On the result equivalence of constraints of an asymptotic nature

2010 ◽  
Vol 268 (S1) ◽  
pp. 32-53
Author(s):  
A. G. Chentsov
Keyword(s):  
Asymptotic Nature

scholarly journals An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions

Stats ◽  
2020 ◽  
Vol 4 (1) ◽  
pp. 1-17
Author(s):  
Samuele Tosatto ◽  
Riad Akrour ◽  
Jan Peters
Keyword(s):  
Upper Bound ◽  
Prediction Error ◽  
Kernel Estimator ◽  
Asymptotic Bias ◽  
Surgical Robots ◽  
Related Literature ◽  
Asymptotic Nature ◽  
Potential Applications ◽  
Self Driving Cars ◽  
Automated Decision Making

The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in several related literature. However, given its asymptotic nature, it gives no access to a hard bound. The increasing popularity of predictive tools for automated decision-making surges the need for hard (non-probabilistic) guarantees. To alleviate this issue, we propose an upper bound of the bias which holds for finite bandwidths using Lipschitz assumptions and mitigating some of the prerequisites of Rosenblatt’s analysis. Our bound has potential applications in fields like surgical robots or self-driving cars, where some hard guarantees on the prediction-error are needed.

scholarly journals On the relation between pole and running heavy quark masses beyond the four-loop approximation

2018 ◽  
Vol 191 ◽  
pp. 04005 ◽  
Author(s):  
A. L. Kataev ◽  
V. S. Molokoedov
Keyword(s):  
Heavy Quark ◽  
Perturbative Qcd ◽  
Mass Relation ◽  
Coupling Constant ◽  
Quark Masses ◽  
Asymptotic Nature ◽  
Asymptotic Expressions ◽  
Loop Approximation ◽  
Euclidean Region ◽  
Direct Calculations

The effective charges motivated method is applied to the relation between pole and M̅S̅-scheme heavy quark masses to study high order perturbative QCD corrections in the observable quantities proportional to the running quark masses. The non-calculated five- and six-loop perturbative QCD coefficients are estimated. This approach predicts for these terms the sign-alternating expansion in powers of number of lighter flavors nl, while the analyzed recently infrared renormalon asymptotic expressions do not reproduce the same behavior. We emphasize that coefficients of the quark mass relation contain proportional to π2 effects, which result from analytical continuation from the Euclidean region, where the scales of the running masses and QCD coupling constant are initially fixed, to the Minkowskian region, where the pole masses and the running QCD parameters are determined. For the t-quark the asymptotic nature of the non-resummed PT mass relation does not manifest itself at six-loops, while for the b-quark the minimal PT term appears at the probed by direct calculations four-loop level. The recent infrared renormalon based studies support these conclusions.

scholarly journals The Maximum Displacement for Linear Probing Hashing

2013 ◽  
Vol 22 (3) ◽  
pp. 455-476
Author(s):  
NICLAS PETERSSON
Keyword(s):  
Probabilistic Model ◽  
Hash Table ◽  
Critical Value ◽  
Hash Tables ◽  
Maximum Displacement ◽  
Asymptotic Nature ◽  
Process Convergence

In this paper we study the maximum displacement for linear probing hashing. We use the standard probabilistic model together with the insertion policy known as First-Come-(First-Served). The results are of asymptotic nature and focus on dense hash tables. That is, the number of occupied cellsnand the size of the hash tablemtend to infinity with ration/m→ 1. We present distributions and moments for the size of the maximum displacement, as well as for the number of items with displacement larger than some critical value. This is done via process convergence of the (appropriately normalized) length of the largest block of consecutive occupied cells, when the total number of occupied cellsnvaries.

scholarly journals On inertial-range scaling laws

1996 ◽  
Vol 306 ◽  
pp. 167-181 ◽  
Author(s):  
John C. Bowman
Keyword(s):  
Steady State ◽  
High Resolution ◽  
Scaling Laws ◽  
State Energy ◽  
Three Dimensional ◽  
Inertial Range ◽  
Energy Spectra ◽  
Two Dimensional ◽  
Unified Framework ◽  
Asymptotic Nature

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's k−5/3 scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

Uncoupled Wave Systems and Dispersion in an Infinite Solid Cylinder

1989 ◽  
Vol 56 (2) ◽  
pp. 347-355 ◽  
Author(s):  
Yoon Young Kim
Keyword(s):  
Cylindrical Surface ◽  
Wave Motion ◽  
Fourier Number ◽  
Large Wave ◽  
Surface Conditions ◽  
Asymptotic Nature ◽  
Wave Numbers ◽  
Complex Wave ◽  
Insight Into ◽  
Free Cylindrical Surface

In this study, it is shown that there exist uncoupled wave systems for general non-axisymmetric wave propagation in an infinite isotropic cylinder. Two cylindrical surface conditions corresponding to the uncoupled wave systems are discussed. The solutions of the uncoupled wave systems are shown to provide proper bounds of Pochhammer’s equation for a free cylindrical surface. The bounds, which are easy to construct for any Fourier number in the circumferential direction, can be used to trace the branches of Pochhammer’s equation. They also give insight into the modal composition of the branches of Pochhammer’s equation at and between the intersections of the bounds. More refined dispersion relations of Pochhammer’s equation are possible through an asymptotic analysis of the itersections of the branches of Pochhammer’s equation with one family of the bounds. The asymptotic nature of wave motion corresponding to large wave numbers, imaginary or complex, for Pochhammer’s equation is studied. The wave motion is asymptotically equivoluminal for large imaginary wave numbers, and is characterized by coupled dilatation and shear for large complex wave numbers.

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