PHP math divide it weird
PHP - Get rid of notation
Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is bijective. By the way, do we need to show that $(0,r^{\frac{n+1}{2}})(1,m)(0,r^{\frac{n+1}{2}})^{-1}=(1,m)$ ? If we can prove this, then what can we conclude? $ a+b+c+d=6 , a^2+b^2+c^2+d^2=12$ $\implies$ $ 36 \leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \leq48 $ Let $a , b, c, d$ be real numbers such that $$ a+b+c+d=6 \\ a^2+b^2+c^2+d^2=12$$  How do we prove that $$ 36 \space \leq\space 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \space\leq48\space$$ ? I have an Android phone that connects with my computer via MTP. This works fine; I can see and transfer files with Nautilus. However, I often want to use the terminal to move large numbers of files, and I cannot seem to find the device anywhere in the filesystem tree. Nautilus reports the location as mtp://[usb:003,007]/, but it's not under /media or /mnt.  Does anyone know where it is? How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$ How to prove $x^a-1\mid x^b-1 \iff a \mid b$, where $x \ge 2$ and $a,b,x \in \Bbb Z$.  I've tried the following in attempting to solve this:  $$a\mid b \Rightarrow aq=b \Rightarrow x^{aq}=x^b \Rightarrow x^ax^q=x^b$$  Because $x^q \in \Bbb Z$, it follows that $x^a\mid x^b$.  This is as far as I have gotten; any help getting further is appreciated.  Note: It may be that this identity is not true at all? Storing C++ template function definitions in a .CPP file I have some template code that I would prefer to have stored in a CPP file instead of inline in the header.  I know this can be done as long as you know which template types will be used.  For example:  .h file  class foo { public:     template &lt;typename T&gt;     void do(const T&amp; t); };   .cpp file  template &lt;typename T&gt; void foo::do(const T&amp; t) {     // Do something with t }  template void foo::do&lt;int&gt;(const int&amp;); template void foo::do&lt;std::string&gt;(const std::string&amp;);   Note the last two lines - the foo::do template function is only used with ints and std::strings, so those definitions mean the app will link.  My question is - is this a nasty hack or will this work with other compilers/linkers? I am only using this code with VS2008 at the moment but will be wanting to port to other environments. Let's say I have a script that will be executed on various machines with root privileges, but I want to execute certain commands within that script without root.   Is that possible? get all page IDs from wp_list_pages Resistance between two points in an infinite metal sphere/cube Let's imagine that we have a tridimensional metal object of infinite size, and decide to calculate the resistance between two arbitrary points. How would we go about doing this?  I have thought of two possibilities for calculating this, the first one being an adaptation of the conductivity (S/m) using the Pythagorean theorem to find a distance and isolating the resistance from there, treating it like we would treat an electrolytic liquid.  The second possibility I have been thinking about would be a 3-dimentional adaptation of the infinite-resistors problem (which I have only seen in 2d).  Would one of these methods result in giving a decent approximation, or would another method give better results? Will moving blog posts from under an existing category to a new SUBcategory have a negative impact from an SEO POV? So my friend has a USB flash drive that isn't working. I don't know a whole lot about "recovery" software so I figured I'd ask you guys.  What steps can I try (I have access to a Linux machine as well) to see....if I can get ANYTHING off the flash drive?  I've google'd around, but most of the stuff seems for windows. Is there a DSLR fixed lens camera? Let $f(z)=\frac{1}{(z-1)(z-2)}$ and $g(z)=\frac{z}{(z-1)(z-2)}$. Let $\gamma$ be the circle of radius 4 centered at the origin and traveled once in the counterclockwise direction. Let $A=\{ z \in \mathbb{C}: |z|&gt;3 \}$  1) I need to show there is no analytic function $G: A \to \mathbb{C}$ such that $G'(z)=g(z)$ for all $z$ in $A$.   2) I also need to show that there is an analytic function $F:A \to \mathbb{C}$ such that $F'(z)=f(z)$ for all $z$ in $A$.  Do I have to integrate to find such function? I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$:  \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2 \mathbb{Z} \end{align}  I'm aware that there are answers to this question on this website. However, almost all of the answers seem to use facts about modules and vector spaces defined over fields. This question is posed in section 7.6 in Dummit and Foote's Abstract Algebra. This section is part of the chapters which introduces the basics of ring theory (rings, homomorphisms, ideals, Chinese Remainder Theorem etc.).  I'm not even sure how to being answering this question. I know that if $R$ were an integral domain, then we would have that $R \cong \mathbb{Z} / 2 \mathbb{Z} $.   Please only give hints, considering that that the statement has to be proved using the material only included up till chapters 9 in the textbook. Reputation in summary tab shows as blank instead of 0, if net result is zero Mean, Average and Expectation What is the precise difference between the Mean, Average and Expectation ? Is there any mathematical formaulation to show the difference ? I see that often they are used interchangeably. Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial?  Or, without loss of generality and more suggestively $$(f(z)=\sum_{k=0}^\infty a_k z^k \quad \land \quad \operatorname{Re}(f(z))\leq |z|^n)\quad \forall z\in \mathbb{C} \qquad \Rightarrow \qquad f(z)=\sum_{k=0}^n a_k z^k \quad\forall z\in \mathbb{C}$$ The documentation for the  function states that you pass it a number, and the positions past the decimal to round. Thus it should do this:  n = 5.59 round(n, 1) # 5.6   But, in actuality, good old floating point weirdness creeps in and you get:  5.5999999999999996   For the purposes of UI, I need to display 5.6. I poked around the Internet and found some  that this is dependent on my implementation of Python. Unfortunately, this occurs on both my Windows dev machine and each Linux server I've tried. .  Short of creating my own round library, is there any way around this? accessing loop variable within a loop Force a rebuild of the start menu program index/database in Windows 10 With Windows 10 Microsoft migrated to a database-ish solution for indexing programs in search and the start menu, rather than just using a /startmenu/ directory. AFAIK the exact implementation is still pretty unclear, but there is something along those lines.  Updating from 8.1 to 10 causes it to migrate everything from /startmenu/ to the new system, which ignoring issues such as a (seemingly arbitrary) cap of 500 applications, works fine.  However this failed for me, and from googling seems to be an issue from upgrading with an older dev build, which then carried on to RTM.  So does anyone know a way to force a rebuild/re-index of the program library? Rebuilding the system wide index through control panel (also in &lt;=8.1) has no effect. How does the Performance Index work? In one of Dota2's recent patches, Valve has added a new feature called "Performance Index" visible in your own in-game profile.  It is not clear how it works. Is there any information available, what "performance" it monitors?  Does it rely on K/D, K/D/A, Gold per minutes, Experience per minute, or all of them?