Author Archives: apgoucher

Self-replicator caught on video

Posted on June 11, 2019 by apgoucher

In a previous article, an announcement was made of a complex self-replicating machine (known as the 0E0P metacell) in a simple 2-state cellular automaton. In the interim between then and now, Thomas Cabaret has prepared a most illuminating video* explaining … Continue reading →

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Five-input Boolean circuits

Posted on May 28, 2019 by apgoucher

Over the past few weeks, I’ve been investigating Boolean optimisation. That is to say, given some circuit of logic gates that implements a particular n-input m-output function, find a more efficient circuit that implements the same function. In practical applications, … Continue reading →

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(W^2 + X^2) / (W^2 + X^2 + Y^2 + Z^2)

Posted on May 24, 2019 by apgoucher

The Box-Müller transform is a method of transforming pairs of independent uniform distributions to pairs of independent standard Gaussians. Specifically, if U and V are independent uniform [0, 1], then define the following: ρ = sqrt(–2 log(U)) θ = 2π V … Continue reading →

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Complexity of integer multiplication almost solved

Posted on March 28, 2019 by apgoucher

Whilst not quite as close as the proofs of the ternary Goldbach conjecture and bounded gaps between primes, there has been a quick succession of two important and somewhat complementary breakthroughs on the computational complexity of integer multiplication: Afshani, Freksen, … Continue reading →

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Fully self-directed replication

Posted on November 12, 2018 by apgoucher

A new form of artificial life has been born — and there are no doubts that it directs its own self-replication: So, what exactly is happening? At 0:06, the organism begins to sequentially construct four identical copies of itself. At … Continue reading →

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6-colourings of subsets of the plane

Posted on September 24, 2018 by apgoucher

There has been further recent activity on the Chromatic Number of the Plane problem, with an eleventh research thread being spawned. Philip Gibbs has been able to 6-colour a large disc (with diameter slightly greater than 4), and Aubrey de … Continue reading →

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Atiyah’s problem

Posted on September 21, 2018 by apgoucher

At the Heidelberg Laureate Forum three years ago, I took lots of selfies with Fields medallists, Abel prizewinners and Turing laureates. This included having a dinner in a castle with Leonard Adleman, pioneer of asymmetric cryptography: …and Endre Szemeredi of … Continue reading →

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Sorting networks

Posted on July 14, 2018 by apgoucher

Important note: whenever log is mentioned in this particular post, it is referring to the ceiling of the base-2 (binary) logarithm. (Elsewhere on cp4space, when there isn’t this disclaimer, it refers to the base-e (natural) logarithm.) For reasons that shall soon become … Continue reading →

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Eurozone’s Lemma

Posted on June 1, 2018 by apgoucher

David Davis has proposed two geopolitical ideas: For Northern Ireland to have dual EU/UK status; For there to be a 10-mile ‘trade buffer zone’ between Northern Ireland and the Republic of Ireland. The second is more interesting from a mathematical … Continue reading →

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Royal Wedding and Polymath16

Posted on May 19, 2018 by apgoucher

Congratulations to Meghan Markle and Prince Harry on what is undoubtedly the most energetic Royal Wedding! In other news, following on from Aubrey de Grey’s 5-chromatic unit-distance graph, there has been an effort to study the algebraic structure of the … Continue reading →

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