A finite, cancellative semigroup is a group
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ For all $a, m, n \in \mathbb{Z}^+$,   $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$ Where should shell tools be installed? Are there Apple guidelines for where user shell tools in Mountain Lion should be installed? Some (e.g. TeX, or Apple's Git) end up in /usr, and others (e.g. X11) seem to end up in /opt (sometimes with a symlink from /usr), while others (e.g., Git) end up in /user/local, and I wonder what the difference is, and whether one is preferred over the other. Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by  $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for all $A\in M_n(\mathbb{C}).$  I want to prove that $\exp$ is a surjective map from $M_n(\mathbb{C})$ to $GL(n,\mathbb{C})=\left\{A\in M_n(\mathbb{C})\,\middle| \det(A)\neq0\right\}$, how do I go about that?   I mean saying that $\exp:M_n(\mathbb{R})\to GL(n,\mathbb{R})$ is an analogous to saying $\exp:\mathbb{R}\to \mathbb{R}_{&gt;0}$ and this is also pretty intuitive, since, in analogy with the case of numbers, $A^0=I\;\forall A$, so $\exp(0)=I+0+\frac{0^2}{2!}+\dots=I$, so even for $A=0$ we get $\det\left(\exp(A)\right)\neq0$ and so because of the first term we can never get a zero determinant. But I have no idea how to prove the subjectiveness.  Thanks in advance. $(i)$ $a&gt;0,b&gt;0,p&gt;0,q&gt;0$   $(ii)$ $\frac{1}{p}+\frac{1}{q}=1$     How can we prove it?   ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$ Where did object dimensions go in 2.8? Anyone seen this panel? Essential for precise modeling. We have a job that is set to run Monday thru Friday. Up until two weeks ago, it ran fine 5 days a week. For the last two weeks it has failed on Monday. I am unable to locate where to find the point of failure.   ### Example Scripts 0 2 * * 1-5 /admin/scripts/example.exp 1&gt;/dev/null 2&gt;&amp;1   We run AIX 7.1.  I've looked in the /var/log/ and there is no cron file there. Looking for advice to add to this so we can troubleshoot.   I found that the log is located in /var/adm/log. Further we are getting this error repeatedly ever since this date/time. How do I clear this max limit?  c queue max run limit reached Fri Nov 25 21:52:00 2016 ! rescheduling a cron job Fri Nov 25 21:52:00 2016 I'm using Maverick with Compiz enabled.  I've searched AskUbuntu and Google. AskUbuntu doesn't have anything on this topic and everything Google turns up is a couple years old.  I'm doing browser tests and my tests are constantly spawning new Firefox windows. The problem is, when the window is spawned it takes focus. This is incredibly annoying as I'm usually in the middle of typing somewhere else. This doesn't only happen with my tests, but is the most common example I run into. Sometimes my IDE decides it wants focus when something special happens (like when something is done compiling) and then later I'll find fragments of IM messages I've typed in my code. This is driving me crazy. I've looked in both Gnome settings and Compiz settings and can't find a way to prevent that from happening.  Does anyone know of a way to prevent an app from stealing focus when it is opened? I'd rather have the taskbar flash at me or something to get my attention. Estimate on the derivative at fixed point Let $D$ be a bounded domain and let $f$ be analytic function from $D$ into $D$. Show that if $z_{0}$ $\in D$ is a fixed point for $f$ , then $|f' (z_{0} )| \leq 1$.  WHAT I WAS THINKING:   To have a conformal mapping from $D$ to the unit disc &amp; after composing it with $f$ ; I was thinking to use Pick's Lemma. So, can anyone tell me any conformal map from any bounded domain to the unit disc?? OR Is there any other way out to solve this problem?? My website's google cache copy looks broken.   Below is link to cache.  http://webcache.googleusercontent.com/search?q=cache:http://paintcollar.com  Can this affect the SERPs and crawling by google?  What can I do to rectify it?  PS. Few pages are being cached properly though.Below is URL to one such page. http://webcache.googleusercontent.com/search?q=cache:http://paintcollar.com/aneel Getting a black screen when installing or Live booting Ubuntu (any version) in my laptop I have an ACER Aspire laptop. Model number is E5-573G-56RG. I bought it only 15 days back. There is no DVD drive in it. I have tried various versions of Ubuntu- 14.04.2, 14.04.3, 15.10, 12.04.5. Tried disabling secure boot, fast startup. I current have Windows 10 Home and Windows 10 PRO (not-activated) installed with GPT partition in UEFI bios. I actually want only the Home edition and Ubuntu 14.04.3 (preferably because of LTS) or 15.10. In case more information is required, kindly ask me..  I tried installing through Wubi, when booting if I choose Ubuntu it automatically restarts and goes into Windows 10 Home! (I didn't have Windows 10 PRO at that time)  NONE OF THE SOLUTIONS I REFERRED HELPED ME.  I have spent 5 days (literally at least 20 hours reading all of them and trying out all kinds of editing in the grub command line, like removing quiet splash, adding and replacing with nomodeset, no splash, acpi_osi=off, acpi_osi=Linux, acpi_backlight=vendor,  acpi_backlight=legacy, noveau.modeset=0, i915.i915_enable_rc6=1 and such things! But I was not able to succeed installing which is why I doubt if there is some more thing to be done or know exactly what has to be done. Equation's symbols or parameters description are shown below, however, the = and the details of parameter are not aligned.     