Spline Method Research Articles

On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m→∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.

Qual é a melhor maneira para definir o tamanho amostral para mudas de couve-flor?

ABSTRACT: This study compared four methods for defining the ideal sample size per experimental unit to estimate the overall experimental mean for the total length, shoot length, root length, and the number of leaves of cauliflower seedlings. An experiment was carried out where the number of leaves, shoot, root, and total length were measured, and the general, perpendicular distance, linear response plateau, and spline methods were tested. While the general method may under or overestimate sample size and the sampling of 10 seedlings suggested by the spline method is still too far from the stabilization point of the curve, the perpendicular distance and linear plateau response methods are recommended to obtain results corresponding to narrower confidence interval widths. Therefore, according to the perpendicular distance method, at least 15 seedlings per experimental unit are required to estimate the overall experimental mean of cauliflower seedlings reliably for the traits here measured.

Establishing mathematical models for determining quantum dot potential for pbs and pbte structures in terms of QD's diameters by numerical methods

The main goal of this paper is to obtain the estimated determination of the quantum dot potential for PbS and PbTe structures (in mV) which are compared to other values at different QD's diameters (in nm). In order to investigate this goal, the study relies on two methods of interpolation such as Neville and Spline methods, as well as it constructs mathematical models that help to find the estimate determination of quantum dot potential for PbS and PbTe structures compared to other values at different QD's diameters. The numerical results were very close to the real results. Finally, we estimated the determinations outside the fields and labs of the measured areas.

URINARY BLADDER CANCER RECURRENCE AND EXPRESSION OF LYNCH AND HER MARKERS: SEARCHING FOR IMMUNOHISTOCHEMICAL PATTERNS AMONG 113 TUMORS FROM 33 PATIENTS.

The aim was to identify immunohistochemical (IHC) markers able to predict recurrence of urinary bladder tumors. The method of multivariate adaptive regression splines (MARS) was applied to IHC data of 33 patients with urinary bladder cancer that relapsed one to six times (24 male and nine female, age 57-87 years). The MARS analysis was used to predict the total number of recurrences and the Ki-67 value by nine IHC markers (epidermal growth factor receptor (EGFR), HER2, HER3, E-cadherin, Ki-67, MLH1, MSH2, MSH6 and PMS2). Data were divided as initial tumors, first and subsequent recurrences, and tumors that relapsed within nine months of previous surgery or later. The IHC markers were semiquantitatively classified into four groups, as follows: 0 means no positive cells; 1, 10% of positive cells; 2, 11%-30% of positive cells; and 3, 31%-100% of positive cells. In predicting the overall number of recurrences, as a surrogate marker of tumor biology, the R2 value for all tumors was 0.423, for initial tumors 0.686, for first recurrence 0.700, and for subsequent recurrences only 0.233. The key predictors for initial tumors were HER2 and MSH2, while for the first recurrence it was EGFR. For quick recurrences (within nine months), the R2 was 0.474 with EGFR and HER3 as predictors, while for slow recurrences R2 was 0.640 due to EGFR and PMS2. In predicting the Ki-67 value of that tumor, the R2 value for all tumors was 0.300, for initial tumors 0.262, for first recurrence 0.360, and for subsequent recurrences only 0.533. The key predictors for first recurrences were EGFR and MSH6, and for subsequent recurrences HER2, EGFR and all Lynch markers. The R2 was 0.266 for quick recurrences and 0.370 for slow recurrences. The finding of E-cadherin was not found relevant by any of these MARS models. In conclusion, the MARS results associated multiple IHC markers with the number of recurrences and with Ki-67 values. It is important that differences in predictive markers were found between initial tumors and first recurrences, and between quick and slow recurrences, thus suggesting that tumor biology is different among these subgroups regarding the total number of recurrences and Ki-67 values.

Statistical Shape Analyses of Pediatric Patients with Scoliosis After Surgical Correction.

