惯性聚合 高效追踪和阅读你感兴趣的博客、新闻、科技资讯
阅读原文 在惯性聚合中打开

推荐订阅源

博客园 - 聂微东
爱范儿
爱范儿
博客园 - 三生石上(FineUI控件)
Recent Announcements
Recent Announcements
大猫的无限游戏
大猫的无限游戏
MongoDB | Blog
MongoDB | Blog
A
About on SuperTechFans
M
MIT News - Artificial intelligence
V
Visual Studio Blog
云风的 BLOG
云风的 BLOG
The GitHub Blog
The GitHub Blog
Jina AI
Jina AI
P
Proofpoint News Feed
博客园_首页
酷 壳 – CoolShell
酷 壳 – CoolShell
T
The Blog of Author Tim Ferriss
G
Google Developers Blog
宝玉的分享
宝玉的分享
Engineering at Meta
Engineering at Meta
Last Week in AI
Last Week in AI
aimingoo的专栏
aimingoo的专栏
罗磊的独立博客
N
Netflix TechBlog - Medium
人人都是产品经理
人人都是产品经理

math.CO updates on arXiv.org

Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
Train Tracks with Gaps: Applying the Probabilistic Method...
William Kuszmaul · 2020-12-01 · via math.CO updates on arXiv.org

We identify a tradeoff curve between the number of wheels on a train car, and the amount of track that must be installed in order to ensure that the train car is supported by the track at all times. The goal is to build an elevated track that covers some large distance $\ell$, but that consists primarily of gaps, so that the total amount of feet of train track that is actually installed is only a small fraction of $\ell$. In order so that the train track can support the train at all points, the requirement is that as the train drives across the track, at least one set of wheels from the rear quarter and at least one set of wheels from the front quarter of the train must be touching the track at all times. We show that, if a train car has $n$ sets of wheels evenly spaced apart in its rear and $n$ sets of wheels evenly spaced apart in its front, then it is possible to build a train track that supports the train car but uses only $Θ( \ell / n )$ feet of track. We then consider what happens if the wheels on the train car are not evenly spaced (and may even be configured adversarially). We show that for any configuration of the train car, with $n$ wheels in each of the front and rear quarters of the car, it is possible to build a track that supports the car for distance $\ell$ and uses only $O\left(\frac{\ell \log n}{n}\right)$ feet of track. Additionally, we show that there exist configurations of the train car for which this tradeoff curve is asymptotically optimal. Both the upper and lower bounds are achieved via applications of the probabilistic method.