Generalization of Vogt's Theorem for curves in higher dimension

Question:
The Vogt's theorem for plane curves states that if A and B are endpoints of a spiral arc,
the curvature nondecreasing from A to B. The angle $\beta$ of the tangent to the arc at B with the chord AB is not less than the angle $\alpha$ of the tangent at A with AB. $\alpha = \beta$ only if the curvature is constant.
Does anyone know of a result which extend this theorem to space curves or curves in higher dimension.
I have the following conjecture for space curves:
Given a regular curve in space $\gamma : [0, l] \rightarrow \mathbb{R}^3$, parametrized by arc-length $s$, let $\kappa$ and $\tau$ denote the Euclidean curvature and torsion respectively. Let us assume that $\kappa$ is non-decreasing and $\tau$ is non-decreasing. Let $A = \gamma(0)$ and $B = \gamma(l)$ and let $\alpha$ be the angle between the tangent plane at $\gamma(0)$ and the chord $AB$ and let $\beta$ be the angle between the tangent plane at $\gamma(l)$ and the chord $AB$. We claim that $\alpha \leq \beta$ and equality holds only if $\gamma$ is a circular helix.
Answer: This is an attempt at an (as yet incomplete) proof of the above claim. I would be happy to receive any corrections and/or comments.
Let us denote the curve $\gamma$ by the following parametrization $\gamma(s) = (x_1(s), x_2(s), x_3(s))$. Without loss of generality let us assume that $A = \gamma(0) = (0, 0, 0)$ and $B = \gamma(l) = (x_1(l), 0, 0)$.
Let $\theta(s)$ denote the angle between tangent plane at $\gamma(s)$ and the chord $AB$, thus we have that:
$\sin \theta(s) = \langle B(s), (1,0,0) \rangle$.
Using Frenet-Serret formulae where :
$T'(s) = \kappa(s) N(s)$ and
$N'(s) = -\kappa(s) T(s)   -\tau(s) B(s)$
we have that :
$\sin \theta(s) = \langle B(s), (1,0,0) \rangle = \frac{1}{\kappa(s)} (\gamma'(s) \times \gamma''(s)) = \frac{1}{\kappa(s)} (x_2'(s) x_3''(s) - x_3'(s) x_2''(s))$.
Claim : $\alpha \leq \beta$
i.e., it is enough to prove that $\int_{\theta(0)}^{\theta(l)} \sin \theta d\theta \geq 0$.
From the equation for $\sin \theta(s)$ we obtain that
$d\theta(s) = \frac{\kappa(s) f'(s) - f(s) \kappa'(s)}{\kappa(s)\sqrt{\kappa(s)^2-f(s)^2}}$,
where $f(s) := x_2'(s) x_3''(s)-x_2''(s) x_3'(s) = \kappa(s) \langle B(s), e_1 \rangle$ and
$e_1:= (1,0,0)$.
On further simplification using Frenet-Serret formulae we get:
$\int_{\theta(0)}^{\theta(l)} \sin \theta(s) d\theta(s) = \int_0^l \frac{\tau(s)}{\kappa(s)} \frac{\langle B(s), e_1 \rangle \langle N(s), e_1 \rangle}{\sqrt{1-\langle B(s), e_1\rangle ^2}} ds$.
From here it is not clear to me that product of the integrand is always positive, or using integration by parts the integrand is always positive.


Is there only one recoverable item per planet?

Question: Whenever I launch a probe and find something on a planet, can I be absolutely sure there isn't nothing else and am I safe to leave orbit, or is there a chance that I might miss something else?
Answer: Here's a Scanning Guide which shows what you can get from each solar system and planet. There does not appear to ever be a case where there's more than one collectible per planet.
Answer: I believe that I collected nearly all items that you can find by scanning in my first plathrough, and there was never a case where a location contained more than one recoverable item.
Answer: You could also find this out by scanning the system and monitoring your % completed statistic.
Both the scanning guide and the personal experience methods could potentially be invalidated by some future DLC and this would provide a fall back method for you.


Can I swim in Iran with my girlfriend in bikini in the Caspian Sea?

Question: Is it possible if she is not Iranian to swim in her swimsuit or bikini in the sea?
She is not from Iran, she is European as me.
Is there some "police" that watches how people dress on the beach?
Comment: Try Azerbaijan. It is close and is much more secular. Many Iranians seem to enter easily. You'll need a visa though.
Answer: The simple answer is no! If you are thinking of the Caspian sea, there may not be any "police" watching you per se, but the people of the North are quite religious, and they police themselves. Bikinis are out of question. If her scarf flies away, they will give you friendly warnings. Don't be scared, they won't harm you, they just will tell you to keep your Hijab on or something like that.
In female only swimming areas she can swim by herself. These areas will be covered (not at top, just at the sides).
In any case, you should not swim in the Caspian sea, it is too polluted. Someone mentioned the Island of Kish being more relaxed. Yes, it is, but not to the point that she can swim with a bikini. The things people get away with in Kish are dancing in public, for example, or singing music that is considered illegal in other parts of Iran, and less policing on Hijab, etc. But again, bikinis are out of question.
Answer: According to the article in the Global Post it is possible as long as she's in a female only area and hence you can't be with her.  This information obviously is a little dated but the only area that is known for laxer regulations would be the island of Kish but that information is even more dated.
The most recent English news I was able to find is an increase in religious police so legally it is highly unlikely that regulations enforced for hijab would be laxer for bikinis.
Answer: Not in public, it is illegal and unaccepted. But in private places, such as private villas connected to sea, you are free.
Answer: According to local rules, women cannot swim in bikini in presence of men around. They are limited places for women to swim in the sea which men are not allowed to enter there. However as a Iranian i recommend you to visit here since there are many other wonderful stuff which will inspire you :)
Answer: It is possible to the extent that she runs the risk of, at best, being deported from the country. However, though the piece below is four years old, it suggests it might be possible to find private beaches on the Caspian:
http://mrzine.monthlyreview.org/2009/pourzal221209.html


How do you convert between SRM and Lovibond?

Question: For instance, if a grain is specified at 300 SRM, what is the equation to convert that to Lovibond? What is the equation to convert it back?
Answer: First, a little history.  Joseph Lovibond developed the Lovibond Scale in the 1860's, as a means of implementing quality control into beer production.  The Lovibond Scale works by having the user visually compare the shade of a substance to the nearest shade of colored disks.  Thus, determination of a color on the Lovibond Scale is a human estimation, rather than a scientific measurement.
SRM, or standard reference method, is a more recent method, which measures the absorption of a substance using a spectrophotometer.  The SRM was developed as an answer to the inaccuracies found in implementation of the Lovibond Scale.
The variance in measuring the color in °L has led different beer experts to develop different methods for converting between the two.  They are as follows:
Morey's Formula
SRM = 1.4922 * (W * L / V) ^ .6859
Daniels' Formula
SRM = (.2 * W * L / V) + 8.4
Mosher's Formula
SRM = (.3 * W * L / V) + 4.7
where
W = weight of malt (in lbs.)
L = color of malt (in °L)
V = volume of wort (in gal.)
Morey's formula gives a better approximation than the others, since it is a nonlinear conversion.  However, if you want simpler math, both Mosher and Daniels' methods are good approximations.
Beer color is usually measured in SRM.  Malts are often given in °L.  In practicality, the discrepancy makes sense.  I've never seen the inner workings at a maltster, but I imagine color is determined by holding a sample of grain next to a color palette and estimating it's darkness by comparison.  On the other hand, if someone wanted to know the darkness of beer, it wouldn't be difficult to run a specimen through a spectrophotometer, and measure it's absorbency at 430 nm.  It wouldn't be possible to run the malts through a spectrophotometer to find the SRM value, so °L is given instead, and human error is deemed acceptable.
Answer: From what I understand, the formula is:
°L = (SRM + 0.6) / 1.35
However, for all values light enough to be visibly different to the human eye, L=SRM.  They only start splitting when you get too dark to be visibly different without a spectrometer.
More Discussion / Source
Answer: There is a one to one correspondence as the SRM is the absorption at 430 nm and one can calculate the density at 430 nm of a Lovibond series 52 glass. The series 52 glasses are 'made up' of Lovibond R and Y glasses and the only people who can tell you what the composition of a series 52 glass are the people at The Tintometer Ltd and they used to do this in a leaflet they included with their instruments but it was demonstrated years ago that what they publish and what they sell are different. Using the published data we can calculate the absorption of the 52 series glasses ant 430 nm and compare to the SRM. Notice that nothing has been said about path or viewing light color both of which are important in the use of a Lovinbond tintometer. We do know that Miller and Stone adjusted the SRM multiplier to get °L and SRM to agree for the light beers they studied. Doing this the formula for SRM in terms of °L is SRM = °L + 0.04662*(°L)^2 and, going in the other direction °L = 0.808*SRM - 0.0083*SRM^2.


Why is IP to MAC address translation needed?

