What is the Riemann-Zeta function?

Question: In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
Comment: A good example of a question that is asked by someone genuinely interested in math but is looking for an accessible way into more advanced number theory that he would otherwise have no other means of finding.
Answer: Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory.  What properties might you want such a distribution to have?  Well, if you're doing number theory then you want to think of the prime numbers as acting "independently": knowing that a number is divisible by $p$ should give you no information about whether it's divisible by $q$.
That quickly leads you to the following realization: you should choose the exponent of each prime in the prime factorization independently.  So how should you choose these?  It turns out that the probability distribution on the non-negative integers with maximum entropy and a given mean is a geometric distribution, as explained for example by Keith Conrad here.  So let's take the probability that the exponent of $p$ is $k$ to be equal to $(1 - r_p) r_p^k$ for some constant $r_p$.
This gives the probability that a positive integer $n = p_1^{e_1} ... p_k^{e_k}$ occurs as
$\displaystyle C \prod_{i=1}^{k} r_p^{e_i}$
where $C = \prod_p (1 - r_p)$.  So we need to choose $r_p$ such that this product converges.  Now, we'd like the probability that $n$ occurs to be monotonically decreasing as a function of $n$.  It turns out (and this is a nice exercise) that this is true if and only if $r_p = p^{-s}$ for some $s > 1$ (since $C$ has to converge), which gives the probability that $n$ occurs as
$\frac{ \frac{1}{n^s} }{ \zeta(s)}$
where $\zeta(s)$ is the zeta function.
One way of thinking about this argument is that $\zeta(s)$ is the partition function of a statistical-mechanical system called the Riemann gas.  As $s$ gets closer to $1$, the temperature of this system increases until it would require infinite energy to make $s$ equal to $1$.  But this limit is extremely important to understand: it is the limit in which the probability distribution above gets closer and closer to uniform.  So it's not surprising that you can deduce statistical information about the primes by studying the behavior as $s \to 1$ of this distribution.

Let me mention two other reasons to care about the limit as $s \to 1$ of the above distribution.  First, the basic reason to think of the primes as acting independently is the Chinese Remainder Theorem.  Second, a natural reason to look at a distribution where the probability that a number has exactly $k$ factors of $p$ is $(1 - p^{-1}) p^{-k}$ is that this is precisely the distribution you get on the residues $\bmod p^n$ for $k < n$.  In fact, I believe this can be upgraded to the corresponding statement about Haar measure on the $p$-adic integers.
Comment: I feel like I've just been let in on a mathematical secret.
Answer: Giving an explanation in layman's terms is always going to be challenging, given that the Riemann-Zeta function (and related hypothesis) inevitably lies in the domain of abstract mathematics, but I shall do my best.
The Riemann-Zeta function is a complex function that tells us many things about the theory of numbers. Its mystery is increased by the fact it has no closed form - i.e. it can't be expressed a single formula that contains other standard (elementary) functions.
Although there are many different ways of expressing the Riemann-Zeta function (the Wikipedia article gives several), it can ultimately be derived from the following simple series of real numbers:
$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s},\quad\Re(s)\gt1$
by extending it into the complex plane.
The reason this strange and esoteric function is so famous and actively discussed in mathematics is due to the Riemann hypothesis - proposed in 1859 by the great Bernhard Riemann and still unsolved. The Wiki article states the problem in quite simple terms:

The Riemann zeta-function ζ(s) is
  defined for all complex numbers s ≠ 1.
  It has zeros at the negative even
  integers (i.e. at s = −2, −4, −6,
  ...). These are called the trivial
  zeros. The Riemann hypothesis is
  concerned with the non-trivial zeros,
  and states that:
<code>The real part of any non-trivial zero of the Riemann zeta function is 1/2.
</code>
Thus the non-trivial zeros should lie
  on the critical line, 1/2 + it, where
  t is a real number and i is the
  imaginary unit.

Although the conjecture (it is only that at the moment) has many consequences for mathematics (number theory in particular), the primary one, at least the one Riemann originally proposed, is about the distribution of prime numbers. In other words, it tells us with great precision what the average gaps between primes are as we move to greater and greater numbers. Many of the other implications are rather more esoteric, though perhaps equally important for pure mathematicians.
Comment: Why does the zeta function have zeros at negative even integers? Wouldn't it simplify to a divergent sum of powers of natural numbers?
Comment: @JacksonFitzsimmons The important part about $\zeta(s)=\sum_{n=1}^\infty\frac1{n^s}$ is the $\Re(s)>1$.  To evaluate when $\Re(s)\le1$, use analytic continuation.
Answer: The above answers give excellent explanations about why the zeta function has close connections to number theory, but I thought I'd mention something about why the Riemann Hypothesis should matter so much.
By taking the logarithm and then differentiating the zeta function, one gets the formula
$$\frac{\zeta'(s)}{\zeta(s)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^s}$$
for $\Re(s)>1$, where $\Lambda(n)$ is the von Mangoldt function which takes the value $\log p$ at powers of primes $p$, and is 0 everywhere else. Think of it as a weighted way of counting the primes (the prime number theorem tells us that $\log p$ is the natural weight to choose).
Much of analytic number theory proceeds by choosing a weight of the set we wish to consider (often the primes), and then encoding this weighting in a so-called Dirichlet series (an infinite sum of the form above). We can then use analysis to study this series and get lots of useful information.
In this case, then, the function we need to study to get information about the primes is $\frac{\zeta'(s)}{\zeta(s)}$, which we can study using complex analysis.
In complex analysis, a good slogan is 'the only things that matter are zeros and poles' (effectively points where the function shoots off to infinity).
Hence to understand the prime numbers, we just need to understand the zeros and poles of $\frac{\zeta'(s)}{\zeta(s)}$ - we know about the simple pole at $s=1$, we know there aren't any other zeros where it counts, and we also know that the only other poles are at zeros of $\zeta(s)$ (roughly because dividing by zero causes infinity).
In other words, if we knew where these zeros are (i.e. the Riemann hypothesis) we can work with $\frac{\zeta'(s)}{\zeta(s)}$ in all kinds of clever ways to get good results on the prime numbers.

More specifically, in the usual contour proof of the prime number theorem, knowing that there aren't any other zeros in $\Re(s)>1/2$ would allow us to shift the contour further to the left, reducing the error term in the result to (roughly) $O(\sqrt{x})$.
Answer: Here there is another attempt at an explanation.
We know that the sum of the inverse of the positive numbers, $1 + 1/2 + 1/3 + \cdots$, diverges. Euler shown that the sum of the inverse of the squares,  $1/(1^2) + 1/(2^2) + 1/(3^2) + \cdots$, has a finite sum, namely $\pi^2/6$. Mathematicians love to generalize things, so they thought at the function
$\displaystyle f(x)=\sum_{n=1}^\infty\dfrac1{n^x}$
which is defined for $x \gt 1$. But this was not enough: they decided that the variable could be a complex number and not a real one. There is a standard tecnique (Analytic continuation) which allows us to extend the function to nearly all the complex plane. So we now have a function which formally is
$\displaystyle \zeta(s)=\sum_{n=1}^\infty\dfrac1{n^s}$
(the variable being $s$ and not $x$ to show that we are dealing with complex numbers) but is not computed in this way. Just to make an example, $\zeta(0)=-1/2$, and sum of an infinity of ones is not $-1/2$. :-)
It may be shown that for $s = -2n$ ($n$ positive integer) $\zeta(s) = 0$. But there are infinite other point $s'=(x,y)$ where $\zeta(s') = 0$. For all of these points, $0 \lt x \lt 1$; Riemann's hypothesis says that for all such points $x = 1/2$. If it were true, we could have the best asymptotic expression to count $\pi(n)$, that is the number of primes below $n$.
Why does the function pop up when we talk about primes? I don't know, but in the case of integer values Euler proved that
$\displaystyle\sum_{n=1}^\infty\frac1{n^s}=\prod_{p \text{ prime}}\frac1{1-p^{-s}}$
Maybe this could be a good start.
Comment: Correction: zeta(0) is -1/2, not 1/2.  Also, I think it is misleading to say analytic continuation is a "standard technique" to extend the zeta-function.  Whether a function has an analytic continuation to some larger region is a property, but checking where that property works often depends on special aspects of the particular function under consideration.
Comment: The product formula is valid for all $s$ with $\Re(s)>1$; it's of *great* importance as to why the zeta-function (and its cousins the L-functions) relates to the distribution of prime numbers.
Comment: @KCd =D Fixed the annoying little error.
Answer: The key point is that the Riemann zeta function is a function whose properties encode properties about the prime numbers.  As mentioned by Noldorin, in order to fully understand the Riemann zeta function you need to "analytically continue it to the complex plane" which is a tricky process which takes serious study.  Fortunately for some easier properties of the primes you can just use the definition of the zeta function for real s.
Claim (due to Euler): The fact that $\zeta(s)$ goes to infinity as $s\to 1$ tells you that there are infinitely many primes.
Sketch of proof: Use the "Euler factorization" mentioned by mau (expand the RHS as a geometric series and then multiply it out using unique factorization into primes):
$\displaystyle\sum_{n=1}^\infty \frac{1} {n^s} = \prod_{p\in\text{prime}} \frac{1} {1-p^-s}$
Now take log of both sides to get:
$\displaystyle\log \zeta(s) = \sum_{p\in\text{prime}} \log \frac{1} {1-p^-s}.$
Now use the taylor series for \log and send s to one.  You'll get that the left hand side goes to infinity (Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?), while the right hand side looks like $\sum 1/p$ + bounded terms.  So there must be infinitely many primes.

What is the Riemann-Zeta function?


What is the best way to show the Facebook "Like" widget?

