Approximate Normal Distribution Research Articles

Droplet size distribution of oil-in-water Pickering emulsions stabilized by cellulose nanofibers using restricted diffusion NMR

In this study, droplet size distributions (DSDs) of oil-in-water Pickering emulsions stabilized by cellulose nanofibers (CNFs) estimated from NMR diffusion measurements in conjunction with cogent mathematical modelling based on the Gaussian phase distribution (GPD) approximation for diffusion in a sphere. The effects of oil types and Pickering stabilizer loadings on DSDs were investigated by using different n-alkanes (octane to hexadecane) and CNF concentrations (0.5 to 1%). The NMR diffusion data of the dispersed oil were well-described by the GPD model with a monomodal log-normal distribution. Pickering emulsions composed of higher n-alkane carbon number and/or CNF concentration were found to have smaller median radii, RMedian, and these RMedian estimates were in good agreement with laser scattering data. The volume-based RMedian against the CNF concentrations or the number of hydrocarbon n-alkanes exhibited an approximate exponential relationship.

An ErdŐs–Kac theorem for integers with dense divisors

Abstract We show that for large integers n, whose ratios of consecutive divisors are bound above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-{\rm e}^{-\gamma})\approx 2.280$ and V ≈ 0.414. This result is then generalized in two different directions.

Group sequential designs for pragmatic clinical trials with early outcomes: methods and guidance for planning and implementation

BackgroundGroup sequential designs are one of the most widely used methodologies for adaptive design in randomized clinical trials. In settings where early outcomes are available, they offer large gains in efficiency compared to a fixed design. However, such designs are underused and used predominantly in therapeutic areas where there is expertise and experience in implementation. One barrier to their greater use is the requirement to undertake simulation studies at the planning stage that require considerable knowledge, coding experience and additional costs. Based on some modest assumptions about the likely patterns of recruitment and the covariance structure of the outcomes, some simple analytic expressions are presented that negate the need to undertake simulations.MethodsA model for longitudinal outcomes with an assumed approximate multivariate normal distribution and three contrasting simple recruitment models are described, based on fixed, increasing and decreasing rates. For assumed uniform and exponential correlation models, analytic expressions for the variance of the treatment effect and the effects of the early outcomes on reducing this variance at the primary outcome time-point are presented. Expressions for the minimum and maximum values show how the correlations and timing of the early outcomes affect design efficiency.ResultsSimulations showed how patterns of information accrual varied between correlation and recruitment models, and consequentially to some general guidance for planning a trial. Using a previously reported group sequential trial as an exemplar, it is shown how the analytic expressions given here could have been used as a quick and flexible planning tool, avoiding the need for extensive simulation studies based on individual participant data.ConclusionsThe analytic expressions described can be routinely used at the planning stage of a putative trial, based on some modest assumptions about the likely number of outcomes and when they might occur and the expected recruitment patterns. Numerical simulations showed that these models behaved sensibly and allowed a range of design options to be explored in a way that would have been difficult and time-consuming if the previously described method of simulating individual trial participant data had been used.

Effect of Prolong-life-with-nine-turn-method (Yan Nian Jiu Zhuan) Qigong on fatigue and gastrointestinal function in patients with chronic fatigue syndrome: Study protocol for a randomized controlled trial.

