Are any font sizes reasonable to use

Question: When I choose a font size for use in a graphic I usually opt for font size menu dropdown defaults like 6, 8, 9, 10, 11, 12, 14, 18, 24, 30, 36, 48, 60, or 72. If I'm working on something large (like an A3 poster) I need much larger fonts. I'm always tempted to keep these as powers of 2 (e.g. 128, 256, or 512). But why? Am I just being superstitious! Does font size have any technical implications? Is choosing 512 in any sense better than 513 or am I actually free to choose whatever looks best?
Comment: But don't forget readability of the text. I recently was given otherwise visually awesome looking layout, but text was so small that it was unreadable. Most of it had to be re-designed with _readable_ text size.
Answer: In the analog days, typefaces came in specific sizes for the simple reason that manufacturing of type in metal and wood required it.
In the digital age, you can pick any sizes you want. You can set your fonts to 512.34492 points if you'd like.
The size is really a visual design issue. Use the size appropriate for the design. The actual numeric size is irrelevant.
Answer: The benefit of that particular set of numbers, and so presumably why those particular numbers became the standard for physical type and persevered as the standard even when it was no longer needed, is that mathematically they share lots of common factors. Choose numbers from this set, and there will be numbers they are all divisible by.
DA01 is right that what looks right and what works on the page should be what drives font size choices, but this can be handy convenience if you are setting up a baseline grid on a complex layout in something like InDesign with various sizes, columns and side bars or blocks. For example, with two levels of header at 24 and 18, lead-in paragraph text at 12 and regular paragraph text at 9, those sizes are all divisible by 3. So you could have a baseline grid at 3 points, and still line everything up quite neatly without too much difficulty, despite the many very varying sizes. More on grid sizes set to increments in this question.
That said, this a) applies more to leading (line spacing) than it does to point size, and b) is only a convenience, not an absolute benefit. Even in a case like the example above, don't be restrained by those arbitrary numbers - if you like those relative proportions but find the whole thing looks better scaled down by 6.4% so the main headers are 23.424 points and the baseline grid snaps text every 2.808 points, then do so.
The only other consideration I'm aware of is (and it's a very small one), if you are designing for a screen and find the font looks bad at a particular size in particular software, it's possible (but unlikely) that there's some hinting threshold or similar set in the font or the software that the decimal puts you just on the wrong side of. So, if you find when testing that a font set to 10.2 is rendered badly in some browser or operating system, the first thing to try is to see if it still looks awful at 10 or 11 (or 10.5), before looking for a replacement font. It's more likely that something about the software or operating system just doesn't look good with that font around that size, but it's worth a try.
So basically, choose whatever size looks best, but remember that a useful set of numbers with lots of useful common factors exists which might be handy if you're struggling to get a grid to work for a complex layout, and that these are the points that the pixel rendering of fonts are most likely to be set to revolve around, which very rarely might be help avoid bad rendering.
Answer:
If I'm working on something large (like an A3 poster) I need much larger fonts. I'm always tempted to keep these as powers of 2 (e.g. 128, 256, or 512). But why? Am I just being superstitious!

Short answer: you're just being superstitious.
The reasons for old type being available only in set sizes is historical.  Modern fonts can reproduce typefaces in any size you like.  You can even have 15.65 point fonts if you like.
The main thing is to be consistent with your own font sizes.  Don't randomly change size, and don't change size for no reason.
Another point to consider is that it can be aesthetically pleasing to adhere to a baseline grid, and doing so many affect the font sizes you choose and guide you in the direction of at least making your large heading type a multiple or near-multiple of your body type.  It makes sense to combine, for example, 12, 24, 36 and 48pt fonts because they are all multiples of 12 (assuming 12pt is your body type size).  But there is nothing conceptually wrong with choosing, say, 13pt or 11.2pt as your body text size and sizing your headings based on those instead.  Also, font size doesn't necessarily have to remain in a fixed relationship with baseline-to-baseline spacing - with increased line spacing (leading), you can use a smaller font in a larger space.
Answer: Just don't use too many different font sizes in a single design, unless confusion is part of your design.
Your font sizes should indicate the hierarchy of information unless disarray is part of your design.
Important copy should be legible unless...you get the idea.
There are a million rules to typography, and almost as many ways to successfully break them. Choose wisely.

Are any font sizes reasonable to use


How to find out main page of a website

Question: When a website's main page cannot be opened as in this example, is it possible to find out the full address? In this example, I tried /index.htm and /index.html without success.
Comment: Site looks fully up and running
Answer: In this particular case, asking Google for all content on that domain comes up with a blank, in other cases searching with site:www.example.com in Google could quite easily come up with the main, or most popular, pages.
Answer: In the case of the above URL, you're seeing the Apache "test page", which is displayed when you start the daemon without either placing files within your web root, or customizing your config file.  It's because the default DirectoryIndex doesn't find a matching file, which would normally be the main page of a URL ending in a slash.
The above file is actually just an "ErrorDocument" rewrite, that says if there is no content the server should display the file under /error/noindex.html; so in the above case you can see the actual file under http://www.cetmiolive.com/error/noindex.html to verify it's HTML.
To normally find the type of engine rendering a page on a website, I usually use the same method as you (try index.htm, index.html, index.php, Default.asp, default.cfm, etc.) to see if I can determine the source.
That does depend on a default installation though - if you change your handler (for example, have html files actually be processed by the PHP engine) then you can mask the engine driving the website - but still determine what the file name for the directory index is.
Now, in the case above, when rewrite rules or ErrorDocument functionality is used, it's anyone's game to find out.  Those mask the URL or change the way the server interprets URL requests, making it extremely difficult to locate the file.
Comment: To figure out what engine a page uses, see also: http://webapps.stackexchange.com/questions/967/determine-which-web-server-is-hosting-a-web-application
Answer: They probably don't have anything there yet, as others have already eloquently pointed out.
However, some other variations you might try include:

index.asp
index.cfm
index.jsp
default.htm
Comment: I would highly suggest Default.aspx (and maybe Index.aspx).  .aspx is for ASP.NET whereas .asp is classic asp.  ASP.NET is a much newer and classic asp is rather old.
Answer: They may actually be under construction and not have an index page as of yet. Try again later.

How to find out main page of a website


Booting Windows 7 in Boot Camp shows a black screen with a blinking cursor

Question: I am trying to restore a DMG image of a Windows 7 Boot Camp partition back to Boot Camp. (I know that I should have used Winclone, but I didn't have that option because the drive was already dying and so I couldn't even access OS X anymore.)
So far, I've gotten to the point where I can start up the computer, hold Option, choose Windows, but then when it boots up, it shows a black screen with a blinking cursor.
I've tried a few things so far to try and fix this, none of which has worked. I have used the Windows 7 install DVD to run bootrec and all its possible arguments. I've ran chkdsk C: /b. I've also used the "Automatically fix startup problems" button, of course, as well.
Anyone know what else I could try to fix this?
Comment: I had the same problem after resizing my disk partitions.
This breaks bootcamp.
I found a way to solve this, I googled and had to do something very elaborate in terminal. Don't remember the details sorry.
Answer: Since it is an Image-File frome a previous (working) Bootcamp Installation it should not be necessary, but there is an upgrade for the graphics and bluetooth driver from Apple with an installation instruction found here "Boot Camp: iMac displays a black screen during installation of Windows 7" which adresses this problem.


What is the origin of the words "put" and "call" that characterize derivatives?

Question: And, in a second round, what is the origin of the "long" and "short" in characterizing the respective positions in assets? Any pointers to etymology, first occurrences, or related literature would be greatly appreciated.
Comment: Are we going to allow history of quantitative finance questions?  I've posted a [meta question](http://meta.quant.stackexchange.com/q/190/1106) on this.
Comment: This question is lacking the "quantitative" component of quantitative finance.
Comment: StackExchange has a site dedicated to the [english language](http://english.stackexchange.com/).  You might get a better answer there.
Answer: The Early History of Option Contracts
Haven't read this so can't vouch for whether they get into etymology.
ABSTRACT:
This chapter discusses the history of option contracts from ancient times until the appearance of
Theorie der Prämiengeschäfte by Vincenz Bronzin in 1908. The history examines the use of
contracts with option features prior to the introduction of trade in free standing option contracts on
the Antwerp bourse during the 16th century. Descriptions of the Amsterdam share option market by
de la Vega in the 17th century and de Pinto in the 18th century are reviewed. The specific language
of a late 17th century English option contract is provided in detail. The development and practice of
option trading in the 18th and 19th centuries, as reflected in merchant manuals of that period, is
examined. The article concludes with an overview of late 19th century option trading in securities
and commodities.


What to do with sore muscles

Question: Say I have done some exercise and the next few days I am rather sore in certain muscles.
What is the best thing to do about that? Let them rest (avoid using them)? Use them a lot? Exercise more? Rub them? Hot bath?
Comment: This thread is littered with upvoted misinformation.  Check out [this](http://fitness.stackexchange.com/questions/11643/my-legs-still-ache-3-4-days-after-legs-day/11687#11687) answer for some information backed up with scientific research.
Comment: Check for a possible D-vitamin defiency with a doctor just in case, especially if you don't get much exposure to sun.
Comment: @Doc Cool, thanks. That was a very interesting read :)
Answer: Prevent them from getting sore in the first place

Bananas

Eat bananas after a workout. A healthy source of potassium is essential to prevent muscle soreness. If you have ever drank (alcohol) heavily you may have also experienced muscle soreness. This is due to losing a lot of electrolytes (salt, potassium) due to the diuretic effects of alcohol.

