This article is a concept paper, which discusses the definition of randomness, and the sources of randomness in the mathematical system as well as in the physical system (the Universe). We document that randomness is an inherited property of mathematics and of the physical world, shaping all ...
The definition of randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as mutual information in individual infinite sequences. Applications to several areas, like probability theory, theory of algorithms, intuitionistic logic are...
Define Mathematical entropy. Mathematical entropy synonyms, Mathematical entropy pronunciation, Mathematical entropy translation, English dictionary definition of Mathematical entropy. n. In information theory, a mathematical measure of the degree of ran
from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce Fun...
2015)). In the 2D setting, we again investigate the emergence of patterns under each definition of the phenotypic switching functions listed in Table 1. Results are displayed in Fig. 5, where in each column from top to bottom we plot the spatial distributions of \(n_0\) and \(n_1\)...
Schnorr, C.P.: A unified approach to the definition of random sequences. Math. Syst. Theory 5, 246–258 (1971) Article MathSciNet MATH Google Scholar Terwijn, S.A., Zambella, D.: Computational randomness and lowness. J. Symb. Logic 66(3), 1199–1205 (2001) Article MathSciNet ...
A Good Definition of Randomness Most mathy people have a pretty good mental model of what a random process is (for example, generating a sequence of 20 independent bits). I think most mathy people also have the intuition that there’s a sense in which an individual string like 1010100111000...
I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is...
“nice point”. 1.1 scaling limit of thick points of planar random walk we will extend the definition of the integer part function by setting for \(x=(x_1,x_2) \in {\mathbb {r}}^2\) , \(\left\lfloor x \right\rfloor = (\left\lfloor x_1 \right\rfloor ,\left\lfloor x_...
loss during communication established a new viewpoint on entropy, which was until then only known as a measure of randomness in thermodynamics [8]. Since then, using entropy as an information measure has attracted much attention not only in signal processing, but also in the field of biology....