Related papers: Normal Cones and Thompson Metric
In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the…
The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in…
A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…
The completeness properties of spaces of immersed curves equipped with reparametrization-invariant Riemannian metrics have recently been the subject of active research. This thesis studies the metric completion of spaces of immersed open…
An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. We propose a theory, DCTC (Definable Completeness/Type Completeness), that describes many properties of o-minimalistic…
Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as…
Asymmetric vector norms are generalizations of asymmetric norms, where the subadditivity inequality is understood in ordered vector space sense. This relation imposes strong conditions on the ordering itself. This note studies on these…
The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the Order Completion Method for nonlinear PDEs. In this regard, we consider an appropriate space ML(X) of normal lower…
It has been known for some time that the topological entropy is a nondecreasing function of the parameter in the real quadratic family, which corresponds to the intuitive idea that more nonlinearity induces more complex dynamical behavior.…
For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in…
Topological measures and quasi-linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set…
Given a conformal metric with finite total Q-curvature, we show that the assumptions on scalar curvature sensitively govern the Q-curvature integral. Additionally, we introduce a conformal mass for such manifolds. Using such mass, we…
For a fixed set $X$, an arbitrary \textit{weight structure} $d \in [0,\infty]^{X \times X}$ can be interpreted as a distance assignment between pairs of points on $X$. Restrictions (i.e. \textit{metric axioms}) on the behaviour of any such…
What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different…
Given a compact K\"ahler manifold $(X,\omega)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $\omega$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass.…
In this paper, we study main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces.
The formula for analytic torsion of a cone in even dimensions is comprised of three terms. The first two terms are well understood and given by an algebraic combination of the Betti numbers and the analytic torsion of the cone base. The…
The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $\Omega$ of a locally convex space $X$ and taking values in a locally convex space…
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
We prove general theorems for isoperimetric problems on lattices of the form ${\mathbb{Z}}^{k} \times {\mathbb{N}}^{d}$ which state that the perimeter of the optimal set is a monotonically increasing function of the volume under certain…