Chronic fatigue syndrome (CFS) is a physical and mental disorder in which long-term fatigue is the main symptom. CFS patients are often accompanied by functional gastrointestinal diseases (FGIDs), which lead to decreased quality of life and increased fatigue. Prolong-life-with-nine-turn-method (PLWNT) is a kind of physical and mental exercise. Its operation includes adjusting the mind, breathing and cooperating with eight self-rubbing methods and one upper body rocking method. PLWNT was used to improve the digestive function in ancient China and to treat FGIDs such as functional dyspepsia and irritable bowel syndrome in modern times. Previous studies have shown that PLWNT can reduce fatigue in patients with CFS. But it is unclear whether the effect of PLWNT on CFS fatigue is related to gastrointestinal function. The aim of this study was to explore the relationship between PLWNT and fatigue and gastrointestinal function in patients with CFS. This study is a non-inferiority randomized controlled trial (RCT). The whole study period is 38 weeks, including 2 weeks of baseline evaluation, 12 weeks of intervention and 6 months of follow-up. Ninety-six CFS patients will be stratified random assigned to the intervention group (PLWNT) and the control group (cognitive behavior treatment) in the ratio of 1:1 through the random number table generated by SPSS. In the evaluation of results, Multidimensional Fatigue Inventory-20 (MFI-20), Gastrointestinal Symptom Rating Scale (GSRS), Bristol Stool Form Scale (BSFS), and Short Form 36 item health survey (SF-36) will be evaluated at week 0 (baseline), week 6 (midterm), week 12 (endpoint) and month 9 (follow up). The intestinal flora will be evaluated at week 0 (baseline) and week 12 (endpoint). The data results will be processed by statistical experts. The data analysis will be based on the intention to treat principle and per-protocol analysis. In the efficacy evaluation, repeated measurement analysis of variance will be used for data conforming to normal distribution or approximate normal distribution. The data which do not conform to the analysis of repeated measurement variance will be analyzed by the generalized estimation equation Linear discriminant analysis will be used to clarify the difference species of intestinal flora. The significance level sets as 5%. The safety of interventions will be evaluated after each treatment session. This trial will provide evidence to PLWNT exerting positive effects on fatigue and gastrointestinal function of CFS. It will further explore whether the improvement of PLWNT on CFS fatigue is related to gastrointestinal function. The trial was registered at Chinese Clinical Trial Registry http://www.chictr.org.cn/showproj.aspx?proj=151456 (Registration No.: ChiCTR2200056530). Date: 2022-02-07.

Numerical analysis on the transport properties and residence time distribution of ribbon biomass particles in a riser reactor based on CFD-DEM approach

A bended ribbon biomass particle model was developed to explore the dynamic transport properties inside a riser reactor. Residence time distribution (RTD) of the particles was analyzed by using the Eulerian-Lagrange method. The effects of sampling height, particle density, particle size and gas-to-solid mass ratio on RTD were investigated. The coupled Computational Fluid Dynamics and Discrete Element Method (CFD-DEM) model was verified firstly by experimental data on pressure drop and residence time distribution density function. The simulation results demonstrated that the ribbon biomass particles display a typical annular-core spatial distribution during transportation. The RTD of particles exhibit an approximate single-peaked normal distribution. The mean residence time (MRT) can reach up to 0.7 s when the particle density is 1200 kg/m3. Particle with higher density has longer mean residence time. The flow patterns are closer to plug flow if particle length over 12 mm. The particle flow pattern is not sensitive to changes in particle density and size, while the gas-to-material mass ratio has a significant impact on it.

Approximate likelihood and pseudo-likelihood inference in meta-analysis of diagnostic accuracy studies accounting for disease prevalence and study design.

Bivariate random-effects models represent a recommended approach for meta-analysis of diagnostic test accuracy, jointly modeling study-specific sensitivity and specificity. As the severity of the disease status can vary across studies, a proper analysis should account for the dependence of the accuracy measures on the disease prevalence. To this aim, trivariate generalized linear mixed-effects models have been proposed in the literature, although computational difficulties strongly limit their applicability. In addition, the attention has been mainly paid to cohort studies, where the study-specific disease prevalence can be estimated from, while information from case-control studies is often neglected. To overcome such limits, this article introduces a trivariate approximate normal model, which accounts for disease prevalence along with accuracy measures in cohort studies and sensitivity and specificity in case-control studies. The model represents an extension of the bivariate normal mixed-effects model originally developed for meta-analysis not accounting for disease prevalence, under an approximate normal within-study distribution for the logit of estimated sensitivity and specificity. The components of the approximate within-study covariance matrix are derived and the likelihood function is obtained in closed-form. The approximate likelihood approach is compared to that based on the exact within-study distribution and to its modifications following a pseudo-likelihood strategy aimed at reducing the computational effort. The comparison is based on simulation studies in a variety of scenarios, and illustrated in a meta-analysis about the accuracy of a test to diagnose fungal infection and a meta-analysis of a noninvasive test to detect colorectal cancer.

