2. An Iterative Decomposition of Global Conformal Invariants: The First Step

Author(s):  
Spyros Alexakis

This chapter fleshes out the strategy of iteratively decomposing any P(g) = unconverted formula 1 for which ∫P(g)dVsubscript g is a global conformal invariant. It makes precise the notions of better and worse complete contractions in P(g) and then spells out (1.17), via Propositions 2.7, 2.8. In particular, using the well-known decomposition of the curvature tensor into its trace-free part (the Weyl tensor) and its trace part (the Schouten tensor), it reexpresses P(g) as a linear combination of complete contractions involving differentiated Weyl tensors and differentiated Schouten tensors, as in (2.47). The chapter also proves (1.17) when the worst terms involve at least one differentiated Schouten tensor.


2001 ◽  
Vol 42 (10-11) ◽  
pp. 1485-1496 ◽  
Author(s):  
M.S. Sahimi ◽  
E. Sundararajan ◽  
M. Subramaniam ◽  
N.A.A. Hamid

2017 ◽  
Vol 29 (02) ◽  
pp. 1750015
Author(s):  
Hui-Chun Wang ◽  
Po-Chou Chen ◽  
Chun-Hsiung Chou ◽  
Cherng-Gueih Shy ◽  
Jo-Chi Jao

Nowadays, magnetic resonance imaging (MRI) has been widely applied for diagnosis of soft-tissue diseases. Most clinical MRI protocols use fat suppression (FS) methods to suppress fat signal, reduce chemical shift artifacts, and increase conspicuity of lesions. To understand the advantages, disadvantages, and clinical applications of the most commonly used FS methods is an important issue. The aim of this study was to evaluate FS performance of six FS methods on a fat-water phantom at 1.5[Formula: see text]T. The six MRI methods included iterative decomposition of water and fat with echo asymmetry and least squares estimation (IDEAL), short inversion time inversion recovery (STIR), and four chemical presaturation (Chem Presat) methods. The phantom was composed of homogeneous oil-in-water emulsions with various fat contents ranging from 0 to 100% in increments of 10%. The difference between the suppressed fat fractions (FS fractions) and the true fat fractions of the phantom was used as an index of FS performance. The correlations and levels of agreement (LOAs) between the FS fractions determined using each FS method and the true fat fractions of the phantom were analyzed. From the phantom study, it was found that FSE T2 FS, STIR and IDEAL could achieve more accurate FS fractions than the other three methods. The FS fractions determined using FSE T2 FS, STIR and IDEAL were in a good agreement. On the contrary, T2-weighted spin echo Chem Presat had the most inaccurate quantification of FS fractions among these six FS methods. Both the ranks of signal-to-noise ratios (SNRs) and contrast-to-noise ratios (CNRs) of the phantom were IDEAL [Formula: see text] FSE T2 FS [Formula: see text] STIR. The FS performance of these six FS methods in clinical use needs further study.


1977 ◽  
Vol 62 (2) ◽  
pp. 316-316 ◽  
Author(s):  
Chuan Chih Hsiung ◽  
Larry R. Mugridge
Keyword(s):  

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