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Like Clockwork: Modulo Addition and Finite Groups of Integers
Modulo arithmetic always scared me in college. (Honestly, it’s still not something I can do as easily as integration or spotting pdfs hidden inside icky integrals.) Then when you add abstract algebra...
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Concatenation as an Operation
Mathematics is like any activity, sport, or skill: it must be honed and practiced. With that in mind, I have been bolstering up my abilities in algebra with a fantastic book A Book of Abstract Algebra,...
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Group Theory, XOR, and Binary Codes: Introducing Coding Theory
Binary codes and bit-wise operations are fundamental in computer science. Whatever device you are running today works because of binary codes, and the bit-wise operations AND, OR, NOT, and XOR. (A fun...
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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...
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Taking Things for Granted: Elementary Properties of Groups
We take a lot of things for granted: electricity, gas at the pump, and mediocre coffee at the office. Many concepts in basic algebra are also taken for granted, such as cancellation of terms, and...
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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...
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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,...
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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...
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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...
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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...
View ArticleIsomorphisms: 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...
View ArticleCayley’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...
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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...
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Like Clockwork: Modulo Addition and Finite Groups of Integers
Modulo arithmetic always scared me in college. (Honestly, it’s still not something I can do as easily as integration or spotting pdfs hidden inside icky integrals.) Then when you add abstract algebra...
View ArticleClik here to view.
Concatenation as an Operation
Mathematics is like any activity, sport, or skill: it must be honed and practiced. With that in mind, I have been bolstering up my abilities in algebra with a fantastic book A Book of Abstract Algebra,...
View ArticleClik here to view.
Group Theory, XOR, and Binary Codes: Introducing Coding Theory
Binary codes and bit-wise operations are fundamental in computer science. Whatever device you are running today works because of binary codes, and the bit-wise operations AND, OR, NOT, and XOR. (A fun...
View ArticleClik here to view.
Equivalence v. Isomorphisms in Category Theory
Introduction Editor’s Note: The article is co-written by Rachel Traylor (The Math Citadel/Marquette University) and Valentin Fadeev (The Open University, UK). Substantial additional review,...
View ArticleApplications of Reflections: Taking a Group to its “Abelian” Form
In continuing the exploration of explicit applications and examples of category-theoretic concepts, we highlight the versatility of reflections and reflective subcategories. This concept can be used...
View ArticleTopologies and Sigma-Algebras
Both topologies and \sigma-algebras are collections of subsets of a set X. What exactly is the difference between the two, and is there a relationship? We explore these notions by noting the...
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