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Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 10, Pages 1661–1669 (Mi zvmmf10280)  

This article is cited in 2 scientific papers (total in 2 papers)

The Dines theorem and some other properties of quadratic mappings

D. Yu. Karamzin

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

Abstract: Real homogeneous quadratic mappings from $\mathbb{R}^n$ to $\mathbb{R}^2$ are examined. It is known that the image of such a mapping is always convex. A proof of the convexity of the image based on the quadratic extremum principle is given. The following fact is noted: If the quadratic mapping $Q$ is surjective and $n>2+\mathrm{dim ker }Q$, then there exists a regular zero of $Q$. A certain criterion of the linear dependence of quadratic forms is also stated.

Key words: quadratic forms and mappings, convexity of image, regular zeros.

DOI: https://doi.org/10.7868/S0044466915100130

Full text: PDF file (125 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:10, 1633–1641

Bibliographic databases:

Document Type: Article
UDC: 519.626
Received: 13.01.2015

Citation: D. Yu. Karamzin, “The Dines theorem and some other properties of quadratic mappings”, Zh. Vychisl. Mat. Mat. Fiz., 55:10 (2015), 1661–1669; Comput. Math. Math. Phys., 55:10 (2015), 1633–1641

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Arutyunov, S. E. Zhukovskiy, “Properties of surjective real quadratic maps”, Sb. Math., 207:9 (2016), 1187–1214  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. A. V. Arutyunov, S. E. Zhukovskiy, D. Yu. Karamzin, “Some properties of two-dimensional surjective $p$-homogeneous maps”, Comput. Math. Math. Phys., 57:7 (2017), 1081–1089  mathnet  crossref  crossref  mathscinet  isi  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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