Wilson Methods Research Articles

Confidence intervals of difference and ratio between two means of delta-Birnbaum-Saunders distributions

During the dry season, the northern region of Thailand consistently grapples with compromised air quality, notably an excessive level of particulate matter with a diameter of less than 2.5 microns (PM2.5), which poses implications for public health. Wind speed emerges as the primary factor that diminishes the concentration of PM2.5, making the evaluation of wind speed changes crucial in understanding and potentially mitigating the impact of air pollution on public health in the region. The wind speed data contain zero and positive values. This can be estimated through the confidence intervals of the difference and ratio between two means of delta-Birnbaum-Saunders distributions using various methods, including the generalized confidence interval (GCI), bootstrap confidence interval (BCI), and generalized fiducial confidence interval (GFCI). These methods rely on variance stabilizing transformations (VST), the Wilson and Hannig methods. According to the simulation, for small sample sizes, the GCI based on the VST and Hannig was the most effective method in terms of coverage probabilities and average lengths. For medium sample sizes, it was found that the performance of GFCI based on the VST method was superior to other methods. Additionally, when confronted with a large sample size, the BCI based on the VST method becomes relevant and will be introduced. To demonstrate the efficacy of our proposed methods, we applied them to wind speed data gathered from Chiang Mai Airport station in Thailand from January to February in both 2022 and 2023.

Estimation methods for the variance of Birnbaum-Saunders distribution containing zero values with application to wind speed data in Thailand.

Thailand is currently grappling with a severe problem of air pollution, especially from small particulate matter (PM), which poses considerable threats to public health. The speed of the wind is pivotal in spreading these harmful particles across the atmosphere. Given the inherently unpredictable wind speed behavior, our focus lies in establishing the confidence interval (CI) for the variance of wind speed data. To achieve this, we will employ the delta-Birnbaum-Saunders (delta-BirSau) distribution. This statistical model allows for analyzing wind speed data and offers valuable insights into its variability and potential implications for air quality. The intervals are derived from ten different methods: generalized confidence interval (GCI), bootstrap confidence interval (BCI), generalized fiducial confidence interval (GFCI), and normal approximation (NA). Specifically, we apply GCI, BCI, and GFCI while considering the estimation of the proportion of zeros using the variance stabilized transformation (VST), Wilson, and Hannig methods. To evaluate the performance of these methods, we conduct a simulation study using Monte Carlo simulations in the R statistical software. The study assesses the coverage probabilities and average widths of the proposed confidence intervals. The simulation results reveal that GFCI based on the Wilson method is optimal for small sample sizes, GFCI based on the Hannig method excels for medium sample sizes, and GFCI based on the VST method stands out for large sample sizes. To further validate the practical application of these methods, we employ daily wind speed data from an industrial area in Prachin Buri and Rayong provinces, Thailand.

What Does the Archaeomagnetic Record of Fired Ceramics Reflect?

A complex of archaeomagnetic and rock-magnetic studies is conducted to compare the archaeomagnetic intensity determined by the Thellier–Coe and Wilson methods with the known “true” value of the geomagnetic field during the firing of the ceramic samples manufactured on February 21, 2017 in Myshkin, Yaroslavl region, Russia. The obtained results show two different values of archaeomagnetic intensity corresponding to two temperature intervals. The values obtained in the low-temperature interval (∼150–350°C) are approximately 13 µT lower than the true value, while the values obtained in the interval of ∼350–600°C are fairly close to the “true” ones. The cause of the phenomenon is likely due to the presence of small magnetic grains in the ceramic under study, which are close in size to superparamagnetic ones and are capable of resuming their growth upon heating and reaching the particle sizes in a single-domain state. The results of rock-magnetic studies suggest that thermoremanent magnetization in the studied ceramics is carried by grains of oxidized magnetite, hematite, and possibly ε-Fe2O3.

High-temperature mass spectrometric study of thermodynamic properties in the TiO2 -Al2 O3 -SiO2 system and modeling.

