I'm trying to prove that the relation R on $\mathbb{Z}$ is an equivalence relation, where R is defined by:  $$\{(x,y)\in \mathbb{Z}\times\mathbb{Z}: x^2\equiv y^2 \!\!\!\mod \!\!4\}$$   I know I need to show that it is reflexive, symmetric, and transitive, but I'm not sure how to do that when working with congruence. Thank you.  I answered my own question below but I'm not sure if I did it correctly. Any feedback would be really helpful. Thanks.
Question 1: Let $x,y \in S$ such that $x\sim y$ if $x^2 =y^2\pmod6 $. Show that $\sim$ is an equivalence relation.    This is what I tried: Reflexive: $x^2\pmod6 = x^2$  implying $x\sim x$ Symmetry: suppose $x\sim y$, then  $x^2 =y^2\pmod6$ Operating by inverse of $x^2$ both sides we have $e=(x^2)^{-1} y^2\pmod6$ Then I have $y^2=x^2\pmod6$, hence $y\sim x$ I would like to get some corrections up to here because am not sure with this     Question 2. Find the orbits of the given permutation:    $$v:\mathbb{Z}\to\mathbb{Z}\quad\text{ defined by }\quad v(n)=n+3$$   Here is my solution which I would also want somebody to correct me! Let the relation $\sim$ be defined on $\mathbb{Z}$ by $a\sim b$ if $b=a+3$ I consider that the orbits of $v$ are the classes of $\mathbb{Z}$ with respect to $\sim$ given by  $\{b\mid b=a+3\}$. Thus I get only three orbits which are $\{1,4,7,10,…,3n-2\mid n\in \mathbb{Z}\}$, $\{2,5,8,…,3n-1\mid n\in \mathbb{Z}\}$, and $\{3,6,9,…,3n \mid n\in \mathbb{Z}\}$.
Let $m$ be the largest real root of the equation $\frac3{x-3} + \frac5{x-5}+\frac{17}{x-17}+\frac{19}{x-19} =x^2 - 11x -4$ find $m$ Let $m$ be the largest real root of the equation $$\frac3{x-3} + \frac5{x-5}+\frac{17}{x-17}+\frac{19}{x-19} =x^2 - 11x -4.$$ Find $m$.   do we literally add all the fractions or do we do something else  I have no clue how to solve this Troubleshooting for biber I'm trying to switch from BibTeX to biber and ran into the problem that biber does not produce any output file. I noticed several questions about problems with biber e.g.,            ,                       But none of them helped me in my particular situation (which was actually just an old version of biber somewhere in the path of my OS). Since I also noticed that several questions regarding problems with biber were closed as too localized since the underlying problem was due to formatting of a bib resource or similar simple issues, I would rather like to ask for a general troubleshooting guideline for using biber (since I saw many hints like "check/delete the cache", "check versions of this and that" etc.).  What steps are to be taken and what things (in particular files) to be checked if biber does not produce an output file (and its output on the command line does not directly give you the advice what to do)? Layover at Tokyo Narita airport: can I travel outside, and what kind of visa would I need? beamer - global transition shifts slide content Using beamer with the , I want to have a global transition effect (a fast fade) which I set using the following code:  \addtobeamertemplate{background canvas}{\transfade[duration=0.2]}{}   as found on  answer.  However, doing it messes up with my slides settings. It doesn't change every slides, only the \section ones that are slightly shifted to the right, adding a white line on the left, as seen here:   I tried using XeLaTeX and LuaLaTeX, as they are the only compilers I could use for my current font, but they both have the same issue.  It probably comes from the theme, however I have no clue why it would happen or how to fix it.  Edit :   \documentclass{beamer} \usetheme{m} \addtobeamertemplate{background canvas}{\transfade}{}  \begin{document} \section{test} \end{document}   Requires the mtheme to work Creating Logo with Fancy Font I am trying to creating a logo that resembles this one shown below, but I do not know the best way of going about it. I know of a fonts package that may be useful, but not sure how to overlay the letters on top of each as so. Would anyone know a neat way of accomplishing this?  Picture of Logo: If $f$ is entire and $|f|\geq 1$, then show $f$ is constant. When searching for the desired SE site, the down arrow key (down) should work in order to choose the site one is looking for and hit enter to visit it, instead of using the mouse point to choose it. Would be pretty convenient. tl;dr  Due to the special nature of Meta and the high rate of "lost souls", should we try out the Ask a Question Wizard here on Meta Stack Exchange too?  General idea  In case anyone doesn't know yet, . It guides new users towards writing better questions.  I think that such a question wizard would be useful on Meta Stack Exchange too, for three main reasons:   . There are four types of questions on Meta, each represented by the different required tags: , ,  and . Especially bug reports can often be split into a certain structure (what's wrong, how to reproduce, why I think it's wrong, examples). The same holds true for feature requests. Such a structure is ideal for a question wizard. Quite a few users  and . While the Meta community is well prepared to deal with these posts quickly, they lead to an less-than-ideal first impression by the new users and they distract the community from more important issues. Site-specific questions are often off-topic. This  since nowhere is it hinted at in the help pages (and there are a lot of old, not-yet-closed SO-specific questions from a prior era of this site) and can lead to unnecessary drama. If we add an honeypot into the wizard to catch this case (and ideally redirect them to the proper meta site), we could prevent this.   As Sonic said, there's an  new users need to read and confirm. However this doesn't really prove effective right now and we might need to try stronger tools (e.g. question wizard).  Therefore I think that we should test the Ask Question Wizard here on Meta too.  