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The following papers are the latest research papers available from the Canadian Journal of Mathematics.
The papers below are all fully peer-reviewed and we vouch for the research inside.
Some items are labelled Author's Draft,
and others are identified as Published.
As a service to our readers, we post new papers as soon as the science is right, but before official publication; these are the papers marked Author's Draft.
When our copy editing process is complete and the paper now has our official form, we replace the
Author's Draft
with the Published version.
All the papers below are scheduled for inclusion in a Print issue. When that issue goes to press, the paper is moved from this Online First web page over to the main CJM Digital Archive.
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| Lagrange's Theorem for Hopf Monoids in Species Aguiar, Marcelo; Lauve, Aaron Author's Draft
Following Radford's proof of Lagrange's theorem for pointed Hopf algebras,
we prove Lagrange's theorem for Hopf monoids in the category of
connected species.
As a corollary, we obtain necessary conditions for a given subspecies
$\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of
any one of the generating series of $\mathbf h$ by the corresponding
generating series of $\mathbf k$ must have nonnegative coefficients. Other
corollaries include a necessary condition for a sequence of
nonnegative integers to be the
dimension sequence of a Hopf monoid
in the form of certain polynomial inequalities, and of
a set-theoretic Hopf monoid in the form of certain linear inequalities.
The latter express that the binomial transform of the sequence must be nonnegative.
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| The Central Limit Theorem for Subsequences in Probabilistic Number Theory Aistleitner, Christoph; Elsholtz, Christian Published: 2011-12-23
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying
\begin{equation}
\tag{1}
f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad
\operatorname{Var_{[0,1]}}
f \lt \infty.
\end{equation}
If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$
the distribution of
\begin{equation}
\tag{2}
\frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}
\end{equation}
converges to a Gaussian distribution. In the case
$$
1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty
$$
there is a complex interplay between the analytic properties of the
function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$,
and the limit distribution of (2).
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| Triangles of Baumslag-Solitar Groups Allcock, Daniel Published: 2011-09-15
Our main result is that many triangles of Baumslag-Solitar groups
collapse to finite groups, generalizing a famous example of Hirsch and
other examples due to several authors. A triangle of Baumslag-Solitar
groups means a group with three generators, cyclically ordered, with
each generator conjugating some power of the previous one to another
power. There are six parameters, occurring in pairs, and we show that
the triangle fails to be developable whenever one of the parameters
divides its partner, except for a few special cases. Furthermore,
under fairly general conditions, the group turns out to be finite and
solvable of derived length $\leq3$. We obtain a lot of information about
finite quotients, even when we cannot determine developability.
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| Finitely Related Algebras in Congruence Distributive Varieties Have Near Unanimity Terms Barto, Libor Published: 2011-12-24
We show that every finite, finitely related algebra in a congruence
distributive variety has a near unanimity term operation.
As a consequence we solve the near unanimity problem for relational
structures: it is decidable whether a given finite set of relations on
a finite set admits a compatible near unanimity operation. This
consequence also implies that it is decidable whether a given finite
constraint language defines a constraint satisfaction problem of
bounded strict width.
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| Corrigendum to ``On $\mathbb{Z}$-modules of Algebraic Integers'' Bell, Jason P.; Hare, Kevin G. Author's Draft
We fix a mistake in the proof of Theorem 1.6 in the paper in the title.
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| Filters in C*-Algebras Bice, Tristan Matthew Published: 2012-02-03
In this paper we analyze states on C*-algebras and
their relationship to filter-like structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the Kadison-Singer conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for q-points in fact also holds for rapid
p-points, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
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| Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters Blomer, Valentin; Brumley, Farrell Published: 2011-12-23
We prove a non-vanishing result for families of
$\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip,
as one factor runs over twists by Hecke characters. As an
application, we simplify the proof, due to Luo, Rudnick, and Sarnak,
of the best known bounds towards the Generalized Ramanujan Conjecture
at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is
the regularization of the units in residue classes by the use of an
Arakelov ray class group.
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| Densities of Short Uniform Random Walks Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim Published: 2011-11-03
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic
to expect similar results for more than five steps. New results are also
presented concerning the moments of uniform random walks and, in particular,
their derivatives. Relations with Mahler measures are discussed.
