How to show: $\psi^{(0)}\left(\frac{1}{n}\right) - \psi^{(0)}\left(1 - \frac{1}{n}\right) = -\pi\cot\left(\frac{\pi}{n}\right)$
Polygamma reflection formula
A virtual file containing the concatenation of other files Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? Postfix Connection Timed out Understanding QGIS buffer tool units Pass An Instantiated System.Type as a Type Parameter for a Generic Class Why was River Song so slow? I'm using post thumbnails to link to a page.  Is it possible to add a class name to the post thumbnail image.  &lt;li&gt;&lt;a href="&lt;?php the_permalink(); ?&gt;" &gt;&lt;?php the_post_thumbnail(); ?&gt;&lt;/a&gt;&lt;/li&gt; Is it correct to say "I wrote him" or "I wrote to him"?  My Mother was a stickler for English grammar and would say "I wrote your Uncle..." rather than "I wrote to your Uncle..." Distribution of the ratio of two i.i.d standard normals I am reading an article in which the author states that given two i.i.d random variables $X,Y\sim\mathcal{N}(0,1)$, we have $\mathbb{P}(\frac{X}{Y}\le t)=\mathbb{P}(\frac{X}{|Y|}\le t)$ since the random varaibles $\frac{X}{Y}$ and $\frac{X}{|Y|}$ are identically distributed by the symmetry of the standard normal distribution.  But I don't really understand why the random variables $\frac{X}{Y}$ and $\frac{X}{|Y|}$ are identically distributed. Can you provide a reason for why these two random variables are identically distributed? Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$. This integral is very closely related to the sophmores dream that states  $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$  For example here   Now I want to bound the integral, and showing that is less that 2.  For the interval $[0,1]$ a good bound is rewriting it to $\exp(x\log x)$ and using the expansion   $$ 1 - x \log(x) + \frac12 (-x \log(x))^2$$  but how does one handle $[1,\infty)$ ? In this answer here     gives bounds to the integral, but they are not tight enough.. So to taste my question again, how does one prove that   $$ \int_0^\infty \frac{\mathrm{d}x}{x^x} \leq 2 $$ I came across following problem     Evaluate $$\int\frac{1}{1+x^6} \,dx$$   When I asked my teacher for hint he said first evaluate     $$\int\frac{1}{1+x^4} \,dx$$   I've tried to factorize $1+x^6$ as  $$1+x^6=(x^2 + 1)(x^4 - x^2 + 1)$$ and then writing  $$I=\int\frac{1}{1+x^6} \,dx=\int\frac{1}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx=\int\frac{1+x^2-x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$ $$I=\int\frac{1}{x^4 - x^2 + 1} \,dx-\int\frac{x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$  However $$x^4-x^2+1=\left(x^2-\frac12\right)^2+\frac{3}{4}$$ But I can't see how it helps  I've also tried to reverse engineer the   And I need to have terms similar to $$\frac{x^2-1}{x^4-x^2+1} \quad , \quad \frac{1}{1+x^2} \quad , \quad \frac{1}{(x+c)^2+1}\quad , \quad \frac{1}{(x+c)^2+1}$$ in integrand, How can I transform my cute looking integrand into these huge terms?  Since in exams I will neither have access to WA nor time to reverse engineer the solution moreover it does not seem intuitive,is there any way to solve this problem with some nice tricks or maybe substitutions? Does gravity affects on electrons in a noticeable way? Is there a way to force a free electron move in a low speed in order to observe it actually hitting 'on the floor'? What is the 'rev' dropdown supposed to do? I only see this on sites where I don't have edit privileges, and only where there have been multiple revisions:        What exactly is this supposed to do?  Intuitively, I would expect it to change the text in the box to what it was at that revision, but it doesn't. And, if that's it, why only where I need to suggest edits? Stack Overflow in ES, PT, RU, JP have mixed-language descriptions How can I get a new browser session when opening a new tab or window on Firefox/Chrome? When opening new Internet Explorer window, a new browser session is issued. Therefore you can login with two different accounts on the same service (let's say ). The same does not apply when opening a new tab in the same window in Internet&nbsp;Explorer.  How can I get a new browser session when opening a new tab or window on Firefox?  How about Google Chrome? I have a common scenario in app design, where I need to store an ordered list, and make the ordering easily accessible and adjustable via an application UI.    The ordering operations are simple and operate on only one item at a time - move to top, move to bottom, move up one position, or down one position.  These positioning operations may be interspersed in with typical add / edit / delete list operations.  For discussion, this is a single-user app and db store.  Because the objects can be highly complex, I'd like to avoid caching the entire list client-side, and tracking newly created objects in memory until a big save operation occurs.    I am using MS-SQL almost exclusively on these projects, and I'm wondering if there are db features, SQL features, or design patterns I should be using to better support the management of ordered lists.    My current approach is to use a DECIMAL column to store the list positions, which become fractional during the item move process.  