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37, the median value for the second prime factor of an integer
Chris Grossack's math blog and professional website.
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A linear algebra trick for computing Fibonacci numbers fast
Fast Fibonacci Computations With a Linear Algebra Twist
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Formalising modern research mathematics in real time
(This is a guest post by Bhavik Mehta) On March 16, 2023, a paper by Campos, Griffiths, Morris, and Sahasrabudhe appeared on the arXiv, announcing an exponential improvement to the upper bound on R…
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When to Shut Up: A Visual Guide (With Included Algorithm)
Use these rules to avoid boring your audience
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What do we mean by "the foundations of mathematics"?
The phrase “foundations of mathematics” is bandied about frequently these days, but it’s clear that there is widespread confusion about what it means. Some say that a proof assistant must be based on a foundation of mathematics, and therefore that the foundations of mathematics refers to some sort of formal system…
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π in Other Universes
How the value of π depends on the definition of distance.
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When gradient descent is a kernel method
Suppose that we sample a large number N N N of independent random functions f i : R → R f_i \colon \R \to \R f i : R → R from a certain distribution F \Fc F and propose to solve a regression problem by choosing a linear combination f ˉ = ∑ i λ i f i . \bar{f} = \sum_i \lambda_i f_i. f ˉ = i ∑ λ i f i . For…
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The convolution empire strikes back
ConvNets Match Vision Transformers at Scale
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Lindenmayer Systems
Originally developed by Aristid Lindenmayer in 1968, an L-System is a formalism to generate self-similar structures. Although, L-Systems are capable of generating famous fractals like the Sierpinski Triangle (Axiom: FXF--FF--FF, F → FF , X → --FXF++FXF++FXF-- , angle = 60 ) or the Gosper Curve (Axiom: FX, X →…
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Why we don’t generate elliptic curves every day
Elliptic curves are standardized, instead of being generated like Diffie-Hellman parameters. There's good reasons!
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Reconciliation is a knapsack problem
Yesterday, we announced adding AI to our recon engine. Today, we're underscoring how reconciliation is a pain point by delving into the math.
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Marvelous Arithmetics of Distance
No reckoning allowed save the marvelous arithmetics of distance (from Smelling the Wind by Audre Lorde) Suppose a child comes up to you and says “I know 1 is odd and 2 is even, but I think 4 is mor…
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Higher quality random floats
Much has been written about how to generate random 64-bit integers; people might argue over which particular (CS)PRNG is best suited for a given task, but there is agreement on the general shape of the solution: keep some state around, and then extract 64 bits of randomness at a time, where each bit is equally likely…
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Fractional Factorial Experiment Design: There Are Too Many Experiments to Do
Have you ever found yourself in a situation where you had a bunch of parameters to tune for a better result, but there were just too many to exhaustively search all possibilities? I recently saw a …
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Circulant Matrices, Eigenvectors, and the FFT
The eigenvectors of a circulant matrix do not depend on the matrix itself, only on the size, and the eigenvectors are columns of the FFT matrix.
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BB(3, 3) is Hard
I’m excited to share a 3 state, 3 symbol Turing Machine that cannot be proven to halt or not (when starting on a blank tape) without solving a Collatz-like problem. Therefore, solving the \(BB(3, 3)\) problem is at least as hard as solving this Collatz-like problem, a class of problem for which Paul Erdős famously…
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Bounty to Recover NIST's Elliptic Curve Seeds
This is a fun challenge: The NIST elliptic curves that power much of modern cryptography were generated in the late ’90s by hashing seeds provided by the NSA. How were the seeds generated? Rumor has it that they are in turn hashes of English sentences, but the person who picked them, Dr. Jerry Solinas, passed away in…
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Decomposing language models into understandable components
Interpretability research decomposing a transformer layer into thousands of interpretable features that can steer model behavior.
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Subtraction is functionally complete
To be precise, IEEE-754 floating point subtraction is functionally complete . That means you can construct any binary circuit using nothing but floating point subtraction.
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Role of Algorithms
This is lobste.rs comment as an article, so expect even more abysmal editing than usual.
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NIST Elliptic Curves Seeds Bounty
Announcing a $12,288 bounty (tripled to charity) for cracking the five seeds selected by the NSA in the '90s for the NIST elliptic curve standard.
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Debunking NIST's calculation of the Kyber-512 security level
[Sidney Harris cartoon used with permission. Copyright holder: ScienceCartoonsPlus.com .]
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The Irish Logarithm
From the highly eclectic blog of Mark Dominus
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An easy-to-implement, arena-friendly hash map
My last article had tips for for arena allocation . This next article demonstrates a technique for building bespoke hash maps that compose nicely with arena allocation. In addition, they’re fast, simple, and automatically scale to any problem that could reasonably be solved with an in-memory hash map. To avoid…
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Inside the Matrix: Visualizing Matrix Multiplication, Attention and Beyond
Use 3D to visualize matrix multiplication expressions, attention heads with real weights, and more. Matrix multiplications (matmuls) are the building blocks of today’s ML models. This note presents mm, a visualization...
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Lean, Coq and other proof assistants: Visualising proofs as trees
This week we (me & Anton Kovsharov) published Paperproof, a Gentzen-tree-like proof interface for Lean 4. In this post I'll review proof visualisations from o...
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Circles do not exist
Many logos, drawings and other graphical designs have the following shape in it. What is this shape? If you thought: "Ah-ha! I'm smart and r...