Question: On networks that use the TCP/IP protocol, why is the IP address translated into a MAC address?
I mean, when every device knows its IP, why further use the MAC address and not use the IP number as the address directly?
Answer: The main difference is that the IP address is used by IP which is a network protocol, and the MAC address is used by Ethernet a link layer ( 2 ) protocol. IP can run over different layer 2 protocols, so it doesn't include the "media access" functionality. You can run IP over Serial lines, Ethernet, DSl, ISDN, etc.
For example, DHCP provides dynamic IP addresses over Ethernet; the DHCP server needs the MAC address so it can assign an IP address to a PC.
Comment: +1 This is really the key issue. IP can run over different physical layers. Different physical layers require different physical addressing. MAC is the addressing used by Ethernet. And there are other Ethernet-based protocols that do not use IP including BootP, DUX, and IPX. The same physical Ethernet may also handle IPv4 and IPv6 simultaneously. But in the end, everything is delivered to a MAC on Ethernet.
Comment: Also important to understand that different times you will see different pieces. Your computer might get a packet from someone that it need to respond to and not know their MAC yet (the mac it see would be from the closest connecting device but that's not important) and need to arp or the reverse and Rarp. Or have the hostname but need the mac or ..... any number of different configs and protocol combos
Answer: Because TCP/IP is used over Ethernet, which uses MAC addresses.
Answer: They are for different layers in the network stack.  The MAC address is about the ethernet networking layer and the IP address is about, well the IP layer.  You can use IP over different transports like SLIP where there is not MAC because there is not ethernet.  You can also use protocol different than IP over ethernet if you wanted to.
If you think about it, if I dial in over a SLIP connection one IP packet could start with not MAC associated and then have a different MAC as it hops from one router to another as it is forwarded across the internet to the final IP destination.
Answer: I could answer your question with another question: You have a phone number, so why do you need a social security number/national identification number?
In other words, they serve different purposes. One is for identifying who you are (SSN, MAC), whereas the other is for defining how to reach you (phone number, IP address).
Further, it's possible to reach someone on the phone if they don't have a SSN (different country, for example) just like it's possible to reach someone on the Internet if they don't have a MAC (different type of link layer). Different link layers have different methods of identifying the destinations and sources of data packets.
Answer: Another way to look at this is to ask the reverse question: Why do we need IP addresses when we already have MAC addresses assigned to the hardware?
This was discussed in episode #29 of the Security Now podcast (HTML transcript).
With MAC addresses, our computers can directly communicate to a single machine. But, in order to know how to get there, we need to have our packets routed to it. It's impractical to have a global MAC address lookup table on each of our computers.
IP addresses, on the other hand, are hierarchically organized. Our computers can determine which IP addresses are on the LAN and the rest get sent to the LAN's gateway (e.g. a home router). After leaving the LAN, successive routers look at the IP address and route the packets on to another router. Eventually, a router which has the destination IP address on its LAN can identify the MAC address and send the packet to the destination computer.
Comment: MAC addresses are assigned by the hardware manufacturers and they have no idea where the network devices will be deployed nor in what configuration. There are lots of sites (such as [this one](http://www.coffer.com/mac_find/)) which allow you to identify the manufacturer of network cards by MAC address.
Comment: Why couldn't MAC addresses have been smart like the IP and have been implemented in such a way that it is hierarchically organized? Were they simply careless or is there a restriction which makes it impossible?
Comment: Good argument (and explanation thx !)
Answer: MAC addresses are guaranteed to be unique to the machine (the network card typically contains the MAC address) and unchanging, whereas you can redefine the IP address on any given machine.
Additionally, public IP addresses can be shared by a whole company using Network Address Translation, so public IP addresses are not guaranteed to have a one-to-one correspondence with a machine or person.
Answer: The IP is either defined in the OS or in the driver software for your network card, or given to you by your network, and can be changed dynamically. The MAC is an identifier for your network card, i.e. the hardware. There is not a 1-to-1 relationship between them.
Answer: Your question is a very legitimate one! The truth is that theoretically an IP only system can be created and function well without the need for MAC addressing system. The Data Link layer can be made to ignore any MAC filtering, leaving this functionality to the Network layer, to filter out according to its own IP address. Such systems existed in the past, such as networks based on linking devices through their UART (e.g. RS232) ports. Obviously, these networking interfaces did not carry MAC addresses or any other form of a unique, physical identity. However, it can make centrally assigning IP addresses, e.g. by a DHCP server, a nightmare, because there would be no way to make sure a device will not get more then the predetermined number of IP addresses allowed to each device. Also, it would be difficult to unambiguously and individually identify each device. The MAC address being unique, at least within a sub-network, serves as a unique identification token. Of course, individually assigning IP addresses to devices may solve the problem, but where large number of devices are present on a network, it becomes impractical to establish and maintain such a scheme.
Answer: You don't know the MAC address of stackoverflow.com because it's not in your local network.
MAC addresses are used as an identifier to the next hop in the chain.
The "IP" is just used in a protocol that has a hint of in what direction it should send your packet (which gateway to use next).
Answer: Because MAC addresses are needed for communication on the Link Layer (IP addresses are used one layer up on the IP layer).
Answer: Lets take a example
If you apply for job in company.. will they give you that without knowing your identity? certainly not.... so your actual identity proof is Mac address
and when you join the company they will give another employee ID  to communicate among that company employees.. this is IP address.
your employee ID can be modified if you change company but your actual info about you can never be changed (age,name,etc)....
Comment: Soooo I can change my actual Identity? :P But this is not really an answer, just a metaphor thats kind of weird and wrong
Answer: IP address are like ZIP codes or like country codes or an ISD codes, wchich simplifies the location address otherwise its is very difficult to sort all the MAC address or to identify the exact location else the computer will take hell of time identifying the address or keeping huge database of MAC address on local computer.


Passing ByteArray from flash (as3) to AMFPHP (2.0.1)

Question: i have a problem passing ByteArray from flash (as3) to amfphp to save an image.
With old version of amfphp, all worked in the past… now, with new version i have many problem.
I'm using version 2.0.1 and the first problem is that i have to do this, for access to my info:
function SaveAsJPEG($json)
    {
        $string =  json_encode($json);
        $obj = json_decode($string);

        $compressed = $obj->{'compressed'};
    }

in the past i wrote only:
function SaveAsJPEG($json)
    {
        $compressed = $json['compressed'];
    }

Anyway… now i can take all data (if i use " $json['compressed']" i receive an error) but i can't receive my ByteArray data.
From flash i write this:
var tempObj:Object = new Object();
tempObj["jpgStream "]=  createBitStream(myBitmmapData);  // return ByteArray
tempObj["compressed"] = false;
tempObj["dir"] = linkToSave;
tempObj["name"] = this.imageName;

So.. in my php class i receive all correct info, except "jpgStream" that seems "null".
Do you have any idea?
Answer: I think you get 'null' because of json_encode/decode. Try using

    $data = (array) $json;
    $compressed = $data['compressed'];

This may help http://www.silexlabs.org/amfphp/documentation/data-types/


cider from fresh apple juice, how to?

Question: I have bought a gallon of Apple juice fresh off the press. I was told to just leave it 'as is' in a container with an air lock/bubbler and that's all I need do.
I usually brew beer and this all seems too simple to me. Can it be that simple.
Comment: It's so simple that a cave man can do it (an probably did!)
Answer: You can certainly make great cider that way, but it's hit or miss.  I've made probably about 6-8 batches using natural fermentation and my record is about 50/50 great cider/vinegar.  A more reliable method is simply to pitch some yeast into it.  Almost any yeast will give good results, but I've made my best ciders with WY4766 cider yeast.
Comment: Thanks for the tip its too late now for the yeast this time. fingers crossed I guessed.
Answer: I just bottled 5 gallons of cider I made (my first batch last year was excellent, and strong).  I've definitely found cider to be easier than beer when it comes to the actual brewing.  That said, if you press the cider yourself and/or add in apple picking for 5 gallons worth of cider, you've got a bit of work to do.
The recipe I used was specified by the gallon, but I just scaled it up for 5 (with the exception of the yeast; I only pitch one vial/packet of yeast since they're supposed to be sufficient for [I believe] up to 6 gallons).  It's basically an apple wine that you then prime with sugar before bottling.  Everything non-apple derived I was able to get from my local brew shop.  Here's what I went with...

Use 1 gallon of unpasteurized cider in your primary fermenter.
Crush 1 Campden tablet and mix into the cider along with 1/2 tsp. of pectic enzyme, 1/2 tsp yeast energizer, 1/3 lb. sugar (I used brown sugar most recently).
Cover primary and let sit 24 hours.
Pitch yeast (a champagne yeast was recommended and worked well for me) and install airlock.
Stir daily (I wasn't always great about this, and it seemed fine with less frequent stirring).
Ferment until SG reaches 1.040 in around 3-5 days (I've been less adherent to this than the stirring and have had good results thus far).
Rack off to secondary leaving behind as much of the suspended pulp as possible.
Ferment until SG reaches 1.000 in around 3 weeks (again, I've been pretty loose on this timeline).
Rack off, prime with 1/4 c. sugar per gallon, and bottle.
Age a few months and then kick back with a cider.

