Semigroup Methods for Evolution Equations on Networks

Abstract

Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems-cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science.Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications.Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.The three major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations, and the "Springer Briefs in Complexity" which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.Future scientific and technological developments in many fields will necessarily depend upon coming to grips with complex systems. Such systems are complex in both their composition -typically many different kinds of components interacting simultaneously and nonlinearly with each other and their environments on multiple levels -and in the rich diversity of behavior of which they are capable.The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors. UCS is explicitly transdisciplinary. It has three main goals: First, to elaborate the concepts, methods and tools of complex systems at all levels of description and in all scientific fields, especially newly emerging areas within the life, social, behavioral, economic, neuro-and cognitive sciences (and derivatives thereof); second, to encourage novel applications of these ideas in various fields of engineering and computation such as robotics, nano-technology...

From Diffusions on Graphs to Markov Chains via Asymptotic State Lumping

“…Let G = (V, E) be a finite geometric graph [31] without loops, where V ⊂ R 3 is the set of vertices, and E is the set of edges of finite length. The number of edges is N , and the edges are seen as C 1 curves connecting vertices.…”

Asymptotics of semigroups generated by operator matrices

“…[10,34,53]. Two classical approaches to this problem have their roots in the somewhat complementary theories of quadratic forms and boundary triples, cf.…”

“…Let more precisely G = (V, E) be a (possibly infinite) directed graph whose V × E incidence matrix we denote by I = (ι ve ): this allows us to introduce the notation E v := {e ∈ E : ι ve = 0}, v ∈ V for the set of edges incident in v. Attach an interval of length ρ e to each edge e: we will assume for the sake of simplicity that ρ e = 1 for all e ∈ E, but all our results prevail if there exist r, R > 0 such that merely r < ρ e < R for all e ∈ E. In this way G turns into a metric graph, in the sense of [34,Chapt. 3].…”

“…It is well-known that A 0 generates an analytic semigroup on X , see e.g. [40] for the case of finite graphs and [34,Chapter 6] for the general case. …”

Explicit formulae for limit periodic flows on networks

“…[1,3,5,6,9,13,15] and also the monograph [16]. Here we focus on transport on networks, modelled by a system of first order transport equations on the edges of the digraph representing the network.…”