says the answer is $1$.  says it's undefined.         I saw an argument made for $0^0$ being $1$, on , along the lines of: if you want the binomial theorem to work for real numbers (in the form $(a + 0)^n$ where $a \in \mathbb{R}$), then you should accept that $0^0 = 1.$
Could someone provide me with a good explanation of why $0^0=1$?  My train of thought:  $x&gt;0$  $0^x=0^{x-0}=0^x/0^0$, so  $0^0=0^x/0^x=\,?$  Possible answers:   $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined   PS. I've read the  on mathforum.org, but it isn't clear to me.
Resolve build errors due to circular dependency amongst classes I often find myself in a situation where I am facing multiple compilation/linker errors in a C++ project due to some bad design decisions (made by someone else :) ) which lead to circular dependencies between C++ classes in different header files (can happen also in the same file). But fortunately(?) this doesn't happen often enough for me to remember the solution to this problem for the next time it happens again.  So for the purposes of easy recall in the future I am going to post a representative problem and a solution along with it. Better solutions are of-course welcome.     A.h  class B; class A {     int _val;     B *_b; public:      A(int val)         :_val(val)     {     }      void SetB(B *b)     {         _b = b;         _b-&gt;Print(); // COMPILER ERROR: C2027: use of undefined type 'B'     }      void Print()     {         cout&lt;&lt;"Type:A val="&lt;&lt;_val&lt;&lt;endl;     } };       B.h  #include "A.h" class B {     double _val;     A* _a; public:      B(double val)         :_val(val)     {     }      void SetA(A *a)     {         _a = a;         _a-&gt;Print();     }      void Print()     {         cout&lt;&lt;"Type:B val="&lt;&lt;_val&lt;&lt;endl;     } };       main.cpp  #include "B.h" #include &lt;iostream&gt;  int main(int argc, char* argv[]) {     A a(10);     B b(3.14);     a.Print();     a.SetB(&amp;b);     b.Print();     b.SetA(&amp;a);     return 0; } Equivalent Definitions of the Operator Norm How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &amp;= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &amp;=\sup\{ \lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert\leq 1\}\\ &amp;=\sup\{\lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert = 1 \}\\ &amp;=\sup\left\{ \frac{\lVert Av\rVert}{\lVert v\rVert}\;\colon\; v\in V\text{ with }v\neq 0\right\}. \end{align*}$$ Getting key with maximum value in dictionary? I have a dictionary: keys are strings, values are integers.  Example:   stats = {'a':1000, 'b':3000, 'c': 100}   I'd like to get 'b' as an answer, since it's the key with a higher value.  I did the following, using an intermediate list with reversed key-value tuples:  inverse = [(value, key) for key, value in stats.items()] print max(inverse)[1]   Is that one the better (or even more elegant) approach? My teacher proposed this question as a challenge proof to do on our own and I can't seem to get it. Was Wondering if anyone could give me a hint or help me on the process to completing it.  Let $a \in \mathbb{R}$. Let $f$ be a function defined, at least, on an interval  centred at $a$, except possibly at $a$. Let $L \in \mathbb{R}$.     If $\lim\limits_{ x→a-} f(x) = \lim\limits_{ x→a+} f(x) = L$   then $\lim\limits_{ x→a} f(x) $ = $L$.    We are supposed to prove this using the "delta-epsilon" proof style. Thanks in advance!! The graduate students at my university (a relatively large state school in the US) are considering joining a union. To help decide whether to sign a union card in support of joining the union, I'm interested in more information about how unions have helped graduate students at other universities. What are some improvements to graduate life gained by graduate students at other universities upon joining a union?  I'm also interested in information about how unions have hurt graduate students at other universities. What are some detriments to graduate life caused by graduate students joining a union?  I'm primarily interested in student-body-wide benefits/detriments, instead of student-specific or faculty/administrator-specific claims (e.g. "my advisor treated me better with the union behind me" or "my students have stopped working since they joined a union" is not what I'm looking for). Meta Stack Exchange should not penalize downvotes When you send a group text message in iOS, it appears as if you automatically opt in to receive any and all replies to that message. Is there a way to avoid being included in group replies? Is there a way to remove oneself from a group chat? Can a black hole absorb photons whose wavelengths are greater than the BH horizon size? If $\Sigma \vdash \varphi$ then $\Sigma \vdash \forall x \varphi$. Why when $(\varphi \implies \forall x \varphi)$ is not true? In the book of first order logic I'm reading they say that if $(\varphi_1,...,\varphi_n)$ is a proof from a set of formulae $\Sigma$, then so is $(\varphi_1,...,\varphi_n,\forall x \varphi_j)$ for all $1\leq j \leq n$. They assume that as an axiom but it doesn't make sense to me. Why if $\varphi$ is true then so is $\forall x \varphi$? For example if $\varphi$ is $x=1$ that wouldn't mean that $\forall x : x=1$ , or would it?  Maybe what I don't understand is: why $(\Sigma \vdash \varphi \implies \Sigma \vdash \forall x \varphi)$ is true but $(\varphi \implies \forall x \varphi)$ is not true?  