Sequential Gaussian co-simulation of tectono-geochemical anomaly for concealed ore deposit prediction

Tectono-geochemical data measured from tectonites host a wealth of information that is closely related to the accumulation and dispersion of ore-forming materials, while structure system is the only pathway to connect the deep orebody with the surficial tectono-geochemical anomaly, and both have correspondence and consistency. Based on the geological precondition that structure system and ore-forming process are closely associated with each other, the study proposed a method of sequential Gaussian co-simulation of tectono-geochemical anomaly, which integrates the tectono-geochemical anomaly with the ore-controlling structure system to solve two basic geological prospecting problems that are closely associated with ore-forming space and ore-forming material migration. A case study for concealed Pb–Zn prospecting based on tectono-geochemical data in the Huangzhuguang ore cluster (China) was employed to examine the potential applications of the proposed method. Firstly, centered logratio transformation was employed to remove the closure effect of tectono-geochemical data and to make the data conform to an approximate normal distribution. Secondly, sequential Gaussian co-simulation was applied to improve the simulation results of the tectono-geochemical anomaly by introducing the ore-controlling structure system. Thirdly, the probability distribution of tectono-geochemical anomalies was quantified by uncertainty modeling technique. The case study demonstrated that the proposed method can significantly improve the performance of the primary variable in predicting concealed ore deposits.

Biometric analysis of cajazeira fruits (Spondias mombin L.) in the northeastern Brazil

The cajazeira (Spondias mombin L.) is a fruit tree belonging to the Anacardiaceae family. Yellow mombins are very common in the caatinga ecoregion of northeastern of Brazil. This study assessed the biometry of cajazeira fruits. The fruits were collected in 2019 within an area of ​​ native vegetation in the campus of the Federal Rural University of the Semi-arid Region (UFERSA), Mossoró-RN. They were then taken to the Plant Science Laboratory to measure the following characteristics: length, width, length/width ratio, fruits thickness, and weight. The variables length, width, length/width ratio, thickness, and weight of the seeds had a low range of variation and low values ​​for relative variation, and therefore high coefficients of Pearson of correlation. The data of length, width, and thickness showed a moderate degree of symmetry and a symmetrical and mesocurtic distribution, while length/width ratio and fruits weight had almost symmetrical and mesocurtic distributions. The variables length and width and width and thickness showed significant positive linear correlations. The biometric data of cajafruits fitted in to the approximate normal distribution of probability.

Scoring from pairwise winning indices

The pairwise winning indices, computed in the Stochastic Multicriteria Acceptability Analysis, give the probability with which an alternative is preferred to another. They are computed taking into account all the instances of the assumed preference model compatible with the preference information provided by the Decision Maker mainly, but not exclusively, in terms of pairwise preference comparisons of reference alternatives. In this paper we present a new scoring method assigning a value to each alternative summarizing the results of the pairwise winning indices. Several procedures of this type have been provided in literature. However, our method, expressing the score in terms of a representative additive value function, permits to disaggregate the overall evaluation of each alternative in the sum of contributions of considered criteria. This permits not only to rank the alternatives but also to explain the reasons for which an alternative obtains its evaluation and, consequently, fills a certain ranking position. We also present a probabilistic model underlying our methodology. This probabilistic model is based on a simple piecewise linear approximation of the cumulative normal distribution, which allows the use of linear programming. To prove the efficiency of the method in representing the preferences of the Decision Maker, we performed an extensive set of simulations varying the number of alternatives and criteria. The results of the simulations, analyzed from a statistical point of view, prove the reliability of our procedure. The applicability of the method to decision making problems is explained by means of a case study related to the evaluation of financial funds.

