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The Dirac equation for spinor-valued fields on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet of the hyperboloid. In particular, we derive an integral formula expressing the value of in a chosen point as an integral over a compact cycle given by the intersection of the null cone with in the Minkowski space .
Integrals of the Cauchy type extended over the boundary of a general compact set in the complex plane are investigated. Necessary and sufficient conditions on are established guaranteeing the existence of angular limits of these integrals at a fixed for all densities satisfying a Hölder-type condition at .
The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity satisfies
are considered on these boundaries. Functions satisfying the Hölder condition of order , , belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise...
The survey collects many recent advances on area Nevanlinna type classes and related spaces of analytic functions in the unit disk concerning zero sets and factorization representations of these classes and discusses approaches, used in proofs of these results.
In the last years the mapping properties of the Cauchy integralCΓf(z) = 1/(2πi) ∫Γ [f(ξ) / ξ - z] dξhave been widely studied. The most important question in this area was Calderón's problem, to determine those rectifiable Jordan curves Γ for which CΓ defines a bounded operator on L2(Γ). The question was solved by Guy David [Da] who proved that CΓ is bounded on L2(Γ) (or on Lp(Γ), 1 < p < ∞) if and only if Γ is regular, i.e.,H1(Γ ∩ B(z0,R) ≤ CRfor every z0 ∈ C, R > 0 and for...
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