The Bezout Number for Piecewise Algebraic Curves

The computation of the Bezout number, the maximum number of intersection points between two piecewise algebraic curves whose common points are finite, is considered. A piecewise algebraic curve is a curve determined by a bivariate spline function. It is found that the maximum number of intersections depends not only on the degrees and the differentiability of the spline functions, but also on the structure of the partition on which the spline functions are defined.  

An analysis of the last round matching problem

The probability distribution of the number of players in the last round of a matching problem is analyzed and the existence of the limiting distribution is proved by using convolution method.  

Inequalities for Alternating Trigonometric Sums

In 1970, J.B. Kelly proved that $$\begin{array}{ll}0 \leq \sum\limits_{k=1}^n (-1)^{k+1} (n-k+1)|\sin(kx)| \quad{(n \in \mathbf{N}; \, x \in \mathbf{R})}.\end{array}$$ We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums $$\begin{array}{ll} & \sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1) | \cos(kx) | \quad {\rm and}\\ & \quad{\sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1)\bigl( | \sin(kx) | + | \cos(kx)| \bigr)}.\end{array}$$   

Quading triangular meshes with certain topological constraints

In computer graphics and geometric modeling, shapes are often represented by triangular meshes (also called 3D meshes or manifold triangulations). The quadrangulation of a triangular mesh has wide applications. In this paper, we present a novel method of quading a closed orientable triangular mesh into a quasi-regular quadrangulation, i.e., a quadrangulation that only contains vertices of degree four or five. The quasi-regular quadrangulation produced by our method also has the property that the number of quads of the quadrangulation is the smallest among all the quasi-regular quadrangulations. In addition, by constructing the so-called orthogonal system of cycles our method is more effective to control the quality of the quadrangulation.  

The generic dimensions of the space of smooth splines of degree 7 on tetrahedral partitions

In this paper, we consider the spaces of once differentiable polynomial splines of degree 7 on tetrahedral partitions and give an expression for their generic dimensions.  

The Haar Wavelet Analysis of Matrices and Its Applications

It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert time-domain problems into frequency-domain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function, a Haar-like wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an $m\times n$ matrix, the computational complexity is $O(mn)$. In addition, when the method is applied to $k$-means clustering, one can obtain that $k$-means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems. In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results.