B Vitamins

B vitamins are also important to eat after strenuous exercise. Especially B1 (thiamine) as a deficiency will cause sore muscles.

A cool after-workout swim (my personal anecdote)

I don't have an official source to back this up but I find that a swim in cool water after a strenuous exercise helps to cut down on the after-workout soreness. The combination of cool water and non-strenuous workout to get a good supply of oxygen/blood to the taxed muscles works wonders.

Properly cool down after a workout

When you do workouts that push your heart into the anaerobic range (think faster than a light jog) your body's oxygen intake will max out and it will start metabolizing anaerobically (without oxygen). The byproduct of anaerobic metabolism is still a controversial topic (due to traditional teaching vs current research) but it's basically what causes the 'burn' you feel from a hard workout. If you don't take the steps to process the byproduct of an anaerobic workout you'll feel more sore the next day. That's why going from a sprint to a dead stop is a bad idea.
So how do you recover after an anaerobic workout? Do a proper cool-down aerobic workout (like a light jog) to provide the oxygen and circulation your body needs to process those byproducts created through the anaerobic (or strenuous) exercise.

Stretching

Stretch before you workout. Not static stretches (long hard stretches in a relaxed state) but dynamic stretches (short quick stretches where you flex the muscles) or else you'll elongate your muscles too much and increase your risk of ligament pain/injury.

Warm up

Properly warm up your muscles. If you have ever worked out with a heart rate monitor you probably already know that your body doesn't go from 0-max very well. It takes some time for your circulatory system to fully adjust to the increased intensity.

Increase your fitness level

The concept is pretty simple. The better your body can circulate and provide oxygen to muscles, the better it will be able to process the byproducts of the strenuous workout.
How you do so depends on what you're building up to. For instance:
If you are training for long distance running, then you'll want to do more workouts in the aerobic range mainly due to the fact that more Type II muscle means more weight to carry and more resources (energy) you'll need to sustain the pace. As you may see on this site, many people like to bring up the topic of 'runners high' where you feel like you can keep going forever without getting tired. This is mainly due to the fact that your body is completely warmed up and effectively metabolizing in the aerobic (and therefore not creating any nasty byproducts) as well as releasing endorphins. As long as you can keep going without any functional issues (muscle strain, ligament pain, boredom) and you can maintain a healthy energy level, theoretically you could go forever. Unfortunately, your body doesn't have the ability to properly maintain itself (digestion, recovery, etc) during exercise. This is due to your body focusing it's resources on the 'sympathetic nervous system' during exercise as opposed to the 'parasympathetic nervous system'.
If you're geared more toward weight lifting, higher intensity, short burst exercises, or other types of anaerobic activities like sprinting; The hard-and-fast rule is, 'the harder you workout the harder you will be able to workout'. This is mainly due to the fact that the more you push into the anaerobic range the better your body will adjust to be able to handle processing of the anaerobic byproducts. Fewer residual byproducts means less after-workout soreness.
If you want specifics on 'why' that rule holds true, google the term 'anaerobic threshold' and spend some time reading up on the topic. Most people (including myself) were brought up to believe 'no pain no gain'. That is only partially true. You will feel some soreness/stiffness when you first start no matter what you do but pain does not equal progress. If you focus on increasing fitness first before strength you will have a much less painful path to progress (and you will probably be more healthy overall).

If you're already sore, don't completely avoid activity. Moving around will help loosen up the muscles a bit and recover faster, just take it easy and let your body recover.
Update:
I just stumbled on a Recovery Guide by the USTA (United States Tennis Association) that covers all of these points and more in detail. Including CNS (Central Nervous System) and psychological fatigue - which I neglected to cover in my answer.
Comment: phew, that was long
Comment: That probably sums up anything I would recommend.
Comment: B vitamins are good for the nervous system when it has been subjected to stress. Also please make sure to check for a possible D-vitamin defiency as that can cause widespread soreness which may have similarities with fybromyalgia.
Comment: Stretching, warming up, and cooling down (literally, with ice/cool water), haven't been shown to have any impact on soreness (see [this](http://fitness.stackexchange.com/questions/11643/my-legs-still-ache-3-4-days-after-legs-day/11687#11687) answer).
Comment: @Doc Your answer that you linked to confuses muscule tissue damage with muscle recovery; injury/inflammation != DOMS. Light dynamic stretches (ie not static stretching as referenced in the study you linked) and a warm-up increase circulation, it's more for preventing injury than reducing muscle fatigue. From your answer, I think you'd agree that good circulation is important to muscle function. The cool-down recovery has actually been proven by Stanford (see http://www.youtube.com/watch?v=e8J6ov48rG0). Recovery doesn't come from icing locally but by bringing core temps back down to normal.
Comment: What about massages? I haven't been properly warming up and cooling down, but I have been doing these other things. Over time my muscles were getting tight to the point of thinking my tendons when injured somehow. Plus every once in a while a knot would show up then very slowly go away. Regardless I was getting much faster (after a limping for the first few minutes) so I didn't want to stop. After the goal race was done I took time off but the problems didn't go away until I started doing recovery runs and massages. Aren't these necessary? Could warming up/ cooling down replace these?
Answer: Muscle soreness after exercise is a very common symptom of a strenuous exercise, and is usually indicative of a "good" workout, i.e. one in which you've stressed the muscles to the point at which they will grow stronger. The cause of the soreness is currently unknown. Previous hypotheses involving lactic acid and muscle spasms have been demonstrated to be incorrect.
You can provide temporary relief using warm compresses and gentle massaging of the affected muscle, but the soreness will persist until about 72 hours. The best thing to do is to stick to a regular routine; soreness is notably worse the first time a muscle group is significantly stressed. Note that while continuing to exercise will stave off soreness, overstressing can lead to sprains and tears.
Answer: There are two major types of soreness

From a very good workout

Over training.

Depending what is the cause of your soreness you have to take appropriate actions.
If its of type 1, give it rest for roughly 4 days(infact it depends on the muscle/muscle group that is/are sore) to recuperate, do not work the muscle and strain it again.
During the recuperation time, actual growth and adaptation(of the muscles to higher levels of stress and strain) takes place. So rest is very important. Which leads to not to train harder or more often.
Type 2 occurs when you work a muscle too often to allow it to fully recuperate, you have to consult a doctor, because he might prescribe a pain killer or some medicine. Take lot of rest, sometimes weeks of it.
One important factor: Trained muscles recover from fatigue faster than untrained muscles.
So the better you get at exercising the faster your recovery rate will be and the more intense your training can become.
Comment: I agree with the last paragraph but you suggest getting a prescription for pain killers to overcome muscle fatigue? Pain is a perfectly natural response that tells you to 'take it easy'. The way to get rid of it is to give your body what it needs to recover faster which includes, rest, water, electrolytes, and nutrients. Munching on pain killers only dulls the symptoms and may produce additional side-effects, it doesn't fix the underlying problem.
Answer: Keep working out according to your schedule (assuming there's a 48-72 hour recovery period after each session -- if not, get a new schedule). It's not harmful, and might even make the soreness go away sooner.
Answer: An ice bath may help against muscle soreness:

The ice bath is thought to:

Constrict blood vessels and flush waste products, like lactic acid,    out of the affected tissues
Decrease metabolic activity and slow down    physiological processes
Reduce swelling and tissue breakdown
Comment: No joke, this answer have substance proven by research. Apparently recovery can be fast-tracked by reducing your body's core temperature. It may also explain why swimming in cool water helps me recover faster. Brought to you by Stanford http://www.youtube.com/watch?v=e8J6ov48rG0.
Comment: Why would an ice bath help relieve muscle soreness? Please update your answer with information from the link you provided.
Comment: Good point there. I've updated it. I'll work on this more later.
Comment: What about a cold shower immediately after a workout ?
Answer: What to do depends on how sore your muscles are, how long they've been sore and your prior experience with sore muscles.  The easy answer is to rest, but it may not be the best advice for your specific situation.  I've always felt that you should continue exercising, but with the focus on areas not directly related to the 'sore' muscle area, this builds an exercise pattern that will provide long term benefit, help increase your pain threshold (pain management is key to any aggressive exercise program) and gives those 'sore' muscles time to heal.
Make sure you're hydrating enough and getting sufficient protein to help reduce the intensity and duration of muscle soreness.
Answer: I started with a trainer a few months ago and starting out, I would get SO sore. Using a foam roller really helped (before and after).
You can find different workouts with a foam roller here.
Comment: It would help if you would list some of the exercises.
Answer: When I am sore I like to recover with a pool session. You only need to get in waist deep and you do a series of stretching exercises (quads, hamstrings, hip flexors, calves, etc.). The water resistance helps with the lactic acid build up. This works the best for me and hopefully may be a good option for your soreness as well.
Comment: Do you happen to have an article to support your claim about water resistance and lactic acid build up?
Answer: I'm surprised no one's mentioned Epsom Salt bath soaks for the mineral content, as Epsom Salt is a dissolvable form of Magnesium. This really helps me with soreness.
Taking vitamin B6 orally may help with uptake, and Calcium works synergistically with the Magnesium.
Here's an article with overall information about minerals and bodybuilding, but you may wish to try and get more updated suggestions for how to take what you need from your doctor.
Comment: It would be good if you cite the crucial parts of the article, content on the Internet often is subject to link rot.
Answer: All types of exercise will eventually make you sore and that is meant to happen. As Mark Ripptoe says, "you don't get strong from lifting weights, you get strong from recovering from lifting weights". An important factor of lifting/exercise is recovery, sleep, and proper nutrition. Your recovery will include things that help relax the muscles such as message, hot baths (switching from hot to cold if you are able to), heating pads and ice, or rubs like tiger balm. Another thing that will really help your recovery is planning to go for walks. Going on a long relaxing but brisk walk at least 1-2 times a week is just as important as planning the days that you lift on. Walking helps circulation and will reduce pain in your muscles. Your nutrition is going to include making sure you are getting the right macro nutrients such as calories, fats, carbs, and proteins as well as making sure you get your micro nutrients from lots of vegetables. As far as supplementation that is very individualized. The only way to know would be to get blood work and have a doctor that specializes in doing full spectrum analysis. This will tell you what you are deficient it. On the other hand it needs to be updated every 6 months to stay relevant which is rather costly. One supplement that can help most people (except those with health issues) is creatine monohydrate. Creatine will help you produce more atp which means more energy for building those gains. Glutamine also pairs well with creatine in maximizing its effects. Lastly getting enough sleep is very important. Lastly, if you are experiencing extreme pain that lasts for days after exercising then you need to be reevaluating your programing and getting advice from a knowledgable strength coach. All the best.

