Can someone explain the difference between symbols:   $\rightarrow$ and $\mapsto$  Thanks.
I am doing , and am having a few problems with function notation. I understand the form f(x).  What does the form f: x ↦ y mean? Could you also give one or two examples?  And, if possible, state your source. Thank you.
If $z_n \to z$ then $(1+z_n/n)^n \to e^z$ We are dealing with $z \in \mathbb{C}$. I know that $$ \left(1+ \frac{z}{n} \right)^n \to e^{z} $$ as $n \to \infty$. So intuitively if $z_n \to z$ then we should have $$ \left(1+ \frac{z_n}{n} \right)^n \to e^{z}. $$ If $z_n \in \mathbb{R}$ I would be happy writing $$ \exp \left(n \log\left(1+ \frac{z_n}{n} \right) \right) = \exp \left(z_n \frac{\log\left(1+ \frac{z_n}{n} \right)-\log(1+0)}{\frac{z_n}{n}-0} \right) \to \exp(z \cdot 1) $$ where here we are using the definition of derivative. But if $z \in \mathbb{C}$ the logarithm is multivalued and I'm not sure that the same calculation is allowed. Is there a different way to show this for complex $z$? Why does the zoom sometimes stop at a point? Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Prove via induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2).  Not sure when to apply the induction hypothesis and how to get the result $1- \frac{1}{(n+2)!}$.  Please help! thanks guys, youre the greatest! I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E).   Reach := {u} LastCardinality := 0 while LastCardinality &lt; |Reach| do   LastCardinality := |Reach|   Next := null   foreach i &lt;- Reach do     foreach j &lt;- V do       if(i,j) exists in E then         Next := Next + {j}   Reach := Reach + Next return Reach   So far I have got that the output needs to be Reach = Goal where Goal = {r | r exists V, there is a path from u to r}. I also have three invariants that I am pretty sure will help prove this.   Reach is always a subset of Goal and Next is always a subset of Goal. After an arbitrary K times through the while loop, for every t that exsits in V, if t has a path of length k+1 from u, then t exists in Reach If Reach is a subset of Goal before the while loop then LastCardinality &lt; |Reach|   I know that to finish up the proof, I need to prove termination, the invariants, and correctness but I have no idea how. I think I need to use induction on the while loop but I am not exactly sure how.   How do I proof those three characteristics about an algorithm? PNG figures in black and white I'm trying to include a png figure and it works when I see it in dvi, but the ps comes in black and white.  If I compile directly to pdf there is no problem. However I'm using psTricks for a different figure, so I need to use ps first.  Any suggestions? Cross product: matrix transformation identity How can one prove the following identity of the cross product?  $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$  $a$ and $b$ are 3-vectors, and $M$ is an invertible real $3 \times 3$ matrix. "All of the ...." or " All the ... "? $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property] Is $$d_1\mid n,d_2\mid n\iff [d_1,d_2]\mid n$$ true ? And if yes, how can I prove it ? I recall that $[d_1,d_2]$ is the least common multiple.  My tries  For the implication : Let $d_1\mid n$ and $d_2\mid n$. Then, $d_1d_2\mid n^2$. By the way, since $d_1,d_2\mid n$, we have that $(d_1,d_2)\mid n$ where $(d_1,d_2)=:\gcd(d_1,d_2)$  and thus $$[d_1,d_2](d_1,d_2)\mid n^2.$$  Question: How can I get $[d_1,d_2]\mid n$ from this?  For the converse, since $d_1,d_2\mid [d_1,d_2]\mid n$, the claim follow. Could not open input file: composer.phar is "I'll" correct as a short answer? Why are there so few robots in Star Trek? Okay, obviously beyond Data and the Soong-type androids. But seriously, we have more robots in our daily life now than there apparently are on Trek. Is there a canon answer why not? It seems like Star Wars is more on the nose in this regard - it’s unclear why all dirty/dangerous/boring work wouldn’t be done by robots (anthropomorphic or not). Who did Khan get his Starfleet insignia from? How do I iterate over the words of a string? I'm trying to iterate over the words of a string.  The string can be assumed to be composed of words separated by whitespace.  