Math ∩ Programming

Last time we saw a number of properties of graphs, such as connectivity, where the probability that an Erdős–Rényi random graph $ G(n,p)$ satisfies the property is asymptotically either zero or one.

Last time we left off with a tantalizing conjecture: a random graph with edge probability $ p = 5/n$ is almost surely a connected graph.

Last time we left off with the tantalizing question: how do you do a quantum “AND” operation on two qubits?

The best place to start our journey through quantum computing is to recall how classical computing works and try to extend it.

Quantum mechanics is one of the leading scientific theories describing the rules that govern the universe.

In the last post in this series we saw some simple examples of linear programs, derived the concept of a dual linear program, and saw the duality theorem and the complementary slackness conditions which give a rough sketch of the stopping criterion for an…

Problem: Alice chooses a secret polynomial $ p(x)$ with nonnegative integer coefficients.

satellite One of the most interesting questions posed in the last thirty years of computer science is to ask how much “information” must be communicated between two parties in order for them to jointly compute something.

I recently wrapped up a fun paper with my coauthors Ben Fish, Adam Lelkes, Lev Reyzin, and Gyorgy Turan in which we analyzed the computational complexity of a model of the popular MapReduce framework.

The Mona Lisa Leonardo da Vinci’s Mona Lisa is one of the most famous paintings of all time.

So far our discussion of learning theory has been seeing the definition of PAC-learning, tinkering with it, and seeing simple examples of learnable concept classes.

Problem: Two players take turns moving a rook on an 8×8 chessboard.

Greedy algorithms are by far one of the easiest and most well-understood algorithmic techniques.

I’m presenting a paper later this week at the Matheamtical Foundations of Computer Science 2014 in Budapest, Hungary.

A while back I announced a preprint of a paper on coloring graphs with certain resilience properties.

Greedy algorithms are among the simplest and most intuitive algorithms known to humans.

Here’s a simple puzzle with a neat story.

Optimization is by far one of the richest ways to apply computer science and mathematics to the real world.

This post is intended for people with a little bit of programming experience and no prior mathematical background.

Graphs are among the most interesting and useful objects in mathematics.

In a previous post we introduced a learning model called Probably Approximately Correct (PAC).

Last time we saw the Diffie-Hellman key exchange protocol, and discussed the discrete logarithm problem and the related Diffie-Hellman problem, which form the foundation for the security of most protocols that use elliptic curves.

Here is a fun puzzle.

So far in this series we’ve seen elliptic curves from many perspectives, including the elementary, algebraic, and programmatic ones.

So here we are.

Two years ago, Erik Demaine and three other researchers published a fun paper to the arXiv proving that most incarnations of classic nintendo games are NP-hard.

Back when I was first exposed to programming language design, I decided it would be really cool if there were a language that let you define your own number types and then do all your programming within those number types.

This is a guest post by my colleague Adam Lelkes.

So far on this blog we’ve given some introductory notes on a few kinds of algebraic structures in mathematics (most notably groups and rings, but also monoids).

Last time we saw a geometric version of the algorithm to add points on elliptic curves.

I’m pleased to announce that another paper of mine is finished.

Last time we looked at the elementary formulation of an elliptic curve as the solutions to the equation $$y^2 = x^3 + ax + b$$ where $ a,b$ are such that the discriminant is nonzero: $$-16(4a^3 + 27b^2) \neq 0$$ We have yet to explain why we want our equa…

This is a guest post by my friend and colleague Adam Lelkes.

Finding solutions to systems of polynomial equations is one of the oldest and deepest problems in all of mathematics.

This is a guest post by my friend and colleague Adam Lelkes.

With all the recent revelations of government spying and backdoors into cryptographic standards, I am starting to disagree with the argument that you should never roll your own cryptography.

A few awesome readers have posted comments in Computing Homology to the effect of, “Your code is not quite correct!”

This post is intended to be a tutorial on how to access the RealityMining dataset using Python (because who likes Matlab?)

A professor at Stanford once said, If you really want to impress your friends and confound your enemies, you can invoke tensor products… People run in terror from the $ \otimes$ symbol.

In tackling machine learning (and computer science in general) we face some deep philosophical questions.

We’ve studied the Fourier transform quite a bit on this blog: with four primers and the Fast Fourier Transform algorithm under our belt, it’s about time we opened up our eyes to higher dimensions.

So far in this series we’ve seen two nontrivial algorithms for bandit learning in two different settings.

For a while I’ve been meaning to do some more advanced posts on optimization problems of all flavors.

In the last twenty years there has been a lot of research in a subfield of machine learning called Bandit Learning.

startups The software world is always atwitter with predictions on the next big piece of technology.

A lot of people who like functional programming often give the reason that the functional style is simply more elegant than the imperative style.

My First Paper I’m pleased to announce that my first paper, titled “Anti-Coordination Games and Stable Colorings,” has been accepted for publication!

During the 1950’s the famous mathematician Paul Erdős and Alfred Rényi put forth the concept of a random graph and in the subsequent years of study transformed the world of combinatorics.

Machine learning is broadly split into two camps, statistical learning and non-statistical learning.

Problem: Prove that for vectors $ v, w$ in an inner product space, the inequality $$\displaystyle |\left \langle v, w \right \rangle | \leq \| v \| \| w \|$$ Solution: There is an elementary proof of the Cauchy-Schwarz inequality (see the Wikipedia articl…