How do I find corrupted PDF files?
Validate/verify PDF files integrity
Can we get rid of the threshold of 15 chars for comments? I understand that there is an upper limit. But why do you have to type at least 15 characters to post a comment? It is easy to work around it anyway (like adding dots or other useless things). Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$? I need to prove  $2n \leq 2^n$, for all integer $n≥1$ by mathematical induction?  This is how I prove this:  Prove:$2n ≤ 2^n$, for all integer $n≥1$  Proof: $2+4+6+...+2n=2^n$  $i.)$ Let $P(n)=1         P(1): 2(1)=2^1\implies 2=2$.        Hence, $P(1)$ is true.  $ii.)$  Assume that $P(n)$ is true for $n=k$, i.e, $2+4+6+...+2k=2^k$, and prove that $P(n)$ is also true for $n=k+1$, i.e, $2+4+6...+2(k+1)=2^{(k+1)}$  from the assumption add $2(k+1)$ on both sides so we have   $2+4+6...2k+2(k+1)=2^k+2(k+1)$  I'm confused with $2^k+2(k+1)$, I don't know how to make $2^k$ be equivalent to $2^{k+1}$.  I feel i'm doing something wrong.   Any help would be appreciated! Using principal component analysis (PCA) for feature selection Unable to start XAMPP using sudo in Desktop Entry SQL Server Escape an Underscore How do I escape the underscore character?  I am writing something like the following where clause and want to be able to find actual entries with _d at the end.  Where Username Like '%_d' Writing to NTFS drive in Linux Which shell interpreter runs a script with no shebang? Suppose the default shell for my account is zsh but I opened the terminal and fired up bash and executed a script named prac002.sh, which shell interpreter would be used to execute the script, zsh or bash? Consider the following example:  papagolf@Sierra ~/My Files/My Programs/Learning/Shell % sudo cat /etc/passwd | grep papagolf [sudo] password for papagolf:  papagolf:x:1000:1001:Rex,,,:/home/papagolf:/usr/bin/zsh # papagolf's default shell is zsh  papagolf@Sierra ~/My Files/My Programs/Learning/Shell % bash # I fired up bash. (See that '%' prompt in zsh changes to '$' prompt, indicating bash.)  papagolf@Sierra:~/My Files/My Programs/Learning/Shell$ ./prac002.sh  Enter username : Rex Rex # Which interpreter did it just use?   **EDIT : ** Here's the content of the script  papagolf@Sierra ~/My Files/My Programs/Learning/Shell % cat ./prac002.sh  read -p "Enter username : " uname echo $uname ExitCodes bigger than 255, possible? If yes, on which operating system, shell or whatever?  Consider the following java program (I'm using java just as an example, any language would be good for this question, which is more about operation systems):  public class ExitCode {     public static void main(String args[]) {         System.exit(Integer.parseInt(args[0]));     } }   Running it on Linux and bash, it returns always values less equal 255, e.g. (echo $? prints the exit code of the previous executed command)  &gt; java ExitCode 2; echo $? 2  &gt; java ExitCode 128; echo $? 128  &gt; java ExitCode 255; echo $? 255  &gt; java ExitCode 256; echo $? 0  &gt; java ExitCode 65536; echo $? 0     EDITED: the (only, so far) answer below fully  explain what happens on UNIXes. I'm still wondering about other OSes. Connection pooling in PHP Why is the prospective new kilogram standard a sphere? I can understand the choice of material, silicon 28, but why is it a sphere rather than (say) a cube? Article   I would have thought that a sphere would have been the hardest shape to machine accurately. Existence of subgroup of order six in $A_4$ Show that  the  alternating  group  $A_4$  of all even permutations of $S_4$ does not contain a  subgroup of order $6$.   For me am thinking to write all elements of $A_4$ and trying to find every cyclic subgroup generated by each element of $A_4$, then I have to check whether there exist such a subgroup or not! This is a long procedure for me, I ask if there is a short way to do this. What is ETTR (Expose To The Right)? [Probability]need help to understand the following expression So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively.  assume c is a constant.  I am having huge trouble to understand the integration of the following expression.  $P(Y&lt;c/u(X))$  $=\int_{t}^{\infty}f_X(x)\int_{0}^{c/u(x)}f_Y(y)dydx +\int_{0}^{t}f_X(x)\int_{c/u(x)}^{\infty}f_Y(y)dydx$  where t is the cross-point that $u(x)$ change sign    here $c/u(x)$ is given by the plot below, t is the point crossing the zero:   Confusion:   I don't understand the second integration of the second term "$\int_{c/u(x)}^{\infty}f_Y(y)dydx$", this isn't right because Y an X only defined for y>0, and x>0. So it's the first quadrant in this plot. Is it possible to plot a graph of any shape? In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$.  A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$. I have also recently learnt about trigonometric graphs and circle equations. In the internet, I have seen users posting graphs of different shapes like a heart-shaped graph and a Batman logo-shaped graph. I am sure there are numerous more graphs that I have yet to see. Seeing that the graphs can be shaped into shapes like the Batman logo and a heart brings me to my question: Is it possible to plot a graph of any shape regardless of its complexity? Perhaps, shaped into an outline of a person or a landmark? Why or why not? How do I achieve this kind of light star effect? I'm kinda new to Blender. Currently playing around with version 2.8 to practice some rendering. Is there a way to achieve this 'light-star' effect for the blooming objects with the emission? If there is an alternative way then it's fine too. See these images below for references. Thanks in advance: Number of total possibilities for an equation I need to find the number of possibilities for which the following equation exists:  $$x_1 + x_2 + x_3 + \cdots + x_{10} \leq 70$$  Each variable is a non-negative integer.  I tried simplifying the question to the point of finding the number of possible solutions to each equation seperatly:  $$x_1 + x_2 + x_3 + \cdots + x_{10} = 70,$$ which is ${79}\choose{10}$.  $$x_1 + x_2 + x_3 \cdots + x_{10} = 69,$$ which is ${78}\choose{10}$.  $$x_1 + x_2 + x_3 + \cdots + x_{10} = 68,$$ which is ${77}\choose{10}$, and so forth.  Then I thought about adding all of these together. This seems a bit too much work considering that the textbook solution is: ${80}\choose{10}$, and I can't seem to figure out the idiomatic way of approaching these kind of questions.   Any help is greatly appreciated. Infinite partition of $\mathbb N$ by infinite subsets I'm looking for a movie that was set in the US, and all the characters (including the robot) spoke English.  I can't recall the exact plot, but mostly it was about an innocent game robot that shoots small objects for fun and becomes dangerous, because its AI made a wrongful choice about 'winning the game'. The robot starts to arm itself with more dangerous objects (not sure but knives and stuff) and starts to attack people.  It was overall supposed to be fun and adventure movie for kids. Color movie, and probably made in 80~90's.   There is a recurring scene when the robot detects an aggressor, it scans and loudly announces "enemy one, enemy one". The robot was somehow looking similar to Wall-E, and had a saucer-like head part. Find Total number of ways out of N Number taking K numbers every M interval I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks. So I know I've heard this saying in American English before but I just can't quite find the original.  I have come up with several made up variations like:   I don't send the Navy (or whole army, or a battalion) when all you need is a diplomat. Don't buy the whole cow when all you need is milk.   What's the more commonly heard saying/phrase for this of expression?