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| Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra Brown, Lawrence G.; Lee, Hyun Ho Published: 2011-12-23
We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$-algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murray-von
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.
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| Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces Calvaruso, Giovanni; Fino, Anna Published: 2011-12-23
We study the geometry of non-reductive $4$-dimensional homogeneous
spaces. In particular, after describing their Levi-Civita connection
and curvature properties, we classify homogeneous Ricci solitons on
these spaces, proving the existence of shrinking, expanding and steady
examples. For all the non-trivial examples we find, the Ricci operator
is diagonalizable.
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| Some Functional Inequalities on Polynomial Volume Growth Lie Groups Chamorro, Diego Published: 2011-08-03
In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups.
We show in particular that improved Sobolev inequalities can be extended to this general framework
without the use of the Littlewood-Paley decomposition.
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| Quantum Random Walks and Minors of Hermitian Brownian Motion Chapon, François; Defosseux, Manon Published: 2011-09-19
Considering quantum random walks, we construct discrete-time
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
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| Compactness of Commutators for Singular Integrals on Morrey Spaces Chen, Yanping; Ding, Yong; Wang, Xinxia Published: 2011-07-15
In this paper we characterize the
compactness of the commutator $[b,T]$ for the singular integral
operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More
precisely, we prove that if
$b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO}
(\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$,
then $[b,T]$ is a compact operator on the
Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and
$0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and
$[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$
for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$.
Moreover, the boundedness of a rough singular integral operator $T$
and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also
given. We obtain a sufficient condition for a
subset in Morrey space to be a strongly pre-compact set,
which has interest in its own right.
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| C$^*$-algebras Nearly Contained in Type $\mathrm{I}$ Algebras Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.; White, Stuart Author's Draft
In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type $\mathrm{I}$ C$^*$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\mathrm{I}$ and use this to obtain an embedding of $A$ into $B$.
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| Level Lowering Modulo Prime Powers and Twisted Fermat Equations Dahmen, Sander R.; Yazdani, Soroosh Published: 2011-09-15
We discuss a clean level lowering theorem modulo prime powers
for weight $2$ cusp forms.
Furthermore, we illustrate how this can be used to completely
solve certain twisted Fermat equations
$ax^n+by^n+cz^n=0$.
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| Poisson Brackets with Prescribed Casimirs Damianou, Pantelis A.; Petalidou, Fani Published: 2011-11-15
We consider the problem of constructing Poisson brackets on smooth
manifolds $M$ with prescribed Casimir functions. If $M$ is of even
dimension, we achieve our construction by considering a suitable
almost symplectic structure on $M$, while, in the case where $M$ is
of odd dimension, our objective is achieved by using a convenient
almost cosymplectic structure. Several examples and applications are
presented.
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| On Flag Curvature of Homogeneous Randers Spaces Deng, Shaoqiang; Hu, Zhiguang Author's Draft
In this paper we give an explicit formula for the flag curvature of
homogeneous Randers spaces of Douglas type and apply this formula to
obtain some interesting results. We first deduce an explicit formula
for the flag curvature of an arbitrary left invariant Randers metric
on a two-step nilpotent Lie group. Then we obtain a classification of
negatively curved homogeneous Randers spaces of Douglas type. This
results, in particular, in many examples of homogeneous non-Riemannian
Finsler spaces with negative flag curvature. Finally, we prove a
rigidity result that a homogeneous Randers space of Berwald type whose
flag curvature is everywhere nonzero must be Riemannian.
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| Universal Families of Rational Tropical Curves Francois, Georges; Hampe, Simon Author's Draft
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 $\mathcal{M}_{n}$.
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| Darmon's Points and Quaternionic Shimura Varieties Gärtner, Jérôme Published: 2011-11-22
In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's
points.
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| Systems of Weakly Coupled Hamilton-Jacobi Equations with Implicit Obstacles Gomes, Diogo; Serra, António Published: 2011-11-03
In this paper we study systems of weakly coupled Hamilton-Jacobi equations
with implicit obstacles that arise in optimal switching problems.
