$-structures on $8$-manifolds. Some of
the results carry over directly, while others do not because of the
increased complexity of the $\Spin(7)$ case.
Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross product Categories:53C26, 53C29 |
20. CJM 2000 (vol 52 pp. 522)
On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems |
On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems We consider the problem
\begin{equation*}
\begin{cases}
\varepsilon^2 \Delta u - u + f(u) = 0, u > 0 & \mbox{in } \Omega\\
\frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega,
\end{cases}
\end{equation*}
where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small
parameter and $f$ is a superlinear, subcritical nonlinearity. It is
known that this equation possesses multiple boundary spike solutions
that concentrate, as $\epsilon$ approaches zero, at multiple critical
points of the mean curvature function $H(P)$, $P \in \partial \Omega$.
It is also proved that this equation has multiple interior spike solutions
which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$.
In this paper, we prove the existence of solutions with multiple spikes
{\it both\/} on the boundary and in the interior. The main difficulty
lies in the fact that the boundary spikes and the interior spikes usually
have different scales of error estimation. We have to choose a special set
of boundary spikes to match the scale of the interior spikes in a
variational approach.
Keywords:mixed multiple spikes, nonlinear elliptic equations Categories:35B40, 35B45, 35J40 |
21. CJM 1999 (vol 51 pp. 673)
Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of $\R^d$ formed in a manner
analogous to the construction of the Sierpinski carpet. We prove a
uniform Harnack inequality for positive harmonic functions; study
the heat equation, and obtain upper and lower bounds on the heat
kernel which are, up to constants, the best possible; construct a
locally isotropic diffusion $X$ and determine its basic properties;
and extend some classical Sobolev and Poincar\'e inequalities to
this setting.
Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions Categories:60J60, 60B05, 60J35 |
22. CJM 1999 (vol 51 pp. 470)
Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations In this article we characterize the univalent harmonic mappings from
the exterior of the unit disk, $\Delta$, onto a simply connected
domain $\Omega$ containing infinity and which are solutions of the system
of elliptic partial differential equations $\fzbb = a(z)f_z(z)$
where the second dilatation function $a(z)$ is a finite Blaschke
product. At the end of this article, we apply our results to
nonparametric minimal surfaces having the property that the image
of its Gauss map is the upper half-sphere covered once or twice.
Keywords:harmonic mappings, minimal surfaces Categories:30C55, 30C62, 49Q05 |
23. CJM 1999 (vol 51 pp. 250)
Convergence of Subdifferentials of Convexly Composite Functions In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'e-Kuratowski
convergence of the graphs of the subdifferentials of convexly
composite functions. We also provide applications to the
convergence of multipliers of families of constrained optimization
problems and to the generalized second-order derivability of
convexly composite functions.
Keywords:epi-convergence, Mosco convergence, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 |
24. CJM 1997 (vol 49 pp. 1281)
Pieri's formula via explicit rational equivalence Pieri's formula describes the intersection product of a Schubert
cycle by a special Schubert cycle on a Grassmannian.
We present a new geometric proof,
exhibiting an explicit chain of rational equivalences
from a suitable sum of distinct Schubert cycles
to the intersection of a Schubert cycle with a special
Schubert cycle. The geometry of these rational equivalences
indicates a link to a combinatorial proof of Pieri's formula using
Schensted insertion.
Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion Categories:14M15, 05E10 |
25. CJM 1997 (vol 49 pp. 887)
Polynomials with $\{ 0, +1, -1\}$ coefficients and a root close to a given point For a fixed algebraic number $\alpha$ we
discuss how closely $\alpha$ can be approximated by
a root of a $\{0,+1,-1\}$ polynomial of given degree.
We show that the worst rate of approximation tends to
occur for roots of unity, particularly those of small degree.
For roots of unity these bounds depend on
the order of vanishing, $k$, of the polynomial at $\alpha$.
In particular we obtain the following. Let
${\cal B}_{N}$ denote the set of roots of all
$\{0,+1,-1\}$ polynomials of degree at most $N$ and
${\cal B}_{N}(\alpha,k)$ the roots of those
polynomials that have a root of order at most $k$
at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$
we show that
\[
\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha
-\beta| \asymp \frac{1}{\alpha^{N}},
\]
and for a root of unity $\alpha$ that
\[
\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}}
|\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil
\frac{1}{2}\phi (d)\right\rceil +1}}.
\]
We study in detail the case of $\alpha=1$, where, by far, the
best approximations are real.
We give fairly precise bounds on the closest real root to 1.
When $k=0$ or 1 we
can describe the extremal polynomials explicitly.
Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separation Categories:11J68, 30C10 |