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.

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.

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 Validation of the procedure: Quantification of the degradation index of Photovoltaic Grid Connection Systems

2021 ◽  
Vol 2 (5) ◽  
pp. 8-13
Author(s):  
Proenza Y. Roger ◽  
Camejo C. José Emilio ◽  
Ramos H. Rubén
Keyword(s):  
Mean Absolute Error ◽  
Absolute Error ◽  
Coefficient Of Determination ◽  
Mean Square ◽  
Degradation Index ◽  
Statistical Parameters ◽  
Grid Connection ◽  
The Mean

The results obtained from the validation of the procedure ‟Quantification of the degradation index of Photovoltaic Grid Connection Systems” are presented, using statistical parameters, which corroborate its accuracy, achieving a coefficient of determination of 0.9896, a percentage of the root of the mean square of the error RMSPE = 1.498% and a percentage of the mean absolute error MAPE = 1.15%, evidencing the precision of the procedure.

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

1975 ◽  
Vol 7 (3) ◽  
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.

The Bordeaux Automatic Meridian Circle

1978 ◽  
Vol 48 ◽  
pp. 227-228
Author(s):  
Y. Requième
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Meridian Circle ◽  
The Mean ◽  
Full Automation

In spite of important delays in the initial planning, the full automation of the Bordeaux meridian circle is progressing well and will be ready for regular observations by the middle of the next year. It is expected that the mean square error for one observation will be about ±0.”10 in the two coordinates for declinations up to 87°.

The mean-square generalized delayed decision feedback sequence detector

2003 ◽  
Vol 14 (3) ◽  
pp. 265-268 ◽  
Author(s):  
Maurizio Magarini ◽  
Arnaldo Spalvieri ◽  
Guido Tartara
Keyword(s):  
Decision Feedback ◽  
Mean Square ◽  
The Mean

Limit tolerances for sums of measurement errors

2018 ◽  
Vol 934 (4) ◽  
pp. 59-62
Author(s):  
V.I. Salnikov
Keyword(s):  
Normal Distribution ◽  
Mean Square Error ◽  
Gaussian Distribution ◽  
Measurement Errors ◽  
Theory And Practice ◽  
Mean Square ◽  
The Mean ◽  
Limiting Values ◽  
Limiting Value

The question of calculating the limiting values of residuals in geodesic constructions is considered in the case when the limiting value for measurement errors is assumed equal to 3m, ie ∆рred = 3m, where m is the mean square error of the measurement. Larger errors are rejected. At present, the limiting value for the residual is calculated by the formula 3m√n, where n is the number of measurements. The article draws attention to two contradictions between theory and practice arising from the use of this formula. First, the formula is derived from the classical law of the normal Gaussian distribution, and it is applied to the truncated law of the normal distribution. And, secondly, as shown in [1], when ∆рred = 2m, the sums of errors naturally take the value equal to ?pred, after which the number of errors in the sum starts anew. This article establishes its validity for ∆рred = 3m. A table of comparative values of the tolerances valid and recommended for more stringent ones is given. The article gives a graph of applied and recommended tolerances for ∆рred = 3m.

scholarly journals Technical note: Inherent benchmark or not? Comparing Nash–Sutcliffe and Kling–Gupta efficiency scores

2019 ◽  
Vol 23 (10) ◽  
pp. 4323-4331 ◽  
Author(s):  
Wouter J. M. Knoben ◽  
Jim E. Freer ◽  
Ross A. Woods
Keyword(s):  
Time Series ◽  
Coefficient Of Variation ◽  
Ad Hoc ◽  
Model Performance ◽  
Technical Note ◽  
Mean Flow ◽  
Model Adequacy ◽  
Robust Model ◽  
The Mean ◽  
Adequacy Assessment

Abstract. A traditional metric used in hydrology to summarize model performance is the Nash–Sutcliffe efficiency (NSE). Increasingly an alternative metric, the Kling–Gupta efficiency (KGE), is used instead. When NSE is used, NSE = 0 corresponds to using the mean flow as a benchmark predictor. The same reasoning is applied in various studies that use KGE as a metric: negative KGE values are viewed as bad model performance, and only positive values are seen as good model performance. Here we show that using the mean flow as a predictor does not result in KGE = 0, but instead KGE =1-√2≈-0.41. Thus, KGE values greater than −0.41 indicate that a model improves upon the mean flow benchmark – even if the model's KGE value is negative. NSE and KGE values cannot be directly compared, because their relationship is non-unique and depends in part on the coefficient of variation of the observed time series. Therefore, modellers who use the KGE metric should not let their understanding of NSE values guide them in interpreting KGE values and instead develop new understanding based on the constitutive parts of the KGE metric and the explicit use of benchmark values to compare KGE scores against. More generally, a strong case can be made for moving away from ad hoc use of aggregated efficiency metrics and towards a framework based on purpose-dependent evaluation metrics and benchmarks that allows for more robust model adequacy assessment.

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