arXiv subjects

Kewei Zhang

Publications and source records attributed to Kewei Zhang.

41 records · Page 3Linked to original sources

arXiv2018

Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kahler-Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita-Odaka coincides with the greatest lower Ricci bound invariant of Tian.

math.AG↗

arXiv2017

Predictability of Irregular Human Mobility

Understanding human mobility is critical for decision support in areas from urban planning to infectious diseases control. Prior work has focused on tracking daily logs of outdoor mobility without considering relevant context, which contain a mixture of regular and irregular human movement for a range of purposes, and thus diverse effects on the dynamics have been ignored. This study aims to focus on irregular human movement of different meta-populations with various purposes. We propose approaches to estimate the predictability of mobility in different contexts. With our survey data from international and domestic visitors to Australia, we found that the travel patterns of Europeans visiting for holidays are less predictable than those visiting for education, while East Asian visitors show the opposite patterns, ie, more predictable for holidays than for education. Domestic residents from the most populous Australian states exhibit the most unpredictable patterns, while visitors from less populated states show the highest predictable movement.

physics.soc-ph↗

arXiv2016

Compensated Convex Transforms and Geometric Singularity Extraction from Semiconvex Functions

We apply upper and lower compensated convex transforms, which are `tight' one-sided approximations of a given function, to the extraction of fine geometric singularities from semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known examples of (locally) semiconcave functions include the Euclidean distance and squared distance functions. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is a singular (non-differentiable) point if and only if it is a scale $1$-valley point, and if $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and $ \lim_{λ\to \infty}λV_λ(f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Fréchet) subdifferential $\partial_- f(x)$ and $V_λ(f)(x)$ is the valley transform at $x$. Thus the limit function $\mathcal{V}_\infty(f)(x):=\lim_{λ\to+\infty}λV_λ(f)(x)=r_x^2/4$ gives a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$, and also provides an asymptotic expansion of the upper transform $C^u_λ(f)(x)$ when $λ\to \infty$. For a locally semiconvex function $f$ with linear modulus we show that the limit of the gradient of the upper compensated convex transform $\lim_{λ\to+\infty}\nabla C^u_λ(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x$, and that for a DC-function $f=g-h$, the scale $1$-edge transform satisfies $\liminf_{λ\to+\infty}λE_λ(f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the convex functions $g$ and $h$ at $x$ respectively.

math.OC↗

arXiv2016

Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Theoretical Foundations

We introduce Lipschitz continuous and $C^{1,1}$ geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in $\mathbb{R}^n$ by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of $C^{1,1}$ functions. We also prove that our approximation methods, which are differentiation and integration free and not sensitive to sample type, are stable with respect to the Hausdorff distance between samples.

math.MG↗

arXiv2015

Compensated Convexity, Multiscale Medial Axis Maps and Sharp Regularity of the Squared Distance Function

We introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale $λ$, and prove a sharp regularity result for the squared-distance function to any closed non-empty subset $K$ of $\mathbb{R}^n$. Our results exploit properties of the function $C^l_λ(dist^2(\cdot;\, K))$ obtained by applying the quadratic lower compensated convex transform of parameter $λ$ to $dist^2(\cdot;\, K)$, the Euclidean squared-distance function to $K$. Using an estimate for the tight approximation of $dist^2(\cdot;\, K)$ by $C^l_λ(dist^2(\cdot;\, K))$, we prove $C^{1,1}$-regularity of $dist^2(\cdot;\, K)$ outside a neighbourhood of the closure of the medial axis $M_K$ of $K$, and give an asymptotic formula for $C^l_λ(dist^2(\cdot;\, K))$ in terms of the scaled squared distance to $K$ and to the convex hull of the set of points that realize the minimum distance to $K$. The multiscale medial axis map, $M_λ(\cdot;\, K)$, is a family of non-negative functions whose limit as $λ\to \infty$ exists and is called the multiscale medial axis landscape map, $M_{\infty}(\cdot;\, K)$. We show $M_{\infty}(\cdot;\, K)$ is strictly positive on the medial axis $M_K$ and zero elsewhere. We give conditions to ensure $M_λ(\cdot;\, K)$ keeps a constant height along parts of $M_K$ generated by two-point subsets with the height dependent on the distance between the generating points, so giving a hierarchy between different parts of $M_K$ that enables subsets of $M_K$ to be selected by thresholding. Given a compact subset $K$ of $\mathbb{R}^n$, while it is well known that $M_K$ is not Hausdorff stable, we prove $M_λ(\cdot;\, K)$ is stable under Hausdorff distance, and deduce implications for localization of the stable parts of $M_K$. Examples are included.

math.MG↗