Higher-order uncertainty relations

Using the non-negativity of Gram determinants of arbitrary order, we derive higher-order uncertainty relations for the symmetric uncertainty matrices of corresponding order n?>?2 to n Hermitean operators (n?=?2 is the usual case). The special cases of third-order and fourth-order uncertainty relations are considered in detail. The obtained third-order uncertainty relations are applied to the Lie groups SU(1,1) with three Hermitean basis operators (K ₁,K ₂,K ₀) and SU(2) with three Hermitean basis operators (J ₁,J ₂,J ₃) where, in particular, the group-coherent states of Perelomov type and of Barut–Girardello type for SU(1,1) and the spin or atomic coherent states for SU(2) are investigated. The uncertainty relations for the determinant of the third-order uncertainty matrix are satisfied with the equality sign for coherent states and this determinant becomes vanishing for the Perelomov type of coherent states for SU(1,1) and SU(2). As an example of the application of fourth-order uncertainty relations, we consider the canonical operators (Q ₁,P ₁,Q ₂,P ₂) of two boson modes and the corresponding uncertainty matrix formed by the operators of the corresponding mean deviations, taking into account the correlations between the two modes. In two mathematical appendices, we prove the non-negativity of the determinant of correlation matrices of arbitrary order and clarify the principal structure of higher-order uncertainty relations.     

Confidence bounds for the reliability of binary capacitated two-terminal networks

Binary capacitated two-terminal reliability at demand level d (2TR_d) is defined as the probability that network capacity, generated by binary capacitated components, between specified source and sink nodes is greater than or equal to a demand of d units. For the components that comprise these networks, reliability estimates are usually obtained from some source of testing. For these estimates and depending on the type of testing, there is an associated uncertainty that can significantly affect the overall estimation of 2TR_d. That is, an accurate estimate of 2TR_d is highly dependent on the uncertainty associated to the reliability of the network components. Current methods for the estimation of network reliability and associated uncertainty are restricted to the case where the network follows a series-parallel architecture and the components are binary and non-capacitated. For different capacitated network designs, an estimate on 2TR_d can only be approximated for specific scenarios. This paper presents a bounding approach for 2TR_d by explaining how component reliability and associated uncertainty impact estimates at the network level. The proposed method is based on a structured approach that generates a α-level confidence interval (CI) for binary capacitated two-terminal network reliability. Simulation results on different test networks show that the proposed methods can be used to develop very accurate bounds of two-terminal network reliability.     

The effect of optical radiation on the semiconductor conductivity in a thin-film ferroelectric-semiconductor structure

We have studied the effect of optical radiation on the conductivity of a thin-film field-effect transistor based on a multilayer structure of the Si-SiO₂-Pt-PZT-SnO_2−x -Pt type. Within the permissible radiation dose, the conducting channel exhibits a residual photoconductivity, with an optical data storage time of no less than 10⁵ s. The effective quenching of the residual conductivity in the structure studied can be produced only by simultaneous action of an alternating electric field and the temperature.     

用改造后的柯氏干涉仪测量量块中心长度的测量不确定度评定

通过用６３３ｎｍ 短频Ｈｅ－Ｎｅ 激光代替氪灯,提高小数测量的准确度等措施,对常规柯氏干涉仪进行了改造,结果获得测量量块中心长度的标准不确定度为ｕｃ＝ （５．５ｎｍ ）２＋ （８．３×１０－ ８Ｌ）２,大大优于一等量块的最大允许不确定度     

Thermophysical Properties of Nitrogen Trifluoride, Ethylene Oxide, and Trimethyl Gallium from Speed-of-Sound Measurements

The speed of sound was measured in gaseous nitrogen trifluoride, ethylene oxide, and trimethyl gallium using a highly precise acoustic resonance technique. The measurements span the temperature range 200 to 425 K and reach pressures up to the lesser of 1500 kPa or 80% of the sample vapor pressure. The speed-of-sound measurements have a relative standard uncertainty of less than 0.01%. The data were analyzed to obtain the constant-pressure ideal-gas heat capacity C ⁰ _p as a function of temperature with a relative standard uncertainty of 0.1%. The values of C ⁰ _p are in agreement with those determined from spectro- scopic data. The speed-of-sound data were fitted by virial equations of state to obtain temperature-dependent density virial coefficients. Two virial coefficient models were employed, one based on square-well intermolecular potentials, and the second based on a hard-core Lennard-Jones intermolecular potential. The resulting virial equations reproduced the sound-speed data to within ±0.02%, and may be used to calculate vapor densities with relative standard uncertainties of 0.1% or less.     

