On the envelope of a Gaussian random field

1978 ◽  
Vol 15 (03) ◽  
pp. 502-513 ◽  
Author(s):  
R. J. Adler
Keyword(s):  
Random Field ◽  
Spectral Theory ◽  
Gaussian Random Field ◽  
Sample Path ◽  
Mean Value ◽  
Two Dimensional ◽  
Qualitative Information ◽  
The Mean

For homogeneous, two-dimensional random field ξ(t), t ∈ R 2 we develop the ‘half' spectral theory sufficient to rigorously define its envelope η (t). We then specialise to the case of ξ Gaussian, which implies η is Rayleigh, and consider the mean value of a certain characteristic of the sets {t:η(t) ≧ u} (u ≧ 0). From this we deduce some qualitative information about the sample path behaviour of the Rayleigh field η .

On the envelope of a Gaussian random field

1978 ◽  
Vol 15 (3) ◽  
pp. 502-513 ◽  
Author(s):  
R. J. Adler
Keyword(s):  
Random Field ◽  
Spectral Theory ◽  
Gaussian Random Field ◽  
Sample Path ◽  
Mean Value ◽  
Two Dimensional ◽  
Qualitative Information ◽  
The Mean

For homogeneous, two-dimensional random field ξ(t), t ∈ R2 we develop the ‘half' spectral theory sufficient to rigorously define its envelope η (t). We then specialise to the case of ξ Gaussian, which implies η is Rayleigh, and consider the mean value of a certain characteristic of the sets {t:η(t) ≧ u} (u ≧ 0). From this we deduce some qualitative information about the sample path behaviour of the Rayleigh field η .

scholarly journals Asymptotics for multilinear averages of multiplicative functions

2016 ◽  
Vol 161 (1) ◽  
pp. 87-101 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Unit Disc ◽  
Convergence Result ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
Multiplicative Functions ◽  
Structural Result ◽  
Convergence Results ◽  
The Mean

AbstractA celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

scholarly journals Three-Dimensional Stochastic Seepage Field Analysis of Multimedia Embankment

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xiaoming Zhao ◽  
Shiyu Shang ◽  
Yuanlin Yang ◽  
Mingming Hu
Keyword(s):  
Hydraulic Conductivity ◽  
Random Field ◽  
Three Dimensional ◽  
Mean Value ◽  
Field Analysis ◽  
Anisotropy Ratio ◽  
Seepage Field ◽  
The Monte Carlo Method ◽  
Total Head ◽  
The Mean

The soil hydraulic conductivity of an embankment has strong spatial variability due to the spatiotemporal variation, both natural and artificial. The strong randomness of the hydraulic conductivity can be expressed by the coefficient of variation (COV) and the fluctuation scale θ. Moreover, different coefficients of variation and fluctuation scales correspond to different random field structures. To study the characteristics of the three-dimensional stochastic seepage field in an embankment under different COVs and fluctuation scales, we generate a three-dimensional random field of the hydraulic conductivity of multimedia embankment based on the local average subdivision technique. In particular, a calculation method for a three-dimensional random seepage field based on the Monte Carlo method combined with a three-dimensional multimedia random field and a deterministic analysis is proposed. The results showed that after three thousand realizations and considering the randomness of the hydraulic conductivity, the position of the free surface of each section in the embankment differed. The mean value of the total head decreased when the COV increased. Furthermore, when the COV was small, the change in the total head with anisotropy ratio was not evident, while the COV was large. The mean value of the total head increased with the anisotropy ratio. When the anisotropy ratio increased, the mean value of the standard deviation of the total head increased first and then decreased.

Pressure and Traction Rippling in Elastohydrodynamic Contact of Rough Surfaces

1974 ◽  
Vol 96 (3) ◽  
pp. 398-406 ◽  
Author(s):  
T. E. Tallian
Keyword(s):  
High Pressure ◽  
Rough Surfaces ◽  
Mean Value ◽  
Surface Plastic ◽  
Two Dimensional ◽  
Plateau Region ◽  
Pressure Ripple ◽  
Ripple Amplitude ◽  
Roll Velocity ◽  
The Mean

