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Category: Algebra

Using Boolean Algebra to Find all Maximal Independent Sets in a Graph

Using Boolean Algebra to Find all Maximal Independent Sets in a Graph

Graph theory may be one of the most widely applicable topics I've seen in mathematics. It's used in chemistry, coding theory, operations research, electrical and network engineering, and so many other places. The subject is mainly credited to have begun with the famous  Seven Bridges of K√∂nigsberg problem posed by Leonard Euler in 1736. Frank Harary…

Cayley’s Theorem: Powerful Permutations

Cayley’s Theorem: Powerful Permutations

We've discussed before how powerful isomorphisms can be, when we find them. Finding isomorphisms "from scratch" can be quite a challenge. Thankfully, Arthur Cayley proved one of the classic theorems of modern algebra that can help make our lives a bit easier. We'll explore this theorem and note the power of groups of permutations.  (more…)

Isomorphisms: Making Mathematics More Convenient

Isomorphisms: Making Mathematics More Convenient

Much of pure mathematics exists to simplify our world, even if it means entering an abstract realm (or creating one) to do it. The isomorphism is one of the most powerful tools for discovering structural similarities (or that two groups are identical structurally) between two groups that on the surface look completely unrelated. In this post,…

All the Same Opposites

All the Same Opposites

Editor's note: see this appendix for supporting proofs. Fields are among the most convenient algebraic structures, preserving much of the arithmetic we know and love from familiar fields like the rationals and the real numbers . Now, it is unnecessary that a set possess infinitely many elements to possibly constitute a field (under the right…

Mailbox Answers: Calculating New Parity After an Overwrite

Mailbox Answers: Calculating New Parity After an Overwrite

I recently did some work for Mr. Howard Marks, an independent analyst and founder of Deep Storage on the subject of data protection and data loss. He e-mailed me with a question regarding calculating the new parity for a stripe of data on a storage system.  Let us consider the case of a RAID 5…

Welcome to GF(4)

Welcome to GF(4)

Everyone has solved some version of a linear system in either high school or college mathematics. If you've been keeping up with some of my other posts on algebra, you know that I'm about to either take something familiar away, or twist it into a different form. This time is no different; we're going to…

A Partition by any Other Name

A Partition by any Other Name

I promise I'm actually a probability theorist, despite many of my posts being algebraic in nature. Algebra, as we've seen in several other posts, elegantly generalizes many things in basic arithmetic, leading to highly lucrative applications in coding theory and data protection.  Some definitions in mathematics may not have obvious "practical use", but turn out to yield theorems and results so…

Reduce the Problem: Permutations and Modulo Arithmetic

Reduce the Problem: Permutations and Modulo Arithmetic

We've all seen permutations before. If you have ten distinct items, and rearrange them on a shelf, you've just performed a permutation. A permutation is actually a function that is performing the arrangement on a set of labeled objects. For simplicity, we can just number the objects and work with permuting the numbers.  (more…)

Theory of Coding, Episode 2: Maximum-Likelihood Decoding

Theory of Coding, Episode 2: Maximum-Likelihood Decoding

The introduction to coding theory in this post will now allow us to explore some more interesting topics in coding theory, courtesy of Pinter's A Book of Abstract Algebra. We'll introduce the notion of a code, informations, and parity check equations. Most communication channels are noisy to some extent, which means that a transmitted codeword may have…