For example:  $(a+b)^2 = a^2 + b^2 + 2ab$  $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$  So my doubt regarding these identities are why does the identity differ when the power is changed and is there any derivation for it?
How was the binomial coefficient of the binomial theorem derived?  $$\frac{n!}{k!(n-k)!}$$
Run Script on Wakeup? I get this warning:     Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and will likely be removed in a future release   What does it mean? What's the difference between Normalization and Standardization? At work we were discussing this as my boss has never heard of normalization. In Linear Algebra, Normalization seems to refer to the dividing of a vector by its length. And in statistics, Standardization seems to refer to the subtraction of a mean then dividing by its SD. But they seem interchangeable with other possibilities as well.   When creating some kind of universal score, that makes up $2$ different metrics, which have different means and different SD's, would you Normalize, Standardize, or something else? One person told me it's just a matter of taking each metric and dividing them by their SD, individually. Then summing the two. And that will result in a universal score that can be used to judge both metrics.  For instance, say you had the number of people who take the subway to work (in NYC) and the number of people who drove to work (in NYC).   $$\text{Train} \longrightarrow x$$ $$\text{Car} \longrightarrow y$$  If you wanted to create a universal score to quickly report traffic fluctuations, you can't just add $\text{mean}(x)$ and $\text{mean}(y)$ because there will be a LOT more people who ride the train. There's 8 million people living in NYC, plus tourists. That's millions of people taking the train everyday verse hundreds of thousands of people in cars. So they need to be transformed to a similar scale in order to be compared.  If $\text{mean}(x) = 8,000,000$  and $\text{mean}(y) = 800,000$  Would you normalize $x$ &amp; $y$ then sum? Would you standardize $x$ &amp; $y$ then sum? Or would you divide each by their respective SD then sum?  In order to get to a number that when fluctuates, represents total traffic fluctuations.  Any article or chapters of books for reference would be much appreciated. THANKS!  Also here's another example of what I'm trying to do.   Imagine you're a college dean, and you're discussing admission requirements. You may want students with at least a certain GPA and a certain test score. It'd be nice if they were both on the same scale because then you could just add the two together and say, "anyone with at least a 7.0 can get admitted." That way, if a prospective student has a 4.0 GPA, they could get as low as a 3.0 test score and still get admitted.  Inversely, if someone had a 3.0 GPA, they could still get admitted with a 4.0 test score.  But it's not like that. The ACT is on a 36 point scale and most GPA's are on 4.0 (some are 4.3, yes annoying). Since I can't just add an ACT and GPA to get some kind of universal score, how can I transform them so they can be added, thus creating a universal admission score.  And then as a Dean, I could just automatically accept anyone with a score above a certain threshold.  Or even automatically accept everyone whose score is within the top 95%.... those sorts of things.  Would that be normalization? standardization? or just dividing each by their SD then summing? I have a few questions regarding CAN bus load  Is there any derived equation/formula to calculate the bus load?  I need transmit below frames(Assume Standard frame format) at baud rate 500 kbps on the bus 5ms  ----- 50 frames. 10ms ----- 10 frames. 100ms ---- 30 frmaes. Event messages ---- 10 frames.   How to calculate bus load, does anyone knows kindly illustrate the same?  What is the maximum bus load on CAN bus? Uniformly Continuous Function sending Bounded Set to Unbounded One Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f: X \to Y$ be a uniformly continuous function. If $A \subset X$ is bounded, must $f(A) \subset Y$ be bounded?  It is clear to me that in metric spaces that satisfy the Heine-Borel property, such as $\mathbb{R}^{n}$, the answer to this question is yes. However, I can see no reason why this should hold for arbitrary metric spaces. Any ideas? Thanks! Is the uncertainty principle a property of elementary particles or a result of our measurement tools? Classical limit of quantum mechanics I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero.  How can this be done?  (Ideally, I would love to see something like: as $\hbar$ goes to zero, the position wavefunction reduces to a delta function and that the Schrodinger equation/Feynman path integral reduces to the Newtonian/Lagrangian/Hamiltonian equations of motion.) Search across Stack Exchange sites in Android app Is it possible to search for a question across all Stack Exchange sites in the Android app?   Right now I can search in only one specific site at a time.   If not, can you add that feature? Prove that for every $a\in A$ $\exists$ $q\in M,s\in A$ s.t. $a=sba+qa$ where $M$ is maximal ideal in $A$. Let A be a commutative ring with unity ring, M be a maximal ideal and $b\notin M$ . Show that there exists $q \in M, s\in A$ such that for every $a \in A$ we have $a=sba+qa$. Show that this is true only when $M$ is a maximal ideal.   