E8 is first of all the largest exceptional root system, which is a set of vectors in an 8-dimensional real vector space satisfying certain properties. Root systems were classified by Wilhelm Killing in the 1890s. He found 4 infinite classes of Lie algebras, labelled An, Bn, Cn, and Dn, where n=1,2,3…. He […]

Let s_v denote the pure power sum \sum_{k=1}^n z_k^v. In a previous paper we
proved that \sqrt n <= \inf_{|z_k| => 1} \max_{v=1,…,n^2} |s_v| <= \sqrt{n+1}
when n+1 is prime. In this paper we prove that \inf_{|z_k| = 1}
\max_{v=1,…,n^2-n} |s_v| = \sqrt{n-1} when n-1 is a prime power, and if 2 <=
i <= n-1 and n => 3 is a prime power then \inf_{|z_k| => 1}
\max_{v=1,…,n^2-i} |s_v| =\sqrt n. We give explicit constructions of n-tuples
(z_1,…,z_n) which we prove are global minima for these problems. These are
two of the few times in Turan power sum theory where solutions in the inf max
problem can be explicitly calculated.

Let $F$ be a global function field of characteristic $p>0$, $\mathcal F/F$ a
Galois extension with $Gal(\tilde F/F)\simeq \mathbb{Z}_p^{\mathbb N}$ and
$E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups
$Sel_E(L)_l$ ($l$ any prime) as $L$ varies through the subextensions of
$\mathcal F$ via appropriate versions of Mazur’s Control Theorem. In the case
$l=p$ we let $\mathcal F=\bigcup \mathcal F_d$ where $\mathcal F_d/F$ is a
$\mathbb{Z}_p^d$-extension. With a mild hypothesis on $Sel_E(F)_p$ (essentially
a consequence of the Birch and Swinnerton-Dyer conjecture) we prove that
$Sel_E(\mathcal F_d)_p$ is a cofinitely generated (in some cases cotorsion)
$\mathbb{Z}_p[[Gal(\mathcal F_d/F)]]$-module and we associate to its Pontrjagin
dual a Fitting ideal. This allows to define an algebraic $L$-function
associated to $E$ in $\mathbb{Z}_p[[Gal(\mathcal F/F)]]$, providing an
ingredient for a function field analogue of Iwasawa’s Main Conjecture for
elliptic curves.

In this paper, the notion of {\em $p$-adic multiresolution analysis (MRA)} is
introduced. We use a “natural” refinement equation whose solution (a
refinable function) is the characteristic function of the unit disc. This
equation reflects the fact that the characteristic function of the unit disc is
the sum of $p$ characteristic functions of disjoint discs of radius $p^{-1}$.
The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real
Haar MRA. But in contrast to the real setting, the refinable function
generating our Haar MRA is periodic with period 1, which never holds for real
refinable functions. This fact implies that there exist infinity many different
2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar
MRA. All of these bases are constructed. Since $p$-adic pseudo-differential
operators are closely related to wavelet-type bases, our bases can be
intensively used for applications.

In this paper we describe the commutant of an arbitrary subalgebra $A$ of the
algebra of functions on a set $X$ in a crossed product of $A$ with the
integers, where the latter act on $A$ by a composition automorphism defined via
a bijection of $X$. The resulting conditions which are necessary and sufficient
for $A$ to be maximal abelian in the crossed product are subsequently applied
to situations where these conditions can be shown to be equivalent to a
condition in topological dynamics. As a further step, using the Gelfand
transform we obtain for a commutative completely regular semi-simple Banach
algebra a topological dynamical condition on its character space which is
equivalent to the algebra being maximal abelian in a crossed product with the
integers.

We provide several equivalent characterizations of Kobayashi hyperbolicity in
unbounded convex domains in terms of peak and anti-peak functions at infinity,
affine lines, Bergman metric and iteration theory.