The TiO2 -Al2 O3 -SiO2 system is the base for various glass-ceramic materials, which have great practical value for a large number of modern technologies. Many TiO2 -Al2 O3 -SiO2 materials are synthesized or applied at high temperatures, which justifies the relevance of the present study. The samples in the TiO2 -Al2 O3 -SiO2 system were synthesized using the method of induction melting in a cold crucible. The thermodynamic properties of the TiO2 -Al2 O3 -SiO2 system were studied using the Knudsen effusion mass spectrometric method. The derived thermodynamic functions were optimized within the generalized lattice theory of associated solutions (GLTAS) approach and compared with the results of calculation using the semiempirical Kohler, Muggianu, Toop, Redlich-Kister, and Wilson methods based on the corresponding data in the binary systems. The SiO2 selective vaporization from the samples under study was shown at temperatures above 1940 K. The thermodynamic properties in the TiO2 -Al2 O3 -SiO2 system, including the TiO2 -SiO2 system, were obtained in the temperature range 1965-2012 K and were optimized using GLTAS to obtain the consistent concentration dependences of the component activities and excess Gibbs energies. Positive deviations from the ideal behavior were observed in the TiO2 -Al2 O3 -SiO2 system at high temperatures. Comparison of these values with the results of the modeling based on the GLTAS approach allowed the recommendations regarding the optimal semiempirical methods for the excess Gibbs energy calculation in different concentration ranges to be made.

High-temperature mass spectrometric study of the thermodynamic properties in the Sm2 O3 -ZrO2 -HfO2 system.

The Sm2 O3 -ZrO2 -HfO2 system is a promising base for the development of a wide spectrum of new refractory materials. Reliable data on thermodynamic properties in this system are of significant importance for planning the preparation and application of high-temperature ceramics. Especially, they can be useful for calculation of the unknown phase equilibria in this system. The thermodynamic properties of the Sm2 O3 -ZrO2 -HfO2 system were studied by the high-temperature mass spectrometric method. The samples in the system under consideration synthesized by the solid-state method were vaporized from a tungsten twin effusion cell using a MS-1301 magnetic sector mass spectrometer. Ionization of the vapor species effusing from the cell was carried out by electrons at an energy of 25 eV. It was shown that, at temperatures below 2500 K, the main vapor species over the ceramics based on the Sm2 O3 -ZrO2 -HfO2 system were SmO, Sm, and O corresponding to vapor composition over pure Sm2 O3 . The SmO, Sm, and O partial vapor pressures over the samples and the Sm2 O3 activities were obtained in the temperature range 2319-2530 K. This allowed the excess Gibbs energy values to be determined. For comparison, the excess Gibbs energies in the Sm2 O3 -ZrO2 -HfO2 system were also calculated by the semi-empirical Kohler, Toop, Redlich-Kister, and Wilson methods and optimized by the statistical thermodynamic Generalized Lattice Theory of Associated Solutions (GLTAS). The thermodynamic data calculated by the semi-empirical approaches at 2423 K were shown to be lower than the experimental values. However, the Toop and Wilson methods were found to be useful for evaluation of the excess Gibbs energy values at the Sm2 O3 mole fraction less and higher than 0.32, respectively. The self-consistent thermodynamic description of the Sm2 O3 -ZrO2 -HfO2 system was derived at high temperatures by optimization of the experimental results using the GLTAS.

A brief review on advanced renewable materials for supporting artificial insemination technology

This paper is a review paper. National meat self-sufficiency program (MSSP), which has been launched since 2000, has been well improved in term of concept, policy instrument, and program management, but the supporting technologies are still as one of some constraints. The study was aimed to metaanalyze comprehensively the frontier technologies supporting the national MSSP, especially advanced materials for artificial insemination technology in order to enhance animal population. The meta-analysis study adopted Wilson and Kelly metanalisis methods consisting of nine steps. Results of the study showed that (1) there were some frontier advanced technologies on animal husbandry in terms of breeding, fattening up and farm management, as well as animal feeds and feedstock’s; (2) those technologies consisted of advancing materials dan methods/process; and (3) particularly, in supporting breeding technology, advancing artificial insemination technologies take important role for significantly enhancing the animal population as well as effectiveness of animal breeding, thus advancing the materials as well as processes involved are highly needed. The study concluded that advanced renewable materials for supporting artificial insemination technology should be taken into account for advancement of animal husbandry technology.

Detection of SARS-CoV-2 Antibodies in Kidney Transplant Recipients.

Detection of SARS-CoV-2 Antibodies in Kidney Transplant Recipients.

