Christoph Schwarzweller (2012)
Formalized Mathematics
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In this article we formalize rational functions as pairs of polynomials and define some basic notions including the degree and evaluation of rational functions [8]. The main goal of the article is to provide properties of rational functions necessary to prove a theorem on the stability of networks
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...
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.
Tomáš Kepka, Petr Němec (1981)
Czechoslovak Mathematical Journal
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Andrew Wells (2010)
Commentationes Mathematicae Universitatis Carolinae
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In “A class of simple Moufang loops”, Proc. Amer. Math. Soc. (1956), 471–482, Paige used the vector matrix construction over fields to produce simple Moufang loops. The purpose of this paper is to generalize the construction to the class of commutative rings, and examine the Moufang loops arising in this fashion. Specific attention is paid to the construction over the ring of integers modulo four.