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Russian Mathematical Surveys, 1973, Volume 28, Issue 1, Pages 69–140
DOI: https://doi.org/10.1070/RM1973v028n01ABEH001397
(Mi rm4835)
 

This article is cited in 42 scientific papers (total in 42 papers)

$J$-expanding mtrix functions and their role in the analytical theory of electrical circuits

A. V. Efimov, V. P. Potapov
References:
Abstract: Chapter I establishes the essential properties of the $\mathscr A$-matrix of a passive multipole depending on the number of its branches. These properties are based on Langevin's theorem. A classification of the basic objects of investigation:$J$-expanding matrix-functions (class $\mathfrak M$), and also positive matrix functions (class $\mathfrak B$ ), is introduced. Chapter II gives an account of a theory of matrix functions of class $\mathfrak M$. It also investigates the simplest (elementary and primary) matrices of this class. The fact is established that elementary (and primary) factors can be split off from a given matrix of class $\mathfrak M$. In particular, the factorizability of a rational reactive matrix of class $\mathfrak M$ is established.
Chapters III–IV set forth a theory of various subclasses of matrix functions of class $\mathfrak M$: $\mathfrak M_{sl}$, $\mathfrak M_{cgl}$, $\mathfrak M_{lr}$. The realizability of the matrix functions of each of these subclasses as $\mathscr A$-matrices of passive multipoles with the corresponding provision for branches is established.
The fact that they are realizable is proved by the construction of a corresponding multipole.
The last chapter is concerned with a generalization of Darlington's theorem, which leads to a realization of functions of the subclasses $\mathfrak M_{clr}$ and $\mathfrak M_{cglr}$ as $\mathscr A$-matrices or $z$-matrices of dissipative multipoles.
Bibliographic databases:
Document Type: Article
UDC: 519.53+512.83
MSC: 15A48, 15A15, 15A23
Language: English
Original paper language: Russian
Citation: A. V. Efimov, V. P. Potapov, “$J$-expanding mtrix functions and their role in the analytical theory of electrical circuits”, Russian Math. Surveys, 28:1 (1973), 69–140
Citation in format AMSBIB
\Bibitem{EfiPot73}
\by A.~V.~Efimov, V.~P.~Potapov
\paper $J$-expanding mtrix functions and their role in the analytical theory of electrical circuits
\jour Russian Math. Surveys
\yr 1973
\vol 28
\issue 1
\pages 69--140
\mathnet{http://mi.mathnet.ru//eng/rm4835}
\crossref{https://doi.org/10.1070/RM1973v028n01ABEH001397}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=394287}
\zmath{https://zbmath.org/?q=an:0268.94009|0285.94009}
Linking options:
  • https://www.mathnet.ru/eng/rm4835
  • https://doi.org/10.1070/RM1973v028n01ABEH001397
  • https://www.mathnet.ru/eng/rm/v28/i1/p65
  • This publication is cited in the following 42 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:856
    Russian version PDF:360
    English version PDF:25
    References:70
    First page:1
     
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