What to do with sore muscles


How to see which accounts contain unread e-mails when they are not expanded?

Question: In earlier versions of Thunderbird you knew which account folders had incoming e-mails even when they are not expanded. Usually the mailbox icon left beside the e-mail-address account changed and showed a little green arrow.
Now this is totally stupid, the icons doesn't change when there are any incoming e-mails. You get a notification, but I even want to see which mailbox have incoming e-mails. Otherwise I need to expand all my e-mail accounts and I don't have so much space on view.
I already tried another theme, but same result. I already tried to disable the default extensions but it didn't help. Any suggestions?
Answer: Ok, it's really a theme issue. Obviously many themes suck and don't display that, but I found a theme which does.


Group as a product of subgroups

Question: Let $G$ be a finite group, and $H,K$ be proper non-trivial subgroups such that $H\cap K=1$, and $HK=G$. Is it necessary that one of these subgroups is normal in $G$?
Answer: No, neither of the subgroups needs to be normal.
Take for example the alternating group $A_5$ which has a subgroup of order 12 (a copy of $A_4$) and one of order 5 (the 5-Sylow).
Now clearly, these intersect trivially, as their orders are coprime. But since the product of their orders is exactly $60 = |A_5|$ this means that their product is all of $A_5$.
Since $A_5$ is simple, neither of these can be normal.
For more information, see the question Has this "generalized semidirect product" been studied? where I got some very nice answers about this particular phenomenon.
Comment: I thought I was the only person who remembered my answers! I'm glad other people don't just read them and forget...
Answer: Please have a look at this link, https://math.stackexchange.com/a/197415/8581. There,  @GeoffRobinson presented another neat example in his first comment. :)
Comment: :-) $\quad +1\quad$
Answer: I added to the link given by Tobias the following:
"These are also called matched pairs of groups. I just did a google search on this term and it gave over 11,400 results, often related to Hopf algebras. The n-fold case is discussed in arXiv:1201.0059 . But I was not aware of the older references given above [i.e. in the link], which are nice to see."

Group as a product of subgroups


Where was Elisha when he sent a son of the prophets to Ramoth Gilead to anoint Jehu?

Question: Where was Elisha when he sent a son of the prophets to Ramoth Gilead to anoint Jehu in 2 kings 9:1?
Answer: We're not told where he was. It's possible that Elisha was still in Damascus when he gave this message, since that is where he was the last time he was mentioned by the author (2 Kings 8:7). I don't think this is likely, given that Elisha is giving a message to one of the sons of the prophets, and its doubtful that this Jewish school had a branch in Damascus.
At this point in the narrative Elisha has largely faded from the scene. Like his predecessor Elijah, the end of his ministry is marked by the anointing of kings. Largely, the focus as shifted off of Elisha's prophetic ministry back to the chronicling of the kings of Judah and Israel. In fact, we don't hear about Elisha again after this until his death (2 Kings 13:14).
I think its important to ask questions like this, but in this case I think the omission is merely a reflection that the narrator has moved on his telling of Israel's history.


Definition of a strange attractor.

Question: May be it's not the right place for this, but I don't know the right definition of a strange attractor. Wikipedia states that "An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it are chaotic." In the Tucker's paper about Lorenz system is written that "an attractor is called strange if for almost all pairs of different points in B( Λf ), their forward orbits eventually separate by at least a constant δ (depending only on Λf )." I feel that this definitions are not equivalent. I also would welcome links to useful literature.
Answer: This is a good question. For some reason, terminology in dynamical systems is not standardized at all--and it's interesting to disentangle various definitions. A good book to look at is Differential equations, dynamical systems, and an introduction to chaos. The authors (Hirsch, Smale, Devaney) are at the center of the field, and they point out there's no standard definition of even "attractor"! (Let alone "strange attractor," which they only use once, informally.)
In my view, definitions based on the shape of an attractor (like the first part of Wikipedia's) are a little odd, since you can have chaotic dynamics on geometrically simple attractors. Think of a map with a circle as an attractor on which the dynamics are like $\theta \mapsto 2\theta$.
The second Wikipedia definition, that the dynamics are chaotic, might imply the Tucker definition--but that in turn depends on your definition of chaos. There's interesting work on how various criteria for "chaos" relate. A good entry point might be "On Devaney's Definition of Chaos" (Banks et al, American Math. Monthly, 1992).
Answer: Ruelle's answer
http://www.ams.org/notices/200607/what-is-ruelle.pdf
Answer: As a note, all strange attractors that have been found have had a fractal dimension.
A strange attractor is: 1. An attractor, and 2. displays sensitive dependence on initial conditions (ie points which are initially close on the attractor become exponentially separated with time), making it "strange".
Apart from those two definitions, there's not much else standardly accepted about them, there are only definitions within specific cases. To have a sensitive dependence on initial conditions implies chaotic behaviour, implying the attractor might not have an integer dimension, as justified by the note. This, however, is not necessarily a certainty, so I think the Wikipedia page is wrong to say this.
The problem is that chaos theory is a relatively new subject, so terminology is not as set in stone as some other topics, and there are still some stuff to work out. As noted, "all strange attracters that have been found have had fractal dimensions" is not to say that all strange attractors that will be found will have fractal dimension, although it is very likely.
Answer: I agree with Martin that there is no accepted definition even of attractor (I like Milnor's paper, "On the concept of attractor", see its Scholarpedia entrance http://www.scholarpedia.org/article/Attractor).
However, it is usually (at least I believe so) said that an attractor is strange if it has positive Lyapunov exponents rather than having sensitive dependence on initial conditions (see for example this paper by Jaeger where he works on "Strange non caotic" attractors http://arxiv.org/abs/0709.0269).
On the fractal dimension mentioned by Padraig, I quite disagree, in fact, an Anosov diffeomorphism on the torus satisfies the attractor conditions (if one is not confortable with the fact that the whole manifold is an attractor, one can multiply by a strong contraction, so a Cr manifold will persist) and has no fractal dimensions.
I also recomend the following paper by Bonatti, Li and Yang ( http://arxiv.org/abs/0904.4393) which discusses possible definitions of attractors related with generic dynamics.
Comment: A reply quite to the point (although I disagree that the reference at the end is helpful in this respect).
Answer: Mainly strange attractors are objects on which you can build the skeleton of dynamics of a chaotic map in case of a dissipative systems. If the system is dissipative, and chaotic, it will generally have a strange attractor. The dimension is non-integer.
Answer: "Sensitive dependence on initial conditions" is the term I remember most clearly.
Which I believe is defined as,
for any epsilon (e) and delta (d)- two positive real values
there exists an n (natural number)s.t.
|x - y| < e => | f^n (x) - f^n (y) | > d
so here it the point is that if e is small and d big, then you can eventually make iterates of x and y far apart.
Unfortunately this obviously holds for functions as simple as f(x) = 2x
so sensitive dependence on initial conditions doesn't imply chaos. but is necessary.
so a "Strange Attractor" must be an attractor, and must be "strange" which is a adjective rather than a mathematical term, which generally means 'chaotic' which will require sensitive dependence on initial conditions.

Definition of a strange attractor.


What needs to be done locally in case of a car accident while travelling in the US?

