Book: Alice’s Adventures in a differentiable wonderland
Book: Alice’s Adventures in a differentiable wonderland Neural networks surround us, in the form of large language models, speech transcription systems, molecular discovery algorithms, robotics, and much more. Stripped of anything else, neural networks are compositions of differentiable primitives, and studying them means learning how to program and how to interact with these models, a particular
Programming vs. mathematical curiosity
Once we hit 6174 the sequence starts repeating, as the result of applying this “biggest digit-arrangement minus smallest digit-arrangement” operation to 6174, is 6174 itself. Or in maths parlance: 6174 is a fixed point of this operation. Now here’s the kicker: as long as the starting number is not a single repeated digit, we can start from any 4-digit number and the sequence will always reach 6174
Integral Calculator • With Steps!
The Integral Calculator lets you calculate integrals and antiderivatives of functions online — for free! Our calculator allows you to check your solutions to calculus exercises. It helps you practice by showing you the full working (step by step integration). All common integration techniques and even special functions are supported. The Integral Calculator supports definite and indefinite integra
Middle-square method - Wikipedia
One iteration of the middle-square method, showing a 6-digit seed, which is then squared, and the resulting value has its middle 6 digits as the output value (and also as the next seed for the sequence). Directed graph of all 100 2-digit pseudorandom numbers obtained using the middle-square method with n = 2. In mathematics and computer science, the middle-square method is a method of generating p
Randomness extractor - Wikipedia
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weak entropy source, together with a short, uniformly random seed, generates a highly random output that appears independent from the source and uniformly distributed.[1] Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible source
Lauren Ipsum and the Unfair Coin
Fair Coins are the currency of Userland. A Fair Coin flips fifty-fifty with mathematical exactness. In the real world most coins are pretty fair, but you never know whether there might be a small bias to one side or another. This presents Lauren Ipsum, a girl who's lost in Userland, with a problem. No one wants her "Quarter Dollars". They might be biased. How can she negotiate an exchange rate? A
カプレカー数 - Wikipedia
カプレカー数（カプレカーすう、Kaprekar number）とは、次のいずれかで定義される自然数である[1]。 2乗して上位の半分と下位の半分とに分けて和を取ったとき、元の値に等しくなる自然数。 桁を並べ替えて最大にした数と最小にした数との差を取ったとき、元の値に等しくなる自然数（カプレカー定数）。 名称は、インドの数学者 D. R. カプレカル（英語表記: D. R. Kaprekar[1][2]）にちなむ[3][4]。カプレカ数[5]、カプリカ数[6]ともいい、原語であるマラーティー語の発音[7]に近づけてカプレカル数[8][9]ともいう。 定義1[編集] 正の整数を2乗し、上位と下位のゼロでない[10]数桁ずつに分けて、それらの和を取る。この操作によって元の値に等しくなる数をカプレカー数と呼ぶ。 例えば、297 はカプレカー数である。2972 = 88209 であり、これを上位の2
6174 - Wikipedia
The number 6174 is known as Kaprekar's constant[1][2][3] after the Indian mathematician D. R. Kaprekar. This number is renowned for the following rule: Take any four-digit number, using at least two different digits (leading zeros are allowed). Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary. Subtract the smaller number
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