Asymptotic Densities in Statistical Ensembles

1964 ◽  
Vol 136 (2A) ◽  
pp. A355-A361 ◽  
Author(s):  
Harvey S. Leff
Keyword(s):  
Statistical Ensembles ◽  
Asymptotic Densities

Density of summable subsequences of a sequence and its applications

2020 ◽  
Vol 70 (3) ◽  
pp. 657-666
Author(s):  
Bingzhe Hou ◽  
Yue Xin ◽  
Aihua Zhang
Keyword(s):  
Prime Numbers ◽  
Linear Operators ◽  
Upper Density ◽  
Asymptotic Densities

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

scholarly journals NMR-Based Study of the Pore Types’ Contribution to the Elastic Response of the Reservoir Rock

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1513 ◽  
Author(s):  
Naser Golsanami ◽  
Xuepeng Zhang ◽  
Weichao Yan ◽  
Linjun Yu ◽  
Huaimin Dong ◽  
...  
Keyword(s):  
Elastic Wave ◽  
Wave Velocity ◽  
Transverse Relaxation Time ◽  
Seismic Attributes ◽  
Elastic Wave Velocity ◽  
Reservoir Rock ◽  
Elastic Response ◽  
Hydrocarbon Reservoir ◽  
Statistical Ensembles ◽  
Pore Types

Seismic data and nuclear magnetic resonance (NMR) data are two of the highly trustable kinds of information in hydrocarbon reservoir engineering. Reservoir fluids influence the elastic wave velocity and also determine the NMR response of the reservoir. The current study investigates different pore types, i.e., micro, meso, and macropores’ contribution to the elastic wave velocity using the laboratory NMR and elastic experiments on coal core samples under different fluid saturations. Once a meaningful relationship was observed in the lab, the idea was applied in the field scale and the NMR transverse relaxation time (T2) curves were synthesized artificially. This task was done by dividing the area under the T2 curve into eight porosity bins and estimating each bin’s value from the seismic attributes using neural networks (NN). Moreover, the functionality of two statistical ensembles, i.e., Bag and LSBoost, was investigated as an alternative tool to conventional estimation techniques of the petrophysical characteristics; and the results were compared with those from a deep learning network. Herein, NMR permeability was used as the estimation target and porosity was used as a benchmark to assess the reliability of the models. The final results indicated that by using the incremental porosity under the T2 curve, this curve could be synthesized using the seismic attributes. The results also proved the functionality of the selected statistical ensembles as reliable tools in the petrophysical characterization of the hydrocarbon reservoirs.

scholarly journals FLUCTUATIONS, CORRELATIONS AND FINITE VOLUME EFFECTS IN HEAVY ION COLLISION

2007 ◽  
Vol 16 (07n08) ◽  
pp. 2229-2234
Author(s):  
LUDWIK TURKO
Keyword(s):  
Finite Volume ◽  
Thermodynamic Limit ◽  
Heavy Ion ◽  
Higher Moments ◽  
Heavy Ion Collision ◽  
Canonical Distribution ◽  
Statistical Ensembles ◽  
Ion Collision ◽  
Rigorous Procedure ◽  
Grand Canonical

Finite volume corrections to higher moments are important observable quantities. They make possible to differentiate between different statistical ensembles even in the thermodynamic limit. It is shown that this property is a universal one. The classical grand canonical distribution is compared to the canonical distribution in the rigorous procedure of approaching the thermodynamic limit.

A Two-Dimensional Cellular Automaton Crystal with Irrational Density

2003 ◽  
Author(s):  
David Griffeath ◽  
Dean Hickerson
Keyword(s):  
Cellular Automaton ◽  
Asymptotic Density ◽  
Coarse Grain ◽  
Constant Density ◽  
Mathematical Framework ◽  
Two Dimensional ◽  
Empty Cell ◽  
Asymptotic Shape ◽  
Conway’S Game Of Life ◽  
Asymptotic Densities

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

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