Let $U$ be a $n\times n$ matrix with complex entries satisfy $$ U^*U=I $$ where the star denotes the conjugate transpose. Can we then deduce that $$ UU^*=I $$ ?
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix.
Preserve blend mode effect when underneath layers are disabled? (Photoshop) Flag indicator present on Meta, but no flags visible It looks as if the 10k-tools flagged post list is b0rken.  Next to the review link in the toolbar there is a flag counter showing there are 10 flags to be reviewed:    Clicking on that flag counter leads to an empty page:        The moment I visit that page (through clicking on the flags counter, or by going to the review queue first then clicking on Tools, selecting the flags tab if not already selected), the flags counter disappears from the toolbar:    but after a while (half an hour, an hour?) the flag count re-appears in the toolbar.  Either the counter is wrong (presumably taken from a Redis cache somewhere) or the 10k flag review page is falling over every time someone tries to load it.  This at first appeared to happen only on Meta, on Stack Overflow the flags page has plenty to review. However, there are now  on Stack Overflow. W.K.T, any user(with more than 2k rep) can review 20 suggested edits per day.  Today I saw the newly appointed moderator  reviewed more than 20.    Is it possible for moderators to review more than the daily limited count?  If yes, can any moderators can do it at any time?  I didn't see this before. How can I save application settings in a Windows Forms application? What's the legal way to stream music online? It's really hard getting a definite answer on this so I'm going to ask it here. Is there a legal way to stream new music on a website without getting artist consent first. There seems to be a lot of popular sites like hotnewhiphop.com doing this that are hosting from both the US and Canada so I'm guessing there is a grey area that allows it. For the record I don't allow downloads of the music and simply stream each song in a player. I have found that for only one page (so far) the spacing between the header and content is different from the spacing on all the other pages. You can see this behavior in the images below (from pages 2 and 3 of the document):     My .tex file contents are shown below.  \documentclass[a4paper]{report} \usepackage{fullpage} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \setlength{\headsep}{0.5cm}  \begin{document} \renewcommand{\headrulewidth}{0.4pt} \renewcommand{\chaptermark}[1]{\markboth{\thechapter. #1}{}} \fancyhead[L]{\bfseries \leftmark} \fancyhead[R]{\bfseries \rightmark}  \chapter{Version Control} \lipsum \lipsum \lipsum \lipsum  \end{document} Upper bound on $\chi(G)$ for a triangle-free graph I'm struggling with the following question;     For every graph $G$ such that $K_3 \not\subseteq G$ (i.e. $G$ does not contain a triangle), prove that $\chi(G) \leq 2\sqrt{n} +1$ (where $\chi(G)$ denotes the chromatic number).   Any hints? I really cannot see the connection between the triangle-free nature of $G$ and a chromatic number being bounded above by $O(\sqrt{n})$. Integrating a function of tan inverse How would I carry out the following integration? $$\int_0^12\arctan x^2$$  I tried to substitute $x^2 = \tan\theta$ From there I wrote the integral as: $$\int_0^{\pi/4}\theta\cdot \frac{\sec^2\theta}{\sqrt{\tan\theta}}\ d\theta$$  Now, is my only option to use integration by parts? Or is there a better method to solve this integral? Why are my textures pink? Basically, I applied my textures by taking each individual object, unwrapping it in the uv editor, then opening up an image in the uv editor and it would apply the texture in the 3D editor. However, when I render my project none of the textures appear and everything looks odd. I don't know how to set up the images properly with the bee.  For example, here's the uv layout image for the bee's Cube.001:     How do I get that applied properly to the Cube.001 in my project?  Here is   Everything appears purple if the images aren't there and I don't know how to fix that either. What is the equivalent of Persian idiom "When the reed blooms"? In Persian, we say "When the  blooms" when we want to  express that an event:   Never happens. (This is only the opinion of the speaker so it's not a fact) It's very unlikely to happen. It's going to take so long to actually happen. (It's so long that you'd be better off giving up expecting it)  3.1. We also use it when someone is always postponing something they have to do so you never actually see it happen.   For example:      Kyle: When are you going to pay my money back?      Cartman: When the  blooms.   Now I don't know if I got the plant's name right because there are some other versions like:        In any event, they have mentioned this plant in the idiom because apparently some versions of it actually never bloom. Others believe it's because farmers harvest them before they get the chance to bloom (e.g to make music instruments), so it never happens. I'd love to know the equivalent idioms or expressions for this.  