Assuming ZFC, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω1 in the Turing degrees.

We prove that if I is a countable ideal in the Turing degrees, then the field RI of real numbers in I is computable from exactly the degrees that list the functions (i.e., elements of ωω) in I. This implies, for example, that the degree spectrum of the field of computable real numbers consists exactly of the high degrees. We also prove that if I is a countable Scott ideal, then it is strictly e...

We study the notion of K-pairs in the local structure of the ωenumeration degrees. We introduce the notion of super almost zero sequences and investigate their structural properties. The study of degree structures has been one of the central themes in computability theory. Although the main focus has been on the structure of the Turing degrees and its local substructure, of the degrees below th...

Parametric-surface tessellation is one of the most important algorithms for CAGD applications. This paper presents a new parametric-surface tessellation method based on degree reduction: (1) a given parametric surface (or NURBS surface) of degrees (p; q) is decomposed into a set of Bezier surfaces, (2) the Bezier surfaces are converted into a set of bilinear surfaces by applying consecutive ste...

In this paper, the issue of multi-degree reduction of Bézier curves with C and G-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method. 2011 Els...

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers.

In this paper, we consider multi-degree reduction of Bézier curves with constraints of endpoints continuity with respect to L2 norm. The control points of the degree reduced Bézier curve can be obtained as a product of the degree reduction matrix and the vector of original control points. We find an explicit form of the multi-degree reduction matrix for Bézier curve with constraints of endpoint...

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a ke...

In this paper, weighted G-multi-degree reduction of Bézier curves is considered. The degree reduction of a given Bézier curve of degree n is used to write it as a Bézier curve of degree m,m < n. Exact degree reduction is not possible, and, therefore, approximation methods are used. The weight function w(t) = 2t(1 − t), t ∈ [0, 1] is used with the L2-norm in multidegree reduction with G-continui...

The issue of Cα-degree reduction of triangular Bézier surfaces is exposed. It is anticipated that both triangular Bézier surfaces are Cα-continuous at the vertices. The Euclidean norm as well as the L2−norm is used. The final solutions are given in terms of the matrix of degree raising, the Gram matrix, and the Bézier points. Moreover, it is shown that the solutions using both norms are equival...