Theory of Branching Processes and Statistics of Rubber Elasticity

1966 ◽  
Vol 39 (5) ◽  
pp. 1472-1488 ◽  
Author(s):  
G. R. Dobson ◽  
M. Gordon
Keyword(s):  
Chain Length ◽  
Branching Processes ◽  
Rubber Elasticity ◽  
Active Network ◽  
Gel Point ◽  
Statistical Parameters ◽  
Polymeric Systems ◽  
Technical Performance ◽  
Cascade Processes ◽  
The Mean

Abstract Recent efforts to reformulate statistical theories of polymeric systems in terms of the theory of branching (“cascade”) processes, are here extended to calculations of statistical parameters for the theory of rubber elasticity, viz., the number and mean length of various forms of active network chains. New simple results are given for random ƒ-functional polycondensation; it is shown that such systems are of interest in rubber elasticity studies in the region just after the gel point, where the concentration of active network chains varies rapidly with conversion, while the mean length of these chains diverges at the gel point. General exact formulas are derived for random crosslinking of arbitrary primary distributions, without assuming that the mean primary chain length is necessarily large; examples are worked out for uniform (homodisperse) chains, random, self-convoluted random, or Poisson primary distributions. Calculations are extended also to the cross-polymerization type of vulcanization reaction attributed to the cure of polybutadiene with peroxides. The results suggest reasons for differences in technical performance observed with such rubbers.

Configurational statistics of copolymer systems

1966 ◽  
Vol 295 (1440) ◽  
pp. 29-54 ◽  
Keyword(s):  
Ad Hoc ◽  
Branching Processes ◽  
Substitution Effects ◽  
Gel Point ◽  
Mean Square ◽  
Statistical Parameters ◽  
Routine Manner ◽  
Isomer Distribution ◽  
Lagrange Expansion ◽  
The Mean

The theory of branching processes allows various weight, size, configuration and network statistics of polymer systems to be derived in a unified and routine manner, fairly free from ad hoc combinatorial investigations. The method is briefly summarized for use in copolymer systems, of which homopolymer systems form a degenerate case. The calculations of statistical parameters are readily restricted to the sol fraction of a system after the gel point, and derivations from randomness of reaction due to chemical substitution effects are easily allowed for. The general randomly branched system of Zimm & Stockmayer is considered in detail. This arises from the vulcanization or radiation crosslinking of randomly distributed primary chains, from ideal copolymerization of monovinyl and polyvinyl monomers, or (most simply) from random condensation of 2-functional and f -functional monomers. Known formulae from several sources are rederived more simply in forms which are variously generalized. In particular, approximations assuming long linear sequences in the system are eliminated, and effects of the different sizes of the 2-functional and f -functional units on the mean square radii are allowed for to a good approximation. Complicated configurational statistics, such as the mean square radius (R 2 x ) of the isomer distribution of the x-mer fraction, are evaluated explicitly. The method of Lagrange expansion of functions of several variables (Poincaré, I. J. Good) is found to be especially useful.

Good’s theory of cascade processes applied to the statistics of polymer distributions

1962 ◽  
Vol 268 (1333) ◽  
pp. 240-256 ◽  
Keyword(s):  
Molecular Weight ◽  
Generating Functions ◽  
Repeat Unit ◽  
Average Molecular Weight ◽  
Stochastic Theory ◽  
Direct Calculation ◽  
Gel Point ◽  
Statistical Parameters ◽  
Cascade Processes ◽  
The Difference

Important statistics of polymerization reactions, whether of the condensation or addition type, can be calculated rather simply and in a standardized way, by an adaptation of Good’s stochastic theory of cascade processes. Examples of such statistics are the various molecular weight averages, the gel point, and the sol fraction. The use of generating functions in the theory greatly reduces the use of probability theory and it allows the direct calculation of the required statistics without the need of explicit expressions for the distributions concerned, or for the summations required in calculating their moments. The generating functions required are mostly combinations of powers of the basic form (1— α) θ 1 + αθ 2 where θ 1 and θ 2 are dummy variables (or unity) and α some parameter measuring the conversion of a functionality. Ordinary (non-vectorial) generating functions suffice when the system contains essentially one type of repeat unit, but for copolymers (in a broad sense) generalization to vectorial generating functions is required. Calculations of the former type, useful in describing the principles involved, include the calculation of new sol-fraction equations for simple polycondensation reactions, and a somewhat more exact sol-fraction equation for the vulcanization of chains initially distributed randomly in length. Copolymerization systems are then exemplified by calculating the weight-average molecular weight and sol fraction for the system glycerol/adipic acid from general formulae derived. Exact allowance is made for the statistical effects due to complete or partial elimination of water and to the difference in rate of esterification of primary and secondary hydroxyls. The gel-point condition for a system involving copolymerization of s different units is generally found by equating to zero a determinant of order s . The mechanism of polycondensation reactions could be elucidated by comparing experimental measurements of the statistical parameters with those calculated from postulated kinetic schemes using the unified and comparatively simple theory here presented.

