Regularization of Positive Definite Matrix Fields Based on Multiplicative Calculus

2012 ◽  
pp. 786-796 ◽  
Author(s):  
Luc Florack
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Multiplicative Calculus ◽  
Matrix Fields

scholarly journals A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients

2019 ◽  
Vol 21 (07) ◽  
pp. 1850057 ◽  
Author(s):  
Francesca Anceschi ◽  
Michela Eleuteri ◽  
Sergio Polidoro
Keyword(s):  
Maximum Principle ◽  
Harnack Inequality ◽  
Weak Solutions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Strong Maximum Principle ◽  
Kolmogorov Equation ◽  
Measurable Coefficients ◽  
Rough Coefficients ◽  
Vector Formula

We consider weak solutions of second-order partial differential equations of Kolmogorov–Fokker–Planck-type with measurable coefficients in the form [Formula: see text] where [Formula: see text] is a symmetric uniformly positive definite matrix with bounded measurable coefficients; [Formula: see text] and the components of the vector [Formula: see text] are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by Golse et al. As a corollary, we obtain a strong maximum principle.

A polynomial algorithm for the quadratic programming problem

2014 ◽  
Vol 29 (3) ◽  
Author(s):  
Valentin A. Bereznev
Keyword(s):  
Computational Complexity ◽  
Quadratic Form ◽  
Quadratic Programming ◽  
Programming Problem ◽  
Polynomial Complexity ◽  
Polynomial Algorithm ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Quadratic Programming Problem ◽  
Convex Polyhedral Cone

AbstractAn approach based on projection of a vector onto a pointed convex polyhedral cone is proposed for solving the quadratic programming problem with a positive definite matrix of the quadratic form. It is proved that this method has polynomial complexity. A method is said to be of polynomial computational complexity if the solution to the problem can be obtained in N

Generalized Inverse as a Weight Matrix

1974 ◽  
Vol 28 (5) ◽  
pp. 698-701 ◽  
Author(s):  
Urho A. Uotila
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Generalized Inverse ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Weight Matrix

In several textbooks, the least-squares solutions are given with generalized matrices, but usually the weight matrix is assumed to be a positive definite matrix especially when weight matrix has off-diagonal elements. A more general solution is presented in which a generalized inverse is used as a weight matrix.

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