Equilibrium pricing in supply chains with discrete stochastic demands: A case study in coffee supply and distribution industry

ABSTRACT This study addresses the optimization of price and production quantity using game theory in the coffee supply chain in the city of Isfahan. The presented model focuses on the competition among members of the chain in a probabilistic environment and is based on discrete selection models. This model examines the impact of inventory and routing costs on the pricing model at the same time. To effectively solve real-world problems, the method of normal distribution approximation is used instead of the binomial distribution. The results of the model indicate that the computational time of the applied approximate model does not change with increasing the number of customers, and the model can determine the optimal price and production quantity. The advantage of the proposed model over other classic models is that it considers the inventory costs that have not yet been considered in price competition models with discrete selection.

Optimal bounds in normal approximation for many interacting worlds

In this paper, we use Stein’s method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert and Wiseman (Phys. Rev. X 4 (2014) 041013). Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin (Ann. Appl. Probab. 26 (2016) 2540–2555).

Research on Spherical Self-rotating Polishing Technology

Polishing technology is widely used in the polishing process of aspheric optical elements, complex curved surface molds, and so on because of its high machining accuracy, good flexibility, and other advantages. Based on the Preston equation, this paper deduces the material removal function of spherical self-rotating polishing by applying the kinematics principle, obtains the removal function with approximate Gaussian distribution by using computer simulation method, conducts processing technology experiments through the spherical self-rotating polishing device, analyzes the influence of polishing pressure and polishing speed on the polishing removal characteristics of ceramic materials by simulation, and verifies the stability of the removal function through experiments. The results show that the removal function graph is Gaussian distribution, which is conducive to the surface shape convergence of airbag machining. The influence of process parameters on material removal rate is in good agreement with the theory and practice and has a good linear relationship. It is a controllable and deterministic processing mode.

Modeling of electroplated diamond wire and its application towards precision slicing of semiconductors

Electroplated diamond multi-wire sawing is becoming the mainstream technique for semiconductor slicing, because of its desirable traits such as thin slice thickness, low kerf loss, and high surface/subsurface quality. Precisely describing the electroplated diamond wire is of significant importance to slicing process analysis. This study presents an electroplated diamond wire model based on statistical measurements of the cutting-edge geometry and scratching direction of abrasives on wire. These measurements are performed on a commercial wire (Diamond Wire-160) after exposing cutting edges from plated nickel layers. Results show that up to 91 % of abrasives have a single cutting edge with the shape of a triangular or quadrangular pyramid. The geometric angles (α1, α2, γ1, γ2) of these pyramids are observed to follow an approximate Gaussian distribution, and the scratching direction follows a uniform distribution. Combined with nanoindentation theory, the developed model is further employed for slicing process analysis, in which slicing forces and bow angles of wire are calculated. Additionally, slicing experiments are conducted on silicon for model verification, and the maximum relative error between predicted and experimental bow angles is 6.25 %. The developed wire model provides an effective tool for slicing process analysis, thereby accelerating the process development for scale-up production of high-precision semiconductor substrates.

The reference intervals for thyroid hormones: A four year investigation in Chinese population.

The reference intervals (RIs) are of great importance for physicans to determine whether or not an individual is healthy. However, many clinical laboratories in China still adopted the default RI provided by the manufacturers; and these "uncalibrated" RIs might lead to the misdiagnosis of diseases. In the present study, we enroll reference people with the purpose of determining the RIs of serum triiodothyronine (T3), thyroxine (T4), free triiodothyronine (FT3), free thyroxine (FT4) and thyroid-stimulating hormone (TSH) in Chinese population, and explore the possible roles of age and sex on the levels of biomarkers. Serum samples from 66,609 individuals who met the inclusion criteria were analyzed using an Roche Cobas E 601 hormone analyzer. The dynamic trends of biomarker were visually assessed by their concentrations over age and sex. Specific partitions were determined by the method of Harris and Boyd. RIs, corresponding to the 2.5th and 97.5th percentiles, as well as the 0.5th, 25th, 50th, 75th and 99.5th percentiles were calculated for each reference partition using a non-parametric rank approach. The serum level of T3, T4, FT4 or TSH showed a right-skewed distribution in both males and females while FT3 presented an approximate normal distribution. Females had a higher mode value of serum T3 or T4, but a lower mode value of serum TSH, FT3 or FT4. All five biomarkers did not need age partitioning according to the approach of harris and boyd, while T3 and FT3 need sex partitioning. The present study not only determined the age- and sex-specific trends of the five thyroid hormones, but provided sex-stratified RIs for T3 and FT3, valuably contributing to the current literature and timely evaluation of thyroid health and disease.

