Correction to: Sobolev Spaces W1, p(ℝn, γ) Weighted by the Gaussian Normal Distribution $$ \gamma (x):= \frac{1}{{\sqrt{\pi}}^n}\exp \left(-{\left|x\right|}^2\right) $$ and the Spectral Theory

Various definitions of capacity of a subset of a domain in Euclidean space have been used in recent times to shed light on the solvability and spectral theory of elliptic partial differential equations and to establish properties of the Sobolev spaces in which these equations are studied. In this paper we consider two definitions of the capacity of a closed set E in a domain G. One of these capacities measures, roughly speaking, the amount by which the set of function in C∞(G) which vanish near E fails to be dense in the Sobolev space Wm, p(G).

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Logarithmic Sobolev Spaces and Their Applications to Spectral Theory

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-71.2.333 ◽

1995 ◽

Vol s3-71 (2) ◽

pp. 333-371 ◽

Cited By ~ 15

Author(s):

D. E. Edmunds ◽

H. Triebel

Keyword(s):

Spectral Theory ◽

Sobolev Spaces

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Spectral theory of Wiener-Hopf type integral operators on generalized Sobolev spaces

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(67)90157-6 ◽

1967 ◽

Vol 17 (2) ◽

pp. 343-372 ◽

Cited By ~ 1

Author(s):

John S Kalme

Keyword(s):

Spectral Theory ◽

Sobolev Spaces ◽

Integral Operators ◽

Generalized Sobolev Spaces ◽

Type Integral

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Spectral theory of nonlinear equations and n-widths of Sobolev spaces

Methods of Approximation Theory in Complex Analysis and Mathematical Physics - Lecture Notes in Mathematics ◽

10.1007/bfb0117471 ◽

1993 ◽

pp. 19-30

Author(s):

A. P. Buslaev ◽

V. M. Tikhomirov

Keyword(s):

Spectral Theory ◽

Nonlinear Equations ◽

Sobolev Spaces

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Scour Induced by Single and Twin Propeller Jets

Water ◽

10.3390/w11051097 ◽

2019 ◽

Vol 11 (5) ◽

pp. 1097 ◽

Cited By ~ 7

Author(s):

Yonggang Cui ◽

Wei Haur Lam ◽

Tianming Zhang ◽

Chong Sun ◽

Gerard Hamill

Keyword(s):

Normal Distribution ◽

Maximum Depth ◽

Scour Depth ◽

Scour Hole ◽

Scour Holes ◽

Ship Propeller ◽

Gaussian Normal Distribution

Single and twin ship propeller jets produce scour holes with deposition dune. The scour hole has a maximum depth at a particular length downstream within the propeller jet. Existing equations are available to predict maximum scour depth and the corresponding scour length downstream. Experiments conducted with various physical propeller models, rotational speeds, propeller-to-propeller distances and bed clearances are presented. The measurements allowed a better understanding of the mechanism of temporal scour and deposition formation for scour caused by single-propeller and twin-propeller. Results show that the propeller jet scour profiles can be divided into three zones, which are the small scour hole, primary scour hole and deposition dune. An empirical 2D scour model is proposed to predict the scour profile for both a single-propeller and twin-propeller using a Gaussian normal distribution.

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Histograms of Gaussian normal distribution for 3D feature matching in cluttered scenes

The Visual Computer ◽

10.1007/s00371-018-1478-x ◽

2018 ◽

Vol 35 (4) ◽

pp. 489-505 ◽

Cited By ~ 4

Author(s):

Wei Zhou ◽

Caiwen Ma ◽

Tong Yao ◽

Peng Chang ◽

Qi Zhang ◽

...

Keyword(s):

Normal Distribution ◽

Feature Matching ◽

Gaussian Normal Distribution

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Pseudodifferential calculus on noncommutative tori, II. Main properties

International Journal of Mathematics ◽

10.1142/s0129167x19500344 ◽

2019 ◽

Vol 30 (08) ◽

pp. 1950034 ◽

Cited By ~ 3

Author(s):

Hyunsu Ha ◽

Gihyun Lee ◽

Raphaël Ponge

Keyword(s):

Spectral Theory ◽

Trace Formula ◽

Sobolev Spaces ◽

Pseudodifferential Operators ◽

Elliptic Operators ◽

Schatten Class ◽

Noncommutative Tori ◽

Dimension Formula ◽

Pseudodifferential Calculus ◽

Negative Order

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

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