The rate of convergence of extremes of stationary normal sequences

1983 ◽  
Vol 15 (01) ◽  
pp. 54-80 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Joint Distribution ◽  
Point Processes ◽  
Rates Of Convergence ◽  
Joint Distribution Function ◽  
Maximal Correlation ◽  
Standard Normal ◽  
Image Position ◽  
The Right ◽  
Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

The rate of convergence of extremes of stationary normal sequences

1983 ◽  
Vol 15 (1) ◽  
pp. 54-80 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Joint Distribution ◽  
Point Processes ◽  
Rates Of Convergence ◽  
Joint Distribution Function ◽  
Maximal Correlation ◽  
Standard Normal ◽  
Image Position ◽  
The Right ◽  
Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r1, r2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

Almost-conservative second-order differential equations

1975 ◽  
Vol 77 (1) ◽  
pp. 159-169 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Differential Equations ◽  
Rate Of Convergence ◽  
Second Order ◽  
Good Function ◽  
Second Order Differential Equations ◽  
Right Hand ◽  
Image Position ◽  
The Right

During the last thirty years an immense amount of research has been done on differential equations of the formwhere ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the formwhere the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assumeand for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

On an inequality for the normal distribution arising in bioequivalence studies

1999 ◽  
Vol 36 (1) ◽  
pp. 279-286
Author(s):  
Yi-Ching Yao ◽  
Hari Iyer
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Normal Random Variable ◽  
Individual Bioequivalence ◽  
Left Hand ◽  
Right Hand ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Image Position ◽  
The Right

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

On an inequality for the normal distribution arising in bioequivalence studies

1999 ◽  
Vol 36 (01) ◽  
pp. 279-286 ◽  
Author(s):  
Yi-Ching Yao ◽  
Hari Iyer
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Normal Random Variable ◽  
Individual Bioequivalence ◽  
Left Hand ◽  
Right Hand ◽  
Standard Normal Random Variable ◽  
Standard Normal ◽  
Image Position ◽  
The Right

For (μ,σ2) ≠ (0,1), and 0 &lt; z &lt; ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 &lt; z &lt; ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

Residual Gas Reactions in the Electron Microscope: I. Qualitative Observations on the Water Gas Reaction

1968 ◽  
Vol 26 ◽  
pp. 292-293
Author(s):  
Richard E. Hartman ◽  
Roberta S. Hartman ◽  
Peter L. Ramos
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Gas Phase ◽  
Absolute Temperature ◽  
Residual Gas ◽  
Gas Reactions ◽  
Image Position ◽  
The Right ◽  
Water Gas ◽  
Thinning Rate

The action of water and the electron beam on organic specimens in the electron microscope results in the removal of oxidizable material (primarily hydrogen and carbon) by reactions similar to the water gas reaction .which has the form:The energy required to force the reaction to the right is supplied by the interaction of the electron beam with the specimen.The mass of water striking the specimen is given by:where u = gH2O/cm2 sec, PH2O = partial pressure of water in Torr, & T = absolute temperature of the gas phase. If it is assumed that mass is removed from the specimen by a reaction approximated by (1) and that the specimen is uniformly thinned by the reaction, then the thinning rate in A/ min iswhere x = thickness of the specimen in A, t = time in minutes, & E = efficiency (the fraction of the water striking the specimen which reacts with it).

scholarly journals Image Restoration by Learning Morphological Opening-Closing Network

2020 ◽  
Vol 4 (1) ◽  
pp. 87-107
Author(s):  
Ranjan Mondal ◽  
Moni Shankar Dey ◽  
Bhabatosh Chanda
Keyword(s):  
Neural Network ◽  
Image Restoration ◽  
State Of The Art ◽  
Source Code ◽  
Back Propagation ◽  
Image Features ◽  
Main Difficulty ◽  
The Right ◽  
Right Order ◽  
Morphological Opening

AbstractMathematical morphology is a powerful tool for image processing tasks. The main difficulty in designing mathematical morphological algorithm is deciding the order of operators/filters and the corresponding structuring elements (SEs). In this work, we develop morphological network composed of alternate sequences of dilation and erosion layers, which depending on learned SEs, may form opening or closing layers. These layers in the right order along with linear combination (of their outputs) are useful in extracting image features and processing them. Structuring elements in the network are learned by back-propagation method guided by minimization of the loss function. Efficacy of the proposed network is established by applying it to two interesting image restoration problems, namely de-raining and de-hazing. Results are comparable to that of many state-of-the-art algorithms for most of the images. It is also worth mentioning that the number of network parameters to handle is much less than that of popular convolutional neural network for similar tasks. The source code can be found here https://github.com/ranjanZ/Mophological-Opening-Closing-Net

scholarly journals Determination of the order of fractional derivative for subdiffusion equations

2020 ◽  
Vol 23 (6) ◽  
pp. 1647-1662
Author(s):  
Ravshan Ashurov ◽  
Sabir Umarov
Keyword(s):  
Fractional Derivative ◽  
Fixed Time ◽  
Time Instant ◽  
Observation Data ◽  
Additional Information ◽  
Fractional Modeling ◽  
Time Fractional Derivative ◽  
The Right ◽  
Right Order

Abstract The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring location, as “the observation data”, identifies uniquely the order of the fractional derivative.

scholarly journals The thermochemistry of carbon: valence states, heats of sublimation and energies of linkage

1946 ◽  
Vol 187 (1010) ◽  
pp. 337-357 ◽  
Keyword(s):  
Ground State ◽  
Electronic Configuration ◽  
Dissociation Energies ◽  
True Value ◽  
Valence States ◽  
Chemical Combination ◽  
The Right ◽  
Right Order ◽  
Unambiguous Assignment ◽  
Apparent Conflict

The controversy which exists at the present time between the figures 125 and 170 kcal./g.- atom for the latent heat of sublimation of carbon into monatomic vapour in the ground state originates largely from the neglect to take into consideration the energy required to raise the carbon atoms from the ground ( 3 P ) state to the lowest tetravalent ( 5 S ) electronic configuration corresponding to that in which it is normally found in chemical combination. Consideration of the energies of removal of a hydrogen atom from the methane and ethane molecules and of the energies of reorganization of the resulting radicals leads to the figure 190 ± about 10 kcal. for L 2 , the heat of sublimation into free atoms in the 5 S state. This in turn leads to a satisfactory and unambiguous assignment of values to bond energies (as distinct from dissociation energies) which can now be expressed with an uncertainty of not more than a few kcal. In the light of the valency distinction there remains no sound evidence to maintain the higher value put forward for L 1 and 125 kcal. is unquestionably of the right order. There are strong indications that an earlier estimate of 100 kcal. for the energy level of the 5 S state above the 3 P (ground) state is about 50 % in excess of the true value. The necessity for establishing this branch of thermochemistry on a sound theoretical and experimental footing has long been a very obvious need. The scheme here suggested reconciles points hitherto in apparent conflict, and brings virtually all established experimental knowledge into alignment.

The focusing of radiation by a random surface when the source is at a finite distance

1960 ◽  
Vol 56 (1) ◽  
pp. 27-40 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Mean Curvature ◽  
Joint Distribution ◽  
Total Curvature ◽  
Statistical Distribution ◽  
Constant Parameter ◽  
Random Surface ◽  
Second Derivatives ◽  
Finite Distance ◽  
The Mean ◽  
Image Position

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

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