Progress Towards an Acoustic Measurement of the Molar Gas Constant at INRIM

Progress in developing an experiment for the determination of the molar gas constant R and the Boltzmann constant k at INRIM is reported. The experiment involves simultaneous measurements of the acoustic and microwave resonance frequencies of a stainless steel spherical resonator for which its hemispheres were deliberately misaligned. For the present work, these frequencies were measured in helium near 273.16 K, in the pressure range from 100 to 800 kPa. From microwave data, the radius of the resonator was determined as a function of pressure with an estimated uncertainty of 6.0 ppm. Using acoustic data and the microwave determination of the resonator radius, the speed of sound in helium was deduced, and these values were compared with those predicted by recent accurate ab initio calculations. Over most of the pressure range, the present values agreed with the ab initio values within the uncertainty of the measurements (standard uncertainty of approximately 7.0 ppm). Many suggestions for reducing the uncertainty are provided.     

International round Robin test for mechanical properties of BSCCO-2223 superconductive tapes at room temperature

International RRT was carried out in order to establish the test method for mechanical properties of commercial BSCCO superconductive tapes under cooperation of seven research laboratories. From the stress versus strain curve, the following quantities were evaluated; modulus of elasticity, 0.2% proof strength, fracture strength and stresses at fixed strains. The scatter of measured values was analyzed by evaluating the relative standard uncertainty (RSU), which is the standard uncertainty divided by the average. The expected value of RSU for N = 3 was derived for each mechanical quantities. In order to make clear the major contribution to the scatterings, the F test was applied. The major source of RSU’s was attributed mostly to the influence of inter-laboratory scattering.     

Cylindrical Acoustic Resonator for the Re-determination of the Boltzmann Constant

The progress towards re-determining the Boltzmann constant k _B using two fixed-path, gas-filled, cylindrical, acoustic cavity resonators is described. The difference in the lengths of the cavities is measured using optical interferometry. Thus, a literature value for the density of mercury is not used, in contrast with the presently accepted determination of k _B. The longitudinal acoustic resonance modes of a cylindrical cavity have lower quality factors Q than the radial modes of gas-filled, spherical cavities, of equal volume. The lower Qs result in lower signal-to-noise ratios and wider, asymmetric resonances. To improve signal-to-noise ratios, conventional capacitance microphones were replaced with 6.3 mm diameter piezoelectric transducers (PZTs) installed on the outer surfaces of each resonator and coupled to the cavity by diaphragms. This arrangement preserved the shape of the cylindrical cavity, prevented contamination of the gas inside the cavity, and enabled us to measure the longitudinal resonance frequencies with a relative standard uncertainty of 0.2 × 10⁻⁶. The lengths of the cavities and the modes studied will be chosen to reduce the acoustic perturbations due to non-zero boundary admittances at the endplates, e.g., from endplate bending and ducts and/or transducers installed in the endplates. Alternatively, the acoustic perturbations generated by the viscous and thermal boundary layers at the gas–solid boundary can be reduced. Using the techniques outlined here, k _B can be re-determined with an estimated relative standard uncertainty of 1.5 × 10⁻⁶.     

Algorithms for analyzing the probability of M-out-of-N events

Simple algorithms are presented to compute both the exact probability of M or more out of N independent input events given unequal probabilities and, when there is uncertainty in the input event probabilities, the associated variance in the M-out-of-N probability. The performance of the M-out-of-N probability algorithm is on the order of N² in time and N in space. The performance of the variance algorithm is N³ in time and N² in space. The algorithms are not based on cut set methodology and, consequently, are not limited by the combinatorial explosion associated with cut set manipulation for the M-out-of-N gate. The algorithms are most useful when N exceeds the limitations of cut set manipulation techniques or the M-out-of-N probability is between 0·1 and 1, such that approximate quantification methods are inaccurate. In addition, the M-out-of-N probability is extended to permit the quantification of standard sensitivity and uncertainty importance measures for both individual input events and groups of input events. Example calculations illustrate the capabilities of the algorithms.     

A Steady-State Apparatus to Measure the Thermal Conductivity of Solids

This paper presents a steady-state apparatus to measure the thermal conductivity of solids based on the use of heat flux sensors. The uncertainty assessment of the experimental apparatus is performed in accordance with the ISO “Guide to the Expression of Uncertainty in Measurement” (GUM): the combined standard uncertainty (k = 1) with reference to the properties of Pyrex Glass CRM 039 at 20°C is ±1.9%. The standard uncertainty has also been assessed in the measurement range from 0.1 to 10W · m⁻¹ · K⁻¹. The experimental apparatus has been used to measure the certified reference material Pyrex Glass CRM 039 in the temperature range from 3.1 to 60.3°C: the standard deviation (k = 1) between the experimental measurements and the certified values is ±1.1% with a maximum deviation of 2.4%. Reasonable agreement is found between the uncertainty assessment and the experimental validation.     