Induced Smectic Phases I. Polymorphism in Binary Mixtures of Alkyloxybenzylidene-alkylanilines with Alkyl- and Alkyloxy- cyanobiphenyls

1981 ◽  
Vol 36 (1) ◽  
pp. 62-67 ◽  
Author(s):  
F. Schneider ◽  
N. K. Sharma
Keyword(s):  
Chain Length ◽  
Binary Mixtures ◽  
Transition Temperatures ◽  
Mean Values ◽  
Smectic Phases ◽  
The Mean ◽  
Induced Smectic Phases

The diagrams of state of mixtures of 4-n-alkyloxybenzylidene-4'-n-butylanilines with 4-n- alkyl- and 4-n-alkyloxy-4'-cyanobiphenyls are studied. The mixtures form induced smectic phases of type SA, SB and SE. In all three smectic phases the thickness of the smectic layers agrees with the mean values of the molecular lengths. In most cases the induced SA phases do not show uninterrupted miscibility with the SA phases of the pure components. For instance, the system 40 • 4/8 CBP exhibits three separate SA phase areas. The maximum transition temperatures of the induced smectic phases increase with increasing chain length of the azomethines, but remain constant in case of the SA and SB phases or even decrease in case of the SE phases with increasing chain length of the cyanobiphenyls

THE TERMINAL GROUPS OF RIBONUCLEATE CHAINS IN EACH OF THE 16S AND 23S COMPONENTS OF ESCHERICHIA COLI RIBOSOMAL RNA

1967 ◽  
Vol 45 (6) ◽  
pp. 937-948 ◽  
Author(s):  
J. L. Nichols ◽  
B. G. Lane
Keyword(s):  
Escherichia Coli ◽  
Chain Length ◽  
Ribosomal Rna ◽  
Gradient Centrifugation ◽  
Sucrose Density Gradient ◽  
Sucrose Density Gradient Centrifugation ◽  
Quantitative Estimates ◽  
16S Rna ◽  
The Mean ◽  
End Group

Ribosomal ribonucleates from Escherichia coli have been resolved into 16S and 28S components by sucrose density-gradient centrifugation, and the chain termini in each of the 16S and 23S RNA components have been analyzed by hydrolysis with alkali. The principal 5′-linked end group of 16S RNA was found to be adenosine, and the principal 5′-linked end group of 23S RNA was found to be uridine. The principal 3′-linked end group of 16S RNA was also found to be adenosine, whereas the principal 3′-linked end group of 23S RNA was found to be guanosine. Quantitative estimates of chain length based on analyses for 5′-iinked terminals indicate that the mean chain length for 16S RNA is about 1.3 × 103nucleotide residues and the mean chain length for 23S RNA is about 2.1 × 103nucleotide residues.

scholarly journals Linking lineage and population observables in biological branching processes

2019 ◽  
Author(s):  
Reinaldo García-García ◽  
Arthur Genthon ◽  
David Lacoste
Keyword(s):  
Population Dynamics ◽  
Generation Time ◽  
Doubling Time ◽  
Branching Processes ◽  
Underlying Mechanism ◽  
Population Doubling ◽  
Population Doubling Time ◽  
Division Rate ◽  
Fluctuation Theorems ◽  
The Mean

Using a population dynamics inspired by an ensemble of growing cells, a set of fluctuation theorems linking observables measured at the lineage and population levels are derived. One of these relations implies inequalities comparing the population doubling time with the mean generation time at the lineage or population levels. We argue that testing these inequalities provides useful insights into the underlying mechanism controlling the division rate in such branching processes.

scholarly journals BatScope manages acoustic recordings, analyses calls, and classifies bat species automatically

2018 ◽  
Vol 96 (9) ◽  
pp. 939-954 ◽  
Author(s):  
M.K. Obrist ◽  
R. Boesch
Keyword(s):  
High Frequency ◽  
Reference Database ◽  
Correct Classification Rate ◽  
Statistical Parameters ◽  
Classification Rate ◽  
Echolocation Calls ◽  
Portable Software ◽  
Automated Processing ◽  
The Mean ◽  
Species Assignment

