Incorrect derivative
Why does putting a 1/2 in front of the squared error make the math easier?
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? I used the prcomp() function to perform a PCA (principal component analysis) in R. However, there's a bug in that function such that the na.action parameter does not work. ; two users there offered two different ways of dealing with NA values. However, the problem with both solutions is that when there is an NA value, that row is dropped and not considered in the PCA analysis. My real data set is a matrix of 100 x 100 and I do not want to lose a whole row just because it contains a single NA value.  The following example shows that the prcomp() function does not return any principal components for row 5 as it contains a NA value.  d       &lt;- data.frame(V1 = sample(1:100, 10), V2 = sample(1:100, 10),                        V3 = sample(1:100, 10)) result  &lt;- prcomp(d, center = TRUE, scale = TRUE, na.action = na.omit) result$x                                # $ d$V1[5] &lt;- NA                           # $ result  &lt;- prcomp(~V1+V2, data=d, center = TRUE, scale = TRUE, na.action = na.omit) result$x   I was wondering if I can set the NA values to a specific numerical value when center and scale are set to TRUE so that the prcomp() function works and does not remove rows containing NA's, but also does not influence the outcome of the PCA analysis.  I thought about replacing NA values with the median value across a single column, or with a value very close to 0. However, I am not sure how that influences the PCA analysis.  Can anybody think of a good way of solving that problem? Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem? Garbage Collection: Is it necessary to set large objects to null in a Dispose method? How to take mysqldump with Generated column? I am getting below error while importing the dump,     ERROR 3105 (HY000) at line 82: The value specified for generated   column 'column_name' in table 'table_name' is not allowed.   When I describe the table with error, there are some generated columns in it.     option_type         | varchar(45)  | YES  | MUL | NULL    | VIRTUAL   GENERATED   What is the correct command to use when taking mysql dump for the database contains generated column?  MySQL server source and destination version:     mysqld Ver 5.7.26 for Linux on x86_64 (MySQL Community Server (GPL))   MariaDB mysqldump CLI version:     mysqldump Ver 10.17 Distrib 10.3.12-MariaDB, for Linux (x86_64) Proof a set cannot contain its own powerset - Russell's Paradox How can I type "lambda-bar"? TimeSpan to String causing an error problem with using loop inside the tabular environment What happens if my ESTA expires on the day of arrival? I'm French and will travel the 14th of June to the US. My ESTA expires the 14th of  June 2013 and I'm wondering if my ESTA will be valid when I'll land in the US on the 14th of June 2013.  Does someone know anything about that? python ImportError: No module named primes Why am I getting black photos when I use the Manual feature on my DSLR? I'm connecting to some enterprise wireless network (eduroam).  The thing is that it saves the User/Password in the wireless settings, anyone can click 'Show password' and see my university account password.  What I want is that it prompts me everytime the credentials inputs when I try to connect to the network and don't save it under the 'settings' page.  Is this possible? If yes, how?  Thanks  Ubuntu 18.04 LTS Undefined reference error for template method In calculating work done by a constant force over a constant distance, why doesn't the subject's initial velocity matter? Assume a point-mass $m$ is travelling in a straight line, and a force $F$ will act on $m$ (in the same direction as $m$'s velocity) over a constant distance $d$; why doesn't $m$'s velocity matter to the calculation of work done on $m$ by $F$? Work is defined such that, in this example, the work done by $F$ on $m$ is equal to $Fd$, but it seems that if $m$ were moving slower, it would spend more time in the field, allowing $F$ more time to act on $m$, thereby doing more work. In fact, if $m$'s velocity were very great, it would hardly spend time in $F$'s field at all (so very little work done). Maybe I misunderstand work; can someone address this confusion of mine? Batch script to resize height of all layers to same height but keeping their original aspect ratio I have hundreds of product image layers and I am trying to resize them all to the same height (ignoring the width) and retaining their individual aspect ratio's.  Is there a script available that does this? Or a feature I am unaware of?  Thanks! Show that the set of functions $\mathbb{N}\to\{0,1\}$ is not countable Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N}\to S$. Now, I consider (0,1). I want to prove by contradiction that $(0,1)$ is not countable.  First, I assume the contrary that there exists a bijection $f$, and I can find an element in $S$, but not in the range of $f$. But I can't find such element. How can you construct such $f$? What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$?  Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle.  But what is the fundamental group of the half circle? Calculate total sum of all numbers in c:forEach loop I have a Java bean like this:  class Person {   int age;   String name; }   I'd like to iterate over a collection of these beans in a JSP, showing each person in a HTML table row, and in the last row of the table I'd like to show the total of all the ages.  The code to generate the table rows will look something like this:  &lt;c:forEach var="person" items="${personList}"&gt;   &lt;tr&gt;&lt;td&gt;${person.name}&lt;td&gt;&lt;td&gt;${person.age}&lt;/td&gt;&lt;/tr&gt; &lt;/c:forEach&gt;   However, I'm struggling to find a way to calculate the age total that will be shown in the final row without resorting to scriptlet code, any suggestions? No continuous function switches $\mathbb{Q}$ and the irrationals