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).

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 .

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Lamb
Keyword(s):  
Standard Deviation ◽  
Design Variable ◽  
Score Function ◽  
Random Variable ◽  
Probability Of Failure ◽  
Design Sensitivity ◽  
Mean Value ◽  
Reliability Based Design ◽  
The Mean ◽  
Sensitivity Method

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

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 On the Accuracy of Error Propagation Calculations by Analytic Formulas Obtained for the Inverse Transformation

2019 ◽  
Vol 64 (3) ◽  
pp. 217
Author(s):  
V. I. Romanenko ◽  
N. V. Kornilovska
Keyword(s):  
Error Propagation ◽  
Random Variable ◽  
Mean Value ◽  
Inverse Transformation ◽  
First Order ◽  
Order Of Magnitude ◽  
The Mean ◽  
Normally Distributed

The accuracy of error propagation calculations is estimated for the transformation x → y = f(x) of the normally distributed random variable x. The estimation is based on the formulas for the error propagation obtained for the inverse transformation y → x of the normally distributed random variable y. In the general case, the calculation accuracy for the mean value and the variance of the random variable y is shown to be of the first order of magnitude in the variance of the random variable x.

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 .

scholarly journals Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

2013 ◽  
Vol 50 (4) ◽  
pp. 909-917
Author(s):  
M. Bondareva
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Step Function ◽  
Cumulative Distribution ◽  
Control Parameters ◽  
Poisson Random Variable ◽  
The Mean ◽  
Standard Deviations

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

scholarly journals A Characterization of the Esscher-Transformation

1982 ◽  
Vol 13 (1) ◽  
pp. 57-59 ◽  
Author(s):  
Erhard Kremer
Keyword(s):  
Distribution Function ◽  
Iterative Methods ◽  
Edgeworth Expansion ◽  
Mean Value ◽  
Risk Theory ◽  
Approximation Methods ◽  
Classical Approximation ◽  
Aggregate Claims ◽  
The Mean

One of the central problems in risk theory is the calculation of the distribution function F of aggregate claims of a portfolio. Whereas formerly mainly approximation methods could be used, nowadays the increased speed of the computers allows application of iterative methods of numerical mathematics (see Bertram (1981), Küpper (1971) and Strauss (1976)). Nevertheless some of the classical approximation methods are still of some interest, especially a method developed by Esscher (1932).The idea of this so called Esscher-approximation (see Esscher (1932), Grandell and Widaeus (1969) and Gerber (1980)) is rather simple:In order to calculate 1 –F(x) for large x one transforms F into a distribution function such that the mean value of is equal to x and applies the Edgeworth expansion to the density of The reason for applying the transformation is the fact that the Edgeworth expansion produces good results for x near the mean value, but poor results in the tail (compare also Daniels (1954)).

scholarly journals The Microphysical Properties of a Sea-Fog Event along the West Coast of the Yellow Sea in Spring

Atmosphere ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 413
Author(s):  
Shengkai Wang ◽  
Li Yi ◽  
Suping Zhang ◽  
Xiaomeng Shi ◽  
Xianyao Chen
Keyword(s):  
Distribution Function ◽  
Yellow Sea ◽  
Droplet Size Distribution ◽  
Liquid Water Content ◽  
Mean Value ◽  
The Yellow Sea ◽  
Sea Fog ◽  
Microphysical Properties ◽  
The Mean ◽  
Microphysical Parameterization

The microphysics and visibility of a sea-fog event were measured at the Qingdao Meteorological Station (QDMS) (120°19′ E, 36°04′ N) from 5 April to 8 April 2017. The two foggy periods with low visibility (<200 m) lasted 31 h together. The mean value of the average liquid water content (LWC) was 0.057 g m−3, and the mean value of the number concentration (NUM) was 64.4 cm−3. We found that although large droplets only constituted a small portion of the total number of the concentration; they contributed the majority of the LWC and therefore determined ~76% of total extinction of the visibility. The observed droplet-size distribution (DSD) exhibited a new bimodal Gaussian (G-exponential) distribution function, rather than the well-accepted Gamma distribution. This work suggests a new distribution function to describe fog DSD, which may help to improve the microphysical parameterization for the Yellow Sea fog numerical forecasting.

scholarly journals Discovery of an extended, halo-like stellar population around the Large Magellanic Cloud

2008 ◽  
Vol 4 (S256) ◽  
pp. 51-56 ◽  
Author(s):  
Steven R. Majewski ◽  
David L. Nidever ◽  
Ricardo R. Muñoz ◽  
Richard J. Patterson ◽  
William E. Kunkel ◽  
...  
Keyword(s):  
Large Scale ◽  
Stellar Population ◽  
Large Magellanic Cloud ◽  
Mean Value ◽  
Magellanic Cloud ◽  
Stellar Surface ◽  
Two Dimensional ◽  
Radial Limit ◽  
Dimensional Distribution ◽  
Red Giant

AbstractWe describe an ongoing, large-scale, photometric and spectroscopic survey of the Large Magellanic Cloud (LMC) periphery. This survey uses WashingtonM,T2+DDO51 photometry to identify distant LMC red giant branch (RGB) star candidates; multi-object spectroscopy is used to confirm the stellar surface gravities of these RGB stars and their association with the LMC (e.g., through radial velocities). The survey now encompasses hundreds of fields ranging from the LMC center with full azimuthal coverage around the LMC and out to 23° from the LMC center. We have confirmed the existence of RGB stars with (the unusual) Magellanic velocities out to the radial limit of this survey coverage. From data in a subsample of these fields, we show that this extended population of stars makes up a diffuse structure enveloping the LMC with a two-dimensional distribution resembling a classical halo with a shallow de Vaucouleurs profile and a broad metallicity spread around a typical mean value of [Fe/H] ~ −1.0.

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