BatScope is a free application for processing acoustic high-frequency recordings of bats. It can import data, including meta-data information, from recorders such as Batlogger. The resulting content can be filtered visually as spectrograms or according to data fields and can be displayed. Automated processing includes detecting and extracting of echolocation calls, filtering noise, and measuring statistical parameters. Calls are classified to species by statistically matching to a reference database. A weighted list of classifiers helps to assign the most likely species per call. Classifiers were trained on 19 636 echolocation calls of 27 European bat species. When classifiers all agree on a species (76.4% of all cases), the mean correct classification rate reaches 95.7%. A sequence’s summary statistic indicates the most likely species occurring therein. Classifications can be verified visually, by filtering, and by acoustic comparison with reference calls. Procedures are available for, e.g., excluding dubious cutouts from the statistics and for accepting or overriding the proposed species assignment. Acoustic recordings can be exported and exchanged with other users. Finally, the verified results can be exported to spreadsheets for further analyses and reporting. We currently reprogram BatScope using Java, PostgreSQL, and R to reach a unified and portable software architecture.

Mathematical-Statistical Methods for the Analysis of Aerial Triangulation Adjustment Errors Using a Computer Program

1975 ◽  
Vol 29 (2) ◽  
pp. 175-188
Author(s):  
M. Mosaad Allam
Keyword(s):  
Computer Program ◽  
Frequency Interval ◽  
Mean Square ◽  
Fisher Criterion ◽  
Statistical Parameters ◽  
Sample Distribution ◽  
Development Section ◽  
The Mean ◽  
Kolmogorov Criterion ◽  
Mean Square Errors

In practice, photogrammetrists use a single statistic reliability interval criterion, based on the mean square errors, to judge the accuracy of adjustment of photogrammetric blocks. Even in some cases, if the practical and theoretical distributions of frequency interval agree, such a test does not make it possible to establish the closeness of their convergence nor the degree of their difference. In other words, to get a complete picture of the character of the distribution of errors in the adjusted photogrammetric blocks, it is insufficient to investigate any single statistic. In the Research and Development Section of the Topographical Survey Directorate, a computer program (SABA) has been designed to analyze the errors of photogrammetric block adjustments, compute various statistical parameters and check the sample distribution using Kolmogorov criterion. Based on the decision taken, the correspondence between the empirical and theoretical distribution series are checked using the criterion χ2. The program divides the adjusted block to make a comparative evaluation of accuracies in the different sub-blocks. In this case, in addition to Kolmogorov and χ2 tests, the program checks the reliability intervals of the means and mean square errors of the samples and uses Fisher criterion ‘F’ to check the hypothesis of the equality of dispersion. SABA is coded in Fortran IV and Compass for the CDC CYBER 74 and requires a central memory of 28K decimal works. SABA is the acronym for Statistical Analysis of Block Adjustment.

Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

1975 ◽  
Vol 7 (03) ◽  
pp. 468-494
Author(s):  
H. Hering
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Transition Function ◽  
Mean Square ◽  
Second Moments ◽  
Mean Square Convergence ◽  
The Mean ◽  
Parameter Dependent ◽  
Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

scholarly journals Modeling Y-Linked Pedigrees through Branching Processes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 256
Author(s):  
Miguel González ◽  
Cristina Gutiérrez ◽  
Rodrigo Martínez
Keyword(s):  
Hearing Loss ◽  
Asymptotic Behavior ◽  
Gibbs Sampler ◽  
Mutant Allele ◽  
Branching Process ◽  
Branching Processes ◽  
Dominant Allele ◽  
Linked Gene ◽  
Different Types ◽  
The Mean

A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.

scholarly journals The Properties of Generalized Collision Branching Processes

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Juan Wang ◽  
Chunhao Cai
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Branching Processes ◽  
Hitting Times ◽  
Conditional Mean ◽  
Basic Properties ◽  
Extinction Probabilities ◽  
The Mean

We consider basic properties regarding uniqueness, extinction, and explosivity for the Generalized Collision Branching Processes (GCBP). Firstly, we investigate some important properties of the generating functions for GCB q-matrix in detail. Then for any given GCB q-matrix, we prove that there always exists exactly one GCBP. Next, we devote to the study of extinction behavior and hitting times. Some elegant and important results regarding extinction probabilities, the mean extinction times, and the conditional mean extinction times are presented. Moreover, the explosivity is also investigated and an explicit expression for mean explosion time is established.

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