Classical Linear Iterations in the Positive Definite Case

2016 ◽  
pp. 35-67
Author(s):  
Wolfgang Hackbusch
Keyword(s):  
Positive Definite ◽  
Definite Case

scholarly journals On the converse of Weyl’s conformal and projective theorems

2013 ◽  
Vol 94 (108) ◽  
pp. 55-65 ◽  
Author(s):  
Graham Hall
Keyword(s):  
Positive Definite ◽  
The Other ◽  
Dimensional Problem ◽  
Projective Tensor ◽  
Definite Case

This note investigates the possibility of converses of the Weyl theorems that two conformally related metrics on a manifold have the same Weyl conformal tensor and that two projectively related connections on a manifold have the same Weyl projective tensor. It shows that, in all relevant cases, counterexamples to each of Weyl?s theorems exist except for his conformal theorem in the 4-dimensional, positive definite case, where the converse actually holds. This (conformal) 4-dimensional problem is then solved completely for the other possible signatures.

scholarly journals A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 213
Author(s):  
Herbert Dueñas Ruiz ◽  
Francisco Marcellán ◽  
Alejandro Molano
Keyword(s):  
Positive Definite ◽  
Linear Functionals ◽  
Definite Case ◽  
Coherent Pairs

In this paper, we study a classification of symmetric ( 1 , 1 ) -coherent pairs by using a symmetrization process. In particular, the positive-definite case is carefully described.

On the global isometric embedding of pseudo-Riemannian manifolds

1970 ◽  
Vol 314 (1518) ◽  
pp. 417-428 ◽  
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
General Result ◽  
Isometric Embedding ◽  
Positive Definite ◽  
Space Time ◽  
Lower Dimension ◽  
Definite Case

It is shown that any pseudo-Riemannian manifold has (in Nash’s sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

scholarly journals Lebesgue perturbation of a quasi-definite Hermitian functional. The positive definite case

2003 ◽  
Vol 369 ◽  
pp. 235-250 ◽  
Author(s):  
A. Cachafeiro ◽  
F. Marcellán ◽  
C. Pérez
Keyword(s):  
Positive Definite ◽  
Definite Case

scholarly journals CONTRA-RIEMANNIAN, NEGATIVE MODULI FOR A LINE

2021 ◽  
Vol 70 (2) ◽  
pp. 35-43
Author(s):  
Demi Caisse
Keyword(s):  
Positive Definite ◽  
Definite Case

Let V be a modulus. We wish to extend the results of[19] to conditionally intrinsic equations. We show that T is not bounded by v. Thus the workin[19] did not consider the positive definite case. Next, unfortunately,

IV.—Studies in Practical Mathematics. V. On the Iterative Solution of a System of Linear Equations

1950 ◽  
Vol 63 (1) ◽  
pp. 52-60 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Linear Equations ◽  
Iterative Solution ◽  
Positive Definite ◽  
System Of Linear Equations ◽  
Characteristic Roots ◽  
New Variant ◽  
Definite Case ◽  
Practical Mathematics

SynopsisThe convergence of customary processes of iteration for solving linear equations, in particular simple and Seidelian iteration, is studied from the standpoint of matrices. A new variant of Seidelian iteration is introduced. In the positive definite case it always converges, the characteristic roots of its operator being real and positive and less than unity.

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