Inspired by methods from the theory of viscosity solutions and
weak KAM theory, we
extend the notion of Aubry set for these
systems. This enables us
to prove a new result on existence and uniqueness of
solutions for the Dirichlet problem, answering a question
of F. Camilli, P. Loreti and N. Yamada.
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| A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path Haglund, J.; Morse, J.; Zabrocki, M. Published: 2011-10-22
We introduce a $q,t$-enumeration of Dyck paths that are forced to touch the main diagonal
at specific points and forbidden to touch elsewhere
and conjecture that it describes the action of
the Macdonald theory $\nabla$ operator applied to a Hall--Littlewood
polynomial. Our conjecture refines several earlier conjectures concerning
the space of diagonal harmonics including the ``shuffle conjecture"
(Duke J. Math. $\mathbf {126}$ ($2005$), 195-232) for $\nabla e_n[X]$.
We bring to light that certain generalized Hall--Littlewood polynomials
indexed by compositions are the building blocks for the algebraic
combinatorial theory of $q,t$-Catalan sequences, and we prove a number of
identities involving these functions.
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| Uniquely $D$-colourable Digraphs with Large Girth Harutyunyan, Ararat; Kayll, P. Mark; Mohar, Bojan; Rafferty, Liam Published: 2011-12-06
Let $C$ and $D$ be digraphs. A mapping $f\colon V(D)\to V(C)$ is a
$C$-colouring if for every arc $uv$ of $D$, either $f(u)f(v)$
is an arc of $C$ or $f(u)=f(v)$, and the preimage of every
vertex of $C$ induces an acyclic subdigraph in $D$. We say
that $D$ is $C$-colourable if it admits a $C$-colouring and
that $D$ is uniquely $C$-colourable if it is surjectively
$C$-colourable and any two $C$-colourings of $D$ differ by an
automorphism of $C$. We prove that if a digraph $D$ is not
$C$-colourable, then there exist digraphs of arbitrarily large
girth that are $D$-colourable but not
$C$-colourable. Moreover, for every digraph $D$ that is
uniquely $D$-colourable, there exists a uniquely
$D$-colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number $r\geq
1$, there are uniquely circularly $r$-colourable digraphs with
arbitrarily large girth.
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| Monodromy Filtrations and the Topology of Tropical Varieties Helm, David; Katz, Eric Published: 2011-09-22
We study the topology of tropical varieties that arise from a certain
natural class of varieties. We use the theory of tropical
degenerations to construct a natural, ``multiplicity-free''
parameterization of $\operatorname{Trop}(X)$ by a topological space
$\Gamma_X$ and give a geometric interpretation of the cohomology of
$\Gamma_X$ in terms of the action of a monodromy operator on the
cohomology of $X$. This gives bounds on the Betti numbers of
$\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the
topology of $\operatorname{Trop}(X)$. We also obtain a description of
the top power of the monodromy operator acting on middle cohomology of
$X$ in terms of the volume pairing on $\Gamma_X$.
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| Balayage of Semi-Dirichlet Forms Hu, Ze-Chun; Sun, Wei Published: 2011-08-15
In this paper we study the balayage of semi-Dirichlet forms. We
present new results on balayaged functions and balayaged measures
of semi-Dirichlet
forms. Some of the results are new even in the Dirichlet forms setting.
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| Hermite's Constant for Function Fields Hurlburt, Chris; Thunder, Jeffrey Lin Published: 2011-08-03
We formulate an analog of Hermite's constant for function fields over a finite field and
state a conjectural value for this analog. We prove our conjecture in many cases, and
prove slightly weaker results in all other cases.
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| Boundedness of Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongong Published: 2011-09-15
Let $({\mathcal X}, d, \mu)$ be a
separable metric measure space satisfying the known upper
doubling condition, the geometrical doubling condition, and the
non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$.
In this paper, we show that the boundedness of a Calderón-Zygmund
operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on
$L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$
to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a
Calderón-Zygmund operator bounded on $L^2(\mu)$,
then its maximal operator is bounded on $L^p(\mu)$
for all $p\in (1, \infty)$ and from
the space of all complex-valued Borel measures on
${\mathcal X}$ to $L^{1,\,\infty}(\mu)$.
