Related papers: On regularities of mass random phenomena
Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a…
One can often encounter claims that classical (Kolmogorovian) probability theory cannot handle, or even is contradicted by, certain empirical findings or substantive theories. This note joins several previous attempts to explain that these…
April 25, 2003, marked the 100th anniversary of the birth of Andrei Nikolaevich Kolmogorov, the twentieth century's foremost contributor to the mathematical and philosophical foundations of probability. The year 2003 was also the 70th…
Pseudorandmness plays an important role in number theory, complexity theory and cryptography. Our aim is to use models of arithmetic to explain pseudorandomness by randomness. To this end we construct a set of models $\cal M$, a common…
This paper provides a systematic yet accessible presentation of the Contextuality-by-Default theory. The consideration is confined to finite systems of categorical random variables, which allows us to focus on the basics of the theory…
We examine several notions of randomness for elements in a given $\Pi^0_1$ class $\mathcal{P}$. Such an effectively closed subset $\mathcal{P}$ of $2^\omega$ may be viewed as the set of infinite paths through the tree $T_{\mathcal{P}}$ of…
Regular variation is a continuous-parameter theory; we work in a general setting, containing the existing Karamata, Bojanic-Karamata/de Haan and Beurling theories as special cases. We give sequential versions of the main theorems, that is,…
We study the probability distribution $P$ of the sum of a large number of non-identically distributed random variables $n_m$. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to…
We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on $2^\omega$, the so-called proper sequences. Our main results are as follows: (1) We…
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it…
We present an improved version of Berry's ansatz able to incorporate exactly the existence of boundaries and the correct normalization of the eigenfunction into an ensemble of random waves. We then reformulate the Random Wave conjecture…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We…
A popular view in contemporary Boltzmannian statistical mechanics is to interpret the measures as typicality measures. In measure-theoretic dynamical systems theory measures can similarly be interpreted as typicality measures. However, a…
Randomness is one of the important key concepts of statistics. In epidemiology or medical science, we investigate our hypotheses and interpret results through this statistical randomness. We hypothesized by imposing some conditions to this…
D. J. Benson conjectures that the Castelnuovo-Mumford regularity of a group cohomology ring is always zero. More generally he conjectures that the cohomology ring always has a system of parameters satisfying a property he calls very strong…
We prove that every connected strongly regular graph on sufficiently many vertices is Hamiltonian. We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a…
We provide a version of the celebrated theorem of Koml\'os in which, rather then random quantities, a sequence of finitely additive measures is considered. We obtain a form of the subsequence principle and some applications.
The social phenomenon of familiar strangers was identified by Stanley Milgram in 1972 with a small-scale experiment. However, there has been limited research focusing on uncovering the phenomenon at a societal scale and simultaneously…
This paper proposes an alternative language for expressing results of the algorithmic theory of randomness. The language is more precise in that it does not involve unspecified additive or multiplicative constants, making mathematical…