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We characterize positive links in terms of strong quasipositivity,
homogeneity and the value of Rasmussen and Beliakova-Wehrli's
$s$-invariant.
We also study almost positive links,
in particular, determine the $s$-invariants of
almost positive links.
This result suggests that all almost positive links might
be strongly quasipositive.
On the other hand, it implies that
almost positive links are never homogeneous links.
It is proved that flatness of an analytic mapping germ from a
complete intersection is determined by its sufficiently high
jet. As a consequence, one obtains finite determinacy of complete
intersections. It is also shown that flatness and openness are
stable under deformations.
We classify and analyze the orbits of the Kepler problem on surfaces
of constant curvature (both positive and negative, $\mathbb S^2$ and
$\mathbb H^2$, respectively) as function of the angular momentum and
the energy. Hill's region are characterized and the problem of
time-collision is studied. We also regularize the problem in
Cartesian and intrinsic coordinates, depending on the constant
angular momentum and we describe the orbits of the regularized
vector field. The phase portrait both for $\mathbb S^2$ and $\mathbb H^2$
are pointed out.
We obtain bounds for the number of variables required to establish
Hasse principles, both for existence of solutions and for asymptotic
formulæ, for systems of additive equations containing forms
of differing degree but also multiple forms of like degree.
Apart from the very general estimates of Schmidt and Browning--Heath-Brown,
which give weak results when specialized to the diagonal situation,
this is the first result on such "hybrid" systems. We also obtain
specialised results for systems of quadratic and cubic forms,
where we are able to take advantage of some of the stronger methods
available in that setting. In particular, we achieve essentially
square root cancellation for systems consisting of one cubic
and $r$ quadratic equations.
We provide a simple, general argument to obtain improvements
of concentration-type inequalities starting from improvements
of their corresponding isoperimetric-type inequalities. We apply
this argument to obtain robust improvements of the Brunn-Minkowski
inequality (for Minkowski sums between generic sets and convex
sets) and of the Gaussian concentration inequality. The former
inequality is then used to obtain a robust improvement of the
Riesz rearrangement inequality under certain natural conditions.
These conditions are compatible with the applications to a finite-range
nonlocal isoperimetric problem arising in statistical mechanics.
We study the regularity of convolution powers for measures supported
on
Salem sets, and prove related results on Fourier restriction
and Fourier multipliers. In particular we show
that for $\alpha$ of the form
${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$-Salem measures
for which the $L^2$ Fourier restriction theorem holds in the
range $p\le \frac{2d}{2d-\alpha}$.
The results rely on ideas of Körner.
We extend some of his constructions to obtain upper regular $\alpha$-Salem
measures, with sharp regularity results for $n$-fold convolutions
for all $n\in \mathbb{N}$.
A fundamental idea in toric topology is that classes of manifolds
with well-behaved torus actions (simply, toric spaces) are classified
by pairs of simplicial complexes and (non-singular) characteristic
maps. The authors in their previous paper provided a new way
to find all characteristic maps on a simplicial complex $K(J)$
obtainable by a sequence of wedgings from $K$. The main idea
was that characteristic maps on $K$ theoretically determine all
possible characteristic maps on a wedge of $K$.
In this work, we further develop our previous work for classification
of toric spaces. For a star-shaped simplicial sphere $K$ of dimension
$n-1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is
$m-n$. We refer to $K$ as a seed if $K$ cannot be obtained
by wedgings. First, we show that, for a fixed positive integer
$\ell$, there are at most finitely many seeds of Picard number
$\ell$ supporting characteristic maps. As a corollary, the conjecture
proposed by V.V. Batyrev in 1991 is solved affirmatively.
Second, we investigate a systematic method to find all characteristic
maps on $K(J)$ using combinatorial objects called (realizable)
puzzles that only depend on a seed $K$.
These two facts lead to a practical way to classify the toric
spaces of fixed Picard number.
We review some simple techniques based on monotone mass transport
that allow to obtain transport-type inequalities for any log-concave
probability measure. We discuss quantitative forms of these inequalities,
with application to the variance Brascamp-Lieb inequality.
