How would you prove that 2 elements of the group $G$ are equal, i.e., $a = b$ if we know that $a$, $b$ belong to group $G$ and that $m$, $n$ are relatively prime (integers) and that $a^m = b^m$ and $a^n = b^n$  Thanks!
Let $D$ be an integral domain and let $a^m=b^m$ and $a^n=b^n$ where $m$ and $n$ are relatively prime integers, $a,b \in D$.   How do I show $a=b$?
How should I take care of my new DSLR? I am getting a new Nikon D5100 (with 18-55mm VR Lens Kit). This is going to be my first DSLR. I wanted to get some tips from you guys on how to maintain it in the long term. I am getting a 52mm UV Protector Filter, as I read that it would do well to keep the lens clean.  Are there any precautions I need to take?  I would also appreciate if someone can suggest some reading material (paperback or online). Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?  It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers. Proofs involving limsup and liminf I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following problem makes sense.  Problem. Let $(s_n)$ and $(t_n$) be sequences and suppose that there exists $N_0$ such that $s_n \leq t_n$ for all $n &gt; N_0$. Show that $\liminf s_n \leq \liminf t_n$ and that $\limsup s_n \leq \limsup t_n$.   The way I approached it was as follows:   Let $N &gt; N_0$. Then $\limsup_{N \rightarrow \infty} \{ s_n : n &gt; N \} \leq t_n$ as $s_n \leq t_n$, and $\limsup s_n $ is the largest possible limit of a subsequence of $s_n$. As $t_n : n &gt; N$ is (by definition) less than $\limsup_{N \rightarrow \infty} \{ t_n : n &gt; N \}$, the proof is complete.  I'm pretty sure this is incorrect, however, and I'm generally unclear about the method behind such a proof. Any help is appreciated! How to parse a month name (string) to an integer for comparison in C#? I need to be able to compare some month names I have in an array.  It would be nice if there were some direct way like:  Month.toInt("January") &gt; Month.toInt("May")   My Google searching seems to suggest the only way is to write your own method, but this seems like a common enough problem that I would think it would have been already implemented in .Net, anyone done this before? Note: this is intended to be a canonical question about whether or not a spell or ability targets so there are multiple sub-questions.   Can Hexproof or Shroud save my creature from ? What about a  that was cast for it's Overload cost? Does shroud prevent me from using the Monstrosity ability of a creature, or from giving it more counters with Proliferate? Can I cast an Aura Enchantment on one of my creatures with Shroud, or an opponent's creature with Hexproof? Or can I use the Equip ability to attach an equipment to one of my creatures with Shroud? Will 's ability trigger when my opponent plays ? And will I have to sacrifice my  if my opponent casts ? Will Shroud or Hexproof prevent an opponent from choosing me with 's ability? Will Hexproof keep one of my creatures from being affected by ? If an Aura or Equipment is attached to a creature with Hexproof, does the Hexproof protect the attached permanent? SQL function as default parameter value? I tried changing a default parameter value with this:  ALTER PROCEDURE [dbo].[my_sp] @currentDate datetime = GETDATE()   and all the SQL pre-compiler gave me was this error:     Msg 102, Level 15, State 1, Procedure my_sp, Line 8 Incorrect syntax near '('.   I have already created the procedure. (I'm not sure if that's relevant.) I was using a null default value and checking for that later, but that doesn't seem proper. Can I do this in one line?   Update: I was going off of :      [ = default ] Is a default value for the parameter. If a default value is defined, the function can be executed without specifying a value for that parameter.            Note:     Default parameter values can be specified for CLR functions except for the varchar(max) and varbinary(max) data types.         When a parameter of the function has a default value, the keyword DEFAULT must be specified when the function is called to retrieve the default value. This behavior is different from using parameters with default values in stored procedures in which omitting the parameter also implies the default value.   Am I reading this wrong?  Many thanks. Evaluation of $\int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$ Evaluation of $\displaystyle \int_{0}^{1}\frac{x\ln (x)}{\sqrt{1-x^2}}dx$   $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{1}\frac{x\ln x}{\sqrt{1-x^2}}dx\;,$ Put $x=\cos \phi\;,$ Then $dx = -\sin \phi d\phi$  and Changing Limit, We get  $$\displaystyle I = -\int_{\frac{\pi}{2}}^{0}\cos \phi \cdot \ln(\cos \phi )d\phi = \int_{0}^{\frac{\pi}{2}} \ln(\cos \phi)\cdot \cos \phi d\phi$$  Now Using Integration by parts, We get  $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{\sin^2 \phi}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\frac{(1-\cos^2 \phi)}{\cos \phi}d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\sec \phi d\phi-\int_{0}^{\frac{\pi}{2}}\cos \phi d\phi$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-\left[\sin \phi\right]_{0}^{\frac{\pi}{2}}$$  So $$\displaystyle I = \left[\ln(\cos \phi)\cdot \sin \phi\right]_{0}^{\frac{\pi}{2}}+\left[\ln\left|\sec \phi+\tan \phi\right|\right]_{0}^{\frac{\pi}{2}}-1$$  Now How can I solve after that, Help Required, Thanks Looking for a quick and dirty way to automate . I presume this could be done with an automation to press the mouse button from two separate pixel coordinates, with delay, so long as the browser window doesn't move -- unless there's a better way.  