To investigate the preoperative and postoperative differences in the upper-body and spinal shapes of patients with scoliosis. Digitized two-dimensional X-ray images were used to obtain the shapes of the upper-body and spine. The preoperative and postoperative mean shapes were compared by using a Generalized Procrustes analysis. The thin plate spline (TPS) method was used to evaluate the spinal shape deformation between the preoperative and postoperative periods. The pre- and postoperative upper-body and spinal shape differences were significant. The TPS graphics showed highlevel deformations between the pre- and postoperative periods. The left superior border of the L4 spinous process showed the highest deformation. The preoperative and postoperative upper-body and spinal shape differences and structural deformations that correlated with scoliosis were shown to be significant.

Research on precise measurement method of full parameters of spline interlocks

This paper mainly introduces the types, characteristics and applications of spline, studies the high-precision measurement technology of different types of spline. Based on gear measuring instrument, optical coordinate measuring machine and coordinate measuring machine, the high-precision full-parameter measurement methods of spline. Developed countries in Europe have spline standards in the 1950s. At present, China has formed a relatively complete system of key and spline linkage standards, but the overall level is still far from developed countries. With the revision of national standards, spline coupling has been applied in more and more fields with the development of industry. It also puts forward higher requirements for high-precision detection methods. Especially for rectangular splines, there is no national calibration specification for the traceability of rectangular splines. We determine the measurement methods and parameters according to the national standards, international standards and the actual production process of the factory. We hope that we can help the factory to improve the precision of splines and prolong their working life.

Application of the modified self-organizing migration algorithm MSOMA in optimal open-loop control problems of switching systems

The article discusses the application of the metaheuristic algorithm of global constrained optimization for solving the problem of finding the optimal open-loop control for nonlinear switching deterministic dynamical systems. The quality of control is assessed by the value of the functional defined on individual trajectories. The optimal control problem is reduced to a parametric optimization problem, which is solved using theMSOMA algorithm, which belongs to the evolutionary group. The MSOMA algorithm is a new algorithm based on the SOMA self-organizing migration algorithm. The modification consists in identifying three leaders among the individuals forming the current population. For each member of the population, two clones are generated with the same position. Thus, in fact, three populations are generated, each of which then realizes a migration cycle (evolutionary process) relative to one of the three selected leaders. A step-by-step algorithm for piecewise-constant, piecewiselinear, quadratic spline and cubic spline methods of control laws approximation is proposed. The effectiveness of the proposed method is demonstrated by the example of solving optimal control problems for switching systems with two and three subsystems. The influence of the parameters of the MSOMA algorithm on the quality of the obtained result is investigated. Comparison of the operation of the algorithm with the known solution is carried out.

Comparing Solutions to the Nonlinear Dissipative Wave Equation

In previous decades, many of the practical problems arising in scientific fields such as physics, engineering, and mathematics have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the dissipative wave equation, has been found to have a plethora of useful applications in different fields. A special class of solutions has been studied for the dissipative wave equation including exact solutions and approximate solutions. The aim of this article is to compare the non-polynomial spline method and the cubic B-spline method with the solution of a nonlinear dissipative wave equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

Interpolasi Curah Hujan Ekstrim Menggunakan Model Spatial Di Provinsi Jawa Barat

Extreme rainfall is an important aspect to study due to its detrimental impacts. The existing constraint is that there is not necessarily an extreme value in every location, so that a zoning is needed that can specifically explain extreme rainfall in terms of location dependency, as well as spatial extreme rainfall patterns. The purpose of this research is to build a special area for extreme rainfall in the West Java region in the period December 2017-February 2018. One of the methods used to estimate incomplete rainfall data is the interpolation technique. There are several methods that can be used to perform interpolation, including Kriging and Thin Plate Splines. In this study, the two methods will be compared based on the GVC value. The results obtained are based on the GCV value, the Thin Plate Splines method is better used to build special areas for extreme rainfall in the West Java region in the period December 2017-February 2018.

