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Keywords:
Colombeau algebra; system of linear equations; generalized real numbers
Summary:
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 $\overline{\Bbb R}$ of generalized real numbers. It is worth mentioning that the algebra $\overline{\Bbb R}$ is not a field.
References:
[1] Colombeau J. F.: Elementary Introduction to New Generalized Functions. North Holland, Amsterdam, New York, Oxford, 1985. MR 0808961 | Zbl 0584.46024
[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. MR 1342577 | Zbl 0837.34026
[3] Mc Cay N.H.: Rings and Ideals. The Carus Mathematical Monographs (Nr. 8), Baltimore, 1948.
[4] Przeworska-Rolewicz D.: Algebraic Analysis. PWN-Polish Scientific Publishers & D. Reidel Publishing Company, Warszawa, 1988. MR 0945395 | Zbl 0696.47002

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