Let $f:U\subset\mathbb{R}^m\rightarrow\mathbb{R}^n$ be a function of class $C^1$ with $m&gt; n$. Prove that $f$ is not an injection function.  I do not know how to solve this problem. Could someone help me with the solution to this problem?
Suppose $m,n&gt;0$, $U$ an open subset of $\mathbb{R}^{m+n}$ and let $f: U \to \mathbb{R}^n$ be continuously differentiable.  Is it possible for $f$ to be injective?    My thinking is that continuous differentiability+injectivity suggests that this is sort of getting at a converse to the Inverse Function Theorem.  We know the Jacobian is not invertible at any point in $U$, and the question is asking if this implies that $f$ cannot be invertible in any neighborhood contained in $U$.
C++ cout hex values? I want to do:  int a = 255;  cout &lt;&lt; a;   and have it show FF in the output, how would I do this? Move the Add Favorite / Ignored Tags boxes to the top It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake. Do Hogwarts Houses have quotas? Not a year goes by where we don't see at least one person Sorted into each House, and it's usually a fairly even distribution.  The Sorting Hat doesn't get a chance to study the mind of each child before Sorting, so it can't Sort them relative to each other (top X in smarts go to Ravenclaw, 4 bravest to Gryffindor, etc)  What would happen if a Sorting occurred where all the witches/wizards were extremely intelligent?  Would they all get Sorted to Ravenclaw?  What if they were all extremely brave?  What if none of them were particularly smart or courageous, but they all wanted power?  We see that there's a fairly small number of beds per room in each dorm, would the Hat stop Sorting people into a House if it knew the House had no more beds?  If so, could Harry Potter have ended up in Hufflepuff if his last name had been 'Zimmers'? What is the difference between `grep`, `egrep`, and `fgrep`? Can anyone tell me the technical differences between grep, egrep, and fgrep and provide suitable examples?  When do I need to use grep over egrep and vice versa? How is a Kelvin Pod different than a standard Escape pod? Can you trip as part of a Tome of Battle Maneuver? inverted breve accent in math mode Are the rules regarding absolute modifiers too absolute? A common grammar lesson that was taught to me in the US and that I've had to teach abroad in EFL classrooms is that we're not to use adverbs of emphasis with absolute modifiers, just as we're not supposed to use them as comparatives or superlatives.  Classic egregious examples of this mistake include very unique and more perfect, which seem obviously flawed to me. Other instances of further modified absolute modifiers are similarly meaningless or contradictory in nature.  I wonder, though, if the rule that I learned isn't overly broad, and whether it's taught similarly in the UK and elsewhere. I find myself modifying absolute modifiers quite regularly, and in some circumstances I think it's not only logical and correct, but quite meaningful. Phrases like almost exactly or virtually all or practically infinite possess a specific meaning that can't really be replicated using nonabsolutes like very nearly or most or quite long.  In my own speech and writing, I use such phrases exactly as I see fit, so this isn't a request for permission, per se. I'm more curious about whether there's a more appropriate rule regarding absolutes that you rely on (and might be used in a classroom). Unable to Retrieve Metadata Imagine your college buddy is now running a multi-billion dollar company, while you're still stuck at the same white-collar day job 5 years after graduation. You chat with him and want to express the idea that you're happy for him and fancy having his lifestyle--something like "I wish I could buy a yacht and cruise around the world like you do; I'm glad you're doing really well!"  Basically, it's the feeling of envy when you take out the resentment from it. Is there a word for it?   (I thought about "admire", but found it too strong and broad) Looking for a book with parallel universes like pages, possibly with "maze" in title, and a black Jewish female character Looking for a book. A series,  actually.  The main characters were a Jewish man and a black woman who was his wife. There was a running gag when she would tell people her last name and they thought it was unusual that a black woman would have a Jewish name.    They went on adventures in parallel dimensions. I believe that each book had a word repeated in the title, maybe "mirror?" Or "maze?" Or "page..."  it seems they jumped into worlds through a tool like a mirror or a book.   The parallel universes were like pages, and the pages closest to you were most similar but the further you got the further they were from your life.  One page over might be that you had something different for breakfast and so now you have a tummy ache, a few more pages over you might have become a prostitute, too far and it was theorized that humans had evolved differently altogether, I didn't get that far in the series.  For adults, and humorous.  I read them about 15 years ago, so they are not new. I would love to find them again. I have seen  and altough I face the same problem, the answer to that question is not helpful to my case.   I have large spreadsheets which are changed frequently. So, according to the solution accepted to that question, it is hard to duplicate all populated columns and it is cumbersome to edit one more cell everytime I wish to make a change on cells background colors.  