Kantor pairs arise naturally in the study of
$5$-graded Lie algebras. In this article, we introduce
and study Kantor pairs with short Peirce gradings and relate
them to Lie algebras
graded by the root system of type
$\mathrm{BC}_2$.
This relationship
allows us to define so called Weyl images
of short Peirce graded Kantor pairs. We use Weyl images to construct
new examples of Kantor pairs, including a class of infinite
dimensional
central simple Kantor pairs over a field of characteristic $\ne
2$ or $3$, as well as a family of forms of a split
Kantor pair of type
$\mathrm{E}_6$.
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 provide the differential equations that generalize the Newtonian
$N$-body problem of celestial mechanics to spaces of constant
Gaussian curvature, $\kappa$, for all $\kappa\in\mathbb R$. In
previous studies, the equations of motion made sense only for
$\kappa\ne 0$. The system derived here does more than just include
the Euclidean case in the limit $\kappa\to 0$: it recovers the
classical equations for $\kappa=0$. This new expression of the
laws of motion allows the study of the $N$-body problem in the
context of constant curvature spaces and thus offers a natural
generalization of the Newtonian equations that includes the classical
case. We end the paper with remarks about the bifurcations of
the first integrals.
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 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.
We correct an error in the proof of a
lemma in
"Translation Groupoids and Orbifold Cohomology",
Canadian J. Math Vol 62 (3), pp 614-645 (2010).
This error was pointed out to the authors
by Li Du of the Georg-August-Universität at Gottingen, who
also suggested the outline for the corrected proof.
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.
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.