In this paper the disc 2 is characterized as the only connected, simply connected domain 3 B with the following property, C 3 : B will cover (by an isometry) any subset P of the plane provided B will cover each 3 points of P.The disc has property C3. For a plane set P can be covered by a p-disc if and only if the members of the family F of p-discs with centers in P have a common point. If now each three points of P are on a p-disc then each three discs of F intersect and by a theorem on convex bodies due to E. Helly 4 there is a point common to all the discs of F.
LEMMA. A bounded, closed subset of the plane contains a largest circle.The proof is accomplished by selecting a sequence of circles from the set whose centers converge to a point and whose radii converge to the least upper bound of the radii of circles in the set and (using the fact that the set is closed) showing that the limiting circle belongs to the set.THEOREM. The disc is the only connected, simply connected domain with property C3. PROOF. We assume B to be the given domain, and show that B 1 Part of a Ph.D. dissertation at University of Missouri, under L. M. Blumenthal, 1940. 2 The disc of radius p and center p is the set of points x of £2 such that px ^ p. 3 Closure of a bounded open subset of JS2. 4 Theorem. If each w + 1 sets of a family of bounded, closed, convex subsets of E n intersect, there is a point common to all the sets, Jahrbuch der Deutschen Mathe-
This paper reviews the status of hollow cathode sputtering as an evolving technology for production of thin-film transparent conducting oxides for PV applications. A large market segment for PV TCOs is represented by thin-film a-Si:H and tandem a-Si:H/nc-Si:H modules. For superstrate devices, textured SnO 2 :F produced on-line by APCVD is currently the market leader, although alternative off-line methods and materials are now emerging. In particular, zinc oxide can be produced by LPCVD, APCVD, magnetron sputtering, and hollow cathode sputtering (HCS). HCS is a stable process featuring low-cost metal targets and a soft deposition process. We discuss the deposition principles and the film results obtained using linear hollow cathodes 0.5 m and 1.0 m in length. We report the direct deposition of highly textured doped ZnO having an electron mobility in excess of 50 cm 2 /Vs. The production cost of textured ZnO is estimated for several competing techniques.
Sophisticated technologies of scientific visualization often require a departure from the standards of mimetic representation. In this paper the authors introduce a set of nine papers derived from the conference on scientific visualization in Norrköping, Sweden in September 2012, which explore problems of scale, color and technology in scientific visualization. These three kinds of problems are common to multiple visualization methods. As a result, this collection constitutes a preliminary exploration of commonalities in various methods of visualization, e.g., from nanoscale images to outerspace pictures of galaxies.
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