Introduction to proofs. I am not at all familiar with mathematical proof-writing and would like to learn how to create my own proofs. So, I was wondering whether it would be possible for you to recommend me to any book or site that would introduce me to proof-writing.  Additionally, I would like this source to not be of a very high mathematical level and not be about computer science based proofs.  Thank you :)
Introduction to proofs with a fair amount of hand-holding? Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff on his own.  He asked me recently if there was a 'mathematical' version of .  The point being, if there was an "introduction to proofs" book which took a more hand-holding approach. He's also expressed fondness for the .  However, metamath's proofs are obviously a bit too low level to learn anything serious.  At one point in our conversation, I mentioned the dependently typed programming language , but I felt it's not quite as helpful until he at least gets some experience with  (for which the target audience of Agda is Haskell programmers, hence the similarity in syntax).  Anyone have any good suggestions on a book?
In SQL Server, how do I "floor" a DATETIME to the second/minute/hour/day/year?  Let's say that I have a date of 2008-09-17 12:56:53.430, then the output of flooring should be:   Year:  2008-01-01 00:00:00.000 Month: 2008-09-01 00:00:00.000 Day: 2008-09-17 00:00:00.000 Hour: 2008-09-17 12:00:00.000 Minute: 2008-09-17 12:56:00.000 Second: 2008-09-17 12:56:53.000 How can I get an underbrace and an overbrace to partially overlap in an equation? Sometimes I want to describe portions of an equation using \underbrace and \overbrace in such a way that the portions are not completely distinct, for example:          /-------\ x = A + B + C + D + E     \-------/   \---/      What is the best way to do this? I see stuff like command 1&gt; out or with 2&gt;&amp;1 to redirect stderr, but sometimes I also see &amp;&gt; by itself, etc.  What is the best way to understand &amp; and what it means exactly? Just downloaded a .zip file from the internet. I want to use the terminal to unzip the file. What is the correct way to do this? Tourist to F1 change of status within 60 days I have a PhD place at a US university starting Aug 21 but want to visit a different university for casual research for 6 weeks from July 1. This is longer than the 30 day grace period for the F1. Can I arrive on the visa waiver program or tourist visa in July and then just stay on for the PhD? I guess I might need to change status. Will this work?  Please don’t suggest exiting and reentering, that’s too expensive.  (British masters student in UK) If you shine a light away from the center of a Schwarzchild black hole, will it stay still? $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? The tablet functionality stops working after I wake my Lenovo Tablet from sleep mode (closing &amp; opening lit). The error message is "Tablet Serial Wacom Tablet (Lenovo X201) removed" (screenshot: ). In the system settings under input devices it states that "No tablet device is detected". Only a restart helps and the touch screen works again.  This error appeared only after upgrading from Kubuntu 12.04 to 14.04. In 12.04 the same error message appeared after waking from sleep mode, but then another followed that the tablet was activated again and it worked properly. Any ideas how this could be fixed? Run PHPUnit Tests in Certain Order Is there a way to get the tests inside of a TestCase to run in a certain order? For example, I want to separate the life cycle of an object from creation to use to destruction but I need to make sure that the object is set up first before I run the other tests. A continuous function that attains neither its minimum nor its maximum at any open interval is monotone Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq  \mathbb R$ , then how to prove that $f$ is monotone ? Is there a way to access QGIS plugins in Python?   I'd like to write a script to do some geoprocessing, and I'd like to use the clip, regular points and convex hull tools in fTools, as well as the Road graph plugin. The latter is probably a different can of worms because it is a C++ rather than Python plugin. I am a physics Master's student.  I have good grades, with CGPA above 9(/10)(However, all courses in my university are graded on a curve so the actual marks are significantly lower than 90%). This has been my marks throughout my undergraduate and graduate courses and I was happy because this was above average. Also, my summer projects in UG were usually computational and any difficulty I had in theoretical aspects I attributed it to my inexperience in advanced Physics.   However, recently I have started to work on pure theoretical project and while reading for the project I realized that I consistently miss some details from each topic. I understand the general idea, can re-derive most of the equations(with some effort) however when I discuss things with my professor or when I read further topics I realize that I have missed certain key-details. This is exactly similar to the case of me missing a few key questions in my exams and getting that 9 grade point instead of 10(or at-least I think so). I was not worried about this before but now I realize that there is some flaw in the way I study (but I do not know what).   Since I want to apply for Physics PhD I realized that I will need to have solid concepts and it would be best to correct my mistakes but I do not know how or even what could be the reason for my imperfect understanding.  Any direct suggestion would help or even certain general guidelines on what I can do.    Things I've tried recently -   1) I've started to written down almost everything I read. This is slightly laborious but is certainly helping. It also helps me to gloss over the things I read previously when I come back the next day.  2) I completely avoid smartphone when I'm library/study room so that I won't get distracted. Not sure if it helps me solve the above problem but certainly helps me study better in general.  