Converting ú to u in javascript
Efficiently replace all accented characters in a string?
Solve the following system of equations  $ \begin{cases}  x_1^{'}(t)=x_1(t)+3x_2(t) \\  x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\  x_3^{'}=-x_2(t)+x_3(t)\end{cases} $.  First, I create the column vectors $X$ and $X^{'}$. Then the matrix $$A= \begin{bmatrix} 1 &amp; 3 &amp; 0  \\ 3 &amp; -2 &amp; -1 \\ 0 &amp; -1 &amp; 1  \\ \end{bmatrix} $$  Now, I find the eigenvalues, $-4,3,1$ and their corresponding eigenvectors $(-3,5,1)^T (-3,-2,1)^T (1,0,3)^T$.   I'm just not sure how to take it from here and solve the system of differential equations. I want a diagonal matrix $D$ so that I can read the solutions easy, but I'm not sure how to do it.  EDIT   Building on @Francisco 's answer, I'd have that:  $$X=c_1  (-3,5,1)^T  e^{-4t}   + c_2 (-3,-2,1)^T e^{3t} + c_3 (1,0,3)^T e^{t} $$.  But I believe this could be written in a simpler form. What is information? Why was the alien threat in The Next Generation episode "Conspiracy" never followed up? In the Star Trek The Next Generation episode  Picard and crew are drawn into an alien conspiracy to take over the Federation. This ends with Picard and Riker destroying an alien "mother" beast, but not before it manages to send a signal to an uncharted sector of space. The feeling at the end of the episode is that this will be a threat that the intrepid crew of the Enterprise would be facing again at some point, but this never happens.   Why was the implied threat at the end of this episode never brought up again on the show? I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. Illogical twoside margins Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)&lt;3$   for all positive integers $n$   This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$ and then proceed by induction.  The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated. Javascript "Variable Variables": how to assign variable based on another variable? I have a set of global counter variables in Javascript:  var counter_0 = 0; var counter_1 = 0; var counter_2 = 0;   etc  I then have a Javascript function that accepts an 'index' number that maps to those global counters. Inside this function, I need to read and write to those global counters using the 'index' value passed to the function.  Example of how I'd like it to work, but of course doesn't work at all:  function process(index) {     // do some processing      // if 'index' == 0, then this would be incrementing the counter_0 global variable     ++counter_+index;       if (counter_+index == 13)     {         // do other stuff     } }   I hope what I'm trying to accomplish is clear. If not I'll try to clarify. Thanks.  EDIT Clarification:  I'm not trying to increment the name of the counter, but rather the value the counter contains. What is the biggest display of power through the Force in Star Wars? Is it possible to enforce unique values in a column, excluding nulls? If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes the cost of moving mass from the source distribution $P_S$ to the target distribution $P_T$,  Do the masses of the source and target distributions have to be identical? If so or not, how are the masses of $P_T$ and $P_S$ accounted for mathematically in, or when minimizing, the shown formula, or what might be missing/overlooked by the Wasserstein approach overall due to the mass differences? What exactly do \csname and \endcsname do? What exactly do \csname and \endcsname do? What are their job? I have glanced at the Texbook and some other books, but none of them was clear enough to me. Can anyone please give a simple example to clarify this issue? Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$ Advantages of ROC curves What is the advantages of the ROC curves?  For example I am classifying some images which is a binary classification problem. I extracted about 500 features and applied a features selection algorithm to select a set of features then I applied SVM for classification. In this case how can I get a ROC curve? Should I change the threshold values of my feature selection algorithm and get sensitivty and specificity of the output to draw a ROC curve?  In my case what is the purpose of creating a ROC curve? How to make a grid-to-sphere animation like this? Why is it called the Millennium Falcon? I'm particularly interested in how the "Millennium" part pertains to the ship itself. It wouldn't hurt to explain the "Falcon" part either though. The distance between touching objects What is the distance between, say, a cup of coffee and the table it rests on? What is the distance between two touching hands? How to add repository from shell in Debian? What does "isomorphic" mean in linear algebra? Top Down RPG Movement w/ Correction? I would hope that we have all played Zelda: A Link to the Past, please correct me if I'm wrong, but I want to emulate that kind of 2D, top-down character movement with a touch of correction. It has been done in other games, but I feel this reference would be the easiest to relate to.  More specifically the kind of movement and correction I'm talking about is:   Floating movement not restricted to tile based movement like Pokemon and other games where one tap of the movement pad moves you one square in that cardinal direction. This floating movement should be able to achieve diagonal motion. If you're walking West and you come to a wall that is diagonal in a North East/South West fashion, you are corrected into a South West movement even if you continue holding left (West) on the controller. This should work for both diagonals correcting in both directions. If you're a few pixels off from walking squarely into a door or hallway, you are corrected into walking through the hall or down the hallway, i.e. bumping into the corner causes you to be pushed into the hall/door.   I've hunted for efficient ways to achieve this and have had no luck. To be clear I'm talking about the human character's movement, not an NPC's movement. Are their resources available on this kind of movement? Equations or algorithms explained on a wiki or something?  I'm using the XNA Framework, is there anything in it to help with this? Type safety: Unchecked cast In my spring application context file, I have something like:  &lt;util:map id="someMap" map-class="java.util.HashMap" key-type="java.lang.String" value-type="java.lang.String"&gt;     &lt;entry key="some_key" value="some value" /&gt;     &lt;entry key="some_key_2" value="some value" /&gt;    &lt;/util:map&gt;   In java class, the implementation looks like:  private Map&lt;String, String&gt; someMap = new HashMap&lt;String, String&gt;(); someMap = (HashMap&lt;String, String&gt;)getApplicationContext().getBean("someMap");   In Eclipse, I see a warning that says:  Type safety: Unchecked cast from Object to HashMap  What did I do wrong? How do I resolve the issue?