Suppose if we take $A=2\mathbb Z$ and $M=4\mathbb Z$, $a=6.$ Then by the given statement $\exists$ $s\in A$, $q\in M$ s.t. $6=6(2s+q)$ $\implies$$s=\frac{1-q}{2}\implies q \text { is odd}$. But $q \in 4\mathbb Z.$  Please someone give some hints. how can solve this.  Thank you.. How do I debug permissions? How do I debug permissions in Drupal 7?  I use core reporting, error logs, devel module error logs, backtrace and node access permission block but sometimes it is not enough. What else should I check to find out why certain fields, views or blocks are not showing to a user?  I also found there is a  for reporting on permissions called  but it is not available for Drupal 7.  I am using various permission related third-party modules:   - devel node access 7.x-1.x-dev   - fast permissions administration 7.x-2.0  - organic groups access control 7.x-1.x   - profile2 group access 7.x-1.x-dev   - ur-node access 7.x-1.x-dev Selecting an ID with a colon in it with jQuery show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$ The space $C^0([0;1],\mathbb{R})$ of all continuous, real-valued functions on $[0,1]$ is not reflexive. Can you stay in Turkey for over 90 days within the 180 day visa period but split over different visits? The Turkish e-visa is valid for a period of 180 days and allows you to stay for a maximum of 90 days. Am I allowed to stay for say 50 days, then another 50 days, over two different visits within the 180 day period?  I'm looking at the e-visa website  and it doesn't seem to be too clear on this. Legally possible to re-create/remake an old game? Recently I started remaking an old game on Unity. Basically I want to remake an old RPG that was released for Playstation 1. I'm not ripping anything from the old game, all the assets are being created by me. I'm creating everything from the scratch, music, 3D models, gameplay however I will keep the (copyrighted ?) names, I will try to create the characters and maps/levels as they were in the original version of Playstation 1 but with better graphics.  I'm not planning on selling it or adding in on a platform like Steam (for obvious reasons, copyrights etc), just giving it for free. But I wanna know when I will finish the development, can I share the game on public with other people so they can download it?  Can the company that made the original game sue me/fine me because of copyrights reasons even though I'm not planning on selling for money?  tl;dr Remake an non-profit game of PS1 from the scratch, can the company that holds the copyrights of the game sue me/fine me? There is this famous example about the order difference between gravitational force and EM force. All the gravitational force of Earth is just countered by the electromagnetic force between the electrons on the surface of my feet and the ground.  But also wave-function of electrons(fermions) do not overlap each other due to Pauli exclusion principle.  So which one is the true reason of me not flopping inside the earth forever? Is it only one of them(my guess is Pauli exclusion) or is it both? I would like to draw an arrow like this using TikZ:    This is taken from . I didn't succeed in reproducing the arrow; is somewhat beyond my (extremely limited) TikZ abilities. I don't need all the workflow stuff. I simply want to draw such an arrow at a certain place in my text, e.g. -->  \documentclass{article} \usepackage{&lt;???&gt;} \usepackage{tikz} \usetikzlibrary{&lt;???&gt;} \begin{document} Donec vehicula augue eu neque. Pellentesque habitant morbi \tikz &lt;???code for the arrow???&gt; ; tristique senectus et netus \end{document}   PS: Some explanations in your solution would be much appreciated, otherwise it will be difficult to customize it. Firefox suspicious subprocess (turned out to be legitimate FF behavior) I've discovered this yesterday, as it was generating system load on otherwise idle Ubuntu 14.04 Desktop machine.  Today, I've confirmed such behavior also on another Ubuntu instance (17.10 in Virtualbox VM)  The sub-process in command line looks like below ( I've put this as picture below intentionally to prevent askubuntu.com from changing/escaping the content )   For me it looks like exploit/malware. Or at least there is something to hide.  this happens even for url No transaction fee ETF trades - what's the catch? I just noticed TD Ameritrade has program where you can trade from a list of ETFs with zero fees.  Sounds like it could save me a bit of money, but my cynical side wonders how they make money off such a program...  Is it a loss leader? (They entice me to have an account there, and then make money off the other trades I make?)  Or do they simply want to grow the size of their asset base -- what is the mechanism by which they make money on the funds that I have in my account?  Edit: in case it wasn't clear in the original question above, this is an ongoing program for free ETF transactions -- not a one-time deal for some number of free trades when you open an account. I have had an account there for years and I'm eligible to participate -- it's not a program explicitly designed just to gain new accounts. In PHP, is there an easy way to convert a number to a word? For instance, 27 to twenty-seven.