Krzysztof Treyderowski, Christoph Schwarzweller (2006)
Formalized Mathematics
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In this article we define the Discrete Fourier Transformation for univariate polynomials and show that multiplication of polynomials can be carried out by two Fourier Transformations with a vector multiplication in-between. Our proof follows the standard one found in the literature and uses Vandermonde matrices, see e.g. [27].
Christoph Schwarzweller, Agnieszka Rowińska-Schwarzweller (2006)
Formalized Mathematics
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A complex polynomial is called a Hurwitz polynomial if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical networks.In this article we prove Schur's criterion [17] that allows to decide whether a polynomial p(x) is Hurwitz without explicitly computing its roots: Schur's recursive algorithm successively constructs polynomials pi(x) of lesser degree by division with x - c, ℜ {c} < 0, such that pi(x) is...
Agnieszka Rowinska-Schwarzweller, Christoph Schwarzweller (2013)
Formalized Mathematics
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A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p)...
Tomáš Kepka, Petr Němec (1981)
Czechoslovak Mathematical Journal
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Tomáš Kepka, Petr Němec (1990)
Acta Universitatis Carolinae. Mathematica et Physica
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Benard M. Kivunge, Jonathan D. H Smith (2004)
Commentationes Mathematicae Universitatis Carolinae
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This note investigates sedenion multiplication from the standpoint of loop theory. New two-sided loops are obtained within the version of the sedenions introduced by the second author. Conditions are given for the satisfaction of standard loop-theoretical identities within these loops.