Functions correspond to one of the key concepts in mathematics and science, allowing the representation and modeling of several types of signals and systems. The present work develops an approach for characterizing the coverage and interrelationship between discrete signals that can be fitted by a set of reference functions, allowing the definition of transition networks between the considered discrete signals. While the adjacency between discrete signals is defined in terms of respective Euclidean distances, the property of being adjustable by the reference functions provides an additional constraint leading to a surprisingly diversity of transition networks topologies. First, we motivate the possibility to define transitions between parametric continuous functions, a concept that is subsequently extended to discrete functions and signals. Given that the set of all possible discrete signals in a bound region corresponds to a finite number of cases, it becomes feasible to verify the adherence of each of these signals with respect to a reference set of functions. Then, by taking into account also the Euclidean proximity between those discrete signals found to be adjustable, it becomes possible to obtain a respective transition network that can be not only used to study the properties and interrelationships of the involved discrete signals as underlain by the reference functions, but which also provide an interesting complex network theoretical model on itself, presenting a surprising diversity of topological features, including modular organization coexisting with more uniform portions, tails and handles, as well as hubs. Examples of the proposed concepts and methodologies are provided respectively with respect to three case examples involving power, sinusoidal and polynomial functions.
翻译:与数学和科学中的关键概念之一相对应的职能与数学和科学中的一种关键概念相对应,允许对几种类型的信号和系统进行表示和建模。目前的工作为分解信号之间的覆盖范围和相互关系制定了一种定性方法,这些信号可以由一套参照功能加以调适,这样就可以对数学和科学中的一种关键概念进行描述和建模。目前的工作为分解信号之间的覆盖范围和相互关系制定了一种定性方法,这些分解信号可以由一套参考功能之间的过渡性信号加以界定。虽然分解信号之间的相近性以各自的欧几里德距离来界定,但参照功能的可调适近性使过渡性网络结构具有惊人的多样性。首先,我们鼓励有可能确定分解连续功能之间的过渡性转变,这一概念随后扩展到离散功能和信号之间的互连性。鉴于在受约束区域中所有可能的分解信号都与有限的案件数量相匹配,因此可以核实这些信号与一套参考功能的相邻性,随后又考虑到这些分解信号之间的接近性,因此有可能获得一个相应的过渡网络,不仅用于研究性质上的连续功能,而且还是一系列的离式结构结构,并且提供了一种不同结构的样本,在结构下的分解模式和结构内提供了一种不同结构的分解的分解的分解的分解的分解的分解的分解的分解的分解特性的分解特性。