All these results generalize the corresponding results of Nazarov et al.
on metric spaces with
measures satisfying the so-called polynomial growth condition.
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| Cubic Polynomials with Periodic Cycles of a Specified Multiplier Ingram, Patrick Author's Draft
We consider cubic polynomials $f(z)=z^3+az+b$ defined over
$\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier
$\lambda$. In the case $N=1$, there are infinitely many such objects,
and in the case $N\geq 3$, only finitely many (subject to a mild
assumption). The case $N=2$ has particularly rich structure, and we
are able to describe all such cubic polynomials defined over the field
$\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.
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| Composition Operators Induced by Analytic Maps to the Polydisk Izuchi, Kei Ji; Nguyen, Quang Dieu; Ohno, Shûichi Published: 2011-11-03
We study properties of composition operators
induced by symbols acting from the unit disk to the polydisk.
This result will be involved in the investigation
of weighted composition operators on the Hardy space on the unit disk
and moreover be concerned with composition operators acting
from the Bergman space to the Hardy space on the unit disk.
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| Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces Kadets, Vladimir; Martín, Miguel; Merí, Javier; Werner, Dirk Author's Draft
We show that for spaces with 1-unconditional bases
lushness, the alternative Daugavet property and numerical
index 1 are equivalent. In the class of rearrangement
invariant (r.i.) sequence spaces the only examples of spaces with
these properties are $c_0$, $\ell_1$ and $\ell_\infty$.
The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$;
the same space is the only r.i. separable function space on $[0,1]$
with the Daugavet property over the reals.
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| Equicontinuous Delone Dynamical Systems Kellendonk, Johannes; Lenz, Daniel Published: 2011-12-23
We characterize equicontinuous Delone dynamical systems as those
coming from Delone sets with strongly almost periodic Dirac combs.
Within the class of systems with finite local complexity, the only
equicontinuous systems are then shown to be the crystallographic
ones. On the other hand, within the class without finite local
complexity, we exhibit examples of equicontinuous minimal Delone
dynamical systems that are not crystallographic.
Our results solve the problem posed by Lagarias as to whether a Delone
set whose Dirac comb is strongly almost periodic must be
crystallographic.
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| Bowen Measure From Heteroclinic Points Killough, D. B.; Putnam, I. F. Published: 2011-11-15
We present a new construction of the entropy-maximizing, invariant
probability measure on a Smale space (the Bowen measure). Our
construction is based on points that are unstably equivalent to one
given point, and stably equivalent to another: heteroclinic points.
The spirit of the construction is similar to Bowen's construction from
periodic points, though the techniques are very different. We also
prove results about the growth rate of certain sets of heteroclinic
points, and about the stable and unstable components of the Bowen
measure. The approach we take is to prove results through direct
computation for the case of a Shift of Finite type, and then use
resolving factor maps to extend the results to more general Smale
spaces.
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| Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields Koh, Doowon; Shen, Chun-Yen Published: 2011-12-23
In this paper we study the extension problem, the
averaging problem, and the generalized Erdős-Falconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.
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| Le lemme fondamental pondéré pour le groupe métaplectique Li, Wen-Wei Author's Draft
Dans cet article, on énonce une variante du lemme fondamental
pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un
ingrédient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractéristique résiduelle suffisamment
grande, on en donne une démonstration à l'aide de la méthode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondéré non standard sur les algèbres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend à ce que cette condition soit
ultérieurement vérifiée.
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| On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops Li, Zhiqiang Published: 2011-09-22
A K-theoretic classification is given of the simple inductive limits
of finite direct sums of the
type I $C^*$-algebras known as splitting interval algebras with
dimension drops. (These are the subhomogeneous $C^*$-algebras, each
having spectrum a finite union
of points and an open interval, and torsion $\textrm{K}_1$-group.)
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| Optimal Polynomial Recurrence Lyall, Neil; Magyar, Ákos Author's Draft
Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$.