Building on our previous work, we study the non-relative homology
of quantum group convolution algebras. Our main result establishes
the equivalence of amenability of a locally compact quantum group
$\mathbb{G}$ and 1-injectivity of
$L^{\infty}(\widehat{\mathbb{G}})$
as an operator
$L^1(\widehat{\mathbb{G}})$-module.
In particular, a locally compact group $G$ is amenable if and
only if its group von Neumann algebra
$VN(G)$
is 1-injective as
an operator module over the Fourier algebra $A(G)$. As an application,
we provide a decomposability result for completely bounded
$L^1(\widehat{\mathbb{G}})$-module
maps on
$L^{\infty}(\widehat{\mathbb{G}})$,
and give a simplified proof that amenable discrete
quantum groups have co-amenable compact duals which avoids the
use of modular theory and the Powers--Størmer inequality, suggesting
that our homological techniques may yield a new approach to the
open problem of duality between amenability and co-amenability.
We study the Bishop-Phelps-Bollobás property (BPBp for short)
for compact operators. We present some abstract techniques which
allows to carry the BPBp for compact operators from sequence
spaces to function spaces. As main applications, we prove the
following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$
has the BPBp for compact operators, then so do $(C_0(L),Y)$ for
every locally compact Hausdorff topological space $L$ and $(X,Y)$
whenever $X^*$ is isometrically isomorphic to $\ell_1$.
If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$
has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$
for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$
has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional
or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any
positive measure $\nu$ and $1\lt p\lt \infty$.
For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact
operators, then so does $(X,L_p(\mu,Y))$ for every positive measure
$\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$
has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$
for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$
for every compact Hausdorff topological space $K$.
By a tiling of a topological linear space $X$ we mean a
covering of $X$ by at least two closed convex sets,
called tiles, whose nonempty interiors are
pairwise disjoint.
Study of tilings of infinite-dimensional spaces initiated in
the
1980's with pioneer papers by V. Klee.
We prove some general properties of tilings of locally convex
spaces,
and then apply these results to study existence of tilings of
normed and Banach spaces by tiles possessing
certain smoothness or rotundity properties. For a Banach space
$X$,
our main results are the following.
1. $X$ admits no tiling by Fréchet smooth bounded tiles.
2. If $X$ is locally uniformly rotund (LUR), it does not admit
any tiling by balls.
3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$
uncountable, do admit
a tiling by pairwise disjoint LUR bounded tiles.
We consider positive solutions of the problem
\begin{equation}
(*)\qquad
\left\{
\begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n},
\\
\frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n},
\\
\end{array}
\right.
\end{equation}
where $a\in (-1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and
$\frac{\partial u}{\partial \nu^a}:=-\lim_{x_{n}\rightarrow
0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case
$b=0$, it is associated to fractional Laplacian equation $(-\Delta)^{s}u=u^{q}
$ in entire space $\mathbb{R}^{n-1}$.
We obtain the existence of positive axially symmetric solutions
to ($*$) for the case $a\in
(-1,0)$ in
$n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a-2},
\;\;q\geq\frac{n-a}{n+a-2}$.
The nonexistence is obtained for the case $a\in (-1,0)$, $b\geq
0$ and any $p,~q\gt 1$ in $n=2$ as well.
We prove the consistency of
$$
\operatorname{add}(\mathcal{N})\lt
\operatorname{cov}(\mathcal{N})
\lt \mathfrak{p}=\mathfrak{s}
=\mathfrak{g}\lt \operatorname{add}(\mathcal{M})
= \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a}
=\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c}
$$
with $\mathrm{ZFC}$, where each of these cardinal
invariants assume arbitrary
uncountable regular values.
The aim of this paper is to prove that a K3 surface is the minimal
model of the quotient of an Abelian surface by a group $G$ (respectively
of a K3 surface by an Abelian group $G$) if and only if a certain
lattice is primitively embedded in its Néron-Severi group.
This allows one to describe the coarse moduli space of the K3
surfaces which are (rationally) $G$-covered by Abelian or K3
surfaces (in the latter case $G$ is an Abelian group).
If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian
surface, this result was already known, so we extend it to the
other cases.