Covert Communications in Intelligent Reflecting Surface-Assisted Two-Way Relaying Networks

Covert communications is critical in many application scenarios for ensuring transmission security and privacy. Two-way protocols are widely adopted in intelligent reflecting surface (IRS) assisted relaying networks to enable covert communications and prevent wireless signals from being overheard. To quantify the performance of covert communications in this special scenario, we derive the closed-form expression of outage probability and its asymptote in this paper. Considering the worst-case of covert communications, the optimal normalized power threshold of warden's detector is analyzed under a complex Gaussian distribution approximation. To meet the requirement of the detection error probability, the ratio of the transmission power between the covert nodes is investigated. Simulation results are provided to validate the theoretical expressions as well as the asymptotic analysis with a small number of reflecting elements.

Genetic dissection of QTLs for starch content in four maize DH populations.

Starch is the principal carbohydrate source in maize kernels. Understanding the genetic basis of starch content (SC) benefits greatly in improving maize yield and optimizing end-use quality. Here, four double haploid (DH) populations were generated and were used to identify quantitative trait loci (QTLs) associated with SC. The phenotype of SC exhibited continuous and approximate normal distribution in each population. A total of 13 QTLs for SC in maize kernels was detected in a range of 3.65–16.18% of phenotypic variation explained (PVE). Among those, only some partly overlapped with QTLs previously known to be related to SC. Meanwhile, 12 genes involved in starch synthesis and metabolism located within QTLs were identified in this study. These QTLs will lay the foundation to explore candidate genes regulating SC in maize kernel and facilitate the application of molecular marker-assisted selection for a breeding program to cultivate maize varieties with a deal of grain quality.

Spatial outlier detection on discrete GNSS velocity fields using robust Mahalanobis-distance-based unsupervised classification

GNSS velocity field filtering can be identified as a multidimensional unsupervised spatial outlier detection problem. To detect and classify the spatial outliers, we jointly interpret the horizontal and vertical velocity fields with the related standard deviations. We also describe the applied feature engineering process, which represents the underlying problem better than the initial attributes. According to this, we discuss the utilized algorithms and techniques, like the spatial- and non-spatial mapping functions, the k-nearest neighborhood (kNN) technique to retrieve the local environment of each GNSS station, as well as the principal component analysis (PCA) as a dimensionality reduction technique. We also assume that regular velocity field samples containing no outliers come from an approximate multivariate normal distribution (MVN) at the local scale. Regarding this, we evaluate the corresponding sample-wise distance related to model distribution, namely the Mahalanobis distance, with the estimation of the robust covariance matrix derived by the minimum covariant determinant (MCD) algorithm. Subsequently, we introduce the applied binary classification on the values of the derived robust Mahalanobis distances (RMD) which follows the χ2distribution. We also present three cases of artificially generated, pre-labeled synthetic velocity field datasets to perform cross-validation and comparison of the proposed RMD approach to other classification techniques. According to this, we found that k = 12 yields > 95% classification accuracy. While the compared methods have a mean classification accuracy of 96.2–99.8%, the advantage of the RMD approach is that it does not require pre-defined labels to indicate regular and outlier samples. We also demonstrate the proposed RMD based filtering process on a real dataset of the EUREF Permanent Network Densification velocity products. The RMD-based approach has been integrated into the EPN Densification as a quality checking algorithm. According to this, we also introduce a co-developed and regularly updated interactive webpage to disseminate the corresponding results.

