This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C1(t) and C2(r) as an implicit curve F(t; r) = 0, where F(t; r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve F(t; r) = 0 has degree 4m 2, whi...

We show that the binary code C of the projective Hall plane Hq2 of even order q where q = 2, for t ≥ 2 has words of weight 2q in its hull that are not the difference of the incidence vectors of two lines of Hq2 ; together with an earlier result for the dual Hall planes of even order, this shows that for all t ≥ 2 the Hall plane and its dual are not tame. We also deduce that dim(C) > 3+1, the di...

In the present paper, we use difference Galois theory to study nature of generating function counting walks with small steps in quarter plane. These series are trivariate formal power Q(x,y,t) that count number confined first quadrant plane a fixed set admissible steps, called model walk. While variables x and y associated ending point path, variable t encodes its length. this prove unweighted ...

The spin degrees of freedom play an essential role in heavy-fermion superconductivity. In UPt3, not only may spin fluctuations provide the pairing mechanism, but the existence of antiferromagnetic order below T„(with a magnetic moment —0.02ptt) ' allows the possibility of multiple superconducting phases. Similarly, recent magnetic x-ray and neutron-scattering studies of URuzSi2 establish the mi...

We characterise t-perfect plane triangulations by forbidden induced subgraphs. As a consequence, we obtain that a plane triangulation is h-perfect if and only if it is perfect.

The special case for θ = π/2 radians is known as Pythagoras’ theorem. The definitions that follow focus on three-dimensional space. Two-dimensional geometry can be derived as a special case when the third coordinate of every point is set to zero. A Cartesian reference system for three-dimensional space is a point in space called the origin and three mutually perpendicular, directed lines though...

Free convection cooling of isothermal circular cylinders positioned above a horizontal plane is investigated numerically, using a commercial Computational Fluid Dynamics (CFD) software package. Computation is performed for high Rayleigh number, in the range 10 − 10. Chien’s turbulence model of low-Reynolds-number K-ε is used, with Prandtl number of 0.707, corresponding to air near standard cond...

A Halin graph, defined by Halin [3], is a plane graph H = T ∪ C such that T is a spanning tree of H with no vertices of degree 2 where |T | ≥ 4 and C is a cycle whose vertex set is the set of leaves of T . In his work, as an example of a class of edge-minimal 3-connected plane graphs, Halin constructed this family of plane graphs, which have many interesting properties. Lovász and Plummer [5] n...

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Ω(m/(n log n)) triangles of T . Eppstein (1993) gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppstein’s argument.