I need to prove that every permutation $\sigma \in S_n, \sigma \neq id$ can be written as a distinct product of disjoint cycles and don't really know where to start. The statement is totally clear, but I cannot think of an elegant way to proof its correctness.
I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes.  Proposition: any permutation in $S_n$ can be written as a product of disjoint cycles.  Proof (attempt): $s\in S_n$. Let $\langle s\rangle$ act on $X=\{1,\cdots,n\}$. Define $x\sim x'\iff x'\in\text{Orb}(x)$ for $x,x'\in X:$ this is clearly an equivalent relation, hence the orbits partition $X$. Each orbit corresponds a cycle in $s$, so we are done.  Does this hold, or have I missed something vital?
How to use an optimization solver to get t-stats and p-values for the estimates? I'm curious to know how important the bias node is for the effectiveness of modern neural networks. I can easily understand that it can be important in a shallow network with only a few input variables. However, modern neural nets such as in deep learning often have a large number of input variables to decide whether a certain neuron is triggered. Would simply removing them from, e.g., LeNet5 or ImageNet have any real impact at all? Google sheets split multi-line cell into new rows (+ duplicate surrounding row entries) I'm compiling a database of business address info, including key staff for each business. The Google spreadsheet I've inherited for this project includes a column for 'key staff' (in column B), with multiple key staff names listed in the same cell, separated by line breaks (ie CHAR(10) ). There is one business per row. The number of lines in the 'key staff' cell varies row-by-row.  My initial sheet looks like this:    I need to do the following to optimize this sheet:   split each multi-line ‘key staff’ cell so each key staff name appears on its own row. This requires that new row/s be inserted beneath the original row. duplicate the data from all other cells on the original row (i.e. from columns A &amp; C:E), so that each new row contains the full data for each business I need an automated process - I'll have about 1000 businesses to process so can't do this with any manual steps   The sheet should then look like this:    Using =TRANSPOSE(SPLIT(B1,CHAR(10))) obviously only goes part of the way - it doesn’t insert new rows and doesn’t duplicate the surrounding column entries. All help gratefully accepted! How to fill gaps in SLC-Off Landsat 7 ETM+ Images with QGIS? Explanation of the problem:     On 31 May 2003 the Landsat 7 Enhanced Thematic Mapper (ETM+) sensor had a failure of the Scan Line Corrector (SLC). Since that time all Landsat ETM+ images have had wedge-shaped gaps on both sides of each scene, resulting in approximately 22% data loss. These images are available for free download from the USGS GloVis website and are found in the L7 SLC-off collection.   source:   Can I use Qgis instead of proprietary software (ENVI, Erdas....) to fill the gaps? How do you run Ubuntu Server with a GUI? Create keyboard shortcuts for touch bar's functions I have broken the touch bar in my MacBook Pro 13". Only the ESC key keeps working.  Replacing the touch bar may be expensive so I'd like to know if it is possible to replace its functions with keyboard shortcuts.  The functions I am highly interested in are:   volume screen backlight keyboard backlight How do I restore an accidentally deleted initrd image? CG series about a beast planet Force applied off center on an object Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. How do I calculate the instantaneous acceleration, rotational acceleration, and trajectory of this object, assuming it starts from rest? Weighting features prior to SVM I'm building an object detector using HOG features and linear SVM. Some of the regions of the object are more "distinctive" so I would like to give more weight to the features extracted from those regions. (e.g. imagine we want to detect between mugs and glasses, then the most "distinctive" part is the mug's handle)  How could I do this? For sure there should be a more intelligent way than just replicating the features of those "distinctive" regions. I have been trying but its not working.  but if i try   &lt;?php echo do_shortcode("[mimo]"); ?&gt; &lt;?php echo do_shortcode("[mibaby]"); ?&gt;   it will work fine.  But to apply it in post content using [mimo] [mibaby]  it won't work.  Here is the Functions i am using...  