Hilbert’s two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher dimensions. In fact, the number of ways in which this may be done ranges from zero to infinite, depending on what properties of the Hilbert curve one considers

The algebraic topology of Collatz attractors (or “Collatz Feathers”) remains very poorly understood. In particular, when pushed to infinity, is their set branches discrete or continuous? Here, we introduce a fundamental theorem proving that the latter true. For any odd x, first define Axn as all numbers with Syr(x) in orbit and up n more digits than x base 2. We then prove limn→∞|Axn|2n+c≥1 c&g...

Consider a set of simple closed curves on a surface of genus g which fill the surface and which pairwise intersect at most once. We show that the asymptotic growth rate of the smallest number in such a set is 2 √ g as g → ∞. More generally, we give a precise asymptotic for filling sets of curves which pairwise intersect at most K ≥ 1 times. We then bound from below the cardinality of a filling ...

The proposed paper p resen ts a v a r i e t y nove l uses o f S p a c e F i l l i n g -Curves (SFCs) f o r Car tes ian mesh methods i n CFD. Whi le thesetechniques w i l l be demonstrated u s i n g n o n b o d y f i t t e d Car tes ianmeshes, most a r e a p p l i c a b l e on genera l b o d y f i t t e d meshes bo ths t r u c t u r e d and uns t ruc tu red . We demonstrate t h e...

We examine several matrix layouts based on space-filling curves that allow for a cache-oblivious adaptation of parallel TU decomposition for rectangular matrices over finite fields. The TU algorithm of [11] requires index conversion routines for which the cost to encode and decode the chosen curve is significant. Using a detailed analysis of the number of bit operations required for the encodin...