I tried building a Watch Me Do workflow, in Automator, but can't seem to get it to work.  I get the error "application Chrome is not running when it is expected to." It also never lists mouse clicks for some reason.  I do have accessibility set up, in system preferences > security &amp; privacy > privacy > accessibility.  Automator, Chrome and System Events are all checked within there (among other stuff of course).  I tried the markashworth workaround as detailed  -- which also doesn't seem to work. Using his method, I get this error:    UPDATE: I just downloaded "cliclick" and figured out how do both clicks as a terminal command, with the proper delays.  BUT -- I don't know code and need a way for this terminal command to repeat "x" amount of times.  Anyone know how I could do that in Script Editor?  The command line is:  cliclick c:132,286 w:100 c:132,286 w:5000 c:1100,238   I would need a few seconds of delay before the loop repeats, which I presume I could do with a final "w:" in that line.  Thanks for the help!! Subsurf on a crown. How can I smooth some edges and keep others sharp? How can I smooth all crown but don't smooth the top of the crown? I did a subdivision surface, but this didn't work. All crown became smooth and changed form. package makeidx: print first letter for each group of keys I've created index with makeidx package. It works. What really need is to print the first letter above every group of keys (i.e. referring to the image I want to print the red letter, that I have manually inserted). How to setup synctex with vim, pdflatex and an open-source PDF-viewer under Linux? I got a problem while making a model of a product. I have tried a lot even seen lots of tutorial but did not get the thing that i want. I am using Blender 9.92 and I am just a beginner I got my hands on Blender 7-8 days ago. I've just basic knowledge about I am trying to model a product &quot;Mattress Topper&quot;   This is the exact look of the product, Help me how i can model this product and render realistic images. A weaker Axiom of Infinity? The coefficient of kinetic friction can be calculated using $\mu_k = F_k/F_n$. A change in angle does not affect the normal force, but doesn't a change in angle affect the friction? As in if the angle is steeper, wouldn't the object have a greater propensity to move down the slope, thus there will be less friction? The lines wrap so much on v1.0.85 that only a few items can be displayed.  Edit: This is on the world's most low end tablet, Azpen A727.  Screenshots of the old:    and new, showing loss of quality Let n be an even integer. How many different ways can I choose subsets of the set {1,2...,n} We already had multiple questions about unbalanced data when using , , ,  and a number of other similar questions, what makes it a very popular topic! Unfortunately, each of the questions seems to be algorithm-specific and I didn't find any general guidelines for dealing with unbalanced data.  Quoting , dealing with unbalanced data     (...) heavily depends on the learning method. Most general purpose   approaches have one (or several) ways to deal with this.   But when exactly should we worry about unbalanced data? Which algorithms are mostly affected by it and which are able to deal with it? Which algorithms would need us to balance the data? I am aware that discussing each of the algorithms would be impossible on Q&amp;A site like this, I am rather looking for a general guidelines on when it could be a problem. How does PK know better than Jaggu? Quickly create a large file on a Linux system How can I quickly create a large file on a Linux () system?   will do the job, but reading from /dev/zero and writing to the drive can take a long time when you need a file several hundreds of GBs in size for testing... If you need to do that repeatedly, the time really adds up.  I don't care about the contents of the file, I just want it to be created quickly. How can this be done?  Using a sparse file won't work for this. I need the file to be allocated disk space. I navigate with Ctrl-n,p,f,b (readline/ keys), so that I don't have to move my hands too much and get  again. Is there a way to disable Ctrl-b being hotkey for bold?  Note that this is not just an Emacs thing. Standard web browsers, like Firefox and Chrome, use things like  Ctrl + K and Ctrl + L to navigate to the Google search bar and address bar respectively.