We estimate exponential sums with the Fermat-like quotients

$${{f}_{g}}(n)=\frac{{{g}^{n-1}}-1}{n}\text{and }{{h}_{g}}(n)=\frac{{{g}^{n-1}}-1}{P(n)},$$

where $g$ and $n$ are positive integers, $n$ is composite, and $P\left( n \right)$ is the largest prime factor of $n$. Clearly, both ${{f}_{g}}(n)$ and ${{h}_{g}}(n)$ are integers if $n$ is a Fermat pseudoprime to base $g$, and if $n$ is a Carmichael number, this is true for all $g$ coprime to $n$. Nevertheless, our bounds imply that the fractional parts $\left\{ {{f}_{g}}(n) \right\}$ and $\left\{ {{h}_{g}}(n) \right\}$ are uniformly distributed, on average over $g$ for ${{f}_{g}}(n)$, and individually for ${{h}_{g}}(n)$. We also obtain similar results with the functions ${{\tilde{f}}_{g}}(n)=g\,{{f}_{g}}(n)$ and ${{\tilde{h}}_{g}}(n)=g{{h}_{g}}(n)$.

We discuss the subRiemannian geometry induced by two noncommutative vector fields which are left invariant on the Lie group ${{\mathbb{S}}^{3}}$.

Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $g$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $g$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $g$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

Dans cet article, on considère un groupe semi-simple $\text{G}$ classique réel et connexe. On suppose de plus que $\text{G}$ possède un sous-groupe de Cartan compact. On définit une famille de sous-algèbres de Lie associée à $\mathfrak{g}=\text{Lie(G)}$, de même rang que g dont tous les facteurs simples sont de rang 1 ou 2. Soit $\mathfrak{g}'$ une telle sous-algèbre de Lie. On construit alors une application de transfert des intégrales orbitales de $\mathfrak{g}'$ dans l’espace des intégrales orbitales de $\mathfrak{g}$. On montre que cette application est définie dès que $\mathfrak{g}$ ne possède pas de facteur simple réel de type CI de rang supérieur ouégal à 3. Si de plus, $\mathfrak{g}$ ne possède pas de facteur simple de type BI de rang supérieur à 3, on montre la surjectivité de cette application de transfert.

On utilise cette application de transfert pour obtenir une formule de réduction de l’intégrale de Cauchy Harish-Chandra pour les paires duales d’algèbres de Lie réductives $(\text{U(}p,q\text{),U(}r,s\text{)})$ et $(\text{Sp(}p,q\text{),O*(2n)})$ avec $p+q=r+s=n$.

We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

A Riemannian manifold $\left( M,\,\rho \right)$ is called Einstein if the metric $\rho $ satisfies the condition $\text{Ric}\left( \rho \right)\,=\,c\,\cdot \,\rho $ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO\left( n \right)/SO\left( l \right)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO\left( n \right)/SO\left( l \right)$ that admits at least $p$$SO\left( n \right)$-invariant Einstein metrics.

Given a complete $\text{CAT}(0)$ space $X$ endowed with a geometric action of a group $\Gamma $, it is known that if $\Gamma $ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the $\text{CAT}(0)$ realization of a Coxeter group $W$, and $\Gamma $ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the $\text{CAT}(0)$ realization of arbitrary Tits buildings.

We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length $t{{R}^{1/3}}$ on a circle of radius $R$, for any given $t\,>\,0$. In particular we prove that any arc of length ${{\left( 40\,+\frac{40}{3}\sqrt{10} \right)}^{1/3}}\,{{R}^{1/3}}$ on a circle of radius $R$, with $R\,>\,\sqrt{65}$, contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points, $\left( {{v}_{1,n}},\,{{v}_{2,n}},\,{{v}_{3,n}},\,{{v}_{4,n}} \right)$, each of which lies on an arc of length ${{\left( 40+\frac{40}{3}\sqrt{10} \right)}^{1/3}}R_{n}^{1/3}\,+\,o\left( 1 \right)$ on a circle of radius ${{R}_{n}}$.

In this paper we prove that a generalized version of the Minimal Resolution Conjecture given by Mustaţă holds for certain general sets of points on a smooth cubic surface $X\,\subset \,{{\mathbb{P}}^{3}}$. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.

For a given de Branges space $\mathcal{H}(E)$ we investigate de Branges subspaces defined in terms of majorants on the real axis. If $\omega $ is a nonnegative function on $\mathbb{R}$, we consider the subspace

$${{\mathcal{R}}_{\omega }}(E)=\text{Clo}{{\text{s}}_{\mathcal{H}(E)}}\{F\in \mathcal{H}(E):\text{there exists }C>0:|{{E}^{-1}}F|\le C\omega \,on\,\mathbb{R}\}.$$

We show that ${{\mathcal{R}}_{\omega }}(E)$ is a de Branges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorants.