Robust numerical method for singularly perturbed differential equations with large delay

Abstract In this paper, a singularly perturbed differential equation with a large delay is considered. The considered problem contains a large delay parameter on the reaction term. The solution of the problem exhibits the interior layer due to the delay parameter and the strong right boundary layer due to the small perturbation parameter ε. The resulting singularly perturbed problem is solved using the fitted non-polynomial spline method. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems of the variable coefficient are considered for numerical experimentation.

Numerical Spline Method for Simulation of Stochastic Differential Equations systems

نقدم في هذا البحث طريقة شرائحية عددية مع وسيطي تجميع لحل نظم من المعادلات التفاضلية العشوائية متعددة الأبعاد. تمت محاكاة عملية وينر العشوائية متعددة الأبعاد المستمرة مع الزمن كعملية منفصلة، ثم دراسة الاستقرار العشوائي بمتوسط المربعات للطريقة عندما تُطَبقْ لحل نظم من المعادلات التفاضلية العشوائية الخطية من البعد الثاني. تبين الدراسة أن الطريقة تكون مستقرة ومتقاربة بمتوسط المربعات من المرتبة الثالثة عندما يتم تطبيقها لحل نظم من المعادلات التفاضلية العشوائية خطية وغير خطية. وقد تم اختبار فعالية الطريقة المقترحة بحل مسألتي اختبار الأولى خطية والثانية غير خطية، وتشير النتائج العددية إلى فعالية وكفاءة الطريقة الشرائحية المقترحة وأن النتائج الحاصلة جديرة بالاهتمام.

Development of Nonparametric Path Model using Multivariate Adaptive Regression Spline (MARS) Method In Compliance Behavior of Paying Credit In Bank

The purpose of this study is to develop a Non-parametric Path with the MARS (Multivariate Adaptive Regression Spline) approach which is applied to the behavior of paying credit compliance at Bank. prospective debtor by a Bank. The data used in this study is primary data using a research instrument in the form of a questionnaire. There are 7 variables, namely 5 exogenous variables in the form of 5C variables (Character (X1), Capacity (X2), Capital (X3), Collateral (X4), Condition of Economy (X5)), and two endogenous variables, namely Punctual Payment (Y1), Obedient Paying Behavior (Y2). Variable measurement technique is done by calculating the average score on the items. Sampling in this study used a purposive sampling technique with the criteria of respondents in the study were mortgage debtors (House Ownership Credit) at Bank X. Respondents obtained in this study were 100 respondents. The analysis used is nonparametric path with Multivariate Adaptive Regression Spline (MARS) approach. The result of this research is the estimation of nonparametric Path function using MARS approach on various interactions. The best estimate of the function of obedient behavior in paying credit is when it involves 4 variables, namely Character (X1), Capacity (X2), Conditions of economy (X5), and On time pay (Y1) with a value of generalized cross-validation The smallest (GCV) obtained is 0.2496. The originality of this research is the development of a nonparametric path with the MARS approach that is able to capture interactions between existing variables and is also able to handle the limitations of the truncated spline to determine the position and number of knot points used when involving many predictor variables. There has been no previous research that has examined the development of a nonparametric path with the MARS approach.

It Is Advisable to Control the Duration of Hypothermia Circulatory Arrest During Aortic Dissection Surgery: Single-Center Experience