Moreover, we prove that a K3 surface $X_G$ is the minimal model
of the quotient of an Abelian surface by a group $G$ if and only
if a certain configuration of rational curves is present on $X_G$.
Again this result was known only in some special cases, in particular
if $G$ has order 2 or 3.
Let $G$ be a locally compact group and let $\omega$ be a continuous
weight on $G$. We show that for each of the Banach algebras $L^1(G,\omega)$,
$M(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$,
the order structure combined with the algebra structure determines
the weighted group.
The dual immaculate functions are a basis of the ring $\operatorname*{QSym}$
of quasisymmetric functions, and form one of the most natural
analogues of the
Schur functions. The dual immaculate function corresponding to
a composition
is a weighted generating function for immaculate tableaux in
the same way as a
Schur function is for semistandard Young tableaux; an "
immaculate tableau" is defined similarly to be
a semistandard
Young tableau, but the shape is a composition rather than a partition,
and
only the first column is required to strictly increase (whereas
the other
columns can be arbitrary; but each row has to weakly increase).
Dual
immaculate functions have been introduced by Berg, Bergeron,
Saliola, Serrano
and Zabrocki in arXiv:1208.5191, and have since been found to
possess numerous
nontrivial properties.
In this note, we prove a conjecture of Mike Zabrocki which provides
an
alternative construction for the dual immaculate functions in
terms of certain
"vertex operators". The proof uses a dendriform structure on
the ring
$\operatorname*{QSym}$; we discuss the relation of this structure
to known
dendriform structures on the combinatorial Hopf algebras
$\operatorname*{FQSym}$ and $\operatorname*{WQSym}$.
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all
of its coefficients in $\{-1,1\}$. There are various old unsolved
problems, mostly due to Littlewood and Erdős, that ask for
Littlewood polynomials that provide a good approximation to a
function that is constant on the complex unit circle, and in
particular have small $L^q$ norm on the complex unit circle.
We consider the Fekete polynomials
\[
f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j,
\]
where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre
symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We
give explicit and recursive formulas for the limit of the ratio
of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive
integer and $p\to\infty$. To our knowledge, these are the first
results that give these limiting values for specific sequences
of nontrivial Littlewood polynomials and infinitely many $q$.
Similar results are given for polynomials obtained by cyclically
permuting the coefficients of Fekete polynomials and for Littlewood
polynomials whose coefficients are obtained from additive characters
of finite fields. These results vastly generalise earlier results
on the $L^4$ norm of these polynomials.
We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that
arises from a weighted graph $(\Gamma, \mu)$, specific cases
of which were previously studied in the context of planar algebras.
We discuss necessary and sufficient conditions of the weighting
which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma,
\mu)$, and study the structure of its positive cone. We then
study the $*$-algebra, $\mathcal{A}$, generated by the generators of
$\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and
techniques of Charlesworth and Shlyakhtenko, as well as Mai,
Speicher, and Weber to show that certain ``loop" elements have
no atoms in their spectral measure. After modifying techniques
of Shlyakhtenko and Skoufranis to show that self adjoint elements
$x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore
some applications to eigenvalues of polynomials in Wishart matrices
and to diagrammatic elements in von Neumann algebras initially
considered by Guionnet, Jones, and Shlyakhtenko.
We continue the investigation of the isomorphism problem for
multiplier algebras of reproducing kernel
Hilbert spaces with the complete Nevanlinna-Pick property.
In contrast to previous work in this area,
we do not study these spaces by identifying them with restrictions
of a universal space, namely the Drury-Arveson space.
Instead, we work directly with the Hilbert spaces and their
reproducing kernels. In particular,
we show that two multiplier algebras of Nevanlinna-Pick spaces
on the same set are equal if and only if the Hilbert
spaces are equal. Most of the article is devoted to the study
of a special class of
complete Nevanlinna-Pick spaces on homogeneous varieties. We
provide a complete
answer to the question of when two multiplier algebras of spaces
of this type
are algebraically or isometrically isomorphic. This generalizes
results of Davidson, Ramsey, Shalit,
and the author.
Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital,
has finitely many extremal traces, and has strict comparison
of positive elements by traces, then its multiplier algebra
$\operatorname{\mathcal{M}}(\mathcal{A})$
also has strict comparison of positive elements by traces. The
same results holds if ``finitely many extremal traces" is replaced
by ``quasicontinuous scale".
A key ingredient in the proof is that every positive element
in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra
can be approximated by a bi-diagonal series.
An application of strict comparison: If $\mathcal{A}$ is a simple separable
stable C*-algebra with real rank zero, stable rank one, and
strict comparison of positive elements by traces, then whether
a positive element is a positive linear combination of projections
is determined by the trace values of its range projection.
Under the local Langlands correspondence, the conductor of an
irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the
Swan conductor of the corresponding Galois representation. In
this paper, we establish the geometric analogue of this statement
by showing that the conductor of a categorical representation
of the loop group is greater than the irregularity of the corresponding
meromorphic connection.
This paper concerns free function theory. Free maps are free
analogs of analytic functions in several complex variables,
and are defined in terms of freely noncommuting variables.
A function of $g$ noncommuting variables is a function on $g$-tuples
of square matrices of all sizes that respects direct sums and
simultaneous conjugation.
Examples of such maps include noncommutative polynomials, noncommutative
rational functions and convergent noncommutative power series.
In sharp contrast to the existing literature in free analysis, this article
investigates free maps with involution --
free analogs of real analytic functions.
To
get a grip on these,
techniques and tools from invariant theory are developed and
applied to free analysis. Here is a sample of the results obtained.
A characterization of polynomial free maps via properties of
their finite-dimensional slices is presented and then used to
establish power series expansions for analytic free maps about
scalar and non-scalar points; the latter are series of generalized
polynomials for which an invariant-theoretic characterization
is given.
Furthermore,
an inverse and implicit function theorem for free maps with
involution is obtained.
Finally, with a selection of carefully chosen examples
it is shown that
free maps with involution
do not exhibit strong rigidity properties
enjoyed by their involution-free
counterparts.
In this paper we introduce a new way of deforming convolution
algebras and Fourier algebras on locally compact groups. We demonstrate
that this new deformation allows us to reveal some information
of the underlying groups by examining Banach algebra properties
of deformed algebras. More precisely, we focus on representability
as an operator algebra of deformed convolution algebras on compact
connected Lie groups with connection to the real dimension of
the underlying group. Similarly, we investigate complete representability
as an operator algebra of deformed Fourier algebras on some finitely
generated discrete groups with connection to the growth rate
of the group.
We explicitly find regulators of an infinite family $\{L_m\}$
of the simplest quartic function fields
with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where
$\mathbb{F}_q$
is the finite field of order $q$
with odd characteristic. In fact, this infinite family of the
simplest quartic function fields are
subfields of maximal real subfields of cyclotomic function fields,
where they have the same conductors.
We obtain a lower bound on the class numbers of the family $\{L_m\}$
and some result on the divisibility
of the divisor class numbers of cyclotomic function fields which
contain $\{L_m\}$ as their subfields.
Furthermore, we find an explicit criterion for the characterization
of splitting types of all the primes
of the rational function field $\mathbb{F}_q (t)$ in $\{L_m\}$.
We study the asymptotic behaviour of the Bloch--Kato--Shafarevich--Tate
group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension
of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$.
In particular, we give upper bounds of these groups in terms
of Iwasawa invariants of Selmer groups defined using $p$-adic
Hodge Theory. These bounds have the same form as the formulae
of Kobayashi, Kurihara and Sprung for supersingular elliptic
curves.
Let $\mathbf{k}$ be an algebraically closed field complete with respect
to a non-Archimedean absolute value of arbitrary characteristic.
Let $D_1,\dots, D_n$ be effective nef divisors intersecting
transversally in an $n$-dimensional nonsingular projective variety
$X$.
We study the degeneracy of non-Archimedean analytic maps from
$\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric
conditions. When $X$ is a rational ruled surface and $D_1$ and
$D_2$ are ample, we obtain a necessary and sufficient condition
such that
there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\setminus
D_1 \cup D_2$.
