Related papers: Plans on measures and AM-modulus
Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring…
New measures for the quantization of systems with constraints are discussed and applied to several examples, in particular, examples of alternative but equivalent formulations of given first-class constraints, as well as a comparison of…
Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…
We study $\infty$-modulus on general metric spaces and establish its relation to shortest lengths of paths. This connection was already known for modulus on graphs, but the formulation in metric measure spaces requires more attention to…
Modular units are functions on modular curves whose divisors are supported on the cusps. They form a free abelian group of rank at most one less than the number of cusps. In this paper we study the group of modular units on $X_{1}( p )$,…
We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit…
This is a survey paper that also contains some new results. It will appear in the proceedings of the AMS summer research institute on Algebraic Geometry at Santa Cruz.
The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the…
In this work, we define the notion of unimodular random measured metric spaces as a common generalization of various other notions. This includes the discrete cases like unimodular graphs and stationary point processes, as well as the…
Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and…
This paper presents new results for the modulus of families of walks on a graph---a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its…
We give an informal survey, emphasizing examples and open problems, of two interconnected research programs in moduli of curves: the systematic classification of modular compactifications of $M_{g,n}$, and the study of Mori chamber…
In this article we study the family of $BMO^p$ spaces, $p \geq 1$, in the general context of metric measure spaces. We give a characterization theorem that allows to describe all possible relations between these spaces considered as sets of…
Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two…
On the basis of an analysis of previous research, we present a generalized approach for measuring the difference of plans with an exemplary application to machine scheduling. Our work is motivated by the need for such measures, which are…
For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of the uniform limit of these mappings. It is proved that, the uniform limit of these mappings is light…
Measurements are shown to be processes designed to return figures: they are effective. This effectivity allows for a formalization as Turing machines, which can be described employing computation theory. Inspired in the halting problem we…
Measurement models (MMs) stand at the highest structural level of quantum measurement theory. MMs can be employed to construct instruments which stand at the next level. An instrument is thought of as an apparatus that is used to measure…
A class of distance measures on probabilities -- the integral probability metrics (IPMs) -- is addressed: these include the Wasserstein distance, Dudley metric, and Maximum Mean Discrepancy. IPMs have thus far mostly been used in more…
We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on…