Objective: The duration of hypothermic circulatory arrest (HCA) is one of the important factors affecting the prognosis of arch surgery, which is still controversial. The purpose of this study was to investigate the effect of HCA duration on early prognosis in type A aortic dissection (TAAD) patients who underwent arch surgery in our center.Methods: All consecutive patients who underwent surgical treatment for TAAD in Fuwai Hospital from January 2013 to December 2018 were included in this study and divided into four quartile groups based on HCA time. Baseline characteristics, perioperative indicators, and early mortality were statistically analyzed by propensity score matching (PSM) and restricted cubic spline (RCS) method. Perioperative adverse events were confirmed according to the American STS database and Penn classification.Results: About 1,018 consecutive patients (mean age 49.11 ± 1.4 years, male 74.7%) with TAAD treated surgically were eventually included in this study. After PSM, with the prolongation of HCA time, the surgical mortality rates of group [2,15], (15,18], (18,22], and (22,73] were 4.1, 6.6, 7.8, and 10.9% with p = 0.041, respectively. As shown in RCS, the mortality rate increased sharply after the HCA time exceeded 22 min. And from the subgroup analysis, the HCA time of 22 min or less was associated with better clinical outcomes (OR 2.09, 95%CI 1.25–3.45, p = 0.004).Conclusions: The early mortality increases significantly with the duration of HCA time when arch surgery was performed. And multiple systems throughout the body can be adversely affected.

Numerical Solution of Fourth Order Homogeneous Parabolic Partial Differential Equations (PDEs) using Non-Polynomial Cubic Spline Method (NPCSM)

Non-polynomial cubic spline functions are already being used in the field of engineering, computer sciences, and natural sciences to solve ordinary differential equations (ODEs) and partial differential equations (PDEs). However, many of the above-mentioned problems do not have an exact, stable, or convergent exact solution. There are different approximations and methods that can be applied to solve these problems. This study implemented the purposed method on homogeneous parabolic PDEs having different dimensions. The results obtained were compared with the exact solution and results of other existing methods in tabular and graphical form. Mathematica was used to find the mathematical and graphical results.EMATICA.
 Keywords: Adomian decomposition method (ADM), non-polynomial cubic spline method (NPCSM), continuous approximation, finite difference approximations, fourth order homogeneous parabolic partial differential equations (PDEs)
 Copyright(c) The Authors

Proposals for Sewage Management at Luruaco Lake, Colombia

This study presents numerical simulations of faecal coliforms dynamics in Luruaco lake, located in Atl\'antico Department, Colombia. The velocity field is obtained through a two-dimensional horizontal (2DH) model of Navier-Stokes equations system. The transport equation of faecal coliforms concentration is provided from a convective-diffusive-reactive equation. The lake's geometry is built through cubic spline and multi-block methods. The discretization method by Finite Differences and the First Order Upwind (FOU) are applied to the 2DH model. The Mark and Cell (MAC) method is used to determine numerically the velocity field of water flow. Numerical simulations are carried out for a 72-hour period in order to understand the influence of faecal coliforms injections from each tributary. From the qualitative and quantitative analysis of the factors that influence faecal coliforms dynamics, proposals are presented, which aim to reduce contamination in some regions of Lake Luruaco. The numerical simulations show that the best option to improve water quality in the lake is the implementation of two actions, the diversion of the Lim\'on stream to the Negro stream and the installation of a sewage treatment plant at the mouth of the Negro stream. Other less expensive proposals are also presented.

Dynamics of notified cases of dengue in Alagoas: Geospatialization and Applied Statistics

The aim of the study was to evaluate the reported cases of dengue in the State of Alagoas via applied statistics and geospatialization.Dengue data were obtained from the DATASUS system between 2000 and 2015. The time series was submitted to descriptive and exploratory statistics and the Pettitt test. In the dengue case maps, the Spline method via QGis version 3.4 was used. In descriptive statistics, only 10 municipalities were evaluated based on the largest case records. All municipalities were above the average of dengue cases in 2010, the year of the biggest recent flood in the state. The boxplot pointed out that all 10 municipalities were positive asymmetric in a few months for dengue cases, the exception was Delmiro Gouveia. The Pettitt test identified biannual cycles (2006/2007 and 2009/2010) associated with the phases of ENSO (El Niño and La Niña) in the moderate category, followed by the months of January, February, April and December, corresponding to the performance of some systems synoptics. In the mapping of annual cases of dengue, the highest records occurred in the Agreste and East Alagoas. Geospatialization and applied statistics are efficient in the spatiotemporal evaluation of dengue cases in Alagoas.

