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Author Archives: apgoucher
The 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’). To summarise, this happens in the following cases: … Continue reading
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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 finite phoenices have period 2, … Continue reading
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Miscellaneous discoveries
Soon after the previous post announcing the discovery of an aperiodic monotile by Smith, Myers, Kaplan, and GoodmanStrauss, the same authors published a second aperiodic monotile which has the property that all of the tiles are of the same orientation: … Continue reading
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Aperiodic monotile
David Smith, Joseph Myers, Craig Kaplan, and Chaim GoodmanStrauss have discovered an aperiodic monotile: a polygon that tiles the plane by rotations and reflections, but cannot tile the plane periodically. Any tiling induced by the monotile is scalemic: the majority … Continue reading
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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 numbertheoretic transforms over the field . Specifically, we have: where the first of these equalities uses the identity that: where rad(k) … Continue reading
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Searching for optimal Boolean chains
I gave a halfhour talk on Tuesday about the project to search for optimal Boolean chains for all equivalence classes of 5input 1output and 4input 2output functions. The talk was not recorded, but the slides and transcript are included here … Continue reading
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The ordered partial partition polytope
In the tensor rank paper we introduced a new family of axisaligned ndimensional polytopes, one for each positive integer n. The vertices are naturally identified with ordered partial partitions (OPPs) of {1, …, n}, and the edges correspond to converting … Continue reading
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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 following results: A new formula for the … Continue reading
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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 multiplication: the best known upper bound for the … Continue reading
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Updates and errata
In the Treefoil article, I erroneously described John Rickard’s length24 cycle in as being the ‘uniquely minimal’ example of a cycle whose three axisparallel projections are all trees (see here for a more detailed history on this problem). Dan Simms … Continue reading
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