$\frac{1}{2}!$ aka $\Gamma(\frac{3}{2})$
Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?
If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$ If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$     My  idea:  $$b_{n}=\min(a_{n},\tfrac{1}{n})\Longrightarrow b_{n}\le a_{n},b_{n}\le\tfrac{1}{n}$$ then I don't know how to go on. Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check? I do not understand much about mine bitcoins, but I downloaded a program that is like a wallet, but I want to know how I do to mine. Has a calling GUIminder but is for windows, someone knows one that is for ubuntu? How can I attach a file to a question? I want to attach a code file to my question, but I am not able to find a link to do it. Is attaching files possible at all? Why aren't people voting for questions? While at least some answers from every question seem to have been up-voted as a "helpful" answer:   about 49% of questions have score of 0 or 1. another 21% of questions have a score of 2.   Most questions seem to have an answer or two with score +5. Is it that we have a lot of fluff questions and not useful ones? Or perhaps it just seems more natural to vote for answers than questions? If we want the most useful info on Stack Overflow to filter to the top, how can this disparity be fixed?  I think this may cause real problems for people getting the badges which require +25 or +100 votes on questions.    I like a lot of the feedback I'm seeing. Two problems are highlighted:   The work flow doesn't promote voting for questions the way it promotes voting for answers. One solution: when a user answers a question, the system should invite them to vote for the question.  If someone cares enough to answer a question, then they care enough to vote for it as well. Users with less than 15 reputation cannot vote for questions or answers. Is this warranted?  Maybe these users could provisionally vote for questions and answers and those ratings would only take effect when the user reaches 15 reputation?   All I ask is this. As you browse Stack Overflow, when you see a good question: vote for it.  If everyone does, then this problem will disappear. I was working quickly, (making mistakes), and suddenly rotation and scale transforms stopped working. I think I hit some shortcut or button by accident. Translate still works though. I checked that the transforms weren't locked and such, to no avail. What could be the cause? FileZilla: How to copy files from one directory to another on the same server? I want to copy files from one directory to another on the same server using FileZilla. Please guide me how can it be done? When starring a message from the star board, it's not reflected in the main chat window "How big of a problem" vs. "how big a problem" Quite a few phrases in English are constructed like so:     How [adjective] a [noun]...?   This is the question form of the construction, which is often answered with the negative:     Not that [adjective] a [noun].   or the positive:     Quite [adjective] a [noun].   However, from time to time I'll hear the word 'of' inserted before the 'a', e.g.:     Not that [adjective] of a [noun].   This usually sounds wrong to me, with the exception of the case where the adjective 'much' is used.  So, this sounds fine to my ear:     Not that much of a problem.   whereas this doesn't:     Not that loud of a noise.   Why is it that 'much' should be used with 'of', and other adjectives not?  Is it because 'much' is seen as measuring a quantity (of something), whereas other adjectives that may be used in this construction are seen as measuring the quality of a whole thing? When a person lifts a book while standing on the earth's surface. Why is his force considered external to the book-earth system? Isn't his force internal like the case of the forces that cause the exploson of a bomb. I think the forces the person exerts on both the book and the earth are internal since they are equal but opposite therefore resulting in the position of the centre of mass of the system being maintained and if i'm right is the mechanical energy of the system constant?  Am i wrong? Please correct me. Deriving even odd function expressions What is the logic/thinking process behind deriving an expression for even and odd functions in terms of $f(x)$ and $f(-x)$?  I've been pondering about it for a few hours now, and I'm still not sure how one proceeds from the properties of even and odd functions to derive: $$\begin{align*} E(x) &amp;= \frac{f(x) + f(-x)}{2}\\ O(x) &amp;= \frac{f(x) - f(-x)}{2} \end{align*}$$ What is the logic and thought process from using the respective even and odd properties, $$\begin{align*} f(-x) &amp;= f(x)\\    f(-x) &amp;= -f(x) \end{align*}$$   to derive $E(x)$ and $O(x)$?    The best I get to is: For even: $f(x)-f(-x)=0$ and for odd: $f(x)+f(-x)=0$  Given the definition of $E(x)$ and $O(x)$, it makes a lot of sense (hindsight usually is) but starting from just the properties. Wow, I feel I'm missing something crucial. How do I put a stencil-like texture across a sectioned object? Why is RSA encryption significantly faster than decryption? I am designing an implementation of RSA . I recorded computation times in Java by using System.currentTimeMillis(). It returned an encryption time of 0.05 ms and decryption time of 0.55 ms, or a ratio of 1:11.   This seems like a large difference. Is it supposed to be this way?  //here my key has 256 bits  for (;;) {             long begin = System.currentTimeMillis();              for (int i = 0; i &lt; num; i++) {              decrypt();             }              long end = System.currentTimeMillis();              long time = end - begin;              if (time &gt;= 10000) {                 System.out.printf("Average Encryption takes: %.2f ms\n",                         (double) time / num);                 break;             }              num *= 2;         }  p = BigInteger.probablePrime(128, random); q = BigInteger.probablePrime(128, random); N=(p.subtract(one)).multiply(q.subtract(one)); e=BigInteger.probablePrime(32, random); d=e.modInverse(N); private void encrypt() {  C= M.modPow(e,N);  }  private void decrypt() {     RM = C.modPow(d, N);     } The assumptions are:    Alice and Bob have perfectly synchronized clocks Alice and Bob have successfully exchanged a pair of entangled photons   The idea is simply to have Alice and Bob perform the Quantum Eraser Experiment (doesn't need to be the delayed choice).   Alice and Bob agree on a specific time when Bob's photon will be between the "path marker" (which is usually just after the slits) and the detector.  If Alice acts collpasing the wave-function on her photon, the interference pattern will disappear. If not it won't.  Alice and Bob can be spatially separated...  What am I misunderstanding?   The only meaningful difference from this spatially separated quantum eraser experiment to one done on tabletop is that you won't be able to use a coincidence detector, but that is not impeditive to identifying the interference pattern, just will make errors more probable. Which we should be able to deal with a appropriate protocol...  There is a experimental paper with a small amount of citations pointing out to the breaking of complementarity in a very similar setup: How to reduce the thickness of letters I installed ubuntu 16.04 LTS, but there is no ubuntu one software, not in ubuntu software center also..Though it is showing selected in unity tweak tool.How can I get it. Please reply. SEO safe way to run beta of new site on seperate domain Test if there are files matching a pattern in order to execute a script I am trying to write an if statement to test whether there are any files matching a certain pattern. If there is a text file in a directory it should run a given script.  My code currently:  if [ -f /*.txt ]; then ./script fi   Please give some ideas; I only want to run the script if there is a .txt in the directory. How should I unit test multithreaded code? How to install a Brother dcp-135c printer? Recently moved over onto Ubuntu. First time for anything like this for me as I'm coming from Windows but I'm here and ready to learn. The only problems Ive experienced so are saving in Libre office doc as a word doc file and then losing the formatting upon receival via email. However the more pressing issue is getting my old Brother DCP 135 c to work. Ive scanned a few help blogs and have found them frankly quite frightening.  Would it be possible for someone to write u a step by step guide (including explanations of terminology)  For those of us new to this magic world?