Algorithmic Symplectic Packing

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2021-06-18 DOI:10.1080/10586458.2022.2041135

Greta Fischer, J. Gutt, M. Junger

引用次数: 0

Abstract

In this article we explore a symplectic packing problem where the targets and domains are 2n-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to Z, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Maley, Mastrangeli and Traynor [MMT00] showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to k = 12 simplices, along with lower bounds for higher values of k. We present a modified algorithmic approach that allows us to determine the k-simplex packing widths for up to k = 13 simplices in dimension four and up to k = 8 simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.

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算法辛包装

本文研究了目标和域为2n维辛流形的辛堆积问题。我们在流形具有等于Z的第一同调群的情况下工作，并且我们需要嵌入来诱导第一同调组之间的同构。在这种情况下，Maley、Mastrangeli和Traynor[MMT00]表明，该问题可以归结为组合优化问题，即将某些允许的单纯形封装到给定的标准单纯形中。他们设计了一个计算机程序并给出了计算结果。特别地，他们确定了高达k=12的单形在维度4中的单纯形填充宽度，以及k的较高值的下界。我们提出了一种改进的算法方法，允许我们确定高达k=13的维度4和高达k=8的维度6的单形的k-单纯形填充宽度。此外，我们的方法确定了所有允许最优包装的单纯形多集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.