Ultra-low palaeointensities from East European Craton, Ukraine support a globally anomalous palaeomagnetic field in the Ediacaran

SUMMARY The time-averaged geomagnetic field is generally purported to be uniformitarian across Earth history—close to a geocentric axial dipole, with average strength within one order of magnitude of that at present. Nevertheless, recent studies have reported that the field was approximately ten times weaker than present in the mid-Palaeozoic (∼410–360 Ma) and late Ediacaran (∼565 Ma). Here we present the first whole-rock palaeointensity determinations of Ediacaran age outside of Laurentia. These were obtained by the Thellier-Coe, Wilson and microwave methods for basaltic rocks of 560–580 Ma age of the Ediacaran traps, southwestern margin of the East European Craton, Ukraine. All four studied sites showed extremely low instantaneous field values of (3–7) μT with corresponding VDMs of (0.4–1) × 1022 Am2. Summarizing all available data, the Ediacaran field appears to be anomalously characterized by ultra-low dipole moment and ultra-high reversal frequency. According to some geodynamo models, this state could indicate a weak dipole field regime prior to the nucleation of the solid inner core. However, given that ultra-low field intensities have also been detected in the Devonian, and that virtually no palaeointensity data exist for the intervening ∼150 Ma, the date of inner core nucleation remains extremely uncertain. Our new evidence of persistent ultra-weak magnetospheric shielding in the Ediacaran may be considered consistent with the recently hypothesized link between enhanced UV-B radiation in this interval and the subsequent Cambrian evolutionary radiation.

Ultrasound findings of the liver of Wilson's disease

Purpose To evaluate and correlate US findings of liver involvement of Wilson's disease Materials and Methods US findings of liver in fourteen patients (9 male, 5 female, between the age of 13-55, median age of 32) with clinically proven Wilson's disease were evaluated. CT and MRI were done in 9 and 5 patients. US findings were correlated with CT and MR images. Results Sonographic features of liver of Wilson's disease included cirrhosis in 8 patients, cirrhosis with normal caudate to right lobe ratio in 7 patients, increased echogenicity in 2 patients, contour irregularity in 8, increased periportal thickness in 11 patients, increased ligamentum teres thickness in 7 patients, heterogeneous parenchyma in 14 patients, hypoechoic nodules in 5 patients, multiple hyperechoic and hypoechoic nodules (mixed pattern) in 5 patients, evidence of perihepatic fat layer in 8 patients. Conclusion Liver involvement of Wilson's disease could be characterized by multiple nodular lesions in the liver, presence of perihepatic fat layer and cirrhosis without caudate lobe hypertrophy which is different from other types of cirrhosis. US could be used to detect early parenchymal changes of liver injury in the progress of Wilson's disease.

Was the Devonian geomagnetic field dipolar or multipolar? Palaeointensity studies of Devonian igneous rocks from the Minusa Basin (Siberia) and the Kola Peninsula dykes, Russia

Abstract Defining variations in the behaviour of the geomagnetic field through geological time is critical to understanding the dynamics of Earth's core and its response to mantle convection and planetary evolution. Furthermore, the question of whether the axial dipole dominance of the recent palaeomagnetic field persists through the whole of Earth's history is fundamental to determining the reliability of palaeogeographic reconstructions and the efficacy of the magnetosphere in shielding Earth from solar wind radiation. Previous palaeomagnetic directional studies have suggested that the palaeofield had a complex configuration in the Devonian period (419–359 Ma). Here we present new high-quality palaeointensity determinations from rocks aged between 408 and 375 Ma from the Minusa Basin (southern Siberia), and the Kola Peninsula that enable the first reliable investigation of the strength of the field during this enigmatic period. Palaeointensity experiments were performed using the thermal Thellier, microwave Thellier and Wilson methods on 165 specimens from 25 sites. Six out of eight successful sites from the Minusa Basin and all four successful sites from the Kola Peninsula produced extremely low palaeointensities (<10 μT). These findings challenge the uniformitarian view of the palaeomagnetic field: field intensities of nearly an order of magnitude lower than Neogene values (except during relatively rare geomagnetic excursions and reversals) together with the widespread appearance of strange directions found in the Devonian suggest that the Earth's field during this time may have had a dominantly multipolar geometry. A persistent, low intensity multipolar magnetic field and associated diminished magnetosphere would increase the impact of solar particles on the Earth's magnetosphere, ionosphere and atmosphere with potential major implications for Earth's climate and biosphere.