Question: My friend is away from home right now; he lives in Ohio and has had a car accident in the state of New Jersey. What does he need to do locally before he can return home? He says a state trooper took his testimony; should he stick around and ensure a police report is filed or can he leave and handle anything else via mail?
Comment: @Karlson No other car involved, just evil deer and bad road conditions. He did report it, the car is totaled.
Comment: What kind of an accident?  Did he report it to his insurance company?  Was any other car involved?  Did the other driver's insurance information known?  How much time is he willing to waste in NJ?
Answer: He should call the courthouse for the relevant jurisdiction.  If it was a state trooper that took down his info, then that means call the state courthouse.  They can probably look up the incident based on his name, driver's license number, etc.  Then ask them what will be required of him, and if he can return home.
Worst case scenario (if he wasn't already arrested for a grievous violation such as a DUI), is he'll be asked to appear in court later, if the police report says he is at-fault, and he is charged with a violation.  He can probably waive his right to a court hearing, by mail (depending on what he is charged with), and pay whatever the default penalties are.
If he is not at-fault, then he probably doesn't need to do anything in New Jersey, from a legal standpoint.  His insurance company will likely want to inspect the vehicle. If it's undrivable, that means in NJ, but he probably doesn't need to be around for that.
The only reason I can think of he may need to stick around (or return) to NJ if he is not at-fault is if the state wants him as a witness in prosecuting the other driver(s). And that will likely take several weeks or longer.
But there are really just way too many variables to provide any sort of definite answer, short of calling the appropriate courthouse and asking.
Answer: First, determine if emergency medical assistance is needed.  Then, take pictures and get contact information from any bystander that witnessed the collision.  Do not assume that the police/trooper will be thorough here -- they have many other accidents to work, crimes to solve etc.  Second, tell the officer exactly what happened and make sure his/her version of events shows up in the police reports.  Call your own insurance carrier to report the accident while on the scene to make sure you've done what your insurer may require.  Typically, it takes 3-5 days for a police report to become available so waiting is usually not an option and many states now have there reports online.  It should not matter that your friend travels out of state as long as this information is secure.
For more useful information, see McAleer Law's page "what to do after a car accident" -- it has a wealth of useful information.
Comment: Great advice. It wouldn't have helped in this case, since he was suffering from injuries and so wasn't able to take photos or anything, but in a smaller accident that'd be a great idea.
Answer: The normal procedure is that trooper or a local police officer write up a report and mail it to their address.  Problem is that they forget so what your friend needed to do was take down the information about the police officer.  Department, Badge Number, and Name.  Once he gets back to Ohio unless the report arrives in 2 weeks he will have to start calling up the department and asking for that report.  The better solution would be to have his insurance company do that if he has comprehensive coverage on his insurance.
Thankfully he didn't total himself which has been known to happen on the back roads of the United States. :)
Comment: The State Trooper took his info, so he's good then?
Comment: @Yamikuronue Did he take trooper's info?  If yes then he's as good as he can be.
Comment: I doubt it, he'd hit his head and was arguing with the paramedics by the time the Trooper finished. I'll tell him to call down to the station and ask about it.


Is it better to have an 'edit mode'?

Question: On a page like a User profile - is it better to allow editing fields by going into an explicit "Edit Mode" where all fields become editable? Or is it better to just have all the fields have 'edit' buttons when you're viewing your own profile?
Comment: There was a [similar question asked earlier](http://ux.stackexchange.com/questions/15840/should-edit-details-form-be-in-a-modal-dialog-or-directly-on-the-page) where a great solution was proposed. This question can be kept because of different wording but doesn't require an answer.
Answer: The answer depends on how your audience is expected to use the site.
Edit Mode
 - If your user is likely to edit their profile only once or twice, as an isolated task - and can easily divide the two modes (view / edit) conceptually - then by all means  make use of an edit mode.
Inline Edit
 - If the editing is likely to be an iterative process - where the use is expected to slowly add to, and amend, information stored in their profile over time - it might be better to allow inline editing.
Answer: I choose option 3) No seperate 'Edit' or 'View' mode at all.
Just give them the ability to go into the user details page and if the user wants to edit something the can.
Provide them with a 'Save Changes' button at the bottom / top of the form that saves any changes they made. If they want to edit then they can, if they just want to View the details then they can just do this too.
Comment: Isn't that the same as 2nd option given in the question?
Comment: I took the second option to mean that you have to select each field, and that each one has it's own 'Edit' button alongside. Either way the two suggestions in the main question both rely on the user entering an 'Edit' state that differs from the default view. I suggest just making the Edit and View states the same.
Comment: I understood that option 1 is different views/states and option 2 is edit mode only for own details.
Comment: Option C: Current profile view is for profile owner or for different user?
Answer: The first option is much better than the second for both (dynamic or static) interfaces. Doubtful convenience in fast access to an editing mode is expressed in ten elements of mode management (probably with different interaction) instead of one. It looks as a TV remote, powered by two AAA-batteries, with separate compartments for each of batteries with the different mechanism of opening of a cover. Considering that the basic part of time you switch channels, instead of be engaged in replacement of batteries - it's much easier to remember one model of interaction.
Answer: I am a developer. I currently develop most stuff in the edit mode, meaning if you are allowed to edit the information, you can edit it from where you can see it.
So, if you are a manager, and you have drilled down to an employee, and you have permission to change the employee info, then you will see text boxes, not just text.  If there are things that you cannot change due to permissions, you will see just text, not text boxes.
I think it's fair to assume that if you can change information and you came to a place to view that information, odds are, you might need to change that information.
And, I loathe creating duplicate display and edit pages. You have to create those pages as well as the navigation between those pages.
I find it much easier to code locking down individual items that to grant 0% or 100% access on whole pages.

Is it better to have an 'edit mode'?


How do I trade the lights that these two light switches control with each other?

Question: I have two light switches in my bathroom.   I'd like to swap the lights that they operate with each other.
I have no experience with this kind of thing, but I'm hoping the proximity of the two switches makes it simpler.
Comment: This is unlikely to be easy for someone with no electrical or cable running experience.  The proximity of them just means there is less drywall to rip up.
Comment: We really need to know what the wires inside of the boxes look like.
Comment: How close are the lights they control? Or do they do something like one turns on the fan and the other turns on the light in a shared vent?
Answer: Your honest statement, "I have no experience with this kind of thing," is both refreshing and cautionary. Electricity can be dangerous. Most of us who dabble in it have learned a good bit by watching those who are much more experienced. When in doubt, call a pro (or at least a well-seasoned amateur). What follows is not a recommendation for you to proceed. Think about getting more experienced help.
The switches can be switched. How difficult it is depends on how they are presently wired.
The switch on the left, Switch A, has a full circuit in the box, that is, a hot, neutral and ground, which powers the outlet. Switch A above it is probably on the same line and works by interrupting the hot current to the light it controls, Light One. The neutral and ground that feed the outlet are probably tied in to the controlled light. The question is, where does the switch on the right, Switch B, and the light it controls, Light Two, get their power?
If Light Two gets its power (hot and neutral) directly from the box that contains Switch A, and the line to Switch B is just a two wire hot line that Switch B either connects or disconnects to Light Two (back through the box containing Switch A), then the lines in Switch A's box that control both lights can be swapped to reverse the switches.
If Light Two draws its power directly from the box on the right (rather than through the Switch A box), you cannot reverse the controls without running a new line.  The most likely solution is to open a channel in the wall between the two boxes and run a new line to Switch B. Then Switches A and B can be reassigned. The channel can then be patched with drywall tape and compound.
The way to tell whether you have the easy or the hard setup is look in the box containing Switch B.  If there is just one line with a black and white wire (and a bare ground), but no other wire, you have the easy setup.  If there are two or more lines in Switch B box, you have the harder setup.
Exactly how to swap the wires is a bit complex, and if you do not already know how to do this, you really should have help.
Answer: If you cant pull down the wall its probably not going to be a friendly task.  there is likely to be 2-4 studs between those 2 switches that you would have to drill through to swap wires. If you can get in to the celing or floor depending on where the wires were run you may be able to do it that way.
step 1 is to find how the wire runs from the switch to the light. if that light is under another level of the house its not going to be fun as it means tearing up the wall and running the wires that way.  if you can get in to a crawl space or attic to get to the wires as they run from the light you can put a new section of wire on the switch end and pull it in to place as you pull up the old wire.
step 2 once you have a feed from the switch to the attic you can run the original wire from the light to the feed from the switch you want to activate that light.  If you have a mobile with a veneer wood wall you may be able to pop the panels off and work on the wiring directly just depends on how the walls are built.
I have not covered the saftey, sizing etc this is just an overview of what would need to be done. most of that would require seeing the actual spot and having an idea of the code restrictions for your area.


Home-made LME&DME-is it achievable?

Question: Pretty often when you brew an all-grain batch you get a quantity of extra wort with no apparent use. A common practice is to keep some of that wort for a starter for the next batch. Could someone instead process it into LME (by boiling it for some more time and storing it in a sanitised vessel) or, even better into DME (by other means) in order to use it as a fermentable in the future?
Comment: I think a good question to ask would be, why do I have wort left over.  If there is any wort leftover its usually very low gravity.  One pound of DME is ~44 gravity points in a gallon.  So a couple gallons of 1.015 wort would be very little DME.  I think this is why people just save it for starters.
Comment: Brewchez makes an excellent point.
Answer: In short, no.  Any LME you end up with will be super-dark.  Dry malt extract requires flash drying with high pressure nozzles and is not a home operation.  Briess has some explanation of the process.
The technical nature of creating DME and LME is more than can be done in the home.  It's even harder to do than malting your own grains, which is difficult enough, and requires a lot of specialized equipment.
Besides, a gallon of wort probably gets you less than a handful of DME.
Answer: If you mean dry extracts, no.  The previous poster has that covered.  It requires freeze-drying and a bunch of other complicated stuff.  However, you can make your own extract syrups.  It's best not to think of them as LME OR DME at this point but more specific extracts (stout,pilsner,pale ale, etc.).  Basically, all you do is mash out your grain and then boil it down until you get a thick syrup.  You're just removing the water from the solution.  Be sure you store it in sterilized airtight containers, though.
Is this efficient? Probably not.  It's just as easy for me to do a full grain extraction every time i brew.  For this to really be worth your time, you'd need to do 10-20 gallons of mash at a time and for that to be time-efficient, you'd need industrial equipment to boil off that much liquid, not to mention mash out that much grain.