Then I sort the list and re-generate integer positions to normalize list.  Following this operation, it's ready for another item move.   In an example implementation, I create a column named Seq for position tracking.  Once decimal place is enough for the .5 storage.   [Seq] [decimal](10, 1) NOT NULL   When an item is re-positioned in the app UI,    I UPDATE that record with new Seq value I retrieve the full table, sorted ascending by Seq and regenerate all Seq values in the table from a 1-based integer series   The new Seq value is easily calculated-    Move to top = 0 Move to bottom = MAX(Seq) + 1 Move up = subtract 1.5 from the current Seq value Move down = add 1.5 to the current Seq value   As lists get longer, updates get slower, since I'm doing N UPDATE statements, where N is the number of rows in the table, so it seems only suitable for small tables.    Is there some kind of in-built capability for managing an ordered list in SQL or MS-SQL? Is there a means to update multiple rows in one call?    A far more performant approach would be to allow more decimal places, and then position at the median of the two adjacent items, with no re-sequencing on the list.     Move to top = Min(Seq) - 1 Move to bottom = MAX(Seq) + 1 Move up = median of the 2 previous item Seq values Move down = median of the 2 following item Seq values   This approach only requires one SELECT (containing 2 rows max) and one UPDATE.    However with enough re-positions I imagine the field would run out of decimal places. For example, if I start with items at positions 10 and 11, and keep moving items down the list from position 9, they'd be given sequences of 10.5, then 10.25, then 10.125.  After 10 moves, the positions get lengthy... 10.0009765625.    UPDATE #1 - IMPROVED RE-SEQUENCING OF THE LIST  Regarding re-sequencing, I'm more caught up on row_number(), OVER, and WITH capabilities in MS-SQL.  It's nice to find that there's a much smoother way to re-sequence the list in a single UPDATE operation;  with OrderedItems as  (     select         ItemID,         Seq,         row_number() over (order by Seq asc) NewSeq     from          Items ) update OrderedItems set Seq = NewSeq Is a small size block cipher usable? Consider a block cipher with a key of size 128 bits but a small block size, say 32 bits.  Is this kind of block cipher secure?  I would like to know to what extent I can use a small block cipher, like a 32 bit block cipher. What are the constraints so that the use is acceptable in a real life application.  For instance, if we use a one-time key, that is, to encrypt only one message. Besides, the message could be small.  Suppose that when we want so send one file, we perform a key agreement so that we use the key for one message. This scenario is not efficient, but the CPA find-then-guess game is not adequate in this case. I have watched the series of Two and a Half Men two times now till the latest episode, but can't understand why either Charlie or Walden just don't kick Alan out? I mean it is a bit cruel but they are shown to put up with him even though he gets up on their nerves (even got on my nerves).    Some examples off the top of my head:   He became the reason of his first engagement with Mia to be broken up.   He once set up his curtains on fire after ruining his day.   He invited his girlfriend/assistant to move in without his permission and started acting like he owned part of the house when he didn't even pay rent. He acted sickly and dying (Walden's time) and tricked him out of money for his care and comfort.   He deliberately tried to break his relations (Walden's) just so he can keep sponging off.   I know not the best examples, but if not in every episode then in every alternate episode he is shown to getting up on their nerves and they just keep on telling him to leave. Why don't they just kick him out? Why put up with such a leach? Now that  is in effect, people who have visited Iraq, Syria, Iran, or Sudan since 2011 are banned from entering the US under the Visa Waiver Program. However it's unclear to me how could the US officials prove one has set foot in those countries, unless:   There's a stamp in one's passport indicating so, which can be side-stepped by getting a fresh passport One mentions the fact on the ESTA application form, which is obviously easy to avoid   Is there any other way the US might be able to identify such individuals?  NB: I'm not eligible for the VWP myself and don't plan to travel to the four countries above, so it's a purely theorethical question Background image shows in front of 3D objects, not behind them. I've done a quick 3D track in blender and I'm trying to render my output, but the plate I used to track is showing up over my 3d objects.  I want to render the objects in the scene with an alpha channel and comp them over the plate I tracked. My renders however, are covered by the aforementioned plate.  Un-checking "Background Images" in the camera properties (only removes it from the viewport) isn't doing it for my render outputs.  Working on some production tests and am not really proficient with the software yet, my apologies in advance. How to get hardware specs of Android device? How can I find information on the hardware of my Android device like processor architecture, RAM, etc?