Index

 

And because arithmetic science and geometric science are connected, and support one another, the full knowledge of numbers cannot be presented without encountering some geometry, or without seeing that operating in this way on numbers is close to geometry; the method is full of many proofs and demonstrations that are made with geometric figures.

Fibonacci, preface to Liber Abaci

(first published 1202, 1228 manuscript translated by Lawrence E. Sigler)

The Spanish Treasure. A story of love and the love of gold, etc

Graphical linear algebra is a work in progress, and there are many open research threads. We are looking for PhD students, so please consider applying!

This blog is written in English. To read and contribute to translations (Dutch, French, German,…) see this page by Vincent Verheyen.


Introduction

Episode 1 – Makélélé and Linear Algebra

Episode 2 – Methodology, Handwaving and Diagrams


Adding and Copying

Episode 3 – Adding (Part 1) and Mr Fibonacci

Episode 4 – Dumbing Down and Magic Lego

Episode 5 – Spoilers, Adding (Part 2) and Zero

Episode 6 – Crema di Mascarpone and Diagrammatic Reasoning

Episode 7 – Copying, Discarding and The Slogan

Episode 8 – When Adding met Copying…

Episode 9 – Natural numbers, diagrammatically


Matrices and PROPs

Episode 10 – Paths and Matrices

Episode 11 – From Diagrams to Matrices

Episode 12 – Monoidal Categories and PROPs (Part 1)

Episode 13 – PROPs (Part 2) and Permutations

Episode 14 – Homomorphisms of PROPs

Episode 15 – Matrices, diagrammatically

Episode 16 – Trust the Homomorphism, for it is Fully Faithful


Integers and Relations

Episode 17 – Maths with Diagrams

Episode 18 – Introducing the Antipode

Episode 19 – Integer matrices

Episode 20 – Causality, Feedback and Relations

Episode 21 – Functions and Relations, diagrammatically

Episode 22 – The Frobenius Equation

Episode 23 – Frobenius Snakes and Spiders


Fractions and Spaces

Episode 24 – Bringing it all together

Episode 25 – Fractions, diagrammatically

Episode 26 – Keep Calm and Divide by Zero

Episode 27 – Linear Relations

Episode 28 – Subspaces, diagrammatically

Episode 29 – Dividing by zero to invert matrices

Episode 30 – The essence of graphical linear algebra


Sequences and Signal Flow Graphs

Episode 31 – Fibonacci and sustainable rabbit farming


Offtopic

Sometimes this blog actually looks like a blog.

16 September 2016 – Leicester and the battle for universities

25 thoughts on “Index

  1. As an electrical engineer, I’ve long enjoyed graphical linear algebra by way of schematics. I’m also working on a graphical algebra for computations. This blog is illuminating and inspiring – thanks for sharing!

    Liked by 2 people

    • Hi Alan,

      Thanks for your comments!

      We have an operational semantics for signal flow graphs, and we wrote it up here: http://users.ecs.soton.ac.uk/ps/papers/popl15.pdf. Ultimately we use a kind of trace equivalence as operational equivalence. For our simple signal flow graphs, where there is one global execution clock, traces are enough. On the other hand, if we were to consider concurrency/multi-threaded executions in this kind of framework then I would expect bisimulation to play a role. It’s definitely interesting the case of Petri nets, which also has a graphical semantics; but also there we got more mileage out of traces because we were interested in using compositionality of the algebra for model checking, and for something like reachability traces are enough. Check out http://users.ecs.soton.ac.uk/ps/papers/rp2014.pdf.

      I took a look at your paper; I like the way it handles both linear and non-linear components: your graphical syntax is very close to what we’ve been doing! Your operational semantics is also not very different from the labelled transition systems that come up in our work, but we typically have two labels on each transition that capture “what is observed on the dangling wires” in that time instant. It seems to me that the main difference between our approaches is that we have not really been dealing very much with traced categories (in fact, these only show up when we start insisting on directed flow); for us compact categories are the norm. For ideological reasons we are against directed flows😉

      It’s interesting that there was so much activity in this area in the late 90s. I’m trying to bring it back in, like flannel shirts😉

      All the best,
      Pawel.

      Liked by 1 person

  2. Even as a grad student in compsci I have to admit – I don’t feel like I’m intuitively comfortable with the idea of matrices, vector spaces and so forth. I understand what they are and how to use them, but the language of linear algebra still often feels foreign to me (why do we care about vector spaces and not some other structure with scalars and vectors?). This blog has been so far giving me what I’ve been looking for for a while. Keep it coming🙂.

    Liked by 1 person

    • Thanks! I’m really looking forward to writing the next episode as well, but I’ve been totally overloaded with work in the last couple of weeks. Luckily it’s looking like this weekend will be relatively free, so there should be an episode next week.

      As far as the reals go, kind of: it will follow the same idea as the continued fraction representation, but I have not written down all the details yet.

      Like

  3. I’m so glad I stumbled upon this blog. I’ve always enjoyed reading about new ways of thinking about traditional mathematics, like Hestene’s Geometric Algebra. Too bad my mathematical maturity is such that I’m comfortable reading the blog and its categorical reasoning, but not at all the paper. I’ll be finishing my undergrad in physics, and it’d be great to be at the point where one can graphically reason through quantum processes in this framework.

    Liked by 1 person

  4. Hi Pawel, can we look forward to another episode in the near future? I don’t mean to rush you (I realize all the work going into actually developing the mathematics behind this blog takes time as well, and is likely a good deal more important), but it has been an enjoyable read so far so I’m hoping there’ll be more!🙂

    Like

    • Hi Rasmus — thanks, it’s nice to hear that someone is looking forward to more!🙂

      It’s been a crazy semester and I’ve been oscillating between super busy and somewhat burnt out.

      I’m hoping to finish my exam marking this weekend though, and I have a half-written episode that I’m pretty excited about. So hopefully, mid next week!

      Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s