Abstract
We introduce a new sequence space hA(p), which is not normable, in general, and show that it is a paranormed space. Here, A and p denote an infinite matrix and a sequence of positive numbers. In the special case, when A is a diagonal matrix with a sequence d of positive terms on its diagonal and p=(1,1,…), then hA(p) reduces to the generalized Hahn space hd. We applied our own software to visualize the shapes of parts of spheres in three-dimensional space endowed with the relative paranorm of hA(p), when A is an upper triangle. For this, we developed a parametric representation of these spheres and solved the visibility and contour (silhouette) problems. Finally, we demonstrate the effects of the change of the entries of the upper triangle A and the terms of the sequence p on the shape of the spheres.
Highlights
We introduce a new sequence space h A ( p), prove that it is a linear metric space with respect to its natural total paranorm, and solve the visibility and contour problems for the visualization of spheres or their parts in three-dimensional space endowed with the relative paranorm of h A ( p)
We show a sphere in the norm of bv0, for the parameters d1 = d2 = d3 = 1, in the middle in the norm of the original Hahn space h, for the parameters d1 = 1, d2 = 2, d3 = 3, and on the right in the norm of the generalized
We introduced the generalized Hahn space h A ( p) for upper triangle matrices A and sequences p =∞
Summary
All the figures in this paper were created by our own software for the visualization of mathematical objects and concepts, in particular of curves and surfaces in three-dimensional space. We represent surfaces, which are spheres in various metrics, by two families of curves, in particular by parameter lines in green and blue. This gives the spheres a natural look. We found values u1 and u2 of the parameters in the parametric representation of the surface and the value of t in the parametric representation of the projection ray In the figures in this paper, the silhouettes are represented by thick black curves
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