The cider took longer than most of my beers for the bottle conditioning to take the hard edges off the flavors, but once it did it was very tasty.  The regular stirring bit has concerned me some (both in terms of possible infection and so many recipes advocating not aerating once fermentation is underway) but the cider I made was quite good.
Comment: Why all the stirring?
Comment: I'm not sure why it calls for all of the stirring, it's just what the recipe I have says.
Comment: I wouldn't risk contamination of stirring so often.  I might swirl the carboy, but even that might not be required.
Comment: I've done it both ways without a problem, but you're probably right about risking contamination.
Answer: Started doing hard ciders (apple, pear, hot pepper, peach, cactus pear, raspberry, pumpkin spice, about 60 gal total to date) last august and have learned a few key things. Using a juicer is just as good as using a traditional cider press. When you use the pressed stuff, you end up having the clarify the hard cider, while juiced apple cider is more clear. The taste between the two cider production methods is not significantly different. You can clarify pressed cider with gelatin just before bottling and it seems to taste just fine.
I've done apples, pears, cactus pears, and peaches and all produced juice with SG's in the 1.050 range that you need for a good hard cider. Even the Mussleman's pasteurized cider (don't use juice, it's too dilute and usually has sorbate added that will inhibit your yeast) from Wally world made a good tasting hard cider and base for some of my freaky brews (made a 1,000,000 scoville unit hot pepper cider that is great with chili or hot wings). Much higher than that SG and the alcohol tends to override the fruit flavor (adjusted my first pear SG to 1.070 and it was almost straight alcohol with no fruit taste). I found that using honey or sorghum to raise the SG for low SG apple juices worked great and gave so good character to the apple cider.
Yeast is important, but not as much with beer. The cheap cotes de blanc or premier cuvee ($0.75 per package) turns out as decent a cider as the more expensive cider yeast from west labs ($5.00 per tube). The primary difference will be in the carbonation during bottling. The red star tends to be more flat than the west labs, but this can be compensated by a larger priming sugar add before bottling). Wyeast 3068 makes a really good apple and pear cider with a little better flavor than the red star varieties, but it is more expensive (less than west labs or saf-brew), about $3 per activator pack. But the advantage here is that with so many more yeast cells, the fermentation takes significantly less time and the fruit flavor comes out faster during aging that with the red star.
As for a recipe, The simplest is for every 5 gallon batch (about 55 pounds of fruit juiced in juicer), add 6 crushed campden tabs to cider and let stand for at least 48 hours. Swirl the carboy hard to let the sulfur dioxide gas to escape, then add 2 Tbl of pectic enzyme, 3 Tbl of yeast nutrient (both have been dissolved into some of the cider) to the carboy and swirl to mix. Check the SG and if it isn't 1.050 add about 2 cups of sorghum or honey (that have been dissolved into some cider) to bring it up. Activate 1 package of cotes de blanc yeast in one cup of non-chlorinated water (Brita filtered or bottled distilled) that has 2 Tbl of sugar dissolved in it. Let the yeast/sugar water mixture stand for about 20 minutes until you see foam starting to form on top. Add the yeast to the carboy and swirl hard for about 5 minutes (a step that I didn't see in other online recipes and learned from my homebrewing soon-to-be-stepson) to introduce air into the cider. Airlock it and let stand out of the light at 65-70 degrees F until the bubble rate in the airlock is less than one per 15 seconds, or until the FG is less than 1.005 (this will be about a 6% abv). Rack into your keg (force carbonating) or priming bucket (traditional bottling). Use the priming sugar calculator (tastybrew.com has a good one) to calculate the amount of sugar to add to the priming bucket then bottle. Should take about 10-14 days to have a nicely carbonated bottled hard cider.
Answer: Your best bet is to pitch yeast into it instead of hoping for spontaneous fermentation.
Keep in mind that the sugars in apple juice are completely fermentable. Which means your end result will be a very dry beverage that will taste much like a dry white wine unless you do something to enhance the body. For a 5 gallon batch add 1-2 lbs of DME and a pound or less of 60L crystal malt. This will give you something that tastes much more like commercial cider.
Comment: you can pitch a low tolerance beer or cider yeast.
Answer: Apples are covered with yeast, and being ground up to make cider really gets that yeast mixed in well. The problem with going wild with cider is that sometimes bruised or wind falls are often also included which improves the flavor of the raw cider, but means some of the apples may have been infected with acetobactor, in which case as the alcohol is being produced by yeast, the acetobactor is just taking the slowly served thank you very much ethanol and turning it into vinegar.

cider from fresh apple juice, how to?


Why did Robert E. Lee stop offensive maneuvers?

Question: According to the Wikipedia article on the Battle of Gettysburg, after that battle Robert E. Lee stopped making offensive maneuvers and rather started reacting to Union offensives.  Is this 100% true, or were there other maneuvers?  Also, why did he stop making any offensive maneuvers after the battle?
Comment: He didn't start reacting to Union offensives then, or stop offensives after Gettysburg. His last offensive was on March 25, 1865.
Comment: upvoted, though I think OP was asking about major offensives
Answer: Gettysburg was pretty much a last ditch effort by Lee and Jefferson Davis to save the Confederacy or at least give it some credibility. There were different objectives that led to the attack in the first place. For one thing, the war in the West was going against the Confederates, and if the West fell, and more importantly access to the Mississippi River, then opportunities for resupply would be lost.
Another consideration was that by obtaining a victory at Gettysburg, Lee and his army would be poised to threaten Philadelphia and New York. They would also have access to fresh farmland and supplies that they could take from the North, rather than being so dependent on the already depleted resources of Virginia and the rest of the South. By threatening two of the largest Northern cities, Lee would have been in a position to petition for a peace plan that would have led to the true and permanent separation of the US into two different countries.
Lastly, by obtaining a major victory in the North, Lee and Davis hoped to draw attention to their cause from external sources. More specifically, the French were close to lending support for the Confederates, and a major victory in the North may well have been enough to encourage the French to throw their gauntlet into the ring. If that had happened, then the French navy would have been able to help relieve the blockades that had been set up by the Northern navy. This in turn would have allowed the Confederates to start receiving supplies from other sources.
The South had reached a point where they recognized that their chances were rapidly diminishing. After losing at Gettysburg, their fate was sealed and it became just a matter of time before the inevitable happened. From that point on, they were mostly hoping to outlast the North. They didn't have the resources or manpower to launch any more offensives, so they could only hope that time would allow them to persuade other foreign powers to come to their aid.
I had some assistance from this source, but most of my answer is based on extensive reading on the American Civil War.
Answer: In June 1864, as General (later President) Ulysses S. Grant was leading the Army of the Potomac against Richmond, Lee DID order an offensive by one of his better remaining subordinates, General Jubal Early.
http://en.wikipedia.org/wiki/Jubal_Early
Early debouched from the Shenandoah Valley, cut a swath across Maryland and southern Pennsylvania, and threatened Washington.
But the South had few punches left after Gettysburg. They had reached their "high water mark" there, losing over 20,000 men, including several generals killed or wounded to no purpose. This, after the loss of 13,000, including that of General "Stonewall Jackson, at the previous battle of Chancellorsville.
The day after Gettysburg, General Ulysses S. Grant had captured Vicksburg, Mississippi, establishing Union control of the whole Mississippi river, cutting off Texas, Louisiana and Arkansas, plus the re-occupied states of Mississippi and Tennessee from the Confederacy. In effect, the latter was now fighting with only six states out of eleven.
These victories soon led to the promotion of Grant to the de facto commander Army of the Potomac, who allowed the Confederates few openings to counter invade the north. The previous invasions had followed Union defeats at the Second Battle of Bull Run (ending at Antietam), and the battle of Chancellorsville, prior to Gettysburg.
Comment: Grant was not the commander of the Army of the Potomac; that was Meade, hero of Gettysburg.  Grant was general in chief, and traveled with the Army of the Potomac, possibly to keep a close eye on Meade.
Comment: @DavidThornley: I added the term "de facto commander," because technically you're right.
Answer: Wikipedia is not correct.  In the Fall of 1863, after Meade sent troops West to match the one's Lee sent west to defeat the Army of the Cumberland at Chickamauga, Lee launched a third offensive campaign that ended in the battle of Bristoe Station.  This was a defeat for Lee.
After this, Lee retired behind the Rappahannock River but left a brigade north of the river in an entrenchment at Rappahannock Station.  Meade's forces make a rapid attack and crushed the bridgehead, capturing most of the brigade.  Lee had to retire behind the Rapidan River.
Comment: I think for the defeat at Bristoe station, AP Hill is the one to blame, since he attacked too early without orders.


When introducing a new interface to managers, do you even show the options that you eliminated?

Question: For a same interface, should I show to managers/clients all the options that came to my mind (even the one that are not appropriated) or should I only show the best one or the one I suggest?
Answer: When I go for a client visit to show my designs or Wireframes, I start by pitching the design which I created first, then the one which I create second last and then finally I show the one which I really liked.
When i show the one which I created 1st and 2nd last, I try to make them understand what mistakes I have made so that i can convince them that there are a lot of thought behind the design I am pitching in. When i arrive at the last one, they are almost confident of my creativity and thought process and they try to understand how I overcame the mistake which I did in the last 2 options. They gives a lot of room to discuss around the same topic rather than discussing about their preferences.
It helps me bring USER perspective to the design table when i am explaining my method of design.
Answer: Usually, for me having a couple of good options is better than just one. If the team rejects one option you still have another one to show. Having multiple options available can help better communicate the pros and cons of the options. Also since most projects have resource constraint having different options can help managers/clients make a decision that works for their constraint (e.g. option 1 is more intuitive but takes twice as much to implement, option 2 is less intuitive but takes half time to implement). The team may go for option 2 knowing that the feature is very rarely used and they better spend time on some other feature that is more important. Of course, you might want to start with showing UI option that in your opinion provides the best UX and then move on to the next options. Since you don't want the teams to always pick "easy to implement" option and sacrifice UX :)
Answer: I agree with Anna: it's important to present options.
Coming from agencyland, clients want to see that you've done the legwork. Don't present all your ideas, just the top three or four. Internal reviews with your team will help to select the leading options.
It's also important to honestly list the pros and cons of each option. This will help your client/manager choose one.
Finally, have an opinion. Make the case for what you think is the best option, but be prepared to gracefully concede should your client choose a different one.
Answer: I always was trained and show two options, no matter the project. It gets into human psychology and saving time more so than creating the "best" design. People tend to want to compare something they are critiquing to another work. If you have just your one option, maybe they'll compare it to your competitor or another work that you can't go ahead and use. Comparing to another option you've created means they can mix and match portions from both.
That said, there is some value to keeping any other versions you created "in your back pocket". If the managers start suggesting something that you've already tried and failed with, showing them that attempt and ruling it out immediately can save a lot of time and aggravation on your part.
No matter what/how many you present, here are some things to keep in mind:

Be able to defend your design decisions, even if it's just, "I wanted to explore another option, but I don't think it works as well."
Have many different talking points to show how much thought went into it.
Have a preference. Even though they may want to play designer (and you can let them to a certain extent), you're the design professional and having an opinion shows you care and are doing the best job you can.
Answer: I think it is critical to share the design rationale with the client.
I like to create a design document which shows the entire history of my design process including all discarded ideas/sketches. I then walk the client through my thinking and why I made the choices that i did. In this way we can identify points where i may have slightly gone of track. I find this an essential part of my design exploration and iteration.
To be honest some clients find this difficult at first but they soon get it. The point is that you need to educate people to understand that design is a process of choosing one solution from many millions of possible ones. Many people think that there is one "right" answer and usually that is not the case! By showing the design rationale you get allow them to understand that and gradually move towards a shared understanding of the problem space and then whereabouts in the "solution space" your ideal final design might lie.