Question: I am working on an e-commerce aggregator site. I need to place a Facebook like widget. Now, I am not sure what is the best way to get the user to like the Facebook page. What would be the best way to get user to like the page on Facebook? Should I go ahead with the traditional Facebook "Like" Widget or would a FB icon linking to the Facebook page be better? Also, what should be the placement for the same?
Answer: There are quite a few questions here.
The best way to get user to like your page would be via the standard like button. Directing them from your website to the facebook page and counting on them to like the facebook page there is a longer flow of action than the user just clicking the like button on your webpage.
While many people like to place facebook like widgets in their headers, I think that sacrifices too much of an important spot - but ultimately it is a design choice and depends on how much stress you want to put on the feature.
As for what kind of button it should be, stay consistent with facebook's guidelines, so that users can easily identify the button and it's standardized functionality they already know from other sites adhering to the same recommendations.
Answer: Unless you have a particular reason for really wanting the visitors to visit the facebook page (such as trying to market your site through facebook), it's much more user-friendly just to let them link it on the site without redirecting to facebook.
If the main point of the page is to get the user to click the "like" button, the lower-right corner of the site would be a good place to put the link.  That's the last place the user will reach, so it's an ideal location for a call to action if you don't have some other call to action already on the page.  Putting the link in a sidebar or heading directly next to the content that you want to be liked could also be effective because it associates the action with the content.
Facebook also allows applications to have built-in likes (where the user's "like" of your custom like icon translates to a "like" on facebook as well) under certain circumstances, so you may find those useful depending on the purpose of your page.  You might be able to associate the custom like button with another function on your site, such as "like item and add to cart" so that you gain an increased rating on facebook even as the users use your site.
Answer: Just to be clear it's the 'Like' function we're talking here?
There's obviously a difference between a 'Like' and a 'Follow'.
Assuming it's Like, I agree with both JonW and 3nafish, use the 'on page' Facebook Like.
Not only does it reduce the friction of the action, is the standard and widely understod behaviour but a visible count offers 'social proof' of the popularity of the item.
You can either use Facebooks default like (link above), of if you want multiple networks you can try AddThis or ShareIt.
The only downside of using Facebook's button or a third party solution is it can often be a performance drag. If possible put it toward the end of your markup and position it with CSS so it doesn't prevent other page elements from loading first.
Alternatively I recently saw this sript; http://socialitejs.com/ which delays the dynamic elements loading until the user interacts with it which might help, though I can't vouch for it's usefulness.


What's the quickest way into Blackreach?

Question: I didn't explore Blackreach much during the Elder Knowledge quest, and I also didn't complete the A Return To Your Roots quest. It took quite a while to fight through Alftand to get there the first time so I'm hoping there is a quicker way to get in?
Answer: Initially, you can only enter via one of the "Great Lifts": Alftand, Mzinchaleft, or Raldbthar, and only with the Attunement Sphere from Septimus Signus. As you adventure below Blackreach, though, you will discover 4 additional entrances via elevators back up to the surface. These supplemental entrances are initially barred with a gate, and therefore inaccessible before you've started spelunking.
As per the UESP, here are the potential entrances into Blackreach:

Great Lift at Alftand - (Elder Knowledge suggests you enter through here)
Great Lift at Mzinchaleft
Great Lift at Raldbthar
Alftand - (outside the rest of the ruins)
Mzinchaleft - (outside the rest of the ruins)
Raldbthar - (outside the rest of the ruins)
Tower of Mzark - Map Marker not visible by default, shows up after you exit with the attuned cube for Discerning the Transmudae

While several of these locations are fast-travelable-to, the Tower of Mzark is likely to be the quickest way into Blackreach, as it's associated "wing" is quite small, with no enemies.
By contrast, the Great Lifts, like Alftand, or Mzinchaleft, still have goodly portions of dwemer dungeon ruin between their elevator and blackreach proper, which will replenish their enemies (and loot) over time. Since Mzark doesn't have enemies to begin with, it will always be enemy free.
Comment: Which of those 7 entrances gets you to the blackreach quickest?
Comment: thanks, but I was sort of trying to hint to you that you should include that in your answer.
Comment: @GmNoob - In my experience, the Tower of Mzark.
Answer: I know this answer comes quite late, but the other answers are wrong.
Alftand isn't the only way in. You can access Blackreach via Alftand, Mzinchaleft or Raldbthar. At the bottom of each Dwemer ruin is the same access mechanism that lowers the stairs to a door that leads to Blackreach, all you need is the Attunement Sphere from Septimus Signus. The quest "Transcending the Transmundane" places a quest marker at Alftand, but you may choose to ignore that and descend another way.
In my experience, Alftand is the longest path down, and Raldbthar is the shortest. However, Alftand is definitely the most rewarding because it's more interesting than the other two ruins.
After you initially access Blackreach, you can exit via the three great lifts (Alftand, Mzinchaleft and Raldbthar) that open up very close to their respective ruins. These are the fastest ways back in, but you can't go down to Blackreach through them until you come up and open the gate that only opens from the inside. There's also the Tower of Mzark lift, but it requires you to go through a short passage instead of leading straight to Blackreach.
Comment: Thank you! I literally just went through Raldbthar tonight after getting to Alftand and thinking "I really don't want to do this super long dungeon again. There has to be a faster way!" I would say Raldbthar is about 10x quicker.
Answer: The Great Lift at Mzinchaleft (it's a separate location from Mzinchaleft, to the south) takes you directly into Blackreach, so is definitely the fastest way in.
Answer: I carry usually 10 wooden plates around because you never know when you have a dwarven ruin you need to explore the think about this is if you happen to find the barred ( caged ) way down to the ruin where your headed to and you don't wanna fight all that shit to get there if you drop the wooden plate and then pick it up by holding A ( Xbox I'm not sure how you hold items on Ps3 ) and you keep it held in front of you as you approach the gate move around against it plate might drop a few times but finally you'll find the right spot and you glitch through the doors that's how I got into blackreach and the rest of the ruins quickly
Comment: This isn't about how to reach blackreach in ruins quick, just the quick passage to get there.
Comment: Nice.  My lifts all reset on me and I didn't think of getting back into them this way. +1


Where do Minecraft texture packs go on OS X?

Question: I play Minecraft on OS X and for some reason clicking this button doesn't do anything.

Where should I be putting my texture packs?
Answer: I poked around and discovered a new folder:
~/Library/Application Support/minecraft/texturepacks/
Putting the texture packs you download in this folder makes them appear in the menu. Don't extract the files, just leave the .zip there.
Answer: Previous to this update, you would put Minecraft.app into a folder called "Minecraft" along with all the files for the texture pack, and launch Minecraft.app from there. However, I don't know if this is still the case with this version.
(The button also doesn't do anything for me.)
Comment: This worked for me. However, with the update, I suspect that Brant's answer would be the place to put them in the future. It's possible that this method may go away in favour of the updated method.
Comment: Yeah, it's probably better to use Brant's method instead.
Answer: How to install Minecraft texture packs - Mac OS:

Open finder.
On the top of the screen click "Go"
There will be a list of items drop down.
Hold down the alt / Option Key.
There should be a "Library" button that appears.
Click on "Library" and hold down alt / Option Key.
Open the "Application Support" Folder
Then open the "Minecraft" Folder.
Then open the "Resourcepacks" Folder.
Put your Minecraft TexturePack in there.

That is where to put the texture pack in.
Answer: I just got busy understanding a bit about what can be done with Mac OS X in terms of resources for changing skins or textures. Sadly most of the resources one finds by googling are for Window OSes. However upon digging deeper (pun intended) to help my 11 yo i found a couple of links that should be of help for Mac players of Minecraft as well a few directions that can help.
Skins:
very simply find any skin that has the .png extension (image file) then log at www.minecraft.net with your account. Once in click on your profile, in there you will find the option to upload  new images for a new skin (i.e. your downloaded/created .png file).
the boave is a summirized version of waht I found at www.tumblr.com/tagged/install+texture+packs+minecraft
Textures:
I am not completely done yet but should you find any texture package you like that is donwloadable as .zip file you can simply take it. By default should end up in your "downloads" folder inside your user "home folder" (look for the little home icon). IN the same user home folder you will find the 'Library' folder. click it open and then locate and open the "applications support" folder inside it. Here once open you will find the Minecraft folder...open it and you will see several folders one of which is the 'texturepacks' folder.
Click and hold your mouse on the downloaded texture .zip file and drag and drop it in the 'texturepacks' folder.
at this point you are in business...just launch minecraft and select the option for textures once you have logged in. In addition to the default texture now you will see the new one(s) you have added and u can simply click on the one you want to try. Once you selected it you can just launch the game and enjoy.
MCPatcher HD
You have probably found out a gazillion sites where to download this, the only issue being it is an .exe file for Window OSes. I still have not figured out if it is really necessary to use it when using the Mac OS version of the game (probably so if the packs you have downloaded require to adjust the tile size to something else rather than the default 16x16 pixels). However here is where to get the .jar version of it if you need it. I tried it and it works although it gives an error message before opening the interface to change tile size and upload the texture files.
http://uploaded.to/file/l4gk5p
Should I find more about anything of the above I will make sure to update this entry.
Good luck and enjoy!
Comment: Once you are in the Minecraft directory it should all be the same as on Windows and Linux.
Answer: I had the same question and found the answer only by try'n'error.
The correct directory is (please forgive me translation errors):
/User/YourName/Library/Application Support/minetest
In this directory you should see cache, client and world directory. Here create a textures directory. Inside of that create another one (mostly they say you should name it "all", I think this is user defined). Anyway, into this directory place all the files in and restart minetest.
Comment: I think you're confused, and this answer does not apply. [Minetest](http://www.minetest.net/) is not Minecraft.
Answer: If you're on a Mac or running OS X, all you have to do is download the pack and put the application into a folder with the texture pack.
Comment: I just tested this out by putting Minecraft.app in a folder with a bunch of texture packs, and it doesn't work. The texture packs don't appear in the menu.
Comment: @brant If there was a `texturepack` directory inside of the Minecraft.app bundle then you could try to put it there.


No sound from iPhone speakers when connected to a dock with line out

Question: I've bought a (very cheap) dock for my iPhone 4, the dock has a line out jack on the back.
I have noticed that when the phone is in the dock, the sound comes out only from the line out and not from the internal speakers.
This makes impossible to use any third party alarm application because no sound plays without connecting some external speaker. Only the ringtones play, so the default Clock app works and plays the alarm sound.
Is there any software option to force the iPhone to use the internal speakers also when connected to a dock with a line out?
Answer: No, it is a feature.
I'm not sure if there is a fix for jailbreaked iphones, but there isn't such an option.
Answer: When docked your the doc connector, pin 21 is connected to a resistor to tell the phone what type of dock you are connected to. There is now way to override this without changing the hardware in your dock.
Pinout Reference
Comment: Thanks for the reference, I'll try to open the dock and remove the connection to pin 21 to the resistor when I'll feel adventurous enough :)
Answer: This question was also asked on another forum. Someone there suggested a trick to opening up the dock and cutting a line, in it.


