Related papers: On a dynamical approach to some prime number seque…

Chaotic dynamical systems are often characterised by a positive Lyapunov exponent, which signifies an exponential rate of separation of nearby trajectories. However, in a wide range of so-called weakly chaotic systems, the separation of…

It is generally believed that the dynamics of simple fluids can be considered to be chaotic, at least to the extent that they can be modeled as classical systems of particles interacting with short range, repulsive forces. Here we give a…

We obtain the posterior distribution of a random process conditioned on observing the empirical frequencies of a finite sample path. We find under a rather broad assumption on the "dependence structure" of the process, {\em c.f.}…

The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the…

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless systems, with say $m$ particles in $N$ single particle states…

We investigate a model of high-dimensional dynamical variables with all-to-all interactions that are random and non-reciprocal. We characterize its phase diagram and show that the model can exhibit chaotic dynamics. We show that the…

We study point sets arising from cut-and-project constructions. An important class is weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern…

We review the occurrence of the patterns of the onset of chaos in low-dimensional nonlinear dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics associated…

Successive pairs of pseudo-random numbers generated by standard linear congruential transformations display ordered patterns of parallel lines. We study the ``ordered'' and ``chaotic'' distribution of such pairs by solving the eigenvalue…

Recurrent Neural Networks (RNNs) frequently exhibit complicated dynamics, and their sensitivity to the initialization process often renders them notoriously hard to train. Recent works have shed light on such phenomena analyzing when…

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences $x_{n+1}=F(x_n)$ generated by such maps display…

In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to $k$-dense and $k$-sparse sets, both commonly called $k$-defective sets. We focus on the…

We propose a robust and computationally efficient algorithm to generically construct first return maps of dynamical systems from time series without the need for embedding. Typically, a first return map is constructed using a heuristic…

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we…

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a…

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 show for several computational problems how classical greedy algorithms for special cases can be derived in a simple way from dynamic programs for the general case: interval scheduling (restricted to unit weights), knapsack (restricted…

A new method involving particle diagrams is introduced and developed into a rigorous framework for carrying out embedded random matrix calculations. Using particle diagrams and the attendant methodology including loop counting it becomes…

The detection of weak and rare effects in large amounts of data arises in a number of modern data analysis problems. Known results show that in this situation the potential of statistical inference is severely limited by the large-scale…