# Article

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Keywords:
Colombeau algebra; singular products of distributions
Summary:
Models of singularities given by discontinuous functions or distributions by means of generalized functions of Colombeau have proved useful in many problems posed by physical phenomena. In this paper, we introduce in a systematic way generalized functions that model singularities given by distributions with singular point support. Furthermore, we evaluate various products of such generalized models when the results admit associated distributions. The obtained results follow the idea of a well-known result of Jan Mikusiński on balancing of singular distributional products.
References:
[1] Colombeau J.-F.: New Generalized Functions and Multiplication of Distributions. North Holland Math. Studies, 84, Amsterdam, 1984. MR 0738781 | Zbl 0761.46021
[2] Damyanov B.: Mikusiński type products of distributions in Colombeau algebra. Indian J. Pure Appl. Math. 32 (2001), 361–375. MR 1826763 | Zbl 1021.46032
[3] Damyanov B.: Modelling and products of singularities in Colombeau algebra $G (R)$. J. Applied Analysis 14 (2008), no.1, 89–102. MR 2444250
[4] Grosser M., Kunzinger M., Oberguggenberger M., Steinbauer R.: Geometric Theory of Generalized Functions with Applications to General Relativity. Kluwer Acad. Publ., Dordrecht, 2001. MR 1883263 | Zbl 0998.46015
[5] Hörmander L.: Analysis of LPD Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin, 1983. MR 0717035
[6] Korn G.A., Korn T.M.: Mathematical Handbook. McGraw-Hill Book Company, New York, 1968. MR 0220560 | Zbl 0535.00032
[7] Mikusiński J.: On the square of the Dirac delta-distribution. Bull. Acad. Pol. Ser. Sci. Math. Astron. Phys. 43 (1966), 511–513. MR 0203392 | Zbl 0163.36404
[8] Nedeljkov M., Oberguggenberger M.: Ordinary differential equations with delta function terms. Publ. Inst. Math. (Beograd) (N.S.) 91(105) (2012), 125 - 135. DOI 10.2298/PIM1205125N | MR 2963815
[9] Oberguggenberger M.: Multiplication of Distributions and Applications to PDEs. Longman, Essex, 1992.

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