Will beer keep in a cask if used gradually?

Question: I've read that beer should keep in a cask for months, but does that only apply if left full? How long do I have to finish once I start drinking from it?
Answer: Untapped casks can be stored for extended periods of times as long and the bungs are secure and they are not leaking.  If you would like to enjoy drinking the beer over an extended period of time you will have to use an aspirator valve in your system set up.  What this does is keep a small amount of CO2 pressure in the head space so the beer is not exposed to O2.  The pressure is low enough not to increase the existing co2 in solution but will allow for extended serving.  This can be installed in an existing co2 line and works independent of the regulator pressure.  Thus you can have your regulator set at whatever pressure you need to serve regular kegs and the valve will restrict the pressure to about 1psi going into the cask.
Comment: Just don't tell CAMRA :)  Good answer, though +1
Answer: Cask beer will last a long time (months), especially if it's a higher gravity beer.  But once you tap a cask, you only have a few days to drink it before it will begin to display signs of oxidation (which is basically "spoiling").  I've had cask beer after 12 or 18 hours, and it was already no good.
The problem is that for every cc of beer you pull out, a cc of air will need to be pulled into the head space of the cask.  That's not quite 100% true at the beginning (just after it's tapped), when some CO2 will come out of solution so that no air needs to be added into the head space.  But after that, every pint of beer means a pint of air that will spoil your cask beer.
If, rather than use a cask, you use a system that dispenses with CO2, then you can drink off that keg for a month with no noticeable degradation in beer quality.  If you want to get out cheap, you can keg into 6 liter bottles that have CO2 on the tap (see this you tube video about Bottling A Five Gallon Batch into Three 6 Liter MillerCoors Bottles).  Not your question, so I'll stop with this line of answering, but it is related in that it does what I think you want to do...drink longer off one batch of beer!


What should I look for when buying a subwoofer?

Question: I've got a pair of studio monitors which I'm very happy with, and am considering matching them with a sub.  Since I can only hear the sub in the store, and without the benefit of hearing how they'd sound with my current monitors, what can I do to make sure that they will sound good together?
Answer: To do this properly, you will need the frequency response curves for both your studio monitors and your subwoofer so that you can get the crossover frequency right:

In the above graph, the red curved line is the subwoofer frequency response.  The green line is the frequency response of your studio monitors.  The crossover frequency is the point where the two lines cross.
In an ideal world, the point where they cross is -3db down for both.
Answer: First you might want to know about the different types of subwoofers, in-depth information on subwoofers.
Guess you should look around through different subwoofers on the internet and read their reviews,
once you have the subwoofer you want you can learn about where you need to place your subwoofer.
Answer: Most studio monitor manufacturers will make subwoofers that match.
Also, many active subwoofers will let you connect your speakers via the subwoofer, via it's own internal cutoff-filter, so you don't get an overlap in output. However, you can still get a mismatch if you instead of an overlap get a gap.
Answer: I have a subwoofer in my home setup; however I have moved away from using it to monitor whilst mixing. If you speak to experienced mix engineers, they will ALWAYS recommend that instead of buying a subwoofer, you sort out your mix position with acoustic treatment - you can do a lot to improve the overall response of your room by installing bass traps (the more the merrier!) and diffusion between and above the mix position. Bass nodes are a massive problem in any room; you'll hear it as a 'lumpy' bass response (or excessively boomy at particular frequencies, in a pattern). As a result, you end up cranking the bass more and more to compensate - or EQing the bass in a weird way which actually sounds awful when you listen on another soundsystem.
If you treat your room well, you can find you can then hear the actual low frequencies - which your nearfield monitors are cranking out! - but which were previously hidden by your room's suboptimal response. It didn't stop me buying a sub as much for listening pleasure as critical mixing :-)
My advice is, generally, buy your sub - any decent one will do, just get one a sweepable crossover, gain and phase adjustments AND A BYPASS FOOTSWITCH! Put the sub in the mix position (where you sit) and adjust it so the rolloff isn't impinging upon your nearfields' rolloff. Then, get on your hands and knees and crawl around the room whilst listening to reference audio! When you find the position where the bass sounds the most natural and consistent, place the subwoofer there. It does mean you have to compromise your room wiring and equipment setup, but that's what you need to do to achieve as consistent a sound as posible (without resorting to loads of bass traps all over the place).
The BBC use dual subwoofer setups in some mix studios, when I was in their Birmingham mix studio for The Archers drama serial a few years back I noted they had dual Genelec subs in opposite corners of the room. However, remember - the more subs you add, the more complexity you add and the more you compromise your room acoustics (so many room nodes!)


Accessing a remote OSX system from OSX, Linux, Windows

Question: Is it possible to connect to a remote OSX machine using OSX, Linux or Windows in a way similar to Windows' remote desktop?
Comment: Of course, MacOSX has a functional unix layer, so ssh is always an option.
Answer: One simple way is to turn on vnc screen sharing by going to System Preferences -> Sharing -> Screen Sharing on the machine you want to share. For client compatibility reasons you  may need to select both "Anyone may request permission to control this screen" and the "VNC viewers may control this screen with a password" checkboxes.
Once you've set up the machine for sharing you can connect to the screen on OSX using the Finder sidebar [SHARED] section or on linux using one of the many vnc clients (vinagre, vnclient etc.).
Answer: X11.app also works well for client/server UI separation if you want the app to run on the Mac side with local access to the filesystem and peripherals, but draw the UI on the remote machine over the network.


08 f150 with vibration in the front driver side tire when turning left only.

Question: 2 months ago I installed a 2 inch suspension lift and replaced upper control arms, tie rods and shocks. The tires were pretty bad and got worse quickly. Vibration and noise started shortly after.Mechanic said it was tires and I replaced them last week. Noise was better at first but already getting worse. any ideas??
Comment: Have you re torqued all the bolts since doing the work? Are your CV joints okay, if applicable? Have you had the tracking/alignment/camber checked by a shop with a laser?
Answer: I had this same issue on an 04 F150 Lariat. After many trips to Ford dealer. Came to find out clutches in limited slip diff were worn out. 4wd, always used 2wd. Replaced the clutches in rear end and had no more issues. I was told tires at first as well.
Answer: It is possible that you have a front wheel bearing/hub that is bad. With the steering wheel locked, safely raise the front wheel off the ground. Grab the tire and rock it back and forth while holding the tire at the left and right sides. Do it again while holding the tire at the top and bottom. There should be no percevible movement of the tire. Slowly rotate the tire to feel for any grinding or rumble in the bearing. Even if you do not feel anything unusual the bearing may only be vibrating while under load. The left front typically sees more load than the right front in 4x4 mode. Being 6 years old I would guess that the right front is also worn, just not as bad. Since you have a lift kit I assume you also have oversized tires. The extra weight and loads from the tires can cause premature bearing wear. They may not cause a failure they will just wear faster. On another note did you get an alignment after the lift kit was installed? Altering the suspension can change the alignment. You may want to check with an alignment shop that works on lifted trucks. Some times after the lift the vehicle requires alignment settings that are different from the stock setting.
Comment: 2 wheel drive truck.. noise started after lift but before larger tires. last alignment guys said was a waste because old tires were shot. going to bearing today but i replaced rotors maybe 6 moths ago and pretty sure bearing assembly was built in with rotor so they should be new. could i have not got the tie rod back in the proper position? tires "look" straight and it dosent pull
Answer: Wheel Bearing seems logical dependent on Mileage, could also be brake caliper issue. is the truck 2 or 4WD.


Simplicity is fundamental in design. How about in writing a novel?

Question: I'm a web designer.  The first rule in design is KISS (Keep It Simple Stupid).
How about writing a novel? Is simplicity fundamental or just an option?
Answer: The answer to this depends a lot on what you're setting out to achieve.
In my experience, clarity is most highly valued in genre writing. Plot is often very important in genre writing. Genre writers often try to keep their writing structurally simple, to the point where the reader can forget that they're parsing words and focus on the story.
Literary writing (as ambiguous as that term is) tends to be less obsessed with simplicity or clarity. A significant percentage of "literary" work is at least partly an experiment in language. When you begin using language in new and strange ways, you're purposely making the text harder to read, but you're doing it to achieve a specific effect.
There's nothing strictly wrong with either approach, and there are even works out there that employ both techniques. Hemingway is the classic example of simplicity to the point of sparseness, but he uses that structural simplicity to achieve profound effects. Joyce employed dense, challenging prose, and he is greatly admired for the layered complexity of his work. Some readers enjoy a variety of styles, others love one and hate the other.
So, to answer your final question: yes, simplicity is an option, not a requirement. Your personal style may lead you in one direction or the other. You might go back and forth between projects, and certain plots or characters may cry out for a particular approach.
Don't be afraid to experiment. Writing multiple drafts of the same story, in different styles, can often be a helpful exercise.
Answer: I think the core of the novel has to be simple - In general, I believe in the rule that you should be able to explain the basic premise of your story in one sentence.  But I think you can build onto that core with as many complications and elaborations as you can come up with, as long as you (and your readers) don't lose sight of the core idea.
Answer: From a purely plot standpoint, it is also relatively dependent on the genre you're writing. People return to certain genres/sub-genres because (and these exist in the first place because) people enjoy having certain specific emotions evoked. Like Sci-Fi/Fantasy = Wonderment and Immersement. Crime/Mystery = that feeling that the reader can "figure it out first," or "see it coming."
So readers of certain "popular fiction" genres generally expect (and enjoy) a high complexity of plot:

Crime/Detective/Mystery (in a labyrinth / plot twist kind of way)
Historical Fiction / Fantasy / Science fiction (in its breadth of scope, vision, and often multiple POVs / plots weaving together)

And the expectation is often for less complexity of plot (as the focus is elsewhere) in:

Action-adventure / Western (more about fun and suspenseful, heroic moments)
Horror (more about setting up terrifying, suspenseful situations)
Romance (the complexity is in the play between different inter-personal relationships)
Inspirational (which is more about hitting heart strings with "universal" experiences)

But these are, of course, not hard-fast rules. And as for literary fiction, the umbrella is just too large to really have any solid answer one way or the other.


