Entries from Stephan Paukner

63. What is ?
64. What are singular matrices?
65. What is the condition number of a matrix?

(Continued from questions 1-62)

In question 15 I want to make a picture how looks like for . I want to take at least two-dimensional vectors, as the inner product is too trivial otherwise. But then is a function on , which is rather difficult to plot. So I want to fix and plot , which is then a function on with values in and therefore easy to plot. As the inner product is , it is in the 2-dimensional case a simple plane-equation ; of course, the inner product is linear and goes through the origin! It cuts the zero-plane in an angle which is dependent on the orientation of the fixed : The zero-line evolves where the vectors are orthogonal to the fixed , i.e., .

This zero-line will now change its angle on the zero-plane while varying the fixed vector . This gives a picture how it behaves as a whole. In Octave, I set

[xx yy]=meshgrid(linspace(-2,2,50));

and then made a plot corresponding to by

mesh(xx+yy)

The resulting 3D-plot shows a boring inclined plane. To better see the location and orientation of the zero-line, I plot the absolute values:

mesh(abs(xx+yy))

To see the orientation even better, I chose

imagesc(abs(xx+yy))
imagesc(abs(xx+2*yy))
imagesc(abs(-xx+4*yy))
corresponding to the further vectors and . This is what I get:

The turning around of that plane now corresponds to the behavior of the mentioned inner product function on . (Beware that the absolute values used in the plots hide the asymmetry of the inner product function!) The values are always real and go into the exponent of the exponential function. then “rotates fastest” as runs along to a fixed , and is constant = 1 if it runs orthogonally; Beware that there’s a difference whether it is constant = 1 or if it reaches that value again and again. So, in the integral of the FT, the exponential function simply covers all frequencies via exhausting all values of , and the FT is then a function in , which takes out a single orientation of those frequencies; As already mentioned some time ago, the term “frequency” isn’t useful anymore in higher dimensions. For two-dimensional signals, i.e. images, a “frequency” can be interpreted as a pattern of lines: The narrower the pattern, the higher the frequency is. The difference to one-dimensional signals is, that those frequencies may now have an angle. It is not enough to just look at a single frequency of that line-pattern, but also to all their orientations! Only with using all orientations of patterns, an image can be analyzed according to the contained frequencies. The 2D-FT of an image therefore tells you what patterns occur in what orientation. To be more precise, you select the frequency by the length and the orientation by its angle , because the inner product as a function in grows faster with growing length of . For me, this is now a milestone in understanding the 2D-FT.

When I started on my way to my Masters degree seriously in January 2006, several questions began to arise in the course of time, and they continue to do so. While there were some very fundamental questions at the beginning (because my last contact to FA and basic analysis had been quite some time ago), they matured to some more advanced questions today. Here, I give a list of them, in their order of appearance. The answers, however, and as far as they have already been provided anyway, did not come up in the same order. Some questions are so general that they are to be understood as a challenge to look into the books and get familiar with it.

This list will continue to grow in separate entries. Also, for the sake of documentation, currently missing answers will only be given separately.

Here I give my personal reading approach to get familiar with the topics which the NuHAG is working on. This is just one way to get started, and I will mention some pros and cons I noticed. There is always some redundancy, but in my eyes that was of great worth.

A good way to get some first impressions is to read A Short Introduction to Gabor Analysis by Thomas Strohmer, which is the introductory chapter of [Feichtinger/Strohmer] and is also available online. You could also read the introductory chapters of [Blatter], which will make clear what the central ideas are in general. After having read that, you also want to understand it all.

Wien kotzt an. Offenbar. Was mir seinerzeit als frisch in die Großstadt gekommener Hinterwäldler gleich aufgefallen ist, sind die dreckigen Gehsteige. Wiens Gehsteige sind dreckig. Sie sind angekotzt. Sie sind angeschissen. Sie sind angepisst. Sie sind aber bei weitem nicht so angeschissen wie angekotzt.

For the usage of a local backup host, I defined the following scenario, which could also be used for a printer server or any other type of host, which shouldn’t run 24/7, but:

• The host should be able to be woken up manually by a special signal on the LAN, i.e., it should be capable of WOL. (This is only a hardware issue)
• If it is woken up, it should run at least for 30 minutes before trying to shutdown again.
• It shouldn’t shut down while certain processes are running, such as a backup, i.e., it should be capable of a shutdown-lock. If the lock is removed, it might shutdown.
• The host shouldn’t shut down as long as a certain client, e.g. a backup client, is (still) up and running (pingable)
• If the criteria above aren’t hurt, the host should finally shut down.

And here’s the shell-script which implements the above. It’s enough to have it run every five minutes. The “echoes” are only interesting for debugging purposes.

Besides my plan for buying a DSLR after finishing my study in spring 2007, I renew my property of a compact digital camera. After my first one, a Canon PowerShot A70 at Christmas 2003, I went over to the Canon Digital IXUS 750 last February, as I wanted to have a more compact camera, which I also could take easily with me on a bike-tour. But I soon noticed that it had to be categorized as a style camera and I was soon missing the enhanced possibilities for photographing, such as time-automatic or aperture-automatic, which now lead me to selling the IXUS at eBay and planning to buy the new Canon PowerShot A710 IS. This is a not-so-compact model again, but—to debilitate the compactness criteria—it won’t waste too much space in my bike-saddle bag anyway if I just wrap it in a plastic bag. What I get is a twice as high zoom (6x instead of 3x) and the very Image Stabilizer for high focal-lengths or long exposures.

I’m starting with the last chapter in Christensen and plan to have finished it on Nov. 5th. I still have some pages left in ch. 11 of Gröchenig as well as the whole ch. 12+13; I plan to have finished them by the mid of November. Then there’s the whole time of the day available for selected chapters of HGFei/Strohmer and—primarily—the GA4IP-stuff.

I also got myself into the MATLAB/Octave toolboxes and went through the diary (i.e. the MATLAB logfile) of HGFei and the examples of P. Søndergaard. (BTW, Søndergaard will be in Vienna at Nov. 8th for a talk at NuHAG.) Tomorrow I’ll visit the IT-staff at the institute to get a mathematics/MATLAB computing account. This term, I’ll visit HGFei’s lecture on Selected Topics in Modern Analysis, which takes place Monday and Thursday 10-11 a.m. Although this hour is “in the middle” of my working time at SOL4-IT, I’ll have to be there regularly, at any cost.