We consider graded connected Gorenstein algebras with respect to the evaluation map $\text{e}{{\text{v}}_{G\,}}\,=\,\text{Ex}{{\text{t}}_{G}}(k,\varepsilon )\,::\,\text{Ex}{{\text{t}}_{G}}(k,G)\,\to \,\text{Ex}{{\text{t}}_{G}}(k,k)$. We prove that if $e{{v}_{G}}\,\ne \,0$, then the global dimension of $G$ is finite.

In the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

Using the theory of $p$-adic Lie groups we give conditions for a finitely generated group to admit a splitting as a non-trivial free product with amalgamation. This can be viewed as an extension of a theorem of Bass.

It is known that a semigroup of quasinilpotent integral operators, with positive lower semicontinuous kernels, on ${{L}^{2}}\,(X,\,\mu )$, where $X$ is a locally compact Hausdorff-Lindelöf space and $\mu $ is a $\sigma $-finite regular Borel measure on $X$, is triangularizable. In this article we use the Banach lattice version of triangularizability to establish the ideal-triangularizability of a semigroup of positive quasinilpotent integral operators on $C(\mathbf{K})$ where $\mathbf{K}$ is a compact Hausdorff space.

The aim of this paper is to characterize those linear maps from a von Neumann factor $A$ into itself which preserve the extreme points of the unit ball of $A$. For example, we show that if $A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital $*$-antihomomorphism.

Our main result is showing the asymptotic existence of tight $\text{OMEPs}$. More precisely, for each fixed number $k$ of rows, and with the exception of $\text{OMEPs}$ of the form $2\times 2\times \cdot \cdot \cdot 2\times 2s\,//\,4s\,\,$ with $s$ odd and with more than three rows, there are only a finite number of tight $\text{OMEP}$ parameters for which the tight $\text{OMEP}$ does not exist.

In Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ${{\mathbb{Z}}^{d}}$, and to other percolation processes, including Bernoulli bond percolation.

In this paper we consider solutions to the free Schrödinger equation in $n+1$ dimensions. When we restrict the last variable to be a smooth function of the first $n$ variables we find that the solution, so restricted, is locally in ${{L}^{2}}$, when the initial data is in an appropriate Sobolev space.

This paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem 1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of ${F}'(\text{x}){F}'{{(\text{x})}^{T}}\,$ are uniformly bounded away from zero for $\text{x}\,\in \,{{\mathbb{R}}^{n}}$, where $F\,:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}^{n}}$ is a class ${{C}^{1}}$ map, then F is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class ${{C}^{1}}$ maps $F$ of ${{\mathbb{R}}^{n}}$ into itself is that all the eigenvalues of ${F}'\,(\text{x})$ are bounded away from zero on ${{\mathbb{R}}^{n}}$. This is stated as Conjecture 2.1. If true, it would imply (via Reduction-of-Degree) injectivity of polynomial maps$F\,:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}^{n}}$satisfying the hypothesis, $\det F'(x)\equiv 1$, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.

A theory of minimal realizations of rational matrix functions $W(\lambda )$ in the “pencil” form $W(\lambda )=C{{(\lambda {{A}_{1}}-{{A}_{2}})}^{-1}}B$ is developed. In particular, properties of the pencil $\text{ }\!\!\lambda\!\!\text{ }{{A}_{1}}\,-\,{{A}_{2}}$ are discussed when $W(\lambda )$ is hermitian on the real line, and when $W(\lambda )$ is hermitian on the unit circle.

We provide more characterizations of varieties with a weak difference term and of neutral varieties. We prove that a variety has a (weak) difference term (is neutral) with respect to the $\text{TC}$-commutator iff it has a (weak) difference term (is neutral) with respect to the linear commutator. We show that a variety $V$ is congruence meet semi-distributive iff $V$ is neutral, iff ${{M}_{3}}$ is not a sublattice of Con $\mathbf{A}$, for $\mathbf{A}\in V$, iff there is a positive integer $n$ such that $V{{\vDash }_{Con}}\alpha (\beta \,o\,\gamma )\le \alpha {{\beta }_{n}}$.