Broadening Microstructure Analysis Program(BMAP): an isotropic and anisotropic crystallite size and strain calculator

BMAPcalculator is a simplified software tool to obtain microstructure characteristics and to investigate their isotropic and anisotropic nature using some common broadening methods. The Scherrer and Stokes–Wilson methods as well as the Williamson–Hall plot (microstructure isotropy) and modified Williamson–Hall models (strain anisotropy) are implemented in the calculations. TheBMAPinput data are the X-ray diffraction peak parameters (the full width at half-maximum or integral breadth and peak position) of a measured line profile and of a reference one. The components of the measured line profile or corrected broadening are based on several approximations (Cauchy, Gauss or Voigt). The linear least-squares fit is used to calculate the size–strain values, and the correlation coefficient for accepting the results. The calculator is applied to an X-ray diffraction experiment of NiO nanoparticles prepared by a sol–gel method for testing. The results ofBMAPcalculator and theMAUDprogram, combined with an anisotropic Popa rules approach, are compared. The uniform stress model based on the Cauchy approximation is the most suitable one in cubic nanostructure nickel oxide, with an average crystallite size of 24 nm and anisotropic strains of 1.54 × 10−3, 0.61 × 10−3, 1.77 × 10−3and 1.49 × 10−3using the {111}, {200}, {220} and {311} reflections, respectively. Generally, for non-Rietveld users, it is useful to collect different methods into one program to depict the anisotropic nature of materials.

Prediction of wave parameters by using fuzzy inference system and the parametric models along the south coasts of the Black Sea

Forecasting of sea-state characteristics has a great importance in coastal and ocean engineering studies. Therefore, the purpose of this study was to investigate performances of Adaptive-Network-Based Fuzzy Inference System (ANFIS) and several parametric methods in the Black Sea. For this purpose, different fuzzy models with different input combinations were developed for two different wind data sources (TSMS and ECMWF) at two offshore buoy stations. It also aimed to apply several approaches to event-based data sets for wave predictions. Generally, in literature the tendency is to use time series data for wave predictions. In this kind of prediction approach, lagged time series data are taken as inputs and current or future variables are taken as output. In this study, event-based data for each independent storm were extracted from time series data. Simultaneous or concurrent data of wind speed, blowing duration, fetch length and wave characteristics were detected for each single storm. These event data were then used to set up models. The hindcast results were validated with significant wave height and mean wave period data recorded in Hopa and Sinop buoy stations. The performance of developed fuzzy models were also compared with that of four different parametric methods (Wilson, SPM, Jonswap, and CEM methods) applied for two wind data sources at both buoy stations. Finally, it was determined that in the prediction of both wave parameters (Hs and Tz) the ANFIS models (R = 0.66, squared correlation coefficient, and MAE = 0.37 m, mean absolute error, for the best model in prediction of Hs) were more accurate than the parametric methods (R = 0.63 and MAE = 0.75 m for the best model in prediction of Hs).

A lemma on binomial coefficients and applications to Lee weights modulo 2 e of codes over $${\mathbb{Z}_4}$$

In this work, we prove that if C is a free $${\mathbb{Z}_4}$$ -module of rank k in $${\mathbb{Z}_4^n}$$ , and $${j\in \mathbb{Z}}$$ and e ? 1, then the number of codewords in C with Lee weight congruent to j modulo 2 e is divisible by $${2^{\left \lfloor \large{\frac{k-2^{e-2}}{2^{e-2}}} \right \rfloor}}$$ . We prove this result by introducing a lemma and applying the lemma in one of the theorems proved by Wilson. The method used is different than the one used in our previous work on Lee weight enumerators in which more general results were obtained. Moreover, Wilson's methods are used to prove that the results obtained are sharp by calculating the power of 2 that divides the number of codewords in the trivial code $${\mathbb{Z}_{4}^k}$$ with Lee weight congruent to j modulo 2 e .

Stratified Wilson and Newcombe Confidence Intervals for Multiple Binomial Proportions

This article proposes the stratified Wilson confidence interval for multiple binomial proportions and the stratified Newcombe confidence intervals for multiple binomial proportion differences. Both confidence intervals are presented in closed forms to facilitate easy calculations. The confidence levels of the proposed intervals are theoretically justified and demonstrated through extensive simulations. The coverage rates are found to be rather satisfactory. When the Wilson and Newcombe methods are used in unstratified analysis, the proposed methods may serve as the counterparts for stratified analysis. The proposed methods are applied to a vaccine trial to compute the stratified sero-conversion rate and rate difference over multiple clinical centers.

