About Dubai, hotel as an unmarried couple but with a twist
Travelling as unmarried couple in Dubai
How prove this two symmetric matrices $AB=0$ Let $A,B$ be real symmetric matrices, and for any $n\in \Bbb N^{+}$, and for all $x,y\in \Bbb R$, we have $$tr(xA+yB)^n=x^ntr(A^n)+y^ntr(B^n).$$  Show that $AB=0$.  My try:since $$(xA+yB)^n=x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n$$ so $$tr\left(x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n\right)=x^ntr(A^n)+y^ntr(B^n)$$  then  $$tr\left(\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}\right)=0$$ so $$\binom{n}{1}x^{n-1}ytr(A^{n-1}B)+\cdots+\binom{n}{n-1}xy^{n-1}tr(AB^{n-1})=0$$ then I can't  I found this is a nice result, maybe this is an odd problem, and I can't solve it. Thank you. Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$ Evaluating $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using polar coordinates. I want to evaluate the integral $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using the substitution $x=r\cos\theta$ and $y=r\sin\theta$ where the Jacobian is given as $|J(r, \theta)|=r$ and thus $dxdy=rdrd\theta$.  I need to set the range for $r$ and $\theta$. For $\theta$, it seems obvious that it ranges from $0$ to $\frac{\pi}{2}$ to cover all parts of the rectangle. But I am confused with the range for $r$. Would it be OK to set minimum $r$ as $0$ and maximum $r$ as $\sqrt{2}$, which is the diagonal of the square domain? I think I am confused with the fundamentals of integration. How to find previous boot log after Ubuntu 16.04+ restarts? My question is, how can I find boot log from previous system boot attempt?  Today when first powering on my PC, boot process stopped on Ubuntu logo, when I pressed Esc I have seen several lines containing some kernel error and restart required at the bottom, so I pressed Ctrl+ALt+Del and next boot went OK without problems.  I have trouble finding messages from the screen I have seen during the first unsuccessful boot. Should I have taken picture to my phone?  /var/log/boot is there but empty, I searched  and  for strings I remembered with today's date like error but found nothing familiar to what I have seen on previous boot screen.  $ journalctl -b -1 gives me only kernel messages during boot, I can find that elsewhere too, and they are not what was appearing on screen during boot, journalctl is useless for me, I am looking for messages appearing on screen during boot time.  For now, only option is take a photo of write the message on paper. How do users navigate to Meta? How do you expect users to:   Know that a site's meta-site exists (and what it's for)? Find the hyperlink to the meta-site?   I know it exists, of course, and how to get there; I'm wondering how (or even whether) you expect new or naive users to do that.  I think there isn't a link to Meta from the main site, am I right (except one link in the "list of all 171 Stack Exchange sites" drop-down menu on the toolbar)?  Sometimes I tell people to post a question on Meta instead of on the main site -- I don't realise that they may not know that Meta exists, and that if they do (because they've been there before) they don't know how to get there (because they haven't yet investigated everything on the toolbar). Atmospheric pressure In FBD doubt Let us say I have a beaker floating in a water tank. Suddenly I put some water in the beaker also. The figure here shows the exact situation   Now I want to draw the FBD of the beaker. I am somehow confused with the role of the atmospheric pressure in drawing the FBD. I thought of two ways of drawing this and I am just unable to figure out which one is right.    Here since the beaker is floating an $F_{buoyant}$ is there in the upward direction which the net mass (beaker+water in beaker) would put its weight downward.  But while solving one question I came to know that total force that a liquid exerts on the bottom of the container is $P_0+m_{liquid}g$ ($P_0$ is the atmospheric pressure). So should the downward force on the beaker be $(M_{beaker}+M_{water in beaker})g+P_0 S$ (S being the base area of the beaker) as in the second diagram down here.    Here the $P_0S$ due to atmospher on the tube gets cancelled by the $P_0S$ exerted by the water in tank on the tube. But what about the $P_0S$ from the water present in the tube? Do the FBD's apply like the one in the new image? Terminating with Master's after several years in a PhD program. How to indicate on CV/Resume? I am not happy with my PhD program, and after several years I am graduating with a master's this month. I earned the M.S. along the way as a PhD student, but have never formally requested it before this semester. What is the best way to indicate this situation on a CV or resume? I don't want to put just M.S. because then it looks like it took me a really long time to get a master's and I can't put that I am a PhD student because I no longer am.  