Wikipedia has that:     Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space.   I wonder if the locality assumption on the structure sheaf (every stalk is a local ring) implies Hausdorffness. If we let the line with two origins have the structure sheaf inherited from the quotient, won't the stalks at each of the origins be $\Bbb R(y)[x]_{(x)}$ and $\Bbb R(x)[y]_{(x)}$, which are local rings?  What about second countability?
Most textbooks introduce differentiable manifolds via . This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and rarely if ever seems to play any role in differential geometry/topology.  There is a much better definition of differentiable manifolds, which I don't know a good textbook reference for, via . This definition does not involve any strange arbitrary choices, and is coordinate free. Paragraph 3 in Wikipedia (which is the actual definition) states:   A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is a sheaf of local $R$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.   This confuses me, because I don't see why such a sheaf should be acyclic, or where conditions like "paracompact" or "complete metric space" or "second countable Hausdorff" are implicit. So either:   The wikipedia entry has a mistake (I would want to sanity-check this before editing the entry, because this is such a fundamental definition which thousands must have read). Somewhere in that definition, the condition that $M$ be paracompact is implicit.   Question: Should the definition above indeed require that $M$ be second-countable Hausdorff or paracompact or whatever? Or is it somehow implicit somewhere, and if so, where?    Also, is this definition given carefully in any textbook?  Update: I have editted the Wikipedia article to require that $M$ be second-countable Hausdorff. But I'm still wondering if there is a textbook covering this stuff, and whether requiring the sheaf to be acyclic might have worked instead as an alternative.
'Just' and 'only' carry a similar meaning, and while my feel for language usually helps me decide which one to use, there are times when I'm at a loss.  From my understanding, 'just' is used as a softener, to add flexibility and politeness. 'Only', on the other hand, is a relatively sharp exclusion. 'Just' also has a wider range of meanings, for instance to denote time.  While this sounds simple enough, it can get messy in practical usage. The sentence that made me stumble was "Not only airports are part of the target customer group, but also other large infrastructures".  Would the meaning of this sentence change when using 'just' instead? The way I see it, there is no difference in this case, except 'just' would make it sound a tad more colloquial. I would like to remove my Raspberry Pi from my iPhone 8 (running iOS 11.2.1) as a connected bluetooth device.  Normally, I would go into my Settings -> Bluetooth and select the Info button (a small blue circled "i") to show the "Disconnect" and "Forget Device" dialogs:    Unfortunately, iOS does not give me this option for my device. The bluetooth device list does not contain an "i" for the item, and there is seemingly no other way to disconnect.    How do I unpair this device if there is no button to trigger to the unpair prompt?  Due to work restrictions I cannot completely reset the device. Why are 15 reputation points required to upvote? I think more reputation points should be required to add a new answer than to upvote an existing answer.  I don't continually watch for new questions, so usually when I find a question it has already been answered.  It seems bit inappropriate to answer an already-answered question in order to contribute to the conversation, but until you earn reputation, you can't upvote or comment; you can only add new answers, which prevents me from contributing at the moment as much as I would like to.  This seems backwards to me. If someone incorrectly upvotes a bad answer, the damage is minimal, as other users can down vote it and upvote the correct answers. On the other hand, an incorrect answer wastes the time of everyone who reads it.  So, IMHO, it should require more reputation points to add an answer than to upvote or comment on an existing answer.  I wouldn't say say that users should be prevented from answering unanswered questions until they have x reputation points, as that might prevent users who have answers from providing them, but perhaps it should be required if the question is already answered, to discourage users from reiterating previously given answers.  What do you think? Turning off "Update page numbers only" in Word When I update the fields document-wide (Ctrl + A, then F9) in a Word document that has a table of contents, a dialog appears asking if I want to update page numbers only, or update the entire table. I never want to update page numbers only if the document headings have changed, nor will I ever.  (Apparently this is given as an option in case someone has manually changed an automatically generated ToC, which is almost always a really bad idea.)   Unfortunately, "Update page numbers only" is the default selection in the dialog, which means that I have to select the other option every time. I update the fields all the time when I'm working in a document that has figures and cross-references, so this is an annoying extra step I'd really like to get rid of.   Is there any way to tell Word to stop asking me this stupid unnecessary question? Make mod_wsgi use python2.7.2 instead of python2.6? i am running Ubuntu 10.04.1 LTS  and it came pre-packed with python2.6 but i need to replace it with python2.7.2.   (The reason is simple, 2.7 has a lot of features backported from 3 )  i had installed python2.7.2 using  ./configure make make altinstall   the altinstall option installed it, without touching the system default version, to /usr/local/lib/python2.7 and placed the interpreter in /usr/local/bin/python2.7  Then to help mod_wsgi find python2.7 i added the following to /etc/apache2/sites-available/wsgisite  WSGIPythonHome /usr/local   i start apache and run a test wsgi app BUT i am greeted by python 2.6.5  and not Python2.7  Later i replaced the default python simlink to point to python 2.7  ln -f /usr/local/bin/python2.7 /usr/bin/python   Now typing 'python' on the console opens python2.7 but somehow mod_wsgi still picks up python2.6  Next i tried,  PATH=/usr/local/bin:$PATH export PATH   then do a quick restart apache, but yet again its python2.6 !!    Here is my $PATH  /usr/local/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games   contents of /etc/apache2/sites-available/wsgisite  WSGIPythonHome /usr/local   &lt;VirtualHost *:80&gt;      ServerName wsgitest.local     DocumentRoot /home/wwwhost/pydocs/wsgi     &lt;Directory /home/wwwhost/pydocs/wsgi&gt;         Order allow,deny         Allow from all     &lt;/Directory&gt;      WSGIScriptAlias / /home/wwwhost/pydocs/wsgi/app.wsgi  &lt;/VirtualHost&gt;   app.wsgi  import sys  def application(environ, start_response):     status = '200 OK'     output = sys.version      response_headers = [('Content-type', 'text/plain'),                         ('Content-Length', str(len(output)))]     start_response(status, response_headers)      return [output]   Apache error.log  'import site' failed; use -v for traceback [Sun Jun 19 00:27:21 2011] [info] mod_wsgi (pid=23235): Initializing Python. [Sun Jun 19 00:27:21 2011] [notice] Apache/2.2.14 (Ubuntu) mod_wsgi/2.8 Python/2.6.5 configured -- resuming normal operations [Sun Jun 19 00:27:21 2011] [info] Server built: Nov 18 2010 21:20:56 [Sun Jun 19 00:27:21 2011] [info] mod_wsgi (pid=23238): Attach interpreter ''. [Sun Jun 19 00:27:21 2011] [info] mod_wsgi (pid=23239): Attach interpreter ''. [Sun Jun 19 00:27:31 2011] [info] mod_wsgi (pid=23238): Create interpreter 'wsgitest.local|'. [Sun Jun 19 00:27:31 2011] [info] [client 192.168.1.205] mod_wsgi (pid=23238, process='', application='wsgitest.local|'): Loading WSGI script '/home/wwwhost/pydocs/$ [Sun Jun 19 00:27:50 2011] [info] mod_wsgi (pid=23239): Create interpreter 'wsgitest.local|'.   Has anybody ever managed to make mod_wsgi run on a non-system default version of python ? I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.  During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voting corrected".     What does this mean, and what caused it?  Did I do something wrong? Why did I lose all of that reputation? Is the system punishing me for leaving too many good answers?  Is there any way to earn it back? Bash Brace Expansion & Variables I have a find command:  find Directory/{Alpha,Bravo,Charlie} arg1 arg2   I want to replace Alpha,Bravo,Charlie with $find_dir  find Directory/{$find_dir} arg1 arg2   however the latter expands to   find Directory/{Alpha,Bravo,Charlie} arg1 arg2   rather then   find Directory/Alpha Directory/Bravo Directory/Charlie arg1 arg2   Why? It's part of a fairly complex bash script that may have more or less directories, not all which are relevant (so globbing Upload/* would not work). So if I setup three now and add another, I'll have to manually add it in later. Plus I need it run from the root of directories to keep finds outputs perspective (./Upload/Dir/file as opposed to ./Dir/file).   But using a variable would permit me to change that as needed and keep it relevant to other parts of the script. I've been using the Linksys USB6300 Adapter but when I switched to Ubuntu it stopped working since its only compatible with Windows. I've tried using ndiswrapper and although it says my windows driver is installed, my device isn't showing up. Anyone know how I could get this wireless network adapter working? Thanks.  lsusb  Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 012: ID 05c6:6766 Qualcomm, Inc.  Bus 001 Device 006: ID 0bda:0153 Realtek Semiconductor Corp. Mass Storage Device Bus 001 Device 005: ID 05e3:0608 Genesys Logic, Inc. USB-2.0 4-Port HUB Bus 001 Device 004: ID 04ca:007d Lite-On Technology Corp.  Bus 001 Device 003: ID 04ca:008a Lite-On Technology Corp.  Bus 001 Device 010: ID 13b1:003f Linksys  Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Why must a computer have a BIOS? The  software is built into the PC, and is the first code run by a   PC when powered on ('boot firmware'). When the PC starts up, the first   job for the BIOS is the power-on self-test, which initializes and   identifies system devices such as the CPU, RAM, video display card,   keyboard and mouse, hard disk drive, optical disc drive and other   hardware. The BIOS then locates boot loader software held on a   peripheral device (designated as a 'boot device'), such as a hard disk   or a CD/DVD, and loads and executes that software, giving it control   of the PC.[2] This process is known as booting, or booting up, which   is short for bootstrapping.   Which requisites determine that a computer system must have a BIOS installed on it?  Why can't the Operational System actually do all of the BIOS work? Easiest way to add a text to the beginning of another text file in Command Line (Windows) What is the easiest way to add a text to the beginning of another text file in Command Line (Windows)? How to remove wine completely How can I remove the wine folder and all the applications in that folder?  I've tried via synaptic but it keeps showing up in the application menu. English (other than American English) has a clear differentiation between the two words. Both are about translocating something. In "bring" the something of somebody is moved to where the speaker is currently situated. "Take" is used to translocate something or somebody to a place that the speaker is not currently at.   