Simpson's Formula Research Articles

Some integral inequalities for functions with ( n−1)st derivatives of bounded variation

In this paper, we generalize Cerone’s results, and a unified treatment of error estimates for a general inequality satisfying f ( n − 1 ) being of bounded variation is presented. We derive the estimates for the remainder terms of the mid-point, trapezoid, and Simpson formulas. All constants of the errors are sharp. Applications in numerical integration are also given.

Nitrous or nitric? Same difference. Molecular formulas in the 1840s

The molecular formulas given by James Y. Simpson for nitrous oxide, diethyl ether, and chloroform are difficult to interpret today. The organic formulas are “incorrect” today because Jean Dumas (the influential chemist who was one of the discoverers of chloroform) used John Dalton's presumption of molecular simplicity. That is, water was long presumed to be HO. The nitrous oxide formula was incorrect owing to confusion with Dalton's nitrous gas, now termed nitric oxide. The Simpson formulas illustrate that inhaled anesthesia arrived at the time of a cusp in the history of chemistry.

The numerical calculus of expectations of fuzzy random variables

In this paper, the concept of the fuzzy Henstock integral on infinite interval is proposed in order to calculate the expectation of fuzzy random variables. Two calculating methods are proposed: one is to calculate directly by the fuzzy Henstock integral proposed in this paper, including quadrature rules and the error estimates such as the midpoint-type rule, trapezoidal-type rule, Simpson's formula, δ -fine quadrature rules and their error estimates; another is to calculate by using the equivalent characteristic of fuzzy Henstock integrability, whose membership function could be obtained by solving a nonlinear programming problem.

Is there any convergence between the differential and integral procedures used in chemical kinetics, particularly in nonisothermal kinetics?

AbstractSome contradictions concerning the classification of the methods to evaluate nonisothermal kinetic parameters into differential and integral ones are analyzed. Using the theorem of media for integrals, the equivalence between the methods that use the integration on low ranges of variables and the differential methods is shown. Concerning the magnitude of the range of change of variables on which the integration of the rate equation is performed, the applicability of the isoconversional methods that use approximate expressions of the temperature integral (Simpson's formula, formula based on the theorem of media for integrals) was analyzed. Our results were verified for simulated data (which are not affected by the experimental errors) as well as for the nonisothermal data for the thermal decomposition of ammonium perchlorate. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 339–344, 2006

Generalisation of a corrected Simpson's formula

AbstractThe results obtained by A. J. Roberts and N. Ujević in a recent paper are generalised. A number of inequalities for functions whose derivatives are either functions of bounded variation or Lipschitzian functions or R-integrable functions are derived. Also, some error estimates for the derived formulae are obtained.

Cardiac Output Measurement by Partial Carbon Dioxide Rebreathing, 2-Dimensional Echocardiography, and Lithium-Dilution Method in Anesthetized Neonatal Foals

The objective of this study was to assess 2 noninvasive methods of measuring cardiac output (CO) in neonatal foals by comparing results to that of the lithium-dilution method. Ten neonatal foals were anesthetized and CO was manipulated by varying the depth of anesthesia and infusion of dobutamine. Concurrent CO measurements were obtained by lithium dilution (reference method), partial carbon dioxide (CO2) rebreathing, volumetric echocardiography (cubic, Teichholz, Bullet, area-length, and single and biplane modified Simpson formulas), and transthoracic Doppler echocardiography. Thirty pairs of lithium-dilution/noninvasive CO measurements were taken from the 10 foals. For each method, relative bias was calculated as a percentage of the average CO. Lithium determinations of CO ranged between 3.09 and 1 1.1 L/min (mean +/- SD = 6.39 +/- 2.1 L/min), resulting in cardiac indices ranging between 79.0 and 209 mL/kg/min (mean +/- SD = 131 +/- 35.9 mL/kg/min). Relative bias of Doppler echocardiography significantly increased (P < .05), whereas that of partial CO2 rebreathing significantly decreased (P = .03) with increasing CO. Other methods were not influenced by the level of CO. Among methods not influenced by the level of CO, relative bias of the Bullet method (-4.2 +/- 20.9%; limits of agreement -45.2 to 36.7%) was significantly lower (P < .05) than that of each of the other noninvasive methods evaluated. Volumetric echocardiography using the Bullet method provides an accurate and noninvasive estimate of CO in anesthetized neonatal foals and warrants investigation in critically ill conscious foals.

