An integral domain $D$ with identity is condensed (resp., strongly condensed) if for each pair of ideals $I,\,J$ of $D,\,IJ\,=\,\{ij\,;\,i\,\in I,j\in J\,\}$ (resp., $IJ=iJ$ for some $i\,\in \,I$ or $IJ\,=Ij$ for some $j\,\in \,J$). We show that for a Noetherian domain $D,\,D$ is condensed if and only if $\text{Pic}\left( D \right)\,=0$ and $D$ is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain $D$ is strongly condensed if and only if $D$ is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension $k\,\subseteq K$, the domain $D=\,k+XK[[X]]$ is condensed if and only if $[K:k]\,\le \,2$ or $[K:k]\,=\,3$ and each degree-two polynomial in $k[X]$ splits over $k$, while $D$ is strongly condensed if and only if $[K:k]\,\le \,2$.

We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma $ such that $FG$ is $\beta $-commutative for a skew-symmetric bicharacter $\beta $ on $\Gamma $ with values in ${{F}^{*}}$.

Soit $K$ un corps de nombres de degré sur $\mathbb{Q}$ inférieur ou égal à 2. On se propose dans ce travail de faire quelques remarques sur la question de l'existence de deux éléments non nuls $a$ et $b$ de $K$, et d'un entier $n\,\ge \,4$, tels que l'équation $a{{x}^{n}}\,+\,b{{y}^{n\,}}=\,1$ possède au moins trois points distincts non triviaux. Cette étude se ramène à la recherche de points rationnels sur $K$ d'une variété projective dans ${{\mathbb{P}}^{5}}$ de dimension 3, ou d'une surface de ${{\mathbb{P}}^{3}}$.

Let $d\,>\,1$ be a square-free integer. Power residue criteria for the fundamental unit ${{\varepsilon }_{d}}$ of the real quadratic fields $\mathbb{Q}(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation ${{x}^{2}}\,-\,d{{p}^{2}}{{y}^{2}}\,=\,-1$. The most important solvability criterion deals with all $d$ for which $\mathbb{Q}(\sqrt{-d})$ has an elementary abelian 2-class group and $p\,\equiv \,5$ (mod 8) or $p\,\equiv \,9$ (mod 16).

Let $U(n)$ be the group of $n\,\times \,n$ unitary matrices. We show that if $\phi $ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is a multiple of $n$, and $\phi $ has the form

$$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$

for some $V,\,W\,\in \,U(m)$. From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

In this paper we formulate and solve Nevanlinna-Pick and Carathéodory type problems for tensor algebras with data given on the $N$-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory.

The $abc$-conjecture is applied to various questions involving the number of distinct fields $\mathbb{Q}\left( \sqrt{f(n)} \right)$, as we vary over integers $n$.

We show that the multi-sided inclusion ${{R}^{\otimes l}}\,\subset \,R$ of braid-type subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$, introduced in Multi-sided braid type subfactors$[\text{E}3]$, contains a sequence of intermediate subfactors: ${{R}^{\otimes l\,}}\subset \,{{R}^{\otimes l-1\,}}\subset \,\,\cdots \,\,\subset \,{{R}^{\otimes 2\,}}\subset \,R$. That is, every $t$-sided subfactor is an intermediate subfactor for the inclusion ${{R}^{\otimes l}}\,\subset \,R,\,\text{for 2}\,\le \,t\,\le \,l$. Moreover, we also show that if $t\,>\,m$ then ${{R}^{\otimes t}}\,\subset \,{{R}^{\otimes m}}$ is conjugate to ${{R}^{\otimes t-m+1\,}}\subset \,R$. Thus, if the braid representation considered is associated to one of the classical Lie algebras then the asymptotic inclusions for the Jones-Wenzl subfactors are intermediate subfactors.

The celebrated theorem of Picard asserts that each non-constant entire function assumes every value infinitely often, with at most one exception. The Riemann zeta function has this Picard behaviour in a sequence of discs lying in the critical band and whose diameters tend to zero. According to the Riemann hypothesis, the value zero would be this (unique) exceptional value.

We consider a class $(A,\,S,\,\alpha )$ of dynamical systems, where $S$ is an Ore semigroup and $\alpha$ is an action such that each ${{\alpha }_{s}}$ is injective and extendible (i.e. it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of $A$. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group $G$ of $S$ constructed in [15, Proposition 6.1]. It turns out that the full crossed product by this coaction is isomorphic to $A\,{{\rtimes }_{\alpha }}\,S$. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.

We define the $\mathcal{M}$-harmonic conjugate operator $K$ and prove that it is bounded on the non-isotropic Lipschitz space and on $\text{BMO}$. Then we show $K$ maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of $\mathcal{M}$-harmonic conjugate operator with ${{L}^{p}}$ symbol.

Define a group $G$ to be in the class $S$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that ${{g}^{n\,}}\in \,K$ for all $g\,\in \,G,\,K$ has finite index in $G$. We show that a free product with amalgamation $A{{*}_{_{C}}}B$ and an $\text{HNN}$ group $A{{*}_{C}}$ belong to $S$, if $C$ is in $S$ and every subgroup of $C$ is finitely generated.

In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci curvature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.

We determine an explicit cell decomposition of the wonderful compactification of a semisimple algebraic group. To do this we first identify the $B\,\times \,B$-orbits using the generalized Bruhat decomposition of a reductive monoid. From there we show how each cell is made up from $B\,\times \,B$-orbits.

A Newton polygon is introduced for a ramified point of a Galois covering of curves over a finite field. It is shown to be determined by the sequence of higher ramification groups of the point. It gives a blowing up of the wildly ramified part which separates the branches of the curve. There is also a connection with local reciprocity.

Fix an elliptic curve ${{y}^{2}}\,=\,{{x}^{3}}\,+\,Ax\,+\,B$, satisfying $A,\,B\,\in \,\mathbb{Z},\,A\ge \,\left| B \right|\,>\,0$. We prove that the $\mathbb{Q}$-torsion subgroup is one of $(0),\,\mathbb{Z}/3\mathbb{Z},\,\mathbb{Z}/9\mathbb{Z}$. Related numerical calculations are discussed.