Let $\mathcal{G}$ be a finite dimensional simple Lie algebra over the complex numbers $C$. Fernando reduced the classification of infinite dimensional simple $\mathcal{G}$-modules with a finite dimensional weight space to determining the simple torsion free $\mathcal{G}$-modules for $\mathcal{G}$ of type $A$ or $C$. These modules were determined by Mathieu and using his work we provide a more elementary construction realizing each one as a submodule of an easily constructed tensor product module.

We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension $E/F$ of $p$-adic fields of characteristic zero. We study the case where the Levi component $M\simeq \text{G}{{\text{L}}_{n}}\left( E \right)\times {{U}_{m}}\left( F \right)$, with $n\,\equiv \,m\,\left( \bmod \,2 \right)$. This, along with earlier work, determines the poles of the local Rankin-Selberg product $L$-function $L\left( s,\,{\tau }'\,\times \,\tau \right)$, with ${\tau }'$ an irreducible unitary supercuspidal representation of $\text{G}{{\text{L}}_{n}}\left( E \right)$ and $\tau $ a generic irreducible unitary supercuspidal representation of ${{U}_{m}}\left( F \right)$. The results are interpreted using the theory of twisted endoscopy.

In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\text{KP}$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau $-functions is given.

We have seen, in the previous works [5], [6], that the argument of a certain product is closely connected to that of the cubic Gauss sum. Here the absolute value of the product will be investigated.

Giordano, Putnam and Skau showed that the transformation group ${{C}^{*}}$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C\left( X \right)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

The concept of $R$-diagonal element was introduced in [5], and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to $R$-diagonality, which relies on freeness with amalgamation. The class of $R$-diagonal elements is enlarged to contain examples living in non-tracial $*$-probability spaces, such as the generalized circular elements of [7].

If $\mathfrak{A}$ is a finite alphabet, $\mathcal{U}\,\subset \,{{\mathbb{Z}}^{D}}$, and ${{\mu }_{\mathcal{U}}}$ is a probability measure on ${{\mathfrak{A}}^{\mathcal{U}}}$ that “looks like” the marginal projection of a stationary stochastic process on ${{\mathfrak{A}}^{{{\mathbb{Z}}^{D}}}}$, then can we “extend” ${{\mu }_{\mathcal{U}}}$ to such a process? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when $D\,=\,1$, we provide some sufficient conditions and some necessary conditions for ${{\mu }_{\mathcal{U}}}$ to be extendible for $D\,>\,1$, and show that, in general, the problem is not formally decidable.

For $c\,>\,1$ we denote by ${{\pi }_{c}}\left( x \right)$ the number of integers $n\,\le \,x$ such that $\left\lfloor {{n}^{c}} \right\rfloor $ is prime. In 1953, Piatetski-Shapiro has proved that ${{\pi }_{c}}\left( x \right)\,\sim \,\frac{x}{c\,\log \,x},\,x\to \,+\infty $ holds for $c\,<\,12/11$. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c\,<\,1.16117\ldots $.

Habib Muzaffar of Carleton University has pointed out to the authors that in their paper $\left[ \text{A} \right]$ only the result

$${{\pi }_{K,d}}\left( x \right)\,+\,{{\pi }_{{{K}^{-1}},d}}\left( x \right)\,=\,\frac{1}{h\left( d \right)}\,\frac{x}{\log x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}x} \right)$$

follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result

$${{\pi }_{K,d}}\left( x \right)\,=\,\frac{1}{2h\left( d \right)}\,\frac{x}{\log \,x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}\,x} \right)$$

stated in Lemma 5.2. This necessitates changes in Sections 5 and 6 of $\left[ \text{A} \right]$. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of $\left[ \text{A} \right]$, each and every occurrence of $o\left( s-1 \right)$ should be replaced by $o\left( 1 \right)$.

Sections 5 and 6 of $\left[ \text{A} \right]$ have been rewritten to incorporate the abovementioned correction and are given below. They should replace the original Sections 5 and 6 of $\left[ \text{A} \right]$.