Why the Heel of the foot?  (Parshas Ekev)

Question: Parshas Ekev starts off

וְהָיָה עֵקֶב תִּשְׁמְעוּן אֵת הַמִּשְׁפָּטִים

Rashi says on this:

אם המצות הקלות שאדם דש בעקביו תשמעון:

What is the significance of the so called lite Mitzvos which people trample with there feet?
Comment: It presumably means; _even_ those mitsvot. Do you have reason to assume it means something else?
Answer: In Sippurei Chassidim (by R' S.Y. Zevin), he brings a story the point of which is that Hashem's desire is that even the person's "heel" - the lowest, crassest, most insensitive part of the body (as the Yiddish expression "Ich her dir mit mayn pyate" - "I hear you with my heel" - i.e., "I am paying absolutely no attention to what you're saying") should be attuned to G-dliness.
Comment: That what the Baal Hatanya h said when he was a child right?
Comment: The Tzemach Tzedek, actually. And he said it in reference to a different verse: עקב אשר שמע אברהם בקולי - "Avraham listened to Hashem even with his heel!"
Answer: Could it be that the heel is considered a weak point like Achillies' heel? Yaakov held Eisav by the heel trying to hold him back. Perhaps it is a notion of letting our guard down and it will lead to a total lapse of doing mitzvot.
Comment: Yussi Abrams, Welcome to mi.yodeya, and thanks very much for your hypothesis! Please consider clicking register, above, to create your account. This will give you access to all of mi.yodeya's features and will allow you to take full credit for your contributions.
Answer: The Gemara in Avodah Zara(ב, ב) says that in the Future the gentiles will have a complaint to Hashem we would have accepted the Torah too for the Jews you held Har Sinai over their heads if you would have done it to us we would of also accepted. Hashem hears there complaint and Gives them one LITE Mitzvah the Mitzvah of Sukkah.  Hashem takes out the sun and makes it extremely hot they go out and Kick the Sukkah and leave. Now we know why it is all dependant on Lite mitzvah which people trample with their feet as this is the test the gentiles will fail when Mosicach Comes and they have their complaint hence  עֵקֶב תִּשְׁמְעוּן the
 Mitzvah people are דש בעקביו.
Comment: But surely, you don't kick the sukkah (which is what those non-Jews end up doing, as narrated in the Gemara) with your heels?
Answer: The Lubavitcher Rebbe writes that the Midrash Tanchuma interprets this verse as saying that this verse refers to mitzvas that people throw under their feet.
The Lubavitcher Rebbe asks what does throwing light Mitzvos have to do with the ankle?
Moreover, it says in Psalms(49:6) that "Why did bad things happen to me? The sin of my ankle surrounds me". The Midrash explains this verse as saying that King David was not worried that he didn't keep the big (important) commandments. He was more worried that he didn't keep the "commandments people throw under their feet". However, there is another verse (19:12) which says "Also your servant guarded them, when he guarded his 'big' heal". ('big' in this context refers to the great reward one gets for the commandments). If so, why was King David worried about the "commandments people throw under their heel".
Therefore, the Midrash means that "worrying about throwing mitzvos under the heel" isn't talking about one who fails to keep the "small" commandment, all the more so it doesn't mean one who embarrasses the commandments (like Rashi's text which refers to the commandments being stepped upon). Rather it refers to a much lighter prohibition.
When one has a choice of how to progress in serving G-d, many times he may think that he has to do so in an orderly progression. First start with the "head" commandments, then proceed down until he reaches the "small" ones. Better, he may say, to first keep the biblical commandments, then the rabbinic, then customs, then one may keep extra "chumras".
One can't begin with "heel" commandments, he claims, before the head commandments. One has to keep Torah and Mitzvos in an order. To keep a stringency before keeping a biblical commandment is like wearing a tie without a shirt, he may claim.
However, even though such reasoning is accurate (even according to "holy intellect", the intellect of Torah), nonetheless, the foundation of all commandments is the "Kabbalas Ol" - acceptance of the yoke. The purpose of a commandment is to connect oneself to Hashem. Therefore, any commandment can accomplish this connection equally, a "big" commandment or a "small" commandment. Whether this mitzvah was written explicitly by the "Great Sanhedrin" or discovered by a modern day Torah Scholar, they are all equal in their effect.
Therefore, King David was worried. Not that he was worried that he didn't keep the "light" commandments. He was just worried that he didn't keep them with the same care as the heavy "head" mitzvos.
It says in the verse in Psalms: "Why did times of evil happen to me? the sin of my heel surrounded me." "In times of evil" refers to behavior (from heaven) that is in a natural way, during the time that Jews are found in exile, as a sheep among seventy wolves, and even (natural) sustenance comes through the supernatural. Therefore, if a Jew thinks (about commandments) in a natural way, which commandment is greater than which, then Hashem looks at him also in a natural way. This results in "days of evil".
However, when one doesn't make such calculations, and one does commandments just to be unified with Hashem, then from above they also treat him in such an unlimited manner.
Therefore, through keeping these "foot" commandments in a happy manner, Hashem gives to the Jews all forms of blessing.


How many effective gears do I have on a 21 speed bike?

Question: I have a 21 speed bike (a hybrid commuting bike). But AFAIK, this does not mean that I have 21 different gear ratios. Is there any way to calculate the different gear ratios?
Answer: Answers for a typical 21-speed commuter I could find specs on: 20, 19, 15, 12.
20 is the technically right answer, 19 is a slightly more technically right answer, 15 is if you take into account gears that are nearly identical, but 12 is the practical usable answer.
Really, it's going to depend on your exact gear setup.
At the very least, you really should avoid the big-front:big-back gear, since that's hardest on your derailer.
So, if you take @jensgram's answer and work it out on a typical bike similar to what you describe ...  I'll take the Trek 7.1FX for my example.  The chainrings in front are 48-38-28t.  In the back it's 14-16-18-20-22-24-34r.  (Numbers are the number of teeth).  I don't know if it's typical on other hybrids, but that jump from 24 to 34 is a little unusual and they market it as a special "megarange" feature.
If you work out all the ratios (I wrote a sort of little computer program up real quick to do it), the only exact duplicate ratios this gear setup has is 28:14 and 48:24, one of which is one of those extreme ranges it's suggested to avoid.  So this bike actually has 20 unique gear ratios, but since one of those unique ranges is 48:34 that you should avoid, it's 19 usable unique gear ratios.
If you round to only one digit after the decimal to give you the nearly identical ratios, you get 15 gears:
0.8 28:34
1.1 28:24 or 38:34
1.2 28:22
1.4 28:20 or 48:34
1.5 28:18 or 38:24
1.7 28:16 or 38:22
1.9 38:20
2.0 28:14 or 48:24
2.1 38:18 or 48:22
2.3 38:16
2.4 48:20
2.6 48:18
2.7 38:14
3.0 48:16
3.4 48:14

But look carefully at what front chainrings you use to get those...  If you use the smallest front chainring with only the 3 biggest rear cogs, the middle front chainring with all but the biggest and  smallest rear, and the big front chainring with only the four smallest rear cogs, you get your whole range of gears in a way you can comprehend, but it's 12 really usable gears (you skip 1.4, 2.0 and 2.7 in my above chart; all close enough to another gear)
How many rear cogs it makes sense to use on each front chainring will depend on the exact gearing setup.  But if you play around with it a little you can probably just feel it.  Switch the front chainring to the middle, switch the back to the smallest and feel that gear; then switch the front to the biggest, switch the back in by 2 or 3 gears and see if it feels about the same.  Do the same thing with the low-end.
Comment: If you use unix and can do basic shell stuff, the "program" was a simple shell script: `for front in 48 38 28; do for back in 14 16 18 20 22 24 34 ; do ratio=$((echo scale=1; echo $front / $back) | bc); echo $ratio $front:$back; done; done | sort -n`
Answer: Excluding the two extremes (see Scott Robinson's answer), you have 19 ratios. Some of these may be very close to each other. The easiest way (although a little cumbersome) is to count the number of teeth (n) on all gears (three in front, seven rear) counting from the centerline of the bike (outwards). The ratio is given as n(front) / n(rear), e.g.:
1st = n(front1) / n(rear1)
2nd = n(front1) / n(rear2)
...
8th = n(front2) / n(rear1)
...
21st = n(front3) / n(rear7)

Now you may find identical (or nearly identical) ratios. The number of unique ratios is the number of effective gears.
Answer: There's no way to calculate the actual gear ratio unless you know the sizes of the cassette and chainrings.
But, you do have 19 effective gear ratios since you shouldn't shift into the two extremes. (Low front/low back, high front/high back)
Answer: The way I visualize this is as follows:
Usually change to a smaller or larger chainring corresponds to roughly 2 jumps at the sprocket cassette (on a road bike "compact double" 34/50, it may be 3 jumps but on triple chainring setups it's almost always 2 jumps, excluding the ridiculous 13-speed cassettes and considering only reasonable 7 or 8 speed cassettes).
Thus, you have 7 ratios on the middle chainring. The small chainring adds 2 ratios and the big chainring adds 2 ratios. So you have 11 ratios.
In theory, you may have more distinct ratios if you want to be accurate, but most of these ratios are very close to each other, some may be actually identical. So in theory the answer "19 ratios" is the correct one, but in practice you have no easy way to access these ratios in sequence, and the jumps between these would not be constantly spaced.
So what matters in practice is how you operate the gears. You either shift on the rear one sprocket up or down, or if you run to the edge of the cassette (or the chainline becomes too crooked), you shift on the front, and then on the rear at the same time in the opposite direction to get effectively one jump. Calculating the practical number of ratios based on this usage pattern, you get 11 ratios.
It's better to visualize the system as consisting of three heavily overlapping ranges, with 7 ratios on each range (minus the ones where the chainline is so crooked you wouldn't use those). Because the ranges are heavily overlapping, you have 11 practically different ratios instead of 21 practically different ratios.
Answer: Here is an excellent gear calculator.  You just need to know a few basic things like crank length, wheel size, etc. and then it will figure everything out for you.
Sheldon Brown's Bicycle Gear Calculator
Answer: I've heard that the you should only use all the rear chain rings when using the middle chain ring on the front.  When using the largest front chain ring, you should only use the bottom 2 or 3 chain rings, and when using the smaller chain ring on the front, you should only use the top 2 or 3 chain rings.  This is to prevent having too much of and angle between in the chain.  Also, you generally don't need to use those gears anyway, since they are usually the same, or in between the range of the other gears, which means they won't help you much. Here's a good explaination  of how to change gears.
So while you may effectively have 21 gears, you shouldn't use them all, as the extremes can cause your chain to strech.  Mostly, you'll use only 3+7+3, giving you 13 gears.  Some people even say to keep you chain really straight really straight, and you should only use 3+3+3 = 9.  Just try to keep your chain as straight as possible. Since there is a lot of overlap, you'll want to shoot for a straight chain as much as possible, over trying to use all the gears.
Answer: There is just another thing to mention. You can't use all combinations else there is too much strain on the system. With my 3x8 I just ignore the extremes:

Front 1: Back 1-6
Front 2: Back 2-7
Front 3: Back 3-8

You loose some combinations, but they are not very useful anyway.
Note: I once had an accident. While driving, the bike suddenly stopped with a metallic klunc. The rear derailleur was caught between the spokes. Ruining about everything. The bike just had its maintenance (replacement of chain and cassette) so it was probably a mistake of the bicycle repairshop. And they agreed). Athough they said that it was because of the extreme settings (Front 3, Back 1). But that was a result of the accident. So they fixed it for free.
Comment: Sounds like the cable might have let go.  Even on my 10 speed rear, which requires full travel, there's sufficient gap to prevent that from happening.  However, if the cable were to break/slip out, it *might* possible hit the spokes.


