V-vertices

Figure 7 The production and emission of NO during denitrification in agricultural soil treated with NO3 fertilizer (KNO3) and the nitrification inhibitor Dyciandiamide (10%) under aerobic (air) and anerobic conditions (N,). Fluxes are means from three soil columns, error bars represent standard deviations from the mean. V = vertical flow through the column H = Horizontal flow over the soil surface.

We consider free centric C-H trees with n vertices of degree 4, that is, with a total of 3 + 2 points [Sec. 36(a)]. Some branch with V vertices of degree 4 has a total of 3v + 1 nodes [Sec. 36(a)]. Thus, we have for a branch originating in the center... 

We consider finite graphs consisting of V vertices connected by B bonds. The V xV connectivity matrix will be denoted by Ct/J Ct/J = 1 when the vertices i and j are connected, and it vanishes otherwise. The bonds are endowed with the standard metric and the length of... 

Minimum oxygen concentration needed for burning at 20°C. h V. vertical burning HB, horizontal burning. 

V] = vertical load from resistance of roof members plus slab dead weight = [(0.5 psi 144 m2/ft2) + (0.5 ft siab)(150 pcf)] (50 ft width)... 

Unless specified, all water is untreated, brackish, bay or sea. Notes H = horizontal, fixed or floating tube sheet, U = U—tube horizontal bundle, K = kettle type, V = vertical, R = reboiler, T=thermosiphon, v = variable, HC = hydrocarbon, C) = cooling range At, (Co) = condensing range At. 

The ( 4,6), 3)-spheres were considered in a chemical setting in [GaHe93]. Theorem 1 of [GrMo63] gives that a ( 4,6), 3)-sphere with v vertices exists only for any even v > 8, except v = 10. 

A ( 2,3), 6)-sphere with v vertices exists for any v > 2 and the proof (in the same spirit as for other parabolic classes) is given below ... 

Theorem 2.0.1 For any v >2, there exists a ( 2,3, 6)-sphere with v vertices. Proof. Consider the regular tiling 3,6 and take the doubly infinite path / of vertices lying on a a straight line in 3,6). If we take another parallel line V at distance t, then l and l bound a domain D, in 3,6). 

Define a 3 -fullerene as a 3-valent map embedded on a surface and consisting of only 5-gonal and 6-gonal faces. Each such object has, say, v vertices, e edges, and / faces of which ps are 5-gons and pf, are 6-gons. 

Note that at least one spherical fullerene with v vertices exists for all even v with v > 20, except for the case v = 22 ([GrMo63]). 

Thus, the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Q, C, h, (m even), Dmh (m even), Dmd (m odd), 7, Oh, and 7. A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list C,-, C2h, Dm, Da, D3d, Du, 7, and /. Clearly, a fullerene with v vertices can be centrally symmetric only if v is divisible by four as p6 must be even. After the minimal case v = 20, the first centrally symmetric fullerenes are at v = 32 (Dm) and v = 36 (Dm). 

It looks too hard to describe all ( 3,4, 4)-spheres that are 3Rq for example, the medial graph of any ( 3,4, 4)-sphere with v vertices is a ( 3,4, 4)-sphere with 21 vertices that are 3Ro. Actually, as the number of vertices goes to infinity, the proportion of ( 3,4, 4)-spheres that are 37 o goes to 1. There are much less (still, an infinity) ( 3,4, 4)-spheres 3/ i, but a classification seems difficult also. 

Here CV(G) stands for a ( 5,6, 3)-sphere with v vertices and symmetry group G. Although this notation is not generally unique, it will suffice for our purpose. 

For a convex polyhedron, topologically like a sphere, with F faces, V vertices and E edges, Euler s law states that V—E+F = 2. A sphere has genus zero and if another more complex polyhedron can be deformed to take the shape of a sphere with N handles, then it has the genusN. This is a useful, but not a complete, characterization... 

Thus, case (a) tells us, for example, that odd monocycles have no Heilbronner modes, odd-odd bicycles have 1 mode, and all-odd trivalent polyhedra on n(v) vertices, such as the tetrahedon and the dodecahedron, have 3n(v)/2 — n(v) = n v)/2 modes, as do the fullerenes. 

Angel, M.V., Vertical migration in the oceanic realm possible causes and probable effects, in Migration Mechanisms and Adaptive Significance, Rankin, M.A., Ed., Port Aransas, Marine Science Institute (Contributions in Marine Science Suppl. to vol. 27), 1986, 45. 

Fig. 6 The barrier factor, which is equal to Ao in LH theory, and the driving force , or survival factor (1 - B/A) of the growth rate as functions of stem length (schematic). Ao, A and B are rates of attachment of the first and subsequent stems, and rate of detachment, respectively. The (1 - B/A) factor is drawn for three temperatures T > T" > V". Vertical rectangles show growth rates for discrete integer folded forms E and F2 of a monodisperse oligomer...

Kroll, V. Vertical distribution of radium in deep-sea sediments. Nature 171, 742 (1953). 

Note V. Vertically S. Sand filling NS. No Sand filling... 

CH, horizontal lifting method V, vertical dipping method. dSilver paste was used as electrodes. 

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