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Series Editors’ NoteWe are pleased to add this typescript to the Bone Marrow Transplantation Statistics Series. We realize the term cubic splines may be a bit off-putting to some readers, but stay with us and don’t get lost in polynomial equations. What the authors describe is important conceptually and in practice. Have you ever tried to buy a new pair of hiking boots? Getting the correct fit is critical; shoes that are too small or too large will get you in big trouble! Now imagine if hiking shoes came in only 2 sizes, small and large, and your foot size was somewhere in between. You are in trouble. Sailing perhaps?Transplant physicians are often interested in the association between two variables, say pre-transplant measurable residual disease (MRD) test state and an outcome, say cumulative incidence of relapse (CIR). We typically reduce the results of an MRD test to a binary, negative or positive, often defined by an arbitrary cut-point. However, MRD state is a continuous biological variable, and reducing it to a binary discards what may be important, useful data when we try to correlate it with CIR. Put otherwise, we may miss the trees from the forest.Another way to look at splines is a technique to make smooth curves out of irregular data points. Consider, for example, trying to describe the surface of an egg. You could do it with a series of straight lines connecting points on the egg surface but a much better representation would be combining groups of points into curves and then combining the curves. To prove this try drawing an egg using the draw feature in Microsoft Powerpoint; you are making splines.Gauthier and co-workers show us how to use cubic splines to get the maximum information from data points, which may, unkindly, not lend themselves to dichotomization or a best fit line. Please read on. We hope readers will find their typescript interesting and exciting, and that it will give them a new way to think about how to analyse data. And no, a spline is not a bunch of cactus spines.Robert Peter Gale, Imperial College London, and Mei-Jie Zhang, Medical College of Wisconsin and CIBMTR.

This research evaluates transverse keys against various spline connections for torque transfer efficiency. Fine splines demonstrate superior performance overall, while transverse keys offer competitive advantages for narrow hubs and cost-sensitive applications. The findings guide designers in selecting suitable connection types based on torque requirements, manufacturing complexity, and practical constraints.

This research aims to explore and compare several nonparametric regression techniques, including smoothing splines, natural cubic splines, B-splines, and penalized spline methods. The focus is on estimating parameters and determining the optimal number of knots to forecast cyclic and nonlinear patterns, applying these methods to simulated and real-world datasets, such as Thailand’s coal import data. Cross-validation techniques are used to control and specify the number of knots, ensuring the curve fits the data points accurately. The study applies nonparametric regression to forecast time series data with cyclic patterns and nonlinear forms in the dependent variable, treating the independent variable as sequential data. Simulated data featuring cyclical patterns resembling economic cycles and nonlinear data with complex equations to capture variable interactions are used for experimentation. These simulations include variations in standard deviations and sample sizes. The evaluation criterion for the simulated data is the minimum average mean square error (MSE), which indicates the most efficient parameter estimation. For the real data, monthly coal import data from Thailand is used to estimate the parameters of the nonparametric regression model, with the MSE as the evaluation metric. The performance of these techniques is also assessed in forecasting future values, where the mean absolute percentage error (MAPE) is calculated. Among the methods, the natural cubic spline consistently yields the lowest average mean square error across all standard deviations and sample sizes in the simulated data. While the natural cubic spline excels in parameter estimation, B-splines show strong performance in forecasting future values.

This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We will identify the strengths and weaknesses of these methods and also offer suggestions for their using in geometric modeling and iso-geometric analysis.

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines with constant polynomial basis is used to approximate solutions of nonlinear time-dependent PDEs. The discretization process is done through a simple implicit time stepping and a collocation technique on a set of points in local domains of influence. Resulted system of nonlinear algebraic equations is solved by Newton’s method. Numerical experiments suggest that the proposed method yields superior accuracy than other RBFs as well as FDM for solving nonlinear PDEs.

In this paper we will show the visualization of the approximations that can be obtained by means of the order 1 spline method for Hermite differential equations with well-interactive examples of GeoGebra applets. Here, we will dedicate ourselves to publisize, the great benefit that can be obtained, in the process of generating new mathematical knowledge for learning and teaching the numerical solutions of differential equations.

Type of the article: Research Article AbstractThis study aims to identify the nonlinear determinants of listing day returns globally using a comprehensive dataset. It further examines the regional distinctions and complexities in the international context using spline regression analysis. It also assesses variation in the linear and nonlinear relationships of listing day returns and their determinants across various geographical regions worldwide. Using a set of 8,914 initial public offerings issued across the globe from January 2011 to October 2024, this study employs a restricted cubic spline methodology. Spline knots for each determinant under study were identified to examine the nonlinear influence of the already studied determinant variables. The results of the analysis depict the offer price and listing delay as major non-linear determinants, whereas issue size and market timing significantly influence listing day returns based on linear analysis. In addition, it was found that the Asia-Pacific market substantially differs from other markets geographically, based on splines. The findings of this study provide valuable insights for associated stakeholders by focusing on issue performance, predictions, and market understanding. There is a substantial presence of nonlinear relations among listing day returns and their determinants worldwide.

This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that includes conditional likelihood, quasi-likelihood and pseudo-likelihood. The theory allows all feasible combinations of the spline degree, the penalty order and the smoothness of the unknown functions. According to this theory, the asymptotic behaviors of the penalized spline estimators depends on interplay between the spline knot number and the penalty parameter. The general theory is applied to obtain results in a variety of contexts, including regression, generalized regression such as logistic regression and Poisson regression, density estimation, conditional hazard function estimation for censored data, quantile regression, diffusion function estimation for a diffusion type process and estimation of spectral density function of a stationary time series. For multidimensional function estimation, the theory (presented in the Supplementary Material) covers both penalized tensor product splines and penalized bivariate splines on triangulations.

Thin-walled shell structures are widely used in aeronautical and aerospace engineering. This paper develops an effective B-spline parameterization method for stiffener layout optimization of shell structures. Height variables are defined by B-spline control parameters to characterize the stiffener layout reinforcing the shell structure. A continuous height field is subsequently generated via B-spline and basis functions. In view of possible curvatures of shell structures, the height field is projected from parametric space onto the shell structure by means of the parametric mapping. In this work, the finite element method is adopted with the solid-shell coupling method used for structural analysis. Pseudo-densities associated with solid elements are determined based on the B-spline parameterization and Heaviside function. Several numerical examples are dealt with to demonstrate the proposed method. Compared with the standard density-based method, the proposed method produces checkerboard-free design results with a clear layout and naturally avoids overhang stiffeners.

The cubic Cardinal spline curve is a fundamental tool in the field of interpolation curve design. However, the cubic Cardinal spline curve cannot adjust its shape locally through the free parameters, and it struggles to accurately represent common engineering curves such as elliptical arcs, circular arcs, and parabolic arcs. To overcome these limitations, a novel quasi‐cubic trigonometric Cardinal spline curve is developed. This new spline curve retains the core advantages of the cubic Cardinal spline curve while introducing significant enhancements. It incorporates free parameters that enable local shape adjustment and is capable of accurately representing elliptical arcs, circular arcs, and parabolic arcs. Additionally, the cubic Cardinal spline surface is introduced, and the schemes for creating fair quasi‐cubic trigonometric Cardinal spline curve and surface are provided.