气相色谱内标法测定塑料制品中乙苯和苯乙烯含量的不确定度评定

目的 对气相色谱内标法测定塑料制品中乙苯和苯乙烯含量进行不确定度评定，为塑料制品中乙苯、苯乙烯含量测定的准确性提供参考。方法 依据GB 31604.16—2016采用气相色谱内标法对塑料制品中乙苯和苯乙烯含量进行测定，根据相应的测量模型，分析不确定度主要影响因素的来源，并对各影响因素引入的相对标准不确定度进行评估。结果 测量塑料制品中乙苯、苯乙烯的扩展不确定度分别为(23.83±1.50)、(23.90±1.26)mg/kg(P=95%、k=2)。结论 结果表明，混标工作液配制、试样重复测量、气相色谱仪性能是影响乙苯和苯乙烯含量测量不确定度的主要因素，应选择精度高且量程合适的量具，加强气相色谱仪的期间核查，提高实验人员操作的熟练水平，进一步减小测量不确定度，使测量结果更为准确、可靠。     

Measurement of the Specific Enthalpy of Vaporization on 1,1,1,2-Tetrafluoroethane in the Temperature Range from 180 to 240 K

An apparatus is described which is capable of measuring the enthalpy of vaporization in the temperature range from 100 to 250 K. The sample (R134a; purity, at least 99.999%) is located in the measuring cell at the saturated vapor pressure, p = p _s. A control circuit allows p to be kept constant by opening a motor-operated valve to a weighing cylinder after having switched on the electrical measuring cell heater. During the experiment, the temperature is kept constant within a 10mK. In the range 180 to 230 K, the data for R134a are compared with calculated values from the fundamental equation given by Tillner-Roth and Baehr, which is recommended by Annex 18 of the International Energy Agency (IEA) as an international standard. Good agreement within a standard uncertainty of 1.6×10^–3 is obtained. At temperatures of only 10 K above the triple-point temperature, the enthalpy of vaporization calculated from the Clausius–Clapeyron equation shows considerable uncertainty due to the determination of the small vapor pressure. It is chiefly in this range that it is advantageous to have the new apparatus.     

The Difference Between GUM and ISO/TC 15530-3 Method to Evaluate the Measurement Uncertainty of Flatness by a CMM

Uncertainty estimation in coordinate measuring machine (CMM) measurements is one of the highest priority tasks in modern industrial world due to its importance in the manufacturing metrology, especially if we take into account the fact that workpieces, according to a new generation of geometrical product specifications, can be accepted/rejected only on the basis of the total measurement results. Evaluation of measurement uncertainty of a CMM is a complex task, especially in the verification of errors of forms. Among several standard methods that can be used to estimate measurement uncertainty, we performed a comparison between two of these for the case of flatness measurement. Optical flat was used in order to exclude a number of essential factors in the case study. The results showed that the expanded measurement uncertainty assessed by GUM is greater than the value of the expanded measurement uncertainty obtained using the ISO 15530 method. However, both methodologies have shown that the maximum permissible error of a CMM has the greatest impact on uncertainty and also, that the total result evaluated by both approaches is close to the maximum permissible error of a CMM.     

Thermophysical Properties of Chlorine from Speed-of-Sound Measurements

The speed of sound was measured in gaseous chlorine using a highly precise acoustic resonance technique. The data span the temperature range 260 to 440 K and the pressure range 100 kPa to the lesser of 1500 kPa or 80% of the sample's vapor pressure. A small correction (0.003 to 0.06%) to the observed resonance frequencies was required to account for dispersion caused by the vibrational relaxation of chlorine. The speed-of-sound measurements have a relative standard uncertainty of 0.01%. The data were analyzed to obtain the ideal-gas heat capacity as a function of the temperature with a relative standard uncertainty of 0.1%. The reported values of C ^o _p are in agreement with those determined from spectroscopic data. The speed-of-sound data were fitted by virial equations of state to obtain the temperature dependent density virial coefficients. Two virial coefficient models were employed, one based on square-well intermolecular potentials and the second based on a hard-core Lennard–Jones intermolecular potential. The resulting virial equations reproduced the sound speed data to within 0.01% and may be used to calculate vapor densities with relative standard uncertainties of 0.1% or less.     

