# quotients Published articles for quotients. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Two's Complement and Group Theory DevFeed: [Two's Complement and Group Theory]() Original publisher: [Read original article]() Published: 2023-07-10T07:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [math](), [Mathematics](), [Computer science](), [Programming]() Tags: [abelian-groups](), [arithmetic](), [bits](), [boolean](), [circuits](), [computer-science](), [group-actions](), [groups](), [math](), [mathematics](), [programming](), [quotients](), [symmetry](), [twos-complement]() ### AI overview The article explains two's-complement signed integer arithmetic using group theory. It presents signed and unsigned n-bit integers as representations of the quotient group of integers modulo 2^n, clarifying why the same arithmetic circuits can operate on both. ### Source excerpt Before I discovered math, I was a first year undergrad computer science student taking Electrical Engineering 101. The first topic I learned was what bits and boolean gates are, and the second was the two's complement representation of a negative n-bit integer. At the time two's complement seemed to me like a bizarre quirk of computer programming, with minutiae you just had to memorize. If the leading bit is 1, it's negative, and otherwise it's positive. ## Fixing Bugs in "Computing Homology" DevFeed: [Fixing Bugs in "Computing Homology"]() Original publisher: [Read original article]() Published: 2014-01-23T23:05:16Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Mathematics](), [Code](), [function](), [Matrix]() Tags: [bugs](), [code](), [computing](), [function](), [homology](), [linear-algebra](), [mathematics](), [matrices](), [persistent-homology](), [programming](), [python](), [quotients](), [row-reduction](), [test]() ### AI overview This article corrects bugs in code for computing homology. It identifies an indexing error, explains a mathematical mistake in simultaneous row and column reduction, and shows how further row reduction is needed to obtain the correct rank for a triangulation of the Möbius band. ### Source excerpt A few awesome readers have posted comments in Computing Homology to the effect of, "Your code is not quite correct!" And they're right! Despite the almost year since that post's publication, I haven't bothered to test it for more complicated simplicial complexes, or even the basic edge cases! When I posted it the mathematics just felt so solid to me that it had to be right (the irony is rich, I know). ## Universal Properties DevFeed: [Universal Properties]() Original publisher: [Read original article]() Published: 2013-05-24T14:53:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Category Theory](), [Mathematics](), [function](), [object]() Tags: [categories](), [category-theory](), [coproducts](), [examples](), [function](), [mathematics](), [ml](), [morphisms](), [object](), [product](), [programming](), [quotients](), [types](), [universal-properties]() ### AI overview This tutorial introduces universal properties in category theory, defining initial, final, and zero objects through unique morphisms. It illustrates the concepts with examples from mathematics and Set, and discusses constructing programs related to these properties. ### Source excerpt Previously in this series we've seen the definition of a category and a bunch of examples, basic properties of morphisms, and a first look at how to represent categories as types in ML. In this post we'll expand these ideas and introduce the notion of a universal property. We'll see examples from mathematics and write some programs which simultaneously prove certain objects have universal properties and construct the morphisms involved. ## Constructing Topological Spaces -- A Primer DevFeed: [Constructing Topological Spaces -- A Primer]() Original publisher: [Read original article]() Published: 2012-11-11T18:13:58Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [spaces]() Tags: [building](), [complex](), [dimension](), [quotients](), [spaces](), [topology]() ### AI overview This primer explains how to construct more complex topological spaces from simpler ones using subspaces and quotients. It introduces the subspace topology and defines spheres, including the circle, as topological spaces. ### Source excerpt Last time we investigated the (very unintuitive) concept of a topological space as a set of "points" endowed with a description of which subsets are open. Now in order to actually arrive at a discussion of interesting and useful topological spaces, we need to be able to take simple topological spaces and build them up into more complex ones. This will take the form of subspaces and quotients, and through these we will make rigorous the notion of "gluing" and "building" spaces.