Cardiac Output Measurement by Partial Carbon Dioxide Rebreathing, 2-Dimensional Echocardiography, and Lithium-Dilution Method in Anesthetized Neonatal Foals

The objective of this study was to assess 2 noninvasive methods of measuring cardiac output (CO) in neonatal foals by comparing results to that of the lithium-dilution method. Ten neonatal foals were anesthetized and CO was manipulated by varying the depth of anesthesia and infusion of dobutamine. Concurrent CO measurements were obtained by lithium dilution (reference method), partial carbon dioxide (CO2) rebreathing, volumetric echocardiography (cubic, Teichholz, Bullet, area-length, and single and biplane modified Simpson formulas), and transthoracic Doppler echocardiography. Thirty pairs of lithium-dilution/noninvasive CO measurements were taken from the 10 foals. For each method, relative bias was calculated as a percentage of the average CO. Lithium determinations of CO ranged between 3.09 and 1 1.1 L/min (mean +/- SD = 6.39 +/- 2.1 L/min), resulting in cardiac indices ranging between 79.0 and 209 mL/kg/min (mean +/- SD = 131 +/- 35.9 mL/kg/min). Relative bias of Doppler echocardiography significantly increased (P < .05), whereas that of partial CO2 rebreathing significantly decreased (P = .03) with increasing CO. Other methods were not influenced by the level of CO. Among methods not influenced by the level of CO, relative bias of the Bullet method (-4.2 +/- 20.9%; limits of agreement -45.2 to 36.7%) was significantly lower (P < .05) than that of each of the other noninvasive methods evaluated. Volumetric echocardiography using the Bullet method provides an accurate and noninvasive estimate of CO in anesthetized neonatal foals and warrants investigation in critically ill conscious foals.

Marked Improvement in Left Ventricular Function and Significant Reverse Left Ventricular Remodeling within 3 Months of Cardiac Resynchronization Therapy in Patients with Dilated Cardiomyopathy

We monitored reverse left ventricular (LV) remodeling and LV function during the first 6 months of cardiac resynchronization therapy (CRT) in 34 patients (mean age = 55.3 +/- 13.6 years, 28 men) with dilated cardiomyopathy (DCM), left bundle branch block, in stable New York Heart Association class III, and on fixed drug regimen who underwent implantation of CRT systems with or without cardioverter defibrillator back-up. QRS-complex duration was reduced from 169.69 +/- 19.6 ms (SD) to 144.1 +/- 23.4 ms during CRT. Parasternal M-mode and apical 2D-echocardiography was performed before and 3 and 6 months after device implantation. LV enddiastolic (EDD) and endsystolic (ESD) diameters were measured, and biplane LV enddiastolic (EDV), and endsystolic (ESV) volumes and ejection fractions (EF) were calculated using a modified Simpson formula. Significant decreases in LVEDD (P = 0.0064 at 3 months and P = 0.021 at 6 months), LVESD (P = 0.023 at 3 months, and P = 0.003 at 6 months), and LVESV (P = 0.006 resp. P = 0.007), and increases in LVEF (P = 0.003 at 3 months and P < 0.001 at 6 months) were observed. Mean LVEF increased from 23% at baseline to 39% at 6 months. CRT induced prominent reverse LV remodeling and significantly increased LVEF within a few months in patients with DCM.

A generalization of Ostrowski's inequality and applications in numerical integration

A new generalization of Ostrowski's integral inequality is established. A consequence of the generalization is that we can derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae. These estimates are improvements of some recently obtained estimates. Applications in numerical integration are also given.

Accurate numerical solution of initial value problems for the time dependent convection-diffusion equation

By using the Fourier transform method and composite Simpson's integration formula, a symbolically computable approximate solution of an initial value problem for the time dependent convection-diffusion equation is constructed. Given an admissible error e and a point ( x, t), the approximation constructed with error less than ϵ at ( x, t) can be symbolically obtained using Mathematics 4.0.

