Related papers: Jumping champions and prime gaps using information…

We study the behavior of a nonlinear semiclassical system using Shannon entropy and two approaches to statistical complexity. These systems involve the interaction between classical variables (representing the environment) and quantum ones.…

We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial…

Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain \cal{M}}_{nn} is known to converge in time \Theta(n^3 \log n) in the uniform case and time \Theta(n^2) in the constant…

Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused…

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 +…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. For $z\geqslant 0$, let $G(n;z)$ denote the number of those indexes $j$ such that $p_{j+1}(n)>p_j(n)^{\exp z}$. We show uniform…

Shannon Information theory has achieved great success in not only communication technology where it was originally developed for but also many other science and engineering fields such as machine learning and artificial intelligence.…

We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.

We study the Shannon entropy of the cluster size distribution in classical as well as explosive percolation, in order to estimate the uncertainty in the sizes of randomly chosen clusters. At the critical point the cluster size distribution…

Information flow, opinion, and epidemics spread over structured networks. When using individual node centrality indicators to predict which nodes will be among the top influencers or spreaders in a large network, no single centrality has…

Feature selection is one of the most fundamental problems in machine learning. An extensive body of work on information-theoretic feature selection exists which is based on maximizing mutual information between subsets of features and class…

To overcome some limits of classical neuronal models, we propose a Markovian generalization of the classical model based on Jacobi processes by introducing downwards jumps to describe the activity of a single neuron. The statistical…

Information propagation characterizes how input correlations evolve across layers in deep neural networks. This framework has been well studied using mean-field theory, which assumes infinitely wide networks. However, these assumptions…

All communication networks comprise of transmission systems and switching systems, even though they are usually treated as two separate issues. Communication channels are generally disturbed by noise from various sources. In circuit…

Emergence is a fascinating property of large language models and neural networks more broadly: as models scale and train for longer, they sometimes develop new abilities in sudden ways. Despite initial studies, we still lack a comprehensive…

Information theory is gaining popularity as a tool to characterize performance of biological systems. However, information is commonly quantified without reference to whether or how a system could extract and use it; as a result,…

We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes…

In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). Determining the number of spikes is a fundamental problem which appears in many scientific…

There is growing evidence regarding the importance of spike timing in neural information processing, with even a small number of spikes carrying information, but computational models lag significantly behind those for rate coding.…

One of the themes of this paper is recent results on large gaps between primes. The first of these results has been achieved in the paper [12] by Ford, Green, Konyagin and Tao. It was later improved in the joint paper [13] of these four…