Home-made LME&DME-is it achievable?


Get CAML from LINQ to SharePoint

Question: Is there an equilivant to GetCommand in LINQ to SharePoint?  I know about ctx.Log, but I'm looking for something to retrieve the CAML from a LINQ statement without executing it.
Answer: Yes, as described in How to: View CAML Generated by LINQ to SharePoint, you just need to access the DataContext.Log property.
Answer: If you want to view your CAML query without deploying and executing the code, you can use LINQPAD with the SharePoint 2010 Data Context Driver (from Codeplex). If you scroll down on the linked documentation page, you see that the CAML is generated under the SQL result tab.

Get CAML from LINQ to SharePoint


Existence of two solutions

Question: I am having a problem with the following exercise.
I need to show the $x^2 = \cos x $ has two solutions.
Thank you in advance.
Comment: In what interval?
Comment: The interval is not specified
Answer: Let $f(x)=x^2-\cos x$. Note that the curve $y=f(x)$ is symmetric about the $y$-axis. It will thus be enough to show that $f(x)=0$ has a unique positive solution. That there is a unique negative solution follows by symmetry.
There is a positive solution, since $f(0)\lt 0$ and $f(100)\gt 0$. (Then use the Intermediate Value Theorem.)
For uniqueness of the positive solution, note that $f'(x)=2x+\sin x$.   In the interval $(0,\pi/2)$, $f'(x)$ is positive because both terms are positive. And for $x\ge \pi/2$, we have $f'(x)\ge \pi-1$.
Answer: $f(x) = x^2 - \cos x$ is a continuous function. Since $f(0) = -1$ and $f(\frac{\pi}{2}) = \frac{\pi^2}{4}$, $f$ has at least one zero in the interval $(0, \pi/2)$. Its derivative $f'(x) = 2x + \sin x$ is strictly positive in the interval $(0,\pi/2)$ so $f$ is strictly increasing and we conclude that $f$ has exactly one root $f(x_0) = 0$ in the interval $(0,\pi/2)$.
For $x \geq \pi/2$, $f(x) \geq \frac{\pi^2}{4} - 1 > 0$, so $f$ has no roots in $[\pi/2, +\infty)$.
Since $f$ is symmetric with respect to the $y$-axis, meaning $f(x) = f(-x)$, $f$ has only two roots -- $x_0$ and $-x_0$.
Comment: I was asked as well to solve this equation using Taylor polynomials (I did that) and found that $x_0=± \sqrt{\frac{2}{3}}$.

But what are the bounds of the error?
Comment: @user43418:   The bound for the error in $y$ is the error in the Taylor series, $\frac {x^4}{4!}$.  You can use the slope of the function at this point to approximate the error in $x$.
Comment: I don't understand...
Answer: Assuming you want to show that $f(x) = \cos x - x^2$ has two solutions over $\mathbb R$, you could try the following approach: Since both $\cos x$ and $x^2$ are even functions, if $f(x)$ has a solution $x > 0$, $-x$ is also a solution. Therefore, it is sufficient to prove that $f(0) \neq 0$ and there is exactly one solution $x \in ]0,\infty[$.
The first part is easy, since $f(0) = \cos 0 - 0^2 = 1$. So, split up the interval $]0,\infty[$ into two parts: $]0,1]$ and $]1,\infty[$. In the following, we will use three well-known properties of $\cos x$, namely:

$\cos x$ is strictly decreasing on $[0,\pi]$.
$|\cos x| \le 1$ for all $x \in \mathbb R$.
$\cos$ is continuous.

In the first interval, we have that $1 = \cos 0 > \cos 1$. On the other hand, $0^2 = 0 < 1$ and $1^2 = 1 > \cos 1$, so $f(0) = 1$ and $f(1) < 0$. By continuity of $\cos$ and $x^2$, $f$ is continuous. Hence by the Intermediate Value Theorem, the equation $f(x) = 0$ has a solution in $[0,1]$, and that solution is not $0$.
Furthermore, let $x > 1$. Then $f(x) \le 1- x^2 < 0$ by property (2), so there are no solutions for $x > 1$ (and, by $f(x)$ even, $x < -1$).
It remains to show that there is no further solution in $[0,1]$. But this follows from the fact that $f(x)$ is strictly decreasing on $[0,1]$.
Answer: Here is a different approach. Consider the function
$$
F(x)=\frac{x^3}{3}-\sin x.
$$
Since $DF(x)=x^2-\cos x$, any critical point of $F$ solves your equation. Now, $F(x) \to -\infty$ as $x \to -\infty$, and $F(x) \to +\infty$ as $x \to +\infty$.  Since $DF(0)=-1$ and $F(0)=0$, there are points $x_1<0$ and $x_2>0$ where $F(x_1)>0$ and $F(x_2)<0$.
Therefore $F$ must have a local maximum point and a local minimum point: two critical points.

Existence of two solutions


Why is lightning more rare during snow storms than rain storms?

Question: Lightning and thunder during a snow storm is uncommon.  As far as I know, more uncommon than during a typical rain storm.  Why is this?  I speculate it might be one, or both, of the following two ideas, one having to do with a change in the dielectric, the other having to do with a change in the catalyst.
Comment: Related: http://physics.stackexchange.com/questions/4006/why-are-cold-thunderstorms-rare
Answer: I can't point to a definitive reference, but my recollection is that thunderstorms are associated with a lower layer of warm air rising rapidly through an upper layer of cold air. It's the rapid vertical transport that generates the static charge and hence the lightening. In winter it's rare to get these atmospheric conditions.
So it's not that there's something special about snow that stops lightening, it's that the atmospheric conditions in very cold weather aren't conducive (no pun intended :-) to lightening.
Answer: The mechanism by which lightning is produced is complex and imperfectly understood, but we know moisture is important in two respects:

Heat is released when water vapor in the air condenses into liquid drops, and this heat helps provide energy to the thunderstorm.
Interactions between supercooled liquid water droplets and ice crystals in the upper atmosphere (about 15,000 to 25,000 feet above sea level) are needed to generate the electrical charges that accumulate until a lightning bolt occurs.

Cold winter air typically doesn't contain a lot of moisture and so isn't conducive to thunderstorms.
Air instability is also important to thunderstorm formation. It commonly occurs when warm air near the surface of the Earth rises due to convection on a sunny summer day when the ground gets hot and warms up the air immediately above it. As the hot air rises, cooler air descends to replace it. If conditions are right, strong updrafts can form that quickly move the warm, humid surface air up to the higher reaches of the troposphere, where the water vapor in the air cools and condenses to fall as rain (or ice, if it's cold enough). These updrafts are a hallmark of thunderstorms--the strong upward motion of the air encourages the interactions between water droplets and ice crystals that can lead to lightning.
In winter, cold surface air temperatures and reduced sunlight mean there's less surface heating, less convection, and thus fewer opportunities for thunderstorms.
Answer: It is simple:-
Snow is solid distilled water (high resistance and dielectric break-down).
Rain water is also distilled water but during its journey to ground, some acidic gases like ( NO2, SO2, SO3 ... etc. ) mix in the rain water droplet and hence becomes better conductor than snow.
I doubt if this question is a Physics question.
Answer: Expounding upon John Rennie's answer since this question is still open. From a thermodynamic point of view: All of the constituents involved in lightning are terrestrial, and we haven't observed anything to have a negative heat capacity naturally so far. We know that if we model a region where a lightning strike forms and travels some, we can say that this approximate region itself has a specific heat capacity $ c_0 $, that is the average SHC for the region. Lightning merits energy transfer, governed by $ \Delta Q = mc\Delta T $.
But let's say that we spike down the temperature of our system to bone chilling temperature. $\Delta T$ is going to be negative, and depending on how far down we spike the temperature, very negative. This will correspond to a change in energy of our system. We know at the very least, that a lightning strike again, is a reaction of energy transfer. At this point, there will be significantly lower amount of energy in the surroundings to even generate a lightning strike ( since all the particles are moving around slower as a result of having less thermal energy ). If a strike were to still occur, in it's travel path, there are yet still regions ( ie in Winter ) that have low thermal energy content. Entropy requires that energy flow from concentrated useful forms, to unordered, less useful forms: Heat! The lightning strike will have to pay a hefty energy toll for the air in the travel path to move, allowing it's wrath, but with a low enough temp or large enough distance, this will either completely disallow a strike, or make it relatively small to the monsters we hear in summer storms.
In the end, it all comes down to moving particles, as the other two responders have stated. Particles with energy close to rest energy are going to be harder to move than those that already have some thermal energy: Something the cold saps out.


What will happen when you reach 1000 saves?

