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Recent Comments
- Meagre sets and null sets | Complex Projective 4-Space on Fat Cantor set
- apgoucher on The neural network of the Stockfish chess engine
- Jeremy Bearimy on The neural network of the Stockfish chess engine
- Johnnycanspell on The neural network of the Stockfish chess engine
- apgoucher on The neural network of the Stockfish chess engine
Author Archives: apgoucher
Meagre sets and null sets
There are two competing notions for describing a subset of the real numbers as being ‘small’: a null set is a subset of the reals with Lebesgue measure zero; a meagre set is a countable union of nowhere-dense sets. Both … Continue reading
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Keep your public keys private
Yes, the title sounds very counterintuitive. After all, don’t digital signature schemes require the general public to know your public key so that they can verify your signatures? That is correct, but importantly they don’t need to know your public … Continue reading
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The neural network of the Stockfish chess engine
Last time, we briefly mentioned the high-level differences between Stockfish and Leela Chess. To recap, Stockfish evaluates about 100 million positions per second using rudimentary heuristics, whereas Leela Chess evaluates 40 000 positions per second using a deep neural network … Continue reading
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Rigid heptagon linkage
In January 2000, Erich Friedman considered the problem of finding a rigid unit-distance graph G containing a regular heptagon as a subgraph. That is to say, the graph is immersed in the plane such that: every edge of G must … Continue reading
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Barreto-Naehrig curves and cryptographic pairings
There’s a very elegant cryptographic construction discovered by Barreto and Naehrig in a 2005 paper. It is beautiful from a pure mathematical perspective, but also has an impressive application: it was* part of the ingenious mechanism by which zCash supports … Continue reading
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Shallow trees with heavy leaves
There are two very different state-of-the-art chess engines: Stockfish and Leela Chess Zero. Stockfish searches many more positions (100 000 000 per second) and evaluates them using computationally cheap heuristics. The tree search methodology is a refinement of alpha-beta pruning. … Continue reading
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Let the circumcentre be your origin
Suppose we have two vectors, u and v, in a Euclidean vector space. If we wanted to somehow quantify the proximity of these two vectors, there are two particularly appealing choices: the squared distance, |u − v|²; the inner product, … Continue reading
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An attempt to understand the Monster group
The Monster group is very large, very complicated, and very mysterious. According to the Classification of Finite Simple Groups that was completed last century, the Monster group is the largest of only 26 finite simple groups that do not fit … Continue reading
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The exceptional Jordan algebra
In the early 1930s, Pascual Jordan attempted to formalise the algebraic properties of Hermitian matrices. In particular: Hermitian matrices form a real vector space: we can add and subtract Hermitian matrices, and multiply them by real scalars. That is to … Continue reading
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Closed-form numbers
Recall that the logarithm function (defined on the punctured complex plane) is multi-valued, with the different solutions to exp(x) = y differing by integer multiples of 2πi. James Propp remarked that the values of the form i^i^i are a … Continue reading
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