Quadratic Spline Research Articles

A study of periodic potentials based on quadratic splines

In this paper, we discuss a method based on a segmentary approximation of solutions of the Schrödinger equation by quadratic splines, for which the coefficients are determined by a variational method that does not require the resolution of complicated algebraic equations. The idea is the application of the method to one-dimensional periodic potentials. We include the determination of the eigenvalues up to a given level, and therefore an approximation to the lowest energy bands. We apply the method to concrete examples with interest in physics and discussed the numerical errors.

Late Gadolinium Enhancement in Patients With Hypertrophic Cardiomyopathy and Preserved Systolic Function

BackgroundA high proportion of patients with hypertrophic cardiomyopathy (HCM) have evidence of late gadolinium enhancement (LGE) on cardiac magnetic resonance (CMR). ObjectivesThis study sought to assess the incremental prognostic utility of LGE in patients with HCM. MethodsWe studied 1,423 consecutive low-/intermediate-risk patients with HCM (age ≥18 years) with preserved left ventricular (LV) ejection fraction (mean age 66 ± 14 years, 60% men) who underwent transthoracic echocardiography (TTE) (including dimensions and LV outflow tract gradients) and CMR (including LGE as a % of LV mass) at our center between January 2008 and December 2015. The primary composite endpoint was sudden cardiac death (SCD) and appropriate implantable cardioverter-defibrillator discharge. The percent 5-year SCD risk score was calculated. ResultsThe mean 5-year SCD risk score was 2.3 ± 2.0. Mean maximal LV outflow tract gradient (TTE) was 70 ± 55 mm Hg (median 74 mm Hg [interquartile range (IQR): 10 to 67 mm Hg]); indexed LV mass and LGE (both on CMR) were 91 ± 10 g/m2 and 8.4 ± 12% (IQR: 0% to 19%); 50% had LGE on CMR. Of these, 458 were nonobstructive and 965 were obstructive (of which 686 were underwent myectomy). At 4.7 ± 2.0 years of follow-up, 60 (4%) met the composite endpoint. On quadratic spline analysis, LGE ≥15% was associated with increased risk of composite events. In the obstructive subgroup, on competing risk regression analysis, ≥15% LGE (subhazard ratio: 3.04 [95% confidence interval: 1.48 to 6.10]) was associated with a higher rate and myectomy (subhazard ratio: 0.44 [95% confidence interval: 0.20 to 0.76]) was associated with a lower rate of composite endpoints (both p < 0.01). Similarly, sequential addition of LGE ≥15% and myectomy to % 5-year SCD risk score improved the log likelihood ratios from −227.85 to −219.14 (chi-square 17) and to −215.14 (chi-square 8; both p < 0.01). Association of %LGE with composite events was similar even in myectomy and nonobstructive subgroups. ConclusionsIn low-/intermediate-risk adult patients with HCM (obstructive, myectomy, and nonobstructive subgroups) with preserved systolic function, %LGE was significantly associated with a higher rate of composite endpoint, providing incremental prognostic utility.

Characterization of the concentration-response curve for ambient ozone and acute respiratory morbidity in 5 US cities.

Although short-term exposure to ambient ozone (O3) can cause poor respiratory health outcomes, the shape of the concentration-response (C-R) between O3 and respiratory morbidity has not been widely investigated. We estimated the effect of daily O3 on emergency department (ED) visits for selected respiratory outcomes in 5 U.S. cities under various model assumptions and assessed model fit. Population-weighted average 8-hr maximum O3 concentrations were estimated in each city. Individual-level data on ED visits were obtained from hospitals or hospital associations. Poisson log-linear models were used to estimate city-specific associations between the daily number of respiratory ED visits and 3- day moving average O3 levels controlling for long-term trends and meteorology. Linear, linear-threshold, quadratic, cubic, categorical, and cubic spline O3 C-R models were considered. Using linear C-R models, O3 was significantly and positively associated with respiratory ED visits in each city with rate ratios of 1.02-1.07 per 25 ppb. Models suggested that O3-ED C-R shapes were linear until O3 concentrations of roughly 60 ppb at which point risk continued to increase linearly in some cities for certain outcomes while risk flattened in others. Assessing C-R shape is necessary to identify the most appropriate form of the exposure for each given study setting.

Approximations for higher order boundary value problems using non-polynomial quadratic spline based on off-step points

Non-polynomial quadratic spline method based on off-step points is used to develop a numerical algorithm for obtaining an approximate solution of higher even order boundary value problems. For the employment of the method, decomposition procedure is used. Higher order boundary value problems are reduced into the corresponding system of second order boundary value problems. Convergence analysis of the method is also discussed. Seven numerical examples are given to illustrate the applicability and efficiency of new method. It is also shown that the new method gives approximations, which are better than those produced by other existing fourth order methods except higher degree splines.