What is the fundamental probabilistic interpretation of Quantum Fields?

Question: In quantum mechanics, particles are described by wave functions, which describe probability amplitudes. In quantum field theory, particles are described by excitations of quantum fields. What is the analog of the quantum mechanical wave function? Is it a spectrum of field configurations (in analogy with QM wave functions' spectrum of particle observables), where each field configuration can be associated with a probability amplitude? Or is the field just essentially a superposition of infinitely many wave functions for each point along the field (as if you quantized a continuous mattress of infinitesimal coupled particles)?
Answer: (Luboš just posted an answer, but I think this is sufficiently orthogonal to post too). The usual wavefunction for a bosonic field is a complex number for each field configuration at all points of space:
$$\Psi(\phi(x))$$
This wavefunction (al) obeys the Schrodinger equation with the field Hamiltonian, where the field momentum is a variational derivative operator acting on $\Psi$. This formulation is fine in principle, but it is not useful to work with this object directly under usual circumstances for the following reasons:

You need to regulate the field theory for this wavefunctional to make mathematical sense. If you try to set up the theory in the continuum right from the beginning, to specify a wavefunction over each field configuration you need to work just as hard as to do a rigorous definition for the field theory. For example, just to normalize the wavefunction overall constant time slice field values, you need to do a path integral over all the constant time field configurations. This is a path integral in one dimension less, but the thing you are integrating is no longer a local action, so there is no gain in simplicity. Even after you normalize, the expectation value of operators in the wavefunctional is a field theory problem in itself, in one dimension less, but with a nonlocal action.
Once you regulate on a lattice, the field wavefunctional is just an ordinary wavefunction of all the field values at all positions. But even when you put it on an infinite volume lattice, a typical wavefunctional in infinite volume will have a divergent energy, because you will have a certain energy density at each point when the wavefuntional is not the vacuum, a finite energy density. Infinite energy configurations of the field theory, those with a finite energy density, are very complicated, because they do not decompose into free particles at asymptotic times, but keep knocking around forever.
The actual equations of motion for the wavefuntional are not particularly illuminating and do not have the manifest Lorentz symmetry, because you chose a time-slice to define the wavefunction relative to.

These problems are overcome by working with the path integral. In the path integral, if you are adamant that you want the wavefunction, you can get it by doing the path integral imposing a boundary condition on the fields at a certain time. But a path integral Monte-Carlo simulation, or even with just a little bit of Wick rotation, will make the wavefunction settle to be the vacuum, and insertions will generally only perturb to finite energy configurations, so you get the things you care about for scattering problems.
Still, the wavefunction of fields is used in a few places for special purposes, although, with one very notable exception, the papers tend to be on the obscure side. There are 1980s papers which attempted to find the string formulation of gauge theories which tried to work with the field Hamiltonian in the Schrödinger representation, and these were by famous authors, but the name escapes me (somebody will know, maybe Lubos knows immediately).
The best example of where this approach bears fruit is when the reduction in dimension gives a field theory which has a relationship with known solvable models. This is the example of the 2+1 gauge vacuum, which was analyzed in the Schrodinger representation by Nair and collaborators in the past decade.
A recent paper which reviews and extends the results is here: http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.6376v1.pdf . This is, by far, the most significant use of Schrodinger wavefunctions in field theory to date.
Answer: It seems you are asking "of what variables is the wave function in quantum field theory a function of?". However, this question doesn't have a unique answer; in contrast with your implicit assertion, it doesn't have a unique answer in ordinary quantum mechanics of particles, either.
The wave function is a set of complex numbers that determine the complex coefficients in front of basis vectors of a particular basis (or a "continuous basis") of the Hilbert space. So quantum mechanics' wave function may be encoded in $\psi(x)$; but it may also be described in the momentum representation, $\tilde \psi(p)$ which is the Fourier transform of $\psi(x)$.
We may also choose different eigenvectors, a different basis. For example, the harmonic oscillator (and many other systems) has a discrete set of energy eigenstates labeled by $n=0,1,2,3,\dots$. In that case, the coefficients $a_n$ completely encode the state vectors – the wave function – as well.
There is a hugely infinite number of possible bases, and even a very large number of bases that are commonly used.
In quantum field theory, the most natural basis (or the most widely used one) is the basis analogous to the harmonic oscillator energy eigenstate example. For each type of a quantum field, each polarization, and each $\vec k$ (the wavenumber, i.e. the frequency and the direction of motion), we define the contribution of this mode of the quantum field to the non-interacting part of the energy. This turns out to be just a rescaled harmonic oscillator with the spectrum $E=n\hbar\omega$ where $n$ is a non-negative integer that may be interpreted as the number of particles.
In this basis, to specify the wave function, we need to determine the complex amplitude in front of the $n_i=0$ "vacuum state", in front of the numerous $n_i=(0,0,0,1,0,0,...)$ states which is equivalent to the non-relativistic wave function (e.g. of positions or, more often, momenta) $\tilde \psi(p_1)$, the complex numbers determinining the complex amplitudes for all two-excitation eigenstates $\tilde \psi(p_1,p_2)$, and so on, indefinitely (increasing number of particles).
However, it's also true that one may choose a "functionally super continuous" basis of all configurations of the quantum fields. In this language, the state vector is a functional
$$ \Psi [\phi(x,y,z), A_i(x,y,z),\dots] $$
which depends on all the fields at $t=0$ but not their time derivatives. A functional is morally equivalent to a function of infinitely many variables (continuously infinitely many, at least in this case). It's not terribly practical to work with such a description of the wave function and it's hard to "measure" the values of the functional at different points (for different configurations of the fields) but it is possible.
The particle-occupation basis mentioned above is more physical because Nature evolves according to the Hamiltonian so all states (objects and their configurations) that tend to be at least a little bit "lasting" are close to energy eigenstates. That's why bases that are close to bases of energy eigenstates (or eigenstates of a "free" part of the Hamiltonian etc.) are generally more useful and natural to work with, especially when one thinks about applications.
Answer: In nonrelativistic quantum mechanics you can think (modulo technicalities with rigged Hilbert spaces) of the coordinate representation of the wavefunction as the projection of the state vector $\Psi$ in the "direction" of the position eigenvector, i.e. $\Psi(x)=\langle x|\Psi \rangle$.
Quantum field theory is traditionally (in textbooks) formulated in the interaction picture.  It can, however, also be formulated in the Schroedinger picture, in which a natural notion of wavefunction emerges.  Instead of being a complex valued function on the space of particle positions, it now becomes a complex valued functional on the space of field configurations.  So the wavefunction(al) is the projection of a Schroedinger picture state $\Psi$ in the "direction" of a field configuration $\phi(\mathbf{x})$
$|\Psi(t) \rangle = \int \cal{D}\phi \Psi[\phi,t]|\phi \rangle$
It satisfies the functional Schroedinger equation
$i\frac{\partial}{\partial t} \Psi [\phi ,t] = H(\phi (\mathbf{x}), -i\frac{\delta}{\delta \phi (\mathbf{x})})\Psi [\phi ,t]$
The only textbook reference treating this that I know of is Brian Hatfield's book.
Comment: D'oh I see some more detailed answers appeared while I was typing mine....
Comment: Maybe a less detailed answer, but I think you most directly address my intended question. Thanks! I will check out the Hatfield book.
Answer: Ron and Luboš's Answers, +1. FWIW, however, I have found it worthwhile to take QFT to be a stochastic signal processing formalism in the presence of Lorentz invariant (quantum) noise.
The devil is in the details, and I cannot claim to be able to say much, or even anything, about interacting quantum fields, but it is possible to construct random fields that are empirically equivalent, in a specific sense, to the quantized complex Klein-Gordon field (EPL 87 (2009) 31002, http://arxiv.org/abs/0905.1263v2) and to the quantized electromagnetic field (http://arxiv.org/abs/0908.2439v2, completely rewritten a few weeks ago). Needless to say, the fact that a random field satisfies the trivial commutation relation $[\hat\chi(x),\hat\chi(y)]=0$ instead of the nontrivial commutation relation $[\hat\phi(x),\hat\phi(y)]=\mathrm{i}\!\Delta(x-y)$ plays out in numerous ways, and this is not for anyone who wants to stay in the mainstream.
I take it as a significant ingredient that we treat quantum fields as (linear) functionals from a Schwartz space $\mathcal{S}$ of window functions into a $\star$-algebra $\mathcal{A}$ of operators, $\hat\phi:\mathcal{S}\rightarrow\mathcal{A};f\mapsto\hat\phi_f$, instead of dealing with the operator-valued distribution $\hat\phi(x)$ directly, even though we may construct $\hat\phi_f$ directly from $\hat\phi(x)$, by "smearing", $\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x$. In terms of these operators, the algebraic structure of the quantized free Klein-Gordon field algebra is completely given by the commutator $[\hat\phi_f,\hat\phi_g]=(f^*,g)-(g^*,f)$, where $(f,g)=\int f^*(x)\mathrm{i}\!\Delta_+(x-y)g(y)\mathrm{d}^4x\mathrm{d}^4y$ is a Hermitian (positive semi-definite) inner product.
A window function formalism is part of the Wightman axiom approach to QFT.
Note that what are called "window functions" in signal processing are generally called "test functions" in QFT. The signal processing community works with Fourier and other transforms in a way that has close parallels with quantum theory and implicitly or explicitly uses Hilbert spaces.
In the vacuum state of the quantized free Klein-Gordon field, we can compute a Gaussian probability density using the operator $\hat\phi_f$,
$$\rho_f(\lambda)=\left<0\right|\delta(\hat\phi_f-\lambda)\left|0\right>=\frac{e^{-\frac{\lambda^2}{2(f,f)}}}{\sqrt{2\pi(f,f)}},$$
which depends on the inner product $(f,f)$ [take the Fourier transform of the Dirac delta, use Baker-Campbell-Hausdorff, then take the inverse Fourier transform]. The reason it's good to work with smeared operators $\hat\phi_f$ instead of with operator-valued distributions $\hat\phi(x)$ is that the inner product $(f,f)$ is only defined when $f$ is square-integrable, which a delta function at a point is not.
We can compute a probability density using an operator $\hat\phi_f$ in all states, but, of course, we can only compute a joint probability density such as
$$\rho_{f,g}(\lambda,\mu)=\left<0\right|\delta(\hat\phi_f-\lambda)\delta(\hat\phi_g-\mu)\left|0\right>$$ if $\hat\phi_f$ commutes with $\hat\phi_g$; in other words, whenever, but in general only when, the window functions $f$ and $g$ have space-like separated supports. At space-like separation, the formalism is perfectly set up to generate probability densities, which is why QFT is like stochastic signal processing, but at time-like separation, the nontrivial commutation relations prevent the construction of probability densities.
I take it to be significant that the scale of the quantum field commutator, the imaginary component of the inner product $(f,g)$, is the same as the scale of the fluctuations in the probability density $\rho_f(\lambda)$, determined by the diagonal component, $(f,f)$. It's possible, indeed, to construct a quantum field state in which the two scales are different (Phys. Lett. A 338, 8-12(2005), http://arxiv.org/abs/quant-ph/0411156v2), so the equivalence could be thought as surprising as the equivalence of gravitational and inertial mass.
Needless to say, there many people who are working away at QFT. The approach I've outlined is only one, with one person working on it, in contrast to string theory, supersymmetry, noncommutative space-time geometry, etc., all of which have had multiple Physicist-decades or centuries of effort poured into them. It's probably better to follow the money. Also, please note that I have hacked this out in an hour, which I have done because rehearsal is always good. Did anyone read all of this?
Comment: This is the description of the vaccuum state of a free field theory, which is always given by a Gaussian wavefunctional, and in the case of the electromagnetic field dates back to Schwinger (or Feynman, I think only Schwinger gave it explicitly). The commutation relation for stochastic fields is _not_ trivial, even though they are classical-looking. I find the most interesting versions are the stochastic 2d fields which reproduce the N=2 SUSY theory by Nicolai map. This is an active field now.
Comment: 's true, Ron. To construct a nontrivial Lorentz invariant vacuum state for a random field that has trivial commutators for the observables $\hat\chi(x)$ in a quantum field formalism requires nontrivial commutators for creation and annihilation operators, which is what makes the connection into QFT. Such an approach effectively puts the dynamical evolution and the description of measurements into the state instead of into the algebra ($[\hat\chi(x),\hat\chi(y)]=0$ gives no information, so information has to come from some other part of the structure).


