I apologize if this is trivial but I have not been able to figure it out.  For a curve $\sigma(t)$, I have a definition for arc length:  $$s(t)=\int_{t_0}^t |\sigma'(t)|dt$$  We reparameterize a curve $\sigma$ by observing that $s$ has an inverse, $t(s)$, and the resulting reparameterized curve has underlying assignment $s \mapsto t \mapsto \sigma(t)$.  A curve is a unit speed curve if  $$\forall t\qquad |\sigma'(t)|=1$$  Thanks
Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie.  length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal to $|b-a|$.  It seems intuitive, but I don't have any idea how can I show this in the formal way.
Two parties are trying to resolve some dispute that involves interest of both. Sometimes it's hard for them to reach an agreement on a fair basis. This is typically when another party who has no interest involved and no bias towards either party comes into play in order to settle the dispute between the two conflicting parties. The term "third party" seems being used quite often. I would also call it a "middle-standing" party or we could say this party is neutral.  These terms seem NOT formal enough to me though. I'd like to see if anybody can come up with a better one. I'm working on a web application that needs to prints silently -- that is without user involvement.  What's the best way to accomplish this?  It doesn't like it can be done with strictly with Javascript, nor Flash and/or AIR.  The closest I've seen involves a Java applet.  I can understand why it would a Bad Idea for just any website to be able to do this.  This specific instance is for an internal application, and it's perfectly acceptable if the user needs to add the URL to a trusted site list, install an addon, etc. Bird flying in a cage Assume that you are holding a cage containing a bird.  Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage? Does it make a difference whether the cage is completely closed or it has rods to let air pass? Where does YouTube's offline feature store video files? I'm using YouTube offline feature on my Android device, so that I can playback the offline videos within the YouTube app even there is no internet connection. When I searched for any of the video files, I can't find them on device/SD card memory. Are they encrypting the file so that it can be played only with YouTube app? Does anyone know the specific folder for offline video data? I want to write a function that returns the nearest next power of 2 number. For example if my input is 789, the output should be 1024. Is there any way of achieving this without using any loops but just using some bitwise operators? I bought a used iPhone 4.  The previous owner re-set the phone, but did not remove the device from their iCloud.  It is asking me for their iCloud information when I try to set-up the phone.  I don't have contact with the previous owner anymore.  Is there a way to remove the device from their iCloud so I can use a new one? In particular, when $x=y=z$ isn't the above expression equal to 1 and not -1? I have an Optimus Elite, and it has a minimum brightness setting. In certain apps, I can reduce the brightness below the safe minimum, but when I leave the app, the brightness is increased to the system-wide minimum.  Using , I can permanently set the brightness below the safe amount, and it will stay even after going to other apps. Or using , I can accomplish almost the same thing even without needing root (it displays a translucent 'filter' over the whole screen, reducing brightness/contrast but not actually changing the backlighting power amount).  What I want to know is where the minimum screen brightness is stored. I have a rooted phone with CyanogenMod 9, so I have permissions to edit anything and could change a config or sqlite entry, but I don't know where this value is stored. What I would like to do is use the status bar brightness slider to be able to slide to the actual minimum of the screen, rather than the OS-controlled minimum. Having an app is useful, but having the functionality without the app seems to be within reach, if only I knew where the OS is getting it's safe minimum setting.  Where is this "safe minimum brightness" setting stored? If I can change it, I can remove my apps and use native controls. Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$   Thank you How to store date/time and timestamps in UTC time zone with JPA and Hibernate Can you make a bonus-action attack from Two-Weapon Fighting before you take your action? The rules on  state:     When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.   However, the description of  states:     You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified   So, can you attack with your bonus action first? What counts as an attack? I was looking over the class feature of the 'fiend' patron warlock in 5e. I'm a bit confused on what actions the player needs to take before they can use the 'Hurl Through Hell' ability on an enemy. The PHB states:     Starting at 14th level, when you hit a creature with an attack, you can use this feature to instantly transport the target through the lower planes.   I'm not clear on what an 'attack' means in this context. Does it have to be a physical attack like a sword or crossbow? Do magical attacks like Eldritch Blast that require an attack roll count? What about spells with no attack roll? What do the numbers in a man page mean? So, for example, when I type man ls I see LS(1). But if I type man apachectl I see APACHECTL(8) and if I type man cd I end up with cd(n).  I'm wondering what the significance of the numbers in the parentheses are, if they have any. Simple Math Problem A bat and ball cost \$1.10.  The bat costs a dollar more then the ball.  How much does the ball cost?  If this is not the correct place to ask a question like this please tell me and I will remove it :)     The ball costs \$0.05, not \$0.10 Get Age of a person in MySQL Literal &#39; appears in flag panel instead of being escaped Recently I have learned how to draw simple stuff with TikZ. I really like it. Now I wonder if it is possible to draw with TikZ on a image. Instead of pasting a image in Illustrator and adding some vectors to point something in the picture I would like to do it using TikZ. Is it possible?   For example, on a picture of a computer (PNG,etc) I would like to draw some arrows and labels saying "Keyboard", "Monitor", etc. Redirecting cout to a console in windows I have an application which is a relatively old. Through some minor changes, it builds nearly perfectly with Visual C++ 2008. One thing that I've noticed is that my "debug console" isn't quite working right. Basically in the past, I've use AllocConsole() to create a console for my debug output to go to. Then I would use freopen to redirect stdout to it. This worked perfectly with both C and C++ style IO.  Now, it seems that it will only work with C style IO. What is the proper way to redirect things like cout to a console allocated with AllocConsole()?  Here's the code which used to work:  if(AllocConsole()) {     freopen("CONOUT$", "wt", stdout);     SetConsoleTitle("Debug Console");     SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE), FOREGROUND_GREEN | FOREGROUND_BLUE | FOREGROUND_RED); }   EDIT: one thing which occurred to me is that I could make a custom streambuf whose overflow method writes using C style IO and replace std::cout's default stream buffer with it. But that seems like a cop-out. Is there a proper way to do this in 2008? Or is this perhaps something that MS overlooked?  EDIT2: OK, so I've made an implementaiton of the idea I spelled out above. Basically it looks like this:  class outbuf : public std::streambuf { public:     outbuf() {         setp(0, 0);     }      virtual int_type overflow(int_type c = traits_type::eof()) {         return fputc(c, stdout) == EOF ? traits_type::eof() : c;     } };  int APIENTRY WinMain(HINSTANCE hInstance, HINSTANCE hPrevInstance, LPTSTR lpCmdLine, int nCmdShow) {     // create the console     if(AllocConsole()) {         freopen("CONOUT$", "w", stdout);         SetConsoleTitle("Debug Console");         SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE), FOREGROUND_GREEN | FOREGROUND_BLUE | FOREGROUND_RED);       }      // set std::cout to use my custom streambuf     outbuf ob;     std::streambuf *sb = std::cout.rdbuf(&amp;ob);      // do some work here      // make sure to restore the original so we don't get a crash on close!     std::cout.rdbuf(sb);     return 0; }   Anyone have a better/cleaner solution than just forcing std::cout to be a glorified fputc? Best Azure Service for Large no of SQLServer Databases I am trying to use netcat to send a simple message over TCP, e.g. I want to send the digit 1.  So I understood that I write the following in the terminal, after installing netcat  netcat [ip-address] [port]  But how do I specify the message to be sent? i.e. where to I write "1" ?