We give some examples of Calabi–Yau 3-folds with $\rho =1$ and $\rho =2$, defined over $\mathbb{Q}$ and constructed as 4-codimensional subvarieties of ${{\mathbb{P}}^{7}}$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top Chern classes from computer based computations over some finite field ${{\mathbb{F}}_{p}}$. Three of our examples (of degree 17 and 20) are new. The two others (degree 15 and 18) are known, and we recover their well-known invariants with our method. These examples are built out of Gulliksen–Negård and Kustin–Miller complexes of locally free sheaves.

Finally, we give two new examples of Calabi–Yau 3-folds of ${{\mathbb{P}}^{6}}$ of degree 14 and 15 (defined over $\mathbb{Q}$). We show that they are not deformation equivalent to Tonoli's examples of the same degree, despite the fact that they have the same invariants $({{H}^{3}},{{c}_{2}}\cdot H,{{c}_{3}})$ and $\rho =1$.

Let $q$ be an algebraic integer of degree $d\ge 2$. Consider the rank of the multiplicative subgroup of ${{\mathbb{C}}^{*}}$ generated by the conjugates of $q$. We say $q$ is of full rank if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\mathbb{Z}[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results.

1. (1) If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\text{disc}\left( \mathbb{Z}\left[ {{q}^{m}} \right] \right)=\text{disc}\left( \mathbb{Z}\left[ {{q}^{n}} \right] \right)$.

2. (2) If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\mathbb{Z}[{{q}^{n}}]=\mathbb{Z}[{{r}^{n}}]$ for infinitely many $n$, then either $q=\omega {r}'$ or $q=\text{Norm}{{(r)}^{2/d}}\omega /r'$ , where $r'$ is some conjugate of $r$ and $\omega $ is some root of unity.

3. (3) Let $r$ be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers $q$ such that $\mathbb{Z}[q]=\mathbb{Z}[r]$.

4. (4) There are only finitely many Pisot-cyclotomic numbers of any fixed order.

Let $(A,\mathfrak{m})$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I)={{\oplus }_{n\ge 0}}{{I}^{n}}/\mathfrak{m}{{I}^{n}}$ be the fiber cone of $I$. We prove certain relations among the Hilbert coefficients ${{f}_{0\,}}(I),\,{{f}_{1}}(I),\,{{f}_{2}}(I)$ of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)$. The periodic ${{\text{C}}^{*}}$-algebras are characterized, and it is shown that ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)\,\simeq \,\text{C}\left( \mathbb{T} \right)\,\otimes \,\mathfrak{U}$ where $\mathfrak{A}$ is a simple ${{\text{C}}^{*}}$-algebra.

Given a positive continuous function $\mu $ on the interval $0\,<\,t\,\le \,1$, we consider the space of so-called $\mu $-Bloch functions on the unit ball. If $\mu \left( t \right)\,=\,t$, these are the classical Bloch functions. For $\mu $, we define a metric $F_{z}^{\mu }\left( u \right)$ in terms of which we give a characterization of $\mu $-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

We obtain a complete description of closed ideals of the algebra $\mathcal{D}\cap \text{li}{{\text{p}}_{\alpha }},0<\alpha \le \frac{1}{2}$, where $\mathcal{D}$ is the Dirichlet space and $\text{li}{{\text{p}}_{\alpha }}$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha $.

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts ${{X}_{n}}$ of the ideal class groups along a ${{\mathbb{Z}}_{p}}$-extension ${{F}_{\infty }}/F$ of a number field $F$ to Iwasawa structural invariants $\lambda $ and $\mu $ attached to the inverse limit ${{X}_{\infty }}=\underleftarrow{\lim }\,{{X}_{n}}$. It relies on “good” descent properties satisfied by ${{X}_{n}}$. If $F$ is abelian and ${{F}_{\infty }}$ is cyclotomic, it is known that the $p$-parts of the orders of the global units modulo circular units ${{U}_{n}}/{{C}_{n}}$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients ${{U}_{n}}/{{C}_{n}}$, so to speak unit class groups, also satisfy good descent properties. We show this directly, i.e., without using Iwasawa's Main Conjecture.

Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\varepsilon ,\mathcal{D}(\varepsilon ))$ on ${{L}^{2}}(E;m)$. For $u\,\in \,\mathcal{D}{{(\varepsilon )}_{e}}$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\hat{X}$ be the dual process of $X$ and $\hat{Y}$ the Girsanov transformed process of $\hat{X} $. We give a necessary and sufficient condition for $(Y,\hat{Y})$ to be in duality with respect to the measure ${{e}^{2u}}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\hat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

In this paper, we mainly study operator spaces which have the locally lifting property $\left( \text{LLP} \right)$. The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the $\text{LLP}$. Several homological characterizations of the $\text{LLP}$ and weak expectation property are given. We also prove that for any operator space $V$, ${{V}^{**}}$ has the $\text{LLP}$ if and only if $V$ has the $\text{LLP}$ and ${{V}^{*}}$ is exact.