The $ n $-th Fubini number counts the number of ordered partitions of a set with $ n $ elements and is the number of possible ways to write the Fubini formula for a summation of integration of order $ n $. Further, Fubini polynomials are natural extensions of the Fubini numbers. Recently, the generalized Fubini polynomials were introduced by Sebaoui-Laissaoui-Guettai-Rahmani, as one of the variants of the Fubini polynomials. The aim of this paper is to study the degenerate generalized Fubini polynomials, which are a degenerate version of those polynomials, and to find an application of them to probability theory in connection with geometric random variable.

On degenerate generalized Fubini polynomials

The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences.

On the divisors of natural and happy numbers: a study based on entropy and graphs

This research proposes a new approach to the Riemann Hypothesis, focusing on the interplay between prime gaps and the non-trivial zeros of the Riemann Zeta function. Utilizing various statistical models and experimental analysis techniques, three important insights are uncovered: 1) Granger causality tests reveal a predictive relationship in which past non-trivial zeros may predict future prime gaps; 2) Complex, nonlinear interactions between prime gaps and non-trivial zeros are identified, challenging simple linear correlations; and 3) Causal network analysis reveals intricate feedback-loop relationships. These findings contribute to a better understanding of prime number distribution and the Zeta function, opening up novel possibilities for further mathematical research. The study aims to motivate mathematicians towards a proof or disproof of the Riemann Hypothesis.

The Riemann Hypothesis: A Fresh and Experimental Exploration

Let a - prime be a positive integer number with divisors. From this definition, the usual prime numbers correspond to the particular case . Here, the seemingly random sequence of gaps between consecutive - primes is numerically investigated. First, the variability of the gap sequences for is evaluated by calculating the informational entropy. Then, these sequences are mapped into graphs by employing two visibility algorithms. Computer simulations reveal that the degree distribution of most of these graphs follows a power law. Conjectures on how some topological features of these graphs depend on are proposed.

/pdf/a-numerical-study-on-the-regularity-of-d-primes-via-ppf036cncn.pdf

A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

A Newman-Watts graph is formed by including random links in a regular lattice. Here, the emergence of synchronization in coupled Newman-Watts graphs is studied. The whole neural network is considered as a toy model of mammalian visual pathways. It is composed by four coupled graphs, in which a coupled pair represents the lateral geniculate nucleus and the visual cortex of a cerebral hemisphere. The hemispheres communicate with each other through a coupling between the graphs representing the visual cortices. This coupling makes the role of the corpus callosum. The state transition of neurons, supposed to be the nodes of the graphs, occurs in discrete time and it follows a set of deterministic rules. From periodic stimuli coming from the retina, the neuronal activity of the whole network is numerically computed. The goal is to find out how the values of the parameters related to the network topology affect the synchronization among the four graphs.

/pdf/on-synchronizing-coupled-retinogeniculocortical-pathways-a-2rc69q9u0x.pdf

On Synchronizing Coupled Retinogeniculocortical Pathways: A Toy Model

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A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

On Synchronizing Coupled Retinogeniculocortical Pathways: A Toy Model

On the divisors of natural and happy numbers: a study based on entropy and graphs