# Topology and First-Order Modal Logic

The normal square root function can be considered to be multi-valued.

Let’s momentarily accept the heresy of saying that the square root of a negative number is $latex 0$, so that our function will be total.

How can we represent the situation of this branching “function” topologically?

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# A LaTeX version of Mumford’s impression of Spec Z[x], or some TikZ tricks

reminds me of the xkcd style graphs competition on stackexchange

Notice: The post An atlas of the affine line over the integers and the page An atlas for $latex \mathrm{Spec}\,\mathbb{Z}[x]$ might be interesting if you came here by your favourite search engine.

As promised on Twitter I decided to draw a TeX version of Mumford’s drawing of an elusive object: $latex \mathrm{Spec}\,\mathbb{Z}[x]$

This can be considered as the first non-trivial example in scheme theory. Mumford’s treasure map tries to depict how we could (but we can’t or shouldn’t for a myriad of reasons) visualize its geometry. The meaning behind it, as found in the mimeographed notes named The Red Book, is discussed thoroughly at neverendingbooks.org’s series on Brave New Geometries.

So in case you don’t know the drawing, we’re talking about

The entire series (and by extension the weblog) is a must-read for every math student by the way! As a matter of fact, it was the strangeness and…

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# An End to Four Years of Prometheus Fusion Perfection

I have decided to end Prometheus Fusion Perfection.

The project is too expensive for me. The Polywell requires at least \$200MM to bring to fruition. I cannot hope to raise that much money.

Although I could raise smaller amounts with Kickstarter, it would not be enough to complete the project.

The blog will remain online indefinitely.

I want to thank everyone that participated in this project: Many people have donated their knowledge, time and money.

Although this project did not solve the energy crisis, it pushed the limits of  DIY science.

Personally, I learned more doing this project than any other endeavor in my life. I’ve plumbed the depths of plasma physics, electrical engineering, hardware hacking and on and on.

It saddens me to end such an amazing project.

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# Use geom_rect() to add recession bars to your time series plots #rstats #ggplot

Zach Mayer’s work reproducing John Hussman’s Recession Warning Composite prompted me to dig this trick out of my (Evernote) notebook.

First, let’s grab some data to plot using the very handy getSymbols() function from Jeffrey Ryan’s quantmod package. We’ll load the U.S. unemployment rate (UNRATE) from the St. Loius Fed’s Federal Reserve Economic Data (src="FRED") and load the time series into a data.frame:

Now FRED provides a USREC time series which we could use to draw the recessions. It’s a bit awkward, though, as it contains a boolean to flag recession months since January 1921. All we really want are the start and end dates of each recession. Fortunately, the St. Louis Fed publishes just such a table on their web site. (See the answer to “What dates are used for the US recession bars in FRED graphs?” on http://research.stlouisfed.org/fred2/help-faq/.) Sometimes it’s still…

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# A Mission Statement

The History, Philosophy and Mythology of Science an Unholy Triumvirate.

Since at least the early 1960s it has been common practice to regard the history and philosophy of science as a specious of Siamese twins somehow joined at the hip, in a well-known (well-known amongst philosophers of science that is!) bon mot Imre Lakatos wrote, “philosophy of science without history of science is empty; history of science without philosophy of science blind” producing a wonderfully Münchhausian definition of the two disciplines and their interdependency. One cannot do history of science without first defining what this thing ‘science’ is that one wishes to investigate historically, a task that definitely belongs to the philosophy of science. On the other hand for the philosopher of science to define ‘science’ he really needs a comprehensive knowledge of how it evolved historically. A classical chicken and egg problem that can only be solved by ‘science’…

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# Deferred evaluation in Renjin, Riposte, and pqR

The previously sleepy world of R implementation is waking up.  Shortly after I announced pqR, my “pretty quick” implementation of R, the Renjin implementation was announced at UserR! 2013.  Work also proceeds on Riposte, with release planned for a year from now. These three implementations differ greatly in some respects, but interestingly they all try to use multiple processor cores, and they all use some form of deferred evaluation.

Deferred evaluation isn’t the same as “lazy evaluation” (which is how R handles function arguments). Deferred evaluation is purely an implementation technique, invisible to the user, apart from its effect on performance. The idea is to sometimes not do an operation immediately, but instead wait, hoping that later events will allow the operation to be done faster, perhaps because a processor core becomes available for doing it in another thread, or perhaps because it turns out that it…

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# Making Money Disappear Through Infinite Iteration

This puzzle is too fun! Comments about infinite greed and Zimbabwean currency make themselves, but the conclusion is too stunning to be believed.

In Joel David Hamkin’s paper Supertasks and Computation, he relates the following puzzle: Suppose that you have a countable infinity of dollar bills, and one day you meet the devil, who offers you the following bargain: In the first half minute from now, the devil will give you two dollar bills, and take one from you in return. In the quarter minute after that, the devil again gives you two dollar bills, and takes one from you in return. And so on, in the eighth of a minute after that, and the sixteenth of a minute after that, etc. After a minute, the whole transaction is complete. Should you take this bargain?

The answer is “no” and the reason is that the devil could do the following: Think of the bills you have at the start as being numbered 1, 3, 5, etc. and imagine that the devil has…

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