Circle-packing representation

Given a map M, its circle-packing representation (see [Moh97]) is a set of disks on a Riemann surface E of constant curvature, one disk D(v, rv) for each vertex v of M, such that the following conditions are fulfilled ... 

Simultaneous circle-packing representations of a map M and its dual M are called primal-dual circle representation of M if it holds ... 

A map M is called reduced (see [Moh97, Section 3]) if its universal cover is 3-connected and is a cell-complex. It is shown in [Moh97, Corollary 5.4] that reduced maps admit unique primal-dual circle packing representations on a Riemann surface of the same genus moreover, a polynomial time algorithm allows one to find the coordinates of those points relatively easily. This means that the combinatorics of the map determines the structure of the Riemann surface. 

FIG. 5 Schematic representation of packing arrangements of natural amphipathic double-chain lipids with different headgroup size in crystalline bilayers. The small filled circles indicate the accommodation of spacer molecules, such as water or ions. (Reprinted by permission from Ref 14, copyright 1992, Elsevier Science.)... 

Figure 2.36(a) Schematic representation of the incommensurate close-packed overlayer of Cu on Au formed in the perchlorate electrolyte The open circles are the gold atoms. Only part of the monolayer is shown in order to exhibit the overlayer-underlayer orientation, (b) Schematic representation of the more open lattice formed in the sulphuric acid electrolyte. From Manne... 

Fig. 4 Schematic representation of possible packing of dye (D, bars) molecules containing long aliphatic groups into the J-aggregates in lipid (L, circles) layer, presented as seen perpendicular top) and along bottom) the lipid layer surface...

Figure 22. Packing arrangement of 7-methoxycoumarin in the crystals—a unit cell representation. Note the two reaction centers (darkened circles) are twisted with respect to each other.

Terminal amino groups are believed to comprise adsorption sites on the surface of amino-phase packings (77). These (1) Eire exposed for direct interaction with adsorbate molecules, (2) are not rigidly positioned on the surface, and (3) have a much lower surface concentration (2-3 /xmol/m ). These characteristics of the various LSC packings so far discussed are summarized in the surface representations shown in Fig. 14 for each adsorbent. The m or adsorption sites in each case are indicated by enclosure within a circle (except the vacancy site for alumina, shown as an asterisk in Fig. 14). 

Fig. 6.7 Representation of the packing of the two forms of 6-VII. Oxygen atoms are denoted by solid circles, (a) The monochnic a form (b) the orthorhombic p form. (From Dromzee 1996, with permission.)...

Figure 32. Space-filling stereoscopic representation of the molecular packing of 33 (P2,2t21) with (001) facing down the more important half-boat conformer out of the 60/40-disorder due to the cyclohexene ring is depicted exclusively 16-unit cells are shown double bond-C with grid, O with circles, N with circled dots.

Figure 2. Schematic representation of principal surface sorption processes, (a) Projection of the CoOOH structure in the ab plane. MSC, multinuclear surface complexation represented by an epitaxy of a-FeOOH (left) ISC, mononuclear monodentate (middle right), mononuclear bidentate (top right) and binuclear bidendate (lower right) inner-sphere complexation OSC, outer-sphere surface complexation (top) LD, lattice diffusion (center), (b) Example of epitaxy without sharing of oxygens (Van der Waals forces). The. ..AB-AB... close-packed anionic layer sequence of Co(OH)2(s) is coherently stacked on the. ..AB-BC-CA... layer sequence of CoOOH. Co(OH)2(s) has a 1H polytypic structure, and CoOOH a 3R. Small circles are...

Figure 101. (A) Packing arrangement in [Cu(TCNPP) Cu]n framework. Separate frameworks are indicated by iight and dark shading. Large circies represent copper ions (B) Schematic representation of extended framework. Cu(ll) ions in center of porphyrin macrocycle represented by small circles. Cu(l) ions coordinating between porphyrin molecules indicated by large circles, Solvate and noncoordinating anions omitted for clarity. Reprinted with permission from Abrahams, B. F. Hoskins, B. F. Michali, D, M. Robson, R. Nature 1994, 369, 727. 1994 Me Millan Publishers Ltd.

Figure 6. Schematic representation of the adsorption of silane coupling agent molecules onto glass fiber surface. A) The silane solution is a low concentration below the onset of association, and B) the silane solution is a high concentration above the onset of association. Open cirde indicates isolated silanes whereas the closed circle denotes hydrogen bonded or oligomeric silanes. Note the disturbance in packing is created at a high concentration.

Fig. 2.11 a Cartoon representation of conjugated polymer film, and its three possible charge transport pathways b Traditional polymers contain one or two alkyl chains on each unit. Steric hindrance may exist when polymers stacked with each other (black circle)-, c Molecular docking strategy to avoid steric hindrance and improve the inter-pol5nner ti-ti interaction. Cartoon representation of the copolymers with different symmetries and their film packings d polymers with centrosymmetric donors and e polymers axisymmetric donors... 

Figure 2 Helical net representation of the synthetic four-helix bundle designed by Kamtekaret al. Polar (P) residues shown with white circles are Lys, His, Glu, Gin, Asp, or Asn. Hydrophobic (H) residues shown with dark circles are Met, Leu, He, Val, or Phe. Helices are bounded with glycine residues for flexibility, to facilitate close packing with different interhelical angles, and terminated with proline as a helix breaker. The broken gray line points to a possible salt bridge. See Figure 4 for a structural model of a four-helix bundle...

FIGURE 8.18 Phase diagram of dipolar hard-sphere particles in the (dipole moment strength y, packing fraction Tj) representation. The circles denote points where the phase boundary was determined and the gray areas denote coexistence regions (where tie lines are vertical). (From Hynninen AP and Dijkstra M. 2005. Physical Review Letters 94 138303. With permission.)... 

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