The Mask of Odd Points n

We present an explicit formula for the mask of odd points n-ary, for any odd n⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-point a-ary schemes introduced by Lian, 2008, and (2m+1)-point a-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd point n-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented

Domain Wall Junctions in Supersymmetric Field Theories in D=4

We study the possible BPS domain wall junction configurations for general polynomial superpotentials of N=1 supersymmetric Wess-Zumino models in D=4. We scan the parameter space of the superpotential and find different possible BPS states for different values of the deformation parameters and present our results graphically. We comment on the domain walls in F/M/IIA theories obtained from the Calabi-Yau fourfolds with isolated singularities and a background flux.Comment: 26 pages, 4 figure

Multilevel refinable triangular PSP-splines (Tri-PSPS)

A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Convexity-preserving scattered data interpolation

This study deals with constructing a convexity-preserving bivariate C1 interpolants to scattered data whenever the original data are convex. Sufficient conditions on lower bound of Bezier points are derived in order to ensure that surfaces comprising cubic Bezier triangular patches are always convex and satisfy C1 continuity conditions. Initial gradients at the data sites are estimated and then modified if necessary to ensure that these conditions are satisfied. The construction is local and easy to be implemented. Graphical examples are presented using several test functions