Tag Archives: abstractmath

abstractmath.org, exposition, math

abstractmath.org beta

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Around two years ago I began a systematic revision of abstractmath.org. This involved rewriting some of the articles completely, fixing many errors and bad links, and deleting some articles. It also involved changing over from using Word and MathType to writing directly in html and using MathJax. The changeover was very time consuming.

Before I started the revision, abstractmath.org was in alpha mode, and now it is in beta. That means it still has flaws, and I will be repairing them probably till I can’t work any more, but it is essentially in a form that approximates my original intention for the website.

I do not intend to bring it out of beta into “final form”. I have written and published three books, two of them with Michael Barr, and I found the detailed work necessary to change it into its final form where it will stay frozen was difficult and took me away from things I want to do. I had to do it that way then (the olden days before the internet) but now I think websites that are constantly updated and have live links are far more useful to people who want to learn about some piece of math.

My last book, the Handbook of Mathematical Discourse, was in fact published after the internet was well under way, but I was still thinking in Olden Days Paper Mode and never clearly realized that there was a better way to do things.

In any case, the entire website (as well as Gyre&Gimble) is published under a Creative Commons license, so if someone wants to include part or all of it in another website, or in a book, and revise it while they do it, they can do so as long as they publish under the terms of the license and link to abstractmath.org.

Previous posts about the evolution of abstractmath.org

Books by Michael Barr and Charles Wells

Toposes, triples and theories

Category theory for computing science

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This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

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math, representations

Riemann clouds improved

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In my post Playing with Riemann Sums I showed a couple of clouds of points, each representing a particular Riemann sum for a particular function.   I have extended the code in a couple of ways.

The new code is in the Mathematica notebook and CDF file called MoreRiemann in the Mathematica section of abstractmath.   The .nb form is a Mathematica Notebook, which requires Mathematica to run and allows you to manipulate the objects and change the code in the notebook as you wish.  In particular, you can rerun the commands generating the clouds to get a new random result.  The .cdf file contains the same material and can be viewed using Mathematica CDF Player, which is available free here.  Both files have several other examples besides the ones shown below.

As always, my code is one-time code to show the ideas, but it is available freely via the Creative Commons Attribution – ShareAlike 3.0 License. I hope people will feel free to develop it further for use in teaching or for their own purposes.

Below is a cloud for \int_0^2 \sqrt{4-x^2} dx, the area of a quarter circle of radius 2, which is \pi.  The blue dots are arbitrary random Riemann sums with mesh shown on the horizontal axis and value on the vertical axis.  The partitions and the point in each subinterval are both random.  The red dots are arbitrary Riemann sums with random partitions but using the midpoint for value.

The next cloud shows random blue dots with the same meaning as above.  The red dots are Riemann sums with uniform subintervals evaluated at midpoints.  Possible discussion question for both of the clouds above:

  • Why do the red dots trend upward?

The following cloud is like the cloud above  with the addition of green dots representing uniform partitions evaluated at the left endpoint or right endpoint. (But the mesh scale is extended, giving different proportions to the picture.)

Of course the left endpoint gives the upper sums and the right endpoint gives the lower sums.

  • Explain the slight downward curvature of both green streaks.
  • Explain the big gap between the blue dots and the green dots.  (Requires some machinations with probability.)
  • Would there be blue dots a lot nearer the green dots if I ran the command asking for many more blue dots?

(These are idle questions I haven't thought about myself, but I'll bet they could be turned into good projects in analysis classes.)

Here is a cloud for \int_0^{\pi}\sin x dc with everything random for the blue dots and random partitions but midpoints for the red dots.

  • Why do these red dots trend upward?

The cloud below is for the same integral but uses uniform subintervals for the midpoint and adds green points for both the left endpoint and the right endpoint of uniform subinterval.

  • Why on earth do all the green dots trend downward???

This is a similar picture for \int_0^1 x^2 dx.  There are red dots but they are kind of drowned out.

And finally, here is \int_{\frac{1}{2}}^2 \frac{1}{x} dx:

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