相关论文: Danzer's Problem, Effective Constructions of Dense…

This work considers a number of optimization problems and reductive relations between them. The two main problems we are interested in are the \emph{Optimal Decision Tree} and \emph{Set Cover}. We study these two fundamental tasks under…

A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…

The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set…

The paper studies a dynamic blocking problem, motivated by a model of optimal fire confinement. While the fire can expand with unit speed in all directions, barriers are constructed in real time. An optimal strategy is sought, minimizing…

Domain generalization is a popular machine learning technique that enables models to perform well on the unseen target domain, by learning from multiple source domains. Domain generalization is useful in cases where data is limited,…

We reformulate and generalize the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed point…

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

It is hard to densely track a nonrigid object in long term, which is a fundamental research issue in the computer vision community. This task often relies on estimating pairwise correspondences between images over time where the error is…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960…

In this paper, we study the complexity of two types of digraph packing problems: perfect out-forests problem and Steiner cycle packing problem. For the perfect out-forest problem, we prove that it is NP-hard to decide whether a given strong…

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under…

This paper presents a novel geometrical approach to investigate the convexity of a density-based cluster. Our approach is grid-based and we are about to calibrate the value space of the cluster. However, the cluster objects are coming from…

Access to below-canopy volumetric vegetation data is crucial for understanding ecosystem dynamics. We address the long-standing limitation of remote sensing to penetrate deep into dense canopy layers. LiDAR and radar are currently…

For a finite set of integers such that the first few gaps between its consecutive elements equal $a$, while the remaining gaps equal $b$, we study dense packings of its translates on the line. We obtain an explicit lower bound on the…

We study the degree of success of a single predator hunting a herd of prey on a two dimensional square lattice landscape. We explicitly consider the self volume of the prey restraining their dynamics on the lattice. The movement of both…

The study of optimal decision trees has gained increasing attention in recent years; however, despite substantial progress, it still suffers from two major challenges: First, trees constructed by existing optimal decision tree (ODT)…

In this work, we consider a class of convex optimization problems in a real Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those…

Dynamical density functional theory (DDFT) is a powerful variational framework to study the nonequilibrium properties of colloids by only considering a time-dependent one-body number density. Despite the large number of recent successes,…