Partial differential equations in "pure mathematics"

Question: One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in  Wikipedia  follow this pattern. I can't help wondering whether there are PDE's arising "naturally" in "pure" mathematics like Geometry, Topology, etc.?  Of course, I can always write an arbitrary surface as one or more PDE's, but they don't seem to drive much research in the subject, afaik. I understand that PDE's originated in Physics, but hasn't the subject grown away from physical models into more abstract examples?
Comment: If you haven't seen PDEs in geometry then you haven't done enough (differential) geometry!
Comment: @Marek: It would be more effective if you flagged the question with that request, rather than leaving it as a comment, since Moderators are the only ones who can CW-ify a question. Cheers.
Comment: I think this should be CW.
Comment: @Willie: ah, I forgot about that. Thanks.
Comment: Related thread: http://mathoverflow.net/questions/74183/applications-of-pde-in-mathematical-subjects-other-than-geometry-topology
Comment: Concerning elliptic differential equations on geometry and topology and the Atiyah-Singer theorem, you may be interested in my recent answer at http://math.stackexchange.com/a/252225/4058
Answer: Some more examples from geometry and topology:
The Atiyah-Singer Index Theorem connects analysis of elliptic differential operators, with the topology of smooth compact manifolds. There are many approaches to proving the Index Theorem, but there is one in particular which does so via studying the heat equation on the compact manifold.
On non-compact Riemannian manifolds, an uniform geometric structure at infinity has deep connections to the spectrum of the associated (conformal) Laplacian, this leads to deep connections with scattering properties of wave equations evolving on such manifolds.
The topology of four dimensional manifolds is connected to their geometries manifesting in Yang-Mills theory, which gives rise to very interesting non-linear elliptic partial differential equations. There's quite a nice book about this whole field.
The study of over-determined systems of partial differential equations (such as this one) can trace a not-insignificant-part of its roots to E. Cartan's program to understand differential geometry using the moving frames method.
On the other direction, PDEs can give rise to interesting problems in other fields. For example, the fundamental solution of a linear constant coefficient hyperbolic partial differential equation has deep connections to topology and geometry of algebraic varieties. And the so-called strong Huygen's principle for the wave equation is closely connected to the geometry of homogeneous spaces.
Answer: I believe that one of the examples can be Ricci flow which are parabolic PDEs describing the deformation of metric for Riemann manifold. The new result is e.g. the solution of Thurston's geometrization conjecture which Perelman did using Ricci flow.
In general as you have mentioned, many PDEs appears from the applied science: physics, finance etc. though I am not sure that PDEs related to Markov processes appeared only after considerations of applied problems.
Finally, there is a theory of Harmonic functions which was based on the Laplace PDE $\Delta u = 0$. These functions are generalizations of linear functions on the real line for the multidimensional case and they inherit many of nice properties of linear 1-dim functions. Although this equation appeared from the physical problem, it has been also important for the needs of the pure mathematics.
Answer: Don't forget the minimal surface equation, one of many PDE's arising in differential geometry as alluded to by Zhen Lin. In his Comprehensive Introduction to Differential Geometry, Spivak includes a chapter on PDE's under the title And Now A Brief Message From Our Sponsor - a nice way of highlighting the many interactions between the two subjects.
Answer: Calabi conjecture stated that on compact Kähler manifold there exists a unique Kähler metric with prescribed Ricci form. This problem can be reformulated in terms of a (non-linear) PDE and has been settled by Shing-Tung Yau who indeed constructed a solution to this PDE (the uniqueness was settled prior to that by Calabi himself).
Answer: The natural linear PDEs that arise in physical situations also arise in "pure math" contexts, for similar reasons, namely, calculus/analysis with symmetry constraints. In my own direct experience:
The spectral theory of automorphic forms, in the simplest/smallest examples exemplified in Iwaniec' "Spectral methods...", and certainly on rank-one spaces, characterizes automorphic forms/functions as being essentially eigenfunctions for a suitable (invariant) Laplace-Beltrami operator. Regularity properties of elliptic operators are used almost without mention. Fundamental solutions, or Green's functions, play the expected geometric-analysis roles in basic estimates on automorphic forms.
Selberg's-and-others' "trace formula(s)" can be viewed as studies of the trace of the resolvent of this Laplace-Beltrami operator on suitable subspaces of $L^2$.
Heat-kernel ideas have been used by many people to prove essential results (Mueller on trace-class of discrete spectrum, Lang-Jorgensen projects, etc.)
Scattering-theory and wave-equation ideas were used by Fadeev and his collaborators, by Lax-Phillips, and others, to study Eisenstein series and the continuous spectrum.
For that matter, the classical linear PDE on Euclidean spaces, Laplace's, heat, wave equation, or one products of circles, or on spheres, seems to me to be primordial analysis of Euclidean geometry, whether or not one is interested in physics or mechanics. After all, I suspect that the aspects of all these things we put in these categories are substantially manifestations of human perceptions and psychology rather than something innately called "physics-or-mechanics", as opposed to "pure mathematics".
Answer: The $\overline{\partial}$-equation can be considered a PDE on a complex maifold. The equation is $\overline{\partial}u = f$ where $u$ and $f$ are complex differential forms, and $f$ is $\overline{\partial}$-closed (i.e. $\overline{\partial}f = 0$). This equation is directly related to the Dolbeault cohomology of the manifold.
In Hörmander's An Introduction to Complex Analysis in Several Variables, he proves Cartan's Theorem B which states that, for a Stein manifold, the $\overline{\partial}$-equation always has a solution. The method he uses has very little to do with complex analysis (at least at first glance), but rather uses standard techniques from PDE's.
Comment: The method does not use complex analysis but the result has a lot of applications in complex analysis (although there are at least a couple of alternatives to this PDE-based approach in several complex variables, as far as I understand).
Answer: One has the classical Gauss-Bonnet Theorem:
If $M$ is a compact surface with Riemannian metric $g$, and if $K_g$ is the sectional curvature of $g$, then $\int_M K_g dvol = 2\pi\chi(M)$.
One can view this as saying that a necessary condition in order for a function $K:M\rightarrow\mathbb{R}$ to be the sectional curvature of some metric on $M$ is that it integrate to $2\pi\chi(M)$.  A natural question arises:  Is it sufficient?
Well, since $K$ is built out of the second partial derivatives of $g$, this amounts to proving the existence or nonexistence of a solution to a PDE.
The answer, as it turns out, is yes:  The necessary condition is also sufficient.  I believe this result is due to Kazdan and Warner but I'm not sure.  (I'll look it up tomorrow morning, if I remember to).
Answer: Another example is the de Rham complex on a manifold, or even simply on $\mathbb{R}^n$. One way to look at it is as a natural set of partial differential equations, in various dimensions; for instance finding closed 1-forms is equivalent to solving the differential equation $\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y} = 0$.
To quote Bott & Tu from "Differential Forms in Algebraic Topology", p. 15: "The de Rham complex may be viewed as a God-given set of differential equations".


Untrusted signature on java app for downloading youtube, harmful?

Question: If you visit most of the "online youtube download" pages, you will get this message:

Why all these sites come up with this message? Is it a worm, or just an unharmfull app?
Answer: Well, there's no way to know that for sure without looking into the application...
Although, it does seem incredibly suspect.
This message lets you know that your browser is trying to launch an applet in trusted mode, which means that the Java code will have full control over your system.
You probably shouldn't accept, unless you know (and trust) the publisher.
...Which is exactly what the warning says.
Answer: To download a YouTube video, you need to make a connection to Youtube masquerading as their Flash-based video player. Flash has some security controls that stop you doing that from inside another Flash file that isn't their video player, and connections from XMLHttpRequest, WebSocket, plain Java applets and Silverlight are deliberately limited in ways that also prevent the right kind of connection being made.
So to successfully download a video, you need to be able to make an arbitrary network connection. This is something that only a full native application, or a Java application with similar permissions, can do. This means there are two ways of providing a YouTube download service:

make connections from software on your own server to download the video;
provide the user with a native app or similarly-permissioned Java app to do it from their desktop.

Some services have provided (1) in the past, but it makes it fairly obvious to YouTube that something's up when loads of connections with an atypical viewing profile come in from a small range of IP addresses. These services tend to get blocked.
So most sites are currently using native-app downloads and Java apps instead. In the Java case that means you have to grant the applet suitable permissions before it will run. There is no guarantee that there isn't malware (most likely: a pay-per-install toolbar) in the applet, so this is exactly as dangerous as downloading and running a .EXE.
Up to you whether you think running it is a chance worth taking. I've always tested these things from a scratch VM, for what it's worth.
The message complaining about the digital signature is incidental. An elevation prompt from a properly-signed applet that does not have this warning, could have malware lurking behind it just the same. All the signature would mean would be that you know who to blame after getting infected. Many sites use unsigned applets because code-signing/time-stamping for Java is historically a bit of a pain.
Answer: The message does not state anything about whether the applet is harmful or not. Java applets, by default, are severely constrained in what they can do, in particular what external hosts they may connect to. This is so that Java applets can be considered as harmless, as in "declawed".
A Java applet may request extended privileges, but the JVM will grant them only if the applet is signed. Moreover, if the signature key is not part of a digital certificate which links to the root certificates that the JVM knows intrinsically, then it will display the message you see.
So the message means that:

the applet is signed, so it probably wants to do some operations that an unsigned Java applet is not permitted to;
the applet developer did not bother buy a 100$ certificate which would have been blindly accepted by the JVM (whether the developer was poor, lazy, or wanted to remain anonymous, is not said);
if you click "run" then the applet may potentially run whatever code it wants on your machine.

So this is like if a masked individual knocked at your door and requested very politely that you give him a copy of your door keys. This does not automatically means that the said individual has burglary in mind -- maybe he is the Masked Plumber, who took an Oath to fix faucets of ordinary citizens at no charge -- but you could be entitled to feel a tiny bit suspicious about the proceedings. If you click "run", you are on your own, and you have been warned.
Answer: In order for an applet to access the local file system (and more permissions) , it needs to be signed and your Java Security Settings need to be at Medium (Lowest leve). When an applet is signed , such a window pops up asking for permissions . Certificated Applets (Virus - Free for sure) run even at the Highest security level.


