We discuss the Eisenbud-Harris theory of limit linear series, and survey a number of applications, ranging from the geometry of moduli spaces of curves to connectedness of certain Hurwitz spaces. We also present a more recent construction of limit linear series, and discuss applications and potential applications both to classical linear series and to their higher-rank generalization. In this s...

In this paper we will prove that if $L$ is a continuous symmetric n-linear form on a Hilbert space and $widehat{L}$ is the associated continuous n-homogeneous polynomial, then $||L||=||widehat{L}||$. For the proof we are using a classical generalized  inequality due to S. Bernstein for entire functions of exponential type. Furthermore we study the case that if X is a Banach space then we have t...

We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...

We study different completeness definitions for two categories of lattice-valued Cauchy spaces and the relations between these definitions. We also show the equivalence of a so-called completion axiom and the existence of a completion.

We consider N single server infinite buffer queues with service rate β. Customers arrive at rate N α, choose L queues uniformly, and join the shortest one. The stability condition is α < β. We study in equilibrium the fraction of queues of length at least k ≥ 0. We prove a functional central limit theorem on an infinite-dimensional Hilbert space with its weak topology, with limit a stationary O...

We calculate the effective action for small velocity scattering of localized 1-branes and 5-branes. Momentum is allowed to flow in the direction along the 1-branes so that the moduli space has only 1/8 of the full supersymmetry. Relative to the more familiar case with the 1-branes delocalized along the 5-branes, this introduces new moduli associated with the motion of the 1-branes along the 5-b...

Customers arrive at rate N α on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate β. We let N go to infinity. We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations, in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenb...

Customers arrive at rate Nα on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate β. We let N go to infinity. We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations, in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenbe...

We consider an estimator b βn(t) defined as the element φ ∈ Φ minimizing a contrast process Λn(φ, t) for each t. We give some general results for deriving the weak convergence of √ n(b βn − β) in the space of bounded functions, where, for each t, β(t) is the φ ∈ Φ minimizing the limit of Λn(φ, t) as n → ∞. These results are applied in the context of penalized M-estimation, that is, when Λn(φ, t...