function wpb_custom_new_menuw() {   remove_filter( 'the_content', 'wpautop' ); } add_shortcode('mibaby', 'wpb_custom_new_menuw');    function wpb_custom_new_menu() {   remove_filter( 'the_excerpt', 'wpautop' ); } add_shortcode('mimo', 'wpb_custom_new_menu'); $D_8$ as a derived subgroup Every undergraduate student knows that there are (exactly) two non-abelian groups of order 8: Dihedral ($D_8$) and Quaternion ($Q_8$). The group $Q_8$ has many interesting properties; simple of them are: it has unique element of order $2$, all subgroups are normal, etc. A property, which is strange and not obvious, the group $Q_8$ has, is that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. On the other hand, $D_8$ has no such property; there are (infinitely many) groups $G$ such that $G/Z(G)$ is isomorphic to $D_8$; example - $D_{16}$.   The question, I would like to ask, is related to a property of $D_8$, which is not shared by $Q_8$. There are (infinitely many) groups whose derived subgroup is $Q_8$; example- $SL_2(3)$. For groups $G$ of order &lt;100, one can check on GAP that $D_8$ is not a derived subgroup of any group. The natural question is then:  Question 1: Does there exist a finite/infinite group whose derived subgroup is equal to $D_8$?  Question 2: Does there exist a finite/infinite group whose derived subgroup is equal to (general) dihedral group $D_{2n}$ of order $2n$ ($n&gt;2$)? So many interesting questions get asked on Stack Overflow and then immediately closed (or worse yet, closed and re-opened for days on end) because they're subjective, argumentative, and fundamentally unanswerable due to our lack of omniscience: without the ability to see into the future and/or know the secret motivations of companies and their developers, we are left to make wildly varying (un)informed guesses. The obvious solution to this is to add a  feature to the Ask a Question page. Similar to the current &quot;Related Questions&quot; feature, this could be used to answer otherwise-unanswerable questions before they're asked, thus saving everyone time and frustration. I propose a simple system whereby a web cam is aimed at an ordinary  device, long recognized as one of the most accessible and accurate fortune-telling mechanisms available to modern society. A simple  could be placed under this rig and triggered in response to the user's typing. Once focus leaves the Title input field, a short delay will be triggered, followed by a bit of logic to capture the 8-ball's result and feed it through a simple OCR routine to produce the result.  So far as I'm aware, SO would be the first major Q&amp;A site to implement such a feature...  Examples:            ...another UV resurrection... How can I go back to the original interface layout? I don't know what the hell I did I just want to get back to modelling Some conditions require that a person lie in bed all the time until recovered. Is there a name for such conditions or people experiencing them at the moment?   For example,      I am currently &lt;bed sick&gt; and can't help you with your homework.   I tried looking for bed sick but hit a dead end. What does functools.wraps do? While building things I want to see the overlays from other modes so I can make better decisions. Can I lock an overlay somehow?  For example: When building roads I want to see the natural resources overlay. When zoning I want to see the wind overlay Equals(=) vs. LIKE When using SQL, are there any benefits of using = in a WHERE clause instead of LIKE?  Without any special operators, LIKE and = are the same, right? Mod shows up with OP background when viewing deleted comments I have been trying to install ubuntu (I tried 12.04LTS, 12.10, 12.10-secure-remix) using the wubi installer on Windows 8 and all the three times ran into the same problem showing some cannot find /ubuntu/winboot/wubildr when I tried to start ubuntu.  I tried to change the boot settings from UEFI to Legacy and then my laptop restarted and showed that "No operating System Found".  I then tried to install Ubuntu using a cd and after a lot of effort and using boot repair,etc. I landed into a new trouble. When I select UEFI on, the laptop loads Windows and windows too does not detect ubuntu. When I select Legacy mode, it starts ubuntu, shows me an option to start windows 8 but gives an error showing windows failed to load.  So in legacy mode, i am able to use only ubuntu and in UEFI mode, I am able to use only Win8. Please explain any process of starting windows in legacy mode (which I tried without any success) or starting ubuntu in UEFI mode.