Using the dictionary between non-Archimedean Nevanlinna theory
and Diophantine approximation that originated in
earlier work with T. T. H. An, %
we also study arithmetic analogues of these problems, establishing
results on integral points on these varieties over $\mathbb{Z}$
or the ring of integers of an imaginary quadratic field.
We consider Schubert problems with respect to flags osculating
the rational normal curve. These problems are of special interest
when the osculation points are all real -- in this case, for
zero-dimensional Schubert problems, the solutions are "as real
as possible". Recent work by Speyer has extended the theory
to the moduli space
$
\overline{\mathcal{M}_{0,r}}
$,
allowing the points to collide.
These give rise to smooth covers of
$
\overline{\mathcal{M}_{0,r}}
(\mathbb{R})
$, with structure
and monodromy described by Young tableaux and jeu de taquin.
In this paper, we give analogous results on one-dimensional Schubert
problems over
$
\overline{\mathcal{M}_{0,r}}
$.
Their (real) geometry turns out to
be described by orbits of Schützenberger promotion and a
related operation involving tableau evacuation. Over
$\mathcal{M}_{0,r}$,
our results show that the real points of the solution curves
are smooth.
We also find a new identity involving "first-order" K-theoretic
Littlewood-Richardson coefficients, for which there does not
appear to be a known combinatorial proof.
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple Hilbert-Lie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$,
which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of
a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism
from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from
the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain
it explicitly
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the
study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain
a complete classification
of the positive energy highest weight representations of affinisations
of $\mathfrak{k}$.
Let $\mathcal A$, $\mathcal B$ be large subsets of $\{1,\ldots,N\}$.
We study the number of pairs $(a,b)\in\mathcal A\times\mathcal B$ such that
the sum of binary digits of $a+b$ is fixed.
We complete Mori's program with symmetric divisors for the moduli
space of stable seven-pointed rational curves. We describe all
birational models in terms of explicit blow-ups and blow-downs.
We also give a moduli theoretic description of the first flip,
which has not appeared in the literature.
We prove that the extremal sequences for the
Bellman function of the dyadic maximal operator behave approximately
as eigenfunctions of this operator for a specific eigenvalue.
We use this result to prove the analogous one with respect to
the Hardy operator.
We study the stability of disjointness preservers on Banach lattices.
In many cases, we prove that an "almost disjointness preserving"
operator is well approximable by a disjointness preserving one.
However, this approximation is not always possible, as our
examples show.
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are
algebraically independent over $\mathbb{C}(x)$. More generally, for
a linear differential or difference equation, it is an important
problem to find all algebraic dependencies among a non-zero solution
$y$ and particular transformations of $y$, such as derivatives
of $y$ with respect to parameters, shifts of the arguments, rescaling,
etc. In the present paper, we develop a theory of Tannakian categories
with semigroup actions, which will be used to attack such questions
in full generality, as each linear differential equation gives
rise to a Tannakian category.
Deligne studied actions of braid groups on categories and obtained
a finite collection of axioms that characterizes such actions
to apply it to various geometric constructions. In this paper,
we find a finite set of axioms that characterizes actions of
semigroups that are finite free products of semigroups of the
form $\mathbb{N}^n\times
\mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$
on Tannakian categories. This is the class of semigroups that
appear in many applications.
The preservation principle of local theta correspondences of reductive dual pairs over
a $p$-adic field predicts the existence of a sequence of irreducible supercuspidal
representations of classical groups.
Adams/Harris-Kudla-Sweet
have a conjecture
about the Langlands parameters for the sequence of supercuspidal representations.
In this paper we prove modified versions of their conjectures for the case of
supercuspidal representations with unipotent reduction.
The initial value problem for a semi-linear fractional heat equation
is investigated. In the focusing case, global well-posedness
and exponential decay are obtained. In the focusing sign, global
and non global existence of solutions are discussed via the potential
well method.
We describe the general form of surjective maps on the cone of
all positive operators which preserve order and spectrum. The
result is optimal as shown by
counterexamples. As an easy consequence we characterize surjective
order and spectrum preserving maps on the set of all self-adjoint
operators.