Paired comparison models with age effects modeled as piecewise quadratic splines

We propose new models for analyzing pairwise comparison data, such as that relating to sports. We focus on changes in players’ strengths and the prediction of future results. Our models are based on the Thurstone-Mosteller and Bradley–Terry models, and make use of the time variation in the parameters. Furthermore, we apply our models to data from the Japanese traditional sport sumo, and analyze this data. The proposed models perform better than the standard Thurstone-Mosteller and Bradley–Terry models according to both the Akaike information criterion and the Brier score. We compare the proposed models in detail by focusing on individual sumo wrestlers.

Building Curves Using Area Preserving Quadratic Splines

We compare two interpolation methods which are widely used to construct discount curves, forecast (projection) curves, basis curves, and other financial curves. We find that the area-preserving, quadratic-spline method is superior to the “smart quadratic” method, yielding smoother, more natural looking forward curves with few of the artifacts exhibited by the smart quadratic curves. We also show how to efficiently implement both methods by an iterative bootstrapping scheme.

&lt;b&gt;Appropriate mathematical models to describe the lactation curves of milk production traits of Iranian Holstein dairy cattle

The objective of this study was to compare the goodness of fit of lactation curve models; Wood, Wilmink, Linear Splines (SPL), Cubic Splines (SPC), Quadratic Splines (SPQ), Cobby and Le Du, Ali Schaeffer and Legendre Polynomial (LEG), in random regression model (RRM) for milk production traits of Iranian Holstein dairy cattle. For this purpose the records obtained from Test-day (TD) regarding milk (928513), fat (788577) and protein (653317) yields related to their first parity were used. These data collected from the years of 2003 to 2011 by the Karaj breeding center of Iran. The genetic parameters were estimated using REML method using WOMBAT software. Based on obtained results, RRM with SPL6 (6,6), SPC6 (6,6) and LEG (3,5) for milk yield, SPL6 (6,6), SPQ6 (6,6), LEG (3,5) for fat yield and SPL5 (5,5), SPQ4 (4,4) and LEG (3,4) for protein yield, were selected as better model to describe the lactation curves. The estimated heritabilities by best models were lower in the beginning lactation than other during lactation. The genetic trend of milk yields was showed an increasing during the 10 past years, which indicated Iranian Holstein dairy cattle population genetically was improved for milk yields.

Convergence of Integro Quartic and Sextic B-Spline interpolation

In this paper, quadratic and sextic B-splines are used to construct an approximating function based on the integral values instead of the function values at the knots. This process due to the type of used B-splines (fourth order or sixth order), called integro quadratic or sextic spline interpolation. After introducing the integro quartic and sextic B-spline interpolation, their convergence is discussed. The interpolation errors are studied. Numerical results illustrate the efficiency and effectiveness of the new interpolation method.

Echocardiographic right ventricular function reference values in pediatrics: Which is the best mathematical model to define Z-scores?

Echocardiographic assessment of right ventricle (RV) function has become essential in pediatric cardiology. The aim of our study was to determine the most appropriate mathematical model for RV function reference values in children. Normal children aged 0 to 18 years referred to the pediatric cardiology consultation were screened in this prospective cross-sectional study. Each RV parameters (S’, E’, A’ waves, TEI index, TAPSE) was modeled with four explanatory variables: age, size, weight and body surface area. Four different mathematical models were applied: a linear model, a polynomial model of degree 2, a linear regression model by spline with free knot, and a polynomial regression model of degree 2 by spline with a free knot. The choice of the best explanatory variable and modelization method was made on the basis of the model's coefficient of determination ( R 2 ). We included 314 normal children aged 2 days to 18 years (46% female, 88 infants, 26 neonates, 226 children). The modelization of the RV variables over all ages did not follow a linear model, with the lowest coefficients of determination in the four explanatory variables. The quadratic spline model best adjusted the modelization of all RV variables. No model accurately explained the TEI index in any of the four explanatory variables. The S’ wave was best modelized by the quadratic spline model with the weight. The knot at 6.86 kg corresponded to the curves’ break. The E’ wave was best modelized by the quadratic spline model with the age. The knot at 0.29 years corresponded to the curve's break. The A’ wave was best modelized by the quadratic model with the size. The knot at 81 cm corresponded to the break of the curve. The TAPSE was best modelized by the quadratic spline model with the weight. The knot at 9.04 kg corresponded to the curve's break. The quadratic spline model best applies to echocardiographic RV reference values in pediatrics.

