By means of the vector calculus, it is proved that the magnitude, orientation, and location of the resultant dipole of a system of sources and sinks inside a finite volume conductor is given by an integration over the bounding surface. The method is applied to finding the ``heart vector,'' or the resultant dipole moment of the human heart. The theory was checked in two- and three-dimensional electrolytic tank models of the human thorax.
We construct a homotopy invariant appropriate for studying the existence of coincidence points of Fredholm operators of nonnegative index and multivalued admissible maps. Cohomotopy methods are used as a more suitable tool than homological ones. Both finite and infinite dimensional cases are investigated.
We study a coincidence problem of the form A(x) ∈ φ(x), where A is a linear Fredholm operator with nonnegative index between Banach spaces and φ is a multivalued A-fundamentally contractible map (in particular, it is not necessarily compact). The main tool is a coincidence index, which becomes the well known Leray-Schauder fixed point index when A = id and φ is a compact singlevalued map. An application to boundary value problems for differential equations in Banach spaces is given.2000 Mathematics Subject Classification: 55M20, 34G20, 47H11, 34B15.
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