Whitelisting Commenters

Question: I have a blog that gets the usual spam, which Akismet is pretty good with. Not good enough for me to turn off moderation just yet.
I am wondering is there a way to "whitelist" a list of readers so their comments can skip moderation and just get posted?
Answer: In
Settings > Discussion
Uncheck An administrator must always approve the comment
Check Comment author must have a previously approved comment
This way comments from people (identified by combination of name, email and site) who have previously approved comments will not require moderation. Rest of comments will.
Answer: I find it better to use a different combination
WP Spam Free - stops most automated spam
hxxp://www.polepositionmarketing.com/library/wp-spamfree/
Moderate first time commenters - stops first time human spammers
For the odd person who has a problem with WP Spam Free (so far 2 people from 1000s) I also have a backup plan http://wordpress.org/extend/plugins/comment-whitelist/
I also blacklist spam pingback domains so I don't get to see their pingbacks, and on rare occasions I also do the same with an email address or IP addresses - for instance recently there was a whole datacenter with multiple servers having a conversations with themselves, so I just blocked based on the first 2 numbers in the IP address which were common to the servers and I made sure were assigned to the one datacenter.

Whitelisting Commenters


Why abstract manifolds?

Question: If we can use Whitney embedding to smoothly embed every manifold into Euclidean space, then why do we bother studying abstract manifolds, instead of their embeddings in $\mathbb{R}^n$? A vague explanation I have heard is that from this abstract viewpoint, we gain understanding into the intrinsic behavior of the manifold, without knowing anything about the ambient space in which it can be embedded. Can anyone give some examples of this, or any other reason why abstraction is necessary?
Comment: If you insist that a manifold has to be embedded in $\mathbb R^n$, then you need to show that whatever property you are interested in is independent of that embedding.
Comment: By the way, PL manifolds DO come embedded in $\mathbb{R}^n$. You need an embedding on $\mathbb{R}^n$ to define a polyhedron. So this is a feature of the smooth and topological categories.
Answer: The same reason we study abstract groups instead of their embeddings into symmetry groups. If we know some properties of a manifold $M$ which are intrinsic and don't depend on embeddings, and we later encounter that manifold $M$ somewhere else in a different embedding than the one we're used to, then we automatically know a list of things we can say about this new copy of $M$. The point is that the study of things like groups and manifolds naturally separates into the study of their abstract structure and the study of their concrete representations, and failing to make this separation is unnecessarily confusing.
For example, topological and smooth manifolds don't have a notion of length of paths or volume: these things are dependent on an embedding into $\mathbb{R}^n$, so if I meet a manifold in a different embedding I can't assume that things like length and volume are the same, and I will only get confused if I do assume this. I can assume, however, that things like the number of connected components, the homotopy groups, the cohomology, etc. are the same, so whenever I meet a manifold whose cohomology groups I know I can always apply that information.
I wouldn't say that the abstraction is "necessary." Nothing in mathematics is necessary. We make the definitions we make so we can more easily understand certain mathematical phenomena, and the phenomena we are interested in with regard to manifolds are easier to understand if we define abstract manifolds first and then worry about their embeddings later.
Comment: Or imagine graphs as being defined as 1-dimensional polyhedra in $\mathbb R^3$.
Comment: The first sentence of this answer is a very nice analogy - kudos.
Answer: The canonical example of the value of the intrinsic viewpoint is probably General Relativity. How would one make physical sense of a 4-dimensional curved space-time if one had to embed it in a higher-dimensional Euclidean space? What would you embed the universe in? It's (by definition) everything.
Answer: A practical point to complement the answers already given: A lot of things can relatively easily be described as manifolds, e.g. as quotient spaces of other manifolds, but their embedding in a higher-dimensional Euclidean space is more difficult to think about or imagine or even to come up with. For instance, the Klein bottle is easily described as a square with certain edge identifications, and projective space is easily described as the set of all directions, and it's rather immediate that these are manifolds, and yet I wouldn't know off the top of my head how to embed either of them in a higher-dimensional Euclidean space. If the theory of manifolds were built on embeddings, you'd first have to figure out how to embed these things before you could start investigating them.
Answer: Many manifolds don't have a nice, "natural" embedding into a Euclidean space.  For example, the usual definition of projective spaces is not as a submanifold of $\mathbb R^n$.  Using the abstract definition, it is easy to see that a projective space is a manifold.  However, it is very non-trivial to find an embedding of it into a Euclidean space and I have never heard of such an embedding being useful.
Comment: Here is such an embedding: $\{ xx^\top : x\in\mathbb{R}^n \textrm{ and } \|x\| = 1 \}$. This is a subset of the symmetric matrices of size $n \times n$. The matrix $xx^\top$ is the orthogonal projector from $\mathbb{R}^n$ to the line spanned by $x$. This establishes a one-to-one map $xx^\top \mapsto \mathrm{range}(xx^\top)$ from that set to real projective space. With the usual structures (embedded for one, the usual charts for the other), the bijection is a diffeomorphism. With the usual Riemannian metrics, it's even an isometry up to a constant.
Answer: Semi-serious: You don't spend time with proving the Whitney embedding theorem although it is a largely useless for any of your purposes.
Serious: You don't know whether such an embedding exists if you impose further structure onto the manifold (such as a metric tensor) and demand a structure-preserving embedding into $\mathbb R^n$. Vice versa, in case you generalize the concept of manifold (say, singular manifolds), it is again unclear whether such an embedding exists or has any meaning, whereas the instrinsic point of view appears much more natural and allows for convenient mathematics.
Comment: nash embedding theorem
Answer: If you restricted the definition of manifolds to be "submanifolds of Euclidean space", you'd run into the problem that sometimes you want to do calculus on an object -- natural objects like homogeneous spaces, Grassmann manifolds, cut-and-paste objects like the "pac man" torus and such.  But the formal restrictions on usage of manifolds would mean you would have to embed the manifold in Euclidean space before you could apply the machinery of calculus to it.  In that sense the submanifold restiction is like a type of formal red-tape that gets in the way of understanding.  Abstract manifolds dispense with that.
Answer: As noted by Yuan, I think, the main reason is intrinsic properties of manifolds. A very nice result related to this is Theorema Egregium of Gauss. There are quantities which can be measured by the people of the manifold without passing to another realms. When you embed the manifold into another space, you also add extra structure to your manifold.
A second factor, I think, is the definition of manifold itself. A manifold is an equivalence classes of atlases. We then think it as a set of points in our head. More precisely, what makes $S^1$ $S^1$ is not the set of points of $S^1$ but the atlas you abstractly consider. Then we say that the point set $S^1$ is a manifold with that atlas since $S^1$ admits that atlas. Since we have an abstract theory that work for our atlas, results are independent of the uniformization.

Why abstract manifolds?


How to set up a virtual machine with VirtualBox

Question: I have decided to set up a virtual console on my laptop to be able to test stuff like other distros. I have never attempted such things before so I'm a bit hesitant and therefore I need a complete step by step guide for such an operation using the virtualbox.deb I have downloaded. Any other tips is also welcome.
Comment: You may also find these answers helpful: on [Unity](http://askubuntu.com/questions/17937/how-can-i-run-unity-in-a-virtual-machine/18556#18556), on [Guest Additions](http://askubuntu.com/questions/22743/how-do-i-install-guest-additions-in-virtual-box/22745#22745), on [Virtual Box PUEL from Oracle repository](http://askubuntu.com/questions/41478/how-do-i-install-the-closed-source-version-of-virtualbox/41487#41487)
Answer: To begin with, you can simply double-click the DEB file you've downloaded to open the Software Center. Then simply click the 'Install' button to install the package:

Once the package is installed, press the super key (typically a button on your keyboard with the Windows logo) and begin typing 'VirtualBox'. When the VirtualBox icon appears, click it to start the application:

At this point you will be presented with the VirtualBox main window:

     * Click for larger image
Note that in the screenshot above, I already have a number of virtual machines - since you are just starting with VirtualBox, you won't have any yet. To create one, click on the "New" button in the toolbar. You will be presented with the "New Virtual Machine Wizard":

     * Click for larger image
The wizard will begin by asking you for a name and OS Type for your virtual machine. You may enter whatever name you like - what is entered here will be displayed in the main window later when you go to start your virtual machine. OS Type consists of a list of operating systems and specific versions / distros underneath. For example, if you are giving Fedora a try, select Linux in the OS drop-down and then Fedora in the version drop-down.
The next step asks you for the amount of memory to assign to the virtual machine. In most cases, VirtualBox picks a sensible default based on the information you provided in the previous page and the amount of memory you have installed. Just accept the value and click "Next".
The next step asks what virtual hard disk should be used for your virtual machine. Since you don't have any virtual hard disks yet, leave the "Create New Hard Disk" option selected and click "Next". You will then be guided through the process of creating a new virtual hard disk.
When creating the virtual hard disk, you will first be asked whether you want to create a dynamically expanding hard disk or a fixed size one. Since the fixed size disk will occupy a huge amount of space, I would suggest going with the "Dynamically Resizing Hard Disk" option. The next step is to give it a name (you can just leave that at its default) and select its size. You will need to check the website of your OS to find out the minimum amount of space required. Generally, 10GB should be enough for most Linux distributions while still providing some room to install applications, etc. Once you have finished with the "Create New Virtual Disk" wizard, you will be returned to the "New Virtual Machine Wizard". Click "Next" and then "Finish" and you're done - you've just created a virtual machine!
We're not ready to boot it quite yet though. We need to install the operating system on it. I'll assume that you want to install a Linux OS using an ISO CD image. You'll likely need to download the 32-bit version of the OS you want to try (look for a "32-bit" or "i386 / i586" option). Once you have the ISO file, click your new virtual machine on the left in the main window and click the 'Storage' section on the right:

Click the CD icon in the Storage Tree and then click the smaller CD icon on the right:

Select "Choose a Virtual CD/DVD Disk File" and browse to the location of the ISO file that you downloaded earlier. Now you can click OK and dismiss all of the dialogs.
Your new virtual machine is ready to boot! Simply double-click its entry in the main window on the left and it will (hopefully) start:

     * Click for larger image
You'll have to wait a minute or two for your new OS's installation menu / installer to load. To interact with your virtual machine, simply click inside the window. VirtualBox will display a dialog explaining how capturing the mouse works.
Hopefully that's enough to get you started.
Comment: thanks a lot dude. Are you a hacker or something?
Comment: You should have mentioned that VirtualBox is aavailable from the official repositories too: `virtualbox-ose`. Installing `.deb` files from third-party websites always involves a risk. (either in terms of safety, otherwise compatibility or stability)
Comment: @Mysterio: No, just a frequent VirtualBox user :)
Comment: @Lekensteyn: the .deb was from the official site. Thanks for the info
Answer: In addition to the installation instructions already provided, a couple of tips on using VirtualBox, especially one on file sharing between the host and guest machines, can be found in a series of blog entries starting at: http://linuxnorth.wordpress.com/2011/05/31/entering-a-virtual-world/   The first blog entry ("Entering a virtual world") discusses installation of VirtualBox directly from the Ubuntu Software Centre.  The two subsequent posts ("Virtually in control" and "Opening a window to the virtual world") are essentially tweaks for using the virtual machine.


