相关论文: Jumping champions and prime gaps using information…
The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A.…
An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally several gaps are equally common. Hence, there can be more than one jumping champion for the same…
Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…
The difference between two consecutive prime numbers is called the distance between the primes. We study the statistical properties of the distances and their increments (the difference between two consecutive distances) for a sequence…
The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…
We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the…
A Prime Difference Champion (PDC) for primes up to $x$ is defined to be any element of the set of one or more differences that occur most frequently among all positive differences between primes $\le x$. Assuming an appropriate form of the…
Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the…
We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\ldots,$ where $q$ and $r$ are coprime integers, $q>r\ge1$. The growth trend and distribution of the first-occurrence gap…
We consider a generalization of an important class of high-dimensional inference problems, namely spiked symmetric matrix models, often used as probabilistic models for principal component analysis. Such paradigmatic models have recently…
In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible…
We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann…
We explore a few common models on how correlations affect information. The main model considered is the Shannon mutual information $I(S:R_1,\cdots, R_i)$ over distributions with marginals $P_{S,R_i}$ fixed for each $i$, with the analogy in…
In recent years, there has been an upswing of interest in estimating information from data emerging in a lot of areas beyond communications. This paper aims at estimating the information between two random phenomena by using consolidated…
The connection between active perception and the limits of performance provide a path to understanding naturalistic behavior. We can take a comparative cognitive modeling perspective to understand the limits of this performance and the…
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…
For any real $x$ and any integer $k\ge1$, we say that a set $\mathcal{D}_{k}$ of $k$ distinct integers is a $k$-tuple jumping champion if it is the most common differences that occurs among $k+1$ consecutive primes less than or equal to…
The data for many classification problems, such as pattern and speech recognition, follow mixture distributions. To quantify the optimum performance for classification tasks, the Shannon mutual information is a natural information-theoretic…
In nervous system information is conveyed by sequence of action potentials (spikes-trains). As MacKay and McCulloch proposed, spike-trains can be represented as bits sequences coming from Information Sources. Previously, we studied…
Statistical Inference is the process of determining a probability distribution over the space of parameters of a model given a data set. As more data becomes available this probability distribution becomes updated via the application of…