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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Systems of division problems for distributions
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by B. Roth PDF
Trans. Amer. Math. Soc. 155 (1971), 493-504 Request permission

Abstract:

Suppose ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ is a $p \times p$ matrix of real-valued infinitely (respectively $m$-times continuously) differentiable functions on an open subset $\Omega$ of ${R^n}$. Then ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ maps the space of $p$-tuples of distributions on $\Omega$ (respectively distributions of order $\leqq m$ on $\Omega$) into itself. In the present paper, the $p \times p$ matrices ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ for which this mapping is onto are characterized in terms of the zeros of the determinant of ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ when the ${f_{ij}}$ are infinitely differentiable on $\Omega \subset {R^1}$ and when the ${f_{ij}}$ are $m$-times continuously differentiable on $\Omega \subset {R^n}$. Finally, partial results are obtained when the ${f_{ij}}$ are infinitely differentiable on $\Omega \subset {R^n}$ and extensions are made to $p \times q$ systems of division problems for distributions.
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 155 (1971), 493-504
  • MSC: Primary 46F10; Secondary 58C25
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0415310-8
  • MathSciNet review: 0415310