Optimal path directed

All locally optimal paths directed from the apex to the base of the delta and their energies are determined by applying the above recurrence relation systematically to all points (x, t) from t = ltot = L with the initial condition... 

A scheme for unifying the different forms of optimal paths—directed and non-directed, self-similar and self-affine—was proposed in Ref. [43]. The bond energies were distributed on the lattice with a density function... 

In principle, it is possible to find the optimal path by direct solution of the Pontryagin Hamiltonian (37), with appropriate boundary conditions. We must stress that even for this relatively simple system, the solution is a formidable, and almost impossible, task. First of all, in general one has no insight into the appropriate boundary conditions, in particular into those at the starting time (which belong to the strange attractor). But even if the boundaries were known, in practice the determination of the optimal path is impossible the functional R of Eq. (36) has so many local minima, that it proved impractical to attempt a (general) search for the optimal path. 

Another permanent marking point is made from the positive direction, stored in the collection S. The adjacent temporary marking points are then modified. If the permanent tag node is already in the collection E, setting the permanent marker as the center, tracing to find an optimal path respectively according to positive direction and reverse direction. If not, going to the next step. 

Going on one step from the positive direction, one step from the opposite direction, repeat the steps (2) and (3), do the loop continuously, until you find the desires of K, the optimal path or the source node and destination node are all permanent marker, the algorithm is end. 

In this paper, a method of the optimal path in the weighted undirected graph and a new method of calculating the K optimal path are proposed based on the traditional weighted directed graph. The K optimal path is convenient for miner to escape, can also assist in the decision-making of leadership. Further studies should focus on the determination of the changing parameters of the roadway to calculate the optimal path. 

DIRECTED OPTIMAL PATHS AT ZERO TEMPERATURE 2.1. Construction of directed paths... 

Determination of the directed optimal path the transfer matrix method for zero temperature... 

Figure 1. The delta geometry in 1 + 1 dimensions obtained by orienting a quadrant of the square lattice so that the origin (0,0) lies at the apex O. The longitudinal direction is chosen along the vertically downward diagonal of the square lattice, thus allowing paths that ate alwj s directed downward along the lattice bonds. The bold line shows a directed path from the apex O to a point A, with coordinates (x, t), on the base of the delta. Since the site A can only be reached from the sites B and C, the directed optimal path from O to A must contain the directed optimal path from 0 to either B or C.

The ultrametricity of the tree is a direct consequence of the non-intersecting property of the locally optimal paths. In ultrametric space, any three points A, A2 and A3 satisfy the inequality [22] ... 

Figure 2. An illustration of the ultrametric tree formed by locally optimal paths from the apex to the base of a 1 + 1 dimensionaJ delta. Three non-intersecting paths are drawn to schematically represent the directed optim ll paths from O to the sites A, A2 and A3. The ultrametric distances between these three sites on the base have the relation U(Ai,A2) = U(A3,Ai) > U(A2, A3) which is the same as the relation in Eq. (22).

Both of these observations were found to be valid only for 1 globally optimal path at a site B on it is defined as the number of sites on the base of the delta that are connected from the site B by directed locally optimal paths [23] W is thus equal to the number of locally optimal paths through B, from the apex of the delta to its base. In 1 -I-1 dimensions, with random bond energies from a Gaussian distribution, the width of the globally optimal path at a height h (i.e., the height /i of a site B on the path) above the base of the delta of linear size L was found to be [23]... 

Perfectly directed optimal paths hardly occur in nature. However the universality class of directed polymers (Eq. 7) is widespread because there are many instances of nondirected optimal paths in which the segments going against the longitudinal direction are insignificant over large distances. 

Figure 3. A non-directed path between the origin 0(0,0) and a site A(xi,X2) on a square lattice. To reach A the path must pass through one of its four neighbouring sites Bi, B2, B3 or B4. This fact leads to a recurrence relation (Eq. 45) for the energy of the optimal path between O and A and forms the basis of the transfer matrix method for determining the non-directed optimal path and its energy.

The non-directed optimal path between the origin and the site x on the >-dimensional lattice is therefore identified as the path with the energy... 

Figure 6.18 Top Chemical wave in the BZ reaction propagating through a membrane labyrinth. The wave was initiated at the lower left corner of the 3.2 x 3.2 cm maze. Bottom Velocity field showing local propagation direction obtained from analysis of fifty images obtained at 50-s intervals. Lines show shortest paths between five points and the starting point S. (Reprinted with permission from Steinbock, O. Toth, A. Showalter, K. 1995. Navigating Complex Labyrinths Optimal Paths from Chemical Waves, Science 267, 868 871.

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