When observations are autocorrelated, standard formulae for the estimators of variance, s2, and variance of the mean, s2 (x), are no longer adequate. They should be replaced by suitably defined estimators, s2a and s2a (x), which are unbiased given that the autocorrelation function is known. The formula for s2a was given by Bayley and Hammersley in 1946, this work provides its simple derivation. The quantity named effective number of observations neff is thoroughly discussed. It replaces the real number of observations n when describing the relationship between the variance and variance of the mean, and can be used to express s2a and s2a (x) in a simple manner. The dispersion of both estimators depends on another effective number called the effective degrees of freedom Veff. Most of the formulae discussed in this paper are scattered throughout the literature and not very well known, this work aims to promote their more widespread use. The presented algorithms represent a natural extension of the GUM formulation of type-A uncertainty for the case of autocorrelated observations.
Prior knowledge of the autocorrelation function (ACF) enables an application of analytical formalism for the unbiased estimators of variance s2a and variance of the mean s2a(xmacr;). Both can be expressed with the use of so-called effective number of observations neff. We show how to adopt this formalism if only an estimate {rk} of the ACF derived from a sample is available. A novel method is introduced based on truncation of the {rk} function at the point of its first transit through zero (FTZ). It can be applied to non-negative ACFs with a correlation range smaller than the sample size. Contrary to the other methods described in literature, the FTZ method assures the finite range 1 < neff ≤ n for any data. The effect of replacement of the standard estimator of the ACF by three alternative estimators is also investigated. Monte Carlo simulations, concerning the bias and dispersion of resulting estimators sa and sa(×), suggest that the presented formalism can be effectively used to determine a measurement uncertainty. The described method is illustrated with the exemplary analysis of autocorrelated variations of the intensity of an X-ray beam diffracted from a powder sample, known as the particle statistics effect.
This paper presents a method of calculation of steady-state processes in threephases matrix-reactance frequency converters (MRFC's), in which voltages and currents are transformed by control signals with two pulsations. A solution of nonstationary differentia equations with periodic coefficients that describe this system is obtained by using Galerkin's method and an extension of equations of one variable of time to equations of two variables of time. The results of calculations are presented in an example of three-phases MRFC with buck-boost topology and compared with a numerical metod embedded in the program Mathematica.
The possibility of the application of nontraditional method of greenhouse gas utilization by the injection of CO2 (sequestration) into porous geological deposits, treated as unconventional gas collectors, requires the fulfillment of basic criteria such as the impact on the environment and long term storage. The important issue is the physical behavior of the deposit during the porous structure saturation phase by carbon dioxide. What should be mentioned first and foremost is: the availability of CO2 transport along the porous structure and adsorption capacity. The work presents the results of water vapor sorption on coal samples from selected Hard Coal Mines of a differentiated carbon content. The received results were presented in the form of sorption and desorption isotherms performed in a temperature of 303 K. It was additionally described with a BET adsorption isotherm. Based on sorption data, a specific surface area was calculated, in accordance with BET theory. The amount of the adsorbed water vapor molecules for the analyzed coal samples was dependent on the degree of metamorphism. The obtained isotherms can be described as type II according to the BET classification. Volumetric type apparatus -adsorption- microburette liquid was used for the sorption experiments. Water vapor sorption in relation to coals allows for the quantitative determination of primary adsorption centers as a measure of adsorbed molecule interconnections with the adsorbent surface. Based on the BET adsorption equation, out of water vapor isotherms, the amount of adsorption active centers, which potentially may take part in CO2 adsorption in coal seams during injection of this gas, was determined. The sorption capacity of coals is determined by the degree of metamorphism, which also has very large impact on the sorption capacity of the deposit.
The high-pressure torsion (HPT) of Ti-Fe alloys with different iron content has been studied at 7 GPa, 5 anvil rotations and rotation speed of 1 rpm. The alloys have been annealed before HPT in such a way that they contained different amounts of α/α' and β phases. In turn, the β phase contained different concentration of iron. The 5 anvil rotations correspond to the HPT steady-state and to the dynamic equilibrium between formation and annihilation of microstructure defects. HPT leads to the transformation of initial α/α' and β-phases into mixture of α and high-pressure ω-phase. The α → ω and β → ω phase transformations are martensitic, and certain orientation relationships exist between α and ω as well as β and ω phases. However, the composition of ω-phase is the same in all samples after HPT and does not depend on the composition of β-phase (which is different in different initial samples). Therefore, the martensitic (diffusionless) transformations are combined with a certain HPT-driven mass-transfer. We observed also that the structure and properties of phases (namely, α-Ti and ω-Ti) in the Ti – 2.2 wt. % Fe and Ti – 4 wt. % Fe alloys after HPT are equifinal and do not depend on the structure and properties of initial α'-Ti and β-Ti before HPT.
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