Asymptotic Study of a Class of Nonlinear Differential Equations Arising from Increasing Combinatorial Structures

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Project title

Code/計畫編號

MOST105-2115-M019-001

Translated Name/計畫中文名

遞增型組合結構上的非線性方程的漸進分析

Project Coordinator/計畫主持人

Hua-Huai Chern

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Computer Science and Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=11897011

Year

2016

Start date/計畫起

01-08-2016

Expected Completion/計畫迄

31-07-2017

Bugetid/研究經費

238千元

ResearchField/研究領域

數學

本次研究計畫旨在針對能將出現在組合結構上幾類間題連成统一刻畫的非線性方程 y 〃 = G(y) 的解之於主奇異（畸）點找出更有故的漸遑分析。最斩的研究遑展上，上述方程除了能提供在如遞迴樹或遞增樹上的特走量的刻畫'在有向非循環圖形（directed acyclic graphs )及甚至如 samplers 的應用上也存在對應的部分。即便在求算上有既成的作法一透過隱含解的型式呈現並加以分析'但如何提出有其舍可有利於更一般化的漸遑求算方法為主要的重點。此外，從線上資料庫OEIS裡能提供不少間題都能分享相同的刻畫方程' 如何釐清可能存在在彼此間的對應關係'亦是此計畫的研究主題之一。 "In this project, the author aims to study the asymptotic behavior near the dominant singularity of the ODE of the type y00 D G.y/ that offers the characterization for many problems in combinatorial structures. Recently, the author finds that the above ODE is not only applicable to the study on recursive trees or increasing trees but also has counterparts for the problems related to directed acyclic graphs or Boltzmann samplers. Although the analysis can be done via the implicit form of the solution, the author intends to find the more generalized method for deriving the asymptotics to serve as an alternative approach. Moreover, there are many well-known examples listed in the on-line database OEIS being linked to the ODE of same type. To get the possible bijections among these examples is also an interesting aspect for research."

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Asymptotic Study of a Class of Nonlinear Differential Equations Arising from Increasing Combinatorial Structures

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