We investigate superdifferentiability of functions defined on regions of the real octonion (Cayley) algebra and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of an octonion variable to be analytic. In particular, the octonion exponential and logarithmic functions are being considered. Moreover, superdifferentiable functions of variables belonging to Cayley-Dickson algebras (containing the octonion algebra as the proper subalgebra) finite and infinite dimensional are investigated. Among main results there are the Cayley-Dickson algebras analogs of Caychy's theorem, Hurtwitz', argument principle, Mittag-Leffler's, Rouche's and Weierstrass' theorems.
We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of a quternion variable to be analytic. In particular, the quaternionic exponential and logarithmic functions are being considered. Main results include quaternion versions of Hurwitz' theorem, Mittag-Leffler's theorem and Weierstrass' theorem.
In this article, nonassociative metagroups are studied. Different types of smashed products and smashed twisted wreath products are scrutinized. Extensions of central metagroups are studied.
Quasi-invariant and pseudo-differentiable measures on a Banach space X over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in R. Theorems and criteria are formulated and proved about quasi-invariance and pseudo-differentiability of measures relative to linear and non-linear operators on X. Characteristic functionals of measures are studied. Moreover, the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems are investigated. Infinite products of measures also are considered. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.
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