2-periodic plane graph

Data for the bulk fluid, line A, indicate that vz varies as a function of z but maintains a value near 0.75 of maximum velocity. The periodicity of vx and vy is clearly evident in the graph of line A and a 1800 out of phase coupling of the components is seen with one positive when the other is negative. This indicates a preferred orientation to the plane of the oscillatory flow and this feature was seen in all the biofilms grown throughout this study. The secondary flow components are 0.1-0.2 of the maximum axial velocity and are spatially oscillatory. The significant non-axial velocities indicate non-axial mass transport has gone from diffusion dominated, Pe = 0, in the clean capillary, to advection dominated, Pe 2 x 103, due to the impact of the biofilm. For comparison, the axial Peclet number is Pe L 2x 10s. Line B intersects areas covered by biomass and areas of only bulk... 

A closed torus M can be represented as a 3-dimensional figure projected on to the plane, but this is not very practical. We represent its universal cover M as a plane having two periodicity directions, i.e. a 2-periodicplane graph. The group ni(M) is isomorphic to Z2 and it is represented on M as a group of translation symmetries. 

Fig. 8.5. Graph of head movement occurring during a functional MRI scan. The graph shows translations in the x, y and z planes during the collection of 18 dynamic images over a period of just under 2 min...

Fig. 3 Radial density in the iy-plane of H2 and the second period hydrides. The central atom is located at (0, 0). The hydrogen nuclei are clearly identified by the distinct depressions (dimples) in the graphs...

Figure 7.6 Field intensity calculation by 2-D FDTD method and examples are as shown in [a-e]. The long axis of the nanobars is normal to the image plane this is an end-on view of the long structures positioned on a Si substrate, with a layer of aluminum oxide between the substrate and the silver bars, and water filling the top portion of the frame. Bright areas show the positions of high local electric field, (f, g) Summary plots show the average electric field intensity 8 nm above the surface (at the location of the dashed lines in (a-e]], compared with fluorescence enhancement along two different polarizations. Excitation polarization compared to bar orientation is indicated by the illustrations within the graphs. Intensity plots (a-e] represent the visualization of calculations for points noted by the labels within graphs (f-g], at periods of 316, 1028, 316, 512, and 1028 nm, respectively. Error bars represent the standard deviation for several measurements of different samples [24].

On the plane (/ii,/i2), there exist two curves Li and L2, corresponding to the homoclinic loops of the saddles 0 and O2 respectively. These curves are the graphs of some smooth functions /ii = hi(/X2) and /X2 = 2(/ i)> respectively, which are defined for positive /x and such that h(0) = 0, h (0) = 0. A stable periodic orbit exists in the region between L and L2- Note that multi-circuit homoclinic loops cannot appear when the saddle values are negative. 

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