How to prevent windows from going to sleep when my c++ application is running?
Prevent windows from going into sleep when my program is running?
Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite? Let $E/K$ be an algebraic extension of fields and denote by $E_s$ the separable closure of $K$ in $E$, i.e. the set of all elements of $E$ that are separable over $K$ (which can be shown is a (separable) subfield of $E$).  I would like to show that the implication     $\left[E:K\right]_s &lt; \infty \implies \left[E_s:K\right] &lt; \infty$   holds, where $\left[E:K\right]_s$ denotes the separable degree of $E/K$ and $\left[E_s:K\right]$ the 'ordinary' degree of $E_s/K$.  In the textbook I am learning with, it is said that one could use the Primitive Element Theorem to do this, but I would also be fine with any other approach, as long as it's not getting too fancy and staying in the realm of 'common' textbook theorems known when dealing with separable extensions (and I didn't see how the Primitive Element Theorem would help).  This is not my focus (I want to write a Bachelor's thesis about polynomial factorizing elements of $\mathbb{Z}[X]$), but I'm in the process of reading into commutative algebra after about 1.5 years break from it.  My idea was something like assuming $\left[E_s:K\right] &lt; \infty$ and $\left[E_s:K\right] = \infty$ to construct infinitely many $K$-homomorphisms $E_s/K \longrightarrow C/K$, where $C$ is some algebraic closure of $E_s$, which would then contradict $\left[E_s:K\right]_s = \left[E_s:K\right] &lt; \infty$. Of course, an infinite basis would yield infinitely many $K$-vector space homomorphisms, but getting (infinitely many) ring morphisms seems to by an entirely different ting (unsurprisingly so...). Thought of using something like     Given two roots of the minimal polynomial of one of the infinitely many elements of a $K$-basis of $E_s/K$, there is a $K$-morphism $E_s/K \longrightarrow C/K$ that sends one to the other.   but this didn't seem very fruitful. I admit I haven't spent hours on this, I just didn't get very far and should focus on moving on, while still getting an answer on the 'howto' along the road would be really nice. The following code snippet (correctly) gives a warning in C and an error in C++ (using gcc &amp; g++ respectively, tested with versions 3.4.5 and 4.2.1; MSVC does not seem to care):  char **a; const char** b = a;   I can understand and accept this. The C++ solution to this problem is to change b to be a const char * const *, which disallows reassignment of the pointers and prevents you from circumventing const-correctness ().  char **a; const char* const* b = a;   However, in pure C, the corrected version (using const char * const *) still gives a warning, and I don't understand why. Is there a way to get around this without using a cast?  To clarify: 1) Why does this generate a warning in C?  It should be entirely const-safe, and the C++ compiler seems to recognize it as such. 2) What is the correct way to go about accepting this char** as a parameter while saying (and having the compiler enforce) that I will not be modifying the characters it points to? For example, if I wanted to write a function:  void f(const char* const* in) {   // Only reads the data from in, does not write to it }   And I wanted to invoke it on a char**, what would be the correct type for the parameter? The  mentions the following matrix identity $$\det\begin{pmatrix}A&amp;B\\B&amp;A\end{pmatrix}=\det(A+B)\det(A-B),$$ valid for squared matrices $A$ and $B$ of the same size.  How is this result proved? How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$ Shape of the universe? How can I find the codes related to a function in GAP? When I use "??" in front of the name of function there is no help, so I want to find the code of function among packages. Erdős-Straus conjecture The  (ESC), states that for every natural number $n \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following equation is satisfied:  $$\frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\tag{1}$$  The basic approach to solving this problem outlined by Mordell [Ref1] is described below  By defining $t$ and $m$ as positive integers greater than zero and $q$ a positive integer greater than one we can observe that  a) There is always a solution for even $n$, since if $n=2^qt$ we have the trivial solution $$\frac{4}{4t}=\frac{1}{t}$$  In the remaining case $n=2(2t+1)$, a solution in the form of two Egyptian fractions can always be found  e.g. $$\frac{4}{2(2t+1)}=\frac{2}{2t+1}=\frac{1}{t+1}+\frac{1}{(t+1)(2t+1)}$$  b) If $(1)$ is a solution for some particular prime $n$ then all composite numbers $mn$ divisible by $n$ are also solutions, thus  $$\frac{4}{mn}=\frac{1}{ma}+\frac{1}{mb}+\frac{1}{mc}$$  will also be a solution. This means that we can simplify the analysis to the cases where $n$ is a prime greater than 2.  Using Mordell's approach we have just shown that we only need to consider the cases where $n$ is prime and where $n \equiv 1 \pmod{2} \;\;[meaning \;\;n=2t+1]$  The argument continues...  Mordell goes on to show in turn that the search can be reduced further to the cases when $$n \equiv 1 \pmod{4} \;\;[meaning \;numbers \;\;n=4t+1]$$ $$n \equiv 1 \pmod{8} \;\;[meaning \;numbers \;\;n=8t+1]$$ $$n \equiv 1 \pmod{3} \;\;[meaning \;numbers \;\;n=3t+1]$$ $$n \equiv 1,2,4 \pmod{7} \;\;[meaning \;numbers \;\;n=7t+1,n=7t+2 \;or\;n=7t+4 ]$$ $$n \equiv 1,4 \pmod{5}  \;\;[meaning \;numbers \;\; n=5t+1 \;or\;n=5t+4]$$  Assembling these results together, Mordell showed that the conjecture can be proved in this context except for the cases when $$n \equiv 1,11^2,13^2,17^2,19^2,23^2 \pmod{840}$$  Mordell stated that since the first prime meeting this condition is 1009, this is proof that the conjecture holds for $n&lt;1009$.  This basic approach can be pursued further. Other workers have shown that the conjecture holds for much higher values of $n$ using similar methods as can be seen on the above Wikipedia page.   