How to switch back from Google+ profile in Blogger

Question: I connected my Blogger profile with Google Plus, but I don't like it anymore. Can I go back and switch to the old Blogger profile?
Answer: Here is a link:
http://www.blogger.com/revert-profile.g
Click this when you are logged into your Google account, and you will land directly on a page where you can switch back to your old Blogger profile.
Answer: This Worked For me..
To Check that your profile name is changed first login to your account follow below link
and check your Blogger Profile Name.
LINK :: https://www.google.com/dashboard [1]
Then Use the link below
LINK :: http://www.blogger.com/revert-profile.g
This will led you to a page where it will ask to "Change the Blogger Profile Display name"
Change the name & Press "Continue To Blogger". The name will be changed. Again
go to link [1] & check that your Blogger Profile Name is Changed.


Can I have multiple concurrent Gravatars associated with the same email address?

Question: If I use the same email address to register at several sites that use Gravatar, is there a way to specify a different avatar for each site? Or do I have to use a different email address for each site?
Answer: With Gravatar you just add a new email to your account using sub-addressing and set up a new image attached with each of the different versions of the same email address.
E.g. Your base email address is firstname.lastname@example.com, with sub-addressing, this can become any of the following:

firstname.lastname@example.com
email@example.com
firstname.lastname@example.com

Set those up in Gravatar and use them to register for the various sites.
If you're using an email provider that allows sub-addressing (or plus-addressing if you're familiar with Gmail) and the site you register at allows you to use an email address with sub-addressing, this will definitely work.
Some sites will consider an email with a plus as invalid, so this won't work for them.


lightdm/unity-greeter wallpaper will not set

Question: I am trying to set the wallpaper on my login screen, but i cant seem to get anything to stick.  I have used gsettings set com.canonical.unity-greeter background '/path/to/pic.jpg' as user 'lightdm'. However, i then get the output (process:6517): dconf-CRITICAL **: unable to create directory '/run/user/1000/dconf': Permission denied.  dconf will not work properly. Any ideas? Thanks
Comment: could be a bug? https://bugs.launchpad.net/unity-greeter/+bug/1266465
Answer: I found a solution.  Though it displays an error message, the lightdm background does in fact change to whatever new background image you set.
Open a terminal and get root
sudo -i

(Type sudo password)
Allow the user lightdm to create a connection to the X server
xhost +SI:localuser:lightdm

Switch user to lightdm.  I had to use sudo to do this, other guides do not do so.
sudo su lightdm -s /bin/bash

Make sure you have turned off the option to dynamically change lightdm's background to match the user's desktop background.
gsettings set com.canonical.unity-greeter draw-user-backgrounds 'false'

Finally, set your desired lightdm background.
gsettings set com.canonical.unity-greeter background '/path/to/your/background.png'

This returns an error about dconf, ignore it. The new lightdm background should be present when you log out.  To double check, you can run
exit

twice, once to get out of lightdm and once to get out of root.  Then run
gsettings get com.canonical.unity-greeter background

Which should return the path of the image you chose.  Similarly, running
dconf-editor

Gets you to a nice GUI.  Choosing the drop-down for com, then canonical, then unity-greeter lets you check the path of lightdm's background image, and should allow you to edit it as well but I was never able to make a valid change from there.
Hope this helps.
EDIT: running
gsettings get com.canonical.unity-greeter background

after logging out does NOT return the new image path, but when lightdm is next used, the correct new image is set.  I know there's a way to run lightdm in test mode, which may be a more accurate way to test if the new background is set. I did not test that method, as the above method I used to change the background image worked for me.


How long can you keep chocolate, and what is the best way to store it?

Question: For instance, I have some bars of Valrhona I use for chocolate tarts and pastries, but I haven't been doing the dessert thing in a while.  How long does chocolate last before losing flavor (or does it)?  And once it gets that white stuff on the outside, is it done?  What's the best way to store it for as long as possible?  Thank you.
Comment: Store chocolate? Chocolate must not be stored, it must be eaten :-p
Comment: Related: http://cooking.stackexchange.com/questions/7392/is-it-better-to-store-chocolate-in-the-fridge-or-at-room-temperature
Answer: Regardless of type, all chocolate should be stored in a cool and low humidity (dry) place away from direct sunlight. It would be best to seal it in an air-tight container, because, as ElendiTheTall said, the cocoa butter in it will absorb flavors.
Dark chocolate will last for years. Milk and white chocolate will last for a much shorter time (a few months), because of their milk content.
Improperly stored chocolate will develop bloom, which shows as a white or grey streaking or spotting on the surface. The spotting or streaking is cocoa butter fat separating and is a sign that the chocolate's temper has been lost. This kind of chocolate is still suitable for any application where the chocolate will be fully melted (most baking). It can even be used as the base (non-seed) chocolate for tempering with the seeding method, but it should not be used for other candy making.
Answer: Stored correctly, chocolate can last for a very long time, due to its extremely low moisture content. Cool and dry are key. Tightly wrapped, it can be frozen, but this seems unnecessary.
4 months, 6 months is no problem. Years might start to be excessive, but in the end the chocolate will still be usable.
Comment: The US military puts chocolate in MREs specifically because it keeps for ages.
Comment: I have heard that chocolate freezes fine but is difficult to defrost without bloom occurring. Any tips?
Answer: When thawing frozen chocolate, keep it wrapped tightly to avoid condensation from forming on it.  Also, it helps to move it from freezer to refrigerator for 24 hours and then to your cupboard keeping it fully covered until it is room temperature.
Answer: The true shelf life on chocolate depends a lot on how it is stored and what type of chocolate we are talking about. For instance, if a chocolate bar is stored in the pantry it will last for 2-4 months - but if stored in the refrigerator it lasts for 4-6 months BEYOND its printed date. The best way to store it is in a cool dry place like the refrigerator. Because chocolate has a low melting point it should not be left in warm places or in the sun for too long. You can freeze it, this lasts the longest!
Source: http://www.eatbydate.com/other/sweets/chocolate-shelf-life-expiration-date/
Comment: Storing chocolate in the refrigerator may work as long as it's packed. Open chocolate will often get ugly condensation when stored in the fridge, or during getting to room temperature afterwards.
Comment: When I store chocolate in the refrigerator, it gets a white layer. Is there any special way to pack it to avoid it?
Answer: The best way to store it is in a cool, dark place with low humidity. You should not store it in the fridge. Chocolate absorbs other flavours like a sponge so keep it away from anything with a strong odour.
The white 'bloom' on chocolate is the cocoa butter seeping out of the surface. It is harmless and will disappear when you melt the chocolate.


How many of you have brewed a 'Triple IPA' and how do you define it?

