About approximation of 3-D random fields and statistical simulation

2003 ◽  
Vol 11 (3) ◽  
Author(s):  
Zoya O. Vyzhva
Keyword(s):  
Random Fields ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Statistical Simulation ◽  
Gaussian Random Fields ◽  
Statistical Characteristics ◽  
Mean Square ◽  
The Mean ◽  
Isotropic Random ◽  
Three Dimensional Space

The estimator of the mean-square approximation of 3-D homogeneous and isotropic random field is investigated. The problem of statistical simulation of realizations of random fields in threedimensional space is considered. The algorithm for the receiving of this realization has been formulated, which has been constructed on the base the mean-square approximation of random fields estimator. It has been constructed the statistical model for the Gaussian random fields in three-dimensional space, which has been given by its statistical characteristics.

scholarly journals The Thinness of the Ocean in S–Θ–p Space and the Implications for Mean Diapycnal Advection

2007 ◽  
Vol 37 (6) ◽  
pp. 1714-1732 ◽  
Author(s):  
Trevor J. McDougall ◽  
David R. Jackett
Keyword(s):  
Dimensional Space ◽  
World Ocean ◽  
Three Dimensional ◽  
Small Scale ◽  
Lateral Motion ◽  
Hydrographic Data ◽  
Mixing Processes ◽  
The World ◽  
The Mean ◽  
Three Dimensional Space

Abstract It is shown that the ocean’s hydrography occupies little volume in the three-dimensional space defined by salinity–temperature–pressure (S–Θ–p), and the implications of this observation for the mean vertical transport across density surfaces are discussed. Although ocean data have frequently been analyzed in the two-dimensional temperature–salinity (S–Θ) diagram where casts of hydrographic data are often locally tight in S–Θ space, the relatively empty nature of the World Ocean in the three-dimensional S–Θ–p space seems not to have received attention. The World Ocean’s data lie close to a single surface in this three-dimensional space, and it is shown that this explains the known smallness of the ambiguity in defining neutral surfaces. The ill-defined nature of neutral surfaces means that lateral motion along neutral trajectories leads to mean vertical advection through density surfaces, even in the absence of small-scale mixing processes. The situation in which the ocean’s hydrography occupies a large volume in S–Θ–p space is also considered, and it is suggested that the consequent vertical diapycnal advection would be sufficiently large that the ocean would not be steady.

scholarly journals Series representations and simulations of isotropic random fields in the Euclidean space

2021 ◽  
Vol 105 (0) ◽  
pp. 93-111
Author(s):  
Z. Ma ◽  
C. Ma
Keyword(s):  
Euclidean Space ◽  
Random Fields ◽  
Spectral Representation ◽  
Series Representation ◽  
Gaussian Random Fields ◽  
Mean Square ◽  
Ultraspherical Polynomial ◽  
Series Representations ◽  
Non Gaussian ◽  
Isotropic Random

This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.

ON ANOMALOUS DIFFUSION OF GROWING CLUSTERS

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 875-880 ◽  
Author(s):  
ADAM GADOMSKI ◽  
JERZY LUCZKA
Keyword(s):  
Anomalous Diffusion ◽  
Materials Science ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Mean Square Displacement ◽  
Polymer Science ◽  
The Mean ◽  
Continuous Random Walk ◽  
Mass Dependent ◽  
Three Dimensional Space

A continuous random walk of a growing cluster of dimension df is studied. It is assumed that its behavior is represented by a standard diffusion equation. A diffusion coefficient is assumed to be a cluster mass dependent power function which in the dilute solution regime takes a form first suggested by Kirkwood. The mass of the cluster increases in time according to another power law. It results in anomalous diffusion of the cluster in a three dimensional space which, in a linear case, is manifested by either power or logarithmic law of the mean square displacement. Some interesting examples and realizations of that process in polymer science, colloid physicochemistry or materials science are pointed out.

scholarly journals Statistical simulation of seismic noise in a multidimensional area in determining frequency characteristics of geological media

2014 ◽  
pp. 62-68
Author(s):  
Z. Vyzhva ◽  
K. Fedorenko ◽  
A. Vyzhva
Keyword(s):  
Random Fields ◽  
Spectral Decomposition ◽  
Seismic Noise ◽  
Statistical Simulation ◽  
Frequency Characteristics ◽  
Partial Sums ◽  
Multidimensional Space ◽  
Mean Square ◽  
Spatial Variables ◽  
The Mean

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

Distribution of random motion at renewal instants in three-dimensional space

2020 ◽  
Vol 17 (4) ◽  
pp. 563-573
Author(s):  
Anatoliy Pogorui ◽  
Ramón Rodrĭguez-Dagnino
Keyword(s):  
Brownian Motion ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Ideal Gas ◽  
Random Motion ◽  
Particle Position ◽  
Brownian Particles ◽  
And Mathematics ◽  
Isotropic Random ◽  
Three Dimensional Space

In physics, chemistry, and mathematics, the process of Brownian motion is often identified with the Wiener process that has infinitesimal increments. Recently, many models of Brownian motion with finite velocity have been intensively studied. We consider one of such models, namely, a generalization of the Goldstein--Kac process to the three-dimensional case with the Erlang-2 and Maxwell--Boltzmann distributions of velocities alternations. Despite the importance of having a three-dimensional isotropic random model for the motion of Brownian particles, numerous research efforts did not lead to an expression for the probability of the distribution of the particle position, the motion of which is described by the three-dimensional telegraph process. The case where a particle carries out its movement along the directions determined by the vertices of a regular $n+1$-hedron in the $n$-dimensional space was studied in \cite{Samoilenko}, and closed-form results for the distribution of the particle position were obtained. Here, we obtain expressions for the distribution function of the norm of the vector that defines particle's position at renewal instants in semi-Markov cases of the Erlang-2 and Maxwell--Boltzmann distributions and study its properties. By knowing this distribution, we can determine the distribution of particle positions, since the motion of a particle is isotropic, i.e., the direction of its movement is uniformly distributed on the unit sphere in ${\mathbb R}^3$. Our results may be useful in studying the properties of an ideal gas.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

A Peptoid with Extended Shape in Water

2019 ◽  
Author(s):  
Jumpei Morimoto ◽  
Yasuhiro Fukuda ◽  
Takumu Watanabe ◽  
Daisuke Kuroda ◽  
Kouhei Tsumoto ◽  
...  
Keyword(s):  
Spectroscopic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Biomolecular Recognition ◽  
Biological Application ◽  
New Class ◽  
Proteolytic Resistance ◽  
Pharmacokinetic Properties ◽  
Optically Pure ◽  
Three Dimensional Space

<div> <div> <div> <p>“Peptoids” was proposed, over decades ago, as a term describing analogs of peptides that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo-(N-substituted glycines) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible and is difficult to achieve a defined shape in water. This conformational flexibility is severely limiting biological application of oligo-NSG. Here, we propose oligo-(N-substituted alanines) (oligo-NSA) as a new peptoid that forms a defined shape in water. A synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. The new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as an ideal scaffold for displaying functional groups in well-defined three-dimensional space, which leads to effective biomolecular recognition. </p> </div> </div> </div>

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