Most of us last saw calculus in school, but derivatives are a critical part of machine learning, particularly deep neural networks, which are trained by optimizing a loss function. This article is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed.

— Permalink

PandaPy has the speed of NumPy and the usability of Pandas (10x to 50x faster).

— Permalink

If you round the result of every computation, then you can get exactly correct financial calculations using floating-point numbers, for realistic ranges of values.

— Permalink

Facebook AI has developed the first neural network that uses symbolic reasoning to solve advanced mathematics problems.

— Permalink

Nota is a nice terminal calculator with rich notation rendering. It is designed for your quick calculations and therefore provides you with a tiny and beautiful language so you can express your ideas easily. Nota is all about beauty and ASCII art.

— Permalink

Ryohei Hisano and Didier Sornette wrote in 2012 a paper titled, “On the distribution of time-to-proof of mathematical conjectures.”

Today Ken and I discuss predicting the end to mathematical conjectures. This is apart from considering odds on which way they will go, which we also talk about.

Nine years ago the Christmas issue of the New Scientist magazine analyzed a small set of solved mathematical conjectures and used it to forecast when the P vs. NP conjecture would be solved. The article estimated that the “probability for the P vs. NP problem to be solved by the year 2024 is roughly 50%”.

— Permalink

Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades.

— Permalink

Recently, I am learning how Elliptic Curve Cryptography works. I searched around the internet, found so many articles and videos explaining it. Most of them are covering only a portion of it, some of them skip many critical steps how you get from here to there. In the end, I didn’t find an article that really explains it from end-to-end in an intuitive way.

With that in mind, I would like to write a post explaining Elliptic Curve Cryptography, cover from the basics to key exchange, encryption, and decryption.

]]>Calculate digits of π. From scratch, that is, using only addition, subtraction, multiplication, and division.

— Permalink

Daniel J. Bernstein, Bo-Yin Yang. "Fast constant-time gcd computation and modular inversion."

— Permalink

elkai is a Python 3 library for solving travelling salesman problems without external dependencies, based on LKH by Keld Helsgaun.

— Permalink

You’ve probably played with model trains, for instance with something like the Brio set shown below.1 And if you’ve built a layout with a model train set, you may well have wondered: is it possible for my train to use all the parts of my track?

]]>Britney Crystal Gallivan (born 1985) of Pomona, California, is best known for determining the maximum number of times that paper or other materials can be folded in half.

]]>Mathigon's innovative new curriculum covers everything from fractions and trigonometry to graph theory, cryptography, prime numbers and fractals.

— Permalink

The Fibonacci numbers are the sequence 1, 1, 2, 3, 5, 8, ..., and satisfy the recurrence F(n) = F(n – 1) + F(n – 2).

They also have a beautiful formula.

My favorite derivation of this formula entirely avoids algebraic manipulation.

— Permalink

Roughly speaking, Gödel’s Incompleteness Theorem states that there are true mathematical statements that cannot be proven. When I was in 11-th grade, my geometry teacher Mr. Olsen, my friend Uma Roy, and I spent five weeks reading through Gödel’s original proof of the theorem. Why did it take so long? Partly because Uma and I were high-school students. Partly because Gödel was a less-than-talented writer. But mostly because the proof is actually pretty hard.

— Permalink

A few months from now, if James Tanton and his Global Math Project co-conspirators have their way, ten million schoolchildren will take a huge mathematical step from the twenty-first century all th…

— Permalink

This is the first million integers, represented as binary vectors indicating their prime factors, and laid out using the UMAP dimensionality reduction algorithm by Leland Mcinnes. Each integer is represented in a high-dimensional space, and gets squished down to 2D so that numbers with similar prime factorisations are closer together than those with dissimilar factorisations.

]]>The purpose of this month's article is to bring once more to the public consciousness some work of Sir Roger Penrose, namely "On the Cohomology of Impossible Figures," which appeared in Structural Topology 17 (1961) pp. 11-16 and was reprinted in Leonardo 25, Nos 3/4 (1992) and then as Chapter 4 of The Visual Mind, Michele Emmer, ed., MIT Press, 1993.

]]>A receiver operating characteristic (ROC) is a graph that illustrates the performance of a binary classifier as its discrimination threshold (cutoff) is changed.

The curve is created by plotting the true positive rate (TPR) against the false positive rate (FPR) at various cutoff settings. The true-positive rate is known as sensitivity, the false-positive rate is known as the fall-out and is calculated as (1 - specificity).

The ROC curve is thus a plot of the true positives (TPR) versus the false positives (FPR). The ROC curve can be generated by plotting the cumulative distribution function (area under the probability distribution from - ∞ to + ∞ ) of the correct detection probability in the y-axis versus the cumulative distribution function of the false-alarm probability in x-axis.

— Permalink

Mathematics is all around us, and it has shaped our understanding of the world in countless ways.

In 2013, mathematician and science author Ian Stewart published a book on 17 Equations That Changed The World. We recently came across this convenient table on Dr. Paul Coxon’s twitter account by mathematics tutor and blogger Larry Phillips that summarizes the equations.

— Permalink

We shall introduce PyDelaunay as a practical spatial construction. It is an efficient Python implementation of Voronoi/Delaunay Tessellation, which can be served as the Basic Map of the neighbourhood concept.

— Permalink

GNU TeXmacs is a free wysiwyw (what you see is what you want) editing platform with special features for scientists. The software aims to provide a unified and user friendly framework for editing structured documents with different types of content (text, graphics, mathematics, interactive content, etc.). The rendering engine uses high-quality typesetting algorithms so as to produce professionally looking documents, which can either be printed out or presented from a laptop.

— Permalink

Joy as mathematicians discover a new type of pentagon that can cover the plane leaving no gaps and with no overlaps. It becomes only the 15th type of pentagon known that can do this, and the first discovered in 30 years.

— Permalink

Mathomatic™ is a portable, command-line, educational CAS and calculator software, written entirely in the C programming language. It is Free and Open Source Software (FOSS), published under the GNU Lesser General Public License (LGPL version 2.1), and has been under continual development since 1986. The software can symbolically solve, simplify, combine, and compare algebraic equations, simultaneously performing generalized standard, complex number, modular, and polynomial arithmetic, as needed. It does some calculus and is very easy to compile/install, learn, and use.

— Permalink

A collection of functions for shaping and tweening signals in the range 0 to 1.

— Permalink

Beautiful math in all browsers.

A JavaScript display engine for mathematics that works in all browsers.

No more setup for readers. It just works.

— Permalink

A probabilistic algorithm is exhibited that calculates the gcd of many integers using gcds of pairs of integers; the expected number of pairwise gcds required is less than two.

— Permalink

This page points to a collection of mathematical quotations culled from many sources.

— Permalink