We show, using Fourier analytic techniques, that if $N\geq
\exp\exp(C\varepsilon^{-1}\log\varepsilon^{-1})$ and
$A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that
\[\frac{|A\cap (A+P(n))|}{N}\gt \left(\frac{|A|}{N}\right)^2-\varepsilon.\]
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| Rectifiability of Optimal Transportation Plans McCann, Robert J.; Pass, Brendan; Warren, Micah Published: 2011-11-03
The regularity of solutions to optimal transportation problems has become
a hot topic in current research. It is well known by now that the optimal measure
may not be concentrated on the graph of a continuous mapping unless both the transportation
cost and the masses transported satisfy very restrictive hypotheses (including sign conditions
on the mixed fourth-order derivatives of the cost function).
The purpose of this note is to show that in spite of this,
the optimal measure is supported on a Lipschitz manifold, provided only
that the cost is $C^{2}$ with non-singular mixed second derivative.
We use this result to provide a simple proof that solutions to Monge's
optimal transportation problem satisfy a change of variables equation
almost everywhere.
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| The H and K Families of Mock Theta Functions McIntosh, Richard J. Published: 2011-10-05
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions.
He observed that as $q$ radially approaches any root of unity $\zeta$ at which
$F(q)$ has an exponential singularity, there is a $\theta$-function
$T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have
been found that possess this property. These functions are related to
a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some
rational number $r$. For this reason we refer to $H$ as a ``universal'' mock
$\theta$-function. Modular transformations of $H$ give rise to the functions
$K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost
notebook. We prove various linear relations between these functions using
Appell-Lerch sums (also called generalized Lambert series). Some relations
(mock theta ``conjectures'') involving mock $\theta$-functions
of even order and $H$ are listed.
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| Salem Numbers and Pisot Numbers via Interlacing McKee, James; Smyth, Chris Published: 2011-08-03
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.
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| C$^*$-Algebras over Topological Spaces: Filtrated K-Theory Meyer, Ralf; Nest, Ryszard Author's Draft
We define the filtrated K-theory of a $\mathrm{C}^*$-algebra over a finite topological space \(X\)
and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over \(X\)
in terms of filtrated K-theory.
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| Fundamental Group of Simple $C^*$-algebras with Unique Trace III Nawata, Norio Published: 2011-08-03
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.
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| Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two Nekovář, Jan Published: 2011-11-15
In this article we refine the method of Bertolini and Darmon
and prove several finiteness results for
anticyclotomic Selmer groups of Hilbert modular forms of parallel
weight two.
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| Note on Cubature Formulae and Designs Obtained from Group Orbits Nozaki, Hiroshi; Sawa, Masanori Published: 2011-11-03
In 1960,
Sobolev proved that for a finite reflection group $G$,
a $G$-invariant cubature formula is of degree $t$ if and only if
it is exact for all $G$-invariant polynomials of degree at most $t$.
In this paper,
we find some observations on invariant cubature formulas and Euclidean designs
in connection with the Sobolev theorem.
First, we give an alternative proof of
theorems by Xu (1998) on necessary and sufficient conditions
for the existence of cubature formulas with some strong symmetry.
The new proof is shorter and simpler compared to the original one by Xu, and
moreover gives a general interpretation of
the analytically-written conditions of Xu's theorems.
Second,
we extend a theorem by Neumaier and Seidel (1988) on
Euclidean designs to invariant Euclidean designs, and thereby
classify tight Euclidean designs obtained from
unions of the orbits of the corner vectors.
This result generalizes a theorem of Bajnok (2007) which classifies
tight Euclidean designs invariant under the Weyl group of type $B$
to other finite reflection groups.
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| The Genuine Omega-regular Unitary Dual of the Metaplectic Group Pantano, Alessandra; Paul, Annegret; Salamanca-Riba, Susana A. Published: 2011-10-22
We classify all genuine unitary representations of the metaplectic group whose
infinitesimal character is real and at least as regular as that of the
oscillator representation. In a previous paper we exhibited a certain family
of representations satisfying these conditions, obtained by cohomological
induction from the tensor product of a one-dimensional representation and an
oscillator representation. Our main theorem asserts that this family exhausts
the genuine omega-regular unitary dual of the metaplectic group.