Effect of thrombolytic therapy on occurrence of complex ventricular arrhythmias in the late hospitalization period in acute myocardial infarct: relation with long-term remodeling of the left ventricle

The impact of thrombolytic therapy on the incidence of complex ventricular arrhythmias is not yet clarified. The aim of this study was to investigate the impact of thrombolytic therapy on the incidence of complex ventricular arrhythmias, as well as long term association between complex ventricular arrhythmias and left ventricular remodelling, and their impact on five-year lethality. Ninety seven consecutive patients with first acute myocardial infarction (streptokinase was administered in 58 patients), underwent 24-hours Holter monitoring at discharge. Ventricular arrhythmias were classified according to Lown classification, and patients were grouped into the group with simple ventricular arrhythmias (Lown class 0 to 2), and the group with complex ventricular arrhythmias (Lown class 3 to 5). Echocardiography was performed at discharge, and six and twelve months after the infarction. Left ventricular volume indexes and ejection fraction was determined using Simpson's biplane formula. In patients with complex ventricular arrhythmias left ventricular volume indexes were higher and ejection fraction was lower throughout the study, whereas wall motion score index was higher one year after the infarction. On the other hand, these variables were similar throughout the follow-up within the groups of patients with and without complex ventricular arrhythmias who received thrombolytic therapy. The incidence of complex ventricular arrhythmias was similar in thrombolyzed and non-thrombolyzed patients (11/58 vs. 5/39). There was no difference in five year lethality between patients with and without complex ventricular arrhythmias (4/16 vs. 13/81 patients). Our data indicated that left ventricular remodelling in patients with complex ventricular arrhythmias was not progressive after hospital discharge. The presence of complex ventricular arrhythmias was not associated with the increased five-year lethality, despite of more pronounced left ventricular remodelling. It occurred that thrombolysis per se had no influence on the incidence of complex ventricular arrhythmias in the late hospital phase after the first acute myocardial infarction.

Two-point formulae of Euler type

AbstractAn analysis is made of quadratureviatwo-point formulae when the integrand is Lipschitz or of bounded variation. The error estimates are shown to be as good as those found in recent studies using Simpson (three-point) formulae.

Computational Aspects of Dual Energy Digital X-ray Photography

The paper is devoted to computational aspects of the dual-energy digital x-ray photography. The efficiency of the most common integration techniques (trapezoid rule, Simpson's, Gregory's, and Gauss's formulas) for given limiting calculation errors has been studied. Recommendations are given for selecting an integration method and a technique for solving systems of integral equations with parameters.

Measurement of left ventricular volume after anterior myocardial infarction: comparison of magnetic resonance imaging, echo and radionuclide ventriculography.

We have compared echocardiography (echo) and radionuclide ventriculography (RNV) with magnetic resonance imaging (MRI) for the measurement of left ventricular (LV) volume and ejection fraction. Seventy asymptomatic patients were studied up to 12 days after first Q wave anterior myocardial infarction and again after 6 months. Each patient had LV volume measured by all three techniques within 24 hours of each other on each occasion. LV end-systolic and end-diastolic volume index (LVESVI and LVEDVI) and LV ejection fraction (LVEF) were measured using the modified Simpson formula (echo), a counts-based method (RNV), and a multislice area summation method (MRI). Radionuclide volumes were measured both with and without correction for attenuation of isotope. Echocardiography overestimated LV volume compared with MRI. Mean (SD) differences (echo–MRI) were: LVEDVI + 10.6 ml/m2 (16.8), LVESVI + 13.7 ml/m2 (12.9), LVEF − 8.5% (11.2). RNV underestimated both volume and ejection fraction compared with MRI. Mean differences (RNV–MRI) were: LVEDVI −25.4 ml/m2 (23.8), LVESVI − 5.0 ml/m2 (18.6), LVEF − 13.8% (10.4). Variability in the difference between echo and MRI and between RNV and MRI was very similar for LVEF (coefficient of variation 23.9% echo, 22.2% RNV) but there was greater variability in the radionuclide than the echo measurements of absolute volume. Variability of the radionuclide measurements was reduced by not correcting for attenuation, and this finding may improve the radionuclide technique for serial measurements of percentage change in volume. Long-term inter-study reproducibility of MRI for LVEF (coefficient of reproducibility) was 10.9%, for echo it was 10.6%, and for RNV it was 14.6%. We conclude that measurements of LV volume depend on the method used and are not interchangeable. Echocardiography agrees more closely with MRI than RNV for the measurement of absolute volume, but the two techniques are similar for the measurement of LVEF.

Preinfarction angina prevents left ventricular remodeling in patients treated with thrombolysis for myocardial infarction.

