scholarly journals Degree estimates of one class cut-offs for improper integrals

2021 ◽  
Vol 2131 (3) ◽  
pp. 032018
Author(s):  
A Pozhidaev ◽  
O Khaustova
Keyword(s):  
Lower Bound ◽  
Gaussian Distribution ◽  
Distribution Density ◽  
Closed Interval ◽  
Concave Function ◽  
Theoretical Studies ◽  
Main Difficulty ◽  
Integrable Functions ◽  
Improper Integrals

Abstract The paper considers a normalized non-integral integral of the first kind with a variable lower bound. In this case the integrand is a generalization of the standard Gaussian distribution density. Such integrals are often called cutoffs or incomplete functions. The purpose of this paper is to obtain power inequalities for this kind of integrals. The necessity of obtaining this type of estimations is due to the fact that incomplete functions have become widespread in applications and theoretical studies. The peculiarity of the results established in the article consists in the fact that arbitrary degrees of a given integral for any value of an argument are evaluated from above not by means, of the value of integrable functions at a certain point, but by the value of the integral in question at some point proportional to this argument. The coefficient of proportionality, a parameter, can take any value from some closed interval. The main difficulty in obtaining these inequalities is that the integrand is a logarithmically concave function, that is, its logarithm is a concave function. The paper also proves that both limits of the closed interval for the parameter cannot be extended. This shows that the obtained estimates are unimprovable.

The Gaussian distribution revisited

1996 ◽  
Vol 28 (2) ◽  
pp. 500-524 ◽  
Author(s):  
Carlos E. Puente ◽  
Miguel M. López ◽  
Jorge E. Pinzón ◽  
José M. Angulo
Keyword(s):  
Fractal Dimension ◽  
Probability Measure ◽  
Gaussian Distribution ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Cumulative Distribution ◽  
Probability Measures ◽  
New Construction

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

The Gaussian distribution revisited

1996 ◽  
Vol 28 (02) ◽  
pp. 500-524 ◽  
Author(s):  
Carlos E. Puente ◽  
Miguel M. López ◽  
Jorge E. Pinzón ◽  
José M. Angulo
Keyword(s):  
Fractal Dimension ◽  
Probability Measure ◽  
Gaussian Distribution ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Cumulative Distribution ◽  
Probability Measures ◽  
New Construction

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let f Θ,D :I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = f Θ,D (X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

CONSTRUCTIVE ESTIMATION OF APPROXIMATION FOR TRIGONOMETRIC NEURAL NETWORKS

2012 ◽  
Vol 10 (03) ◽  
pp. 1250021
Author(s):  
JIANJUN WANG ◽  
WEIHUA XU ◽  
BIN ZOU
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Lower Bound ◽  
Upper Bound ◽  
Intrinsic Property ◽  
Order Of Approximation ◽  
Integrable Functions ◽  
Hidden Layer ◽  
Hidden Neurons ◽  
Bound Estimation

For the three-layer artificial neural networks with trigonometric weights coefficients, the upper bound and lower bound of approximating 2π-periodic pth-order Lebesgue integrable functions [Formula: see text] are obtained in this paper. Theorems we obtained provide explicit equational representations of these approximating networks, the specification for their numbers of hidden-layer units, the lower bound estimation of approximation, and the essential order of approximation. The obtained results not only characterize the intrinsic property of approximation of neural networks, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of neural networks.

scholarly journals A nonlinear Lazarev–Lieb theorem: L2-orthogonality via motion planning

2020 ◽  
pp. 1-17
Author(s):  
Florian Frick ◽  
Matt Superdock
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Unit Circle ◽  
Smooth Function ◽  
Unit Interval ◽  
Inner Product ◽  
Lower Bounding ◽  
Integrable Functions ◽  
Equivariant Topology ◽  
Norm Bound

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to [Formula: see text] can be simultaneously annihilated in the [Formula: see text] inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain [Formula: see text]-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the [Formula: see text]-coindex of a space.

The Heat Equation on the Spaces of Positive Definite Matrices

1992 ◽  
Vol 44 (3) ◽  
pp. 624-651 ◽  
Author(s):  
P. Sawyer
Keyword(s):  
Lower Bound ◽  
Heat Equation ◽  
Heat Kernel ◽  
Symmetric Spaces ◽  
Positive Definite ◽  
Explicit Formulas ◽  
Positive Definite Matrices ◽  
Inverse Transform ◽  
Integrable Functions ◽  
Asymptotic Development

AbstractThe main topic of this paper is the study of the fundamental solution of the heat equation for the symmetric spaces of positive definite matrices, Pos(n,R).Our first step is to develop a “False Abel Inverse Transform” which transforms functions of compact support on an euclidean space into integrable functions on the symmetric space. The transform is shown to satisfy the relation is the usual Laplacian with a constant drift).Using this transform, we find explicit formulas for the heat kernel in the cases n = 2 and n = 3. These formulas allow us to give the asymptotic development for the heat kernel as t tends to infinity. Finally, we give an upper and lower bound of the same type for the heat kernel.

Structure images of Si, Ge and Mos2 crystals and some application to radiation damage studies

1978 ◽  
Vol 36 (1) ◽  
pp. 292-293 ◽  
Author(s):  
K. Izui ◽  
T. Nishida ◽  
S. Furuno ◽  
H. Otsu ◽  
S. Kuwabara
Keyword(s):  
Radiation Damage ◽  
Exposure Time ◽  
Gaussian Distribution ◽  
Intensity Distribution ◽  
Chromatic Aberration ◽  
Magnetic Lens

Recently we have observed the structure images of silicon in the (110), (111) and (100) projection respectively, and then examined the optimum defocus and thickness ranges for the formation of such images on the basis of calculations of image contrasts using the n-slice theory. The present paper reports the effects of a chromatic aberration and a slight misorientation on the images, and also presents some applications of structure images of Si, Ge and MoS2 to the radiation damage studies.(1) Effect of a chromatic aberration and slight misorientation: There is an inevitable fluctuation in the amount of defocus due to a chromatic aberration originating from the fluctuations both in the energies of electrons and in the magnetic lens current. The actual image is a results of superposition of those fluctuated images during the exposure time. Assuming the Gaussian distribution for defocus, Δf around the optimum defocus value Δf0, the intensity distribution, I(x,y) in the image formed by this fluctuation is given by

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