Sebastiaan's Subspace

My journal about mathematics and other interests

Author: S.G. Janssens

Calculating an auto-convolution integral by Fourier transforms

Recently I came across a forum contribution, where the author Charles Link describes how to use the unitarity of the Fourier transform on $L^2(\RR)$ to compute the definite integral

\begin{equation}\label{eq:I}
I(t_0) \DEF \int_{-\infty}^{\infty}{\frac{\sin(\tau – t_0)}{(\tau – t_0)}\frac{\sin(\tau + t_0)}{(\tau + t_0)}\,d\tau},
\end{equation}

where $t_0 \in \RR$ is a constant. In the comments it is also argued that one may alternatively use contour integration. I liked the article, but wondered whether the symmetry of the problem would perhaps admit a simpler approach.

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From offline LaTeX to online QuickLaTeX

For writing mathematics in WordPress using LaTeX I have been using Pavel Holoborodko’s nice QuickLaTeX plugin. It makes it very easy to convert offline LaTeX source code, possibly including custom macros, into (part of) a WordPress post. To reduce the effort a bit further, I wrote a tiny Python 3 script.

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Saved by a pipe smoking penguin

In the late 1990s I used Linux for the first time. Back then, Slackware was one of a limited number of distributions on the market. Installation took very long and failed repeatedly. Partly this was due to my slow and unreliable internet connection, but my inexperience and incompetence were likely at least as detrimental to my progress. From what I remember I managed to set up X11 with the aid of a configuration script that asked me many questions to which I guessed most answers. I also found out how to make Linux talk to my external dial-up modem. No longer chained by evil Microsoft, I felt very proud and particularly hackerish.

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Continuity of eigenvalues

As a simple corollary to the result from my previous post I would like to show you how to obtain continuous dependence of the eigenvalues of a matrix on its entries.

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Continuity of polynomial roots

It was recently brought up how to show that the n roots of a real or complex polynomial depend continuously on the polynomial’s coefficients. Although I have used this proposition numerous times, implicitly and explicitly, I realized that I never saw a proof of it.

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Laatste-wil-pil

Gisterenavond las ik op NU.nl en de website van dagblad Trouw dat het voor senioren mogelijk moet worden een eind aan hun leven te maken door inname van een “laatste-wil-pil”. Aldus is althans het streven van de Nederlandse Vereniging voor een Vrijwillig Levenseinde en de Coöperatie Laatste Wil, die hiervoor steun vinden bij Tweede Kamerlid Pia Dijkstra (D66). Zij schrijft momenteel aan een wetsvoorstel van dergelijke strekking. De doelgroep bestaat nadrukkelijk uit ouderen die volgens de huidige euthanasiewet niet in aanmerking komen voor hulp bij zelfdoding, maar wel hun leven willen beëindigen omdat “ze er zelf helemaal klaar mee zijn en er echt niets meer van kunnen verwachten.”

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Troy Davis and the censorship of doubt

Troy Davis (1968 - 2011)

Since September I have been a member of a well-known internet forum on science, mathematics and engineering. Overall, this was an enjoyable and stimulating experience. At some point I made it to “science adviser”, which I believe is a term forum staff reserves for members that display a better than average understanding of the topic at hand in technical discussions. Unfortunately, not much later something unpleasant happened.

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