On systems of linear algebraic equations in the Colombeau algebra

Jan Ligęza; Milan Tvrdý

Mathematica Bohemica (1999)

  • Volume: 124, Issue: 1, page 1-14
  • ISSN: 0862-7959

Abstract

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From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra ℝ ¯ of generalized real numbers. It is worth mentioning that the algebra ℝ ¯ is not a field.

How to cite

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Ligęza, Jan, and Tvrdý, Milan. "On systems of linear algebraic equations in the Colombeau algebra." Mathematica Bohemica 124.1 (1999): 1-14. <http://eudml.org/doc/248461>.

@article{Ligęza1999,
abstract = {From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra $\overline\{\mathbb \{R\}\}$ of generalized real numbers. It is worth mentioning that the algebra $\overline\{\mathbb \{R\}\}$ is not a field.},
author = {Ligęza, Jan, Tvrdý, Milan},
journal = {Mathematica Bohemica},
keywords = {Colombeau algebra; system of linear equations; generalized real numbers; Colombeau algebra; system of linear equations; generalized real numbers},
language = {eng},
number = {1},
pages = {1-14},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On systems of linear algebraic equations in the Colombeau algebra},
url = {http://eudml.org/doc/248461},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Ligęza, Jan
AU - Tvrdý, Milan
TI - On systems of linear algebraic equations in the Colombeau algebra
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 1
SP - 1
EP - 14
AB - From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra $\overline{\mathbb {R}}$ of generalized real numbers. It is worth mentioning that the algebra $\overline{\mathbb {R}}$ is not a field.
LA - eng
KW - Colombeau algebra; system of linear equations; generalized real numbers; Colombeau algebra; system of linear equations; generalized real numbers
UR - http://eudml.org/doc/248461
ER -

References

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  1. Colombeau J. F., Elementary Introduction to New Generalized Functions, North Holland, Amsterdam, New York, Oxford, 1985. (1985) Zbl0584.46024MR0808961
  2. Ligęza J., Generalized solutions of boundary value problems for ordinary linear differential equations of second order in the Colombeau algebra, Dissertationes Mathematicae (Different aspect of differentiability) 340 (1995), 183-194. (1995) Zbl0837.34026MR1342577
  3. Mc Cay N.H., Rings and Ideals, The Carus Mathematical Monographs (Nr. 8), Baltimore, 1948. (1948) 
  4. Przeworska-Rolewicz D., Algebraic Analysis, PWN-Polish Scientific Publishers & D. Reidel Publishing Company, Warszawa, 1988. (1988) Zbl0696.47002MR0945395

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