I've been writing some simple machine learning algorithms and looking at time-series data. In doing so, I have come across the use of feature scaling: rescaling and standardizing as they are referred to on .   For something like k-Nearest Neighbors, I imagine rescaling to be the best scaling to use so there's a completely level playing field between variables' influence. Is that true?   When would you apply standardizing to data instead? What are the advantage of standardizing and rescaling?
In some literature, I have read that a regression with multiple explanatory variables, if in different units, needed to be standardized.  (Standardizing consists in subtracting the mean and dividing by the standard deviation.)  In which other cases do I need to standardize my data?  Are there cases in which I should only center my data (i.e., without dividing by standard deviation)?
Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$ Deteremine the sum of the series $$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$$  So I first write down the $n^{th}$ term $a_n=\frac{\frac{n(n+1)}{2}}{n!}=\frac{n+1}{2(n-1)!}$.  So from there I can write the series as $$1+\frac{3}{2}+\frac{4}{2\times 2!}+\ldots +\frac{n+1}{2(n-1)!}+\ldots $$  I am quite sure I can do some sort of term by term integration or differentiation of some standard power series and crack this. Any leads? How to remove film from stainless steel pan I have an unsightly film around my stainless pan as shown here:    What is causing this to happen?  And how can I remove the film? I have soaked it in soapy water, scraped it with a plastic scraper, and run it in the dishwasher several times, but I can't seem to get rid of the film.   UPDATE: the pan is an All-Clad stainless, fully-clad frypan: Ain't and negatives Pick random row from a file What is the best way to find out a researcher's current affiliation? Let's say I know a name and a field, but not much more.   A google search seems like the most obvious approach, but unfortunately university personnel or research group pages often seem to be quite poorly maintained, and it is not uncommon that someone is listed even if they left more than a year ago.  Early-career researchers are particularly difficult to track down since they often have less of a "presence" on institution webpages and are likely to be moving relatively frequently. An idempotent operator is compact if and only if it is of finite rank Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank. Let $R$ be a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal. Prove: Let $R$ a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal.     I got stuck on this. I understand that every ideal is generated by finitely many elements. Here is my approach:  It's enough to prove the following implication: $$ R/I \ \text{is a domain} \quad \Rightarrow \quad R/I \ \text{is a field}$$ Then only finding an inverse is left. So let $x \in R$ in arbitrary. We have to find an inverse element of $x+I \in R/I$, that as an element y so that $x \cdot y \in I+1$. We denote $I = (x_1, x_2, \cdots, x_n)$.  Now I have to find y so that $xy -1 = r_1 x_1+ r_2 x_2 + \cdots + r_n x_n$. This is where I no more knew what to do.   Your advise will be appreciated. In X-Men Origins: Wolverine,      Wolverine is shot through the head with what appears to be a vibranium bullet by William Stryker. He remarks that the brain will grow back but the memories won't.   So the projectile penetrated the adamantium skull and damaged the brain. Adamantium is an inorganic substance. It will survive anything that would otherwise destroy the rest of Wolverine's body. But if anything (such as vibranium) manages to damage the adamantium, it is not covered by Wolverine's healing abilities.  Does this mean that Wolverine has an Achilles' heel on his forehead? If so, how did he take a bullet to the exact same spot in X-Men: United (at Ice Man's home)? What it says on the tin, a foreign friend of mine has asked and I can't tell him; apart from it sounding horrible.  For example:  "Why is it raining today?"   Instead of:  "Why it is raining today?" When Cantor's Intersection theorem won't work with closed sets Give an example to show that Cantor's Intersection Theorem would not be true if compact sets were replaced by closed sets.    Compact set is closed and bounded, so what I'm going to find is something that is closed but not bounded.  By the Cantor theorem, which says that a decreasing sequence of non-empty compact subset will have a non-empty intersection.  What I was trying to approach is how two sets just "touched" and as if they have not bounded, two sets "touched" nothing. But I cant give an example for that I'm using a BooleanToVisibilityConverter in WPF to bind the Visibility property of a control to a Boolean. This works fine, but I'd like one of the controls to hide if the boolean is true, and show if it's false. I was reading an example about describing homomorphisms and I'm having a bit of trouble with this one.  Show there is no group homomorphism $f : \mathbb{Z}_{10} \to \mathbb{Z}_{25}$ such that $f(1) = 3$. The first step of the proof is  $|1| = 10$ in $\mathbb{Z}_{10}$, so $|f(1)| \in\{ 1, 2, 5, 10\}$.  Does this mean that $|f(1)|$ can be $1$, $2$, $5$ or $10$? I see that $|1| = 10$ since $1$ is a generator of $\mathbb{Z}_{10}$, and that $|f(1)|$ seems to be a factor of $10$. If $f(1)$ is the image of $1$ under $f$, how would I find the order of $f(1)$?  Thanks for your help. Outcast boy meets girl underwater How to use sed to replace only the first occurrence in a file? I would like to update a large number of C++ source files with an extra include directive before any existing #includes. For this sort of task, I normally use a small bash script with sed to re-write the file.   How do I get sed to replace just the first occurrence of a string in a file rather than replacing every occurrence?  If I use  sed s/#include/#include "newfile.h"\n#include/   it replaces all #includes.   Alternative suggestions to achieve the same thing are also welcome. Should I not answer my own questions? Check if macro is fully expandable Question  Is there a way to check whether a macro is fully expandable (or rather "safe in an expansion-only context" )?    Consider this code:  \def\a{Just a string} \def\b{\a} \def\c{\def\unsafe} \def\d{\c}   How could I check which of the macros (a-d) are safe in an expansion-only context? By looking at them I know that a and b are whereas c and d are not but if I wanted to know the same for a macro I haven't written myself this could get quite useful.      Background  I am working on a way to detect whether some input is a valid number in PGF. For this I developed  which makes use of passing the input into \pgfmathfloatparsenumber. The problem I have run into is that said macro appears to somehow manages to expand the input until there is an error (if the input is in fact not safe in an expansion-only-context). I tried using protected, noexpand and similar but somehow PGF manages to circumvent those.    So the idea is to check whether the input is safe before actually passing it to PGF. The problem is: I don't know how I'd go about that... C99 stdint.h header and MS Visual Studio Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant. While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated.  Let $U ⊂ \mathbb{C}$ be a domain and $f_1, . . . , f_n : U \rightarrow \mathbb{C}$ holomorphic functions, such that $\sum_{k=1}^n \left| f_k(z) \right|$ is constant on $U$. Show that $f_1, . . . , f_n$ are constant. Give users a way to see their "Positive Question Record" score Is there any reason to use this->