Is there 28 hole front/rear drum brake hub

Question: No time research this now, so has there ever been made a 28 hole front or rear drum brake hub? I know that Sachs and Sturmey-Archer have done a lot of drum brakes. Neither to my knowledge have such a hub in production currently. The reason for asking this is that I have 2010 version of Kona Band Wagon and I'd like to move to using drum brakes, but also retain the current rims that are 28 hole rims.
Comment: You do realise that it's not likely to be much cheaper unless you rebuild the wheels yourself?
Answer: I've seen 36 and 48 hole brakes, but not 28. 28 strikes me as an unlikely number because hub gears/brakes are not usually made for especially lightweight bikes. Harris Cyclery only have 32 hole and 36 hole in their parts list, and I can't really make enough sense of the Shimano web site to really be clear on the options. Note that Shimano make "roller brakes" rather than drums, and those clip onto the side of the hub. Sturmey Archer drums all seem to be 36H.
If you are willing to really work at building the wheel you could cross-lace a 36H hub into a 28H rim, but you wouldn't get an especially strong wheel and it would take a lot of work (I expect you'd have to do it experimentally and that really means having a range of spokes sizes available on a "pay for what you use" basis).
Answer: Yes there have been 28 hole rear drum brake hub by STURMEY ARCHER. you can only get used, which means you have to rebuild it yourself. Plenty of them on ebay such as this one http://www.ebay.com/itm/ws/eBayISAPI.dll?ViewItem&item=220912258562&ssPageName=ADME:B:BOC:US:1123
Comment: Amazing. Not quite useful for the original purpose, but such a thing actually exists and you found it, so +1 from me.
Answer: This depends on where you live, however, in the UK the only widely available hub brakes are the Shimano Nexus series. These are 36 hole and can be retro-fitted to an existing bike designed for rim brakes.
Given that you will be needing spokes as well as the hubs, consider getting some rims that will work with your hub brakes, i.e. not needing the surface for the rim-brake. Have fun building them up or pay the small fee that a professional wheel builder will charge.
Visit a good shop and go out on a bike that does have Shimano Nexus - try before you buy. They may not be what you are looking for - with performance benefit in the wet but not with the feel that you are used to.
Generally 36H is a good idea for wheels that have the stopping gubbins in the centre as the forces are different to that of a rim-brake wheel - you are decelerating the centre and exposing the spokes to more force.
By building a new set of wheels you will be able to keep the old wheels - or find a friend to take ownership of them - and not have a bunch of left over hubs 'n' spokes kicking around.
Answer: Shimano actually made a 28H drum brake (model number looks to be Y.L 802-60 shimano m system rotatex).
I have what looks to be a virtually new one as part of a 16" x 1.75" rim ... I'm actually trying to "acquire" a 20" x 1.75" 36H version if anyone is interested in an exchange.


How can I identify which file/function will be used to respond to a particular URL?

Question: I'm a Drupalista who occasionally works in WordPress so this might be obvious :) I was helping a colleague debug some AJAX behaviour, and having got the URL of the callback from Firebug, I wasn't sure how to translate that into a file and function server-side. Can someone enlighten?! TIA
Comment: Can you please tell us the link you're trying to debug?
Comment: @Brady I'm looking for a general solution for working out the file and function from the URL (maybe WP just doesn't work that way). I know exactly how I'd do it in Drupal, and maybe that's leading me to assume things work similarly in WP.
Answer: WP doesn't quite work that way. How WP does work is a long and involved discussion, but here's the short version.
(BTW, AJAX is a whole other thing entirely. There's a whole system in WP just for doing AJAX requests easily, and most people are doing-it-wrong. If your question is AJAX specific, none of the below may apply.)
The first thing to understand is the rewrite system. WordPress uses a class called WP_Rewrite which essentially implements rewrite rules similar to how you'd do it in an .htaccess file. It uses regular expressions to convert "pretty" URLs into query variables.
For example, http://example.com/author/otto would be converted into http://example.com/index.php?author_name=otto internally. Similarly, http://example.com/2011/08/04 would be converted into http://example.com/index.php?year=2011&monthnum=08&day=04.
There's a rather large set of these rules (50+) that are generated to handle these URL cases, and what these rules actually are depends heavily on your settings, what custom post types you're using, etc. Plus, themes and plugins and such can adjust them too.
Those query variables are standardized and named in the WP_Query system, which controls what is being queried for in the database. So that year/monthnum/day thing gets read and turned into $wp_query->year and $wp_query->monthnum and so on. This works based on a whitelist system, so the variables have to be defined beforehand by either the core code or plugins/themes adjusting them.
Anyway, the main WP_Query gets initialized and populated by the WP_Rewrite system. When the query runs, it determines what it's pulling from the database based on what variables are set. It sets a bunch of flags, such as is_author and is_archive and is_home and is_single and similar, to denote various types of things. In broad terms, these are used to define the Template Hierarchy.
After this is done, the template loader is called to determine what part of the theme to display. Using these is_* rules and a series of template determination functions, one of the theme's template files is chosen and then called upon to display the page. The template is expected to implement something called The Loop, which basically pulls the posts from the main $wp_query global and displays them properly, according to however it wants to do that.
So it's not really a matter of files and functions, it's a matter of procedures based on variable inputs. All calls to WordPress basically take the same sequence of events:

The URL is parsed into query variables.
The main Query is executed based on those variables.
The template hierarchy determines what theme file to use.
The theme file displays the page.

Generally, if you want to change things on existing pages, you either change the proper file in the theme (or make a new one based on the Theme Hierarchy), or you create a new "type" of thing (such as a Custom Post Type) which can automatically adjust both the query variables and rewrite rules appropriately, or if you're doing something totally custom then you do-it-yourself and modify the rewrite rules directly.
The point being that in order to modify things in the easiest way, you have to really stop thinking about connecting URLs to functions, but instead start thinking about "what type of things am I using, and what do I want them to actually do and how do I want them to live inside the system?"
More specific questions will get you more specific answers. :)
Comment: Thanks so much for such a comprehensive answer! So taking Milo's and your answer together, when doing a normal page request I should think in terms of what's being displayed and The Loop; when dealing with an AJAX call (that's done properly) I can check which functions are bound to the relevant action. Does that sound about right?
Answer: The problem is that many people do ajax their own way rather than using WordPress provided ajax.
If it's done the proper WordPress way, the js will call /wp-admin/admin-ajax.php via an ajaxurl var and pass an action, which will map to an add_action('wp_ajax_{$action}', 'some_function'); somewhere in the theme's functions.php or a plugin, which will call some_function when the action matches.
Comment: Thanks Milo, that's exactly what I was looking for! In my case the plugin wasn't calling that URL (you _are_ saying that the JS itself calls that URL - it doesn't get rewritten or anything?) Also I've put a comment under Otto's answer and would appreciate your feedback there as well :)


What does beating eggs actually do (chemically speaking)?

Question: When I look at a lot of recipes, any eggs usually have to be beaten before they're added. If it's all going to be mixed anyway (and well, in dough for example) is it really necessary?
Answer: The main purpose of beating an egg is to "denature" the protein within the egg.  Proteins are long chains of amino acids and they have lots of internal chemical bonds, which hold them together into tightly contained units.  When a protein is denatured, those internal bonds break and the amino acid chains unravel and become elongated.  At the same time, atoms that were previously bonded (as part of the internal bonds I mentioned) become available to bond with other molecules.
When an egg is heated to 40 degrees celsius, its protein chains become denatured and elongate, which allows chemical reactions to occur.  The egg changes from a liquid solution of protein into a solid mass.
A similar thing happens with you beat an egg.  The physical act of beating causes the protein strands to stretch, thereby causing the protein to denature (the internal bonds are broken as a result of the application of physical force).  So, rather than the protein chains being bound up into tight balls, they become long strands.  This is similar to gluten when it is developed.  These strands form structures that allow the trapping of air, which results in a lighter texture.
Beating eggs is not just about mixing.  It's about changing the structure of the eggs to produce a physical effect.  To read more about this, see "The Science of Cooking" by Peter Barham.
Comment: This is exactly what I was looking for! Thank you!
Comment: what does this mean in regard to the nutrition content of the egg?
Comment: @Fitri: Beating an egg does not change any nutritional values.
Answer: When you cracked the egg and it was in the bowl, you can see that its snuggled together. not mixed.
When beating it, the egg expands which allows the air to be trapped which makes the texture lighter.
This shows that the protein molecules aren't snuggled together but its preformed as an expanded strand egg, which shows the protein molecules every where from small to expanded.
This is about when beating the egg when used as a force called physical.


What does a 'nano crystal coat' do on this lens?

