Trapezoidal Rule Research Articles

Investigating the Response Variability of Statically Determined Sandwich Beams considering two Random Fields of Elastic Modulus

In this paper, the displacement variation in sandwich beams is determined by employing a semi-analytical approach. The classical displacement is calculated by integration using Mohr’s equation, although the integration is complicated due to the inclusion of random fields in the inertial moment term. Using the trapezoidal rule to compute these integrals, the random fields are discretized into random variables at the nodal point of the beam segments. Thus, the expected displacement, standard deviation, and coefficient of variation can be computed. To validate the results, the random fields are simulated using a previously described spectral method. The results of numerical examples were compared with the semi-analytical method and the Monte Carlo simulation demonstrating the high accuracy of the proposed method. The results also illustrate the influence of the parameters of the random fields of elastic modulus on the variability of displacement.

On the integral solution of hyperbolic Kepler’s equation

In a recent paper of Philcox, Goodman and Slepian, the solution of the elliptic Kepler’s equation is given as a quotient of two contour integrals along a Jordan curve that contains in its interior the unique real solution of the elliptic Kepler’s equation and does not include other complex zeroes. In this paper, we show that a similar explicit integral solution can be given for the hyperbolic Kepler’s equation. With this purpose, we carry out a study of the complex zeros of the hyperbolic Kepler’s equation in order to define suitable Jordan contours in the integrals. Even more, we show that appropriate elliptic Jordan contours can be defined for such integrals, which reduces the computing time. Moreover, using the ideas behind the fast Fourier transform (FFT) algorithm, these integrals can be approximated by the composite trapezoidal rule which gives an algorithm with spectral convergence as a function of the number of nodes. The results of some numerical experiments are presented to show that this implementation is a reliable and very accurate algorithm for solving the hyperbolic Kepler’s equation.

IN VIVO EVALUATION OF OPTIMIZED FORMULATION OF DAPAGLIFLOZIN AND SAXAGLIPTIN BILAYERED TABLETS

Combining dapagliflozin and saxagliptin presents a promising strategy for managing type 2 diabetes mellitus by leveraging their distinct, yet complementary, mechanisms of action. A novel bilayer tablet with a 5 mg sustained release layer of saxagliptin and a 10 mg immediate release layer of dapagliflozin was developed. Live animal studies (in vivo) were conducted on rabbits to evaluate the effects of an optimized formulation. Based on in vivo performance, the novel bilayer tablets demonstrated greater bioavailability. A new, easy-to-use technique was created to simultaneously measure the two drugs (dapagliflozin and saxagliptin) in rabbit blood plasma. Evaluation of the technique’s parameters were done on rabbit plasma in accordance with ICH guidelines. The parameters for pharmacokinetic analysis were (AUC0-∞), (Cmax), Tmax and others. Direct calculations of the Cmax and Tmax were made using experimental plasma concentration versus time data. The AUC0-∞ was produced by adding the computed AUC0-24h using the trapezoidal rule. Using sample analysis of variance or independent sample t tests, average data variation was compared (one way analysis of variance). Statistical significance (p=0.05) was assessed using a 95% confidence interval.

Postprandial glucose variability and clusters of sex hormones, liver enzymes, and cardiometabolic factors in a South African cohort of African ancestry

IntroductionThis study aimed to, first, determine the clusters of sex hormones, liver enzymes, and cardiometabolic factors associated with postprandial glucose (PPG) and, second to evaluate the variation these clusters account...

Numerical analysis of COVID-19 model with Caputo fractional order derivative

This paper focuses on the numerical solutions of a six-compartment fractional model with Caputo derivative. In this model, we obtain non-negative and bounded solutions, equilibrium points, and the basic reproduction number and analyze the stability of disease free equilibrium point. The existence and uniqueness of the solution are proven by employing the Picard–Lindelof approach and fixed point theory. The product–integral trapezoidal rule is employed to simulate the system of FODEs (fractional ordinary differential equations). The numerical results are presented in the form of graphs for each compartment. Finally, the sensitivity of the most important parameter (β) and its impact on COVID-19 dynamics and the basic reproduction number are reported.

Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions

This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently was introduced for boundaries discretized using composite Gauss–Legendre quadrature. The original method builds on swapping the target singularity for its preimage in the complexified space of the curve parametrization, where the source panel is flat. This allows the integral to be efficiently evaluated using an interpolatory quadrature with a monomial basis. In this extension, we use the target preimage to swap the singularity to a point close to the unit circle. This allows us to evaluate the integral using an interpolatory quadrature with complex exponential basis functions. This is well-conditioned, and can be efficiently evaluated using the fast Fourier transform. The resulting method has exponential convergence, and can be used to accurately evaluate layer potentials close to the source geometry. We report experimental results on a simple test geometry, and provide a baseline Julia implementation that can be used for further experimentation.

Distribution and elimination kinetics of midazolam and metabolites after post-resuscitation care: a prospective observational study

Administration of sedatives for post-resuscitation care can complicate the determination of the optimal timing to avoid inappropriate, pessimistic prognostications. This prospective study aimed to investigate the distribution and elimination kinetics of midazolam (MDZ) and its metabolites, and their association with awakening time. The concentrations of MDZ and its seven metabolites were measured immediately and at 4, 8, 12, and 24 h after the discontinuation of MDZ infusion, using liquid chromatography-tandem mass spectrometry. The area under the time-plasma concentration curve from 0 to 24 h after MDZ discontinuation (AUClast) was calculated based on the trapezoidal rule. Of the 15 enrolled patients, seven awakened after the discontinuation of MDZ infusion. MDZ and three of its metabolites were major compounds and their elimination kinetics followed a first-order elimination profile. In the multivariable analysis, only MDZ was associated with awakening time (AUClast: R2 = 0.59, p = 0.03; AUCinf: R2 = 0.96, p < 0.001). Specifically, a 0.001% increase in MDZ AUC was associated with a 1% increase in awakening time. In the individual regression analysis between MDZ concentration and awakening time, the mean MDZ concentration at awakening time was 16.8 ng/mL. The AUC of MDZ is the only significant factor associated with the awakening time.

A novel estimator for the two-way partial AUC

BackgroundThe two-way partial AUC has been recently proposed as a way to directly quantify partial area under the ROC curve with simultaneous restrictions on the sensitivity and specificity ranges of diagnostic tests or classifiers. The metric, as originally implemented in the tpAUC R package, is estimated using a nonparametric estimator based on a trimmed Mann-Whitney U-statistic, which becomes computationally expensive in large sample sizes. (Its computational complexity is of order O(nxny)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n_x n_y)$$\\end{document}, where nx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_x$$\\end{document} and ny\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_y$$\\end{document} represent the number of positive and negative cases, respectively). This is problematic since the statistical methodology for comparing estimates generated from alternative diagnostic tests/classifiers relies on bootstrapping resampling and requires repeated computations of the estimator on a large number of bootstrap samples.MethodsBy leveraging the graphical and probabilistic representations of the AUC, partial AUCs, and two-way partial AUC, we derive a novel estimator for the two-way partial AUC, which can be directly computed from the output of any software able to compute AUC and partial AUCs. We implemented our estimator using the computationally efficient pROC R package, which leverages a nonparametric approach using the trapezoidal rule for the computation of AUC and partial AUC scores. (Its computational complexity is of order O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n \\log n)$$\\end{document}, where n=nx+ny\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = n_x + n_y$$\\end{document}.). We compare the empirical bias and computation time of the proposed estimator against the original estimator provided in the tpAUC package in a series of simulation studies and on two real datasets.ResultsOur estimator tended to be less biased than the original estimator based on the trimmed Mann-Whitney U-statistic across all experiments (and showed considerably less bias in the experiments based on small sample sizes). But, most importantly, because the computational complexity of the proposed estimator is of order O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n \\log n)$$\\end{document}, rather than O(nxny)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n_x n_y)$$\\end{document}, it is much faster to compute when sample sizes are large.ConclusionsThe proposed estimator provides an improvement for the computation of two-way partial AUC, and allows the comparison of diagnostic tests/machine learning classifiers in large datasets where repeated computations of the original estimator on bootstrap samples become too expensive to compute.

