How do you treat gradient as a vector?
Creating the Nabla operator (also known as Del operator) as an operator
When writing questions and answers here about the Harry Potter series, I often need a word for "someone who's not a Muggle". People often refer to "the wizarding world" as opposed to the Muggle word, so "wizard" would be one choice - but it seems slightly sexist to exclude witches in that way. On the other hand, "witch or wizard" is a bit cumbersome, much longer than the simple word "Muggle".  Is there a sensible word to use which is neither sexist nor cumbersome, that makes a good equivalent to "Muggle"? What would JK Rowling say to fill in a sentence such as this:     Muggles use technology, while [...] use magic. Perform feature normalization before or within model validation? A common good practice in Machine Learning is to do feature normalization or data standardization of the predictor variables, that's it, center the data substracting the mean and normalize it dividing by the variance (or standard deviation too). For self containment and to my understanding we do this to achieve two main things:   Avoid extra small model weights for the purpose of numerical stability. Ensure quick convergence of optimization algorithms like e.g. Conjugate Gradient so that the large magnitude of one predictor dimension w.r.t. the others doesn't lead to slow convergence.   We usually split the data into training, validation and testing sets. In the literature we usually see that to do feature normalization they take the mean and variance (or standard deviation) over the whole set of predictor variables. The big flaw I see here is that if you do that, you are in fact introducing future information into the training predictor variables namely the future information contained in the mean and variance.  Therefore, I do feature normalization over the training data and save the mean and variance. Then I apply feature normalization to the predictor variables of the validation and test data sets using the training mean and variances. Are there any fundamental flaws with this? can anyone recommend a better alternative? Looking for a book series about a girl and her friends captured and sold as slaves I'm trying to find the name of a book series I read about 10 years ago. It might be a children/young-adult series since I was fairly young. There is also a small chance it was written by a french author, as I read it in french. The series is composed of at least three books, maybe more. Here is what I remember :   The main character is a poor girl/young woman, but there are several point of view characters At the beginning of the first book the plot is set in a city where rich people live in the city center and the further you go from the center the poorest people are. Moreover, the houses are painted according to the wealth of the owner. It was something like : the richer had red houses, the poorer blue houses, and the moderately wealthy purple (it was not those colors in particular but it was something like that) At the end of one of the book there is some kind of uprising and the people paint all the houses in multiple colorful shades (to symbolize the new found equality between the citizen) At some point the main character and other peoples of the city get captured (maybe after some kind of war the city was involved in), separated and sold to slavery One of them, an older boy, become a soldier and learn a special fighting technique that looks like dancing (and he grows to like it) Another of the captives, a young girl (the sister of the main character if I recall correctly) get marked with a fire-heated mark to show her condition of slave   That is all I remember, I have no clue about the time or geographic setting nor the story details.  I know it's not much to go by, any help would be greatly appreciated, as I would like to find these books and read them again. Why shouldn't root be allowed to login via ssh? why should root access be disabled for ssh? I always hear that's for security reasons. But I don't get it. What's different to login as non-root and then sudo su -? What's the preferred way to handle tasks that need privileged permissions?  Thanks  Thomas What translation occurs when writing to a file that was opened in text mode that does not occur in binary mode? Specifically in MS Visual C.  unsigned char buffer[256]; for (int i = 0; i &lt; 256; i++) buffer[i]=i; int size  = 1; int count = 256;   Binary mode:  FILE *fp_binary = fopen(filename, "wb"); fwrite(buffer, size, count, fp_binary);   Versus text mode:  FILE *fp_text = fopen(filename, "wt"); fwrite(buffer, size, count, fp_text); One-dimensional [Noetherian] UFD is a PID Convergence of an alternating series : $ \sum_{n\geq 1} \frac{(-1)^n|\sin n|}{n}$ Study the convergence of $$\displaystyle \sum_{n\geq 1} \frac{(-1)^n|\sin n|}{n}.$$   I am stuck with this series, we need probably some measure of irrationally of $\pi$, unfortunately I am unfamiliar with this. So here is my attempt :  Let $f(x) = \sum \frac{|\sin{n}|}{n} x^n, |x| &lt; 1$  It's not difficult to compute the Fourier series of $|\sin(x)|$ :  $$ \displaystyle|\sin(x)|=\frac{2}{\pi}-\frac{4}{\pi}\sum_{n=1}^{+\infty}\frac{\cos(2nx)}{4n^2-1} $$  Then Fubini's theorem () works very well (because the previous series converges absolutely at $x$ fixed ) and all calculations made, we find that for all $x\in( -1,1)$:  $$ \displaystyle f(x)=\frac{2}{\pi}\sum_{n=1}^{+\infty}\frac{x^n}{n}-\frac{4}{\pi}\sum_{p=1}^{+\infty}\frac{x^2-2x\cos(p)}{(4p^2-1)(x^2-2x\cos(p)+1)} $$  However, the second sum I have not been able to show the convergence. I feel the series diverge because the following series $$ \displaystyle\sum\frac{1}{p^2\sin^2\left(\frac{p}{2}\right)} $$  diverge because  $0$ is an accumulation point of $\displaystyle (n\sin(n))$ sequence.  Any ideas (for the original series) ? Searching for a string on multiple zip files How to create Custom Fields on Standard Objects in the package.xml Implementing i18n string translation in a custom module? I have a custom module I've built which essentially acts as a repository of custom CTools keywords that users can arbitrarily define. The issue I'm facing is I need to be able to translate these user-defined strings, so I think I need to implement string translation using the i18n_string module.  