Suppose that $f:    [0,1 ] \to \mathbb{R}$ has a continuous second derivative, $ f(0)=f(1)=0$ and $f(x)&gt;0$ for all $x\in (0,1)$. Prove that $\int_0 ^1 |f''(x)/f(x)|dx&gt; 4.$ I have no idea how to proceed with this problem, how does one approach such a problem? Any hints will be appreciated.
Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints.    I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx &gt;4. $$  I could not make any progress. Since the unit interval is compact, we know that the continuity of $f, f'$ and $f''$ are uniform and they attain their maximum and minimum values. Moreover, I tried to apply the Mean Value Theorem but I failed.
What does "it" refer to in "it's raining"? I wanted to leave the question title as is so as not to take away from my amusement :).  Anyway,     It's raining.   What is raining? Is it the sky? The clouds? The weather? The rain? What is "it"? Any historical insights on the statement? How do we know for certain that space is expanding? Move rigidbody using WASD keys but rotate it based on camera/mouse movement I am very new to unity and my scripts are based on some scripts found on the web.   I want to make a first person view so the camera is attached to my rigidbodys head. Then i want to move my rigidbody based on keyboard input (wasd) and rotate it using the mouse movement.  Until now, i used the following code for moving my rigidbody:  void Update() {      /* calculate rigidbody movement */     movement = new Vector3( Input.GetAxisRaw("Horizontal"), 0, Input.GetAxisRaw("Vertical")).normalized;      /* calculate mouse look to rotate the rigidbody*/     mouseDelta = new Vector2(Input.GetAxisRaw("Mouse X"), Input.GetAxisRaw("Mouse Y"));     mouseDelta = Vector2.Scale(mouseDelta, new Vector2(sensitivity * smoothing, sensitivity * smoothing));     smoothV.x = Mathf.Lerp(smoothV.x, mouseDelta.x, 1f / smoothing);     smoothV.y = Mathf.Lerp(smoothV.y, mouseDelta.y, 1f / smoothing);     mouseLook += smoothV;  }  void FixedUpdate() {         /* apply rigidbody movement from Update() */     transform.Translate( movement * moveSpeed * Time.deltaTime );             /* apply rigidbody rotation from Update() */     transform.localRotation = Quaternion.AngleAxis(mouseLook.x, transform.up);             /* apply camera rotation from Update() */     Camera.main.transform.localRotation = Quaternion.AngleAxis(-mouseLook.y, Vector3.right);    }   I learned today that i dont have to move a rigidbody's transform but now i am totally confused and it seems i have to restart my whole (first!) project.  So when i use MovePosition and MoveRotation, how do i rotate it based on the camera perspective/mouse movement?  I think the movement Vector is ok so i can simply rewrite the FixedUpdate to something like  rigidbody.MovePosition(movement * speed * Time.deltaTime)   but i am totally stuck on how to apply the camera rotation to my rigidbody to use it like  void FixedUpdate() {      /* move the rigidbody */                rigidbody.MovePosition(movement * moveSpeed * Time.deltaTime);     /* rotate the camera */     camera.transform.localRotation = Quaternion.AngleAxis(-mouseLook.y, Vector3.right);      /* but how can i rotate the rigidbody based on the cameras rotation??? */     rigidbody.MoveRotation(?????);  }   I hope its clear what i want, thank you! My code runs inside a JAR file, say foo.jar, and I need to know, in the code, in which folder the running foo.jar is.  So, if foo.jar is in C:\FOO\, I want to get that path no matter what my current working directory is. Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective?  I tried the following: $$f:\mathbb{R}\rightarrow \mathbb{R^{+}} $$ $$f(x)=x^{2}$$  and  $$g:\mathbb{R^{+}}\rightarrow \mathbb{R}$$ $$g(x)=\sqrt{x}$$  $f$ is not injective, and $g$ is not surjective, but $f\circ g$ is bijective  Any other examples? php String Concatenation, Performance In languages like Java and C#, strings are immutable and it can be computationally expensive to build a string one character at a time.  In said languages, there are library classes to reduce this cost such as C# System.Text.StringBuilder and Java java.lang.StringBuilder.  Does php (4 or 5; I'm interested in both) share this limitation?  