Writing in a text file using java swing
How do I get a platform-dependent new line character?
euler's theorem of homogeneuos function suppose that $f$ is a mapping from $\mathbb R^n \to \mathbb R$.  a positively homogeneous of degree $n$ and suppose $f_1, f_2, .... f_N$ are continuous for $a \neq 0$. then   $$\sum_{i=1}^n a_if_i(a)= n*f(a)$$   now, they asked me to prove this. (Which is a total no no to me)  I write my proof as;  $f$ is a mapping from $\mathbb R^n \to \mathbb R$  A homogeneous function of degree $n$ if for all $a \in \mathbb R^n$ and all $L&gt;0$, $f(La)=L^n\,f(a)$. let $a \in \mathbb R^n$ and define function $g$ be a mapping from $[0,\infty) \to \mathbb R$. so I write $g(L) = f(La)-L^nf(a)$. then $L≥0$ let $L=0$ then, $g(L)=0$ and therefore $g'(x) =0$?  then i don't know what's goes next. can anyone help me about this? Evaluation of $ \sum_{k=0}^n \cos k\theta $ I just wanted to evaluate  $$ \sum_{k=0}^n \cos k\theta $$  and I know that it should give  $$  \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)}   $$  I tried to start by writing the sum as  $$ 1 + \cos\theta + \cos 2\theta + \cdots  + \cos n\theta $$  and expand each cosine by its series representation. But this soon looked not very helpful so I need some clue about how this partial sum is calculated more efficiently ... I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear electrical circuits, linear filters, linear electrical elements, linear approximation, linear optimization.  I'm getting crazy trying to understand the application of the "linear concept" to all this aspects (function, equation, mapping, system, operator, transformation, algebra, etc.) and I wish to know the one essence that is to be linear. What something has to be to be linear? If I say something is linear, what do I know for sure about that something (no matter what the something is)?       I heard a definition of linearity is by homogeinity (scaling the input results in a scaled output) and addition (summing the inputs results in summing the outputs). Can I apply this simple definition to all the branches (operator, mapping, system, transformation, algebra, ...) I mentioned ? Do they all behave like a line ?  y = ax + b, for example, is a line but doesn't behave like a line because y is not linear. Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? I have Blender 2.79 on my Windows 7 system.    I want to install 2.8, but I'm completing some courses which use 2.79.  If I install 2.8, will it automatically replace 2.79?  Will they interfere with one another?  Thanks. Can QGIS export Geospatial PDF for Avenza PDF map app? Since QGIS uses GDAL, does any map exported as a PDF become a Geospatial PDF? I am hoping to import maps from QGIS into the Avenza PDF Map app on my phone. Previous posts mentioned exporting as Geospatial PDF's and GeoTiff's, but none had an exact answer or process. How to select html nodes by ID with jquery when the id contains a dot? If my html looked like this:  &lt;td class="controlCell"&gt;     &lt;input class="inputText" id="SearchBag.CompanyName" name="SearchBag.CompanyName" type="text" value="" /&gt; &lt;/td&gt;   How could I select #SearchBag.CompanyName with JQuery? I can't get it to work and I fear it's the dot that's breaking it all. The annoying thing is that renaming all my id's would be a lot of work, not to mention the loss in readability.  Note: Please let's not start talking about how tables are not made for lay-outing. I'm very aware of the  and try hard to use it as much as possible. What happens when a moderator clicks 'leave open' in the close-vote queue? It takes  clicking "leave open" in the close-vote queue to remove the question from the queue.  How about diamond mods?  Do they remove it from the queue in one click, or not? No \xrightrightarrow[a]{b}? How to Connect & Share your Internet Connection (Wired & Wireless) I have Ubuntu 12.04 on a HP 430 notebook and this has a single wired internet connection and I would like to share this with a LG Optimus One with Android 2.3.3. I followed  instructions and created a hotspot as shown below, but UbuntuHost is not visible on the LG Optimus One when I enable the Wifi on it.  To summarize, I want my Laptop with an internet connection to act as a WiFi router and access the internet from other devices (LG Optimus One with Android in this case).  What additional steps have to be done? How do I send a file with FileZilla from the command line? I have a batch file that builds an application, and then I want to upload it to an FTP server from the command line.   Considering I do the upload manually with , is there a command line for it to upload files?  I am limited to Windows. How to choose the default gcc and g++ version? Suppose John has 5 sweets. Is there any difference between the following two sentences?     Jack has 3 times as many sweets as John.      Jack has 3 times more sweets than John.   I prefer the first construction and would know unambiguously that Jack has 15 sweets in this case. However in the second construction I would be inclined to think that Jack has 20 sweets, since it seems to suggest 15 sweets in addition to the original 5. Computer unstable at lower temperatures My computer is playing some strange joke at me, and I'm at a loss.  Everybody knows that hot temperatures causes many components to become unstable and even cause damage. When I built this machine, I was careful to do a bunch of experiments with different airflow configurations, with CPU-and-GPU intensive benchmarks running, and measuring the temperature at different points to find the best configuration.  Generally, the internal temperature of this machine is very close to ambient temperature (with 2 exceptions: the GPU always runs ~10°C above ambient, and the CPU goes to at most ~12°C above ambient when running CPU-intensive applications).  Now comes the problem: at really cold mornings. At cold mornings (~10°C (50°F) or below), the system shows really strange general instability. Beeps from the BIOS, the operating system displays strange messages about memory corruption and PCI devices that failed to initialize, and then, everything hangs. The computer hangs to the point that all USB devices turn of. Hitting the Reset button blanks the screen but the mainboard fails to proceed to the boot.  I never thought that low temperatures could cause computer instability. And 10°C is not that cold, it is still within operating temperatures of all components as far as I checked.  I'm quite sure it is related with the temperature. There were too many cold and warm mornings that demonstrated that the problem only showed in the cold ones. I'm working around this problem by letting the computer sit at the GRUB menu for some 5-10 minutes before letting the OS start, or blowing hot air inside the case if I'm in a hurry.  I don't know what should I approach next to find the source of this problem. Since it is difficult to reproduce (I need a cold morning, and the computer must be cooled down (powered off for many hours)), I need to plan ahead the experiment, since it stabilizes after a few minutes on.  Machine is an ASUS P5VDC-X equipped with Intel P4 64-bit 3GHz, 2GB of DDR2 RAM, a 500W PSU and an nVidia GeForce 9600GT. Everything else was already disconnected/removed before and the problem persisted, so the problem should be in one of these remaining parts.  Any help is appreciated! access parent object in javascript When voting to migrate an off-topic question, why is the list of sites limited? Add Area 51 Meta to the app Does $\exp(2ir\pi)$ equal $1$? What's wrong? Since $e^{ix}=\cos x+i\sin x$, thus  $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$     Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$    But this cannot be true since $\cos2\pi r+i\sin2\pi r$  isn't necessarily $1$ for arbitrary $r$. Where did it go wrong? Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$ What is the exact reason that normal matter can not exist within an event horizon?  I can understand how a super-dense object like a neutron star could accrete mass until its physical radius is less than its Schwarzschild radius and an event horizon forms around it.   But why can't the neutron star remain inside the event horizon as it was before the event horizon formed? Aside from the escape velocity at the surface reaching $c$, what actually changed?  Why must all matter within an event horizon undergo complete physical collapse to a point?