Orientable double cover

Given a space X with an orientation double covering... 

The relative -j-groups <5, (X,Y) of a pair of spaces (X,Y) (equipped with an orientation double covering w) are defined to fit into a commutative diagram of abelian groups with exact rows and columns... 

Spontaneous aggregation of lipids — double chain amphiphiles — in aqueous medium (smectic phase). Hydrophobic core of long alkyl chains (lamellar orientation) is covered by hydrophilic headgroups (see Figs. 2, 15). 

The transformation from P to P is therefore not an identity transformation, but rather an involution, with P and F as conjugate points. The identity transformation corresponds to a double rotation of 27t along the Mobius surface. The two sides of the paper corresponds to a double covering of the non-orientable topological Mobius surface. 

Figure 5.4 signifies more than elemental or nuclide periodicity. It summarizes the appearance of ponderable matter in all modifications throughout the universe. Following the extended hemlines from top left at Z/N = 1.04 — bottom left at 0 —> top right at Z/N = 1.04 bottom right at 0, and back to top left, the involuted closed path, which is traced out, is mapped to the non-orientable surface of a Mobius band in Figure 5.7. The two sides of the double cover are interpreted to represent both matter and antimatter.

The complete process for the fabrication of the proton-conducting membranes, previously reported in [69,73], can be described as follows. A 4-inch 520 /itm thick n+ (100)-oriented double-side polished silicon wafer is first thermally oxidized in an oven at 1000 °C under O2 and water steam flows to obtain a 2 nm thick Si02 layer on both sides of the substrate. These layers will allow the electrical insulation between the two parts of the future fuel cell. Then these previous layers are covered with sputtered Cr-Au layers on both sides. The Cr layers are used as adherence layers for the Au layers and are relatively thin (30 nm). The Au layers are 800 nm thick and will serve as current collector layers for the fuel cells. They are also useful as masking layers for the next different etchings since Au is not etched neither by KOH solution nor by HF solution, the two wet etchants used in the next steps. 

So far no account has been taken of stress distributions. The experimental evidence, de.scribed in the 3rd paper in this volume (fig. 3), is that there is ductile fracture with a crack which progressively opens into a V-notch until catastrophic failure occurs when the notch covers about half the fibre cross-section. If there is a defect, usually on the surface but sometimes internally (when the V-notch becomes a double cone), the stress concentration will lead to the start of the rupture, although it has a negligible effect on the mean fibre stress at which this occurs. If there is no defect, the evidence is that an initial crack will form by a coalescence of voids that form under high stress. Variation in the degree of orientation across a fibre may well play a part. If the skin of the fibre is more highly oriented, it will reach its limiting extension before the core. 

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