Bayesian-based method for the simultaneous identification of structural damage and moving force

The input force is usually difficult to measure directly in the damage identification due to the system uncertainty and noise uncertainty. A Bayesian method for the simultaneous identification of damage and moving force is proposed. The moving force on the structure are transformed into equivalent nodal forces through the element function, and these equivalent nodal forces are expanded by using Chebyshev orthogonal polynomials. The posterior probability density function of the damage parameter and the force orthogonal parameter under Bayesian theory is deduced in time-domain. The prior information of the Chebyshev orthogonal polynomials adopts the uniform prior, and the damage parameters adopt the Gaussian prior in accordance with the characteristics of each parameter. The Laplace approximation method for time domain data is used to determine the hyperparameters of elemental damage and moving force. The first-order and second-order sensitivities of the dynamic response with damage and force parameters are derived to update the hyperparameters and to accelerate gradient-based system identification. The uncertainty of the identified results of damage parameters is quantified by using an approximate Gaussian distribution. Two numerical cases of a simply supported beam and the Tanjiang Bridge on the Shen-Mao Railway show that the proposed method can simultaneously identify the elemental damage and moving forces with uncertainty quantified.

Measuring sub-surface spatially varying thermal conductivity of silicon implanted with krypton

The thermal properties of semiconductors following exposure to ion irradiation are of great interest for the cooling of electronic devices; however, gradients in composition and structure due to irradiation often make the measurement difficult. Furthermore, the nature of spatial variations in thermal resistances due to spatially varying ion irradiation damage is not well understood. In this work, we develop an advancement in the analysis of time-domain thermoreflectance to account for spatially varying thermal conductivity in a material resulting from a spatial distribution of defects. We then use this method to measure the near-surface (≲1 μm) thermal conductivity of silicon wafers irradiated with Kr+ ions, which has an approximate Gaussian distribution centered 260 nm into the sample. Our numerical analysis presented here allows for the spatial gradient of thermal conductivity to be extracted via what is fundamentally a volumetric measurement technique. We validate our findings via transmission electron microscopy, which is able to confirm the spatial variation of the sub-surface silicon structure, and provide additional insight into the local structure resulting from the effects of ion bombardment. Thermal measurements found the ion stopping region to have a nearly 50× reduction in thermal conductivity as compared to pristine silicon, while TEM showed the region was not fully amorphized. Our results suggest this drastic reduction in silicon thermal conductivity is primarily driven by structural defects in crystalline regions along with boundary scattering between amorphous and crystalline regions, with a negligible contribution being due to implanted krypton ions themselves.

Age-related normal limits for spatial vision.

To establish age-related, normal limits of monocular and binocular spatial vision under photopic and mesopic conditions. Photopic and mesopic visual acuity (VA) and contrast thresholds (CTs) were measured with both positive and negative contrast optotypes under binocular and monocular viewing conditions using the Acuity-Plus (AP) test. The experiments were carried out on participants (age range from 10 to 86 years), who met pre-established, normal sight criteria. Mean and ± 2.5σ limits were calculated within each 5-year subgroup. A biologically meaningful model was then fitted to predict mean values and upper and lower threshold limits for VA and CT as a function of age. The best-fit model parameters describe normal aging of spatial vision for each of the 16 experimental conditions investigated. Out of the 382 participants recruited for this study, 285 participants passed the selection criteria for normal aging. Log transforms were applied to ensure approximate normal distributions. Outliers were also removed for each of the 16 stimulus conditions investigated based on the ±2.5σ limit criterion. VA, CTs and the overall variability were found to be age-invariant up to ~50 years in the photopic condition. A lower, age-invariant limit of ~30 years was more appropriate for the mesopic range with a gradual, but accelerating increase in both mean thresholds and intersubject variability above this age. Binocular thresholds were smaller and much less variable when compared to the thresholds measured in either eye. Results with negative contrast optotypes were significantly better than the corresponding results measured with positive contrast (p < 0.004). This project has established the expected age limits of spatial vision for monocular and binocular viewing under photopic and high mesopic lighting with both positive and negative contrast optotypes using a single test, which can be implemented either in the clinic or in an occupational setting.