In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the Alexander-Taylor-Ullman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and inner-outer factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.
If the Hasse invariant of a $p$-divisible group is small enough,
then one can construct a canonical subgroup inside its $p$-torsion.
We prove that, assuming the existence of a subgroup of adequate
height in the $p$-torsion with high degree, the expected properties
of the canonical subgroup can be easily proved, especially the
relation between its degree and the Hasse invariant. When one
considers a $p$-divisible group with an action of the ring of
integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$,
then much more can be said. We define partial Hasse invariants
(they are natural in the unramified case, and generalize a construction
of Reduzzi and Xiao in the general case), as well as partial
degrees. After studying these functions, we compute the partial
degrees of the canonical subgroup.
We will give a representation-theoretic proof for the multiplication
formula
in the Ringel-Hall algebra
$\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first
application, we see immediately the existence of Hall polynomials
for cyclic quivers, a fact established
by J. Y. Guo and C. M. Ringel,
and derive a recursive formula
to compute them.
We will further use the formula and the construction of a certain
monomial base for $\mathfrak{H}_\Delta(n)$ given
by Deng, Du, and Xiao
together with the double Ringel--Hall algebra realisation of
the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$
given by
Deng, Du, and Fu
to develop some algorithms and to compute the canonical basis
for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples,
we will show explicitly the part of the canonical basis
associated with modules of Lowey length at most $2$ for the quantum
group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho
x+d_{j}$ and
probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested
in the
self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of
finite type.
In this paper we study the multi-fractal analysis of such measures,
extending the theory to measures arising from non-regular probabilities
and
whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists
a subset
of supp$\mu $ of full $\mu $ and Hausdorff measure, called the
truly
essential class, for which the set of (upper or lower) local
dimensions is a
closed interval. Within the truly essential class we show that
there exists
a point with local dimension exactly equal to the dimension of
the support.
We give an example where the set of local dimensions is a two
element set,
with all the elements of the truly essential class giving the
same local
dimension. We give general criteria for these measures to be
absolutely
continuous with respect to the associated Hausdorff measure of
their support
and we show that the dimension of the support can be computed
using only
information about the essential class.
To conclude, we present a detailed study of three examples. First,
we show
that the set of local dimensions of the biased Bernoulli convolution
with
contraction ratio the inverse of a simple Pisot number always
admits an
isolated point. We give a precise description of the essential
class of a
generalized Cantor set of finite type, and show that the $kth$
convolution
of the associated Cantor measure has local dimension at $x\in
(0,1)$ tending
to 1 as $k$ tends to infinity. Lastly, we show that within a
maximal loop
class that is not truly essential, the set of upper local dimensions
need
not be an interval. This is in contrast to the case for finite
type measures
with regular probabilities and full interval support.
Let $k$ be a number field. We describe the category of Laumon
$1$-isomotives over $k$ as the universal category in the sense
of Nori associated with a quiver representation built out of
smooth proper $k$-curves with two disjoint effective divisors
and a notion of $H^1_\mathrm{dR}$ for such "curves with modulus".
This result extends and relies on the theorem of J. Ayoub
and L. Barbieri-Viale that describes Deligne's category
of $1$-isomotives in terms of Nori's Abelian category of motives.
In this paper, we prove the spherical fundamental lemma for
metaplectic group $Mp_{2n}$ based on the formalism of endoscopy
theory by J.Adams, D.Renard and Wen-Wei Li.
We prove that for every surface $\Sigma$ of Euler genus $g$,
every edge-maximal embedding of a graph in $\Sigma$ is at most
$O(g)$ edges short of a triangulation of $\Sigma$. This provides
the first answer to an open problem of Kainen (1974).
In this paper we study a class of second order fully nonlinear
elliptic equations
containing gradient terms on compact Hermitian manifolds and
obtain a priori estimates under
proper assumptions close to optimal.
The analysis developed here should
be useful to deal with other Hessian equations containing gradient
terms in other contexts.