Two Apple ids and just want one
stop Ipod from asking for old Apple ID password
Volumes of n-balls: what is so special about n=5? The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$  For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ $$V_3\approx 4.18$$ $$V_4\approx 4.93$$ $$V_5\approx 5.26$$  $$V_6\approx 5.16$$ $$V_7\approx 4.72$$ It is not difficult to prove that $V_n$ assumes its maximal value when $n=5$.   Question. Is there any non-analytic (i.e. geometric, probabilistic, combinatorial...) demonstration of this fact? What is so special about $n=5$? I also have a similar question concerning the $n$-dimensional volume $S_n$ ("surface area") of a unit $n$-sphere. Why is the maximum of $S_n$ attained at $n=7$ from a geometric point of view?   note: the question has also been  for those curious to other answers. I will be landing at New Delhi airport. The worth of new laptop  (Lenovo Yoga Idea Pad 13) is about $1000 and old one (Macbook Pro) is > 1 lakh INR. I want to leave the new laptop in India. I have a copy of bill with me but not the original bill. I want to know if I have risk of being charged with custom duty. Advantages/Disadvantages of Partitioning a Drive I have a 100GB hard drive in my Win XP Pro desktop. I use an external 250GB drive for backups, and am wondering if there would be any advantages to partitioning my internal hard drive.   I understand that some people like to keep their OS and programs on one partition, and their data on another partion on the same hard drive.  I'd like people to list the advantages, or note any disadvantages.  Thanks. Background: This question relates to versions of Delphi below 2009 (ie without Unicode support built in). I have a specification that requires me to transmit a Unicode encoded string over a TCP connection but I do not have Delphi 2009.   Question Is there a single function or very small library (I don't need too much bulk) that I can use to encode a single string into UTF-8 immediately prior sending over the wire? As a second part of my question: if there are UTF-8 encoded strings being sent back as a response, I guess I would then need another function to get it back into a Delphi string format. I understand the limitations of such Unicode support in this way. I was thinking about this issue from time to time but I didn't know whether it really exists or just an speculation on my side until today when I woke up [!], I saw  downvoted (in the meantime I'll leave that answer unedited for the sake of discussion; note that this is just a sample. I'm not asking "how to deal with this specific answer." The question is completely general as it's nearly impossible to keep track of all of your answers to see whether they require updating or not). While the downvoter didn't explain a thing, I suspect the reason is that the answer is no longer applicable (It's one of the new features in iPhone OS 3.0). As we go to celebrate a long sequence of birthdays for Stack Overflow (hopefully!), we're going to see lots of answers become obsolete. How we're going to deal with this issue? Short story about a crew finding an ancient alien ship filled with tons of artifacts from their society Graph for relationship between two ordinal variables What is an appropriate graph to illustrate the relationship between two ordinal variables?  A few options I can think of:   Scatter plot with added random jitter to stop points hiding each other. Apparently a standard graphic - Minitab calls this an "individual values plot". In my opinion it may be misleading as it visually encourages a kind of linear interpolation between ordinal levels, as if the data were from an interval scale. Scatter plot adapted so that size (area) of point represents frequency of that combination of levels, rather than drawing one point for each sampling unit. I have occasionally seen such plots in practice. They can be hard to read, but the points lie on a regularly-spaced lattice which somewhat overcomes the criticism of the jittered scatter plot that it visually "intervalises" the data. Particularly if one of the variables is treated as dependent, a box plot grouped by the levels of the independent variable. Likely to look terrible if the number of levels of the dependent variable is not sufficiently high (very "flat" with missing whiskers or even worse collapsed quartiles which makes visual identification of median impossible), but at least draws attention to median and quartiles which are relevant descriptive statistics for an ordinal variable. Table of values or blank grid of cells with heat map to indicate frequency. Visually different but conceptually similar to the scatter plot with point area showing frequency.   Are there other ideas, or thoughts on which plots are preferable? Are there any fields of research in which certain ordinal-vs-ordinal plots are regarded as standard? (I seem to recall frequency heatmap being widespread in genomics but suspect that is more often for nominal-vs-nominal.) Suggestions for a good standard reference would also be very welcome, I am guessing something from Agresti.  If anyone wants to illustrate with a plot, R code for bogus sample data follows.  "How important is exercise to you?" 1 = not at all important, 2 = somewhat unimportant, 3 = neither important nor unimportant, 4 = somewhat important, 5 = very important.  "How regularly do you take a run of 10 minutes or longer?" 1 = never, 2 = less than once per fortnight, 3 = once every one or two weeks, 4 = two or three times per week, 5 = four or more times per week.  