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### Knot intervals and multi-degree splines

Thomas W. Sederberg, Jianmin Zheng, Xiaowen Song
2003 Computer Aided Geometric Design
Using knot interval notation, the paper introduces MD-splines, which are B-spline-like curves that are comprised of polynomial segments of various degrees (MD stands for "multi-degree").  ...  This paper studies the merits of using knot interval notation for B-spline curves, and presents formulae in terms of knot intervals for common B-spline operations such as knot insertion, differentiation  ...  This paper introduces such a piecewise curve, and we refer to it as a multi-degree spline, or MD-spline. Figure 1 shows three examples of MD-splines.  ...

### On multi-degree splines [article]

Carolina Vittoria Beccari, Giulio Casciola, Serena Morigi
2017 arXiv   pre-print
Multi-degree splines are piecewise polynomial functions having sections of different degrees.  ...  We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines.  ...  InFig. 1, the sequence of breakpoint intervals on ∆ and the associated extended knot partitions ∆ Figure 2 : 2 Example 1: Multi-degree B-spline basis N i,n , for n = 0, . . . , 4, and i = 5 − n, . .  ...

### Page 6227 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews
65D07 Sederberg, Thomas W. (1-BYU; Provo, UT); Zheng, Jianmin |Zheng, Jian Min'|; Song, Xiaowen Knot intervals and multi-degree splines.  ...  Using knot interval notation, the paper introduces MD-splines, which are B-spline- like curves that are comprised of polynomial segments of various degrees (MD stands for ‘multi-degree’).  ...

### Computation of multi-degree B-splines [article]

Hendrik Speleers
2019 arXiv   pre-print
This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines.  ...  These basis functions are called multi-degree B-splines (or MDB-splines).  ...  ACKNOWLEDGMENTS This work was partially supported by the Mission Sustainability Programme of the University of Rome Tor Vergata through the project IDEAS (CUP E81I18000060005) and by the MIUR Excellence  ...

### The Joint Estimation of Term Structures and Credit Spreads

Patrick Houweling
1999 Social Science Research Network
We present a new framework for the joint estimation of the default-free government term structure and corporate credit spread curves.  ...  The obtained spread curves are smooth functions of time to maturity, as opposed to the twisting curves one gets from the traditional method, and are less sensitive to model specifications.  ...  The authors thank Ton Vorst, the editor, the associate editor, and two anonymous referees for valuable suggestions and critical remarks; Michel van der Spek and Chios Slijkhuis for  ...

### The joint estimation of term structures and credit spreads

Patrick Houweling, Jaap Hoek, Frank Kleibergen
2001 Journal of Empirical Finance
We present a new framework for the joint estimation of the default-free government term structure and corporate credit spread curves.  ...  The obtained spread curves are smooth functions of time to maturity, as opposed to the twisting curves one gets from the traditional method, and are less sensitive to model specifications.  ...  The authors thank Ton Vorst, the editor, the associate editor, and two anonymous referees for valuable suggestions and critical remarks; Michel van der Spek and Chios Slijkhuis for  ...

### Developing an Approach to Redesign Freeform Surfaces Using B-Spline Technique

2018 Engineering and Technology Journal
format, and then construction the B-Spline surface of the model that contain multi patches by using basis function of the surface.  ...  This paper reported a new approach to design 3D freeform CAD model and then represented this model mathematically by using B-Spline technique.  ...  Acknowledgement This Work is supported by the Industrial Department of Production and Metallurgy Engineering, University of Technology; the authors also gratefully acknowledge the support of the technology  ...

### An Algorithm for Direct Multiplication of B-Splines

Xianming Chen, R.F. Riesenfeld, E. Cohen
2009 IEEE Transactions on Automation Science and Engineering
B-spline blossoming provides another direct approach that can be straightforwardly translated from mathematical equation to implementation; however, the algorithm does not scale well with degree or dimension  ...  B-spline multiplication, that is, finding the coefficients of the product B-spline of two given B-splines is useful as an end result, in addition to being an important prerequisite component to many other  ...  represented by multi-dimensional Bsplines of any degrees, with appropriate knot vectors and Therefor, if the same knot vector 0012345 · · · is chosen for all three dimensions, the quadratic trivariate  ...

### Changeable degree spline basis functions

Wanqiang Shen, Guozhao Wang
2010 Journal of Computational and Applied Mathematics
A B-spline basis function is a piecewise function of polynomials of equal degree on its support interval.  ...  This paper extends B-spline basis functions to changeable degree spline (CD-spline for short) basis functions, each of which may consist of polynomials of different degrees on its support interval.  ...  Acknowledgements We are very grateful to the referees for their helpful suggestions and comments. We also want to thank Zijin Lu and Lincong Fang for their help in improving the language. This  ...

### Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion

Jesse Chan, John A. Evans
2018 Computer Methods in Applied Mechanics and Engineering
We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions.  ...  We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces result in less stringent CFL restrictions than equivalent piecewise continuous or discontinuous  ...  However, the locations of interior knots and Greville abscissae are squeezed closer towards the center of the interval, as shown in Figure 4 . 3 Because the spline basis depends on the knot vector,  ...

### A Tchebycheffian extension of multi-degree B-splines: Algorithmic computation and properties [article]

Rene R. Hiemstra, Thomas J. R. Hughes, Carla Manni, Hendrik Speleers, Deepesh Toshniwal
2020 arXiv   pre-print
The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval.  ...  The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes.  ...  multi-degree polynomial splines in [2, 42] .  ...

### Random regression analyses using B-spline functions to model growth of Nellore cattle

A. A. Boligon, M. E. Z. Mercadante, R. B. Lôbo, F. Baldi, L. G. Albuquerque
2011 Animal
A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and  ...  The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials.  ...  degree basis functions, already evaluated, and the width of the adjoining intervals between knots (Meyer, 2005b) .  ...

### A practical method for computing with piecewise Chebyshevian splines [article]

Carolina Vittoria Beccari, Giulio Casciola, Lucia Romani
2021 arXiv   pre-print
This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines.  ...  A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion.  ...  Acknowledgements The authors are grateful to Marie-Laurence Mazure for her comments and useful suggestions.  ...

### Spline surfaces with T-junctions [article]

Kestutis Karciauskas, Daniele Panozzo, Jörg Peters
2017 arXiv   pre-print
GT-splines are B-spline-like and cover T-junctions by two or four patches of degree bi-4. They complement multi-sided surface constructions in generating free-form surfaces with adaptive layout.  ...  Since GT-splines do not require a global coordination of knot intervals, GT-constructions are easy to deploy and can provide smooth surfaces with T-junctions where T-splines can not have a smooth parameterization  ...  This is the (geometric) input mesh both for the GT-spline and for the T-spline, but the knot-intervals differ: for the GT-spline they can all be chosen equal, while the horizontal T-spline knot intervals  ...

### Functional Coefficient Regression Models for Non-linear Time Series: A Polynomial Spline Approach

Jianhua Z. Huang, Haipeng Shen
2004 Scandinavian Journal of Statistics
The estimated model is used to produce multi-step-ahead forecasts, including interval forecasts and density forecasts. The methodology is illustrated by simulations and two real data examples.  ...  We propose a global smoothing method based on polynomial splines for the estimation of functional coefficient regression models for nonlinear time series.  ...  Acknowledgment We wish to thank the editor, an associate editor, two referees, and Paul Shaman for helpful comments.  ...
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