Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.
Title: Subforms of norm forms of octonion fieldsJournal: Archiv der Mathematik, Volume 110, Issue 3, Pages 213-224, March 2018
Authors: Norbert Knarr, Markus J. Stroppel
We characterize the forms that occur as restrictions of norm forms of octonion fields. The results are applied to forms of types E6, E7, and E8 and to positive definite forms over fields that allow a unique non-split octonion algebra, e.g., the field of rational numbers.
Title: Stability of quaternion-valued impulsive delay difference systems and its application to neural networks
Journal: Neurocomputing, Volume 284, Pages 63-69, April 2018
Authors: Jingwen Zhu, Jitao Sun
In this paper, we make first attempt to investigate exponential stability for quaternion-valued impulsive delay difference systems. By employing Lyapunov methods and quaternion-modulus inequality technique, sufficient conditions for the exponential stability are presented. In addition, as a subsequent result, the obtained theory is successfully applied to study a class of discrete-time quaternion-valued impulsive neural networks with delay. Finally, a numerical example is given to illustrate the effectiveness of theoretical results.
Title: Principle of transference – An extension to hyper-dual numbers
Journal: Mechanism and Machine Theory, In Press, Available online 27 December 2017
Authors: Avraham Cohen, Moshe Shoham
The algebra of hyper-dual numbers and hyper-dual vectors of order n, developed in this paper, follows the same rules as those of dual numbers and dual vectors. By showing that the basic formulae of vectors scalar and vector multiplication, hold for dual vectors of order n and that the basic trigonometric formulas hold for dual angles of order n, we concluded, that all formulae of vector algebra and trigonometric functions that are based on the above identities also hold for dual numbers of order n. This, as a result, extends Kotelnikov’s “principle of transference” developed for dual numbers, to hyper-dual numbers of order n.
Source: Email from N. Valous, email@example.com, 16 Feb. 2018.