It has been shown that preinfarction angina may have beneficial effects on infarct size and mortality. However, there are no studies that have serially assessed the impact of preinfarction angina on left ventricular (LV) function in a large series of patients. The study was undertaken to determine whether preinfarction angina (within 7 days before infarction) influences LV remodeling. In all, 119 consecutive patients with acute myocardial infarction were serially evaluated by 2-dimensional echocardiography (on Days 1, 2, 3, and 7; at 3 and 6 weeks; and at 3, 6, and 12 months following infarction). Left ventricular volumes were determined using Simpson's biplane formula and normalized for body surface area. Wall motion score index and sphericity index were calculated for each study. Coronary angiography was performed before discharge. Preinfarction angina was detected in 39 of 119 patients. Initial echocardiographic and clinical data as well as the incidence of patent infarct-related artery and collaterals were similar for patients with and without preinfarction angina. In the subset of thrombolysed patients, patients with preinfarction angina showed decrease of LV end-diastolic and end-systolic volumes during the follow-up period (p = 0.033 and p = 0.001, respectively), and improvement of wall motion score index (p < 0.001) and ejection fraction occurred (p = 0.001), without changing of LV shape (p > 0.05); in addition, patients with preinfarction angina had smaller LV volumes and higher ejection fraction than did those without angina, from 3 weeks onward. These favorable effects were not detected in patients not treated with thrombolysis. These data indicate that preinfarction angina has an inhibiting effect on long-term LV remodeling in patients who underwent thrombolysis for first acute myocardial infarction. It appears that preinfarction angina has no impact on infarct size and early postinfarction LV function.

Simpson's formula for functions whose derivatives belong to Lp spaces

In [1], Dragomir gave an inequality of Simpson's type for functions whose derivatives belong to L p spaces. Here, we generalize his results using functions whose n th derivatives, n ∈ {2, 3, 4}, belong to L p spaces.

A new generalization of Ostrowski's integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means

In this paper, we establish a new inequality of Ostrowski type for functions with bounded derivatives. This has immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid, mid-point, and Simpson formulae. Application to special means are also investigated.

ON SIMPSON'S QUADRATURE FORMULA FOR MAPPINGS OF BOUNDED VARIATION AND APPLICATIONS

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Comparison of echocardiographic acoustic quantification system and radionuclide ventriculography for estimating left ventricular ejection fraction: Validation in patients without regional wall motion abnormalities

Echocardiographic automated border detection of blood-endocardium interface is made on the basis of the principle of acoustic quantification. The automated border system is capable of providing on-line left ventricular (LV) cavity area and function. Recently, ABD algorithms have been devised to estimate LV volume on line from a long-axis image, calculated by established area-length method or Simpson's formula. To test the clinical validity of this newly developed echocardiographic method, LV volumes and ejection fraction measured by real-time acoustic quantification were compared with radionuclide ejection fraction in 24 subjects on the same day. Patients were included in the study if ≥75% of their endocardium was visualized with conventional two-dimensional echocardiography. Sixteen (66%) of 24 patients had a technically adequate conventional echocardiogram with a broad range of ventricular dimensions and systolic function. None of the study patients had regional wall motion abnormalities. Echocardiographic measurements were obtained from the LV apical four-chamber, long-axis view. Ejection fraction, determined by the acoustic quantification and by radionuclide ventriculography, showed a strong linear relation ( r = 0.92, standard error of the estimate = 4.4, p < 0.05). However, acoustic quantification overestimated the radionuclide ejection fraction with rather wide limits of agreement (3.8% ± 16.4%; bias ± 2 SD). Thus echocardiographic automated border detection technique is a reasonably accurate method for on-line assessment of LV function. (Am Heart J 1997;133:359-63.)

Accurate determination of antenna directivity

The derivation of a formula for accurate estimation of the total radiated power from a transmitting antenna for which the radiated power density is known in a finite number of points on the far-field sphere is presented. The main application of the formula is determination of directivity from power-pattern measurements. The derivation is based on the theory of spherical wave expansion of electromagnetic fields, which also establishes a simple criterion for the required number of samples of the power density. An array antenna consisting of Hertzian dipoles is used to test the accuracy and rate of convergence of the derived formula. Its performance is compared to that of numerical integration using Simpson's formula.