PS: I only know "When pigs fly" but I'm not a fan of it because the word pig sounds a bit rude to me and partly because it implies a strong impossibility which doesn't work for the #3 use. I use Ubuntu 12.04 64-bit, I did the following: sudo gedit /etc/apt/apt.conf and added  APT::Install-Recommends "false"; APT::Install-Suggests "false";    But it did not work. When I try to install a package, it still wants to install the suggested and recommended packages. How can I solve this? Hereditary properties with nested frames (mdframed and tikz) Show OPs if someone is answering, live I ask a lot of question on SO, so this recommendation/idea comes from that experience. I'm not really sure how this would fit in with the current structure of SE.   Oftentimes after asking a question, maybe after an hour or so, I end up finding the answer I needed somewhere. And it comes time for me to answer the most difficult question of all: do I delete this question, or do I keep it in case someone answers and others may benefit from it. The problem is, if I don't delete it and no one answers, my acceptance rate goes down. So usually I'll just delete it.  However, sometimes I fear that someone is typing a really really long answer to my question since itäs only been an hour, and so I imagine the nightmare that person experiences when they find out their essay will be gone to waste. Overall, this never-to-be-seen-again answer could have benefitted many people had the question remained.  Since SE has gone "live-happy" lately, where answers and comments refresh live, I thought it might be interesting if the original asker of the question (or anyone else for that matter) could see how many people are currently responding to the question (just count, not necessary to display their names). This way, if I know someone is replying, I won't delete the question, and others can benefit from it. Why do people risk death by joining Starfleet if not for money or preservation of their homes? Watching Star Trek made me think of a few "deeper" questions about Starfleet, which are all related to the Star Trek TV series.  So, let's take for example the Dominion War: people are dying every second, ships get destroyed and cadets are still joining although they know they are probably going to die unless they are with Picard or Sisko.  As I read and got to understand, money has no value anymore, latinum is the general trading currency in the universe. But people on Earth don't trade and don't earn their paychecks. Earth is a much better place than today and yet young boys and girls from every culture/civilization are joining Starfleet as they are willingly signing their death note.  Why are they joining, what motivates them? They might know that they're probably going to die.  They don’t earn paychecks. They could be doing any other job, yet they choose this. And what's more interesting how many died on battlefields must be lined up in front of StarFleet every day?  What do they receive after service? Why are they doing it other than preserving their homes? Raise bounty on question What is Thanos's motivation for his action? If I had a gauntlet with essentially unlimited power, and my goal was to help make sure life in the universe was able to prosper, then my first thought toward this goal probably would not be      to extinguish half of all life.   This doesn't directly help anyone, and it is, at best, a temporary solution, since      populations will eventually grow back.   Assuming that I am not, in fact, secretly in love with an avatar of death itself, why would I choose to      commit universal genocide   instead of directly solving the limited resource problem?  So, to be specific, my question is: In-universe, what is Thanos's motivation for attempting to do this instead of more reasonable alternatives? What's the name of this flat illustration style? I want to buy some stock photos like the ones below. What is this design called?  I don't even know how to start searching... What are the differences between "su", "sudo -s", "sudo -i", "sudo su"? I already read it from manual but I can't see difference..  su - change user ID or become superuser  sudo -s [command]   The -s (shell) option runs the shell specified by the SHELL environment variable if it is set or the shell as specified in passwd(5).  If a                    command is specified, it is passed to the shell for execution.  Otherwise, an interactive shell is executed.  sudo -i disappear description in manual Why do computers only use 0 and 1? Why do computers only use 0 and 1? Won't the addition of other numbers such as 2 or 3 speed up computers? Also, 2 and 3 can be used to shorten the bit-length of integers (2 and 3 can be used to end an integer, so that the number 1 only needs one two bits.)..  Why is binary computer more preferred? Let $\mu(\Omega)$ be a probability measure (i.e. $\mu(\Omega) = 1$), and let $f,g$ be non-negative measurable functions on $\Omega$ such that $fg \geq 1$. Show that $1 \leq (\int f^p )(\int g^p)$ for all $0 &lt; p &lt; \infty$   I'm a bit at a loss of what to do. I know that since $fg \geq 1$, we may obtain:  $$\int_{\Omega} d\mu = \mu(\Omega) = 1 \leq \int_{\Omega} fg = \int_{\Omega} |fg|$$ The right-most integral is true since $fg$ is positive and it is equal to $||fg||_1$.  By Hölder's inequality, we obtain:  $$1 \leq \int |fg| \leq ||f||_p ||g||_q = (\int f^p)^\frac{1}{p}(\int g^q)^\frac{1}{q}$$  where $p$ and $q$ satisfy $\frac{1}{p} + \frac{1}{q} = 1$. Raising the left-most and right-most sides of this inequality to the power $p$, I may obtain:  $$1 \leq (\int f^p)(\int g^q)^\frac{p}{q}$$  I now wish to show that $(\int g^q)^\frac{p}{q} \leq \int g^p$, but I've been at a loss of how to do it. Any tips or hints would be greatly appreciated.