I want to make a parameters list for a equation like this, in which symbols, = and detail information are aligned in terms of vertical position.  where    ...    P   =   notional permeability factor    N   =   number of waves    Sd  =   damage level    ...   A tabular environment will produce more spacing before and after text. So how to make the = vertical aligned?   The codes is provided:  \begin{equation} \frac{H_s}{\Delta D_{n50} } = 1.0~ P^{0.13}~ \left(\frac{S_d}{N} \right)^{0.2} \xi_m^P~ \sqrt{\cot \alpha} \end{equation} where:   $H_s$ = significant wave height, equal to the average of the highest 1/3 of the waves  $\Delta$  = relative buoyant density, equal to $\rho_r / \rho_w - 1$, where     $\rho_w$ is the water density  $D_{n50}$ = nominal  diameter defined in Equation (2)  $P$ = notional permeability factor  $S_d$ = damage level   $N$ = number of waves  $\xi_m$ = breaker parameter based on mean wave period $T_m$  $\alpha$ = slope angle How should a physics student study mathematics? Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house).  This question is broad, involved, and to some degree subjective.  (I started out as a physics-only student, but eventually decided to add a mathematics major. I am greatly interested in mathematics; the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. Naturally, I enjoy  immensely.)  This question asks mainly of undergraduate-level study, but feel free to discuss graduate-level study if you like.  Please do not rush your answer or try to be comprehensive. I realize the StackOverflow model rewards quick answers, but I would rather wait for a thoughtful, thorough (on a point) answer than get a fast, cluttered one. (As you probably know,  produces clear, useful writing; and a properly-done comprehensive answer would take more than a reasonable amount of time and effort.) If you think an overview is necessary, that is fine.  For a question this large, I think the best thing to do is focus on a specific area in each answer.    Update: To Sklivvz, Cedric, Noldorin and everyone else: I had to run off before I could finish, but I wanted to say I knew I would regret this; I was cranky and not thinking clearly, mainly from not eating enough during the day. I am sorry for my sharp responses and for not waiting for my reaction to pass. I apologize.  Re: Curricula:   Please note that I am not asking about choosing your own curriculum in college or university. I did not explicitly say that, but several people believed that was my meaning. I will ask more specific questions later, but the main idea is how a physics student should study mathematics (on his or her own, but also by choosing courses if available) to be a competent mathematician with a view to studying physics.  I merely mentioned adding a mathematics major to illustrate my conclusion that physics student need a deeper mathematical grounding than they typically receive.   And now I have to run off again. Starting eclipse from unity launcher How to find the shortest path between opposite vertices of a cube, traveling on its surface? I am stuck with the following problem that says:      Let $A,B$ be the ends of the longest diagonal of the unit cube . The length of    the shortest path from $A$ to $B$ along the surface is :     $\sqrt{3}\,\,$  2.$\,\,1+\sqrt{2}\,\,$  3.$\,\,\sqrt{5}\,\,$ 4.$\,\,3$    My Try:   So, the length of the longest diagonal $AB=\sqrt{3}$. If I reach from $A$ to $B$ along the surface line $AC+CD+BD$, then it gives $3$ units. But the answer is given to be option 3.   Can someone explain? Thanks in advance for your time. What equipment should I have to kill the Eye of Cthulhu? I just manually summoned the Eye of Cthulhu, going into battle with copper armour, tungsten broadsword, gold bow, jester arrows and plenty of lesser healing potions. I came rather close to death but I managed to stay alive. However, the Eye took off when I had his health down to ~1700. I'm assuming this means you only have a few minutes to finish it off after a certain point. Is there any equipment I'm missing to help kill the Eye faster and to help my survivability? If it's relevant, my current life is 120. How does the Schengen 90/180 rule work? find() function in python2.7.5 As I'm relatively new to the electronics world, I was wondering if you can connect multiple  ground terminals (corresponding to several different voltage outputs) to the same ground. Or do they each need their own?  In my case I have 5 lines, one 12V, one 3.3V, two 5V and one variable from a 12V in using a  voltage regulator that already has its own ground. So I'm guessing that each would need its own ground, but it is never bad to ask. Principal component analysis (PCA) is usually explained via an eigen-decomposition of the covariance matrix. However, it can also be performed via singular value decomposition (SVD) of the data matrix $\mathbf X$. How does it work? What is the connection between these two approaches? What is the relationship between SVD and PCA?  Or in other words, how to use SVD of the data matrix to perform dimensionality reduction? Conveying "under the radar" in a less colloquial way I am searching for a word to contrast with iconic  I want to characterize views of the city as both iconic and under the radar, maybe "non-iconic."  I was thinking about saying that a view of the city was undiscovered, but I think that maybe that overstates what I'm think-- I more mean under-appreciated. Are there any other words that come to mind?