Impairment localization and quantification using noisy static deformation influence lines and Iterative Multi-parameter Tikhonov Regularization

Abstract Bridge structures decay throughout their lives even under nominal operating conditions. As bridge infrastructure ages and wears naturally or under extreme load, there is a need to monitor and evaluate bridge performance in an efficient way. This paper presents an impairment detection method that assesses the curvature of noisy static deformation influence lines to predict the location and severity of structural damage. In this method, a parametric approximation and two direct regularization methods i.e., Tikhonov Regularization (TR) and the proposed Iterative Multi-parameter Tikhonov Regularization (IMTR), are implemented to reduce the impact of measurement noise on flexural rigidity estimations. While the TR method assumes one regularization parameter for all unknowns of the optimization problem, the IMTR method has individual, iteratively optimized, regularization parameters for each unknown. To evaluate the performance of the presented method, four nominally identical beam structures with four different damage scenarios and multiple levels of measurement noise are studied. Combining the quadratic spline parametric approximation with either direct regularization method produces adequate curvature estimates that are subsequently used to predict the location and severity of damage. In cases of deep, sharp damage, the IMTR method improves the prediction performance by reducing percent error 4–32%, for noise levels ranging from 0% to 5%, when compared to prediction results from the conventional TR. Both regularization methods give comparable results for shallow, wide damage. A laboratory experiment is included that presents the FRE on a statically indeterminate system; both TR and IMTR provide reasonable estimations of the location and severity of damage.

A numerical approach for solving a class of variable-order fractional functional integral equations

This paper proposes new discretization techniques to estimate variable-order fractional integral operators based on the piecewise integro quadratic spline interpolation. The proposed methods are modified to solve a class of variable-order fractional functional integral equations. Moreover, we investigate the performance of the proposed methods by solving the variable-order fractional pantograph and Emden–Fowler functional integral equations.

Postprocessing Galerkin method using quadratic spline wavelets and its efficiency

The wavelet-Galerkin method is a useful tool for solving differential equations mainly because the condition number of the stiffness matrix is independent of the matrix size and thus the number of iterations for solving the discrete problem by the conjugate gradient method is small. We have recently proposed a quadratic spline wavelet basis that has a small condition number and a short support. In this paper we use this basis in the Galerkin method for solving the second-order elliptic problems with Dirichlet boundary conditions in one and two dimensions and by an appropriate post-processing we achieve the L2-error of order Oh4 and the H1-error of order Oh3, where h is the step size. The rate of convergence is the same as the rate of convergence for the Galerkin method with cubic spline wavelets. We show theoretically as well as numerically that the presented method outperforms the Galerkin method with other quadratic or cubic spline wavelets. Furthermore, we present local post-processing for example of the equation with Dirac measure on the right-hand side.

Quadratic spline wavelets with short support satisfying homogeneous boundary conditions

In this paper, we construct a new quadratic spline-wavelet basis on the interval and on the unit square satisfying homogeneous Dirichlet boundary conditions of the first order. The wavelets have one vanishing moment and the shortest support among quadratic spline wavelets with at least one vanishing moment adapted to the same type of boundary conditions. The stiffness matrices arising from the discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers, and the condition numbers are small. We present some quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis require fewer iterations than methods with other quadratic spline wavelet bases. Moreover, due to the small support of the wavelets, when using these methods with the new wavelet basis, the system matrix is sparser, and thus one iteration requires a smaller number of floating point operations than for other quadratic spline wavelet bases.

Solution of Thermoelectricity Problems Energy Method

On the basis of the fundamental laws of conservation of energy in conjunction with local quadratic spline functions was developed a universal computing algorithm, a method and associated software, which allows to investigate the Thermophysical insulated rod, with limited length, influenced by local heat flow, heat transfer and temperature

Echocardiographic right ventricular function reference values in pediatrics: Which is the best mathematical model to define Z-Scores?

The aim of our study was to determine the most appropriate mathematical model for right ventricular (RV) function normal values in children. Normal children aged 0 to 18 years referred to the pediatric cardiology consultation were screened in this prospective cross-sectional study. Each RV parameters (S’, E’, A’ waves, TEI index, TAPSE) was modeled with four explanatory variables: age, size, weight and body surface area. Four different mathematical models were successively applied: a linear model, a polynomial model of degree 2, a linear regression model by spline with free knot, and a polynomial regression model of degree 2 by spline with a free knot. The choice of the best explanatory variable and the best modelization method was made on the basis of the model's coefficient of determination (R 2 ). We included 314 normal children aged 2 days to 18 years (46% female, 88 infants, 26 neonates, 226 children). The modelization of the RV variables over all ages did not follow a linear model, with the lowest coefficients of determination in the four explanatory variables. The quadratic spline model best adjusted the modelization of all RV variables. No model accurately explained the TEI index in any of the four explanatory variables. The S’ wave was best modelized by the quadratic spline model with the weight. The knot at 6.86 kg corresponded to the curves’ break. The E’ wave was best modelized by the quadratic spline model with the age. The knot at 0.29 years corresponded to the curve's break. The A’ wave was best modelized by the quadratic model with the size. The knot at 81 cm corresponded to the break of the curve. The TAPSE was best modelized by the quadratic spline model with the weight ( Fig. 1 ). The knot at 9.04 kg corresponded to the curve's break ( Fig. 1 ). The quadratic spline model best applies to echocardiographic RV reference values in pediatrics.

