Given an L-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve L-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This c...

We prove some general results on the existence of partitions of unity in Sobolev type spaces on various innnite dimensional manifolds. As special cases we obtain in particular, (continuous) partitions of unity a) in the Malliavin test functions on an abstract Wiener space; b) in the rst order Sobolev space on pinned and free loop spaces; c) in the rst order Sobolev spaces associated with revers...

this formula defines a Schwartz function, and hence the solution u = uf ∈ S also, and the mapping f 7→ uf is continuous in the sense that if fn is a sequence of Schwartz functions such that ‖fn − f‖α,β → 0 for every Schwartz semi-norm ‖ · ‖α,β , then also ‖un − u‖α,β → 0 for every Schwartz semi-norm, where un = ufn , u = uf . In fact the formula above extends to define a distributional solution...

ABSTRACT. Previous work by Rubinstein [Rub] and Gao [Gao] computed the n-level densities for families of quadratic Dirichlet L-functions for test functions φ̂1, . . . , φ̂n supported in ∑ n i=1 |ui| < 2, and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce hi...

We prove a characterization theorem for the test functions in a CKS-space. Some crucial ideas concerning the growth condition are given.

{ For 2 ? 1 n < p < n we prove existence of a distributional solution u of the p-harmonic system ? div(jruj p?2 ru) = in ,

The Weibull function is widely used to describe skew distributions observed in nature. However, the origin of this ubiquity is not always obvious to explain. In the present paper, we consider the well-known Galton-Watson branching process describing simple replicative systems. The shape of the resulting distribution, about which little has been known, is found essentially indistinguishable from...

Volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, volatility is unobservable and only the price is known. Diffusion theory has many common points with the research on volatility, the key of the analogy being that volatility is a time-dependent...