Let $p$ be a prime and $\vartheta$ an integer of order $t$ in the multiplicative group modulo $p$. In this paper, we continue the study of the distribution of Diffie–Hellman triples$(\vartheta^x, \vartheta^y, \vartheta^{xy})$ by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie–Hellman triples as the pair $(x,y)$ varies over small boxes.

Let $(R, {\frak m})$ be a local Cohen–Macaulay ring and let $I$ be an ${\frak m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring ${\mathcal F}$ of $I$ in terms of other well-known invariants of $I$. Special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen–Macaulayness of ${\mathcal F}$.

We define torsors for sheaves of simplicial groups and sheaves of groupoids enriched in simplicial sets, and give classification results for these torsors in terms of the homotopy theory of simplicial sheaves. The proofs of the classification results use a new, general approach to cocycles taking values in simplicial sheaves. We prove a homotopy classification result for gerbes locally isomorphic to a fixed sheaf of groups.

Recent computations of UNil-groups by Connolly, Ranicki and Davis are used to study splittability of homotopy equivalences between 4-dimensional manifolds with infinite dihedral fundamental groups.

Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson–Walker–Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points and an integral homology sphere as the quotient.

Topological surgery in dimension four is used to show a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

We consider the functions $T_n(x)$ defined as the $n$th partial derivative of Lebesgue's singular function $L_a(x)$ with respect to $a$ at $a=\frac{1}{2}$. This sequence includes a multiple of the Takagi function as the case $n=1$. We show that $T_n$ is continuous but nowhere differentiable for each $n$, and determine the Hölder order of $T_n$. From this, we derive that the Hausdorff dimension of the graph of $T_n$ is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for $T_n(x)$ using the binary expansion of $x$, and use this to find the sets of points where $T_2$ and $T_3$ take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of $T_2$ and $T_3$.

We derive universal Diophantine properties for the Patterson measure $\mu_{_{\Gamma}}$ associated with a convex cocompact Kleinian group $\Gamma$ acting on $(n+1)$-dimensional hyperbolic space. We show that $\mu_{_{\Gamma}}$ is always an ${\cal S}$-friendly measure, for every $(\Gamma,\mu_{_{\Gamma }})$-neglectable set ${\cal S}$, and deduce that if $\Gamma$ is of non-Fuchsian type then $\mu_{_{\Gamma}}$ is an absolutely friendly measure in the sense of Pollington and Velani. Consequently, by a result of Kleinbock, Lindenstrauss and Weiss, $\mu_{_{\Gamma}}$ is strongly extremal which means that $\mu_{_{\Gamma}}$-almost every point is not very well multiplicatively approximable. This is remarkable, since by a well-known result in classical metric Diophantine analysis the set of very well multiplicatively approximable points is of $n$-dimensional Lebesgue measure zero but has Hausdorff dimension equal to $n$.

We consider the class $S_{p}$ of functions \[ f(z) = z + a_{p+1}z^{p+1} +\cdots \] univalent in the unit disk $\triangle$. We show that, if $f\in S_{p}$ and $p$ is large, \[ \alpha (f) = \lim\limits_{n{\to}\infty}\frac{|a_{n}|}{n} \le \alpha (p) = \frac{2(\log p\log\log p)^2}{p^4}\]. We also obtain estimates for $|a_{n}|$ when $n\in S_p$.

In a companion paper [1] it will be shown that there exists $f$ in $S_{p}$ for $p=1,2,\ldots$ such that \[ \alpha (f) \ge \frac{C_{0}}{p^4},\] where $C_{0}$ is a positive absolute constant.

We give a simple constructive proof of a geometric form of Hahn-Banach theorem, simplifying a proof in [5].

The classical result of Gohberg, Markus and Feldman states that, when $E$ is one of the classical Banach sequence spaces $E=l^p$ for $1\leq p<\infty$ or $E=c_0$, the only closed, two-sided, non-trivial ideal in $\cal B(E)$, the Banach algebra of operators on a Banach space $E$, is $\cal K(E)$, the ideal of compact operators. Gramsch and Luft completely classified the closed, two-sided ideals in $\cal B(H)$ for an arbitrary Hilbert space $H$ through the idea of $\kappa$-compact operators, for infinite cardinals $\kappa$. This paper presents an extension of this result to the non-separable versions of $l^p$ and $c_0$.

We develop the theory of contractive Markov systems initiated in [8]. We construct a coding map for such systems and investigate some of its properties.

A wide class of scalar first order linear autonomous neutral delay differential equations with distributed type delays is considered. By the use of two distinct real roots of the corresponding characteristic equation, a new result on the behavior of the solutions is established.

We develop new criteria for the stability of a periodic solution of a given newtonian equation based on the $L^1$-norm of the coefficients of the third order approximation, by proving that the twist coefficient is different form zero.