Related papers: Spectral Methods in Complex Systems
This work deals with two groups of spectral analysis results for matrices arising in fully implicit Runge-Kutta methods used for linear time-dependent partial differential equations. These were applied for different formulations of the same…
Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. Many special functions are eigenfunctions to explicit operators, such as difference and…
Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…
Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets…
Recent results in control systems and numerical integration literature utilize invariant set theory to lift dynamical systems evolving on nonlinear manifolds to those evolving on vector spaces. We leverage this technique to propose an…
We show that an intensity speckle can be directly interpreted as the properties of incident light - amplitude, phase, polarization, and coherency over spatial positions. Revisiting the speckle-correlation scattering matrix (SSM) method [Lee…
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…
The complexity of condensed matter arises from emergent behaviors that cannot be understood by analyzing individual constituents in isolation. While traditional condensed-matter approaches-developed primarily for ideal crystalline…
We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…
Spectral variability is one of the major issue when conducting hyperspectral unmixing. Within a given image composed of some elementary materials (herein referred to as endmember classes), the spectral signature characterizing these classes…
Economic and ecological models can be extremely complex, with a large number of agents/species each featuring multiple interacting dynamical quantities. In an attempt to understand the generic stability properties of such systems, we define…
Accurate and efficient fluid flow models are essential for applications relating to many physical phenomena including geophysical, aerodynamic, and biological systems. While these flows may exhibit rich and multiscale dynamics, in many…
Large Language Models (LLMs) have demonstrated their transformative potential across numerous disciplinary studies, reshaping the existing research methodologies and fostering interdisciplinary collaboration. However, a systematic…
Understanding the structural complexity and predictability of complex networks is a central challenge in network science. Although recent studies have revealed a relationship between compression-based entropy and link prediction…
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of…
We present a technique novel in numerical methods. It compiles the domain of the numerical methods as a discretized volume. Congruent elements are glued together to compile the domain over which the solution of a boundary value problem of a…
Spectrum sensing and analysis is crucial for a variety of reasons, including regulatory compliance, interference detection and mitigation, and spectrum resource planning and optimization. Effective, real-time spectrum analysis remains a…
The multiplicity of software projects' stakeholders and activities leads to the multiplicity of software specification views and thus creates the need to establish mutual consistency between them. The process of establishing such…
Network theory provides tools which are particularly appropriate for assessing the complex interdependencies that characterise our modern connected world. This article presents an introduction to network theory, in a way that doesn't…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…