Question: In Skyrim, I make backups of my quicksaves very often, and I have a question: what will happen if you save the game more than 999 times (with overwriting, while I made 150 saves in Skyrim, I have only about 30 individual save files)?
Comment: Note that in Skyrim each save has a size of ~14.5 MB, that'd be much in total.
Answer: I just hit 1000 save games in Skyrim yesterday and I can tell you that nothing bad happens. The counter is able to handle 4 digits save game files without any problem. I am currently at save game number 1030 or so.
Comment: Thanks for your answer! What about FNV?
Comment: I have over 1200 saves for FNV on Xbox 360 without any issue.  I can only assume the PC version is the same.
Comment: Yes, I have over 1000 saves in Fallout with no issue.
Answer: Skyrim has a major bug in it (this same behavior is found in Oblivion as well) where after you reach 1000+ saved games, it very frequently stops saving games on user demand, but auto save continues to work.  I have experienced this bug on the Xbox 360.
This issue has been reported by another player over at http://www.gamefaqs.com/boards/615803-/61870496 and I would have loved to add my notes there but the idiots oddly archived the thread so that no one can comment on the problem.
The real pain here is that the issue seems to have a spillover effect that also causes problems for other Skyrim accounts on the same Xbox even though they have far fewer than 1000 saved games. For example, my daughter has 200 saved games and started having problems saving her games after I hit the 1000+ mark.
In Oblivion, the manual game saves would actually work but they would appear at the bottom, or near the bottom, of your saved game list and you could go find them with effort. But this same behavior does not appear to be happening with Skyrim. It is simply not saving the games when you tell it too. The interface will give the appearance of the game being saved, but it will not be saved. Only auto saves work and we're now having to go through backflips to trigger auto saves when we want to save the game.
In Oblivion I was able to make things behave better by going through the horrible Xbox interface and spend hours deleting single game saves, file, by, file, over, and, over, for, hours, because there is no bulk delete functionality on the Xbox, but I have not yet tried this this pain in the ass solution with Skyrim.
Bottom line, because of Skyrim's poor game design where game saves are involved, you should develop a habit of capping your total game saves to well below 1000 by saving over previously saved games.
To my knowledge, there is no known fix for this profoundly bad flaw in the Skyrim game save architecture.
Answer: In my experience, at least with regard to the PS3, having over 999 saves from either Skyrim and FO:NV hasn't led to any problems.
Answer: After I passed 1000 saves while playing Skyrim on Xbox 360 the system seemed to be saving but when I would quit the game and come back it would load a previous save. The later saves were not visible when trying to load. When I deleted several of the earlier saves I saw that the later saves started showing up and were available to load. Now when I save, the system acts normally. So it seems that after 999 saves the system does still save but it cannot show or load them until you reduce the number of saves. I hope this helps someone else.


Preposition used after "do"

Question: I am an English native speaker working as a teacher in Germany.
When marking my pupils' essays I often encounter the phrase "to do something against something", which is as far as I know simply a direct translation of a German phrase. But I cannot find any answers in my dictionaries.
One current example: "The USA never did anything against violence."
My gut feelings says the preposition has to be about instead of against here. Is that correct?
If not, fine; but further, can against ever be used after do something?
Comment: "My gut feelings says it has to about here" - I'm not sure what that sentence is lacking, but I can't decipher it. What did you mean, there?
Comment: I meant the preposition should be "about" and not "against" shouldn't it?
Comment: It might be, Naomi, but it surely doesn't need to be.

Further, against can be and often is used after do something?
Answer: It is alright to use against after do. Against simply means in opposition to, so

The USA never did anything against violence

means

The USA never did anything in opposition to violence.

On the other hand about means with regard to, so

The USA never did anything about violence

means

The USA never did anything with regard to violence.
Answer: The answer to the last part of the question is clearly that against can be used after do. Never do anything against the law is one example. I never did anything against you is another. Why, then, does The USA never did anything against violence sound odd, when it is certainly grammatical? We can, after all, say The USA intends to act against violence wherever it occurs. The reason is semantic rather grammatical. To do X against Y requires Y to be, at least, a neutral entity. When, as with violence, it has negative connotations, there is a mismatch. Compare The USA never did anything against violence with The USA never did anything against any country that sought peace and stability. In the first, against needs to be substituted with about to effectively convey the intended meaning. In the second, against cannot be substituted with about without changing entirely the intended meaning.
Comment: So the implication is that it sounds wrong because of the shared negative connotations of *violence* and *against*? Or am I not following you?
Answer: Different prepositions get used in different ways with do. I realize this is not a helpful comment, but at least it's true. Of the Wikipedia list, about seems to me the right single preposition for the cited sentence, though, as noted, against is possible. There is a lot of individual variation on this topic in English, but do something about X is an idiom for pretty much everybody.
The problem is that this do is an Active do, a Pro-verb (analogous to Pronoun, but usually spelled with a hyphen to distinguish it from proverb) that can stand for any semantically active verb (i.e, one that denotes an action, requires an agent subject, and can occur in the progressive construction).
Often some NP appears as the direct object of Active do in order to clarify the action. This is where the Negative Polarity Item anything, triggered by never, fits in; this is one of those places where you can't use something very easily.
Further clarification of do can use prepositional phrases. Virtually any preposition allowing motion, like about, against, beyond, on, and toward(s) can be used to relate the action of do metaphorically to some topic. Some others, specifically targeted at this, include modulo, concerning, and regarding (note that the last two are reduced participles).
Finally, of course, the usual adverbial prepositions can occur here as elsewhere, but are not governed by do specifically, and only indicate where, when, how, etc.
Answer: I think whilst using the word against in that context is understandable, it's not the best word to use. Something is telling me against shouldn't be use when discussing an abstract noun (such as love, freedom, happiness). Instead I would use in opposition to, and leave being against for concrete nouns
I can't find a rule anywhere online supporting this, so it's just my gut feeling really.
Comment: I feel exactly the same, but it's driving me nuts that there is no clearcut answer.

Preposition used after "do"


Applications of Gauss sums

Question: For proving the quadratic reciprocity, Gauss sums are very useful. However this seems an ad-hoc construction. Is this useful in a wider context? What are some other uses for Gauss sums?
Comment: [Related HSM.SE question](http://hsm.stackexchange.com/questions/2133/what-motivated-gauss-quadratic-sums)
Answer: Gauss sums are not an ad-hoc construction!  I know two ways to motivate the definition, one of which requires that you know a little Galois theory and the other which is totally mysterious to me.
Here is the Galois-theoretic explanation.  Let $\zeta_p$ be a primitive $p^{th}$ root of unity, for $p$ prime.  The cyclotomic field $\mathbb{Q}(\zeta_p)$ is Galois, so one can define its Galois group, the group of all field automorphisms which preserve $\mathbb{Q}$.  Such an automorphism is determined by what it does to $\zeta_p$, and it must send $\zeta_p$ to another primitive $p^{th}$ root of unity.  It follows that the Galois group $G = \text{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$ is isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{\times}$, which is cyclic of order $p-1$.
Now suppose $p$ is odd.  As a cyclic group of even order, $G$ has a unique subgroup $H$ of index two given precisely by the multiplicative group of quadratic residues $\bmod p$, so by the fundamental theorem of Galois theory the fixed field $\mathbb{Q}(\zeta_p)^H$ is the unique quadratic subextension of $\mathbb{Q}(\zeta_p)$.  And it's not hard to see that this unique quadratic subextension must be generated by
$$\sum_{\sigma \in H} \sigma(\zeta_p) = \sum_{a \text{ is a QR}} \zeta_p^a = \frac{1}{2} \left( \sum_{a=1}^{p-1} \zeta_p^{a^2} \right)$$
which you will of course recognize as a Gauss sum!  So the Gauss sum generates a quadratic subextension, and any of various methods will tell you that this subextension is precisely $\mathbb{Q}(\sqrt{p^{\ast}})$ where $p^{\ast} = (-1)^{ \frac{p-1}{2} } p$.  (This does not actually require any computation: if you know enough algebraic number theory, it follows from a consideration of which primes ramify in cyclotomic extensions.)
The totally mysterious explanation is that Gauss sums naturally appear when you start thinking about the discrete Fourier transform.  For example, the trace of the DFT matrix is a Gauss sum.  But more mysteriously, Gauss sums are eigenfunctions of the DFT in a certain sense.  (I sketch how this works here.)  There is a sort of mysterious connection here to the Gaussian distribution, which is an eigenfunction of the continuous Fourier transform; see this MO question.  Again, I don't know what to make of this.  There is a book by Berg called The Fourier-analytic proof of quadratic reciprocity and it may or may not be about this construction.
Answer: Not just quadratic reciprocity, one can use them to prove higher
reciprocity laws: see Ireland and Rosen's A Classical Introduction
to Modern Number Theory. They also turn up in the functional
equation for Dirichlet L-functions (and are massively generalized in the
topic of root numbers).
Answer: It can be used to prove a certain theorem called the Stickelberger's theorem, the likes of which can in turn be used for proving things like Catalan's Conjecture.
Answer: They are also used to describe something called the Talbot Effect:
http://www.phy.bris.ac.uk/people/berry_mv/research.html
look at #8 in the list. I attended a seminar by Mike Berry about 12 years ago where he claimed that the Talbot Effect was a physical manifestation of Gauss Sums.
Answer: Gauss sums and exponential sums in general are particularly useful for determining the size of certain algebraic varieties in finite fields or even in general abelian groups.  If one defines
$$
A_t = \{x \in \mathbb{F}_q^d : f(x) = t\}
$$
where $t \in \mathbb{F}_q\setminus\{0\}$, then by orthogonality we have
$$
|A_t| = q^{-1} \sum_{s \in \mathbb{F}_q} \sum_{x \in \mathbb{F}_q^d} \chi(s (f(x) - t)),
$$
where $\chi$ is any nontrivial additive character on $\mathbb{F}_q$.
For example, if one considers $x = (x_1, \dots , x_d) \in \mathbb{F}_q^d$ and defines $f(x) = x_1^2 + \dots + x_d^2$, then $A_t$ would be some finite field analogue of a sphere.  Bounding such a set would then be equivalent to bounding
$$
q^{-1}\sum_{s \in \mathbb{F}_q} \left(\sum_{x \in \mathbb{F}_q} \chi(sx^2) \right)^d \chi(-st).
$$
Gauss Sums and in particular well-known bounds for Gauss sums imply that such a sum is of size $q^{d-1}(1 + o_d(1))$ as $q \to \infty$.
As Qiaochu points out above, such bounds are nice to have when one works with the discrete Fourier transform.
Answer: Srinivasa Ramanujan actually had discovered some definite integral formulas related to the Gauss sums. Please see the below article:

Some definite integrals connected with Gauss sums. Messenger of Mathematics XLIV, $1915$, $75-85$

From Wikipedia: ( Sorry, I can't explain this.)

