☮ Vašek

In geometric optics, one comes across compositions of matrices of the following form:

M(u,v) =

[1+uv u]

[-uv^2 1-uv]

where u and v are arbitrary real numbers. I am suspecting that one can use products of these matrices to reach any 2x2 real unimodular matrix, but I can't seem to prove it.

Is it true? If not, what are the further constraints on the reachable set of matrices besides having a determinant of 1?

Here's what I have so far:

the set of

A(a) =

[1 a]

[0 1], a ∈ R

together with

B(b) =

[1 0]

[b 1], b ∈ R

does generate all real unimodular matrices but neither of the two subsets alone does. Setting v = 0 in the matrix of my question, we retrieve the first subset with A(a) = M(u,0). So the question is whether B(b) can also be written as a product of some number of matrices if the form M(u,v) for any given b.

Here's the effect I am interested in:

http://1x.com/photo/46277/category/macro/popular-e...

I am using a Nikon D3200 with the stock 18-105 mm lens. I have noticed that I can't focus really precisely. If I take a picture of a structured surface at an angle, I can clearly see the limits of the depth of field, but even at its centre, the pixels are not really sharp, I can see a circle of confusion of about 5 px at the full resolution. See the image (F/5.6, ISO 100, 2s, f=66 mm, cropped):

http://i50.tinypic.com/slqjb6.jpg

Is that something a prime lens would solve, or a property of the chip?

I have a Nikon D3200. I noticed that shots I made with exposure times in seconds also take longer to write to the SD card *after the exposure is finished*. Images of dark scenes taken with a maximum exposure of 30 seconds take almost one more 30 extra seconds to save before the camera is ready to take another picture. I don't understand it – shouldn't the CCD be in about the same state after the mirror jumps back regardless of how long it had been opened before?? Is the camera post-processing anything?

It's in some old form of Cyrillic. Picture taken in a monastery in Sinaia, Romania.

http://i46.tinypic.com/1y2yhd.jpg

Modern transcription and/or translation welcome.

I am using a chain of pipes to extract a set of links from a downloaded HTML. However, in the one before last step, I have a list of relative URLs and I would like to prepend the address of the server. I would love to insert that in the chain using a command like "prepend" in this example:

grep "my regexp" file.html | more commands | prepend "http://www.server.com/%22 | wget -i

More generally, it would be cool to have a command which would insert the contents of each input line at a specified place in a pattern, like

sandwich -c _ abc_def:

input line 1

input line 2

->

abcinput line 1def

abcinput line 2def

But a command for only prepending or only appending something to each line would suffice. I mean, it's easy to cut out a part of each line using cut, there should be a reverse command available, too.

Please note before answering that I'm sure that

1) sed or awk could handle this with ease (but I don't feel like learning them), and

2) writing a Perl script for this would take 10x less time than typing out this question,

but I think I am missing some really neat standard way of doing this straight from the command line without any extra gear and I want to know. So, how do I go about doing so?

I read that all of these mean, in some sense, "someone" or "somebody", so I guess the usage is different. Could you please give me examples of sentences where each of them is adequate? If any of these words are actually freely interchangeable, please let me know which ones.

I'm reading a book by Darya Dontsova and it seems like quite a common practice in it. However, I am not sure to what extent to take that literally, given the amount of sarcasm and exaggerration in most of the other situations. It sounds quite rude to me but on the other hand, I may am biased, coming from a Slavic country myself where the habit may be different (a Czech doctor would never ever call a patient "ty" unless it's a small child).

Also, are there any translations? Somehow I can't find out on English websites and my Russian is not good enough yet... but I'd love to watch these!

I am a huge fan of traditional British Christmas carols but when learning more about them, I found that some of my favourite ones actually come from the East: for example, "Carol of the Bells" is originally Ukrainian and "Good King Wenceslas" carries a significant Czech influence. So I decided I'd love to expand my knowledge of originally Slavic carols.

What other Slavic carols would you recommend to me? They can be old or modern but should not be just adaptations of the "mainstream" ones. Links to your favourite rendition (YouTube, mp3, choir sheets) are most welcome, too.

How is it explained that the word for 40 suddenly becomes dissimilar to the name of any other multiple of 10? What is so important about 40 that it deserves a special word? Unlike 12, 20, 60, or 144, which have special names in some languages I am aware of, I can neither think of anything in Nature of which there is 40 nor of any mathematical rarity making this number practical for calculations.

I am sorry if I am missing an important cultural or religious connection, I just don't know about these things (and they might very well be the answer, too).

Also, give an example of a sequence of elementary rotations of an actual Rubik's cube that gives an element of this order.

Toying around with the puzzle, I have found a simple one with an order of 64, but I have made no rigorous attempts on proving or disproving that I could do better.

Studying university-level physics, there's still something on the most elementary level that eludes my intuition, namely how the Coulomb model of friction could be so widely applicable in mechanics while representing a blatant exception from the linear or exponential (let alone continuous) nature ubiquitous in most of physics. Why does the coefficient of friction change suddenly when the relative speed of two surfaces becomes nonzero, but does not significantly change with speed after that? Shouldn't there be a smooth transition, somehow, especially if we consider the relative velocity as a 2-dimensional vector?

I find it somewhat hard to put the sounds M+G+N in a single consonant block - is there any trick to it or do I just have to practice harder?

Also, what is the historical origin of the need for characters denoting two phonemes instead of one? According to Wikipedia, the predecessors of Greek script did not display this phenomenon, which is now quite widespread among various Europen languages and scripts.

Suppose that two people pick random permutations of n elements, n > 1. Is it more probable that they agree on exactly one element or none at all?

Provide a proof to support your answer.

"You can be good at understanding all the formulas but you do not really understand physics until you realise _______________."

Let f be a holomorphic function on the unit disk, let the value of f is zero on some arc of the unit circle. Is necessarily f=0 for all z in the unit disk?

I think the answer is yes because something like that would hold for any set with an accumulation point within the disk. I just can't think of any similar theorem if the arc is a part of the boundary of the function's domain.

I have a 64-bit HP laptop. I have been using a 64-bit operating system and most applications for maybe 3 years now. However, I am going to reinstall my system from scratch soon and I consider going back to a fully 32-bit version.

Here are my arguments for doing so:

I couldn't really see any advantages it was supposed to give me, and as I have only 1 GB of memory, I have no need for 64-bit memory addressing or memory maps. I think that in my situation, it only means that most applications consume twice as much memory for pointers and integers, which is not very good.

Frankly speaking, I see no arguments why I should NOT downgrade to 32 bits. But I don't want to have a bad surprise in the end. Computer specialists, are there any reasons for staying with 64 bits?

What is the difference in altitude between George Square and, say, the main entrance of John Anderson Building?