Correct regalia when collecting a masters degree but you already hold a doctorate
What graduation gown should a PhD-holder wear when later receiving a Master's in another field?
Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative test and it seems to suggest that $\Delta u(\vec x) \leq 0$ if $\vec x$ is a min and $\Delta u(\vec x) \geq 0$ if $\vec x$ is a max. However, I'm not convinced of this result. Can somebody point me towards a statement (theorem, etc.) involving the above. I appreciate the help. Java Round up Any Number I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far.  To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two conditions:   $\forall a, b \in \mathbb{Z}[i] \backslash {0} \hspace{2 mm} a \mid b \hspace{2 mm} \rightarrow N(a) \leq N (b)$ $\forall a, b \in \mathbb{Z}[i] \hspace{2 mm} b \neq 0 \rightarrow \exists c,r \in \mathbb{Z}[i] \hspace {2 mm}$ so that $\hspace{2 mm} a = bc + r \hspace{2 mm} \text{and} \hspace{2 mm} (r = 0 \hspace{2 mm} \text{or} \hspace{2 mm} r \neq 0 \hspace{2 mm} N(r) \lt N (b) )$   I have that the application meant to be the "norm" goes: $N(a +b i) = a^2 + b^2$, and I've managed to prove the first condition, given that N is a multiplicative function, but I can not find a way to prove the second condition.   I've search for a similar question but I have not found any so far, please redirect me if there's already a question about this and forgive me for my poor use of latex. How was the Cloak of Invisibility Created? Is is possible that a software issue could be causing spotty USB-C connections on 2017 MacBook Pro? I have a 2017 MacBook Pro. It has 2 USB-C ports on the left side and I use an Aukey hub to provide more ports (4 USB, 1 HDMI, 1 USB-C) and always to connect two monitors.   I connect one monitor through the hub with the HDMI port and another directly to the MBP with a USB-C adapter. Sometimes when I connect the hub to the Mac and the 2nd monitor to the Mac, it works as it should with no problem...but more than half the time I have to connect, disconnect, jiggle, jangle, switch ports, reconnect HDMI, etc. and hopefully they will both work. OFTEN, I can only get one monitor to fire up.  Is there any kind of software fix or reset that might solve this issue? Any other thoughts welcome. Thanks. How gently could a comet/asteroid/meteorite “hit” Earth? What are "Community Wiki" posts? Whats this white stuff on my board? I'm not an engineer, but my monitor stopped working yesterday (it just wouldn't display anything but still gets power) so I figured I'd like to take it apart and see what went wrong.  At the very least I'd get to see what the inside of a monitor looks like :).  Anyway, I took it apart, and also read around about my issue and it seems the capacitors have gone bad, at least some of them.  However, I also noticed some white elastic stuff all over the board.  Some of them are sticking two capacitors together and to the board, but I also see some from where power socket connects to the board.  My question is, what is the white stuff?  Should I replace all capacitors that have white stuff around it, and should I also replace the power socket part?  Or should I just replace the bulging capacitors and leave it at that?  UPDATE: What is the exact difference between fluid and responsive design? Java: how to parse double from regex Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$  How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).  Any guidelines please? This  gives this comparison chart for the four Windows 8 Editions (Core is usually just called Windows 8). I notice specifically, about half-way down, the max physical processors is listed as 1 for "Core" and 2 for all other.  In the second image i see physical processors listed as 8 in a Windows 8 Task Manager.  My question is, what is the limit then? Can anyone verify the article or the screenshot?  Was this a mock-up? Another Edition? (Server perhaps?)      To clarify, i am looking for links to official microsoft.com articles detailing these specifications.  I have been unable to find any. I want to install Win Xp on my Ideapad Z 570 laptop but i always get blue screen with message with some drive related problem which asks me to run CHKDSK command. I tried with that but no luck. Is there any way i can install Win Xp with SP 2 ?  Edit: i searched across the sites and came to know there could be some Compatibility problems, and i could fix it by setting HDD mode to IDE; will it solve the problem? Why is this limit $\lim_{n \to \infty}(1 + \frac{1}{n})^n$ equal to $e$? I dont get it, if I look at $$\lim \limits_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e$$    It seems like it should be $1$ because $1 \over n$ is almost zero and $1^\infty$ is still $1$ From SL Loney Trigonometry:     Prove that   $$r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$$   for a triangle with sides $a$, $b$, $c$, circumradius $R$, inradius $r$, and exradii $r_1$, $r_2$, $r_3$.   This one is a really dirty question.   I have tried a lot of stuff like $r=4R\times \Pi(\sin\frac{A}{2}$) and  $r_1=4R\times\sin\frac{A}{2}\cos\frac{B}2\cos\frac{C}2$ etc..  There are other ways like $-r+r_1+r_2+r_3=4R$,  but all these are way too cumbersome or clumsy.   Anything more intuitive? Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. unbalaced data set in classification tasks I just stumbled upon a :     My wife and I's seafood collaboration dinner. How does it look?   Sure enough, the top comment immediately points out that it should be "my wife's and my". However, a cross-post to the Grammar subreddit produced :     It's fine as it is written. "my wife and I" is a noun phrase, functioning as a subjective pronoun in the singular and made possessive with the apostrophe. It is exactly the same as "our".      It seems weird because you would never use "I's" on its own but it is not on its own here - it is part of a noun phrase.   That's a rather intriguing argument. Does it hold any water? Recovering broken or deleted NTFS partitions Keeping the time stamp when syncing with Ubuntu One? I would like to keep the original time stamp of a file when synchronizing with Ubuntu One.  Now Ubuntu One is messing with my time stamps, and I no longer know when I last worked on the file.  Ubuntu One also seem to mess up time stamps on the original source host.  This really is a show stopper, and we are just about to implement Ubuntu One across the many users in the organization.