Relative economic value of global ensemble prediction system of NCMRWF, India, for extreme weather events

AbstractAn ensemble prediction system quantifies the uncertainty in the forecast by a numerical weather prediction model and helps in decision‐making. The global ensemble prediction system (NEPS‐G) of NCMRWF, India, has 1 control and 22 perturbed ensemble members. In the present study, the relative economic value of NEPS‐G forecast has been investigated based on a simple decision‐analytic model. Temperature at 850 hPa (T850), maximum surface temperature (Tmax) and precipitation (rf) over the Indian domain have been studied. Ensemble mean forecast offers a larger value to the users than the control forecast. For rare extreme events, the area under the curve (T‐AUC) of a relative operating characteristic (ROC) obtained using the trapezoidal approximation for 22‐member NEPS‐G (NEPS‐22) is significantly smaller than the area under the curve (Z‐AUC) of the full ROC obtained using binormal model and Z‐transformation. As a result, the discrimination ability and relative economic value of NEPS‐G are underestimated. A method (SUB‐EM) which subdivides the lowest probability category into several subcategories and uses ensemble mean forecast as the secondary decision variable to enhance T‐AUC and improve relative economic value of NEPS‐G has been investigated. Tmax &gt; 45°C and rf &gt; 65 mm·day−1 have been considered as extreme events. The impact of increasing the number of perturbed ensemble members from 22 to 33 (NEPS‐33) by running additional 11 perturbed members from 0000 UTC initial condition has also been investigated and compared with the results obtained using SUB‐EM. SUB‐EM offers larger economic value to the users with small cost–loss ratio (α) without engaging any extra computational resources. An experiment is also carried out to study the impact of using different ensemble summary statistics as a secondary decision variable. Ensemble Maximum provides maximum value and benefits the largest range of users.

Intraoperative blood pressure and cardiac complications after aneurysmal subarachnoid hemorrhage: a retrospective cohort study.

Previous studies report that intraoperative hypotension worsens outcomes after aneurysmal subarachnoid hemorrhage (aSAH). However, the hypotensive harm threshold for major adverse cardiovascular events (MACE) remains unclear. The authors included aSAH patients who had general anesthesia for aneurysmal clipping/coiling. MACE were defined by a composite of acute myocardial injury, acute myocardial infarction, and other cardiovascular complications identified by electrocardiogram and echocardiography. The authors initially used logistic regression and change-point analysis based on the second derivative to identify mean arterial pressure (MAP) of 75mmHg as the harm threshold. Thereafter, our major exposure was MAP below 75mmHg characterized by area, duration, and time-weighted average. The area below 75mmHg represents the severity and duration of exposure and was defined as the sum of all areas below a specified threshold using the trapezoid rule. Time-weighted average MAP was derived by dividing area below the threshold by the duration of anesthesia. All analyses were adjusted for baseline risk factors including age greater than 70 years, female sex, severity of intracerebral hemorrhage, history of cardiovascular disease, and preoperative elevated myocardial enzymes. Among 1029 patients enrolled, 254 (25%) developed postoperative MACE. Patients who experienced MACE were slightly older (59±11 vs. 54±11 years), were slightly more often women (69 vs. 58%), and had a higher prevalence of cardiovascular history (65 vs. 47%). Adjusted cardiovascular risk increased nearly linearly over the entire range of observed MAP. However, there was a slight inflexion at MAP of 75mmHg. MACE was significantly associated with area [adjusted odds ratios (aOR) 1.004 per 10mmHg.min, 95% CI: 1.001-1.007, P =0.002), duration (aOR 1.031 per 10min, 95% CI: 1.009-1.054, P =0.006), and time-weighted average (aOR 3.516 per 10mmHg, 95% CI: 1.818-6.801, P <0.001) of MAP less than 75mmHg. Lower blood pressures were associated with cardiovascular complications over the entire observed range, but worsened when MAP was less than 75mmHg. Pending trial data to establish causality, it may be prudent to keep MAP above 75mmHg in patients having surgical aSAH repairs to reduce the risk of MACE.

Estimates of linear expressions through factorization

The aim of this paper is to establish various factorization results and then to derive estimates for linear functionals through the use of a generalized Taylor theorem. Additionally, several error bounds are established including applications to the trapezoidal rule as well as to a Simpson formula-type rule.

Some notes on the trapezoidal rule for Fourier type integrals

This paper deals with the error analysis of the trapezoidal rule for the computation of Fourier type integrals, based on two double exponential transformations. The theory allows to construct algorithms in which the steplength and the number of nodes can be a priori selected. The analysis is also used to design an automatic integrator that can be employed without any knowledge of the function involved in the problem. Several numerical examples, which confirm the reliability of this strategy, are reported.