A numerical method for factorizing the rational spectral density matrix

Improving Rozanov (1967, Stationary Random Processes. San Francisco: Holden-day.)’s algebraic-analytic solution to the canonical factorization problem of the rational spectral density matrix, this article presents a feasible computational procedure for the spectral factorization. We provide numerical comparisons of our procedure with the Bhansali's (1974, Journal of the Statistical Society, B36, 61.) and Wilson's (1972 SIAM Journal on Applied Mathematics, 23, 420) methods and illustrate its application in estimation of invertible MA representation. The proposed procedure is usefully applied to linear predictor construction, causality analysis and other problems where a canonical transfer function specification of a stationary process in question is required.

A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads

Abstract A mixed method is presented to study the dynamic behavior of functionally graded (FG) beams subjected to moving loads. The theoretical formulations are based on Euler–Bernoulli beam theory, and the governing equations of motion of the system are derived using the Lagrange equations. The Rayleigh–Ritz method is employed to discretize the spatial partial derivatives and a step-by-step differential quadrature method (DQM) is used for the discretization of temporal derivatives. It is shown that the proposed mixed method is very efficient and reliable. Also, compared to the single-step methods such as the Newmark and Wilson methods, the DQM gives better accuracy using larger time step sizes for the cases considered. Moreover, effects of material properties of the FG beam and inertia of the moving load on the dynamic behavior of the system are investigated and analyzed.

Incidence and Prevalence of Patients with Myelodysplastic Syndromes (MDS) in Düsseldorf 1996-2005.

Abstract 1774Poster Board I-800 BackgroundPopulation-based data on myelodysplastic syndromes (MDS) are scarce. Since its inception in 1982, the Düsseldorf MDS Registry has captured 3598 patients with MDS, of whom 21 % live in the town district of Düsseldorf. Between 1996-2005 we are confident that all MDS patients residing in the town district of Düsseldorf were entered and followed in the registry through regular follow-ups. As yearly age- and sex-specific population counts are also available for Düsseldorf, the Registry provides a unique opportunity to estimate MDS disease frequency. Here we aim to quantify the incidence and prevalence of MDS disease subtypes. MethodsPatients were identified from the MDS Registry. Age- and sex-specific yearly population counts were obtained from the Statistical Office of North-Rhine Westphalia. The number of residents in Düsseldorf minimally increased during the study period from 1996 to 2005 (mean: 571,000 residents). The number of patients with a first-time MDS diagnosis in a given calendar year (incident cases) was divided by the total Düsseldorf population in that year to estimate incidence. Prevalence was estimated by dividing the number of existing plus incident MDS patients in a given year by the total population of Düsseldorf that year. Incidence (per 100,000 person-years (PY) and prevalence per 100,000 persons are presented with corresponding 95% confidence intervals (CI) calculated using the delta and Wilson methods, respectively. Low-risk MDS was defined as WHO subtypes RA, RCMD or MDS with del(5q). High-risk included subtypes RAEB-I and RAEB-II. Results344 MDS patients were included in our analyses (279 incident patients). Incidence and prevalence of low-risk MDS was 2.87 (95%CI 2.46–3.35) per 100,000PY and 12.4 (95%CI 11.5–13.4) per 100,000 persons, respectively, with no difference between men and women (Table). High-risk disease was less common. The incidence of high-risk disease in men appeared to be higher (1.22 (95%CI 0.87–1.72) per 100,000PY) than in women (0.63 (95%CI 0.40–0.99) per 100,000PY). Prevalence of high-risk disease among men was statistically significantly higher in men compared to women (2.93 (95%CI 2.35 – 3.65) per 100,000 persons) and 1.56 (95%CI 1.17 – 2.08) per 100,000 persons), respectively (Table). ConclusionIn this population-based study we found that MDS more common in men than in women. The majority of MDS cases had low-risk disease. Incidence and prevalence of high risk disease (but not low risk) is higher in men compared to women.Financial disclosures: This study was supported by Amgen Inc.Table 1Incidence and prevalence of myelodysplastic syndromes (MDS) by disease subtype in Düsseldorf 1996 - 2005MDS subtypeIncidencePrevalenceNIncidence rate1 (95% CI)N2Prevalence3 (95% CI)Overall42794.88 (4.34–5.49)34417.8 (16.7–18.9)Male1555.74 (4.90–6.72)19119.1 (17.5–20.8)Female1244.12 (3.45–4.91)15316.6 (15.2–18.2)Disease risk groupLow-risk disease51642.87 (2.46–3.35)20112.4 (11.5–13.4)Male833.07 (2.48–3.81)10112.5 (11.3–13.9)Female812.69 (2.16–3.34)10012.3 (11.1–13.6)High-risk disease6520.91 (0.69–1.19)642.21 (1.85–2.63)Male331.22 (0.87–1.72)412.9 (2.35–3.65)Female190.63 (0.40–0.99)231.56 (1.17–2.08)WHO classification4RA/RARS370.65 (0.47–0.89)503.92 (3.44–4.47)5q-50.09 (0.04–0.21)70.42 (0.28–0.63)RCMD/RCMDRS1222.14 (1.79–2.55)1438.07 (7.37–8.84)RAEB I210.37 (0.24–0.56)281.16 (0.91–1.47)RAEB II310.54 (0.38–0.77)361.05 (0.82–1.35)CMML I300.53 (0.37–0.75)401.96 (1.63–2.36)CMML II70.12 (0.06–0.26)80.26 (0.16–0.43)RAEB-T260.46 (0.31–0.67)310.93 (0.71–1.21)1) Incidence per 100,000 person-years2) Prevalent patients alive during >1 calendar year counted once each year3) Prevalence per 100,000 persons4) Including CMML and RAEB-T5) WHO subtypes RA,RARS,RCMD,RCMDRS and del (5q)6) WHO subtypes RAEB-I, RAEB-II DisclosuresNeukirchen:Amgen Inc.: Research Funding. Schoonen:Amgen Inc.: Research Funding. Aul:Amgen Inc.: Research Funding. Haas:Amgen Inc.: Research Funding. Gattermann:Amgen Inc., Celgene, Novartis: Honoraria, Research Funding. Germing:Amgen Inc., Celgene, Novartis: Honoraria, Research Funding.