Implementation details (help welcome)  I think the wizard should have six main entry points:   I want to discuss a specific post or action from any Stack Exchange site. (-> honeypot, redirects to the appropriate per-site meta) I have a question regarding the use of Stack Exchange sites (-> support mode) I think I found a bug while using the Stack Exchange sites  (-> bug mode) I have an idea for improving the Stack Exchange sites  (-> feature request mode) I have a discussion I want to start about Stack Exchange communities or their rules  (-> discussion mode) I have some other (e.g. programming) question  (-> honeypot, redirects to )   Coming from these choices, the user would either be redirected to a landing page explaining, why this is not the right community and offering some alternative choices or they would follow the guided assistant.  For 1. the user might be asked on which site the post/action is. Maybe the system could show the three sites where the user was recently active/has the highest reputation on.  support mode would ask the user for these details:   what do you want to achieve? what have you tried? what is your current problem?   bug mode would ask the user for these details:   bug description how to reproduce the problem why do you think that this is wrong? (possibly) examples   feature request mode would ask the user for these details:   what is your goal? how do you think could this be achieved? why will the community benefit from this change? (possibly) details   discussion mode would ask the user for these details:   which rule do you want to propose/clarify (optionally) examples/why is this rule required?   Note: the above implementation details probably require some revision. Feel free to edit/suggest improvements for both wording and details. How to derive variance-covariance matrix of coefficients in linear regression How to extend summations to fractional orders? What does "Perform GC" button do in jconsole? Which command should I use for textual subscripts in math mode? Textual subscripts like in W_{total} should be typeset in an upright text font (so that code isn't good practice), as has been  in   . Now there are several ways to achieve this, and I'm somewhat confused which I should use:   W_{\rm total} is nice and short, and appears to work in all circumstances, but I expect LaTeX people will frown upon it :-) Is there any good reason against this? W_{\mathrm{total}} &ndash; is this the LaTeX equivalent of the \rm approach? W_{\textnormal{total}}  as total is text and not math&nbsp;... In , Ulrike Fischer suggests a more complicated approach. Even W_{\operatorname{total}} appears to work, but it seems inappropriate.   Note: I always assume here that amsmath is loaded.  I've already asked some questions above, but my main question is: which of these options is the best practice for textual subscripts?    Let me point out why I haven't mentioned \text, \textrm or \textup above. Those commands (mostly) keep the font of the context they're used in, e.g., in an italic context, \text and \textrm would give italic subscripts, and all three yield bold subscripts in a bold context.  Final note: This question is similar to, but not the same as all those  about the  between , , , ,  that have been asked before. Maybe it's still a duplicate&nbsp;... I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry:     If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.    First of all, since the momentum energy tensor of dust $T^{\mu \nu}=\rho U^\mu U^\nu$ is symmetric, one has the energy-momentum conservation law $$ \partial_\nu T^{\mu \nu} = \partial_\nu T^{\nu \mu}  =0.$$ A vector $Q$ is constant in any frame related by a Lorentz transformation, while this isn't the case for the components $Q^\mu$. But here $Q=0$ and hence $Q^\mu=0$ also for every component $\mu =0,\cdots,4$?   Even if a straightforwardly apply the equation: $$ Q^i = \partial_\nu T^{i \nu} = (\partial_\nu \rho) U^i U^\nu + \rho(\partial_\nu U^i)  U^\nu + \rho U^i(\partial_\nu  U^\nu ),$$ doesn't seems to yield something...   Is my first reasoning right ? Otherwise: can someone give me a hint ?  Thanks How to change radius/weight of curve bezier? try scaling, change radius, but its dont work Evaluate forecasting ability of GARCH models with RMSE and MAE Why can't an enum value be fully qualified in a switch statement? Sort Order on the User Filter is Surprising To Me When filling in the "Type to Find Users" , it seems to sort by exact match and then by reputation when I am on the reputation tab.   For example, please consider the following search results:    This threw me off since I thought that the sort order is reputation. I guess this often might give people what they want, but when picking a popular name is shows a lot of 0 rep users. Maybe adding a filter for low rep to the exact match might give more expected results, or just sorting on Rep? How to set a random value for multiple shape keys? My question is related to my previous asked one:   I have about 300 shapekeys and want to set the random value for all of them . How to do it using scripting? Doom-style 2.5D Movement Animations I've started working on a demo for my 2.5D game. For a basic scene I figured I would just use the good old fashioned doom sprites textured to a double sided plane. Simple enough. But I've been scratching my head as to how to display the proper sprite animation relative to the player viewing the sprite...if that makes since.  For example suppose I have three players in the scene Players A, B, and C as such:    Player C should see player A as: Here's an example of what I think player A should look like from C's perspective:    While player B would see player A as follows:    My current idea is that for any given player, I will need to know the current direction of each other player, and where they are located/moving to relativly. From this, I could calculate which sprite image is required.  Is this correct? What improvements/optimizations could be made?  ** EDIT ** Total number of directions will be 8 as depicted below:    For each direction there are two sprites (to show walking movement, sprites for the right hand directions are mirrors of the left):    ** EDIT 2 ** Got this working and have a simple javascript/threejs example for anyone interested: Invalid Data Source when importing geojson to QGIS Im trying to import geojson, I add data, select protocol and use the uri  QGIS says "Invalid Data Source:  is not a valid or recognized data source"  I've followed the tutorial here  but I cant get the data to come in, any ideas? I'm using QGIS version 2.18.13