9 research outputs found
Universal valued Abelian groups
The counterparts of the Urysohn universal space in category of metric spaces
and the Gurarii space in category of Banach spaces are constructed for
separable valued Abelian groups of fixed (finite) exponents (and for valued
groups of similar type) and their uniqueness is established. Geometry of these
groups, denoted by G_r(N), is investigated and it is shown that each of
G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are
Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are
studied and it is proved that every separable metrizable topological vector
space may be enlarged to G_r(0) with a `linear-like' structure which extends
the linear structure of the given space.Comment: 60 page
Hodge and Gelfand theory in Clifford analysis and tomography
2022 Summer.Includes bibliographical references.There is an interesting inverse boundary value problem for Riemannian manifolds called the Calder贸n problem which asks if it is possible to determine a manifold and metric from the Dirichlet-to-Neumann (DN) operator. Work on this problem has been dominated by complex analysis and Hodge theory and Clifford analysis is a natural synthesis of the two. Clifford analysis analyzes multivector fields, their even-graded (spinor) components, and the vector-valued Hodge鈥揇irac operator whose square is the Laplace鈥揃eltrami operator. Elements in the kernel of the Hodge鈥揇irac operator are called monogenic and since multivectors are multi-graded, we are able to capture the harmonic fields of Hodge theory and copies of complex holomorphic functions inside the space of monogenic fields simultaneously. We show that the space of multivector fields has a Hodge鈥揗orrey-like decomposition into monogenic fields and the image of the Hodge鈥揇irac operator. Using the multivector formulation of electromagnetism, we generalize the electric and magnetic DN operators and find that they extract the absolute and relative cohomologies. Furthermore, those operators are the scalar components of the spinor DN operator whose kernel consists of the boundary traces of monogenic fields. We define a higher dimensional version of the Gelfand spectrum called the spinor spectrum which may be used in a higher dimensional version of the boundary control method. For compact regions of Euclidean space, the spinor spectrum is homeomorphic to the region itself. Lastly, we show that the monogenic fields form a sheaf that is locally homeomorphic to the underlying manifold which is a prime candidate for solving the Calder贸n problem using analytic continuation
Notes in Pure Mathematics & Mathematical Structures in Physics
These Notes deal with various areas of mathematics, and seek reciprocal
combinations, explore mutual relations, ranging from abstract objects to
problems in physics.Comment: Small improvements and addition