From the view of differential geometry, a straight line is a geometric curve with the curvature κ(s) = 0. A plane curve is a family of geometric curves with torsion τ(s) = 0. Helix is a geometric curve with non-vanishing constant curvature κ and non-vanishing constant torsion τ [4]. The helix may be called a circular helix or W -curve [9]. It is known that straight line (κ(s) = 0) and circle (κ...

We present new computations of “intrinsic” symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group MCG(S, L) or Sym(L) of the pair (S, L). Elements in this symmetry group can (and often do) fix the link and act nontrivially only on its complement. We ignore such elements and focus on the “intrinsic” symmetry group of a link,...

In this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres S n ⊂ R n+1 using spherical basis functions. Error estimates in the Sobolev norm are derived.

Traditionally, optimality and uniqueness of an (n, N, t) spherical code is proved using linear programming bounds. However, this approach does not apply to the parameter (4, 10, 1/6). We use semidefinite programming bounds instead to show that the Petersen code (which are the vertices of the 4dimensional second hypersimplex or the midpoints of the edges of the regular simplex in dimension 4) is...

We developed an open source software for crustal strain analysis using the least-squares collocation method based on the spherical model of the earth. The software is able to simultaneously determine the signal and noise of the velocities at the observation points with the best possible removal of the observational errors, or at any other position with no velocity observation. Furthermore, the ...

New spherical codes called laminated spherical codes are constructed in dimensions 2–49 using a technique similar to the construction of laminated lattices. Each spherical code is recursively constructed from existing spherical codes in one lower dimension. Laminated spherical codes outperform the best known spherical codes in the minimum distance sense for many code sizes. The density of a lam...

We prove that the D4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in S, based on numerical computations suggesting that every 5-design consisting of 24 points in S is in a 3-parameter family (which we describe explicitly, based on a constr...

in this paper we will determine the multiple point manifolds of certain self-transverse immersions in euclidean spaces. following the triple points, these immersions have a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to dold manifold $v^5$ or a boundary. we will show there is an immersion of $s^7times p^2$ in $mathbb{r}^{13...

in this research, the effect of the first order magnetic field on the ground-state of a centered hydrogenic donor impurity in a gaas/alas spherical quantum dot has been calculated. the perturbation method has been used within the framework of effective mass approximation for these calculations. overall, the analysis shows that a proper choice of quantum dot radius and magnetic field can signifi...