You cannot “bring” your books to school if you are currently at home. You can only take them to school. Most of the time one can get the meaning from the context of the sentence but it can get very confusing when the other party is on the telephone and you do not know their location.   Why does American English not differentiate and when did it lose the differentiation? Algebraic closure of $\mathbb{C}(x)$ is isomorphic to $\mathbb{C}$ Let $x$ be transcendental over $\mathbb{C}$. Let $K$ be the algebraic closure of $\mathbb{C}(x)$. How to show that $K$ is isomorphic to $\mathbb{C}$? How to disable notifications for a single mailbox in Mail? When managing multiple mailboxes, is it possible to disable the notifications for certain mailboxes? UITableViewCell content overlaps delete button when in editing mode in iOS7 Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials.  We have that $R[[x]]$ is a ring containing $R[x]$ as a subring.  How to prove that if $a_0\in R$ is a unit, then  $\displaystyle\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$? How to change the size of desktop icons? I tried gconf-editor as given at  .  I am not getting the option for changing size at app &gt; nautilus &gt; icon_view.  I am using Ubuntu 11.10. Kindly help me in changing default icon size.   I want to make them small. Symbolic coordinates for a hyperbolic grid? Rephrasing     (one year later)    (original question is below)   Apparently the original question wasn't clear, or nobody knows an answer (or both).  So I will try to rephrase it.   Look at your favorite .  This question asks for a labeling system for the points.  Of course we could draw the grid on the Poincare disk, and use real numbers (the (x,y) coordinates of each point), but in practice that would raise all sorts of precision issues that could surely be avoided if one had a proper symbolic coordinate system.  You shouldn't need real numbers if you're staying on grid points!  Just like (2,3)+(0,1) is trivial to do by hand (or in your head) for the square grid, we'd like some form of symbolic coordinates (i.e. consisting of finite sequences of symbols) for a hyperbolic grid, in which simple operations on small grid vectors are similarly easy.  If you take some sequence of left and right turns in a hyperbolic grid city, what should you keep track of in your head so you can take the optimal route back to your starting point?  In a Euclidean square grid, it is {(int)X, (int)Y, (enum)direction you are facing}, and we all know how to use this.  Is there any hyperbolic grid where this question has a nice answer?    Original Question  If we consider the Euclidean plane, then a square grid works well with using two integers (the coordinates) as a representation for vectors that lie on the grid.  Does anybody know of a similarly nice representation for working in the hyperbolic plane?  Desirable features of any such system:  $-$ It should be easy to add two vectors.  $-$ It should be easy to see if two vectors are the same.  $-$ It should be easy to see if two vectors are close.  $-$ It should be easy to rotate a vector by an angle that is a symmetry of the grid.  $-$ Every point in the plane should be near some vector. (The grid shouldn't have large empty regions.)   Note that "addition" using continuous coordinates would most naturally be defined in terms of parallel transport, and it is neither commutative nor associative.  On a grid, however, the issue of orientation needs to be addressed differently, since parallel transport along arbitrary grid vectors will introduce rotations that are not a symmetry of the grid, so to remain on the grid, addition will need to be defined using something other than parallel transport.  Since hyperbolic grids have an exponential number of points (as a function of distance from the origin), our symbolic coordinates will need to have a length proportional to the length of the vector.  Don Hatch has a web page which gives some .  ("Easy" means an efficient algorithm that is doable by hand, like how the algorithms for addition and subtraction using the standard digit-sequence representation of integers can be used for the Euclidean square grid. Using hyperbolic functions of real numbers, always keeping track of enough digits to identify the nearest grid point, does not count as easy!)    Addendum: Many readers seem to be disturbed by a little voice saying "But vectors don't make sense in the hyperbolic plane!": What that voice is really saying is that the hyperbolic plane doesn't give you all the nice properties you are used to in Euclidean space or in vector spaces.  So if you define vectors as objects having specific properties you are used to (such as having a direction that is unaffected by parallel transport), then that definition may not correspond to anything in the hyperbolic world.  You could use the term isometry, but that is typically a fixed mapping from the space to itself, and in the hyperbolic plane the isometry corresponding to a vector would depend on the starting point. (However, isometries will be central to any solution, I assume.) And a geodesic is a fixed path within the space, no different from a hyperbolic line or segment.  The paper  offers one solution.  offer another.  I deliberately leave them undefined in this question, as I am looking for any workable system. Let $A = \mathbb{Z}$, $B = [−1, \pi]$, $C = (2, 7)$. List all elements of $A \cap (B^c \cap C)$. After working it out on a number line, I got: $\{4, 5, 6\}$.  As it stands, the expression contains the integers that do not belong to the set $B$ that cross into $C$. This would result in $4, 5, 6$. Number $7$ would not be included because of the soft brackets surrounding $C = (2, 7)$. Is this correct? Making leaflet control open by default I'm using leaflet to display a map.  I have a control section that I'd like to be open by default, but I can't find a way to do it through the . Is there a way to do this?