Vertex line

J. A.F. Plateau, who first studied their properties. It is the Plateau borders, rather than the thin Hquid films, which are apparent in the polyhedral foam shown toward the top of Figure 1. Lines formed by the Plateau borders of intersecting films themselves intersect at a vertex here mechanical constraints imply that the only stable vertex is the one made from four borders. The angle between intersecting borders is the tetrahedral angle,... 

Figure 4 A representative step m the downhill simplex method. The original simplex, a tetrahedron in this case, is drawn with solid lines. The point with highest energy is reflected through the opposite triangular plane (shaded) to form a new simplex. The new vertex may represent symmetrical reflection, expansion, or contractions along the same direction.

Scheitel-. vertical peak, maximum (Anat.) parietal, -ausschlag, m. amplitude. -Knie,/ vertical line, -punkt, m. vertex zenith. 

Two or more lines are concurrent if there is a single point which lies on all of them. The three altitudes of a triangle (if taken as lines, not segments) are always concurrent, and their point of concurrency is called the orthocenter. The angle bisectors of a triangle are concurrent at a point equidistant from their sides, and the medians are concurrent two thirds of the way along each median from the vertex to the opposite side. The point of concurrency of the medians is the centroid. 

In these line-angle formulas it is understood that there is a carbon atom at each vertex of the hexagon hydrogen atoms are not shown. This model is consistent with many of the properties of benzene. The molecule is a planar hexagon with bond angles of 120°. The hybridization of each carbon is sp2. However, this structure is misleading in one respect Chemically, benzene does not behave as if double bonds were present... 

In this shorthand, they assume that any vertex between two lines contains a carbon atom unless specified otherwise. If there are several carbons in a row, they will make the lines join at angles, so that they can count the carbons if need be. 

Figure 55 Line drawing of structurally characterized Djf, and 2 isomers 16 and 17 of T)4H)4, and D4J isomer 18 of Each vertex represents an Si-H unit and each edge contains the...

In the case of a function of two variables the direction is from the rejected vertex through the middle of the line of the triangle that is opposite to this point. The new point together with the previous two points define a new equilateral triangle. 

Figure 4.19 (a) Sets of parallel lines that constitute the ring and equatorial C-H bonds of the chair conformation, (b) The axial bonds are all vertical. When the vertex of the ring points up, the axial bond is up and vice versa. 

Fig. A3.1. Some lowest-order diagrams for the temperature GF (A3.6). The dashed and solid lines correspond to the GF for high-frequency and resonance low-frequency vibrations of a molecular planar lattice in the harmonic approximation (see Eq. (A3.9) and (A3.10)). Each vertex is associated with the factor -y/N, the integration and summation being performed over each vertex coordinates r, from 0 to / , and over all internal wave vectors K. At ptiClK 1, the main contribution is provided by a-type diagrams.184...

Consider a v = 3 vertex out of which a bond of length p emerges and waves can imping on the vertex from two lines and be either transmitted or reflected. The vertex scattering matrix in this case is 2 x 2 and it reads... 

For Figure 7.1, this point occurs for c = 5, and the optimal values of x are x1 = 0.5, x2 = 1.5. Note that the maximum value occurs at a vertex of the constraint set. If the problem seeks to minimize/, the minimum is at the origin, which is again a vertex. If the objective function were / = 2x1 + 2jc2, the line / = Constant would be parallel to one of the constraint boundaries, x1 + x2 = 2. In this case the maximum occurs at two extreme points, (xx = 0.5, x2 = 1.5) and (xx = 2, x2 = 0) and, in fact, also occurs at all points on the, line segment joining these vertices. 

With these definitions, it is very easy to draw the diagram corresponding to a given term of the expansion (41) reading a contribution from right to left (i.e. in the arrow of time), the lines of the initial state p ipl 0) are first represented then each of the interactions which lead the system to the intermediate states k", (k ". .. are indicated by the corresponding elementary vertex of Fig. 3 until the final state k is reached. These vertices are ordered from right to left. 

Fig. 1-4. Generation of the 13-vertex polyhedron found in 1,2-,2-C2Bii Hio-3Ph by breaking a single edge (hashed line) in a 13-vertex deltahedron.

Polygons are made up of angles and line segments called sides. Each angle is made up of two sides and the point at which they meet is called the vertex. 

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