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This thesis concerns optimal packing problems of tubes, or thick curves, where thickness is defined as follows. Three points on a closed space curve define a circle. Taking the infimum over all radii of pairwise-distinct point triples defines the thickness ...
Given a two-dimensional smooth manifold M and a bijective pro jection p from M on a fixed plane (or a subset of that plane), we explore systematically how a wavelet transform (WT) on M may be generated from a plane WT by the inverse projection. Examples whe ...
We present the hardware and reconfiguration experiments for an autonomous self-reconfigurable modular robot called Roombots (RB). RB were designed to form the basis for self-reconfigurable furniture. Each RB module contains three degrees of freedom that ha ...
We investigate numerical simulations and visualizations of the problem of tying a knot in a piece of rope. The goal is to use the least possible rope of a fixed, prescribed radius to tie a particular knot, e.g. a trefoil, a figure eight, and so on. The rop ...
The Apollonian packings (APs) of spheres are fractals that result from a space-filling procedure. We discuss the finite size effects for finite intervals s is an element of [s(min), s(max)] between the largest and the smallest sizes of the filling spheres. ...
In this paper, we derive an explicit form of the convolution theorem for functions on an n-sphere. Our motivation comes from the design of a probability density estimator for n-dimensional random vectors. We propose a probability density function (pdf) est ...
Institute of Electrical and Electronics Engineers2010
What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simpl ...
We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness ...
Let A be a commutative noetherian ring of Krull dimension 3. We give a necessary and sufficient condition for A-projective modules of rank 2 to be free. Using this, we show that all the finitely generated projective modules over the algebraic real 3-sphere ...
The Apollonian packings (APs) of spheres are fractals that result from a space-filling procedure. We discuss the finite size effects for finite intervals s is an element of [s(min), s(max)] between the largest and the smallest sizes of the filling spheres. ...