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| Optimal Roughening of Convex Bodies Plakhov, Alexander Published: 2011-11-03
A body moves in a rarefied medium composed of point particles at
rest. The particles make elastic reflections when colliding with the
body surface, and do not interact with each other. We consider a
generalization of Newton's minimal resistance problem: given two
bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2
\subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the
resistance in the class of connected bodies $B$ such that $C_1 \subset
B \subset C_2$. We prove that the infimum of resistance is zero; that
is, there exist "almost perfectly streamlined" bodies.
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| Lifting Quasianalytic Mappings over Invariants Rainer, Armin Published: 2011-07-15
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.
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| A Stochastic Difference Equation with Stationary Noise on Groups Raja, Chandiraraj Robinson Edward Published: 2011-12-23
We consider the stochastic difference equation $$\eta _k = \xi _k
\phi (\eta _{k-1}), \quad k \in \mathbb Z $$ on a locally compact group $G$
where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued
random variables and $\eta _k$ are unknown $G$-valued random variables.
This equation was considered by Tsirelson and Yor on
one-dimensional torus. We consider the case when $\xi _k$ have a
common law $\mu$ and prove that if $G$ is a distal group and $\phi$
is a distal automorphism of $G$ and if the equation has a solution,
then extremal solutions of the equation are in one-one
correspondence with points on the coset space $K\backslash G$ for
some compact subgroup $K$ of $G$ such that $\mu$ is supported on
$Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide
a necessary and sufficient condition for the existence of solutions
to the equation.
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| Classic and Mirabolic Robinson-Schensted-Knuth Correspondence for Partial Flags Rosso, Daniele Published: 2011-12-31
In this paper we first generalize to the case of
partial flags a result proved both by Spaltenstein and by Steinberg
that relates the relative position of two complete flags and the
irreducible components of the flag variety in which they lie, using
the Robinson-Schensted-Knuth correspondence. Then we use this result
to generalize the mirabolic Robinson-Schensted-Knuth correspondence
defined by Travkin, to the case of two partial flags and a line.
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| $p$-adic $L$-functions and the Rationality of Darmon Cycles Seveso, Marco Adamo Published: 2011-10-05
Darmon cycles are a higher weight analogue of Stark--Heegner points. They
yield local cohomology classes in the Deligne representation associated with a
cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$.
They are conjectured to be the restriction of global cohomology classes in
the Bloch--Kato Selmer group defined over narrow ring class fields attached
to a real quadratic field. We show that suitable linear combinations of them
obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real)
quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express
the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the
weight variable in terms of a global cycle defined over a quadratic
extension of $\mathbb{Q}$.
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| Holomorphic Mappings between Domains in $\mathbb C^2$ Shafikov, Rasul; Verma, Kaushal Published: 2011-08-15
An extension theorem for holomorphic mappings between two domains in
$\mathbb C^2$ is proved under purely local hypotheses.
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| On Cardinal Invariants and Generators for von Neumann Algebras Sherman, David Published: 2011-07-15
We demonstrate how most common cardinal invariants associated with a von
Neumann algebra $\mathcal M$ can be computed from the decomposability number,
$\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating
set, $\operatorname{gen}(\mathcal M)$.
Applications include the equivalence of the well-known generator
problem, ``Is every separably-acting von Neumann algebra
singly-generated?", with the formally stronger questions, ``Is every
countably-generated von Neumann algebra singly-generated?" and ``Is
the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we
determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M),
\operatorname{dens}(\mathcal M) \bigr)$,
which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq
\mathfrak C^{\operatorname{gen}(\mathcal M)}$.
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| Pure Infiniteness of the Crossed Product of an AH-Algebra by an Endomorphism Thomsen, Klaus Published: 2011-10-22
It is shown that simplicity of the crossed product of
a unital AH-algebra with slow dimension growth by an endomorphism
implies that the algebra is also purely infinite, provided only that
the endomorphism leaves no trace state invariant and takes the unit
to a full projection.
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| On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms Van Order, Jeanine Author's Draft
We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel
weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston
and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies
in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same
time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions.
It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique
of Skinner and Urban.
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