$\alpha$ is an arbitrary (random) circle that passes through the origin.   Show that there are real numbers $s$ and $t$ such that $\alpha$ is the graph of $r = 2s \cos (\theta + t)$.  I believe that $r = 2s \cos (\theta +t)$ can be simplified down to $r = x^2+y^2$ using Cartesian coordinates. What next?
Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$  I know this has to be converted to a cartesian equation, but how do I do this, and what do I do after?  Thanks
How do you do AppBar docking (to screen edge, like WinAmp) in WPF? Is there any complete guidance on doing AppBar docking (such as locking to the screen edge) in WPF? I understand there are InterOp calls that need to be made, but I'm looking for either a proof of concept based on a simple WPF form, or a componentized version that can be consumed.  Related resources: I understand there is a method for solving simultaneous modular equations. For example; $$x = 2 \mod{3}$$ $$x = 3 \mod{5}$$ $$x = 2 \mod{7}$$ We find numbers equal to the product of every given modulo except one of them - giving $5 \cdot 7$, $3 \cdot 7$ and $3 \cdot 5$. We then find the multiplicative inverses of these numbers with modulo equal to the number missing from the product. The numbers found are then 2, 1 and 1 in this case. The value of x is then given by: $$x = 2 \cdot (5 \cdot 7) \cdot 2 + 3 \cdot (3 \cdot 7) \cdot 1 + 2 \cdot (3 \cdot 5) \cdot 1 = 233 = 23 \mod{3\cdot5\cdot7}$$  But I do not understand how this method correctly gives the value of $x$. I understand that the Chinese remainder theorem proves that there is a unique value of $0\le x \lt 3\cdot5\cdot7 \mod{3\cdot5\cdot7}$ but can someone please explain why this method finds this value of x? The abstract environment changes the \linewidth for the whole document In the article class, it seems that using the abstract environment has an effect to the rest of the document. More precisely, it seems that it redefines the \linewidth, which equals \textwidth without abstract, is smaller with the abstract. My feeling is that \linewidth is redefined for the abstract environment, but the value is not reseted at the end of the abstract.  Here is a minimal example:  \documentclass{article} \begin{document} \abstract{This is an abstract. This is an abstract.}\bigskip  This is text. This is text. This is text. This is text. This is text. This is text.\\  \rule{\textwidth}{3em} \rule{\linewidth}{3em} \end{document}   The output is not the same if you comment the \abstract line or not.  Do you have the same problem as I do when you compile (if not, I will upload some PDF files to show)? Is this a (known) bug in LaTeX2e? How can we circumvent the problem?  Note: I compile with the pdflatex command, and I use pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian). I saw this question  but it doesn't have a full solution. What is the full solution to the 99 bottles of beer on the wall for Vim adventures?  Context: for the sake of being a good SO member I will give more details but a full solution will answer my question. It seems from reading the other post that my register is incorrect for some reason. I can't get the &lt;NL&gt;. I think that's the issue since pasting won't work... Jquery - change event firing on page load Beamer and pseudocode How can I manage the "Open with Explorer" button based on user permissions? How to mitigate the Spectre and Meltdown vulnerabilities on Linux systems? Looking for a scifi novel with advanced pheromone technology I've been studying the Huddleston and Pullum book for four months now. So far only one thing confuses me: the identity of gerund. Is it a noun or a verb?        His constant smoking upset me.  smoking seems  noun because of adjective constant.   Him/His constantly smoking upset me. smoking seems to be verb because of adverb constantly.   Him/His smoking cigars upset me. smoking seems to be a verb because of object cigars.      "His smoking upset me." So is this smoking gerund a noun or verb? Because there is no differentiation, is it both verb and noun? Maybe it's new word category?  Also his seems to be both  sometimes a subject and at other times a possessive determiner. Is that correct? Should the results of a calculation happen automatically, or require a button push? This is a problem I've ran into several times, and I imagine that others have ran across it too.    Basically, I have a complicated equation, and I want the user to type in the values needed to calculate the results.  (For what it's worth, there are good default values for the variables in the equation.)  I've constructed a GUI that shows the user the names of the variables/fields that need to be provided, along with a control (like a numeric up/down, slider, etc.) to enter the values of the field.  Below that, I have an area that shows the output of the equation.  So my question is this.  Should my results area automatically change the calculated results whenever the user makes any changes to any of the input fields, or should they push a "Calculate" button to generate the results?  One other thing that probably applies is that the user is not trying to "get the right answer" out of the results.  They shouldn't be using the output to then go back and modify the input to get a better answer.  Are there any benefits to requiring the button press?  Is the extra click to see the results worth those benefits? How can I monitor internet usage in my network? I need to know which computer is consuming my bandwidth. Is there a tool to do this? Frying Oil Reuse What are some guidelines or rules with regard to filtering and reusing (vegetable/canola/sunflower) oil that has been used for deep frying?  Is it safe and acceptable under some circumstances to filter and keep oil? If so, does it depend on which foods were fried? How long can the filtered oil for be kept for?  Or, is it never appropriate? If not, are the reasons health or flavour related? or both? Intuition for multiple temporal dimensions It’s easy, relatively speaking, to develop an intuition for higher spatial dimensions, usually by induction on familiar lower-dimensional spaces. But I’m having difficulty envisioning a universe with multiple dimensions of time. Even if such a thing may not be real or possible, it seems like a good intellectual exercise. Can anyone offer an illustrative example? So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. However, I do not know what $\lfloor i\rfloor$ equals. Is there a clearly defined definition of $\lfloor i\rfloor$? Why can't I find a USB-C hub with multiple USB-C ports? My computer only has one USB-C port and I want to connect multiple USB-C devices to it.  I have trawled through Amazon but am amazed that I cannot find a single one. There are many hubs that have USB 3.1 USB-A ports but not USB-C ones.  Is there a limitation in the spec that prevents this being possible? Change the location of saved screenshot when screenshot taken by printscreen button in GNOME 3 How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected?    Note that $X$ is a topological space with subspace topology $\tau = \left\{ U \cap X \mid U\text{ is open in }\mathbb{R}^2\right\}$.  I only know two things (with respect to path connectedness):   A topological space $X$ is path connected iff for every $p,q \in X$, there is a continuous function $f: [a,b] \to X$ with $f(a)=p$ and $f(b) = q$.    Continuity $f$ is defined as for every open set in $X$, its preimage is open in $[a,b]$ (with subspace topology induced by $\mathbb{R}$ (with standard topology) )   Suppose $X$ and $Y$ are two topological spaces and $f: X \to Y$ is a continuous surjection. If $X$ is path connected, then $Y$ is path connected. In Minecraft, you can only build up to a certain height limit (which is something around 256). However, if a player uses mods/hacks to go above the limit (Or simply goes into Creative Mode), then they can go above the height limit.  How high is the player height limit in Minecraft? Is there even a limit to it? I've tried testing this myself, but my thumb started hurting when I reached Y= 1000. I got my Galaxy S4 today (i337M), it's wayyy better than my old SII hehe.  I immediately removed the stock Bell firmware and put the latest snapshot available from CyanogenMod.  I'm wondering if it's possible to convert apps like S Planner, S Note, S Health to CyanogenMod.  Current stock android version available for i337M is 4.2.2. Latest Android version for CM is 4.4.  If I install CM 4.2, can I import/install the APKs and then upgrade to 4.4?  Hope you'll be able to help me :)