scholarly journals Jacob’s Ladder: Prime Numbers in 2D

2020 ◽  
Vol 25 (1) ◽  
pp. 5
Author(s):  
Alberto Fraile ◽  
Roberto Martínez ◽  
Daniel Fernández
Keyword(s):  
Computer Simulations ◽  
Exponential Decay ◽  
Prime Numbers ◽  
Two Dimensions ◽  
Gap Size ◽  
Property A ◽  
Simple Representation ◽  
Jacob's Ladder

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 variety of problems. In this work, we present a simple representation of prime numbers in two dimensions that allows us to formulate a number of conjectures that may lead to important avenues in the field of research on prime numbers. In particular, although the zeroes in our representation grow in a somewhat erratic, hardly predictable way, the gaps between them present a remarkable and intriguing property: a clear exponential decay in the frequency of gaps vs. gap size. The smaller the gaps, the more frequently they appear. Additionally, the sequence of zeroes, despite being non-consecutive numbers, contains a number of primes approximately equal to n / log n , n being the number of terms in the sequence.

scholarly journals Mermin–Wagner fluctuations in 2D amorphous solids

2017 ◽  
Vol 114 (8) ◽  
pp. 1856-1861 ◽  
Author(s):  
Bernd Illing ◽  
Sebastian Fritschi ◽  
Herbert Kaiser ◽  
Christian L. Klix ◽  
Georg Maret ◽  
...  
Keyword(s):  
Computer Simulations ◽  
System Size ◽  
Amorphous Solids ◽  
Two Dimensions ◽  
Density Fluctuations ◽  
Long Wavelength ◽  
Broken Symmetries ◽  
Lindemann Criterion ◽  
The Mean ◽  
2D Crystals

In a recent commentary, J. M. Kosterlitz described how D. Thouless and he got motivated to investigate melting and suprafluidity in two dimensions [Kosterlitz JM (2016)J Phys Condens Matter28:481001]. It was due to the lack of broken translational symmetry in two dimensions—doubting the existence of 2D crystals—and the first computer simulations foretelling 2D crystals (at least in tiny systems). The lack of broken symmetries proposed by D. Mermin and H. Wagner is caused by long wavelength density fluctuations. Those fluctuations do not only have structural impact, but additionally a dynamical one: They cause the Lindemann criterion to fail in 2D in the sense that the mean squared displacement of atoms is not limited. Comparing experimental data from 3D and 2D amorphous solids with 2D crystals, we disentangle Mermin–Wagner fluctuations from glassy structural relaxations. Furthermore, we demonstrate with computer simulations the logarithmic increase of displacements with system size: Periodicity is not a requirement for Mermin–Wagner fluctuations, which conserve the homogeneity of space on long scales.

NULL RECURRENT UNIT ROOT PROCESSES

2011 ◽  
Vol 28 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Terje Myklebust ◽  
Hans Arnfinn Karlsen ◽  
Dag Tjøstheim
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Unit Circle ◽  
Unit Root ◽  
Markov Models ◽  
Two Dimensions ◽  
Property A ◽  
Starting Point ◽  
Recurrent Markov Chain ◽  
Null Recurrence

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and β-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is β-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made β-null recurrent for any β ∈ (0, 1) and can also be made null recurrent without being β-null recurrent.

Aggregation kinetics in two dimensions: Real experiments and computer simulations

2001 ◽  
Vol 114 (1) ◽  
pp. 520 ◽  
Author(s):  
Attila Vincze ◽  
Attila Agod ◽  
János Kertész ◽  
Miklós Zrı́nyi ◽  
Zoltán Hórvölgyi
Keyword(s):  
Computer Simulations ◽  
Two Dimensions ◽  
Aggregation Kinetics

scholarly journals Symmetric solutions for a 2D critical Dirac equation

2021 ◽  
pp. 2150019
Author(s):  
William Borrelli
Keyword(s):  
Dirac Equation ◽  
Variational Methods ◽  
Exponential Decay ◽  
Two Dimensions ◽  
Effective Model ◽  
Sobolev Embedding ◽  
Honeycomb Structures

In this paper, we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we also prove smoothness and exponential decay at infinity.

On the Validation of the Monte Carlo Technique in Simulation of Grain Growth in Small, Two-Dimensional Systems

2007 ◽  
Vol 558-559 ◽  
pp. 1087-1092
Author(s):  
Ola Hunderi ◽  
Knut Marthinsen ◽  
Nils Ryum
Keyword(s):  
Monte Carlo ◽  
Grain Boundary ◽  
Grain Growth ◽  
Computer Simulations ◽  
Two Dimensions ◽  
Theoretical Treatment ◽  
Two Dimensional ◽  
Simulation Techniques ◽  
Boundary Curvature ◽  
Kinetics Of

The kinetics of grain growth in real systems is influenced by several unknown factors, making a theoretical treatment very difficult. Idealized grain growth, assuming all grain boundaries to have the same energy and mobility (mobility M = k/ρ, where k is a constant and ρ is grain boundary curvature) can be treated theoretically, but the results obtained can only be compared to numerical grain growth simulations, as ideal grain growth scarcely exists in nature. The validity of the simulation techniques thus becomes of great importance. In the present investigation computer simulations of grain growth in two dimensions using Monte Carlo simulations and the grain boundary tracking technique have been investigated and compared in small grain systems, making it possible to follow the evolution of each grain in the system.

Computer simulations of auxetic foams in two dimensions

2013 ◽  
Vol 22 (8) ◽  
pp. 084009 ◽  
Author(s):  
A A Pozniak ◽  
J Smardzewski ◽  
K W Wojciechowski
Keyword(s):  
Computer Simulations ◽  
Two Dimensions

Self-assembly of molecular tripods in two dimensions: structure and thermodynamics from computer simulations

RSC Advances ◽  
2013 ◽  
Vol 3 (47) ◽  
pp. 25159 ◽  
Author(s):  
Paweł Szabelski ◽  
Wojciech Rżysko ◽  
Tomasz Pańczyk ◽  
Elke Ghijsens ◽  
Kazukuni Tahara ◽  
...  
Keyword(s):  
Computer Simulations ◽  
Self Assembly ◽  
Two Dimensions
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