Question: I am doing one for a competition -I hopped it like a double or Imperial IPA in the boil, but I am hopping it with three dry hop additions added in separate increments.
Comment: Sounds like a double IPA to me.  Or maybe you've crossed over into American Barleywine.
Comment: I second brewchez' comment.  The DIPA style covers anything stronger than an IPA, up to the alcohol concentration limits of the yeast.
Answer: Triple IPA is more marketing buzz than actual style.  You could characterize it as anything on the extreme high end of the alcohol scale (>10% ABV) for Imperial IPA. Pliny the Younger would be the prototypical version of this.
The trick is to make sure you are still in IPA territory and not barley wine.  To ensure this you need to use a large amount of simple sugars -- or "White Malt" as Vinnie calls it.  And then just hop the hell out of it.  If you ever ask yourself "Is there too much hops in this?" the answer is NO.
Comment: Just to add to this a little, the simple sugars will ferment out entirely, making the beer much dryer. If you can gain 10% with a FG of 1.015 you'd have a Triple IPA.  If you end up with 10% and a FG of 1.030, you have an American Barleywine.
Answer: Double IPA gets its name either because there are two I's (IIPA), for imperial IPA, or just double because of having roughly double the alcohol.
For what you describe, I would call it a Triple Dry Hops Brewed Beer.
What was your reasoning for putting the dry hops in at different intervals, and which hops did you use?
Answer: It's one more than a double IPA.


Why does the iterative version take longer?

Question: I was looking over at http://programming.lispdream.com/blog/2011/06/recursion-vs-iteration/ and I saw that on his implementation of the recursive and iterative implementations of the factorial function, the iterative actually takes longer given n = 1,000.  I can't figure out why (he doesn't explain, but says that it is an exercise for the reader).  Sorry for my newness to all of this.
Answer: The two programs aren't equivalent.  The recursive version is computing
(...((1*2)*3)*4 ... *n)
while the iterative one is computing
(...((n*(n-1))*(n-2) ... * 1)
thus intermediate quantities are growing more rapidly for the iterative version and big num computation is quicker when the numbers involved are small  (Computing 1000! without big num has no sense and lisp dialect switch to big num automatically).
Answer: When you make a recursive algorithm iterative, you have to explicitly implement the stack that tracks results. This act adds extra operations dealing with pushing and popping the stack that the recursive algorithm gets for free (well not quite for free but the extra operations add up to more than the cost of recursion).
Comment: Did you look at the programs?  The iterative factorial doesn't manipulate a stack at all.
Answer: I can only guess, I am not even sure if those benchmarks are from the C or from the SBLC code. My guess is the culprit is mutating the variable. 1000! is a pretty big number, maybe it is faster to populate stack with intermidiaries and clean up than to create a copy and overwrite.
Comment: Those were from the SBCL code, I think.


Shemona Veshishim - mi yodeya?

Question: Who knows sixty-eight?
Please cite/link your sources, if possible. At some point in the next few days, I will:

Upvote all interesting answers.

Accept the best answer.

Go on to the next number.
Comment: Prev: https://judaism.stackexchange.com/questions/2339/shiva-veshishim-mi-yodeya?r=SearchResults
Comment: Next: https://judaism.stackexchange.com/questions/2357/tisha-veshishim-mi-yodeya?r=SearchResults
Answer: According to Rabbi Shimon (Sanhedrin 17a), the number of zekainim that assumed their posts in the ohel moed.  (70 minus Eldad and Meidad who did not feel they deserved the position and were zochu to another nevuah)
Answer: There are 68 bones that make up the essential structure of the human body (the parts that are essential for life, apparently): 22 ribs, 18 spinal vertebrae, 9 bones of the skull, 8 cervical vertebrae, 6 in the "key of the heart" (sternum?), and five at the orifices. (Malbim to Song of Songs 3:7; he explains that the "sixty mighty men standing guard around Shlomo's bed" mentioned there are these approximately sixty organs that guard the soul and keep it from leaving the body.)
Answer: CE Year of the Churban Bayis Sheini
Comment: Not according to all opinions; the Rambam and Rabbeinu Tam place it in 69.
Comment: True, it is a machlokes.  (Although I have heard differing approaches to Rambam).  You can catch it at tisha v'shishim!
Answer: The number of Yaakov and his family members that were yordim to Mitzrayim at that time according to Rashi's pshat of the Shiv'im -Rashi includes Yocheved ("bein hachomos"), I am including Yehudah (who was sent ahead) and Yaakov as well.  The Torah doesn't include wives.
Essentially, 71-3
Answer: This one was a toughie!
The Mishnah in Succah 19b states that the Sages allow one to construct a succah by leaning s'chach against a wall. The length of this s'chach must be enough to accommodate the minimum measure of a succah, which is 10 tefachim high and 7 tefachim wide. Some Rishonim say that this minimum measure needs to fit under the s'chach. Others, however, say it is sufficient if the s'chach is just 17 tefachim long, so that 10 tefachim will serve as a wall and 7 tefachim as a roof. Hence, if one fabricates a succah in the form of a 4-sided pyramid, the total length of s'chach he will need (as measured from the base of each side to the top) will be 17 x 4, or 68 tefachim.
(The truth is that in this case it might be enough to have 13.5 on each side, so that each two opposing sides will together contribute 7 tefachim to the roof. I'm not sure how that would work.)

Shemona Veshishim - mi yodeya?


Wearing a kippa at work

Question: If one has a job where his employer informs him that if he wears a kippa he will be fired, does he have to wear a kippa?
Comment: Call his bluf, but only after you contact a competent attorney.
Comment: Sounds like a time to start looking for a different job. But take the Kippa off in the mean time
Answer: Firstly, I don't know of anyone who requires a kippa as opposed to some other head covering.  So if at work he could wear a beret, hard-hat, baseball cap, coonskin cap, deerstalker, you name it, by all means do that.
There's the issue of head coverings for praying; for making blessings; for eating; and then at all other times.
Much of yarmulka as we know it is a very, very strong and well-founded custom (minhag), based on a Gemara about how a head covering kept a rabbi out of trouble.
As a theoretical concept, if keeping a custom means losing your job, you don't have to lose your job.  (Heard from Rabbi Welcher; minhag is overridden l'tzorech parnasa).
But practically today, it's advisable for a person to talk with his rabbi regarding his individual situation.  We don't want people shrugging off deep-seated customs willy-nilly.  It's also a good opportunity to have a spiritual checkup with the rabbi, make sure you still feel (and live) Jewish with this different situation.
Comment: "for eating" Where does it say in Tanach, Mishnah, Bavli, Yerushalmi, Sifra, Sifrei, Tosefta, Mechilta, Gaonim, Rambam, Rishonim, or Shulchan Aruch, that one must wear a head covering to eat?
Answer: I work at a job in sales where I deal with contracts and large amounts of money. I am also one of the few people on the team who are Jewish.
My rav reasoned that due to the fact that some people are unhappy with the service we provide (and would immediately blame the fact that I was Jewish on their dissatisfaction) that wearing a kippah would cause a chillul hashem, especially with the demographics of the neighborhood where I work. It is not the most Jewish friendly area of town. In addition we thought, though did not prove, that I would lose business by wearing my kippah. He paskined that I should not wear my kippah at work and I havn't had any issues.
Comment: zenmastertaco, thank you for sharing your experiences, and welcome to mi.yodeya.
Answer: I wrote a piece on another blog about 3 years ago, that I think is still valid as a general rule with regard to wearing a Kippah at job interviews or at work.  And I still believe that if an employer won't tolerate your wearing a Kippah you may not want to work there, but everyone needs to make his own decision.  However, in your particular case, it sounds like your employer is violating Title VII of the Civil Rights Act of 1964, which ensures religious freedom (among other freedoms) in the workplace in the U.S. Of course, I don't know if you are located/working in the U.S., but if you are, it might be worth reporting it to the United States Equal Employment Opportunity Commission.
Just to paraphrase the closing paragraph from the other piece, what I have to say on the subject in general is this: it's a personal choice. If you feel wearing a Kippah is hurting you in your employment or job search, you have every right to take it off - and you have every right to put it back on at any point. However, be prepared to be made uncomfortable for doing so, and begin looking for a new job with an employer that can see past it - all while diligently doing your job. On the other hand, if you choose to wear it, I will give you the Berachah (blessing) given to me by a rabbi I consulted on the subject: You will get a job that is good for you (either by this one changing to become friendlier or another one), and being Moser Nefesh will come back to reward you, either in this world or the next. Even if it isn't about observance/Halachah, but you simply wear your Kippah as a badge of honor and Jewish pride, continue to do so because you will ultimately find something suitable, and your colleagues – particularly in an environment where they are willing to hire someone who is proud of his heritage and beliefs – will respect you for it.
Answer: Sh"t Igrot Moshe (C"M 1:93 and O"C 4:2) writes that since wearing a Kippah is neither a positive or negative mitzvah one need not loss a large portion of his money by not taking such a job. However since they don't care if you wear it in another room or in the marketplace one must wear it in such places. Sh"t Igrot Moshe Y"D 4:11(3) adds that even according to the Taz who holds there's a prohibition of Chukot HaGoyim (that the practice of Goyim used to be to always wear hats and when they saw down to eat or something similar they would take off their hat) nowadays it wouldn't apply since people don't wear hats at all. Thus, one can take a job at a place where they don't allow one to wear a Kippah. However, if they allow one to wear a regular hat (not a Kippah) one should wear a hat he wears regularly.
Halachipedia.com
Answer: This would be permissible because it is only Midat HaHasidut (see Bet Yosef O"H 91, Rama Darke Moshe 2:3). Although in other sources (siman 46, 8) it may seem that he says it is Asur- Rav Dawid Yosef Shelita writes in Oserot Yosef O"H 4 that the opinion in siman 91 is the correct. According to most Poskim it is only Midat HaHasidut (see Birke Yosef 2:2, Bah siman 2, Perisha 2:8, Magen Abraham 91:5) However, those who follow the Taz it may be Asur.
Since there are so many Matirim, and opinions that it is only Midat HaHassidut, I presume that your Rav will give you a Heter.
HOWEVER, when blessing, Lechatehila you must cover your head, and whenever you say H's Name.
Answer: I have worn a Kippah to every job interview and job I've ever held and have never had an issue. However, a colleague of mine works as a public school teacher in PA where (apparently) the education laws forbid him from wearing a kippah because it would violate the prohibition of public school employees making displays of religion. I don't think that going without a head covering during work hours is such a big deal next to losing your parnasa.
Comment: Wow. The schools in PA sound like they're violating some basic rights.

