Ideals in $\mathbb{Z}[X]$ with three generators (and not with two)
Construct ideals in $\mathbb Z[x]$ with a given least number of generators
Show $a^p \equiv a \mod 2p$ I've stucked at the following problem:  Let $p$ be an odd prime number. Show that for all $a\in\mathbb{Z}$  $$a^p \equiv a\ (\mathrm{mod}\ 2p)$$  How can I prove it? I know within Fermat's theorem that $a^p \equiv a\ \mathrm{mod}\ p$. Why can't humans throw objects at bullet speed when the force involved seems so small? The force that propels a bullet should at least be similar in intensity to the force experienced as recoil and the force exerted on the target. But these forces seem really low (the equivalent of a shove).  If all it takes to make a small object achieve bullet speed is the equivalent of a shove, why can't humans throw objects at these speeds? How to animate A curve segment? Total combinations from number of unique combination? How can you find count of the total combinations possible if you have the count of unique combinations? Suppose I need to form a sum of $3$ using $\{1,2\}$ Unique combn. $(1+1+1)$, $(1+2)$ hence '$2$' unique but total is '$3$' i.e, $(1+1+1)$, $(1+2)$, $(2+1)$ I know the elements used to make the combination Note: Number of unique combinations (ie. 2 here) is known. How can I find the total number of combination given the unique combinations? Android: Using default styles Can't append  element Any idea why the piece of code below does not add the script element to the DOM?  var code = "&lt;script&gt;&lt;/script&gt;"; $("#someElement").append(code); Can I publish parts of the Ph.D thesis as a paper in a journal? I would like to hide all items that I don't need from appearing in App Lens.  For example: XTerm, File Roller and the Home Folder.  Is this possible? How can I make a logo in Adobe Illustrator like the AntiParticle logo below, from ?   It is a large number circles of different, varying sizes, which form the shape of a text character. Why Can't I duck reverb on the mix bus in Unity? i'm surprised there is so little documentation on this issue. But i'm trying to duck the audio on all the mix buses in Unity so that you can clearly hear the VO of the game.  Now it works great except that I have some reverb zones in the game and any sound that plays in these reverb zones, the actual reverb itself is NOT DUCKED.  So the VO is accompanied with a huge wall of wishy washy muddy noise.  Anyone deal with this before? How can we deal with this? Migrated question's answers can be upvoted twice by the same user What is the preferred way to remove spaces from a string in C++? I could loop through all the characters and build a new string, but is there a better way? Only Icons Missing from desktop I opened the PC in the morning, and this is what I got:      As is visible the icons of the UWA apps were no affected and the start menu also is unaffected.    When Control Panel is in category view the entire page is empty and in large/small icons the name is visible, no icons.  I deleted the iconcache.db in %USERPROFILE%\AppData\Local folder. Following the instructions here: . But to no avail and now windows is not rebuilding the cache.   I also ran sfc /scannow but no changes took place.  What should I do?  (01/01/2016 8:00PM GMT +5:30) Edit: I tried opening the system properties and I got 0x8000ffff error and this:        Then I closed the popup and pressed the OK button on the System Properties Windows and I got 0x8007057A error:      Opening the Image Backup Control Panel page I again got an 0x8007000E error:        I tried using system restore via advanced startup but once again I got 0x800FFFF error.  P.S.: I did try restarting, didn't work. Why can I only change my Stack Overflow name once every 30 days? Do GTA 5 saves back-up to a cloud service? For GTA 5 (on PC), if I have a save file, will it automatically back-up to either the Steam or Rockstar cloud, or do I have to save it manually? If it have to save it manually, will you please provide the default directory for saves? Prove that if W is a subspace of a finite dimensional vector space V, then dim(W) ≤ dim(V). My attempt at a solution:  Proof  Given a vector space $V$, let $S={v_1,...,v_n}$ be a basis for $V$. Consequently $\dim(V)=n$. Now given $W$, a subspace of $V$, it will be shown that $\dim(W)\le \dim V$   First, consider the case where $W=${0}. Then $dim(W)=0 \le\ n$,   Subsequently, consider the case for $W=V$. Then it follows that $dim(W)=dim(V)=n$.  Finally, consider the case where $W \subset V$ and without loss of generality assume $v_n \notin W$. Let $T={v_1,...,v_k}$ where $k\lt n$ be a basis for the subspace W. Now, since S is a basis for V, for every $v_i \in S$,  $c_1v_1+...+c_nv_n=0$ has only the trivial solution, thus none of the vectors in S can be expressed as linear combinations of the other vectors in S. It follows that the set T is not a basis for V as $v_n$ cannot be generated from T. The $dim(W)$ is given by the number of vectors in the basis. Therefore, $dim(W)=k \lt n$  Therefore, it has been shown that if $W \subseteq V$ then $dim(W) \le dim(V)$ Did your team win the football match yesterday?         The weather was very bad, so we weren't playing.   vs     The weather was very bad, so we didn't play.   Is the first one not correct? Why is it? Sublime Command not working in SSH server on Ubuntu 14.04? I am new to this and not sure what to make of all the options available. I am interested in putting a machine-learning-related preprint, which will potentially be submitted to places like NIPS/ICLR/ICML in the future.   arXiv.org perpetual, non-exclusive license to distribute this article (Minimal rights required by arXiv.org)  Creative Commons Attribution license (CC BY 4.0)  Creative Commons Attribution-ShareAlike license (CC BY-SA 4.0)  Creative Commons Attribution-Noncommercial-ShareAlike license (CC BY-NC-SA 4.0)  Creative Commons Public Domain Declaration (CC0 1.0) How do I use reflection to call a generic method? What's the best way to call a generic method when the type parameter isn't known at compile time, but instead is obtained dynamically at runtime?  Consider the following sample code - inside the Example() method, what's the most concise way to invoke GenericMethod&lt;T&gt;() using the Type stored in the myType variable?  public class Sample {     public void Example(string typeName)     {         Type myType = FindType(typeName);          // What goes here to call GenericMethod&lt;T&gt;()?         GenericMethod&lt;myType&gt;(); // This doesn't work          // What changes to call StaticMethod&lt;T&gt;()?         Sample.StaticMethod&lt;myType&gt;(); // This also doesn't work     }      public void GenericMethod&lt;T&gt;()     {         // ...     }      public static void StaticMethod&lt;T&gt;()     {         //...     } }