For a lattice $L$ of $\mathbb{RR}^n$, a sphere $S(c,r)$ of center $c$
and radius $r$
is called empty if for any $v\in L$ we have $\Vert v -
c\Vert \geq r$.
Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay
polytope}
$P=\operatorname{conv}(S(c,r)\cap L)$.
A Delaunay polytope is called {\em perfect} if any affine transformation
$\phi$ such that $\phi(P)$ is a Delaunay polytope is necessarily
an isometry
of the space composed with an homothety.
Perfect Delaunay polytopes are remarkable structure that exist
only
if $n=1$ or $n\geq 6$ and they have shown up recently in covering
maxima studies.
Here we give a general algorithm for their enumeration that relies
on
the Erdahl cone.
We apply this algorithm in dimension $7$ which allow us to find
that there are only two perfect Delaunay polytopes: $3_{21}$
which
is a Delaunay polytope in the root lattice $\mathsf{E}_7$ and
the
Erdahl Rybnikov polytope.
We then use this classification in order to get the list of all
types
Delaunay simplices in dimension $7$ and found $11$ types.
Let $\operatorname{P} = \operatorname{M} \operatorname{N}$ be a Levi decomposition of a maximal parabolic
subgroup of a connected
reductive group $\operatorname{G}$ over a $p$-adic field $F$. Assume that there
exists $w_0 \in \operatorname{G}(F)$ that normalizes $\operatorname{M}$ and conjugates $\operatorname{P}$
to an opposite parabolic subgroup.
When $\operatorname{N}$ has a Zariski dense $\operatorname{Int} \operatorname{M}$-orbit,
F. Shahidi and X. Yu describe a certain distribution $D$ on
$\operatorname{M}(F)$
such that,
for irreducible unitary supercuspidal representations $\pi$ of
$\operatorname{M}(F)$ with
$\pi \cong \pi \circ \operatorname{Int} w_0$,
$\operatorname{Ind}_{\operatorname{P}(F)}^{\operatorname{G}(F)} \pi$ is
irreducible
if and only if $D(f) \neq 0$ for some pseudocoefficient $f$ of
$\pi$. Since
this irreducibility is conjecturally related to $\pi$ arising
via
transfer from certain twisted endoscopic groups of $\operatorname{M}$, it is
of interest
to realize $D$ as endoscopic transfer from a simpler distribution
on a twisted
endoscopic group $\operatorname{H}$ of $\operatorname{M}$. This has been done in many situations
where $\operatorname{N}$ is abelian. Here, we handle the `standard examples'
in cases
where $\operatorname{N}$ is nonabelian but admits a Zariski dense
$\operatorname{Int} \operatorname{M}$-orbit.
In this paper, we introduce the anisotropic
Sobolev capacity with fractional order and develop some basic
properties for this new object. Applications to the theory of
anisotropic fractional Sobolev spaces are provided. In particular,
we give geometric characterizations for a nonnegative Radon
measure $\mu$ that naturally induces an embedding of the anisotropic
fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into
the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0\lt \beta\le
n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter.
Such a geometric quantity can be used to approximate the anisotropic
Sobolev capacity with fractional order. Estimation on the constant
in the related Minkowski inequality, which is asymptotically
optimal as $\alpha\rightarrow 0^+$, will be provided.
We give a survey on Moeglin's construction of representations
in the Arthur packets for $p$-adic quasisplit symplectic and
orthogonal groups. The emphasis is on comparing Moeglin's
parametrization of elements in the Arthur packets with that of
Arthur.
In this paper, we prove that, on a compact, $n$-dimensional Alexandrov
space with curvature $\geqslant -1$, the Wasserstein barycenter of
Borel probability measures $\mu_1,...,\mu_m$ is absolutely continuous
with respect to the $n$-dimensional Hausdorff measure if one
of them is.
We study the Chern-Ricci flow, an evolution equation of Hermitian
metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which
are non-Kähler compact complex manifolds with negative Kodaira
dimension. We prove that, after an initial conformal change,
the flow converges, in the
Gromov-Hausdorff sense, to a torus with a flat Riemannian metric
determined by the OT-manifolds themselves.