Wearing a kippa at work


Storing bottle conditioned beers and carbonation

Question: I store all my beer at 55°F(12.7°C). Do I need to worry about the beers that are bottle conditioned not being carbonated, or not being at the proper carbonation level? Should they already be carbonated when I purchase them? The main reason for this question is the issues Lost Abbey and the Bruery were having with carbonation. A friend suggested I let all my Lost Abbey stuff sit around 70°F (21°C) and shake it up a bit every once in a while for a few months. Is this a good suggestion?
Comment: Relevant http://brewadvice.com/questions/35/how-to-store-beer/36#36
Answer: When bottle conditioned beers leave the brewery they are already at the proper CO2 levels, unless otherwise specified.
It's better to store beer at a cooler temperature.  The 3 things that destroy beer are light, heat, and oxygen.
Don't shake your beers, you will re-suspend everything that has dropped out over time.
Answer: Without a major increase in heat, I can't see how your bottle conditioned beer would get much better or worse.  Isn't that the purpose of conditioning?
Answer: What style beers are they?  Lost Abbey and The Bruery (both great breweries) have a lot of quads and unique styles that aren't going to be as carbonated as a pilsner or IPA.
I wouldn't shake it up right before drinking. Let the bottles sit right before opening then gently pour so you don't get all the yeast into your glass.


Should I formally study to make games?

Question: I don't have formal studies in programming nor in graphic design, I'm self-taught, you know I have learned some things here and there. So far have been doing amateur stuff, sort of hobbyist. I would like at some point of my life be doing professional games, don't know if that would mean acquiring formal knowledge at some gaming college. It is something even desirable or advisable?
Comment: From a programmer's perspective: http://gamedev.stackexchange.com/questions/1611/game-development-degree-vs-computer-science-degree
Comment: From an artist's: http://gamedev.stackexchange.com/questions/4438/should-i-go-to-a-university-or-trade-school-to-be-a-game-artist
Answer: If you think you'd enjoy making games then go for it!
It's always best to do what makes you happy.
But really, there are two routes you could take. You could take a generalised course that included computer programming for example a computer science course. Then maybe learn game programming afterwards. That way, if the game programming career falls through, you have other options.
If you look at the money some of the recent games made especially with the financial recession, then I suppose it would be an advisable career. Notch's minecraft game is a good example. He did that near enough on his own and he now has nearly 1 million paying customers. It pays off if you hit a niche. It all comes down to luck like a lot of things in life.
So definitely just go with it if you think you'll enjoy it.
Answer: When thinking about C/C++ (because these are the industrial standard languages), I learned most of C by myself, but I admit that school helped a lot for C++: C++ is really a fat subject when used for game programming, but that doesn't necessarily mean I couldn't learn things by myself; school is a very efficient shortcut, but like every shortcut, it's harder and can cost money, and even worst, sometimes you might feel you are learning nothing.
Don't forget there are 2 sides for being a game developer; first you have to be able to use C++ fluently, and not just classes and virtual function: also the basic STL containers. Secondly, since game programming is all about code and nothing else, you have to learn about design patterns, but often that might not be enough.
I'm sure you could do some Ogre programming for example if you have some project in mind and time to spare. Don't forget that it's much much better to learn something new to solve a problem when you explicitly know what it's for: learning the theory while practicing is more logical because you have a concrete example which makes it better to remind you what it's for. It's more reassuring than school.
On the other hand, you have to have a pretty good amount of free time and self-confidence, because at school you have help not only from teachers but also from students, but don't forget that there's also pretty good help on forums and IRC channels.
To answer your question, I'd say it mainly depends on what you need to learn, since the basics are quite important, but it also depends on your curiosity and how autonomous you can be.
I have taken the opportunity to get into a private school because I had funds at the moment, I don't know if I would have risked to make a loan for that. On the other hand I met people, which is an important thing, but that won't change the way I conceive programming and how I like to learn things: most of what I see are people unable to seek the information for themselves and are too much used to being teach planned things rather than asking questions or asking themselves questions, after to watch them complain about the program.
I guess the answer would have been very much different would you have asked it 10 or 15 years ago, but nowadays multi-core processors and giga-powerful consoles require a game programmer to be able to get hold of bigger data structures and algorithms and design patterns etc etc.
Of course I'm not talking about hobbyist/low-project game programming, which can be done with unity for example, but more about professional/know-how-it-s-done/from-scratch game programming, which gives you a lot more.
Answer: Straight answer: it is very advisable. I'll explain you what I've learned, and hope it helps.
Formal studies always help when you want to take things to the professional level. However, the entertainment industry is very big and there is an advantage/disadvantage when it comes to media: everything comes down to portfolio and imagination.
Let's say you want to work on EA. Big thing, huh? They ask for formal studies; sure, but then they see your portfolio (and you need a strong one, not necessarily a big one). So... it helps, but imagination and portfolio helps better.
Now, let's say you want to go indie. Good thing. People don't ask for code standards, design documents or formal education; they want games. Good, funny, maybe free ones.
If you have the time, and the budget, to afford formal education; go for it. Why? You'll be on the shoulder of giants and you'll have subjects to focus your attention on; so learning will be kind of faster. But always remember that all of the theoretical material is worthless unless you make games, show it to people and get feedback from them to get the hang of it.
Finally, keep self-teaching yourself. That's key on being successful. Users/clients/bosses don't care too much if you were taught about a subject, but rather whether you actually know about it.
PS: As David pointed out, it's recommended to go for a specialized curriculum. Computer Science or Arts and getting a game development major. But that's up to you to decide and there are pros and cons that don't belong to this subject.

Should I formally study to make games?


How to learn guitar without physical lessons?

Question: I have an acoustic guitar that I've tried to learn to play many times. I'd like to learn to play by ear... I'm an accomplished musician on several other instruments but those are all woodwinds and I've used music reading. What's the best way to really start out? I've used books to no avail. Thank you!
Comment: It would be helpful to know more about your goals in order to recommend specific books/videos, etc.  What style of music are you interested in playing on the guitar?  Also, if you could expand on your desire to learn to play by ear and not use notation on the guitar - what is your thinking there?  Do you feel those are mutually exclusive?
Answer: Getting guitar lessons with a good teacher would be the first choice, but if you really can't have a teacher, then watching guitar video lessons online would be the second choice. You have to be careful though, there are many guitar players posting lessons on Youtube, and not all of them are good teachers or even good players. One of the best online teachers would be Justin Sandercoe. He has a complete beginner's and intermediate course, plus he's a pro musician and has been teaching guitar for many years. His website is http://www.justinguitar.com. Also check out the links given by Dr Mayhem, they contain some useful information as well. One more thing, you mentioned that you would like to learn to play by ear. Justin Sandercoe explains it on his website, but to play be able to by ear, you have to start by learning to transcribe music, which basically means listening to a song or a solo and figuring out the chords and notes by ear.
Comment: +1. Re: transcribing: The guitar isn't like a woodwind instrument in that it has multiple places on the neck to play the same pitches, with each position having a different timbre. Learning to hear those differences takes a while but is important when trying to figure out chord changes and solos.
Comment: Justin publishes an amazing amount of free educational material ranging from folk to blues to jazz and more. He provides sound advice as well. I wish that I had these resources as a young student.
Answer: You'll probably want to look at

Need help in starting to learn guitar
What is the first scale one should learn on the guitar
Is there any guitar tuition software...?

And there are a lot of other good questions tagged guitar and learning
Answer: Music is communication. They've found that when you play with other players, the parts of the brain that deal with speech light up. You can get somewhere playing by yourself, but you get farther faster when you play with someone else. It doesn't have to be in a lesson context.
Plus, when you play with someone else, you get a strong sense of what needs attention. You may think you want to do fancy flatpicking runs, but you'll know you need work on timing and changing chords cleanly before picking up double-down-up picking ever need show up.
Answer: I think that with your background, you can educate yourself. But at first it might be helpful to get a few lessons from a professional guitar teacher in order to teach you how to place the fingers of each hand to execute the notes correctly and produce the proper tone. You will also need a critique of how you sit, stand and hold the instrument to enable playing with the least amount of fatigue, and factors like that.
A surprisingly difficult problem for many players who come from an instrument where you only play one note at a time, such as woodwinds, is this: You can form one chord on a guitar, and you can form another chord on a guitar, but how to you get your left hand to move from one chord to the next chord in the tempo of the song? You want to hold one chord for its full rhythmic value and then, in an instant, form the next chord just a fraction of a second before the next chord needs to sound. This is something else that a teacher could help drill you on.
Answer: In this day and age - learning from youtube is not a bad idea. There are many helpful tutorials you can find there.
Also, try learning a song that you really like and that has easy chords. This way you'll be motivated to practice more. I would recommend beginning with chords and then moving onto finger-picking, unless you wanted to play classical guitar.
Answer: I saw a really great set of videos on how to play the guitar from Brazil on Y-Tube. Subtitled in English -- it's free form and breaks it various styles down in rhythm and chords for you -- perhaps a different style for learning that you might find nice as an alternative.
Answer: If your already a player of other instruments, and an accomplished musician, the the first advice i would give you is to transpose some of the music you know on the other instruments onto the guitar.
You should be able to do this by ear, note by note if need be.
Transposing scales/arpeggios and such by ear shouldn't be a problem for you, just remember that once you have transposed a scale in a particular key, to remember the shape, as on the guitar the shapes stay the same across all keys.
Your major learning curve will be chords, fretting them on a guitar can be quite difficult to begin with.  In your position, I would choose a simple song/piece i was familiar with, and learn/apply the chord shapes for the guitar, if you have a good ear this shouldnt be a problem.  Then build from there with another song, or write something with what you have learned etcetera.
Same applies with chords, once you have learnt a shape, say one of the minor triad shapes, it stays the same regardless of the root.
Answer: Learn and Master Guitar goes through the basic to the fundamentals of playing by ear.
http://www.learnandmaster.com/guitar/
Answer: One idea that I don't see mentioned here often for people just getting started on guitar is to enroll in a community college beginning guitar class. The cost is much lower than individual lessons, basic questions get answered, you have a chance to get to know others who are also just beginning, you won't be the only one making mistakes and having trouble, they only last for a semester, and they can be a lot of fun to participate in. And they usually have evening classes for working adults. You won't be an accomplished player when the semester is over, but you'll have the option to continue advanced classes or study on your own.