The absolute value of Gauss sums is usually found as an application of Plancherel's theorem on finite groups.

Another application of the Gauss sum: How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$

http://www.isibang.ac.in/~sury/luckyoct10.pdf
Answer: A small additional note, in line with an earlier answer: Gauss sums are, literally, the Lagrange resolvents obtained in the course of expressing roots of unity in terms of radicals. (Yes, then the Kummer-Stickelberger business can be used to effectively obtain the actual radical expressions...:  here .)

Applications of Gauss sums


Safe to Repair NTFS Drive/Partitions When Paragon Is Installed

Question: Is is safe to repair NTFS drive/partitions with Paragon installed? I don't have access to windows anymore and can't format them
Comment: Disk utility is not designed to fix NTFS volumes.
Answer: It is safe to use Disk Utility to add/delete partitions on a drive that is partitioned in MBR or GUID. As long as paragon followed the standards, you won't have any issues.
Do note that NTFS volumes are left alone by Disk Utility so you won't be changing them - just deleting and erasing or working in the free space around them.
Answer: Do not try it. I did it and now I have a found.000 folder on my external drive that replaced everything that was on it.


How can I check the health of my SSD disk

Question: I replaced my disk with an SSD. Smooth transition, everythng fine. Now, I want to keep the disk health checked, since I know that SSD tend to degrade over time. Which utilities can I use for this task ?
Answer: DiskTester from diglloydTools and its "Recondition" seems like one of the only options.
Answer: there is no one way to diagnose this info because it depends on the controller type used in the SSD. It is possible that the vendor for your controller or the manufactureer of the controller chip have released software, and this is where you should look. What might they tell you? Summary:
an SSD (made with NAND Flash) is composed of "blocks - each one from 4k to 16k usually - that add up to the amount your SSD can hold. Each block can be erased or read from all it once, (although it is possible to write them little bits at a time) but only a certain number of times. It is common to see NAND that has 5,000 to 10,000 times it can be erased. That doesn't sound like a lot (when we are used to talking about gigabytes and gigahertz) but the way in which the SSD writes data to itself - called "wear leveling" - can make this 5k to 10k last a very long time.
The only metric worth measuring becomes how many times each cell has been written to (and which cells are dead altogether, which controllers expect and anticipate),  but because each controller is different how to get that number (if possible) and what that number means in terms of percent depleted becomes a highly specific question.
Bottom line, ask your manufacturer for diagnostic tools.


Suitable ways to get around the Darién Gap?

Question: I'm planning a backpacking trip from Central America down to South America.  Does anyone have a recommended way of getting around the Darién Gap?  I read that you can fly, or you can take a boat but the information seems extremely limited.  How much does it cost?  Can I book alternative routes ahead of time?
Answer: I was in South America (Argentina to Colombia) and in Ecuador & Colombia met a lot of people coming down from (and afterwards friends I'd met travelled up into) Central America.
With just one exception (who flew), the others had all gone via the San Blas Islands.
http://wikitravel.org/en/San_Blas_Islands
Which to me look simply stunning!  I'd seen several of their pictures, and if I wasn't hopping across to Florida for the shuttle launch (which didn't happen, sigh) I'd have done this too - it sounds like the perfect stress-free break and fun way to get around 'the Gap'.
Comment: Yes, correct. :)
Comment: Do you mean they sailed to the San Blas Islands I assume?
Answer: Several people already mentioned the sail boat option, but as none of them has done it themselves, I will add my experience from May 2009:
As said elsewhere the hostels in Panama city act as intermediaries between travelers and the boats. They have a list of boats leaving on certain dates with prices which ranging from 250 to 350 US dollars.
While some of the boats seem to do the trip from San Blas to Cartagena (usually five days) in Colombia all the time and you can find out about them online, others (like mine) didn't bring up any results on Google. So you never know what you get.
There are bigger boats for up to 30 passengers, they have a proper crew and you are just a passenger.
In my case we had a single Australian boat owner and a beautiful mid 1960s yacht. We were six passenger turned crew members as we had to do all kinds of work on the boat. From cooking and cleaning to setting sails and navigating, this added to the experience though and we all loved it. If you don't want to do any work, make sure you take a bigger boat and check in advance. The hostels can answer your questions by talking to the boat owners.
The hostel in Panama city usually organises transport to the East coast in 4 wheel drives and also take a good cut of your full price. We left at 5 in the morning in Panama city and the last part of the drive was on muddy roads through the jungle. We then transferred to a river boat for an hour and then onto another boat bringing us to one of the San Blas islands where you get on your sail boat.
As with any tour, whether you really enjoy it depends a lot of your fellow travelers, so until you have a group already, you never know what you get.
You usually spend the first three days cruising around the small islands. Some have little villages on them which you can visit for shopping and you also get your Panama exit stamp at some immigration hut. However we spent more time snorkeling and swimming to some uninhabited islands including some tiny ones with a single palm tree.

Photo credit: Peter Hahndorf
Local people come alongside the boat to offer freshly caught fish and other food. You can then have a barbecue on an island. This is all pretty close to paradise.
During the last two days you are actually getting out to the open sea to cross over to Colombia. Cartagena is a big city but the old town is very beautiful.
Even if it may actually be more expensive than a flight, I strongly recommend the sailing trip if you have enough time. It was such a nice break from using buses all the time.
I would say both San Blas and Cartagena are worth visiting on their own anyways, so why not connect the two with a nice cruise?
The boats also sail back from Cartagena to San Blas, so this works in both directions.
Answer: When I was in Panama City about 4 or 5 years ago all the hostels had ads posted by private people, sometimes families, that sailed back and forth in their yachts and wanted travellers to come along to help with costs. There was a waiting list for each yacht.
At the time the price to sail was pretty much the same as a flight which was in the region of $300 USD.
Some people insist that you can cross easily on foot if you stick to the Caribbean coast, but all of the printed travel guides warn at length against even this option, and some of them go into some detail about why it's a bad idea.
Answer: I've heard several reports that the Darien Gap is quite dangerous due to drug trafficking. Personally, I would fly although I understand it's possible to cross by hopping on small boats on the Caribbean side. One of the Lonely Planet books (sorry I forgot which one) has more details, but they also don't recommend it.
Comment: It's not only the drug trafficking. There are also two separate guerilla groups, the FARC and the ELN. Even the indigenous villages in the jungle may not be friendly since the guerilla groups punished them severely for helping perceived enemies. Also it is not unusual for your hired guides to disappear. There is a very good episode of the reality travel show "Banged Up Abroad" aka "Locked Up Abroad" with a reenactment of one group's misadventure trying to cross the Darien.


Can I import my IMAP sent folder from another account without creating duplicates?

Question: I have an email account that supports IMAP and I also have a Gmail account. I want to import a folder from the email account that supports IMAP to Gmail (my sent messages folder).
This Lifehacker article suggests a solution for that. Basically, you take a mail client, set up IMAP for both Gmail and the account, and then drag the folder into the Gmail account. This sounds okay, but the thing is that I want the Sent folder to be synchronized with the Gmail account.
I don't mind every once in a while dragging the Sent folder to Gmail folder manually, but my question is - will it create duplicates from the Sent folder in the Gmail account?
Comment: looks like you've already got your answer down there... and probably by trying it yourself too (since you never came back)! :D
Comment: i tried this method many times, and gmail will not create duplicate.
In fact I had tons of dups in my old outlook db, and instead of trying buggy software to remove dups, I just moved everything to gmail, which cleaned everything for me.
Answer: Google's duplicate matching is based on the Message-ID: header. As long as they're truly the same message duplicates will not be created.
Answer: Gmail is pretty good at preventing duplicate messages. In fact it is so good that it will even sometimes eliminate non-duplicate messages. For example, if you send the same exact email twice (same 'from', 'to', 'subject', and 'body') to a Gmail account (within a few minutes or so), it will only appear once in the receiving Gmail account.
So my guess is that it will not produce duplicates. However, I haven't tried lifehacker's method, so I can't say for sure.
Answer: For the specific problem of keeping a Gmail-searchable archive of sent mail, I suggest an alternate solution:
Use Gmail as your SMTP server. You can do this entirely independently of using a separate IMAP server for your incoming mail. I have done this for years. Your mail client will still store a copy of every sent message in the sent folder on your IMAP account, just as before, but now Gmail keeps every sent message just as though you sent it from Gmail.
You can still set your "From" address to any address which is registered as an alternate in your Gmail account.


