Related papers: Plans on measures and AM-modulus
This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then…
The notion of $p$-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for $p$-modulus. We show that minimal subfamilies…
Models of physical systems are used to explain and predict experimental results and observations. When students encounter discrepancies between the actual and expected behavior of a system, they revise their models to include the newly…
The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for…
Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…
By studying the minimum resources required to perform a unitary transformation, families of metrics and pseudo-metrics on unitary matrices that are closely related to a recently reported quantum speed limit by the author are found.…
We study homeomorphisms of controlled $p$-module by certain integrals. In this way, we establish various properties of mappings and show that their features are close to quasiconformal and bilipschitz mappings.
Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The…
We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of…
We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of…
We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure $\mu$, for $p \geq 1$ and $\lambda>0$ we consider the functional \[ E(\gamma) =…
Traditional antenna pattern measurements involve minimizing the impact of multipath propagation in the measurement environment. In contrast, this work introduces a measurement approach that uses rather than mitigates multipath propagation.…
Due to considerable recent interest in the use of density matrices for a wide variety of purposes, including quantum computation, we present a general method for their parameterizations in terms of Euler angles. We assert that this is of…
A generic physical situation is considered where Im $\Pi$, the imaginary part of polarization operator (generalized susceptibility), can be measured on a finite interval and the high frequency asymptotics (up to a few orders) of $\Pi$ can…
The approximation of probability measures on compact metric spaces and in particular on Riemannian manifoldsby atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of…
Open discrete mappings with a modulus condition in metric spaces are considered. Some results related to local behavior of mappings as well as theorems about continuous extension to a boundary are proved.
We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the…
In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn't rely on the…
A generalization of the amoeba and the Ronkin function of a plane algebraic curve for a pair of harmonic functions on an algebraic curve with punctures is proposed. Extremal properties of $M$-curves are proved and connected with the…
Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…