scholarly journals Extremizers of a Radon Transform Inequality

2014 ◽  
Author(s):  
Michael Christ
Keyword(s):  
Critical Points ◽  
Radon Transform ◽  
Sobolev Inequality ◽  
Affine Group ◽  
Steiner Symmetrization

This chapter discusses the extremizers of a Radon transform inequality. The model for this analysis is Lieb's characterization of extremizers for the Hardy–Littlewood–Sobolev inequality for certain pairs of exponents. The chapter first introduces the four main steps of this model and sets up an endpoint inequality, before developing the identities to be used for the analysis in the remainder of this chapter. It then discusses some preliminary facts concerning extremizers and brings up direct and inverse Steiner symmetrization. Finally, the chapter returns to the inequality described in the first part of the chapter and begins the process of identifying extremizers for it. It concludes with further discussion on compact subgroups of the affine group as well as critical points.

Kleinian characterization of asymptotic symmetries of real general relativity

1993 ◽  
Vol 441 (1913) ◽  
pp. 465-471 ◽  
Keyword(s):  
General Relativity ◽  
Automorphism Group ◽  
Group Action ◽  
Radon Transform ◽  
Symmetry Groups ◽  
Asymptotic Symmetry ◽  
The Real ◽  
The Given ◽  
Asymptotic Symmetries

According to Klein’s Erlanger programme, one may (indirectly) specify a geometry by giving a group action. Conversely, given a group action, one may ask for the corresponding geometry. Recently, I showed that the real asymptotic symmetry groups of general relativity (in any signature) have natural ‘projective’ classical actions on suitable ‘Radon transform’ spaces of affine 3-planes in flat 4-space. In this paper, I give concrete models for these groups and actions. Also, for the ‘atomic’ cases, I give geometric structures for the spaces of affine 3-planes for which the given actions are the automorphism group.

The d-plane radon transform on the torus $$ \mathbb{T}^n $$

2011 ◽  
Vol 14 (2) ◽  
Author(s):  
Ahmed Abouelaz
Keyword(s):  
Radon Transform ◽  
Inversion Formula ◽  
Flat Torus ◽  
Smooth Functions ◽  
Suitable Function

AbstractWe define and study the d-plane Radon transform, namely R, on the n-dimensional (flat) torus. The transformation R is obtained by integrating a suitable function f over all d-dimensional geodesics (d-planes in the torus). We specially establish an explicit inversion formula of R and we give a characterization of the image, under the d-plane Radon transform, of the space of smooth functions on the torus.

Applications of the Global Coordinate System

2011 ◽  
Author(s):  
Yves Balasko
Keyword(s):  
Coordinate System ◽  
Critical Points ◽  
Global Coordinate System ◽  
Coordinate Systems ◽  
Analytical Characterization ◽  
Equilibrium Manifold ◽  
System P ◽  
Definition Of

The global coordinate system for the equilibrium manifold follows from: (1) the determination of the unique fiber F(b) through the equilibrium (ρ‎, ω‎) where b = φ‎((ρ‎, ω‎) = (ρ‎, ρ‎ · ρ‎1, …, ρ‎ · ρ‎m); and (2) the determination of the location of the equilibrium (ρ‎, ω‎) within the fiber F(b) viewed as a linear space of dimension (ℓ − 1)(m − 1) and, therefore, parameterized by (ℓ − 1)(m − 1) coordinates. If there is little leeway in determining the fiber F(b) through the equilibrium (ρ‎, ω‎), there are different ways of representing the equilibrium (ρ‎, ω‎) within its fiber F(b). This leads to the definition of coordinate systems (A) and (B) for the equilibrium manifold. This chapter defines these two coordinate systems and applies them to obtain an analytical characterization of the critical equilibria, i.e., the critical points of the natural projection.

A Remark on Extensions of CR Functions from Hyperplanes

2008 ◽  
Vol 51 (1) ◽  
pp. 21-25
Author(s):  
Luca Baracco
Keyword(s):  
Radon Transform ◽  
Integral Geometry ◽  
Holomorphic Functions ◽  
Holomorphic Extension ◽  
General Statement ◽  
Cr Geometry ◽  
Cr Functions ◽  
Natural Context ◽  
Extension Of Functions

AbstractIn the characterization of the range of the Radon transform, one encounters the problem of the holomorphic extension of functions defined on ℝ2 \ Δℝ (where Δℝ is the diagonal in ℝ2) and which extend as “separately holomorphic” functions of their two arguments. In particular, these functions extend in fact to ℂ2 \ Δℂ where Δℂ is the complexification of Δℝ. We take this theorem from the integral geometry and put it in the more natural context of the CR geometry where it accepts an easier proof and amore general statement. In this new setting it becomes a variant of the celebrated “edge of the wedge” theorem of Ajrapetyan and Henkin.

scholarly journals Ultrasound evaluation of inguinoscrotal pain: an imaging-based review for the ultrasonographer

2018 ◽  
Vol 51 (3) ◽  
pp. 193-199 ◽  
Author(s):  
Eduardo Kaiser Ururahy Nunes Fonseca ◽  
Milena Rocha Peixoto ◽  
Francisco de Assis Cavalcante Júnior ◽  
Antonio Rahal Júnior ◽  
Miguel José Francisco Neto ◽  
...  
Keyword(s):  
Critical Points ◽  
Emergency Care ◽  
Low Cost ◽  
Accurate Diagnosis ◽  
Complication Rates ◽  
Imaging Findings ◽  
Ultrasound Evaluation ◽  
Ultrasound Findings ◽  
Vast Range

Abstract Emergencies involving the inguinal region and scrotum are common and can be caused by a plethora of different causes. In most cases, such conditions have nonspecific symptoms and are quite painful. Some inguinoscrotal conditions have high complication rates. Early and accurate diagnosis is therefore imperative. Ultrasound is the method of choice for the initial evaluation of this vast range of conditions, because it is a rapid, ionizing radiation-free, low-cost method. Despite the practicality and accuracy of the method, which make it ideal for use in emergency care, the examiner should be experienced and should be familiarized with the ultrasound findings of the most common inguinoscrotal diseases. On the basis of that knowledge, the examiner should also be able to make an accurate, direct, precise report, helping the emergency room physician make decisions regarding the proper (clinical or surgical) management of each case. Here, we review most of the inguinoscrotal conditions, focusing on the imaging findings and discussing the critical points for the appropriate characterization of each condition.

scholarly journals Distance functions and umbilic submanifolds of hyperbolic space

1979 ◽  
Vol 74 ◽  
pp. 67-75 ◽  
Author(s):  
Thomas E. Cecil ◽  
Patrick J. Ryan
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Hyperbolic Space ◽  
Critical Points ◽  
Morse Function ◽  
Complete Riemannian Manifold ◽  
Distance Functions

In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em.

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