I have never worked with translation or localization before, so I'm a bit lost with regards to going about how to implement this. I think I'm on the right path with i18n_string though. Can someone experienced with i18n provide examples that I can look at so I can learn the proper way to implement this? How can I install apt packages non-interactively? How to make the Java.awt.Robot type unicode characters? (Is it possible?) We have a user provided string that may contain unicode characters, and we want the robot to type that string.  How do you convert a string into keyCodes that the robot will use? How do you do it so it is also java version independant (1.3 -> 1.6)?  What we have working for "ascii" chars is  //char c = nextChar(); //char c = 'a'; // this works, and so does 'A' char c = 'á'; // this doesn't, and neither does 'Ă' Robot robot = new Robot(); KeyStroke key = KeyStroke.getKeyStroke("pressed " + Character.toUpperCase(c) ); if( null != key ) {   // should only have to worry about case with standard characters   if (Character.isUpperCase(c))   {     robot.keyPress(KeyEvent.VK_SHIFT);   }    robot.keyPress(key.getKeyCode());   robot.keyRelease(key.getKeyCode());    if (Character.isUpperCase(c))   {     robot.keyRelease(KeyEvent.VK_SHIFT);   } } finite and infinite product of discrete space How does one prove the following limit? $$   \lim_{n \to \infty}   \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$ Recommended ways to upgrade AlwaysOn AG from 2014 to 2016 What is the recommended process of upgrading Availability Groups with one Sync Replica from SQL 2014 to 2016?  My understanding is that in-place upgrades are not really liked among database professionals. Is there a way to avoid in-place upgrades in Availability Groups? Is there a possibility to join a 2016 server into an existing 2014 availability group and to failover there, then "kill" one of the other instances and upgrade it? (e.g. by removing/re-installing SQL side-by-side on the same host)  Are there any other upgrade scenarios that we should consider?  The ideal solution would be that we always have 2 online replicas, and that we don't need to do any in-place upgrades. What's better here - a single column or multicolumn primary key? Let's assume you have a table groups and a table item. Each item belongs to exactly one group. It is an inherent part of that group. An item cannot exist outside of a group and it cannot be moved to another group.  When trying to decide on a primary key for the item table, what should I use?  Should I make up an artificial global serial key like this:  CREATE TABLE items (     item serial PRIMARY KEY,     group integer NOT NULL REFERENCES groups(group), );   ... or should I rather use a composite key and per-group item serial like this:  CREATE TABLE items (     group integer NOT NULL REFERENCES groups(group),     item integer NOT NULL,      PRIMARY KEY(group, item) );   The reason why I'm leaning more towards the second solution is that the post URL will always show the group and item, so it makes sense to have both of them as the composite primary key. In case of the first solution, the URL contains superfluous information because the group ID can already be deduced from the item ID alone. The URL structure is given, however, and cannot be changed.  The disadvantage of the second solution is that you have to manage a per-group serial (i.e. each item integer should start from 0 for each group).  What's better in terms of best practices, normalization and performance? Or is it simply a matter of taste? Put two figures side by side There are already several threads that deal with this: they suggest having one figure and putting putting minipages inside, or using the subcaption package and having subfigure environments. However, I've tried both of these approaches and I'm still going wrong:  \begin{figure}[h]  \begin{minipage}[b]{0.2\linewidth}  \begin{tikzpicture}[sibling distance=0pt] \Tree [.TP         [.DP \edge[roof]; {a 'we'} ]         [.T$'$             [.T \node(li){li 'eat$_i$'}; ]             [.VP                 [.Spec t ]                 [.V$'$                      [.DP \edge[roof]; {saka 'rice'} ]                     [.V \node(t){t$_i$}; ] ] ] ] ] \draw [-&gt;] (t) .. controls +(south:3) and +(south:3) .. (li); \end{tikzpicture}  \end{minipage}\hfill  \begin{minipage}[b]{0.2\linewidth}  \begin{tikzpicture}[sibling distance=0pt] \Tree [.TP         [.DP \edge[roof]; {a 'we'} ]         [.T$'$             [.T \node(li){la 'have'}; ]             [.VP                 [.Spec t ]                 [.V$'$                      [.DP \edge[roof]; {saka 'rice'} ]                     [.V \node(t){li 'eaten'}; ] ] ] ] ] \draw [red,-&gt;] (t) .. controls +(south:3) and +(south:3) .. (li); \end{tikzpicture}  \end{minipage}  \end{figure} If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$. The proof is already given in the textbook but I tried other way around.   Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So $\gcd(p,a)=1$ and $\gcd(p,b)=1$. Given that we can construct linear combinations   $sp+ta=1$ together with $up+wb=1$. Multiplying the left and the right sides of the equations we get  $spup+spwb+taup+tawb=1$ or $p(sup+swb+tau)+ab(tw)=1 \implies \gcd(p,ab)=1$. This is a contradiction.   Is it rigorous and sound proof? shell scripts are still working without #!(sha-bang line) I am new to shell scripts and many books have written that use #!(sha-bang) line in starting of the script to invoke the interpreter .And this will invoke a new shell for script and do the interpretation line by line.But one of my basic script is still running without the magic line.  so my questions are:   from where did my basic script get the interpreter. how did the script manages to locate the interpreter.   now let me tell you about my basic script it just contains the following line:  echo "basic script without the magic lines" “situation where” vs. “situation in which”