Question: The Nikkor 85mm f/1.4G AF-S has a 'Nano Crystal Coat'...what is that and why is it on most of their most expensive lenses?  Obviously its a coating and some links seem to indicate it may increase transmission and sharpness, but it seems counter intuitive to me that coating something will make it sharper.
Answer: Nanocoating: New and Different!
To more specifically address the "Nano Crystal Coating" type of lens coating, as other answers seem to be either addressing multicoating in general or think nanotechnology coating is just a marketing term.
Nanocoating is actually NOT the same as multicoating, it is very different in design, and affects light in a different way. The use of the term "Nano Crystal Coating" is most definitely not just a marketing term! To start as simple as possible:

Multicoating is an advancement on the concept of singlecoating, and is designed on the basis of waveform interference.

Works by "tuning" reflected light in such a way that the reflected
particles waveforms cancel each other out.

Nanocoating is a much newer concept, intriguingly based on the structure and design of moth eyes (which barely reflect any light at all.)

Designed to avoid reflection in the first place, and guide light rays into the lens without allowing them to reflect at all.

Multicoating and Waveform Interference
Light exhibits both particle and waveform properties. As such, two photons can interact in such a way as to cancel each other out. This is best demonstrated with illustration, and I'll borrow a wikipedia image for that purpose. Below is an example of a single-coated lens, and how the coating produces reflected photon waveforms that are in opposition to each other (and therefor capable of canceling each other out):

The anti-reflective coating is designed to be exactly as thick as half the wavelength of the frequency of light. Light will reflect at every intersection of material, such as between the air & coating as well as coating & lens. Since the coating is as thick as half the wavelength of light, the reflection from the air/coating interface negatively interferes with the reflection from the coating/lens interface, and the two cancel each other.
Multicoating works the same way, however with multiple layers of coating at different thicknesses. Since the color of light is determined by its wavelength, coating a lens with several layers of exactly half the wavelength of key frequencies of light (such as violet, blue, blue-green, green, yellow-green, yellow, orange, red) will cancel out considerably more light than a simple single-coating will. Single coatings were generally designed in the green to yellow-green band of light, as they tend to be most prevalent in sunlight and daylight. Multicoating is intended to work on a the full spectrum as much as possible.
Deficiencies of Multicoating
The advent of multicoating was a huge breakthrough in terms of lens transmission (the amount of light they allow to pass), reaching levels as high as 99%. Multicoating is not ideal, though. When strong flare and ghosting do occur, they are only capable of entirely filtering out light reflected at the exact wavelengths each layer is designed to filter out. Wavelengths near the intended frequencies will be mitigated, however they will not be entirely canceled. A bright off-axis non-incicent beam of light, such as from the sun in the corner of a frame, can still create large, bright, and very detrimental flare, ghosting, and contrast reduction even on a lens with multicoating.
Additionally, multicoating is simply taking advantage of a property of light to use a negative property of lenses...reflectance...to minimize the impact that reflectance has on image quality. As such, transmission is not ideal, and up to several percent of incident light can be lost for any given wavelength, usually resulting in 1-2% total loss in transmission PER COATED ELEMENT/GROUP. Granted, that is far lower than the 8-10% that used to exist with single coating and uncoated lenses, however in complex lenses with many elements, a considerable amount of light can still be lost overall (i.e. a complex 15 group telephoto lens could end up with 15-30% loss in total transmission in the face of strong flare.)
Improvements with Nanocoating
Nanocoating, unlike multicoating, is not a continued evolution of a previous technology...it is indeed an entirely new approach to solving an old problem. Nanocoating is based on the design of moth eyes, which are known in the scientific community to have one of the lowest reflectance indexes of any material. The general design is based on nano-scale roughly dome/spike-like structures intended to guide as much light as possible into the lens, avoiding reflectance entirely whenever possible.

If and when flare or ghosting does occur, since nanocoating is not designed to work on any given wavelength of light but light in totality, the resulting artifacts or loss of contrast is considerably less than with a multicoated lens. In many cases, careful and close scrutiny is required to find small elements of flare and ghosting in photo taken with a nanocoated lens, and when it does exist, it often does not detrimentally affect IQ.
Transmission levels for nanocoating are at least 99.95% PER COATED ELEMENT/GROUP. At a lost of 0.05% or less, the grand total transmission loss for any lens, even complex lenses with many element groups, will remain very low (i.e. a complex 15 group telephoto lens would end up with a total of 0.75% transmission loss.)
Design of a Lens Nanocoating
(NOTE: The exact nature of light that passes through a nanocoat is not widely publicized, so I can only base my explanation here off of what I have seen and read. I am not claiming 100% accuracy, however I think it is generally accurate enough.)

The design of the illustration above is taken from a few of the SWC, or Subwavelenth Structure Coating, diagrams I have found on Canon's websites. In comparison to Nikon's Nano Crystal Coating, Canon's SWC is the same thing, although their specific implementation may differ in the details. Canon explicitly calls out the "wedge shape" of the nano-scale structures, and calls out the pseudo-layered nature with wedges of differing size and height. The size and thickness of the structure layer is explicitly designed to be considerably smaller than the wavelengths of visible light used for most photography (around 200nm at the largest, where as the wavelengths of visible light range from 380nm to 790nm or so).
The technological purpose for using such a structure is to eliminate the primary cause of reflection: Large changes in refractive index at material boundaries. Replacing layered multicoating, which creates many interfaces where there could be large changes in refractive index, with a structured coating where there is no single interface  thereby creating a "smooth transition" layer. The thickness of the layer is kept small, presumably to minimize the impact to angle of incidence of rays that pass through it (don't actually have any specifically concrete information about why wedges are kept so small.)
Light is effectively "guided in" through the nanostructure layer into the lens element. The ultimate goals is for light to pass through the nano-structure elements and enter the lens element in the spaces between wedges, largely "unscathed". The amount of reflection is minimal, and what reflection does occur usually reflects off of the nano-structure/element interface where one exists. When light reflects off of an internal lens element and back up to a previous element, the same nano-structure coating will have the same effect on that reflected light, helping it pass through the internal elements to either diffuse harmlessly off the low-reflectivity innards of the lens, or right back out the front element...little to no harm done.
Better Sharpness?
Regarding whether nanocoating allows for improved sharpness. I would not be inclined to say that the nanocoating itself can really improve sharpness a whole lot. It certainly improves transmission, such that in lenses with lots of element groups, the total transmission loss is reduced from several percent to usually under, often well under, one percent. In terms of overall IQ improvement, the improved transmission should also improve contrast, even at a microcontrast level. Improved microcontrast will lead to improvements in sharpness, to some degree.
The claim to improved sharpness is more likely due to more freedom in lens design, and the ability to utilize more lens elements a lens designer might otherwise be limited to due to transmission requirements. If you can only use 8 lens elements with multicoating because more would reduce overall light transmission too much, you might be able to use 15 or more with a nanocoating and still have far better transmission characteristics. That offers lens designers the freedom to implement greater control over image reproduction than in the past, which should ultimately lead to improved sharpness.
I believe that is exactly the case with newer Canon lenses, largely "Mark II" generation or "new entrants" such as the EF 8-15mm f/4 L Fisheye lens. It is probably also the case with Nikon lenses with NCC. Canon's newer lenses are significantly outperforming their predecessors in the area of MTF (Modulation Transfer Function, a way of measuring the sharpness and contrast of a lens). Almost all of Canon's L-series lenses introduced since around the middle of 2008 (possible a little earlier than that) that use SWC have theoretical MTF's (most lens manufacturers these days generate MTF charts from computer models of lenses) demonstrating significant jumps on overall resolution, sharpness, and contrast, with some demonstrating nearly "perfect" results according to the criterion of their MTF (which is admittedly lower than most of their lenses should actually be capable of resolving, but consistent in terms of comparison with MTF's of older lenses.)
So, technically, it is not the coating itself that directly improves sharpness (although as it improves contrast it may have a slight direct impact). Improvements to sharpness are more likely due to the ability to make improvements in lens design without as much concern for transmission as in the past. (I guess that could be either corroborated or refuted by comparing the lens designs of new lenses with nanocoatings vs old lenses without.)
Comment: wow. Now THATS how to do an answer!!!
Comment: Great post about nano crystal coat can do in lens.
Answer: Here's a description of Pentax's implementation of nanocoating, called Aero Bright Coating (source):

...[the] PENTAX-original Aero Bright Coating...ensures outstanding anti-reflection performance over a broader wavelength range to deliver images that are brighter and of higher quality. Created using PENTAX's advanced nanotechnology, this exclusive coating reduces lens reflections and greatly improves light transmittance by forming a silica aerogel layer with uniform pores on the surface of the optical elements.

Note that Aero Bright Coating is only used in a few select lenses, including the DA* 55mm and DA645 25mm lenses.
Answer: I don't think the Nano crystal coat improves sharpness in and of itself. What it does, however, if give the lens designer more freedom in designing the lens.
Before coatings came into use, practical lens designs were limited to about 5 element groups (at very most). Single coating increased that to around 7 or 8. Multicoating increased it to around a dozen or fifteen.
Each of these allowed the lens designer to do a better job of correcting aberrations. Not only could they use more elements if necessary, but were more free to separate elements into separate groups, rather than grouping the elements just to minimize reflections.
Offhand, I'm not sure exactly how many more groups the nano crystal coatings (or equivalents from other vendors, which are coming into use) allow, but almost certainly at least a few. I'd guess it also allows a bit more freedom not just in to add more elements/groups, but also to arrange them a bit more freely to concentrate on reducing aberration without worrying (nearly as much) about how much flare/ghosting it might introduce.
Answer: I guess the most important reason for using subwavelength anti-reflection coating are the problems related with strong curvature of lenses. Multi-layer AR coatings work perfect for flat surfaces and lenses which are not too much bended. For the strong aspheres used for example by Nikon in all new zoom lenses like 14-24 f/2.8 see Nikon Precision Glass Mold the standard deposition methods for AR coatings fail to produce the correct multi-layer thickness at the steep slope areas. Especially in these steep regions the light loss from Fresnel reflection becomes a problem, even more severe are multiple reflections within the lens. Nano crystal coatings perfectly match the refractive index from air to glass. Thus flare, glare and the overall noise level in the image is very much improved. This leads to a much better contrast and resolution. Nano crystal coatings and the ability to make strong aspheres for reasonable price due to glass molding are a perfect combination. This offers the optical designer full freedom to design perfect lenses with very low noise.
The future is bright!
Reinhard
Answer: Some light that hits glass gets reflected back, not through the lens. Anti-reflection coatings lessen this and allow the light to pass through the lens. You see this a lot in telescopes, binoculars, and eyepieces where light gathering is critical.
The wikipeda article has a pretty good explanation.
As for the "Nano" part, aside from adding to the "price" of the lens, it is probably referring to some form of consumer nano-technology to make the coating better, or at least appear better. Given the price of the lens, well, I'd certainly hope it makes things better!


