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A combinatorial proof of Houston’s identity
Robin Houston recently discovered a rather interesting formula for the determinant of an n-by-n matrix. In particular, the formula improves upon the best known upper bound for the tensor rank of the...
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Updates and errata
In the Treefoil article, I erroneously described John Rickard’s length-24 cycle in as being the ‘uniquely minimal’ example of a cycle whose three axis-parallel projections are all trees (see here for a...
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Matrix multiplication update
At the end of the recent post on a combinatorial proof of Houston’s identity, I ended with the following paragraph: This may seem paradoxical, but there’s an analogous situation in fast matrix...
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Tensor rank paper
Robin Houston, Nathaniel Johnston, and I have established some new bounds on the tensor rank of the determinant over various fields. The paper is now available as an arXiv preprint and contains the...
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The ordered partial partition polytope
In the tensor rank paper we introduced a new family of axis-aligned n-dimensional polytopes, one for each positive integer n. The vertices are naturally identified with ordered partial partitions...
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Searching for optimal Boolean chains
I gave a half-hour talk on Tuesday about the project to search for optimal Boolean chains for all equivalence classes of 5-input 1-output and 4-input 2-output functions. The talk was not recorded, but...
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The Osmiumlocks Prime
A couple of years ago I described a prime p which possesses various properties that renders it useful for computing number-theoretic transforms over the field . Specifically, we have: where the first...
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Aperiodic monotile
David Smith, Joseph Myers, Craig Kaplan, and Chaim Goodman-Strauss have discovered an aperiodic monotile: a polygon that tiles the plane by rotations and reflections, but cannot tile the plane...
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Miscellaneous discoveries
Soon after the previous post announcing the discovery of an aperiodic monotile by Smith, Myers, Kaplan, and Goodman-Strauss, the same authors published a second aperiodic monotile which has the...
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Every finite phoenix has period 2
A phoenix is an oscillator in Conway’s Life where every cell dies in every generation. The smallest example is Phoenix 1, which oscillates with period 2 and has a constant population of 12: All known...
View ArticleThe minimal infinite threeld
In the post on threelds, we investigated under what conditions the additive group of one field (the ‘inner field’) could be isomorphic to the multiplicative group of another field (the ‘outer field’)....
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