Order sharp stability estimates for the decompositions of the normal distribution in the L�vy metric

1994 ◽  
Vol 72 (1) ◽  
pp. 2848-2871 ◽  
Author(s):  
L. B. Golinskii ◽  
G. P. Chistyakov
Keyword(s):  
Normal Distribution ◽  
Stability Estimates ◽  
Sharp Stability

scholarly journals A PRIORI ERROR ESTIMATES FOR ENERGY-BASED QUASICONTINUUM APPROXIMATIONS OF A PERIODIC CHAIN

2011 ◽  
Vol 21 (12) ◽  
pp. 2491-2521 ◽  
Author(s):  
CHRISTOPH ORTNER ◽  
HAO WANG
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Stability Estimates ◽  
Norm Estimates ◽  
Priori Error Estimates ◽  
Sharp Stability ◽  
Consistency Error ◽  
Periodic Chain ◽  
Homogeneous Strain

We derive a priori error estimates for three prototypical energy-based quasicontinuum (QC) methods: the local QC method, the energy-based QC method, and the quasi-nonlocal QC method. Our analysis decomposes the consistency error into modeling and coarsening errors. While previous results on estimating the modeling error exist, we present a new and simpler proof based on negative-norm estimates. Our stability analysis extends previous results on sharp stability estimates under homogeneous strain to the nonlinear setting. Finally, we present numerical experiments to illustrate the results of our analysis.

scholarly journals On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 2014
Author(s):  
Junjiang Lai ◽  
Hongyu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
A Priori ◽  
Time Derivative ◽  
Stability Estimates ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Sharp Stability

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

scholarly journals High order numerical schemes for transport equations on bounded domains

2021 ◽  
Vol 70 ◽  
pp. 84-106
Author(s):  
B. Boutin ◽  
T.H.T. Nguyen ◽  
A. Sylla ◽  
S. Tran-Tien ◽  
J.-F. Coulombel
Keyword(s):  
Boundary Data ◽  
Convergence Rates ◽  
Summer Camp ◽  
Stability Estimates ◽  
Bounded Domains ◽  
Finite Difference Approximations ◽  
Gravity Flows ◽  
Optimal Convergence Rates ◽  
Sharp Stability ◽  
Geophysical Fluids

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

scholarly journals Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature

2018 ◽  
Vol 2018 (741) ◽  
pp. 275-294 ◽  
Author(s):  
Giulio Ciraolo ◽  
Alessio Figalli ◽  
Francesco Maggi ◽  
Matteo Novaga
Keyword(s):  
Decay Rate ◽  
Mean Curvature ◽  
Lipschitz Constant ◽  
Classical Case ◽  
Stability Estimates ◽  
Single Sphere ◽  
Boundary Controls ◽  
Bounded Smooth ◽  
Sharp Stability

Abstract We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its {C^{2}} -distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

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