From rigid rod to random walk. Some exact results

1993 ◽  
Vol 30 (5) ◽  
pp. 617-620 ◽  
Author(s):  
J. Rieger
Keyword(s):  
Random Walk ◽  
Exact Results ◽  
Rigid Rod

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

scholarly journals Exact results for the roughness of a finite size random walk

2006 ◽  
Vol 370 (1) ◽  
pp. 127-131 ◽  
Author(s):  
V. Alfi ◽  
F. Coccetti ◽  
M. Marotta ◽  
A. Petri ◽  
L. Pietronero
Keyword(s):  
Random Walk ◽  
Finite Size ◽  
Exact Results

scholarly journals Some approximate results in renewal and dam theories

1971 ◽  
Vol 12 (4) ◽  
pp. 425-432 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walk ◽  
Sequential Analysis ◽  
Random Variables ◽  
Renewal Theory ◽  
Slight Modification ◽  
Exact Results ◽  
Fundamental Identity ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Two Barriers

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

Morphology of High-Performance Rigid-Rod Polymer Fibers and Molecular Composites

1989 ◽  
Vol 47 ◽  
pp. 338-339
Author(s):  
W.W. Adams ◽  
S. J. Krause
Keyword(s):  
Tensile Strength ◽  
High Performance ◽  
Liquid Crystalline ◽  
Molecular Level ◽  
Synergistic Effects ◽  
Tensile Modulus ◽  
Commercial Development ◽  
The Matrix ◽  
Molecular Composites ◽  
Rigid Rod

Rigid-rod polymers such as PBO, poly(paraphenylene benzobisoxazole), Figure 1a, are now in commercial development for use as high-performance fibers and for reinforcement at the molecular level in molecular composites. Spinning of liquid crystalline polyphosphoric acid solutions of PBO, followed by washing, drying, and tension heat treatment produces fibers which have the following properties: density of 1.59 g/cm3; tensile strength of 820 kpsi; tensile modulus of 52 Mpsi; compressive strength of 50 kpsi; they are electrically insulating; they do not absorb moisture; and they are insensitive to radiation, including ultraviolet. Since the chain modulus of PBO is estimated to be 730 GPa, the high stiffness also affords the opportunity to reinforce a flexible coil polymer at the molecular level, in analogy to a chopped fiber reinforced composite. The objectives of the molecular composite concept are to eliminate the thermal expansion coefficient mismatch between the fiber and the matrix, as occurs in conventional composites, to eliminate the interface between the fiber and the matrix, and, hopefully, to obtain synergistic effects from the exceptional stiffness of the rigid-rod molecule. These expectations have been confirmed in the case of blending rigid-rod PBZT, poly(paraphenylene benzobisthiazole), Figure 1b, with stiff-chain ABPBI, poly 2,5(6) benzimidazole, Fig. 1c A film with 30% PBZT/70% ABPBI had tensile strength 190 kpsi and tensile modulus of 13 Mpsi when solution spun from a 3% methane sulfonic acid solution into a film. The modulus, as predicted by rule of mixtures, for a film with this composition and with planar isotropic orientation, should be 16 Mpsi. The experimental value is 80% of the theoretical value indicating that the concept of a molecular composite is valid.

Elements of the Random Walk

2004 ◽  
Author(s):  
Joseph Rudnick ◽  
George Gaspari
Keyword(s):  
Random Walk
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