A while ago I linked to Steven Strogatz’s series on mathematics in the New York times. The series seems to have come to an end, with some interesting new stories, on integration, differentiation. and probability among others. Here’s an interesting quote on his essay on change:

There’s a more general principle at work here — things always change slowest at the top or the bottom. It’s especially noticeable here in Ithaca. During the darkest depths of winter, the days are not just unmercifully short; they barely improve from one to the next. Whereas now that spring is popping, the days are lengthening rapidly. All of this makes sense. Change is most sluggish at the extremes precisely because the derivative is zero there. Things stand still, momentarily.

Recently, I have read somewhere about Daniel Fleisch and his book “A Student’s Guide to Maxwell’s Equations“. At first glance, I have nothing to do with Maxwell’s equations. However, I thought that a field is a field, be it electric potential or water pressure. First, this is the first time I read a book that was that clearly structured. Essentially each of the four laws (Gauss laws for electric and magnetic fields, Faraday’s law, and Ampere-Maxwell law) cover one chapter, and the final chapter is about how Maxwell “developed his comprehensive theory of electromagnetism” by considering these four equations in combination. This seems not like anything special. However, the structure is in each chapter essentially identical: first the terms of the mathematical equation are briefly explained, then there’s a section on what this is about, the physical meaning, and then there are sub-chapters on every term, sometimes “only” on the sign (in the case of Lenz’s law). I think the critical factor that makes this approach superior to other approaches is how Fleisch links between the equations. Where are similarities in equations? Are they because of similar physical processes? And how are physical processes covered mathematically?

Why do I think all this is relevant? In most disciplines I have worked in, this way of thinking is important: How is a process covered in mathematics. Even more, the way of thinking is essentially the same — it doesn’t matter what kind of field you’re looking at. There is even a quote attributed to Maxwell in the preface: “… analogies are useful not because the quantities are alike but because of the corresponding relationships between quantities“. And by Fleisch: “… although nothing is actually flowing in a static electric field, you’re likely to find the analogy between a faucet (a source of fluid flow) very helpful in understanding the nature of the electrostatic field“. And I think the reverse holds true too. Additionally, Fleisch does a really excellent job in explaining details of the Nabla operator, divergence, curl, and gradient.

Here is one example taken from Fleisch’s book, which is directly relevant to statistics:

Figure 1: Line integral for a scalar function

Figure 1a (Figure 3.2 in Fleisch’s book) shows a wire with variable density. Here is how Fleisch describes what has to be done to determine the mass of the wire. In other words, what has to be done to find the distribution function from its density function.

To determine the total mass of the wire, imagine dividing the wire into a serios of short segments over each of which the linear density is approximately constant, as shown in Figure 1b. The mass of each segment is the product of the linear density of that segment times the segment length dx, and the mass of the entire wire is the sum of the segment masses

via Amphibol

update Saturday; September 4, 2010 The paper “Nürnberger Nachrichten” published an article today pointing out the geological risks of Stuttgart 21.

Has it disappeared? Is it still there? Lots of people, lots of thought. And those bacteria sure must have been hungry!

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Now, what if you wanted, for whatever reason, to estimate the underlying distribution based on the individual sets of your samples? What you could do is estimate a distribution based on each sample. Then you would have to somehow average the distributions. How would you do that? By multiplication? Turns out – no. In fact, the average of the density functions of each sample seems to be a pretty good estimator of the underlying density function.

using 10 sets of samples with 10 samples each

using 25 sets of samples with 100 samples each

• my list of rcommended hydrogeology books;
• individual book reviews of “bottled and sold”, “bottlemania” “numbers rule your world” or “just enough”;

These recommendations so far have all been for books written in english. They are not all purely technical books or particularly textbooks. However, they are all related to the theme of “water”, despite my hands-on quantitative-statistical take on that theme.

Two things happened in the recent past: I realized that I use wikipedia for surprisingly many tasks. I realized that I haven’t used a book in ages for example to look up a distribution function. The other thing that happened, just the other day, was that I took a fairly old programming book, because I wanted to re-read on a certain topic. And I knew it was covered in that (old) book well.

That’s when I thought, that I really like well written books, despite the wickedness of the internet (despite the fact that it recently was claimed to be dead – or not). Then I wondered, what are the best German written books related to the subject I am working in and which also is the topic of this blog? Of course, this is a tough question. And the two books I am going to name in a second are most likely not the only ones. However, they are the two that came to my mind immediately:

• “Statistik — Lehr- und Handbuch der angewandten Statistik” by Joachim Hartung

• “Hydrogeologie — Einführung in die Allgemeine und Angewandte Hydrogeologie” by Bernward Hölting and Wilhelm G. Coldewey

The statistics book appears a bit daunting at first with its 1145 pages. I don’t believe it is possible to read the entire book and neither it is recommended by the authors. However, it has happened to me many times that I was looking for a certain topic, like “How does the Wilcoxon Rank Test work?”, and sure enough I found theory for the test as well as an example in the book.

In contrast to statistics, hydrogeology is not as clearly defined. It’s much harder to distinguish a given topic to be or not to be “a hydrogeology topic”. Hence it would seem much harder to write a book on “hydrogeology”, because it would have to cover many topics. However, Hölting and Coldewey have done an excellent job to explain in short and precise language the basics of hydrogeology, from geology, well hydraulic to chemical aspects. It deserves special mentioning that there is a nice chapter on practical aspects, for example on how to get water out of the ground or groundwater protection.