The book is this: Would it be possible to generate the original data from a SHA-512 checksum? So, I have a file that is XY Exabytes big. I create a SHA-512 hash from it.    Are there any theoretical chances that I can retrieve the original data just from the sha512 hash?    Also, are there any better hashes then sha512? How can I identify an anonymous inner class in a NotSerializableException How to deal with high correlation among predictors in multiple regression? How can I Remove .DS_Store files from a Git repository? How can I remove those annoying Mac OS X .DS_Store files from a Git repository? How to install the OS with a new Hard drive? I have a specific requirement (probably silly!) for testing a browser of Chrome. I realize this requirement isn't the most logical, but politics being what they may I'd like to know the technical workaround.  I need to test a site in version, let's say, 28 of Google Chrome. Obviously Google runs a silent updater which can be disabled, but at this point they don't (to my knowledge) release old archives of the MSI installers (which I can easily use to reinstall and force a no-update option for testing). If we can get this working on a single instance then there's a possibility that a virtual machine or some other workaround will give us an easy restore point to meet this test.  If anyone has a good idea of where to start here that would be amazing.  I've already floated the "why are we doing this question?" and the answer is that it is part of the requirements so we need to make a best effort.  UPDATE:   FileHippo appears to only archive the online installer (automatic update)  OldApps doesn't archive Chrome anymore  OldVersion doesn't appear to have any Chrome Archives   UPDATE 2:  - The accepted answer's server is the correct solution here at this point in time. Basically speaking you're looking for the full install (30 or so MB). If you have an 800K file it is the online installer which automatically updates. Photoelectric effect intensity I understand the PE effect quite well but I'm failing to understand one thing. Intensity is the amount of energy per second incident to a given area.   So can you can increase the intensity by either increasing the electrical energy or making the area smaller? By increasing the electrical energy, what exactly is happening?  I assume that by increasing intensity you are increasing the number of photons incident to an area of the metal thus increasing the rate of photoelectric emission. But why doesn't increasing intensity via more electrical power lead to greater energy of the electrons, if by increasing intensity you are increasing the energy of the incident waves?  I understand the results but I'm trying to understand why the intensity works the way it is thanks How can I hire a professor to answer my advanced technical questions in a niche area? In short, How does one go about hiring a professor or someone really well established in their field for a private tutoring session?  I tried looking online but can’t find much, mainly websites for tutors that are practitioners or “experts” as in with with n years of experience at best. This doesn’t cut it at my level as I am a researcher with a PhD and already have published papers on my topic.   However, I am very passionate about my subject and always looking to grow. So I do have questions, very technical (in my case computer science) and would happily pay a fee for consulting with a professor in the field.  Any advice? In the movies, the Death Eaters leave a smoky trail behind them as they fly. As the Order arrive in the Ministry of Magic in Order of the Phoenix, they leave a silvery trail behind them. In the books, there’s no mention of how they’re doing this. And J.K. Rowling said in the books that only Voldemort and Snape could fly like this; first seen in the Battle of the Seven Potters.  Please can somebody explain how the Death Eaters are able to fly like this? Do different "Picture styles" affect RAW output? I was  the manual of my camera (Canon 50D, but I am sure that each DSLR offer the same kind of functionality, albeit I don't know the exact names adopted) and noticed the issue of Picture styles which I had overlooked in my first reading.   In short I think that they are a collection of various calibration settings (more or less enhanced colours, things like that...). Those that are available on my camera are:   Standard Portrait Landscape Neutral Faithful  Monochrome 3 user definable modes.   These styles are so important to deserve a direct button just under the LCD screen of the camera (I guess this is useful for simplifying the use of the camera in the basic modes as opposed to manual / less automated ones).  And now for the question:   These settings obviously affect the appearance of the picture "as it comes out of the camera". But given the raw file I assume that they are all equivalent, a style is not applying a destructive transformation to the raw data, isn't it?  Shooting raw, is there a reason to favor one of these styles (Neutral?) over the other? Up to now I have been using Standard.   As a bonus (don't know if it deserves a separate question, in which case I will ask it again), what is the difference between Neutral and Faithful? Why does my ArrayList contain N copies of the last item added to the list?