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Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ How far can you go with rm -rf / as root Insert a paragraph into a list that uses the whole text width How do I escape a backtick ` within in-line code in Markdown? After what speed air friction starts to heat up an object? I understand that air friction cools off an object at low speeds. For example, if you blow on a spoon of hot soup, it cools off. Or if you swing a hot frying pan in the air, it cools off faster.  But at higher speeds, the situation changes in the opposite. For example, consider a meteor falling on Earth. It is so fast that it heats up to such a high temperature that it burns into ashes.  What is the critical speed in which an object starts to heat up?  For example, consider a spherical object with radius of 1 meter. Let its density be 1 g/cm3 if needed. And let the air pressure be 1 atm. Assume that the temperature of the object is 400 K and the temperature of the air is 300 K. Also assume that the specific heat constant of the material is 1 cal/(gr.K). What is the critical speed for this object after which its temperature starts to rise above 500 K? Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$ When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi = 1.61803398875...$. What if I were to plug in $x=2$? $$2+\cfrac{1}{4+\cfrac{1}{8+\cfrac{1}{16+\ddots}}}$$ Using a caluclator, it seems to converge upon the irrational number: $2.24248109286...$ Is there any way I can represent this irrational number algebraically? Config files for assemblies in GAC In which cases is a word, or a group of words written in italics? Is italics used in specific contexts, or it is quite normal to write words in italics? How to add a Login / Logout link in Main navigation menu? How can I add a Login link to the "Main navigation" menu?  When a user clicks on the link, it should take them to the /user/login page. Once they login, the link should change to say "Logout" instead of "Login".  Is there any way to add such a link? the teacher gave me this problem to solve. So i hope you can help me with it. Here is the problem: John poured 150ml ob boiling water in 2 cups made out of polystyrene. Cups are open on the top so heat goes out there. Temperature of the outside air is 20°C. In 1 cup he add 5 mm thick layer of hot oil. He measured starting temperatures and then he measured temperatures again after 10 min. He measured the mass of water and water+oil before and after 10 min.  Here are the resaults:   -cup 1 (only water)  starting temp: 96°C  finnish temp: 64°C  starting mass: 149.4g  finnish mass: 142.42g  -cup 2(with oil on top)  starting temp.: 96°C  finnish temp.:91°C  starting mass: 164.72g  finnish mass: 164.70g  The surface of the both cups is 0.003 m2  Questions: 1.Why did the water covered with oil cooled down slower that the one with no oil? Show with calculations. 2.Why did John use oil with temperature around 100°C and not oil with room temperature? How would the resaults change if he used room temeprature oil? 3. Which other physical values, which are relevant, can you calculate from this informations? Basiccaly i need the physics behing why does this happen. Questions about a commutative ring with exactly three ideals Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$      1) Show that if $a \in R-I$, then $a$ is a unit in $R$.      2) Let $a,b\ne0$ in $I$. Show that $ab=0$. Use variable with TOP in select statement in SQL Server without making it dynamic How to position the paper's authors' addresses on the front page (maketitle)? When I use the following code for the title page of a paper (please see below), the authors' addresses and emails appear on the last page of the paper. However I need them to appear on the first-title page (after the authors' names before the abstract). How can I do this?  \documentclass [reqno, 12pt ]{amsart} ... \begin{document}  \title[paper]{Nice paper}  \author{Me} \address{University} \email{mmm@lala.com} \thanks{.} \author{my friend} \address{Institute} \email{kkk@lala.edu} \thanks{.}  \date{\today} \keywords{XXX }  \begin{abstract} Lalala \end{abstract}  \maketitle I have a list with 15 numbers in, and I need to write some code that produces all 32,768 combinations of those numbers.   I've found  (by Googling) that apparently does what I'm looking for, but I found the code fairly opaque and am wary of using it. Plus I have a feeling there must be a more elegant solution.  The only thing that occurs to me would be to just loop through the decimal integers 1–32768 and convert those to binary, and use the binary representation as a filter to pick out the appropriate numbers.   Does anyone know of a better way? Using map(), maybe? Margin on child element moves parent element Let's say we have $x^2=25$ So we have two real roots ie $+5$ and $-5$.  But if we were to differentiate on both sides with respect to $x$ we'll have the equation $2x=0$ which gives us the only root as $x=0$.  So does differentiating on both sides of an equation alter it? If it does, then how do we conveniently do it in Integration by substitutions? If not then what exactly is going on here ? What does "What are you up to?" mean? I overheard my manager asking "What are you up to?"  What does that idiom mean? Is it an informal/negative way of asking?? Is the ArcGIS Server JavaScript API free to use? The 2008  announcing the ArcGIS Server JS API says:     Everyone can use the ArcGIS JavaScript API. There is no fee for using the ArcGIS JavaScript API or deploying an application built with it.   But Esri's  page says "Contact us for pricing".  Which is correct? country code TLD vs second level ccTLD I'm really worried about professionalism.  My site would be something like http://mywebsite.uy instead of http://mywebsite.com.uy.  What do you think about this practice? What should I have in mind at the moment of choosing one? How do you differentiate the likelihood function for the uniform distribution in finding the M.L.E.? There is a classic problem:  Suppose that $X_1,\ldots,X_n$ form an i.i.d. sample from a uniform distribution on the interval $(0,\theta)$, where $\theta&gt;0$ is unknown. I would like to find the MLE of $\theta$.  The pdf of each observation will have the form: $$ f(x\mid\theta) =  \begin{cases} 1/\theta\quad&amp;\text{for }\, 0\leq x\leq \theta\\ 0 &amp;\text{otherwise}. \end{cases} $$ The likelihood function therefore has the form: $$ L(\theta) = \begin{cases} 1/\theta^n \quad&amp;\text{for }\; 0\leq x_i \leq \theta\;\; \text{for all }i,\\ 0 &amp;\text{otherwise}. \end{cases} $$ The general solution is usually that the MLE of theta must be a value of $\theta$ for which $\theta \geq x_i$ and which maximizes $1/\theta^n$ among all such values.   The reasoning is that since $1/\theta^n$ is a decreasing function of $\theta$, the estimate will be the smallest possible value of $\theta$ such that $\theta\geq x_i$.  Therefore, the mle of $\theta$, $\hat{\theta}$, is $\max(X_1,\ldots,X_n)$.  Here, I do not understand why we cannot just differentiate the likelihood function with respect to theta and then set it equal to $0$?  Thanks!