Order of $xy$ in a group G I'm studying abstract algebra alone as my doctor needs a doctor haha.  Ok, I'm studying about cyclic groups now. I read that:  if 2 elements $x$ and $y$ commute in a group $G$, then the order of $xy$ is a divisor of $l.c.m(o(x),o(y))$,   then I read that it's exactly equal to $l.c.m(o(x),o(y))$, so got confused.  Can anybody explain this to me? Also, can anybody advise me to read some detailed lectures about cyclic groups, as I feel like if I need much more to be strong at this lesson.
Order of products of elements in a finite Abelian group We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have  $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$.   How I approached this question was by saying let $\operatorname{LCM}(|a|,|b|) = L$. Then if we show that $L$ divides $|ab|$ and $|ab|$ divides $L$ then $|ab| = L$. Showing that $|ab|$ divides $L$ was fine. But then I am having trouble with the second part that is showing $L$ divides $|ab|$. For this, so far I have:  Consider $(ab)^{|ab|} = e = a^{|ab|}b^{|ab|} $  since we know $a \neq b^{-1}$ by assumption then we can conclude that $|a|$ divides $|ab|$ and $|b|$ divides $|ab|$. We know that $L = \frac{|a||b|}{\operatorname{gcd}(|a|,|b|)}$ from here can we conclude that $L$ divides $|ab|$?
Non in front of hyphenated adjective If one wishes to add "non" in front of a hyphenated adjective, should one add a hyphen after "non?" Simple undivided plane UV map not behaving as expected (Google) Title of page not updated, everything else did My website changed significantly this month.  Amongst the changes was the website's title.  Two weeks ago, I finally used Webmaster Tools and submitted the page.  So far, Yahoo and Bing have updated the title in the search results, but not Google.  I've resubmitted via Webmaster Tools the website, and the bots have crawled the page, and have added "content keywords".  But the title still doesn't match.  The "content keywords" lists many of the words in the new title (but contains keywords of the existing title also).  Must say that 4 of the top 5 key words in the "significance" list are the new title's.  Still, no change.  Any remedies for this problem?  Is adding the "alt" tag for the logo significant enough to trigger the change?  There isn't an appropriate place to put the company name between  tags on the front page.  Any suggestions?  Thanks. How to combine complex powers? Convergence in measure and almost everywhere In a finite measure space, let $\{f_{n}\}$ be a sequence of measurable functions. Show that $f_{n}  \rightarrow f$ in measure if and only if every subsequence $\{f_{n_{k}}\}$ contains a subsequence $\{f_{n_{k_{j}}}\}$, that converges almost everywhere to f.   I can prove from converge in measure to converge almost everywhere, but I don't know how to write it down for the other direction. The staircase paradox, or why $\pi\ne4$ What is wrong with this proof?    Is $\pi=4?$ How can I disable landscape mode in Android? How can I disable landscape mode for some of the views in my Android app? Should I always disconnect event handlers in the Dispose method? Consequences of a master thesis with errors I finished my master thesis 6 years ago, since then I gained a lot of experience on my field and also in academia. I can say that in that time I was a "child" , very inexperienced. I think that also my advisor did not help a lot, and I tried to do  a lot of things by my own, without someone to really review my things. Despite that, I still received a good mention by the master committee on final results.   Some days ago, I just decided to read part of master thesis, and I just realized a big nightmare coming. I identified a lot of typos, some references errors, and some improvements that I should do, but I did not. The worst thing is that I wrote a paper to a conference with my master thesis results (and some errors are there). The conference is not a high impact conference, but still the paper has my name and is available for everyone...   As I am finishing my PhD studies for now, I am concerning a lot that situation. My dream and goal is to pursue a Professor career. So, my questions are: What are (or could be) the consequences of a master thesis and paper with errors (some minor and some major) in academia? Revoke degree after years by errors in master is possible ( I am concerning a lot this situation that I thought on it)? How can I minimize it (I thought that I can put a fixed version of my thesis and paper in the future, for instance on my personal website)? How those errors can impact me in a Professor recruitment process?  Sorry for the long text, I am blaming myself all the time, and really afraid about the possibilities since I know that Academia career is really competitive. I am currently working on a model which takes two parameters and produces a measurement statistic. Think of it as Z = f(X,Y).   Z is a matrix of my statistics and I am creating a surface plot of it in matlab. Basically, I am looking for a mathematical/analytical way of determining if the surface is smooth, or if it is jagged. Do large values tend to be clustered together or are they dispersed throughout the matrix? - that is my question. Basically, how mixed up are the values of my matrix?  