K-quasiderivations

Caleb Emmons; Mike Krebs; Anthony Shaheen

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 824-834
  • ISSN: 2391-5455

Abstract

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A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss several interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extending a previous result. In particular, we show that any such K-quasiderivation must be linear over R. We then discuss two previously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K-quasiderivations which includes the ordinary derivative as a special case.

How to cite

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Caleb Emmons, Mike Krebs, and Anthony Shaheen. "K-quasiderivations." Open Mathematics 10.2 (2012): 824-834. <http://eudml.org/doc/269700>.

@article{CalebEmmons2012,
abstract = {A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss several interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extending a previous result. In particular, we show that any such K-quasiderivation must be linear over R. We then discuss two previously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K-quasiderivations which includes the ordinary derivative as a special case.},
author = {Caleb Emmons, Mike Krebs, Anthony Shaheen},
journal = {Open Mathematics},
keywords = {K-quasiderivation; Polynomial ring; Derivation system},
language = {eng},
number = {2},
pages = {824-834},
title = {K-quasiderivations},
url = {http://eudml.org/doc/269700},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Caleb Emmons
AU - Mike Krebs
AU - Anthony Shaheen
TI - K-quasiderivations
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 824
EP - 834
AB - A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss several interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extending a previous result. In particular, we show that any such K-quasiderivation must be linear over R. We then discuss two previously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K-quasiderivations which includes the ordinary derivative as a special case.
LA - eng
KW - K-quasiderivation; Polynomial ring; Derivation system
UR - http://eudml.org/doc/269700
ER -

References

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  13. [13] Nöbauer W., Derivationssysteme mit Kettenregel, Monatsh. Math., 1963, 67(1), 36–49 http://dx.doi.org/10.1007/BF01300680 Zbl0107.02902
  14. [14] Stay M., Generalized number derivatives, J. Integer Seq., 2005, 8(1), #05.1.4 
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  16. [16] Westrick L., Investigations of the number derivative, preprint available at http://www.plouffe.fr/simon/OEIS/archive_in_pdf/intmain.pdf Zbl06349659

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