Note that I'm not interested in C string functions or that kind of character manipulation/access. Also, please give precedence to elegance over efficiency in your answer.  The best solution I have right now is:  #include &lt;iostream&gt; #include &lt;sstream&gt; #include &lt;string&gt;  using namespace std;  int main() {     string s = "Somewhere down the road";     istringstream iss(s);      do     {         string subs;         iss &gt;&gt; subs;         cout &lt;&lt; "Substring: " &lt;&lt; subs &lt;&lt; endl;     } while (iss); }   Is there a more elegant way to do this? How can top principal components retain the predictive power on a dependent variable (or even lead to better predictions)? Suppose I am running a regression $Y \sim X$. Why by selecting top $k$ principle components of $X$, does the model retain its predictive power on $Y$?   I understand that from dimensionality-reduction/feature-selection point of view, if $v_1, v_2, ... v_k$ are the eigenvectors of covariance matrix of $X$ with top $k$ eigenvalues, then $Xv_1, Xv_2 ... Xv_k$ are top $k$ principal components with maximum variances. We can thereby reduce the number of features to $k$ and retain most of the predictive power, as I understand it.   But why do top $k$ components retain the predictive power on $Y$?   If we talk about a general OLS $Y \sim Z$, there is no reason to suggest that if feature $Z_i$ has maximum variance, then $Z_i$ has the most predictive power on $Y$.   Update after seeing comments: I guess I have seen tons of examples of using PCA for dimensionality reduction. I have been assuming that means the dimensions we are left with have the most predictive power. Otherwise what's the point of dimensionality reduction? Why XML-Serializable class need a parameterless constructor How to pass javascript value to controller..? I want to pass the java-script value to controller.I am using repeat on VF page.  VF page      &lt;apex:repeat value="{!listAdj}" var="Adj" id="hrs"&gt;       &lt;td&gt;&lt;div class="requiredInput"&gt;&lt;div class="requiredBlock"&gt;&lt;/div&gt;     &lt;apex:inputField id="decimal" value="{!Adj.Hours_Decimal__c}" onchange="setHidden5(this);" &gt;       &lt;/apex:inputField&gt;     &lt;/div&gt;&lt;/td&gt;             &lt;apex:inputField id="hours" value="{!Adj.Hours__c}" /&gt; &lt;apex:inputField id="decimal" value="{!Adj.Hours_Decimal__c}" onchange="setHidden5(this);"/&gt;              &lt;/apex:repeat&gt;   Javascript code  function setHidden5(element){     //split the id of the inputField by ':' and then use the 5th index in the array to exact position of the inputField in the apex:repeat.   var idArray = element.id.split(':');     var hiddenRep1 =document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":decimal");          var hiddenRep2 = document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":hours");       if(element.value.length &gt;=1){        document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":hours").disabled = true; **hiddenRep2.value=hidden1.value;**                }                             }     }   I want to pass the hiddenRep2.value to controller.How to pass that. Hidden2.value is same as hidden1.value + some addition subtraction.i dont know how to pass that value. Can someone me sort out this.  Thanks Attempting official deploy process with single sandbox may reset/remove all existing work? Is there an comparison of the various mapping libraries for Android? I am currently looking at the various libraries that I can use to show a map in my custom Android App. I have found several like Mapsforge, OSMDroid, ArcGIS runtime for Android, Google Maps for Android and so on. I'm sure that I am missing a few.   Is there a comparison chart somewhere about the differences in functionalities and pros and cons of each one?   If there isn't one, we could make this question community wiki, to serve alongside  &amp; . What is the most efficient way to deep clone an object in JavaScript? What is the most efficient way to clone a JavaScript object? I've seen obj = eval(uneval(o)); being used, but . I've done things like obj = JSON.parse(JSON.stringify(o)); but question the efficiency.  I've also seen recursive copying functions with various flaws.  I'm surprised no canonical solution exists. Bring Debian to second Position in the GRUB Menu at startup