Ann-Christin Rommen in conversation with Maria Shevtsova Experiencing the Movement: Working with Robert Wilson

Ann-Christin Rommen began to work with Robert Wilson in 1983 with The CIVIL warS. Her long collaboration with him since then has included more than thirty productions, most recently her co-direction of Dreamplay (Stockholm, 1998), Woyzeck (Copenhagen, 2000), Peer Gynt (Oslo, 2005), and The Winter's Tale (Berlin, 2005). She also has her own company in Germany, Codanza, which combines contemporary dance and early music, and whose work has been seen in Europe, Asia, and Australia. In this interview with Maria Shevtsova, which began in Oslo on 7 February 2005, Ann-Christin Rommen offers insights into Wilson's working methods, with particular reference to his Oslo Peer Gynt. Maria Shevtsova, Co-Editor of NTQ, is the author of Robert Wilson for the Routledge Performance Practitioners Series (forthcoming 2007).

Kondo effect with noncollinear polarized leads: A numerical renormalization group analysis

The Kondo effect in quantum dots attached to ferromagnetic leads with general polarization directions is studied combining poor man scaling and Wilson's numerical renormalization group methods. We show that polarized electrodes will lead in general to a splitting of the Kondo resonance in the quantum dot density of states except for a small range of angles close to the antiparallel case. We also show that an external magnetic field is able to compensate this splitting and restore the unitary limit. Finally, we study the electronic transport through the device in various limiting cases.

Isothermal Vapor Liquid Equilibrium for Binary 2-Methylpropene + Methanol to Butanol Systems

Vapor−liquid equilibria measured with a static apparatus are reported for binary mixtures of 2-methylpropene and the following C1−C4 alcohols at the corresponding temperatures: methanol (323.15 K), ethanol (323.15 K), 2-propanol (323.15 K), 2-butanol (323.15 K), and 2-methyl-2-propanol (313.15 K and 322.77 K). Measured pTz (pressure−temperature−total composition) data were reduced to liquid- and vapor-phase compositions using Barker's method. Binary interaction parameters were optimized for the Legendre polynomial, Wilson, UNIQUAC, and nonrandom two-liquid methods. An error analysis for all measured results was performed. All systems indicated a positive deviation from Raoult's law. Azeotropic behavior was observed for 2-methyl propene + methanol binary.