Extreme values of the resurgence for homogeneous ideals in polynomial rings

We show that two ostensibly different versions of the asymptotic resurgence introduced by E. Guardo, B. Harbourne and A. Van Tuyl in 2013 are the same. We also show that the resurgence and asymptotic resurgence attain their maximal values simultaneously, if at all, which we apply to a conjecture of E. Grifo. For radical ideals of points, we show that the resurgence and asymptotic resurgence attain their minimal values simultaneously. In addition, we introduce an integral closure version of the resurgence and relate it to the other versions of the resurgence. In closing we provide various examples and raise some related questions, and we finish with some remarks about computing the resurgence.     

Transseries for a class of nonlinear difference equations *

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? ₀ we associate with it a stable manifold of solutions with asymptotic expansion ? ₀. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? ₀ in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]     

基于红外热成像的林地余火死灰复燃点诊断方法研究

林地余火阴燃特性具有隐蔽性强、持续时间长、目测难度大、且具有死灰复燃等特点,从而一直困扰森林火灾的彻底扑灭。为了及时、高效地发现林地余火阴燃点,探索林地余火死灰复燃的特征及规律,在南京森林警察学院点烧基地里进行测试实验,以无人机搭载热红外成像系统、气象采集系统等为工具,把火源点设置在杨树林内,人为干预进行点烧、熄灭、复燃等重复实验,实验包括白天和夜晚两个时间段,用安装于无人机上的红外热成像仪对火源进行观测。实验表明：林地余火死灰复燃的温度在500～600℃,离散程度较大;林地余火死灰复燃在白天的时间普遍短于夜晚时间,表明外界温度越高,越会促进林地余火死灰复燃的速率;在森林中,不同地点、不同时间段的森林背景温度标准差比较稳定,主要处于1～9 之间;林地余火红外图像的温度数据的标准差值分布在30～85之间;红外图像的温度数据的标准差值分布在55～85 之间,则可定为死灰复燃可疑阶段。该方法量化了死灰复燃的火环境及温度参数阈值,明确林地余火阴燃点引燃特征值。该研究成果将推动森林防火技术的发展,为安全扑火提供重要的方法和资料。     

Multi-resurgence and connection-to-Stokes formulæ for some linear meromorphic differential systems

In this article, we consider a linear meromorphic differential system with several levels $r_1<...<r_p$. For any k, we prove that the Borel transforms of its $r_k$-reduced formal solutions are resurgent and we give the general form of all their singularities. Next, under some convenient hypotheses on the geometric configuration of singular points, we display exact formulæ to express some Stokes multipliers of level $r_k$ of initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already obtained by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one numerical example.     

Resurgent transseries & Dyson–Schwinger equations

We employ resurgent transseries as algebraic tools to investigate two self-consistent Dyson–Schwinger equations, one in Yukawa theory and one in quantum electrodynamics. After a brief but pedagogical review, we derive fixed point equations for the associated anomalous dimensions and insert a moderately generic log-free transseries ansatz to study the possible strictures imposed. While proceeding in various stages, we develop an algebraic method to keep track of the transseries’ coefficients. We explore what conditions must be violated in order to stay clear of fixed point theorems to eschew a unique solution, if so desired, as we explain. An interesting finding is that the flow of data between the different sectors of the transseries shows a pattern typical of resurgence, i.e. the phenomenon that the perturbative sector of the transseries talks to the nonperturbative ones in a one-way fashion. However, our ansatz is not exotic enough as it leads to trivial solutions with vanishing nonperturbative sectors, even when logarithmic monomials are included. We see our result as a harbinger of what future work might reveal about the transseries representations of observables in fully renormalised four-dimensional quantum field theories and adduce a tentative yet to our mind weighty argument as to why one should not expect otherwise.