The state space of the stabilization of a C*-algebra

Question: Given a $C^*$-algebra $A$, I wonder up to what extent we can describe the state space of the stabilization $A\otimes K$ of $A$ in terms of the state space of $A$. Of course, the "tensor-product" states are the most obvious ones in general. But this seems to be far from the whole
state space of the stabilization of $A$.
Comment: I haven't got a reference to hand, but I think some of E.C. Lance's early papers on nuclear $C^*$-algebras have some discussion of this, because a key tool used in his proofs is a careful analysis of states on tensor products
Comment: @Martin: yes, I think that's fairly standard. It's certainly the usual definition of the stabilization of a $C^*$-algebra.
Comment: K are the compact operators?
Comment: I fixed a typo and removed some pointless LaTeX in the title.
Answer: From the general point of view of $C^*$-algebra theory, Bill Paschke mentioned to me today an interesting classification of states (or even, positive functionals) on the stabilization of $A$ as follows:
Proposition. Let $A$ be unital $C^{\ast}$-algebra, $H$ be a Hilbert space and $\varphi$ be a positive linear functional on $A\otimes\mathcal{K}(H)$. Let ${\cal L}^1(H)$ denote the ideal of trace-class operators on $H$. Then there exist a Hilbert space $H_{\varphi}$, a representation $\pi:A\rightarrow B(H_\varphi)$ and an operator $S:H\rightarrow H_{\varphi}$ such that $S^*S\in{\cal L}^1(H)$ (so $S^{\ast}\pi(a)\,S\in\mathcal{L}^1(H)$ for $a\in A$), and $\varphi(a\otimes K)={\rm tr}(S^{\ast}\pi(a)\,S\,K)$ for all $a\in A$ and $K\in\mathcal{K}(H)$.
Moreover, $\varphi$ is a state iff ${\rm tr}(S^{\ast}S)=1$.
Proof. Since ${\cal K}(H)^{\ast}\cong{\cal L}^1(H)$, any positive linear functional $\varphi$ can be regarded as a completely positive map $T:A\rightarrow{\cal L}^1(H)$ with $\varphi(a\otimes K)={\rm tr}(T(a)K)$. Define a sesquilinear form on the algebraic tensor
product $A\odot H$ by $$\langle a\otimes\xi,b\otimes\eta\rangle_{\varphi}:=\langle T(b^{\ast}a)\xi,\eta\rangle_{H},$$ set $N_{\varphi}={\rm span}\{a\otimes\xi\in A\odot H\mid\langle T(a^{\ast}a)\xi,\xi\rangle_{H}=0\}$ and $H_{\varphi}:=\overline{A\odot H/N_{\varphi}}$. Then we can define a representation $\pi:A\rightarrow B(H_\varphi)$ by
$$\pi(a)(b\otimes\eta+N_{\varphi}):=ab\otimes\eta+N_{\varphi}.$$ Now, let $S:H\rightarrow H_{\varphi}$ be the operator defined by $$S\xi:=1\otimes\xi+N_{\varphi}.$$ Then $S^{\ast}:H_{\varphi}\rightarrow H$ is given by $$S^{\ast}(b\otimes\eta+N_{\varphi})=T(b)\eta,$$ and we have $S^{\ast}S=T(1)\in{\cal L}^1(H)$. Moreover, $S^{\ast}\pi(a)\,S=T(a)$,
which is what we need. It is easy to see that $||\varphi||={\rm tr}(S^{\ast}S)$. Q.E.D
Answer: Thanks, Yemon! I haven't found exactly what I am looking for in the papers you mentioned or some others yet. Part of the problem is how often we can find a separable state on the stabilization of $A$. Here, a separable state means one which can be expressed as a convex combination of the tensor-product states on $A\otimes{\cal K}$. Any other state is called entangled. (Apparently,  this terminology comes from Quantum Information Theory.) The study of separable states is still under progress for the tensor product of matrix algebras (cf. [1]).
[1]: E. Alfsen F. Shultz, Unique decompositions, faces, and automorphisms of separable states, http://arxiv.org/abs/0906.1761v3.


Are the chargers the same on the iPad and the iPad 2?

Question: Is the charger of iPad and iPad 2 same? I want to buy i extra charger for my iPad but in future I can buy iPad 2 also. So I want to buy a charge which should be compatible with iPad and ipad 2 both.
Answer: The charger is 10W like the old iPad. It's called iPad 10W USB Power Adapter and has product number MC359LL. Specs and QnA are available at the Apple US online store.
Answer: I have an iPad and my wife has an iPad 2. It's too dark in my house to look at the power adapters right now but they are either identical, or close enough to not matter. They are both 10W. We interchange them all the time.
Answer: It is probably also useful to know that you can also charge an iPad using any iPhone or iPod Touch charger. It will charge a lot slower when not using the iPad specific charge, but it will still charge it.
The non-iPad chargers don't however provide enough power to charge the iPad while it is using used, they will provide enough to stop it discharging though.


How to gather and weight stakeholder ideas

Question: When you interview stakeholders in order to gather business requirements – do you also document content ideas such as product benefits or ideas for sections on the web site? Do you put them in the same requirement document and
how much do you follow those ideas? Do you make some kind of thinning?
Answer: By stakeholder, I assume that you mean the client/product manager because in my opinion, even the dev team is also a stakeholder. Anyhow, it is usually a good idea to keep requirements (the 'what' part) separate with the implementation details (the 'how' part). Sometimes the client/business analysts think that they know the best way how a system should be developed but actually it is not correct. I think you can document content ideas but keep them out of the business requirements.
Comment: Yes, I'm talking about the client and people in the clients organisation that the web site will have an impact on such as the customer service division or the sales division. When I talk to those people about how the web site could help them they usually come up with a lot of content ideas and stuff that the think could be improved such as search. - Can I transform those ideas into business requirements instead?
Comment: Yes, you can but be aware of the project charter and scope. Staff may give you their wish-list of items that they would like to be developed but are those tasks approved or not by the client management (project sponsor).
Answer: First, I think this questions belongs more to a site like Programmers.
At my job we put all bugs and suggestions into the same list. Its then up to the product manager and the developer team to evaluate it.
We give it a description including both What and Why, in viewpoint of the user scenario. Technical details go into a notes section.
We encourage anyone including users to report feature suggestions. They should get documented to understand the needs of the user, not to dictate the details of how to get there. The suggestion evaluation process takes place in the development team, and it is basically up to the developers to decide how to implement it.
And don't forget Ford: "If I would ask what my customers wanted, they would say a faster horse."
Comment: The problem is that when you mix requirements with implementation details, requirements docs stop saying "Achieve X", and start saying "Achieve X by using button Y in position Z". Then when someone realizes that X can be achieved better in a different way, or that the original idea doesn't actually make sense in practice, the statement of works has to get renegotiated.
Comment: I see now I got quite off topic. I'll edit it to try to focus on the question.
Answer: You're doing user experience design. It's not called stakeholder experience design for a reason.
So, your primary goal is to grab all those stakeholders who actually have something to do with the system: those who will actually use it. These are the customers, the administrators, the whatever, called together users.
Your customers ain't your users. Usually your customers are people with too much money who employ those people who are actually your users.
Usually there's only one need for your customers: they want the users to behave in a way that brings them more revenue.
The unfortunate thing is that you're employed by your customer, not by your users.
Also, usually the programmers are also stakeholders: some want easy implementation, others want challenge, others want to try out some new technologies.
Your priority is simple at first: make users have the best experience possible.
Last priority: eliminate everything which the users won't use. So, if the customer insists that (s)he wants to have his/her puppy on every single keyscreen on a website which has nothing to do with puppies, you can easily dismiss it. I'm not saying you shouldn't do it if it's their mania, but it's your last priority.
Middle ground: whenever there's a contradiction between users and the customers, choose a solution which brings the least pain for the users and the most revenue for the customer. The order is important: least pain first. You're, again, a User Experience designer.
You can draw a simple chart where you say:

download bmml source – Wireframes created with Balsamiq Mockups
As a user experience designer, your job is to create low-pain solutions to users, but as an employee of the customer, you have to bring revenue. Your priority is about the users - the profession which concentrates on revenue a bit more is called "marketing"
If there's a problem of cost, again, you should aim for the best user experience within the available resources. Resources include cost, technology, time and so on. As an employee, you have to stay within budget, and as a fellow employee, you shouldn't make too much trouble for the developers. However, you should push this to reasonable limits: your job is about user experience, so as long as the developers are still happy with their job, try to ask them to do what benefits the users, but let them have their creative playground sometimes.
If there's a problem within different user groups, paying users come first: while you shouldn't make a nightmare out of the job of customer service, customers (of the product) are keeping the company alive. So if it takes 10 minutes more every morning for the customer service to look at a given feature which makes life of paying users of the software easier, they won't die in it. However, if you can stick in a feature to have less pain for the service clerk, do it: the customer service clerk is still your user.
Your dimensions are the following:

external users who are keeping the company alive
internal users who are keeping the product running
developers and other stakeholders which make the product available
customers who are paying for you and for everyone else in the previous two groups

Your priorities are similar. As a User Experience Designer, you should focus on the first two groups more than the last two. Whenever there's a clash of interests, you try to bring minimal loss to the lower-priority partner while having the best for the higher-priority one.
That said, of course it's a hard balance and it has to be decided case-by-case.
Comment: Well, you had me at stakeholder experience design.