1,456 research outputs found
Differentiable functions of quaternion variables
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 quaternion 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.Comment: 48 pages, Late
Clifford structures on Riemannian manifolds
We introduce the notion of even Clifford structures on Riemannian manifolds,
a framework generalizing almost Hermitian and quaternion-Hermitian geometries.
We give the complete classification of manifolds carrying parallel even
Clifford structures: K\"ahler, quaternion-K\"ahler and Riemannian products of
quaternion-K\"ahler manifolds, several classes of 8-dimensional manifolds,
families of real, complex and quaternionic Grassmannians, as well as
Rosenfeld's elliptic projective planes, which are symmetric spaces associated
to the exceptional simple Lie groups. As an application, we classify all
Riemannian manifolds whose metric is bundle-like along the curvature constancy
distribution, generalizing well-known results in Sasakian and 3-Sasakian
geometry.Comment: Final version, 28 page
Horizontal variation of Tate--Shafarevich groups
Let be an elliptic curve over . Let be an odd prime and
an embedding. Let
be an imaginary quadratic field and the corresponding Hilbert class
field. For a class group character over , let be
the field generated by the image of and the prime
of above determined via . Under mild
hypotheses, we show that the number of class group characters such that
the -isotypic Tate--Shafarevich group of over is finite with
trivial -part increases with the absolute value of the
discriminant of
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