Diagnostic method for a piezoelectric injector using the Newton-Cotes formula

The article presents an original method for regenerating common rail injectors, which involves extending the standard diagnostic procedure with a phase of analytical calculations. These calculations were conducted based on the closed Newton-Cotes formula (commonly known as the trapezoidal rule), allowing for the estimation of the resulting fuel injection surface areas and their comparison with the manufacturer's reference. The presented example demonstrates that the proposed solution is particularly useful in challenging situations where a clear assessment of the injector's technical condition is difficult. Several other advantages have been highlighted, making this method suitable for application in laboratory and workshop conditions.

First-year cumulative myeloperoxidase-ANCA titres are associated with all-cause mortality in patients with microscopic polyangiitis.

We investigated whether first-year cumulative myeloperoxidase (MPO)-antineutrophil cytoplasmic antibody (ANCA) and proteinase 3 (PR3)-ANCA titres were associated with all-cause mortality and relapse during follow-up in patients with microscopic polyangiitis (MPA) and granMETHODS: Altogether, 74 patients with MPA and 40 with GPA were included in this study. Their clinical data at diagnosis were collected. First-year cumulative ANCA titres were defined as the area under the curve (AUC) of ANCA titres during the first year after MPA or GPA diagnosis, which was obtained using the trapezoidal rule. All-cause mortality and relapse were considered poor outcomes of MPA and GPA. The median ages of patients with MPA and GPA were 65.5 and 60.5 years, respectively. No significant correlation was observed between ANCA titres at diagnosis and concurrent MPA and GPA activity or the inflammatory burden. First-year cumulative MPO-ANCA titres exhibited a significant AUC for all-cause mortality during follow-up in patients with MPA. The optimal cut-off of first-year cumulative MPO-ANCA titres for all-cause mortality was determined as 720.8 IU/mL using receiver operating characteristic curve analysis. MPA patients with first-year cumulative MPO-ANCA titres ≥720.8 IU/mL exhibited a significantly higher risk for all-cause mortality than those without (relative risk 13.250). Additionally, MPA patients with first-year cumulative MPO-ANCA titres ≥720.8 IU/mL exhibited a significantly lower cumulative patients' survival rate than those without. This is the first study to demonstrate the association between first-year cumulative MPO-ANCA titres and all-cause mortality during follow-up in patients with MPA.

Investigation of junction capacitance in Zener tunnelling tunnel diode partially depleted silicon on insulator

Junction capacitance variation in Zener Tunnelling Tunnel Diode Partially Depleted Silicon On Insulator (ZT-TDPDSOI), due to the geometry effects of source and drain are discussed in this paper. Electric field profile and a capacitance equivalent circuit are used to reveal the gate-source capacitance dependence on the P+ region in ZT-TDPDSOI. It is demonstrated how the inclusion of a P+ region below the source/drain region impacts the gate substrate capacitance using the hole concentration and valence band energy at the source channel junction. A decrease of 42 % was noted in gate-substrate capacitance for a P+ doping of 2*1020/cm3 and thickness of 0.05 μm under fixed biased condition. The use of the trapezoidal approximation is also suggested as an extraction technique for the depletion width.

APPLICATION OF SINGLE STEP EULER AND TRAPEZOID METHODS OF SOLVING TRANSIENT DISTRIBUTION IN MARKOV CHAIN

The computation of state probability distributions at an arbitrary point in time, which in the case of a discrete-time Markov chain means finding the distribution at some arbitrary time step n denoted π^((n) ), a row vector whose i^th component is the probability that the Markov chain is in state i at time step n, and this is the iterative solution methods for transient distribution in Markov chain. In this study, the solutions of transient distribution in Markov chain using Euler and trapezoid methods have been investigated, in order to provide some insight into the solutions of transient distribution in Markov chain, which produce a significantly more accurate response in less time for some types of situations and also tries to get to the end result as quickly as possible with the solution conform with a specified number of well-defined stages is computed for large space Markov chain. Matrices operations, such as the product and matrix inversion are performed and Markov chain laws, theorems, formulas with MATLAB software are utilized. For illustrative examples, the transient distribution vector's π_((i+1) ),i=0,1,2,…, ; is computed for both Euler and Trapezoid methods and their corresponding error when compared with different method. The effect of decreasing the step size using the same infinitesimal generator is examined, also, by taking π_((0) )=(■(1&amp;0&amp;0)) and the length of the interval of integration to be τ = 1, it is observed that the accuracy achieved with the explicit Euler method has a direct relationship with the step size. It is also concluded that the much more accurate results are obtained with the trapezoid method.

