Torus, tori

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

Wang J, Sorensen P G and Flynne F 1994 Transient period doublings, torus osoillations and ohaos in a olosed ohemioal system J. Phys. Chem. 98 725-7... 

Torus (obtained by rotating a circle of radius / about a line whose distance is ft > / from the center of the circle)... 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

On the expander side, the expander wheel is surrounded by the nozzle vanes. The nozzle vanes, in turn, reeeive gas from a toroidal spaee that is eonneeted to tlie expander inlet piping. Any non-uniformity in the torus spaee and/or in the nozzle vane design may result in a non-uniform pressure distribution around the expander wheel. Non-uniform gas pressure around the expander wheel will result in a non-uniform load and, henee, produee a gas dynamie radial load on the bearing. In the expander ease, however, the nozzle throat flow resistanee is mueh larger than the easing peripheral pressure nonuniformity. The latter aets as a buffer making the expander wheel eireumferential pressure variations smaller than those of the eompressor side. This smaller pressure variation produees mueh less radial load when eompared to that of the eompressor side. 

Under the hghtest winds, the air rises over the warmest part of the urban core, drawing cooler air from all directions from the surroundings (Fig. 17-21). Subsidence replaces this air in rural areas, and a closed torus (doughnut)-shaped circulation occurs with an outflow above the urban... 

There are two major types of elastomer compression couplings. One is the torus type in which the elastomer is attached directly to the coupling hubs (see Figure 9-18). The other is the compression type with the elastomer held in compression by the hub geometry (see Figure 9-19). 

Figure 9-16. Torus type coupling. Courtesy of Falk Corporatioii ...

Key Words —Carbon, molecular dynamics, torus, helix, graphitic forms. 

Here, the topological nature of the tori will be discussed briefly. Figure 1 shows the five possible prototypes of toroidal forms that are considered to be related to fullerenes. These structures are classified by the ratios of the inner and outer diameters r, and r, and the height of the torus, h. (Note that is larger than / ,) As depicted in Fig. 1, if r, = r, and h r, and h = — r,) then the toroidal forms are of type... 

Fig. 1. Five possible simple prototypes of the toroidal forms of graphitie carbon. All cross-sections of the tube are square. Here r , r,, and h are the outer and inner radii and height of the torus, respectively.

Fig. 2. Optimized toroidal structures of Dunlap s tori (a) torus C540 and (b) torus 05, pentagons and heptagons are shaded.

Fig. 3. Pentagon-heptagon transformation (a) five-fold rotational surface of C , (b) negatively curved surface created by pentagon-heptagon transformation (c) a part of the remaining surface in creating the torus.

The relaxed structures of the various (rotational) symmetric toroidal forms were obtained by steepest decent molecular-dynamics simulations[15]. For the elongated tori derived from torus C240, the seven-fold rotational symmetry is found to be the most stable. Either five-fold or six-fold rotational symmetry is the most stable for the toroidal forms derived from tori Cjyo and C540, respectively (see Fig. 5). 

Because the cohesive energy of the fullerene Cyo is —7.29 eV/atom and that of the graphite sheet is —7.44 eV/atom, the toroidal forms (except torus C192) are energetically stable (see Fig. 5). Finite temperature molecular-dynamics simulations show that all tori (except torus Cm2) are thermodynamically stable. 

Fig. 4. Optimized toroidal structures (a) torus and (b) torus C24o Pentagons and heptagons are shaded. The diameters of the tube of the stable torus determined by optimization using molecular dynamics with Stillinger-Weber poiemial[211, is 8.8 A. The diameter of the hole is 7.8 A, which is quite close to the diameter of fullerence Qy.

Elongated tori. The experiments, at the present time, suggest that the torus of type (D) with parallel fringes at a separation of 3.7 A, such as C240, is likely to exist. Thus, the type (C) structures having height of 3.7 A could exist. See Fig. 6. 

Fig. 6. Part of the elongated torus here, 2. sad L are the number of the hexagons along the inner circle, outer circle, and height of the torus, respectively this figure is for the case of i = 12, = 6, and L =. ...

Fig. 8. Optimized shallow toroidal structures the subscripts indicate the number of the carbon atoms in the torus pentagons and heptagons are shaded.

Table I. Cohesive energies of shallow tori the parameters n, and ii, are the number of hexagons along the outer and inner eirele, respeetively (see Fig. 6). Here Af is the tiumber of atoms in a torus...

The properties of optimized helical structures, which were derived from torus C54D and Cs7a, >yps (A), (proposed by Dunlap) and torus C ,o> Dpe (B), (proposed by us) by molecular dynamics were compared. (see Figs. 9 (a) and 10). (Although the torus Cs7f, is thermodynamically stable, helix 57 was found to be thermodynamically unstable 14]. Hereafter, we use helix C to denote a helix consisting of one torus (C ) in one pitch. 

Fig. 10. Ffelically coiled form 540 one pilch contains a torus...

From elongated tori, such as type (C), type (D), and type (E), helical structures are derived. For example, from the type (C) elongated torus of mentioned in 3.2.2, helix C756 (/t = 6, /t2 = 3, L = 1) and... 

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