Given an integral functional defined on ${{L}_{p}}$, $1\le p<\infty $, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand.

Over a decade ago, this author produced class number one criteria for real quadratic fields in terms of prime-producing quadratic polynomials. The purpose of this article is to revisit the problem from a new perspective with new criteria. We look at the more general situation involving arbitrary real quadratic orders rather than the more restrictive field case, and use the interplay between the various orders to provide not only more general results, but also simpler proofs.

For each $n\ge 4$ we construct a class of examples of a minimal $C$-dependent set of $n$ automorphisms of a prime ring $R$, where $C$ is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that the preceding examples are completely general, whereas for $n=6$ an example is given which fails to enjoy any of the nice properties of the above example.

We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.

The aim of this article is to obtain an upper bound for the exponential sums $\sum{e(f(x)\,/\,q)}$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$ and $e(\alpha )$ denotes $\exp (2\pi i\alpha )$.

We shall show that the upper bound depends only on the values of $q$ and $s$, where $s$ is the number of terms in the polynomial $f(x)$.

The spectra of the Toeplitz operators on the weighted Hardy space ${{H}^{2}}\,(W\,d\theta \,/\,2\pi )$ and the Hardy space ${{H}^{p}}\,(d\theta \,/\,2\pi )$, and the singular integral operators on the Lebesgue space ${{L}^{2}}\,(d\theta \,/\,2\pi )$ are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

In this paper we prove the following:

1. 1. Let $m\ge 2,\,n\ge 1$ be integers and let $G$ be a group such that ${{(XY)}^{n}}\,=\,{{(YX)}^{n}}$ for all subsets $X,Y$ of size $m$ in $G$. Then

   1. a) $G$ is abelian or a $\text{BFC}$-group of finite exponent bounded by a function of $m$ and $n$.

   2. b) If $m\ge n$ then $G$ is abelian or $|G|$ is bounded by a function of $m$ and $n$.

2. 2. The only non-abelian group $G$ such that ${{(XY)}^{2}}\,=\,{{(YX)}^{2}}$ for all subsets $X,Y$ of size 2 in $G$ is the quaternion group of order 8.

3. 3. Let $m$, $n$ be positive integers and $G$ a group such that

   $${{X}_{1}}\cdot \cdot \cdot \,{{X}_{n}}\,\subseteq \,\bigcup\limits_{\sigma \in {{S}_{n}}\,\backslash \,1}{{{X}_{\sigma (1)}}\cdot \cdot \cdot \,{{X}_{\sigma (n)}}}$$

for all subsets ${{X}_{i}}$ of size $m$ in $G$. Then $G$ is $n$-permutable or $|G|$ is bounded by a function of $m$ and $n$.

The function $\Delta (x,N)$ as defined in the title is closely associated via $\Delta \,(N)\,=\,{{\sup }_{x}}\,|\,\Delta (x,N)|$ to several problems in the upper bound sieve. It is also known via a classical theorem of Franel that certain conjectured bounds involving averages of $\Delta (x,N)$ are equivalent to the Riemann Hypothesis. We improve the unconditional bounds which have been hitherto obtained for $\Delta (N)$ and show that these are close to being optimal. Several auxiliary results relating $\Delta (Np)$ to $\Delta (N)$, where $p$ is a prime with $p\nmid N$, are also obtained and two new conjectures stated.

We show that for certain compact right topological groups, $\overline{r(G)}$, the strong operator topology closure of the image of the right regular representation of $G$ in $L(H)$, where $H\,=\,{{L}^{2}}\,(G)$, is a compact topological group and introduce a class of representations, $R$ , which effectively transfers the representation theory of $\overline{r(G)}$ over to $G$. Amongst the groups for which this holds is the class of equicontinuous groups which have been studied by Ruppert in [10].We use familiar examples to illustrate these features of the theory and to provide a counter-example. Finally we remark that every equicontinuous group which is at the same time a Borel group is in fact a topological group.