Which inkjet-printable labels perform well?

Question: I've never bothered to label my beer before, but I think I want to start.  Have you ever used inkjet-printable labels on your brew before?  How did they perform?  Are there any brands of labels that you could either recommend or advise to avoid?
Answer: I use Avery 3/4" round labels on the caps and that works great for me. Run a couple of sheets (24 labels per sheet) through the printer for each 5 gallon batch. I can easily identify what is in the bottles, and no clean-up when I re-use the bottles for the next batch.
Comment: This is a great suggestion for getting the desired functionality - being able to identify your beer.  It would never be mistaken for a case of form-over-function, though.  :D
Answer: I used some Avery labels one time. I don't recall which number. They work great, except they don't come off. The adhesive used is pretty solid. When you soak a bottle in water, the top portion with the ink sort of crumbles, but the adhesive sticks. It's not ideal for reuse.
Lately, when I've been sharing brews, I put a vinyl sticker on the bottle. These don't come off easily at all, but they also don't fall apart when you're cleaning them.
I'm sure there's a better option, like what the "good" breweries in this list use, but I don't know what they are.


Is there any site with photos of photography masters?

Question: Is there some online collection of really great photos of great photographers, maybe with some comments giving hints on the reason of their greatness?
Answer: If you want well-known masters of photography, then here are two sites.
Profotos has a lot of narrative about the photographers, with links to photographs.
Masters of Photography has a lot of images, but the site is littered with pop-up ads, so may put you off
Atget Photography is a good combination of image galleries and short biographies on a number of master photographers.  It also includes links to worldwide museums and galleries.
Comment: All great sites, thanks. I choose this one because actually about photography masters.
Answer: 1x is known for strict selection of photos.
Feature Shoot is a blog with frequent articles on interesting photographers, mostly consisting of photos, but often also featuring a short interview.
Answer: 500 Photographers Blog!
What genre of photography do you like? There are SO many photographers out there.
Also, visit your local art school, and go to their library, visit photography section, get some coffee and get going...
Comment: I do not konw :) I still am searching around for what fits me better.


How many skills/items/upgrades can I buy in a single playthrough?

Question: I just started Dishonored last night and I was wondering, just how far can I get in the tech tree in a single playthrough? Can I buy everything? Are there branches that cut each other off or simply not enough coin to buy it all?
Answer: Definitely possible with upgrades, but it's not necessarily a huge amount of fun spending a lot of time searching for coin loot. Runes, definitely not but you can get at least half. It's worth doing two playthroughs anyway (one ghosting, one hyperviolent vengeance spree).
Answer: I was able, by being thorough in my loot collecting, to have more than enough coin to buy all the upgrades that I had available after collecting all the blueprints I found. Although you won't actually need to buy all the upgrades if you stick to one play style through the entire game.
Comment: This covers gadget upgrades, but not powers, which require runes, of which I don't think there are enough.


How to obtain true probabilities from Black-Scholes?

Question: How to obtain true probabilities from Black-Scholes option pricing equation?
Suppose, that we know risk adjusted discount rate for the underlying asset (the drift term in the physical measure) and risk free rate. The task is to find a real (not risk neutral) expected payoff for a call option.
Answer: You cannot deduce the real-world probabilities from the option prices.
It may seem strange, but here is a simple example which might help you to understand.
Suppose that everyone in the market agrees on the real-world probabilities, and that they are not changing for any external reason.
Then suppose that the investment board of a large pension fund decides that they need to increase the amount of options they have bought because they get a feeling that they would like to hold more protection against an adverse move (and since most pension funds are net long equities, this is likely to mean that they want to buy out-of-the-money equity put options to protect against a sell off in the equity market).
The pension fund will come to the dealers (investment banks probably) and will buy a whole load of put options, say. Naturally the price in the market will go up (simple law of supply/demand, and demand has increased), which implies that the implied vols will go up.
In summary: no change in the real-world probabilities, but a big change in the implied volatilities which will in turn lead to a change in the implied underlying probability distribution.
Answer: You cannot get "true probabilities" (empirical distribution) from the BS model. Option price is required initial investment, which is risk neutral expectation of payout. "True probabilities" are irrelevant in Black-Scholes. However, you can estimate the risk-neutral probability distribution (i.e. implied risk-neutral density) of the stock returns through Breeden-Litzenberger formula.
Answer: The true probabilities underlying the B-S equation are actually postulated.
The pricing process is assumed to follow the stochastic process $d S_t =\mu S_t d t + \sigma S_t dW_t$, where $W_t$ is the Wiener process.
It means that (for simplicity, let's talk about European call)  $\ln S_T$ is distributed as $N(ln(S_0)+(\mu-\frac{1}{2}\sigma^2)T, \sigma^2 T)$
Correct me if I'm wrong, you'd like to find $E_P(C) = e^{-rT} E_P[max(S_T-K,0)] $, where P is a "physical" probability measure. Just to make sure, this expected value won't represent the fair price of the option.
If my calculations are correct, this expected value is equal to $S_0 N(d_1(\mu)) e^{(\mu-r)T} - K N(d_2(\mu))e^{-rT}$
the terms $d_1$ $d_2$ are from the B-S formula, with the adjustment to replace risk-free rate $r$ there with "risky" $\mu$

Now, I write down some derivation steps, please check them.
Let's rewrite expectation as follows, $E_P[...]=E_P[\textbf{I}(S_T\geq K)(S_T-K)]$, where $\textbf{I}(.)$ is the indicator function.
Notice that the inequality $S_T\geq K$ is equivalent to $\ln S_T \geq \ln K$
Then, $... = E_P[S_T \textbf{I}(\ln S_T \geq \ln K)]-E_P[K \textbf{I}(\ln S_T \geq \ln K)] $
$= E_P[e^{\ln S_T} \textbf{I}(\ln S_T \geq) \ln K)]- K N(d_2(\mu))$
To calculate the first term, use the following lemma: if X distributed as $N(a,s^2)$ then $E(e^X\textbf{I}(l<X))=e^{s+\frac{1}{2}s^2} N (\frac{\mu+s^2-l}{s})$
Take $\ln S_T$ as $X$ and $l$ as $\ln K$, obtain $E_P[S_T \textbf{I}(\ln S_T \geq \ln K)]=e^{\ln S_0 + (\mu - \frac{1}{2}\sigma^2)T + \frac{1}{2}\sigma^2 T}N(\frac{\ln S_0 + (\mu-\frac{1}{2}\sigma^2)T + \sigma^2 T - \ln K}{\sigma\sqrt T}) = S_0 e^\mu N(d_1(\mu))$
Finally, discount it with the risk-free rate $r$ and we get the result.
Comment: You need to discount at the option's actual expected return, not the risk-free rate in the last step, to get the correct price.
Answer: It is true that you cannot infer the real World probabilities from the BSM formula directly. It is also equally true that the "right value" of the option in the real world is obtained by replacing the risk free rate with the expected return of the stock.
Another example of this is simply to look at the real world price of a forward on the stock. If arbitrage, or replication, is not allowed would you systematically buy or sell a ten year forward on a stock where the forward is priced using the risk free rate?
As long as the expected return on the stock is above the risk free rate, which is consistent with most classical financial assumptions, you would chose to buy the forward. The equilibrium price for the forward when arbitrage is not allowed is obtained when the forward is priced using the expected return of the stock as the rate. Then you would be indifferent to buying or selling the forward.
The simple example of the forward pricing did not use any real world probabilities in pricing the forward. The probabilities might be behind how you came up with the expected rate of return, or the drift, of the stock if you wish but all you need to know to price the forward is actually the expected return.
Derivatives people, like myself, might complain that this is truoblesome because you need to make an assumption of the drift of the stock to arrive at the answear. Well tough luck, that is what equities people need to do every day when pricing stocks.
Answer: I think the simplest answer is just to understand that Black and Scholes uses risk neutral probabilities, vis-a-vis "d1" and "d2" in commonly cited derivations. These are very similar to "Z-scores" in statistics, but they are derived with the assumption that a dynamically risk-free portfolio can be constructed -- this allows us to derive a closed form solution for the differential equation. Just as how a standard Z-score can be converted to a probability, the Black and Scholes converts d1 and d2 to a "risk neutral" probabilities through the cumulative distribution function (CDF) of the standard normal ("i.e., Gaussian) variety.
If you want to find the raw probability of an options expiring above or below a given strike, it would be so much easier to just utilize the Z-score formula, as in:
$$Z = \frac{\ln(S / K) - m(T-t)}{s \sqrt{T-t}}$$
$$p = \phi(Z)$$
where $S$ is the stock price, $K$ is the arbitrary strike price, $m$ is expected annualized drift, $s$ is the annualized volatility, and $T - t$ is the time till expiration in years, $p$ is the probability that the underlying will expire at or above the strike at time $T - t$, and $\phi$ is the CDF of the Gaussian distribution.
Comment: I think you are missing a $(T - t)$-factor multiplying $m$ in the nominator of $Z$. Also - $m$ is the drift of the *logarithmic* stock price. I.e. if $\mathbb{E} \left[ S_t \right] = S_0 e^{\mu t}$, then $m = \mu - s^2 / 2$.