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 ${{\tilde{h}}_{g}}(n)=g{{h}_{g}}(n)$ .

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.

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.

We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ${{\mathbb{C}}^{n}}$ . We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\,\,\|z\|)$ on ${{B}^{n}}\backslash \{0\}$ , where $f(z,t)$ is a convex subordination chain over $(0,1)$ .

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\ge 0$ , we conjecture that $L_{n}^{(-1-n-r)}(x)=\Sigma _{j=0}^{n}\left( _{n-j}^{n-j+r} \right){{x}^{j}}/j!$ is a $\mathbb{Q}$ -irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows fromrecent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\ge 5$ . Here we verify it in three situations: (i) when $n$ is large with respect to $r$ , (ii) when $r\le 8$ , and (iii) when $n\le 4$ . The main tool is the theory of $p$ -adic Newton Polygons.

Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight ${{\aleph }_{1}}$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

In this paper we study square integrable representations and $L$ -functions for quasisplit general spin groups over a $p$ -adic field. In the first part, the holomorphy of $L$ -functions in a half plane is proved by using a variant form of Casselman's square integrability criterion and the Langlands–Shahidi method. The remaining part focuses on the proof of the standard module conjecture. We generalize Muić's idea via the Langlands–Shahidi method towards a proof of the conjecture. It is used in the work of M. Asgari and F. Shahidi on generic transfer for general spin groups.

For an isotropic submanifold ${{M}^{n}}(n\underline{\underline{>}}3)$ of a space form ${{\tilde{M}}^{n+p}}(c)$ of constant sectional curvature $c$ , we show that if the mean curvature vector of ${{M}^{n}}$ is parallel and the sectional curvature $K$ of ${{M}^{n}}$ satisfies some inequality, then the second fundamental form of ${{M}^{n}}$ in ${{\tilde{M}}^{n+p}}$ is parallel and our manifold ${{M}^{n}}$ is a space form.

An unsettled conjecture of V. Bergelson and I. Håland proposes that if $(X,\,\mathcal{A},\,\mu ,\,T)$ is an invertible weak mixing measure preserving system, where $\mu (X)<\infty $ , and if ${{p}_{1}},{{p}_{2}},...,{{p}_{k}}$ are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no ${{p}_{i}}$ , nor any ${{p}_{i}}-{{p}_{j,}}i\ne j$ , is constant on a set of positive density, then for any measurable sets ${{A}_{0}},{{A}_{1}},...,{{A}_{K}}$ , there exists a zero-density set $E\subset Z$ such that

1

$$\underset{n\notin E}{\mathop{\underset{n\to \infty }{\mathop{\lim }}\,}}\,\mu ({{A}_{0}}\cap {{T}^{p1(n)}}{{A}_{1}}\cap \ldots \cap {{T}^{pk(n)}}{{A}_{k}})=\prod\limits_{i=0}^{k}{\mu ({{A}_{i}}).}$$

We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families $\left\{ {{p}_{1}},{{p}_{2}},\,.\,.\,.\,,\,{{p}_{k}} \right\}$ satisfying the hypotheses of this theorem.

We give a constructive proof, in the special case of $\text{G}{{\text{L}}_{3}}$ , of a theorem of Ash and Stevens which compares overconvergent cohomology to classical cohomology. Namely, we show that every ordinary classical Hecke-eigenclass can be lifted uniquely to a rigid analytic eigenclass. Our basic method builds on the ideas of M. Greenberg; we first form an arbitrary lift of the classical eigenclass to a distribution-valued cochain. Then, by appropriately iterating the ${{U}_{p}}$ -operator, we produce a cocycle whose image in cohomology is the desired eigenclass. The constructive nature of this proof makes it possible to perform computer computations to approximate these interesting overconvergent eigenclasses.

Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$ . In this paper we determine all the cases in which Lie $(N)$ is prehomogeneous under $\text{Ad}\left( m \right)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$ .

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following.

1. (a) Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets.

2. (b) If $G$ is a countably infinite Abelian group with finitely many elements of order 2 and $\beta G$ is the Stone–Čech compactification of $G$ as a discrete semigroup, then for every idempotent $p\,\,\in \,\,\beta G\backslash \{0\}$ , the subset $\{p,-p\}\subset \beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and $\{-p\}$ .