Note that other intermediate results can be constructed from the above congruence's, e.g. $n \equiv 1 \pmod{24}$.  The question is:  Are there any other elementary approaches to solving this problem than the one outlined by Mordell (and described above)?  [Ref1] Louis J. Mordell (1969) Diophantine Equations, Academic Press, London, pp. 287-290. Multivariate differentiability verification I have tried an attempt on the following:  Let $F:U\in \Bbb{R}^n\to \Bbb{R}$ and $f:\Bbb{R}\to \Bbb{R}$ where $f$ is an even function. Now $F(\mathbf{x})=f(|\mathbf{x}|)$, where $|\ . |$ is the Euclidean norm. Given $f$ is $C^r$, I have to show that $F$ is $C^r$.  My attempt was to argument by induction. I showed that $F$ is $C^1$, basically using univariate chain rule on $\frac{\partial f\left(\sqrt{x_1^2+\dots+x_n^2}\right)}{\partial x_1}$ and arguing that it would be continuous using the fact that $f$ is $C^r$. Since the argument works for all $x_i$, it establishes first-differntiability for $F$ and continuity of the operator $DF(\mathbf{x})$. Then I moved on to show $F$ is $C^r$ assuming $F$ is $C^{r-1}$ and $f$ is $C^r$. Now what I did was partially differentiate:  $$ {\partial \over \partial x_2} \left[\frac{\partial ^{k-1}f}{\partial x_1^{k-1}}\right] = \underbrace{{\partial \over \partial |\mathbf{x}|} \left[\frac{\partial ^{k-1}f}{\partial x_1^{k-1}}\right]}_{(1)} . \underbrace{{\partial |\mathbf{x}| \over \partial x_2}}_{(2)} $$  My argument runs as follows: this partial derivative is continuous since $f$ is $C^r$ and hence $(1)$ is continuous. Continuity of $(2)$ is establishd in a similar manner, since $\sqrt{\mathbf{x}}$ is $C^r$ if $\mathbf{x} \neq \mathbf{0}$. Then the product would be continuous. I am uneasy on multivariate arguments and dont know if my approach is completely off track. I didnt use the even function property, so I am sure I might me missing something. Any hints and references would be welcome. Thanks! I put all my music on a playlist seperate from the rest of the music on the iTunes library, so I could easily get to what music I wanted to listen to, instead of my brother or sister's music. My brother recently got an iPod nano, and didn't know how to use iTunes, after explaining it to him, he deleted my whole entire playlist. I didn't know, so when I hooked my phone up to the computer to sync some songs I was going to buy, all the music on my phone disappeared. I've tried re-doing the playlist and syncing all of it to my phone again, but it won't work. I've also tried turning off my phone for certain periods of time, still nothing.  I lost about 200 songs (thankfully not more), and I don't know how to get it back. Site Analytics header does not show the link to "traffic sources" when in that section I was seeing the Site Analytics page (/site-analytics)  (>25K users) and also  and found out that the header is not consistent:   When in history, the header shows "history | traffic sources" When in traffic sources, the header just shows "history"   Graphically: Why and how does C# allow accessing private variables outside the class itself when it's within the same containing class? How can I edit/create new launcher items in Unity by hand? Will Unity allow making custom launcher icons from .desktop files or via menu editing system? (Right now the launcher doesn't give the option to "keep in launcher" on all programs.  For some programs I use, I have to make custom launchers or .desktop files.  For instance, daily blender builds are generally just folders with an executable.  In basic Gnome or KDE, I can make a new menu entry with the menu editing system. Then, I can also add it to Docky either from the menu or by dragging a .desktop file to it. Unity launcher doesn't support drag and drop, so thats not a bug or anything, but when I open a .desktop file, it has unpredictable results. Most time it will not have "keep in launcher". Sometime it will have a pinnable item without the .desktop's icon, and if I pin the item to the launcher, it will not call upon the program again after closing it. I've also gotten it to just work with a .desktop file for celtx. Prime ideal and nilpotent elements Where does console output go Extracting only the BibTex records relevant to the paper My colleague asked me to write a page for a paper he writes. I wrote the page in LyX, compiled it to LaTeX, and sent him the TeX files. Then he reminded me that I forgot to send the .bib file with all the references.  But, my .bib file is huge - it contains over 1000 references (I have a global BiBTeX file for all my papers). The page I wrote uses only about 20 of them.  Is there an automatic way to extract from the bib file, only the records relevant to the current paper? Galois extension of intersection of fields I have finite Galois extensions: E/K and E/L. $$M:=K \cap L$$ I am trying to prove that if the extension E/M is finite then it is also Galois. Any suggestions? Thanks How can I define a math operator behaving like \lim or \sum with limits? Do I have to move swap partition to the right side? According to  I just expand my sda4 to include unallocated space, my question is after create 2 new partitions sda7, sda8, do I need to move the swap partition to the most right of the disk? (I really don't know this, I saw  which moving the swap partition to the right side)  Thanks. Arrays, linked lists and other data structures in cmd.exe (batch) script Are the WotC archives of d20 Character Optimization forum forever lost? I was looking for the Wizard Handbook but all the links were broken.  Somewhere on the handbooks of other classes (on other forums) I saw a      This was archived by WotC, here's the link   But the link leads to the welcome page of WotC (since it's broken).  Is there a way to get all that info back? Do you know where to get it from? The Wayback Machine did not archive the forums.