I need to run the model over different parameter sets and I want to be able to analytically determine which one of my surfaces is the smoothest, has the greatest clustering of large values, and ideally, has no negative values.   Any help will be greatly appreciated and please let me know if you need any further information.   Cheers Is Using Digital Signatures to prove identity a zero knowledge proof? Suppose Alice publishes a book with a public key in it, and later wants to prove that she wrote the book. She could sign challenge messages with her private key, and others could verify those signed messages. Alice is proving she knows a private key without revealing anything about it.  Is this an example of an interactive zero knowledge proof?   Or does this example somehow not apply to zero knowledge proofs? Suppose I wish to allow only users who have scored at least 15 in the particular tag to answer my question, Is it possible to set a condition like this while asking a question? So that I can avoid strangers/answers from dummy users. Mavericks input source is set to US keyboard in login screen I'm from Belgium, and on my own user account, the input source is set to a Belgian (actually French) keyboard. The input sources menu is enabled in the menu bar, so I can switch it to the Belgian keyboard before entering my password, but how can I permanently change the input source used for login? I'm the system admin, and I know how to get root access if necessary.  It worked perfectly in Mountain Lion. When should I fudge dice rolls? When I started my first D&amp;D campaign as a DM, I accidentally rolled and confirmed a crit on a PC from a goblin. It was an instant kill as our campaign had no cleric. That time I decided to fudge the dice. Now, there's a similar problem, but this time the PCs are at 3rd level and they are dealing with a very important NPC. If they fail this one diplomacy check, the entire game world will fall apart, as the king would then send his entire army against the PC's homeland and will destroy it, plunging my entire game world into chaos, at which time drow that I wanted to save for a later adventure will have to come out, which will instigate a wizard's guild to release their secret ultra-deadly virus, and everything in my game world, including the PCs, will die. So should I fudge the dice roll? When is it good to fudge it? I need to find node voltage at A. This seems like a trivial question, but when I set up the resistive voltage divider equations they are not equal to each other. Am I doing something wrong?  My equations are \$V_A = V_1(R3/(R3+R1)) = V_2(R3/(R3+R2))\$      I tried solving this problem with an LTSpice simulation and with the superposition method, but my answers did not match (although very close to each other). Is this because the simulator uses an approximation? Can someone compare results? Connectedness of the Hausdorff distance. Does anyone know a proof of connectedness of the Hausdorff distance?  I mean a proof of the following:  Theorem  If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected.  Remark:  $F(X) = \{ A \subseteq X | A \text{ is nonempty, closed and bounded } \} $, and $d_h$ stands for the Hausdorff distance. Force a Blender operator to always start with default values I'm writing an operator for which it is crucial that every time it gets called all the properties are set to their default values.   The ops.object.delete has 1 param. use_global. It is always false, no matter what the user put in last time it got called. I would like to replicate exactly that.   (For example: I want to modify delta scale. The default val for my_factor is 1, so nothing happens until the user alters that number. If the user calls the operator again, it will have stored whatever number the user has used last, however I want it to always start with 1.) I've bought a  to use with my EOS 200D and while I love the wide aperture, it appears to be quite difficult to achieve sharp focus on people's faces when shooting at f/1.8. I've tried both using Dual Pixel focus in Live View and using the center focusing point in the viewfinder, but it doesn't get me perfect results.  Is this simply a characteristic of cheap wide-aperture lenses? Or is it impossible to get sharp focus with a fully open aperture?  When I'm at at home using a tripod and focusing on a still subject I do get good focus (so it's not a camera or lens issue). But it's hard to do with people who are slightly moving between shots. I was just wondering why Fresnel Lenses are not widely used in the production of solar electricity. Their use there would mean that you could produce heat within a fraction of a second, up to a few minutes and run a turbine to produce electricity.  Though it is used in welding, I am not sure what are the problems in producing electricity, as stated by :     New applications have appeared in solar energy, where Fresnel lenses can concentrate sunlight (with a ratio of almost 500:1) onto solar cells. Group Membership Test for Permutations