Related papers: Dynamics and computation in functional shifts
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
The goal of data-driven learning of dynamical systems is to interpret time series as a continuous observation of an underlying dynamical system. This task is not well-posed for a variety of reasons - such as multiple co-existing…
The study of neural computation aims to understand the function of a neural system as an information processing machine. Neural systems are undoubtedly complex, necessitating principled and automated tools to abstract away details to…
Economic transformation -- change in what an economy produces -- is foundational to development and rising standards of living. Our understanding of this process has been propelled recently by two branches of work in the field of economic…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
We consider the category of partially observable dynamical systems, to which the entropy theory of dynamical systems extends functorially. This leads us to introduce quotient-topological entropy. We discuss the structure that emerges. We…
Many complex systems satisfy a set of constraints on their degrees of freedom, and at the same time, they are able to work and adapt to different conditions. Here, we describe the emergence of this ability in a simplified model in which the…
A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to…
We investigate the computability (in the sense of computable analysis) of the topological pressure $P_{\rm top}(\phi)$ on compact shift spaces $X$ for continuous potentials $\phi:X\to {\mathbb R}$. This question has recently been studied…
We study the dynamical properties of certain shift spaces. To help study these properties we introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified…
The question What is Complexity? has occupied a great deal of time and paper over the last 20 or so years. There are a myriad different perspectives and definitions but still no consensus. In this paper I take a phenomenological approach,…
Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on…
We consider large-dimensional dynamical systems involving a linear force and a random force comprising both potential and non-conservative contributions. Such systems are known to exhibit a topological trivialization phase transition as the…
Complexity science offers a wide range of measures for quantifying unpredictability, structure, and information. Yet, a systematic conceptual organization of these measures is still missing. We present a unified framework that locates…
Classification and invariants, with respect to basis changes, of finite dimensional algebras are considered. An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with…
A mathematical concept is identified and analyzed that is implicit in the 2012 paper Turing Incomputable Computation, presented at the Alan Turing Centenary Conference (Turing-100, Manchester). The concept, called dynamic level sets, is…
Many potentially non-terminating functions cannot be directly defined in a logic of total functions, such as HOL. A well-known solution to this is to define non-terminating functions using a clock that forces termination at a certain depth…
We study the parametrized complexity of fundamental relations between multidimensional subshifts, such as equality, conjugacy, inclusion, and embedding, for subshifts of finite type (SFTs) and effective subshifts. We build on previous work…
A central question in cognitive science is whether conceptual representations converge onto a shared manifold to support generalization, or diverge into orthogonal subspaces to minimize task interference. While prior work has discovered…
We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a…