Find all the possible functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(1)=2$$ and $$f(xy)=f(x)f(y)-f(x+y)+1\text.$$  I managed to find the function for the set of natural number, by putting $x=1$ and $y=n$. From it, I have that $$f(n+1)=f(n)+1\text,$$ which means that $f(n)$ is an arithmetic progression of natural numbers. I am stuck with generalizing this into the set of rational number, any hit will be appreciated.
I'm working on this problem already, the domain and co-domain are the set of Rational Numbers. we see that setting $x = 1$ and $y = n$, we obtain $f(n + 1) = f(n) + 1$  But I'm trying to prove that $f(n) = n + 1$ for all $n$, the case when $n = 1, 2$ is obviously true, how do I approach this via induction?
Objective-C for Windows What would be the best way to write Objective-C on the Windows platform?  Cygwin and gcc? Is there a way I can somehow integrate this into Visual Studio?  Along those lines - are there any suggestions as to how to link in and use the Windows SDK for something like this. Its a different beast but I know I can write assembly and link in the Windows DLLs giving me accessibility to those calls but I don't know how to do this without googling and getting piecemeal directions.  Is anyone aware of a good online or book resource to do or explain these kinds of things? How to evaluate $\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2} \, \mathrm{d}\theta$ how is this possible? I know there is some trick, should someone please explain?! Why does this cloth go down past the top surface of the cube? I was following a , and the presenter said that he did not change the default options other than enabling self-collision and increasing the collision object's friction from 5 to 10. His looks like the first image below, but when I tried that, the cloth goes down past the top surface and stops at the bottom surface (second and third images below). Why is it so?   I had a previous problem with cloth, and it was due to the objects' small size. This time, I made sure that both the cloth and the cube are large enough. (Cloth=7.68*8.22*1.99m, cube = 7.68*7.68*0.897m)  The project file can be . I would like to update VirtualBox on my 32 bit Ubuntu 15.04 machine from 4.3.26 to 4.3.28, using a Debian package I manually downloaded from . But simply double-clicking the package opens up the Software Centre which gives an error that installing it would break an existing package (the exact message was:   Breaks existing package 'virtualbox' conflict: virtualbox-4.3 ()   and it does not appear to have an option to ignore this and proceed with the installation anyway. Is there a command-line work-around this problem, perhaps? Clarify or visualize the parameters involved for receiving the Curious badge MSBuild exec task with for I am trying to run the following commands as part of an MSBuild script:  for /R . %f in (*.targets) do copy /Y "%f" "C:\Program Files (x86)\MSBuild\Microso ft\VisualStudio\TeamBuild"   The commands is implemented in an exec the following way:  &lt;Exec WorkingDirectory="$(SolutionRoot)" Command="for /R . %f in (*.targets) do copy /Y &amp;quot;%f&amp;quot; &amp;quot;$(MSBuildExtensionsPath)\Microsoft\VisualStudio\TeamBuild&amp;quot;" /&gt;   The command works fine from console, but when trying to run it from MSBuild I get the error:   Task "Exec"   Command:   for /R . %f in (*.targets) do copy /Y "%f" "C:\Program Files (x86)\MSBuild\Microsoft\VisualStudio\TeamBuild"   f" "C:\Program was unexpected at this time. C:\Users\rd-build\AppData\Local\Temp\OH Test2\Continuous.BuildTargets\BuildType\TFSBuild.proj(98,5): error MSB3073: The command "for /R . %f in (*.targets) do copy /Y "%f" "C:\Program Files (x86)\MSBuild\Microsoft\VisualStudio\TeamBuild"" exited with code 255. Done executing task "Exec" -- FAILED.   Any suggestions? Can't remove URL on ALL profiles There is a bug removing URL from the profile.   Website link has some URL. Remove URL (empty field). "Save and copy changes to all Stack Exchange communities" button pressed.   Observed result: Empty URL only on the current SE site.    Expected result: Empty URL on all Stack Exchange sites. Recently, my iPhone 5 was stolen, and while I have a new iPhone 6, I received a message from a number of someone claiming to be in possession of the device, with a new number.   All images, contacts, Outlook mail, you name it, are still on the device.  While I realize it's my fault for not having it connected to iCloud to wipe manually, is there anything I can do to have this sensitive information wiped from the device? Here's an example:  Double d = (1/3); System.out.println(d);   This returns 0, not 0.33333... as it should.  