Trees randomized

Amit Y, Geman D. Shape quantization and recognition with randomized trees. Neural Comput 1997 9 1545-88. 

Hollas, B. (2005a) Asymptotically independent topological indices on random trees. /. Math. Chem., 38, 379-387. 

We presume you might be interested in going through the same kind of proof for a swollen coil (which we discussed in Chapter 8) we proved then that R = hN /, where b is proper N-independent, that is, associated with the monomer, length scale — see formula (8.14)), as well as for a random tree R = feiV / ). You would then be able to see for yourself that the power laws do indeed correspond to self-similar objects, that is, to those which have, say, a g-unit organized in the same way as the whole thing (obeying the same power law as the whole chain). 

Note Node numbers correspond to those on the chronogram (Figure 17.1). Point estimates are from analyses of the all-compatible majority rule consensus tree and posterior probability values are rep ted. The Mode value represents the most likely divergence lime value under the specified model (obtained by local density estimation calculated over the 100 random frees drawn from the posterior distribution of trees and parameters), and the HPD values limits the confidence interval for the estimates. ( node constrained j age estimates show bimodal distribution across the 100 random trees > age distribution with a pronounced right tail across the 100 random frees)... 

The number of random trees in which the NRI and NTT measures were lower than the actual estimate (9750 and above is significant). 

The value V = 0.5 is consistent with a random-walk structure for the gel, as well as any other structure of fractal dimension 2. This result was frequently found in colloidal silica aggregates and was confirmed by X-rays experiments performed on dry silica. Structures of fractal dimension 2 include random-trees (lattice animals) with excluded volume. The random-tree would seem a likely suggestion for the short-scale structure of a silica gel. 

This is a technique developed during World War II for simulating stochastic physical processes, specifically, neutron transport in atomic bomb design. Its name comes from its resemblance to gambling. Each of the random variables in a relationship is represented by a distribution (Section 2.5). A random number generator picks a number from the distribution with a probability proportional to the pdf. After physical weighting the random numbers for each of the stochastic variables, the relationship is calculated to find the value of the independent variable (top event if a fault tree) for this particular combination of dependent variables (e.g.. components). 

The fault tree (Figure 7.4-1) has "Pre.ssure Tank Rupture" as the top event (gate G1). This may result from random failure of the tank under load (BEl), OR the gate G2, "Tank ruptures due to overpressure" which is made of BE6 "Relief valve fails to open" AND G3, "Pump motor operates too long." This is made of BE2, "Timer contacts fail to open," AND G4, "Negative feedback loop inactive" which is composed of BE3, "Pressure gauge stuck," OR BE4, "Operator fails to open switch," OR "BE5, "Switch fails to open,"... 

In Section 20.2, equations for tlie reliability of series and parallel systems are established. Various reliability relations are developed in Section 20.3. Sections 20.4 and 20.5 introduce several probability distribution models lliat are extensively used in reliability calculations in hazard and risk analysis. Section 20.6 deals witli tlie Monte Carlo teclinique of mimicking observations on a random variable. Sections 20.7 and 20.8 are devoted to fault tree and event tree analyses, respectively. 

Exploratory studies indicate that 64 mature Valencia oranges per sample, when picked in a consistent manner with 16 fruits from each of 4 randomized normal trees, suffice to give fair (10%) agreement between replicates such a sample ordinarily weighs from 18 to 24 pounds. Very cursory tests with Thompson seedless grapes demonstrate... 

Approximately eight pounds of fruit constituted each analytical sample 11, 14), the number of fruits per sample varied from ten to thirty. Fruits were selected at random from within a peripheral band around the tree 3 to 6 feet above the ground. Individual samples were constituted with fruits from six to eight trees and replicate samples were taken from different groups of trees. In some cases, samples of deciduous fruits were collected from three trees and additionally involved a portion of fruit from the upper quarter of the tree. Duplicate, or more generally triplicate, samples were utilized for analyses. All fruits for penetration studies were collected in paper bags, which were immediately stapled to ensure sample integrity. 

Apples. The Rome Beauty apples used in the wash tests were sampled from trees that had received varying amounts of DDT mixtures in as many as six cover sprays. Duplicate or triplicate samples of 30 apples each were taken at random for the residue analyses from the fruit passed through each experimental wash mixture. Additional lots of 30 washed apples each were placed in cold storage for subsequent examinations. Unless otherwise indicated, all washing tests were run in a flood-type washer of recent design (a BADD washer with a heated prewash tank unit, an unheated main tank unit, a water rinse tank unit, and a velour roller dryer unit, manufactured by the Bean-Cutler Division, Food Machinery Corporation, San Jose, Calif.). Surface deposits of DDT were determined as described (10, 12) on samples taken just before and immediately after the washing treatments. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

Sicherer, S.H., Munoz-Furlong, A. and Sampson, H.A., Prevalence of peanut and tree nut allergy in the United States determined by means of a random digit dial telephone survey A 5-year follow-up study. J. Allergy Clin. Immunol., 112, 1203, 2003. 

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