If so, are there similar solutions to the problem available? Sometime in the 1970s (I think) I saw a really awful film set in a small town by a lake. The film opened with some people camping out in the woods. One of them was zipped up in a sleeping bag. Something (as yet unidentified) moved in between the sleeping camper and the camera, and we heard a "crunch." When the creature moved off, we saw that it had bitten the head off the camper, causing a fountain of blood to spout of of the neck. Far from being shocking or gory, it was so badly done as to be laughable.  The main story was that these huge and nasty mutated creatures were coming out of the lake and eating people. A local industrial company was under suspicion for poisoning the lake, but repeated water tests showed it to be clean.  In the end, someone had the idea of testing the water at the bottom of the lake, which revealed it to be heavily poisoned with mercury. As mercury is much heavier than water, it had sunk, so the previous tests (done from a small row boat) had missed it as they only sampled the water at the surface.  They got it all sorted out eventually, and the bad guys ended up in prison.  After the credits had rolled, and you thought it was all over, another monster reared up in front of the camera and roared. I think we were supposed to go "Oh no, it's not all over!" However, the few people stupid enough not to have left by now laughed at the ridiculous sight and left, shaking their heads wondering why they had bothered!  I have a morbid fascination to see it again. Anyone have any idea what it was called? Keep "Employee" on the profile of SE employees on meta Currently, SE employees are given diamonds wherever they need them. I agree with  on this to some degree, employees shouldn't be treated any different.  But... on meta, employees are different. They have the power to dole out official responses, and thus we should be able to differentiate between an employee and a normal user. Currently, only some employees have a diamond on MSO (see , ). This makes it confusing, especially when you see official-sounding posts like .  Now, giving diamonds to all employees on meta may not be the best solution to this. I don't think all employees want their precious inboxes spammed with all the suspensions that are doled out on meta, and other mod stuff. They won't really need the diamond powers either (though it wouldn't hurt).  So, I propose this:  Similar to how moderators have "Moderator" clearly on their profile    display "employee" on an employee's profile, along with the diamond if they have one. At least on meta.    It would imho be even better to display it on their usercards in some form, but that has already been shot down.  It would be even even better if non-MSO mods had greyed out diamonds, similar to , but that's just me wanting more shiny stuff ;-) How to redirect output of a running program to /dev/null I know that in order to suppress the output of a program I can redirect it to /dev/null, for example to hide all error and warning messages from chromium I can start the program like this  chromium-browser 2&gt; /dev/null &amp;   However, if I happen to forget about the error messages and type  chromium-browser &amp;   (which is quite annoying when they appear in the middle of a command) I don't know what to do except for stopping the application and starting it again properly.  Can I somehow redirect error output without restarting the application? Why is Euler's Totient function always even? I have a class like this:  public class myClass {   public List&lt;myOtherClass&gt; anewlist = new List&lt;myOtherClass&gt;;    public void addToList(myOtherClass tmp)   {     anewList.Add(tmp);   }  }   So I call "addToList" a hundred times, each adding a unique item to the list.  I've tested my items to show that before I run the "addToList" method, they are unique.  I even put a line in to test "tmp" to make sure it was what I was expecting.  However, when I do this (lets say myClass object is called tmpClass):  int i = tmpClass.anewList.Count(); for (int j = 0; j&lt;i; j++) {    //write out each member of the list based on index j... }   I get the same exact item, and it's the last one that was written into my list.  It's as if when I add, I'm overwriting the entire list with the last item I've added.    Help?  This makes no sense. I've also tried List.Insert, where I'm always inserting at the end or at index 0.  Still no dice.  Yes, I'm doubly source my indexing is correct and when I do my test I'm indexing through each of the elements.  :)  UPDATE: Okay, I tried this and still had the same problem:  foreach(myOtherClass tmpC in tmpClass.anewList) {         Console.WriteLine(tmpC.theStringInMyClass.ToString()); }   and still for each of the 100 items, I got the same string output... I'm sure I'm doing something completely stupid, but I don't know what yet. I'm still 100% sure that the right string is getting passed in to begin with.  -Adeena    Okay, I tried this and still had the same problem:  foreach(myOtherClass tmpC in tmpClass.anewList) {     Console.WriteLine(tmpC.theStringInMyClass.ToString()); }   and still for each of the 100 items, I got the same string output...  I'm sure I'm doing something completely stupid, but I don't know what yet. I'm still 100% sure that the right string is getting passed in to begin with.  -Adeena It was a useful answer. I want to ask about two sentences, and what would be the correct way to form a question:   It's very important that we eat healthy food, ...........?  Which would be the correct question tag: (isn't it) or (don't we)? It's a must that all applicants follow the rules of the competition, ..........?  Which would the correct question tag: (don't they) or (mustn't they)?   Please could you tell me the correct answers, and, more importantly, why those are correct? Drawing visibility polygons in Unity (for vision cones with occlusion) I'm currently trying to create an effect like the one displayed below in Unity:    However, it seems that I can't do this with any of Unity's built-in light objects, and I simply cannot find anything online that talks about this. Is there a way to achieve this in Unity? 2D or 3D solutions are fine, as I'm sure that they're both very similar to each other, Using \@-ified punctuation while using \frenchspacing After reading  about spacing and punctuation, I was left wondering about an issue that didn't seem to be addressed: using \frenchspacing together with the modified punctuation \@., \@?, etc.  As far as I can tell visually, using \frenchspacing together with these commands works correctly (i.e., it seems like nothing breaks, and the extra space is suppressed as normal), but are there problems that can arise from doing so?   Moreover, when using \frenchspacing, "should" I still use \@. and its friends where it would ordinarily be necessary, in case someone wants to remove \frenchspacing later? Would accommodating (hypothetical) future editors in this way be "standard", "above and beyond the call of duty", or "crazy"? How to grep for text in a file and display the paragraph that has the text? Below is the text in the file:  Pseudo name=Apple Code=42B state=fault  Pseudo name=Prance Code=43B state=good   I need to grep for "42B" and get the output from the above text like:   Pseudo name=Apple Code=42B state=fault   Does anyone have idea on how to achieve this using grep/awk/sed? Is there a standard naming convention for XML elements? Is there any standard, de facto or otherwise, for XML documents? For example which is the "best" way to write a tag?  &lt;MyTag /&gt; &lt;myTag /&gt; &lt;mytag /&gt; &lt;my-tag /&gt; &lt;my_tag /&gt;   Likewise if I have an enumerated value for an attribute which is better  &lt;myTag attribute="value one"/&gt; &lt;myTag attribute="ValueOne"/&gt; &lt;myTag attribute="value-one"/&gt; If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$ I'm at a loss with “Dungeons and Dragons.” How does one play it, anyway? I'm no gamer at all, so this could be like a very silly, basic question for you guys. The thing is that I always read about "Dungeons and Dragons" game (not the video game, but the normal game to play with friends that are actually gathered together). And I would like to know how to play it.  I've read that it is a book (or several ones, which is more confusing for me to understand), but I've read somewhere that it can be played as a normal "table" game, with a board, and pieces, and somewhat weird dice (or sets of dice).  Is it like a normal game, where you pull the stuff out of a box and start to play?  I don't know if any of you watch the show "The Big Bang Theory", but I think they have played the game in a recent episode, and it looks just like a regular game.  I've Googled it for more information, but I haven't found anything as basic as I need, because I've never played a role game (except in a video game).  I just want to learn how to play, so I can get my friends into it to play it each time we get together. How do i show that $w=z^2$ maps $|z-1|\leq 1$ onto the cardioid $R=2(1+\cos \theta)$?