Relationship between residual feed intake and carcass composition, meat quality and size of small intestine in a population of F 2 chickens

Feed represent about 70% of the total costs in poultry production. These costs can be reduced by improving feed efficiency through genetic selection. Selection for improving feed efficiency may have correlated effects on other economically important traits such as carcass composition and meat quality. Therefore, the aim of the present study was to evaluate the relationships between the breeding values of cumulative residual feed intake (RFI) and carcass weight, carcass compositions, meat quality, and size of small intestines in chicken. The population studied was an F2 derived from a cross between an Arian fast-growing broiler line and Iranian native fowl. Individual feed intake and body weight measured every week from 2 to 12 weeks and carcass and meat quality traits were assessed after slaughtering at 12 weeks of age. Breeding values for cumulative RFI (RFIBV) were calculated using a quadratic spline with four knots and heterogeneous residual variance. Traits were analyzed using analysis of covariance with a model including RFIBV and sex. No significant relationship was observed between RFIBV with live body weight and eviscerated carcass weight (P > 0.05). RFIBV had negative association with breast muscle percentage, gizzard percentage and Ultimate pH of breast meat (P < 0.05). A positive relationship existed between RFIBV and abdominal fat percentage, skin percentage, lightness and yellowness of breast muscle color. No significant relationship was detected between RFIBV and back and neck, wing, liver, lung, spleen bursa and heart percentages. Also a non-significant association was observed between RFIBV and cooking loss percentage, shear force, drip loss percentage and small intestinal morphometric measurements. In general, the results suggest that selection on RFI will improve the feed efficiency of chickens without impairing carcass compositions and meat quality characteristics.

Non-polynomial quadratic spline method for solving fourth order singularly perturbed boundary value problems

In this paper, a non-polynomial quadratic spline method is described for solving fourth-order boundary value problems whose highest-order derivative is multiplied by a small perturbation parameter. This method is applied directly to the solution of the problem without reducing the order of the problem. Convergence analysis of the fourth order method is discussed. To illustrate the efficiency of the method, a boundary value problem is considered with different type of boundary conditions and obtained numerical results are compared with the existing methods.

A Study on the Modelling of the State Variables for Superheated and Saturated Region

The state variables of saturated and superheated region in the steam table were simultaneously modeled using the neural networks. And the results were compared with quadratic spline interpolation and Lagrange interpolation. Two input data without distinguishing parameter were used in the neural networks. For comparison, quadratic spline interpolation method for superheated region and Lagrange interpolation method for saturated region were applied. The overall results revealed that the neural networks were greatly superior to quadratic interpolation method or Lagrange interpolation method.

Erratum to: Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in $$\mathbb {R}^3$$ R 3

The aim of this paper is to present spline methods for the numerical solution of integral equations on surfaces of $$\mathbb {R}^3$$ , by using optimal superconvergent quasi-interpolants defined on type-2 triangulations and based on the Zwart–Powell quadratic box spline. In particular we propose a modified version of the classical collocation method and two spline collocation methods with high order of convergence. We also deal with the problem of approximating the surface. Finally, we study the approximation error of the above methods together with their iterated versions and we provide some numerical tests.

A Quadratic Trigonometric Nu Spline with Shape Control

A significant curve modeling technique has been introduced with a view to its applications in geometric modeling, computer graphics and computer-aided design. It is a new spline method using quadratic trigonometric functions with well-controlled shape influences of parameters introduced through geometric continuity of order two. The proposed curve model owns the best possible geometric properties such as convex hull, partition of unity, affine invariance and variation diminishing. A quadratic normally has three control points giving lesser flexibility to one piece of curve. However, a cubic has four control points giving higher flexibility to one piece of curve. In the proposed scheme, we have introduced a quadratic trigonometric with four control points. Thus, the proposed quadratic trigonometric has embedded geometric features of cubic/cubic trigonometric. The proposed spline method, constrained with Nu spline like GC2 smoothness, produces a quadratic trigonometric Nu spline (QTNS) with interesting shape control locally and globally. The method is helpful for a variety of shape effects, like point tension, interval tension or global tension. It also produces a quadratic trigonometric alternative to cubic/cubic trigonometric spline because of having four control points in its piecewise description.