Does anyone know? Why is my Nikon 50mm f/1.8 giving me an fEE error? I have a AF Nikkor 50mm f/1.8. When I have the aperture setting on the lens to 22 I can take photos. But when I change the dial to anything other than 22(lowest f/stop) then I get a flashing fEE message and r09 where the number of exposure left is.  I don't have the manual for this lens but am just wondering if anyone knows what the issue is here.  My camera is a Nikon D70S. Place each foot on a scale: can you add the two to find your weight? I frequent a blog from a British psychologist, and every Friday he likes to pose an interesting puzzle or riddle. The Monday after that he posts the answer. They're good fun, and IANAP but  made my it-might-not-be-quite-as-simple-as-that detector go off.  My question boils down to this: let's say I have two identical scales, and I stand on the scales with one foot on each scale. The scales read W1 and W2. Does my weight equal W1 + W2? Prove $f(x) = \sqrt x$ is uniformly continuous on $[0,\infty)$ I have seen a proof in   Below shows an alternative proof. Please correct me if im wrong.  Proof:  For any given $\varepsilon &gt;0$,  Let $\delta_1 = \frac{\varepsilon}{2}$, $\forall x,y \in [1,\infty)$ with $|x-y|&lt;\delta_1$  Since $|\sqrt x + \sqrt y|\geq2$  $$|\sqrt x - \sqrt y|  = \frac{|x-y|}{|\sqrt x+\sqrt y|} &lt; |x-y| &lt; \delta_1 = \frac{\varepsilon}{2}$$  Hence, $\sqrt x$ is uniformly continuous on $[1,\infty)$.  $\sqrt x$ is continuous on [0,1] , so $\sqrt x$ is uniformly continuous on [0,1].  So, there exist $\delta_2 &gt; 0$ such that $\forall x,y \in [0,1], |x-y|&lt;\delta_2$, $|\sqrt x -\sqrt y| &lt;\frac{\varepsilon}{2}$  Let $\delta = \min{(\delta_1,\delta_2)}$  $\forall x,y \in [0,\infty)$ with $|x-y|&lt;\delta$,  Case 1: $x,y \in [0,1]$ Proven above as $|x-y| &lt; \frac{\varepsilon}{2} &lt; \varepsilon$  Case 2: $x,y \in [1,\infty)$ Proven above as $|x-y| &lt; \frac{\varepsilon}{2} &lt; \varepsilon$  Case 3: $x \in [0,1] , y \in [1,\infty]$  $$|\sqrt x-\sqrt y| = |\sqrt x -1+1-\sqrt y| \leq |\sqrt x-1| + |\sqrt y -1| &lt; \frac{\varepsilon}{2} + \frac{\varepsilon}{2}= \varepsilon$$  by applying case 1 and case 2. How do I disable overlay scrollbars? It looks like Ubuntu 11.04 will come with  by default. I do like them, but I don't like the current mix of scrollbar types as many applications don't yet use those overlay scrollbars.  Is there a way to disable overlay scrollbars (without removing the overlay-scrollbar package)? What is the difference between Parse() and TryParse()?  int number = int.Parse(textBoxNumber.Text);  // The Try-Parse Method int.TryParse(textBoxNumber.Text, out number);   Is there some form of error-checking like a Try-Catch Block? Prove that: $2^a+3^b<3a+4b$ What can I do if a repository/PPA does not have a Release file? When updating, I get the following error message:   W: The repository 'http://ppa.launchpad.net/mc3man/trusty-media/ubuntu xenial Release' does not have a Release file.   Here, I find another statement on this error:   This recommends removing certain PPAs; and, I'm not sure if I should do that since it might mean not getting the updates that I need.    Is this what I should do? Preview of CAD files in format .dwg I'm looking for a utility or a program for Ubuntu, and possibly even for Windows that is able to convert a CAD file format .DWG to PDF.   My need is to provide a preview of CAD files, and I thought that the easiest way is to convert them into PDF format, but I'm open to suggestions, even if there is only the possibility to convert them to JPG image file type or PNG would be fine.  What interests me is only to find some solution, no matter if it is commercial or free Using a java library from python I have a python app and java app. The python app generates input for the java app and invokes it on the command line.  I'm sure there must be a more elegant solution to this; just like using JNI to invoke C code from Java.  Any pointers? (FYI I'm v. new to Python)    Clarification (at the cost of a long question: apologies) The py app (which I don't own) takes user input in the form of a number of configuration files. It then interprits these and farms work off to a number of (hidden) tools via a plugin mechanism. I'm looking to add support for the functionality provided by the legacy Java app.  So it doesn't make sense to call the python app from the java app and I can't run the py app in a jython environment (on the JVM).  Since there is no obvious mechanism for this I think the simple CL invocation is the best solution. The new profile shouldn't mistakenly imply that you can earn the Informed badge on a meta site