Influence of Shunt Type on Survival and Right Heart Function after the Norwood Procedure for Aortic Atresia

The study objective was to compare the results after Norwood procedure between modified Blalock-Taussig shunt (MBTS) and right ventricle-to-pulmonary artery conduit (RVPAC) according to Sano in patients with hypoplastic left heart syndrome (HLHS) and aortic atresia (AA). A total of 146 neonates with HLHS and AA who underwent the Norwood procedure at our institution between 2001 and 2020 were divided into 2 groups according to shunt type (MBTS or RVPAC). Survival after the Norwood procedure was compared between the groups. Longitudinal right ventricular and tricuspid valve function in each group were evaluated using cubic splines method. RVPAC was performed in 103 patients and MBTS in 43 according to surgeon preference. There were no differences in the 30-day mortality rates (16.5% vs 16.3%, P =0.973). Survival at 0.5, 1 and 3 years was 79.6%, 74.6%, and 68.9% in RVPAC and 66.8%, 64.3%, and 58.5% in MBTS (P = 0.293). Among 23 patients undergoing tricuspid valve procedure, different mechanisms of tricuspid regurgitation were observed between the groups. Longitudinal analysis revealed greater prevalence of late right ventricular dysfunction in RVPAC patients. In 77 patients who completed Fontan procedure, the postoperative N-terminal pro B-type natriuretic peptide value was significantly higher in RVPAC vs MBTS (554 vs 276 ng/L, P = 0.007). No survival advantage of RVPAC over MBTS was observed in neonates with HLHS and AA undergoing the Norwood procedure. Longitudinal analysis demonstrated a greater prevalence of right ventricular dysfunction and higher N-terminal pro B-type natriuretic peptide values during late follow-up in patients with RVPAC.

New Spline Method for Solving Linear Two-Point Boundary Value Problems

In this research, second order linear two-point boundary value problems are treated using new method based on hybrid cubic B-spline. The values of the free parameter,Gamma , are chosen via optimization. The value of the free parameter plays an important role in giving accurate results. Optimization of this parameter is carried out. This method is tested on four examples and a comparison with cubic B-spline, trigonometric cubic B-spline and extended cubic B-spline methods has been carried out. The examples suggest that this method produces more accurate results than the other three methods. The numerical results are presented to illustrate the efficiency of our method.

Non-polynomial Cubic Spline Method for the Solution of Second-order Linear Hyperbolic Equation

Second-order linear hyperbolic equations are solved by using a new three level method based on nonpolynomial spline in the space direction and Taylor expansion in the time direction. Numerical results reveal that three level method based on non-polynomial spline is implemented and effective

Exploring Optimality of Piecewise Polynomial Interpolation Functions for Lung Field Modeling in 2D Chest X-Ray Images

In this paper, a landmark based approach, using five different interpolating polynomials (linear, cubic convolution, cubic spline, PCHIP, and Makima) for modeling of lung field region in 2D chest X-ray images have been presented. Japanese Society of Radiological Technology (JSRT) database which is publicly available has been used for evaluation of the proposed method. Selected radiographs are anatomically landmarked using 17 and 16 anatomical landmark points to represent left and right lung field regions, respectively. Local, piecewise polynomial interpolation is then employed to create additional semilandmark points to form the lung contour. Jaccard similarity coefficients and Dice coefficients have been used to find accuracy of the modeled shape through comparison with the prepared ground truth. With the optimality condition of three intermediate semilandmark points, PCHIP interpolation method with an execution time of 5.04873 s is found to be the most promising candidate for lung field modeling with an average Dice coefficient (DC) of 98.20 and 98.54% (for the left and right lung field, respectively) and with the average Jaccard similarity coefficient (JSC) of 96.47 and 97.13% for these two lung field regions. While performance of Makima and cubic convolution is close to the PCHIP with the same optimality condition, i.e., three intermediate semilandmark points, the optimality condition for the cubic spline method is of at least seven intermediate semilandmark points which, however, does not result in better performance in terms of accuracy or execution time.