We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$-function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$-adically analytic phases, which can be naturally seen as a $p$-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

We extract a quantitative variant of uniqueness from the usual hypotheses of the implicit function theorem. Not only does this lead to an a priori proof of continuity, but also to an alternative, full proof of the implicit function theorem. Additionally, we investigate implicit functions as a case of the unique existence paradigm with parameters.

This paper focuses on the analyticity of the limiting behavior of a class of dynamical systems defined by iteration of non-expansive random operators. The analyticity is understood with respect to the parameters which govern the law of the operators. The proofs are based on contraction with respect to certain projective semi-norms. Several examples are considered, including Lyapunov exponents associated with products of random matrices both in the conventional algebra, and in the (max, +) semi-field, and Lyapunov exponents associated with non-linear dynamical systems arising in stochastic control. For the class of reducible operators (defined in the paper), we also address the issue of analyticity of the expectation of functionals of the limiting behavior, and connect this with contraction properties with respect to the supremum norm. We give several applications to queueing theory.

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn, and independent copies A1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

We consider continuous Gaussian stochastic process indexed by a compact subset of a vector space over a local field. Under suitable conditions we obtain an asymptotic expression for the probability that such a process will exceed a high level. An important component in the proof of these results is a theorem of independent interest concerning the amount of ‘time’ which the process spends at high levels.