Related papers: A Topological and Operator Algebraic Framework for…

In this work a theory is developed for unifying large classes of nonlinear discrete-time dynamical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces…

Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system…

A coupled map lattice whose topology changes at each time step is studied. We show that the transversal dynamics of the synchronization manifold can be analyzed by the introduction of effective dynamical quantities. These quantities are…

Mechanical lattices support topological wave phenomena governed by geometric phases. We develop a compact Hilbert space description for one-dimensional elastic chains, expressing intra-cell motion as a normalized superposition of orthogonal…

Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…

In this paper, we aim to investigate the synchronization problem of dynamical systems, which can be of generic linear or Lipschitz nonlinear type, communicating over directed switching network topologies. A mild connectivity assumption on…

Over the past two decades, complex network theory provided the ideal framework for investigating the intimate relationships between the topological properties characterizing the wiring of connections among a system's unitary components and…

We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…

Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the…

We present a new method for the statistical process control of lattice structures using tools from Topological Data Analysis. Motivated by applications in additive manufacturing, such as aerospace components and biomedical implants, where…

In this article, we study some parallel processing algorithms for multiplication and modulo operations. We demonstrate that the state transitions that are formed under these algorithms satisfy lattice-linearity, where these algorithms…

We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to…

Understanding the global dynamics of a robot controller, such as identifying attractors and their regions of attraction (RoA), is important for safe deployment and synthesizing more effective hybrid controllers. This paper proposes a…

In topological phases of matter, the interplay between intrinsic topological order and global symmetry is an interesting task. In the study of topological orders with discrete global symmetry, an important systematic approach is the…

We propose a functional description of rewriting systems on topological vector spaces. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting…

We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that…

Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical…

We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple…

Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system…

We present a class of mechanical lattices based on elliptical gears with quasiperiodic modulation and geometric nonlinearity, capable of exhibiting topologically protected modes and amplitude-driven transitions. Starting from a…