If it  would be natural to treat "often" as a dependent variable and "importance" as an independent variable, if a plot distinguishes between the two.  importance &lt;- rep(1:5, times = c(30, 42, 75, 93, 60)) often &lt;- c(rep(1:5, times = c(15, 07, 04, 03, 01)), #n=30, importance 1            rep(1:5, times = c(10, 14, 12, 03, 03)), #n=42, importance 2            rep(1:5, times = c(12, 23, 20, 13, 07)), #n=75, importance 3            rep(1:5, times = c(16, 14, 20, 30, 13)), #n=93, importance 4            rep(1:5, times = c(12, 06, 11, 17, 14))) #n=60, importance 5 running.df &lt;- data.frame(importance, often) cor.test(often, importance, method = "kendall") #positive concordance plot(running.df) #currently useless   A related question for continuous variables I found helpful, maybe a useful starting point: Simple approximation of Inverse Incomplete gamma function I have built my house in Minecraft PE, and I have a ton of fences and sticks, but I need to know how big of a perimeter I need to let me sleep at night? For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$ Integral of $\sin x \cos x$ using two methods differs by a constant? $$ \int \sin \theta\cos \theta~d \theta= \int \frac {1} 2  \sin 2\theta~ d \theta=-\frac {1} 4  \cos 2\theta$$ But, if I let  $$ u=\sin \theta , \text{ then }du=\cos \theta~d\theta $$ Then $$ \int \sin \theta\cos \theta~d \theta= \int u ~ du =\frac { u^2 } 2 =\frac {1} 2 \sin^2 \theta $$ Since $$ \sin^2 \theta =\frac {1} 2 - \frac {1} 2 \cos 2\theta$$ The above can be written as $$ \int \sin \theta\cos \theta~d \theta= \frac {1} 2 \sin^2 \theta =\frac {1} 2 \left( \frac {1} 2 - \frac {1} 2 \cos 2\theta \right)=\frac {1} 4-\frac {1} 4 \cos 2\theta   $$ Why are the two results differ by the constant $1/4$? Thank you. I have a StreamReader object that I initialized with a stream, now I want to save this stream to disk (the stream may be a .gif or .jpg or .pdf).  Existing Code:  StreamReader sr = new StreamReader(myOtherObject.InputStream);    I need to save this to disk (I have the filename). In the future I may want to store this to SQL Server.   I have the encoding type also, which I will need if I store it to SQL Server, correct? How do we escape the cycle of bad college eating? Mouseover text: And yet I never stop thinking, 'sure, these ingredients cost more than a restaurant meal, but think how many meals I'll get out of them! Especially since each one will have leftovers!'    Randall Munroe really hit the nail on the head with this comic and especially the mouseover text. There's this sort of cycle that I seem to follow when I try cooking, and I need to get my way out of it. It goes like this:   Decide I should cook something. Thumb through my recipe books, find one or two recipes I like. Write down the ingredients for the recipes in a list. Go to the grocery store, stumble around for a while and eventually find exactly those ingredients. Go back home; at some point in time soon after, cook each recipe. Whew, that was a lot of work! Spend the next week or two eating junk again, until I decide I should cook something. (repeat)   Very similar to the comic; sometimes I too let the leftovers go bad (usually not). And every time I rack up a huge bill at the supermarket and think to myself it's justified because of the leftovers or the amount of food. Not to mention, that's a lot of work for just a couple of meals, and a long list of ingredients.  I love cooking fresh meals but that's just so much overhead and so time consuming! So I always resort back to Easy Mac, Ramen, frozen dinners and frozen pizza. Yuck.  How can I escape this cycle and cut down on the overhead? How can I focus on more than 2 recipes at a time without buying giant lists of ingredients and spending tons of money? Is it really worth the trouble or should I just chock it up to being a college student and wonder about the years ahead of me where I'll have much more time to spend cooking one meal at a time? Do graduate schools care more about my grades in math courses or my general GPA? I'm currently a junior in mathematics at UIUC. I have been doing pretty well in my math courses (mostly A and A+). I will be completing a graduate course in topology, a second course in abstract algebra, and a complex analysis course before I graduate.  I feel that I can work hard to maintain good grades in my math courses, but my non-math courses are not that good (some B and C's).    Do graduate schools care more about my math courses grades more or my general GPA more?  Will poorer marks in non-math courses hurt my chances of getting into good grad schools? "Writ large", "Writ small", and others I have seen this used with other words besides "large" and "small".  What other words can be used? Can this structure be used more generally with other adjectives? Can I use variables inside {} expansion without `eval`? Can I use variables inside {} expansion without appealing to eval? If so, how?  This doesn't work:  $ touch 1.foo 1.bar $ ls 1.{foo,bar} 1.bar  1.foo $ extensions=foo,bar $ ls 1.{$extensions} ls: cannot access 1.{foo,bar}: No such file or directory   It works with eval:  $ eval ls 1.{$extensions} 1.bar  1.foo Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ __NSArrayM insertObject:atIndex:]; object cannot be nil Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$. I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number.   I found this result by exploring the series on wolframalpha and looking up the sequence of numbers generated. I have no experience with Bell numbers other than knowing that they represent the number of ways of partitioning a set.   What I tried  Looking at the base case we can verify $$\sum_{n=0}^\infty \frac{n}{n!} = \sum_{n=1}^\infty \frac{1}{(n-1)!} = e. $$  But then I realize that $n^2$ is going to be difficult.  Specifically what I want  I am doing this for fun, so I just want to glean some sort of lesson out of this. Resources that let me learn for myself are just as good!   An accepted answer will have any one of the following:   A proof that $\sum_{n = 0}^{\infty} \frac{n^2}{n!} = 2e$ (this interests me a lot) A description of why this series is related to Bell numbers and the number of partitions of a set A link to some resource that introduces Bell numbers and/or their connection to this series. I have been around on Twitter for awhile and I regularly find people using "Please to retweet this" or "Please to help me", etc. Is this proper English? I do not think prefixing the infinitive form with a 'to' is necessary here at all.