508,549 research outputs found
On the Four-Color-Map Theorem
Coloring planar Feynman diagrams in spinor quantum electrodynamics, is a non
trivial model soluble without computer. Four colors are necessary and
sufficient.Comment: 8 page
A factorization of a super-conformal map
A super-conformal map and a minimal surface are factored into a product of
two maps by modeling the Euclidean four-space and the complex Euclidean plane
on the set of all quaternions. One of these two maps is a holomorphic map or a
meromorphic map. These conformal maps adopt properties of a holomorphic
function or a meromorphic function. Analogs of the Liouville theorem, the
Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem,
the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and
branch points of a super-conformal map are obtained.Comment: 21 page
Absolute Continuity Theorem for Random Dynamical Systems on
In this article we provide a proof of the so called absolute continuity
theorem for random dynamical systems on which have an invariant
probability measure. First we present the construction of local stable
manifolds in this case. Then the absolute continuity theorem basically states
that for any two transversal manifolds to the family of local stable manifolds
the induced Lebesgue measures on these transversal manifolds are absolutely
continuous under the map that transports every point on the first manifold
along the local stable manifold to the second manifold, the so-called
Poincar\'e map or holonomy map. In contrast to known results, we have to deal
with the non-compactness of the state space and the randomness of the random
dynamical system.Comment: 46 page
G. Birkhoff's problem in irreversible quantum dynamics
We study operator spaces determined uniquely by a completely positive map and
find a complete invariance upto cocycle conjugacy for an extremal element in
the convex set of unital trace preserving completely positive map on matrix
algebra over the field of complex numbers. We prove a Hann-Banach-Arveson's
type of theorem on operator spaces which made it possible to use dynamical
systems approach to arrive at our main result via P. Jordan's theorem on order
isomorphisms on algebras.Comment: This paper has been withdrawn by the author as it needs substantial
changes and some end results are wrong in its present for
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