Deriving Potential Energy Density for Longitudinally Vibrating String
Potential energy of an infinitesimal length of elastic rod
More numbers between $[0,1]$ or $[1,\infty)$? Example 1) Let's say I have some random creature with protection from white, and my opponent plays , a white card with the effect "Destroy all creatures." Can my creature with protection from white be destroyed by a white card that isn't actually targeting it specifically?  Example 2) I was playing a match with a friend where I had  on the field against his 3 zombies. Is he able to declare his zombies as blockers even though Elite Inquisitor has "protection" from zombies?  If he can declare them as blockers, does protection mean that no damage is dealt by the zombies to  Elite Inquisitor? Theoretically possible to run two operating systems simultaneously? A friend of mine asked a question that got me thinking. Is it possible (theoretically, I assume it hasn't been done) to run two operating systems simultaneously on the same computer? Not in the sense of multitasking in one window, but being able to quickly switch from one to the other without restarting. Could something low-level be created that would intercept system calls (or whatever needs to be done) to make each OS think it is running alone? Or hardware itself that would accomplish this?  I'm not talking about dual booting or using virtual machines, but actually running two OS's side-by-side. date difference in days, in Android what are all the open subgroups of $(\mathbb{R},+)$ What is the principle of underdetermination? While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. Drawing arrows on an HTML page to visualize semantic links between textual spans I have an HTML page with some textual spans marked up something like this:  ... &lt;span id="T2" class="Protein"&gt;p50&lt;/span&gt; ... &lt;span id="T3" class="Protein"&gt;p65&lt;/span&gt; ... &lt;span id="T34" ids="T2 T3" class="Positive_regulation"&gt;recruitment&lt;/span&gt; ...   I.e. each span has an ID and refers to zero or more spans via their IDs.  I would like to visualize these references as arrows.  Two questions:   How can I map an ID of a span to the screen coordinates of the rendering of the span? How do I draw arrows going from one rendering to another?   The solution should work in Firefox, working in other browsers is a plus but not really necessary. The solution could use jQuery, or some other lightweight JavaScript library. I have a population of phone calls - 200,000.  There are different reasons for each call, but lets assume the number of reasons is known. i.e. 7 different call reasons: 1) Check on order 2) Cancel order 3) Billing information 4) Account information etc...  My question is, I want to get a statistically significant sample size so the number of calls I listen to in my sample are representative of the distribution of the reasons in the overall population of 200,000 calls.  What should the sample size be? Which methods to use to calculate the sample size? Task manager for Ubuntu What is the first movie to show this hard work message at the very end? What is the first movie to show this hard work message at the very end? I haven't logged onto my blog for years, and have, of course, forgotten the password. I click the 'Forgot' button and get a mail with a link, but that mail already says     Sorry, that key does not appear to be valid.   And sends me round the click to reset, receive a mail, click the link again etc. As if the timeout for password resets has been set to 0.5s or something.  This is a shared hosted blog, so I have full access to cPanel and all of the WP files via FTP, if that helps anything. Deleted object would be re-saved by cascade (remove deleted object from associations) Cycles Normal Bake does'nt bake indents that are not tapered Prove that $f$ has a fixed point . For $f:[a,b]\rightarrow [a,b]$ is a continuous . Prove that $f$ has a fixed point . Is that true if we change $[a,b]$ by $[a,b)$ or $(a,b)$. What is $\arctan(x) + \arctan(y)$ I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$  which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by which I mean, has different definitions for different domains:  $$g(x) = \begin{cases}\arctan\left(\dfrac{x+y}{1-xy}\right), &amp;xy &lt; 1 \\[1.5ex]  \pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &amp;x&gt;0,\; y&gt;0,\; xy&gt;1 \\[1.5ex]  -\pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &amp;x&lt;0,\; y&lt;0,\; xy &gt; 1\end{cases}$$  Furthermore, When I plot the function $2\arctan(x)$, it turns out that the book definition is correct. I don't understand how such peculier definition emerges. Thank you. How can I format my active hard drive to NTFS? I'm trying to install Windows 7 with a 4GB flash drive, but my error that comes up is that my hard drive I'm trying to install on is in ext4. I need to format it to read NTFS. I can't seem to find any topics on how to format an active hard drive. I found a topic that explains how to move Ubuntu to a new drive, but it's a bit confusing to me. Flatten an irregular list of lists How to show path-connectedness of $GL(n,\mathbb{C})$ Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to show a path from $A$ to $I$ then define $f(t)=At+(1-t)I$  for $t\in[0,1]$ which is possible continous except where the $\operatorname{det}{f(t)}=0$ i.e. which has $n$ roots and I can choose a path in $\mathbb{C}\setminus\{t_1,\dots,t_n\}$ where $t_1,\dots,t_n$ are roots of $\operatorname{det}{f(t)}=0$, is my thinking was correct? Could anyone tell me the solution? I would like to filter the markers on my (geojson) layer according to two features: year and type of event, ideally using both filters (year and eventType) as check-boxes without using layerGroup at all. I would like to be able to click on a checkbox and update dynamically the markers by filtering them according to feature.properties.eventType too, in addition to the year displayed, something like    &lt;input type="checkbox" id="all" name="all" value="all"&gt;   &lt;label for="all"&gt;All&lt;/label&gt;   &lt;input type="checkbox" id="funfair" name="funfair" value="newsletter"&gt;   &lt;label for="funfair"&gt;Funfair&lt;/label&gt;   &lt;input type="checkbox" id="trade" name="trade" value="trade"&gt;   &lt;label for="trade"&gt;Trade&lt;/label&gt;   &lt;input type="checkbox" id="exhibition" name="exhibition" value="exhibition"&gt;   &lt;label for="exhibition"&gt;Paid exhibition&lt;/label&gt;   etc... which would trigger a  filter: function(feature, layer) {   return feature.properties.eventType == value-of-checkbox; }   I have five types of eventType. So, for instance, if 1890 is checked, the map is showing all 1890 events, no matter what kind of event it is (because by default it would show all events). If the user checks the 'funfair' and the 'trade' checkbox, though, I want the map to show only the 1890 events of the funfair and trade kind, and not other kind of events.  This is my complete script, which so far is working fine with filtering by year.  var oldmap = L.tileLayer('address', {   maxZoom: 12 }),  var modernmap = L.tileLayer('https://api.tiles.mapbox.com/v4/{id}/{z}/{x}/{y}.png?access_token={accessToken}', {   maxZoom: 12,   id: 'mapbox.streets',   accessToken: 'mytoken' });  var map = L.map('map', {   center: [40.1857864,-4.5536861],   zoom: 7,   layers: [oldmap, modernmap] });  var baseMaps = {   "Old map": oldmap,   "Modern map": modernmap };  var jsontest = {"type": "FeatureCollection",   "features": [     {       "type": "Feature",       "geometry": {         "type": "Point",         "coordinates":  [ -16.5471268,28.4136726 ]       },       "properties": {         "Name":"Point1",         "year":'1892',         "eventType":"funfair",         "notes":""       }     },     {       "type": "Feature",       "geometry": {         "type": "Point",         "coordinates":  [ 2.1753895,41.3767925 ]       },       "properties": {         "Name":"Point2",         "year":'1890',         "eventType":"trade demonstration",         "notes":""       }     }   ]};  var mapLayerGroupsYear = [];  function onEachFeature(feature, layer) {   var year = mapLayerGroupsYear[feature.properties.year];   if (year === undefined) {     year = new L.layerGroup();      //add the layer to the map     year.addTo(map);      //store layer     mapLayerGroupsYear[feature.properties.year] = year;   }    //add the feature to the layer   year.addLayer(layer);    }   var myLayer = L.geoJSON(jsontest, {   onEachFeature: onEachFeature })  L.control.layers(baseMaps, mapLayerGroupsYear).addTo(map);   I am a beginner with leaflet so please forgive the dumb question, in case it is. How to prove reflexivity, symmetry and transitivity for the following relation?