A hybrid rotordynamic modeling method for a rotor system with flexible foundation and nonlinear support force: Numerical and experimental investigation

The dynamic characteristics of a flexible foundation would greatly affect the dynamic behavior of the rotor system. In this paper, a hybrid dynamic model method in time domain combines the finite element (FE) method and the subspace-based identification method to model a rotor system with the consideration of a flexible foundation and nonlinear support forces. The state-space model of flexible foundation is identified by the subspace state-space system identification algorithm with measured frequency response functions (FRFs). To solve the calculation problem of the hybrid dynamic model in time domain, a general algorithm named the hybrid model fusion solution (HMFS) method is proposed, which combines the linear and nonlinear nodes separation method used in the FE model and the trapezoidal rule used in the state-space model, improving the calculation efficiency of solving transient response with nonlinear forces. Simulations for the hybrid dynamic model under constant speed and speed-up cases were carried out, resulting that the dynamic characteristics of the entire system can be represented when the nonlinear factors of rolling bearings and gravity were also taken into consideration. Additionally, owe to the consideration of tested dynamic behaviors of flexible base, experimental results in the laboratory test rig are well agreed with the numerical ones under two different test configurations. The investigation provides a complete modeling and highly efficient solution framework for the rotor system with a flexible foundation and nonlinear rolling bearings, with the abilities of further consideration of more nonlinear components such as squeeze film dampers and rubbing forces.

Volume Calculation of Sidoarjo Mud Using TLS Data and Numerical Methods

Sidoarjo Mud is one of the phenomena of mud volcanoes located in the northern part of Java Island. A mud volcano is a phenomenon where mud material is released from the earth due to pressure (eruption). The mud continuously flows and accumulates around the central embankments. This phenomenon has caused the embankments to break several times due to the uncontainable mud volume. Observing the volume of Sidoarjo Mud is expected to help mitigate this problem. The Sidoarjo Mud volume was calculated using point cloud data obtained from Terrestrial Laser Scanner (TLS). Numerical methods such as the Trapezoidal and Simpson rules were employed to determine the mud volume. These numerical methods involve approximating the volume of irregularly shaped earthwork surfaces using mathematical techniques. The numerical method approximates the area under a trapezoid or curve representing the cross-section of the earthwork. In this study, kriging interpolation was conducted with a grid size of five meters. The relative difference between the numerical methods was found to be within acceptable limits, not exceeding 2.78% based on the American Society for Testing and Materials (ASTM) 2002 and not exceeding 2% based on ASTM 1998 as an industry acceptance error. Considering the relative difference results and the institutional report, the Trapezoidal method with a 5-meter grid size was considered the accurate volume calculation method. Based on the processed data, the total volume of Sidoarjo Mud within the embankment was 32,793,917.171 m3 in February 2022, 32,889,012.793 m3 in April 2022, and 33,390,051.407 m3 in June 2022. The volume calculation results show a difference of 95,095.622 m3 between February-April and a difference of 501,038.613 m3 between April-June. The mud flow rate for the February-April period was calculated as 1,872.114 m3/day, while for the April-June period was determined as 9,278.492 m3/day.

Richardson Extrapolation Technique for Singularly Perturbed Convection-diffusion Problem with Non-local Boundary Conditions

This study presents the Richardson extrapolation techniques for solving singularly perturbed convection -diffusion problems (SPCDP) with non-local boundary conditions. A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) mesh. To handle the non-local boundary conditions, the trapezoidal rule is applied. The study establishes an error bound for numerical solutions and determines the numerical approximation for scaled derivatives. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from firs-order to second order convergence

Estimation of quadrature errors for layer potentials evaluated near surfaces with spherical topology

Numerical simulations with rigid particles, drops, or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper, we present quadrature error estimates for layer potentials evaluated near surfaces of genus 0, parametrized using a polar and an azimuthal angle, discretized by a combination of the Gauss-Legendre and the trapezoidal quadrature rules. The error estimates involve no unknown coefficients, but complex-valued roots of a specified distance function. The evaluation of the error estimates in general requires a one-dimensional local root-finding procedure, but for specific geometries, we obtain analytical results. Based on these explicit solutions, we derive simplified error estimates for layer potentials evaluated near spheres; these simple formulas depend only on the distance from the surface, the radius of the sphere, and the number of discretization points. The usefulness of these error estimates is illustrated with numerical examples.