Isn't there a way to sum data based on a condition regarding cell background color in Google Spreadsheets? How to disable Ctrl+Shift+U? Ctrl+Shift+U is a combined shortcut in Eclipse. But when I press it, it shows up a u, just like input with an underline. I guess this shortcut has been declared by Ubuntu, so I can't use it. I used to solve this problem by typing Caps Lock first. Is there any better method? Convert track bike to geared Is there any way to convert a track bike to a geared bike? I have a track bike with only the rails for the wheel but I've moved out of the city and there are more hills where I am now living so would like to convert to gears if possible and rather than buy a new bike, save some cash. Where does the extra kinetic energy come from in a gravitational slingshot? Change this behavior to allow for spelling corrections and the like: "Edits must be at least 6 characters" I'm making an app with a custom list in multiple languages.  Is there an easy way so that I only have to create the site columns, content type and list once? So I don't have to create them for each language?  My app will be in English, French and Dutch.  My lists elements.xml looks like this  &lt;?xml version="1.0" encoding="utf-8"?&gt; &lt;Elements xmlns="http://schemas.microsoft.com/sharepoint/"&gt;     &lt;!-- Do not change the value of the Name attribute below. If it does not match the folder name of the List project item, an error will occur when the project is run. --&gt;     &lt;ListTemplate         Name="List1"         Type="10000"         BaseType="0"         OnQuickLaunch="TRUE"         SecurityBits="11"         Sequence="410"         DisplayName="List1"         Description="My List Definition"         Image="/_layouts/15/images/itgen.png"/&gt;   &lt;Field       Name="Adres"       ID="{2a32d9eb-ad3b-4dbb-abba-66dd01ca7c6e}"       DisplayName="$Resources:Adres"       Type="Text"       Required="TRUE" /&gt;   &lt;Field       Name="Naampje"       ID="{ac47be45-2f43-4d16-9a06-162c85945ec9}"       DisplayName="$Resources:Naampje"       Type="Text"       Required="FALSE" /&gt;   &lt;Field     Name="Nickname"     ID="{6B0A2CD7-A7F9-41ca-B932-F3BEBB603793}"     DisplayName="$Resources:Bijnaam"     Type="Text"     Required="FALSE" /&gt;  &lt;/Elements&gt;   Well if i change my language in Sharepoint (Office 365) it's not changing the titles from dutch to the other languages i have added. Where does the question mark go — inside or outside the parentheses? I know that when brackets enclose part of a sentence, the full stop goes outside.     I tripped over this morning. I was distracted by a plane (which turned out to be Superman).   If it's a full sentence, the full stop goes inside.     I tripped over. (I was distracted by a plane that turned out to be Superman.)   What happens when there's a question mark and only part of the sentence is a question?     I tripped over this morning. I was distracted by a plane (or was it Superman?)      (or was it Superman?).      (or was it Superman)?   What happens to the full stop that belongs to the rest of the sentence, and why? Does Lin's  assume that the 2 data sets have linear or monotonic tendencies? Or, can I measure the concordance between 2 data sets that have a sinusoidal tendency? Just ran the latest batch of updates on 20.04 (Xubuntu), and now I'm getting a GRUB error: symbol 'grub_calloc' not found  I'm dropped into the 'grub rescue' shell, but have no idea what to do there that might be useful. To me, 'symbol not found' implies some sort of build error with the grub package, but I don't really know how grub works. I noticed that this update also included 'firmware', not sure if that could be related. Is my best bet just to boot from a live CD and see if I can roll back the update to grub somehow? Edited to add: OK, thanks to lots of people ! Here's what I think I now understand.  On 'non-UEFI' systems, grub is installed in two separate parts. The first, most basic, part is the part that is started on bootup. But for most of it's functionality, it needs the second part. These parts must be aligned - neither part must require any functionality from the other part which is not actually there. The visible, run-time problem occurs when these parts are not aligned, and the function grub_calloc is not supplied. It's not 100% clear to me if grub_calloc belongs in the second, larger part or the first. I would have expected the second, but the grub build system is a work of considerable art, so I don't know :).  The root cause of the problem is that the grub update has not ensured that both parts have been updated. Ideally, failure to do this should cause grub installation to fail, and the system should be reverted to a safe state. This does not happen. This is actually still a bit of a mystery to me. All that the update needs to do by default is put each part where the current parts are, because obviously that worked. If the install locations/drives are configuration - driven, and one of these locations can't be reached, then somehow a mismatch has arisen between that configuration data and reality. This might not show up as a problem as long as no new dependency was introduced between the parts.   All flavours of solution involve reinstalling grub to ensure that the two parts are aligned. It's not actually necessary to go back to the previous version ( although that will work ), because it's not the grub runtime per se that is broken. There are numerous ways to achieve this, depending on your environment, but running the Boot-repair live disk worked for me. It may be useful, for the purpose of avoiding such a misalignment in future, to ensure that the grub installer on your system is configured to install to the correct devices. This update resolves some important bugs (). If you have reverted grub to resolve this problem, be aware that you are exposed to these issues.