IntMath Newsletter: Sampling, svgPHPGrapher
By Murray Bourne, 30 Sep 2015
30 Sep 2015
In this Newsletter:
1. Sampling
2. Resource: svgPHPGrapher
3. Application of hyperbolic decay
4. Math puzzle
5. Math movie: Jim Simons
6. Final thought: Haze
1. Sampling to produce mathematical function graphs
I've been developing a new graphing system for IntMath (see the following article on svgPHPGrapher).
It's been an interesting challenge and I've learned a lot doing it. One of the issues I constantly grappled with was that of sampling. To produce a smooth curve that goes to the edge of the graph boundaries (especially when it involves discontinuities), you need to sample a sufficient number of points. But if you take too many, it takes too long for the computer to create the graph, and ends up inflating the size of the graph file.
I thought while doing this that there are a lot of parallels with the way students learn how to sketch graphs. The age-old technique of drawing up a table, substituting in some values, putting dots on the graph paper and then joining them with a curve is not always a good idea, since it can trap those who aren't careful, or who don't have a sense of what the graph should look like in the first place.
This article is about sampling, and includes a bunch of examples of the new graphs I've been developing.
While I obviously believe students should use computers to draw graphs, it's important even then to have a sense of what it should look like first.
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When sketching mathematical functions on paper or computers, it's a good idea to have a sense of what the general shape of the curve is before you start. Sampling can lead to erroneous results. |
2. Resource: svgPHPGrapher - a new math plotter
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svgPHPGrapher produces clean math graphs with no javascript, and all the processing happens on the server. This article also talks about SVG - a mathematical approach to crisp images. |
You may also find the syntax example page interesting (not necessarily for the syntax, but for the graphs on the page). Here's the link to the syntax examples page:
3. Application of hyperbolic decay
A few months back I wrote an article about Hyperbolic and Exponential Discounting. A reader kindly sent me a journal article that describes an application of hyperbolic decay, involving capacitors.
Almost all electronic devices contain capacitors, which have the ability to hold an electrical charge for a short time.
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The article talks about the case where the capacitor is charged for a period of time, then discharged, then charged again in a pattern that resembles the hyperbolic decay graph. Here's the paper: On hyperbolic laws of capacitor discharge through self-timed digital loads |
4. Math puzzles
The puzzle in the last IntMath Newsletter asked about cards on a table.
Correct answers with explanation for the main question were given by: Henry, Isabel, Tomas, and Don. For the bonus question, kudos to Henry, Isabel, Don, Tomas, and Pritesh.
As usual, it was interesting to observe the very different approaches each of you took.
New math puzzle
In the set of digits below, you need to circle 6 of them that sum together to give 21.
1 1 1
3 3 3
5 5 5
9 9 9
You can leave your responses here.
5. Math movie - Jim Simons: The mathematician who cracked Wall Street
There's several things in this video that made me uncomfortable (he sees little ethical dilemma in the hedge fund industry siphoning off the best and brightest, or in living much of his life with the main aim of making mountains of money), but he redeems himself at the end with his philanthropic work.
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It's a long interview, but gives an interesting insight into a mathematician who has had a broad range of experiences. Go to: Jim Simons: A rare interview with the mathematician who cracked Wall Street |
6. Final thought: Haze
It's that time of the year in SE Asia when millions of people (including me) suffer due to the smoke and haze created by burning forests for paper and palm oil. Spare a thought for the Indonesian villagers nearby the fires, who at times are faced with a PSI (Pollutant Standards Index) of 2000+, when anything over 100 is considered unhealthy. Such pollutant indices include factors such as the concentration of 2.5 μ (micron) fine particles - the ones that can easily pass through your cell walls.
You can see hourly PSI trend graphs for Singapore here: PSI Trend Graphs
Come on, Indonesian farmers - show some responsibility for the environment, for your own people, and for the vast number of people who are really suffering due to your "cash before life" approach.
Until next time, enjoy whatever you learn.
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