Packing Density of Combinatorial Settlement Planning Models

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-09-28 DOI:10.1080/00029890.2023.2254181

Mate Puljiz, Stjepan Šebek, Josip Žubrinić

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引用次数: 4

Abstract

AbstractWe consider a combinatorial settlement model on a rectangular grid where each house must be exposed to sunlight from east, south, or west. We are interested in maximal configurations, where no additional houses can be added. Once the settlement is completely built, it seems natural to consider the building density of the obtained maximal configuration. In this article we consider two different random models which produce maximal configurations and, using simulations, we plot an estimate of the distribution of the building density (actually, the occupancy—the total number of houses built) and we conjecture that the means of these distributions converge to a certain limit as the grid dimensions grow to infinity.MSC: 60C0590C27 AcknowledgmentsWe thank the anonymous referees for helpful comments that have led to improvements of the presentation of the article. We also wish to thank Professors Tomislav Došlić and Pavel Krapivsky for fruitful and stimulating discussions.Notes1 Our Southern Hemisphere friends are welcome to turn the page upside down when inspecting the figures in our paper.2 We write X=(d)Y if two random variables X and Y are equal in distribution. Since we are dealing with discrete random variables, this is the same as requiring P(X=z)=P(Y=z) for all z∈R.Additional informationNotes on contributorsMate PuljizMATE PULJIZ was born in Croatia in 1988. He received his master’s degree from the University of Zagreb, Croatia, in 2012 and his Ph.D. in Pure Mathematics from the University of Birmingham, United Kingdom, in 2017. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include abstract dynamical systems, particularly the topology of their orbits, particle models, and fractional calculus. mate.puljiz@fer.hrStjepan ŠebekSTJEPAN ŠEBEK was born in Croatia in 1990. He received his master’s degree in 2014 and his Ph.D. in Mathematics in 2019, both from the University of Zagreb, Croatia. He is currently an Assistant Professor with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include geometry and potential theory of random walks.Josip ŽubrinićJOSIP ŽUBRINIĆ was born in Croatia in 1993. He received his master’s degree in 2016 and his Ph.D. in Mathematics in 2022, both from the University of Zagreb, Croatia. He is currently a Postdoc with the Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Zagreb, Croatia. His research interests include homogenization and dimension reduction in the theory of elasticity. josip.zubrinic@fer.hr

查看原文本刊更多论文

组合聚落规划模型的充填密度

摘要我们考虑了一个矩形网格上的组合沉降模型，其中每个房子必须从东、南或西暴露在阳光下。我们感兴趣的是最大的配置，没有额外的房子可以添加。一旦聚落完全建成，考虑得到的最大配置的建筑密度似乎是很自然的。在本文中，我们考虑了产生最大配置的两种不同的随机模型，并使用模拟，我们绘制了建筑密度分布的估计值(实际上，占用率-建造的房屋总数)，我们推测这些分布的均值随着网格尺寸增长到无穷大而收敛到某个极限。我们感谢匿名审稿人提供的有益意见，这些意见使文章的表达得到了改进。我们还要感谢托米斯拉夫教授Došlić和帕维尔·克拉皮夫斯基教授进行了富有成果和鼓舞人心的讨论。1 .南半球的朋友们在查阅我们的论文数据时，可以把页面倒过来看如果两个随机变量X和Y在分布上相等，则写成X=(d)Y。因为我们处理的是离散的随机变量，所以对于所有z∈R，这与要求P(X=z)=P(Y=z)是一样的。马特·普尔吉兹马特·普尔吉兹1988年出生于克罗地亚。2012年在克罗地亚萨格勒布大学获得硕士学位，2017年在英国伯明翰大学获得纯数学博士学位。他目前是克罗地亚萨格勒布电子工程与计算学院应用数学系的助理教授。他的研究兴趣包括抽象动力系统，特别是它们的轨道拓扑，粒子模型和分数阶微积分。mate.puljiz@fer.hrStjepan ŠebekSTJEPAN ŠEBEK, 1990年出生于克罗地亚。2014年获得克罗地亚萨格勒布大学数学硕士学位，2019年获得数学博士学位。他目前是克罗地亚萨格勒布电子工程与计算学院应用数学系的助理教授。他的研究兴趣包括几何和随机游走的势理论。Josip ŽubrinićJOSIP ŽUBRINIĆ 1993年出生于克罗地亚。2016年获得克罗地亚萨格勒布大学数学硕士学位，2022年获得数学博士学位。他目前是克罗地亚萨格勒布电子工程与计算学院应用数学系的博士后。主要研究方向为弹性理论中的均质化和降维。josip.zubrinic@fer.hr

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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