Modeling of uncertainty for flood wave propagation induced by variations in initial and boundary conditions using expectation operator on explicit numerical solutions

This study proposes a framework for uncertainty analysis by incorporating explicit numerical solutions of governing equations for flood wave propagation with the expectation operator. It aims at effectively evaluating the effect of variations in initial and boundary conditions on the estimation of flood waves. Spatiotemporal semivariogram models are employed to quantify the correlation of the variables in time and space. The 1D nonlinear kinematic wave equation for the overland flow (named EVO_NS_KWE) is applied in the model development. Model validation is made by comparison with the Monte Carlo simulation model in the calculation of statistical properties of model outputs (ie, flow depths), that is, the mean, standard deviation, and coefficient of variation. The results from the model validation show that the EVO_NS_KWE model can produce excellent approximations of the mean and less satisfactory approximations of the standard deviation and coefficient of variation compared with those obtained by using the Monte Carlo simulation model. It concludes that the uncertainties of flow depths in the domain are significantly affected by variations in the boundary condition. Future application of the EVO_NS_KWE model enables the evaluation of uncertainty in model outputs induced by the initial and boundary condition subject to uncertainty and will also provide corresponding probabilistic information for risk quantification method.     

A Gas Pressure Scale Based on Primary Standard Piston Gauges

The National Institute of Standards and Technology (NIST) has redefined its gas pressure scale, up to 17 MPa, based on two primary standard piston gauges. The primary standard piston gauges are 35.8 mm in diameter and operate from 20 kPa to 1 MPa. Ten secondary standard piston gauges, two each of five series of the Ruska 2465 type, with successively smaller diameters form the scale extending up to 17 MPa. Six of the piston gauges were directly compared to the primary standards to determine their effective area and expanded (k = 2) uncertainty. Two piston gauges operating to 7 MPa were compared to the 1.4 MPa gauges, and two piston gauges operating to 17 MPa were compared to the 7 MPa gauges. Distortion in the 7 MPa piston gauges was determined by comparing those gauges to a DH Instruments PG7601 type piston gauge, whose distortion was calculated using elasticity theory. The relative standard uncertainties achieved by the primary standards range from 3.0 × 10⁻⁶ to 3.2 × 10⁻⁶. The relative standard uncertainty of the secondary standards is as low as 4.2 × 10⁻⁶ at 300 kPa. The effective areas and uncertainties were validated by comparison to standards of other National Metrology Institutes (NMIs). Results show agreement in all cases to better than the expanded (k = 2) uncertainty of the difference between NIST and the other NMIs, and in most cases to better than the standard (k = 1) uncertainty of the difference.     

Comparison of the dimensions of electrical resistance units, supported on the basis of the hall quantum effect, in the All-Russia Research Institute of Metrological Service and the Czech Metrological Institute

VNIIMS (Russia) and CMI (Czech Republic) quantum Hall resistance standards are compared using a VNIIMS portable resistance standard of 1 and 10 kΩ. Conformity is established for the dimensions of units within the limits of relative expanded (k = 2) uncertainty of 10⁻⁷. Translated from Izmeritel’naya Tekhnika, No. 12. pp. 58–61, December, 2008.     

Isochoric p–ρ–T Measurements for 2,2-Dichloro-1,1,1-Trifluoroethane (R123) at Temperatures from 176 to 380 K and 1-Chloro-1,2,2,2-Tetrafluoroethane (R124) from 104 to 400 K at Pressures to 35 MPa

The p––T relationships were measured for 2,2-dichloro-1,1,1-trifluoroethane (R123) and 1-chloro-1,2,2,2-tetrafluoroethane (R124) by an isochoric method with gravimetric determinations of the amount of substance. Temperatures ranged from 176 to 380 K for R123 and from 104 to 400 K for R124, while pressures extended up to 35 MPa. Measurements were conducted on compressed liquid samples. Most published p––T data are in good agreement with this study. The uncertainty is 0.03 K for temperature and 0.01% for pressure at p>3 MPa and 0.05% at p<3 MPa. The principal source of uncertainty is the cell volume (28.5 cm³), with a standard uncertainty of 0.003 cm³. When all components of experimental uncertainty are considered, the expanded relative uncertainty (with a coverage factor k=2 and, thus, a 2-SD estimate) of the density measurements is estimated to be 0.05%.     

一种准确可靠的期间核查方法

通过对选定的核查标准的测量,利用被核查标准的最大允许误差和校准证书给出的校准结果及其不确定度,来确定核查测量控制图的“中心线”、上下控制界限和警戒线,利用Excel电子表格绘出这些控制界限,使每次核查测量结果在图中直观准确地反映被核查标准的误差或量值的变化方向和大小。该结果也可作为计量标准稳定性考核的依据。     

自动标准压力发生器动态控制准确度分析

自动标准压力发生器(以下简称发生器)性能的一个重要指标是准确度。当发生器处于动态控制模式及使用面板显示时需要考虑附加控制准确度。本文通过对PACE6000系列的发生器进行比对实验,以活塞式压力计为标准器,高准确度数字压力计为参考对象,得出了其控制准确度,最后给出了以0.01级数字压力发生器为标准器、0.05级数字压力计示值误差的测量不确定度分析,其内置压力传感器的最大允许误差须为±0.008%。