scholarly journals Comparison between W2 distance and Ḣ−1 norm, and Localization of Wasserstein distance

2018 ◽  
Vol 24 (4) ◽  
pp. 1489-1501 ◽  
Author(s):  
Rémi Peyre
Keyword(s):  
Compact Support ◽  
Wasserstein Distance ◽  
Sobolev Norm ◽  
Bump Function ◽  
Small Perturbations ◽  
Localization Phenomenon ◽  
Comparison Results

It is well known that the quadratic Wasserstein distance W2(⋅, ⋅) is formally equivalent, for infinitesimally small perturbations, to some weighted H−1 homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the W2 distance exhibits some localization phenomenon: if μ and ν are measures on ℝn and ϕ: ℝn → ℝ+ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between ϕ ⋅ μ and ϕ ⋅ ν by an explicit multiple of W2(μ, ν).

scholarly journals On the excursion area of perturbed Gaussian fields

2020 ◽  
Vol 24 ◽  
pp. 252-274
Author(s):  
Elena Di Bernardino ◽  
Anne Estrade ◽  
Maurizia Rossi
Keyword(s):  
Limit Theorem ◽  
Random Perturbation ◽  
Wasserstein Distance ◽  
Expansion Formula ◽  
Kinematic Formula ◽  
Small Perturbations ◽  
Asymptotically Normal ◽  
Excursion Sets ◽  
Growing Domain ◽  
Non Gaussian

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2 under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

scholarly journals Sobolev-Kantorovich Inequalities

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Michel Ledoux
Keyword(s):  
Pattern Formation ◽  
Probability Density ◽  
Riemannian Manifolds ◽  
Wasserstein Distance ◽  
Sobolev Norm ◽  
Heat Flows ◽  
Interpolation Inequalities ◽  
Harnack Inequalities ◽  
Negatively Curved ◽  
Reference Measure

Abstract In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.

Perceptions of Stuttering of Different Age Groups

2019 ◽  
Vol 4 (6) ◽  
pp. 1311-1315
Author(s):  
Sergey M. Kondrashov ◽  
John A. Tetnowski
Keyword(s):  
Age Groups ◽  
School Age Children ◽  
Clinical Implications ◽  
School Age ◽  
Fourth Edition ◽  
Self Perception ◽  
Behavioral Measures ◽  
Self Rating ◽  
Comparison Results ◽  
Subject Pool

Purpose The purpose of this study was to assess the perceptions of stuttering of school-age children who stutter and those of adults who stutter through the use of the same tools that could be commonly used by clinicians. Method Twenty-three participants across various ages and stuttering severity were administered both the Stuttering Severity Instrument–Fourth Edition (SSI-4; Riley, 2009 ) and the Wright & Ayre Stuttering Self-Rating Profile ( Wright & Ayre, 2000 ). Comparisons were made between severity of behavioral measures of stuttering made by the SSI-4 and by age (child/adult). Results Significant differences were obtained for the age comparison but not for the severity comparison. Results are explained in terms of the correlation between severity equivalents of the SSI-4 and the Wright & Ayre Stuttering Self-Rating Profile scores, with clinical implications justifying multi-aspect assessment. Conclusions Clinical implications indicate that self-perception and impact of stuttering must not be assumed and should be evaluated for individual participants. Research implications include further study with a larger subject pool and various levels of stuttering severity.

Energy Efficiency Analysis in ERA using NFL-BA Algorithm

2020 ◽  
pp. 340-348
Author(s):  
Palky Mehta ◽  
H. L. Sharma
Keyword(s):  
Cluster Head ◽  
Vital Role ◽  
Wireless Sensor ◽  
Energy Usage ◽  
Cluster Heads ◽  
Comparison Results ◽  
Current Scenario ◽  
Energy Efficiency Analysis ◽  
Selection Of

In the current scenario of Wireless Sensor Network (WSN), power consumption is the major issue associated with nodes in WSN. LEACH technique plays a vital role of clustering in WSN and reduces the energy usage effectively. But LEACH has its own limitation in order to search cluster head nodes which are randomly distributed over the network. In this paper, ERA-NFL- BA algorithm is being proposed for selects the cluster heads in WSN. This algorithm help in selection of cluster heads can freely transform from global search to local search. At the end, a comparison has been done with earlier researcher using protocol ERA-NFL, which clearly shown that proposed Algorithm is best suited and from comparison results that ERA-NFL-BA has given better performance.

Gluing Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Stress Tensor ◽  
Compact Support ◽  
Smooth Solution ◽  
Continuous Solution ◽  
Total Momentum ◽  
Frame Of Reference ◽  
Galilean Transformation ◽  
Hölder Continuous ◽  
Pressure Form ◽  
Original Frame

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

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