Optimizing your Wordle Strategy
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Wordle seems to have quickly become quite popular. As word games go, it’s a good option if you want a quick dose. And as I’m beginning to find, it can be rather addictive.
Posts
Catalan Numbers in Perturbation Theory
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Catalan numbers are everywhere. Quite famously, in volume 2 of his Enumerative Combinatorics, Richard P. Stanley gives 66 different interpretations of these numbers. They are given by
Jacobi’s Triple Product
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There are some surprising connections between bosonization in quantum field theory, classical theorems in number theory, and characters of the affine algebra $A_1^{(1)} = \widehat{\mathfrak{su}}(2)_1$. All of them connect to the Jacobi triple product identity:
Continued Fractions 2: The Final $F$-coalgebra
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In a previous post, we addressed some questions whether continued fractions can be considered a natural and practical way to represent real numbers. From a decimal standpoint, continued fractions can at first seem awkward, but soon some nice properties start to emerge. For instance, a number is rational if and only if its continued fraction expansion terminates. This is much simpler than in decimal expansions, when rational numbers can either have finite expansions or infinite but eventually periodic ones. Better yet, infinite but eventually periodic expansions also have their place in the land of continued fractions: they correspond to quadratic irrationals, a beautiful fact that is somewhat harder to prove.
Tropical Geometry of ReLU Neural Nets
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A rectified linear unit (ReLU) is a neuron with activation function
Continued Fractions: Introduction and Gosper Arithmetic
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Continued fractions are expressions of the form \[\lbrack a_0;a_1,a_2,\ldots\rbrack = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots}}}.\] Every real number has a unique expression as a continued fraction, so long as we exclude representations which end in 1, since \[a_0;a_1,\ldots,a_n,1] = [a_0;a_1,\ldots,a_n+1].\] Note that this is no worse (and probably quite a bit better) than the ambiguity in decimal expansions, for which \[1 = 0.999\cdots.\]