Right now, I am thinking something like  M.S. in &lt;field&gt;, &lt;University&gt;, May 2016 PhD student in &lt;field&gt;, &lt;University&gt;, Aug. 2011 - May 2016 B.S. in &lt;field&gt;, &lt;Undergrad University&gt;, Aug. 2007 - May 2011   Would that be clear what's going on? Alternatively, should I list PhD above M.S. even though I didn't complete it? Should I list both M.S. and PhD in same line since it is the same university and span of time? When Apple was in the high $600 or so range, it split 6 times and made it more affordable to purchase more shares for less, even though the value was technically the same. I see companies like Google are extremely high per share and am wondering why or why not they would or wouldn't split their stock right now. If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$? Let $f(x)$ be a continuous real-valued function on $[a,b]$ and $M=\max\{|f(x)| \; :\; x \in [a,b]\}$. Is it true that: $$ M= \lim_{n\to\infty}\left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n} ? $$  Thanks! Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$. Let $a \in \mathbb {Z}$, $n &gt; 1$ a natural number with $\gcd(a, n) = 1$, and let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove that $r \mid \phi(n)$.  Euler's generalization of Fermat's Little Theorem: If $n \in \mathbb {N}$ and $m \in \mathbb{Z}$ such that $\gcd(m, n) = 1$, then $m^{\phi(n)} \equiv 1\bmod(n)$.  By Euler's generalization I would have $a^{\phi(n)} \equiv 1\bmod{n}$.  Then I could say $a^r \equiv a^{\phi(n)} \equiv 1 \bmod{n}$. At this point I can see that $r$ clearly must be some multiple of $\phi(n)$.  If I want to show that $r \mid \phi(n)$, then I can show that $\phi(n) = rk$, for some $k \in \mathbb {Z}$.  However, I am a bit stuck about how to make this last step showing that  $\phi(n) = rk$, where $k \in \mathbb {Z}$. Where to get 2010 Census Block data? I would like to download 2010 Census Block Data for several towns and I can't seem to find it on American Factfinder.   Is it there? And if not, does anyone know where I could find it? Is there a way to create synchronization profiles for Google Chrome? For example I want to tag some of my browsers as "home browser"s and others as "work browser"s and I want:   my bookmarks to be synced with all browsers, my extensions to be synced only within browsers of the same type (home with home, work with work) "Optimiser" vs "optimizer"? I have seen words which are spelled either with a "z" or with a "s"  like "optimizer" and " optimiser".  I thought that the ones with "z" are wrong but on searching the web I found that both are correct. Why so? Is there any difference in the use also? In calculating work done by a constant force over a constant distance, why doesn't the subject's initial velocity matter? Assume a point-mass $m$ is travelling in a straight line, and a force $F$ will act on $m$ (in the same direction as $m$'s velocity) over a constant distance $d$; why doesn't $m$'s velocity matter to the calculation of work done on $m$ by $F$? Work is defined such that, in this example, the work done by $F$ on $m$ is equal to $Fd$, but it seems that if $m$ were moving slower, it would spend more time in the field, allowing $F$ more time to act on $m$, thereby doing more work. In fact, if $m$'s velocity were very great, it would hardly spend time in $F$'s field at all (so very little work done). Maybe I misunderstand work; can someone address this confusion of mine? Attributive nouns vs. of-genitive I would be very grateful for some advice on how to decide whether to use an attributive noun or the "of genitive" (periphrastic genitive).  It seems to me that an attributive noun is generally preferable, except when it leads to ambiguity, often due to the combination of several modifiers with mixed dependent relationships.  Additionally, the of-genitive, when used stand-alone, seems to impart a more formal tone, a higher register.  Simple examples (preferred first, less-preferred second):  The department secretary vs. the secretary of the department.  But:  The excellent secretary of the department vs. the excellent department secretary.  Any additional guidelines would be most welcome. Matrices that commute with all matrices Add-on for blocking sites in Google Search I noticed that my American friends tend to use the past perfect form when they use the verb mention in its past form, almost despite of the context of the sentence. I hear them say  "you had mentioned" a lot more often even in situations where the use of simple past would have been perfectly fine. Have you noticed it too and is it one of those unwritten rules, or is there a more formal explanation to it?  It seems like there was more verbs like that, but I am having a hard time remembering it now. If I do, I will edit the post and likely change the title to accommodate other verbs. Generalized Second Borel-Cantelli lemma Let $A$ and $B$ be two constant matrices and let $x$ and $ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables and $A$ and $B$ are constants, but what about the matrix case?