Related papers: Chaos at Fifty
Fuzziness and randomicity widespread exist in natural science, engineering, technology and social science. The purpose of this paper is to present a new logic - uncertain propositional logic which can deal with both fuzziness by taking…
In 1987, Pheidas showed that the field of Laurent series $\mathbb{F}_q((t))$ with a constant for the indeterminate $t$ and a predicate for the natural powers $\{t^n \mid n > 0\}$ of $t$ is existentially undecidable. We show that the same…
Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical…
Deterministic inference is a comforting ideal in classical software: the same program on the same input should always produce the same output. As large language models move into real-world deployment, this ideal has been imported wholesale…
Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this…
This article is a brief personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and…
A new concept, primitive chaos, was proposed, as a concept closely related to the fundamental problems of sciences themselves such as determinism, causality, free will, predictability, and time asymmetry [{\em J. Phys. Soc. Jpn.} {\bf…
This two-parts paper offers a survey of linear logic and ludics, which were introduced by Girard in 1986 and 2001, respectively. Both theories revisit mathematical logic from first principles, with inspiration from and applications to…
In this paper, we present a unified framework of multiple attractors including multistability, multiperiodicity and multichaos. Multichaos, which means that the chaotic solution of a system lies in different disjoint invariant sets with…
In this paper, we demonstrate, first in literature known to us, that potential functions can be constructed in continuous dissipative chaotic systems and can be used to reveal their dynamical properties. To attain this aim, a Lorenz-like…
Weather and climate forecasts are inherently uncertain due to chaotic dynamics, imperfect initial conditions, and incomplete representation of the underlying physical processes. Operational ensemble forecasts aim to represent these…
The mathematical analysis was conceived in XVII century in Newton and Leibniz works. The problem of logical rigor in definitions was considered by Arnauld and Nicole in "Logique ou l'art de penser". They were the first, who distinguished…
The idea of predicting the future from the knowledge of the past is quite natural when dealing with systems whose equations of motion are not known. Such a long-standing issue is revisited in the light of modern ergodic theory of dynamical…
The paper presents a general introduction to the astonishing method for deriving probability approximations that was invented by Charles Stein around 50 years ago.
Einstein in 1905, his year of miracles, wrote five papers which mark a watershed between classical physics and modern physics. They dealt with problem of reality of atoms, theory of special relativity which overthrew Newtonian conceptions…
Existing theories for the evolution of aging and death treat senescence as a side-effect of strong selection for fertility. These theories are well-developed mathematically, but fit poorly with emerging experimental data. The data suggest…
The sciences of complexity present some recurrent themes: the emergence of qualitatively new behaviors in dissipative systems out of equilibrium, the aparent tendency of complex system to lie at the border of phase transitions and…
It is well known that the Lorenz system has $Z_2$-symmetry. Using introducted in math.DS/0105147 topological covering-coloring a new representation for the Lorenz system is obtained. Deleting coloring leads to the factorized Lorenz system…
In 1927 Heisenberg discovered that the ``more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa''. Four years later G\"odel showed that a finitely specified, consistent formal…
In classical probability theory, the convergence of empirical frequencies to theoretical probabilities: as captured by the Law of Large Numbers (LLN): is treated as axiomatic and emergent from statistical assumptions such as independence…