Local variations in asperity dimensions (rippling) of elastohydrodynamic (EHD) pressure are calculated using Christensen’s stochastic model of the hydrodynamics of heavily loaded two-dimensional contacts between rough surfaces. Pressure ripple amplitudes of the order of the maximum Hertz pressure, i.e., well in excess of 105 psi (69.107 N/m2) are predicted at the inlet perimeter of the EHD contact plateau and at the upstream slope of the exit constriction for heavily loaded contacts, if the plateau film thickness to rms roughness ratio is h/σ = 2. Pressure ripple amplitudes in excess of 104 psi (69.106 N/m2) are probable even for the thick film condition h/σ = 10. Sliding traction rippling is calculated for small slide/roll velocity ratios in the same type of contact, and ripple amplitudes in excess of the mean value of the traction are predicted in the high pressure EHD plateau region of the contact for h/σ = 3. The predicted traction ripple amplitude exceeds 30 percent of the mean traction, even for h/σ = 6. Rippling increases the average traction over that for smooth surfaces. Both the pressure and traction rippling may contribute to surface plastic flow and fatigue.

A VARIATIONAL FORMULA FOR SOME RANDOM FIELDS: AN ANALOGUE OF ITO'S FORMULA

1999 ◽  
Vol 02 (02) ◽  
pp. 305-313
Author(s):  
SI SI
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
White Noise ◽  
Random Fields ◽  
Gaussian Random Field ◽  
Nonlinear Function ◽  
Variational Formula ◽  
Canonical Representation ◽  
Two Dimensional ◽  
Dimensional Parameter

We shall first establish a canonical representation of a Gaussian random field X(C) indexed by a smooth contour C in terms of two-dimensional parameter white noise. Then, we take a nonlinear function F(X(C)) of the X(C) and obtain its variation when C deforms slightly. The variational formula is analogous to the Ito formula for a stochastic process X(t), but somewhat simpler.

Excursions above a fixed level by n-dimensional random fields

1976 ◽  
Vol 13 (2) ◽  
pp. 276-289 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
Random Fields ◽  
Mean Value ◽  
Regularity Conditions ◽  
Gaussian Field ◽  
Differential Topology ◽  
One Dimensional ◽  
Excursion Set ◽  
The Mean

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

The crystal and molecular structure of quaterrylene

1960 ◽  
Vol 257 (1291) ◽  
pp. 477-490 ◽  
Keyword(s):  
Molecular Structure ◽  
Crystal Structure ◽  
Crystal And Molecular Structure ◽  
Current Theory ◽  
Mean Value ◽  
Unit Cell ◽  
Two Dimensional ◽  
Monoclinic System ◽  
Space Group ◽  
The Mean

Quaterrylene (III) is isotypic with perylene (I); it crystallizes in the monoclinic system with a = 11·25, b = 10·66, c = 19·31 Å, β = 100·6°, and with four molecules per unit cell and the space group P 2 1 / a . The crystal structure has been determined by two-dimensional methods. In the b -axial projection a majority of the carbon atoms are resolved, and, since the molecule lies with its greatest length almost exactly perpendicular to this axis, the lengths of the peri -bonds connecting the naphthalenic residues have been determined with moderate accuracy. The mean value is assessed at 1·53 ± 0·01 Å, which is significantly larger than current theory predicts. Possible reasons for this difference are discussed.

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

Excursions above a fixed level by n-dimensional random fields

1976 ◽  
Vol 13 (02) ◽  
pp. 276-289 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
Random Fields ◽  
Mean Value ◽  
Regularity Conditions ◽  
Gaussian Field ◽  
Differential Topology ◽  
One Dimensional ◽  
Excursion Set ◽  
The Mean

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

scholarly journals Sur les processus arithmétiques liés aux diviseurs

2016 ◽  
Vol 48 (A) ◽  
pp. 63-76 ◽  
Author(s):  
R. de la Bretèche ◽  
G. Tenenbaum
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Mean Value ◽  
Two Dimensional ◽  
Limit Laws ◽  
Uniform Probability ◽  
Dimensional Distribution ◽  
The Mean ◽  
Generalized Moments

AbstractFor natural integer n, let Dn denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(Dn≤nt) (0≤t≤1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the generalized moments of arbitrary orders for the limit laws. We also evaluate the mean value of the two-dimensional distribution function ℙ(Dn≤nu, D{n/Dn}≤nv).

Sign in / Sign up

Export Citation Format

Share Document