3) I've tried to read the topics from several sources. The books I am reading is somewhat like a collection of research papers and hence does not have a exercises. So I though reading from various sources might help get better ideas but it's not really helping because most other sources are too high level as an introductory text. But this will probably help me in cases where it's practical to do this.  4) Somewhat similar to point (1)- I am trying reread, whenever I can, what I've done until that point. But not sure how much this will help me in filling the gaps. Copy files excluding x,y,z causing error in shell script Change footer in beginning of chapters I want the footer of all pages to be "Page N°X" on the right. I used this:  \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} % clear all header and footer fields \fancyhead[R]{\leftmark} \fancyfoot[R]{\footnotesize Página N$^\circ$ \thepage\ } \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt}   It works on every page but the chapter beginning ones. how to use these hash functions in python? grub2 wont load Ubuntu on dual boot Having an issue with a dual boot setup that is being run from Windows boot manager. I can successfully load into grub2 from Windows boot manager, however it seems that grub2 cannot locate to the Kernel or something, as it is dropping me right into a shell with the following output:  [ Minimal BASH-like line editing is suported. For the First word, TAB list the posible command completion. Anywhere else tab list the posible completions of a device/filename,]  grub&gt;   I have used Easybcd in the past and successfully loaded a Linux grub through Windows bootloader, but this was when both operating systems were on the same partition, but since my OS installationd are now on separate partitions it seems EasyBcd cant work its magic.   ONe encouragement is that I was able to get Easybcd to load a working Grub if I used the Neo grub bootloader and edit the confg with:  title Ubuntu 14.04 find --set-root /boot/vmlinuz-3.19.0-61-generic kernel /boot/vmlinuz-3.19.0-61-generic ro root=/dev/sdc initrd /boot/initrd.img-3.19.0-61-generic   however this loads in Grub4DOS which is very slow, and as of today this method stopped working See  for details).  Here is the output of EasyBcd Settings for all the different methods I have tried for the Ubuntu 14.04 installation:  Default: Windows 7 Timeout: 30 seconds Boot Drive: C:\  Entry #1 Name: Windows 7 BCD ID: {current} Drive: C:\ Bootloader Path: \Windows\system32\winload.exe  Entry #2 Name: Ubuntu 14.04 Legacy BCD ID: {a4f127cf-3150-11e6-8aaf-408d5cb9e442} Drive: C:\ Bootloader Path: \NST\nst_linux.mbr  Entry #3 Name: Ubuntu 14.04 Grub2 BCD ID: {a4f127d0-3150-11e6-8aaf-408d5cb9e442} Drive: C:\ Bootloader Path: \NST\AutoNeoGrub0.mbr  Entry #4 Name: Ubuntu Neo Grub BCD ID: {a4f127d1-3150-11e6-8aaf-408d5cb9e442} Drive: C:\ Bootloader Path: \NST\NeoGrub.mbr   *It seems that none of the paths seen above are pointing to my dev/sdc2/ partition which would be considered DISK1 on my Windows Volume manager.     EDIT - In the confusion of attempting to get a working bootmanager for Ubuntu/Windows, you will see that grub and Windows Boot manager have all been installed in numerous locations. Below are the present locations and contents of all bootmanager &amp; Grub installations:  dev/sdb Windows7 drive  /dev/sdb1 - 512 MB fat32 partition which is currently empty  /dev/sdb2 - 110 GB ntfs partition containing Windows7 installation. This partition contains a 'Boot' folder which contains BCD files and a whole wack of langauage folders. THIS IS The FOLDER THAT WINDOWS BOOTLOADER uses.  /dev/sdb4 - 121.53 ntfs partition containing storage for media    dev/sdd - Ubuntu Drive  dev/sdd1 - 512 MB partition containing a 'EFI' folder, inside of which are two folders 'grub' and 'Ubuntu', both of which contain the exact same files (grub.cfg, grubx64.efi, MokManager.efi, shium64.efi)  dev/sdd2 - 48.83 GB ext4 partition that contains the '/' folder and Ubuntu instalation.   dev/sdd3 - 69.91 GB ntfs partition containing storage for media  what do I need to do so that the Grub shell I am being dropped into will load up Ubuntu? HOw can I get this machine to dual boot out of Grub? Smooth rendering of bitmap fonts I would like to use the Concrete Math font in LaTeX, which is a bitmap font. The rendering with pdflatex is naturally poor-quality with jagged edges.  My solution so far has been to generate a DVI file and then convert to PDF using Linux's dvipdf. This tool effectively traces and smooths the edges of the glyphs and the end result is beautiful. However, using the DVI output format (with latex) causes a host of other problems with including figures, etc. For instance, you can't include .png or .pdf figures when using plain LaTeX.  Is there a way to somehow get the same bitmap font-smoothing effect in pdflatex? or via an intermediate format that still allows you to include at least .png figures? How to fix "error: unknown filesystem. grub rescue> I get this error:  Error: unknown filesystem. grub rescue&gt;   I'm not exactly sure what I did. I believe I installed Ubuntu over Windows on my Acer netbook. I then tried to install , but it wouldn't start installing.  I turned the netbook off and back on. Now I get the error.  I have read a lot of other questions like this, but in my case I cannot boot a CD. If I put a Ubuntu CD or a Linux Mint CD into my external CD/DVD drive and change my BIOS to boot the CD-ROM first, it just gives me the same error screen.  Update copied from comment dated 2012-05-26 02:54:29Z  Here are some results from my commands:   grub rescue&gt;ls (hd0) (hd0,msdos5) (hd0,msdos1)  grub rescue&gt;ls (hd0,msdos5) unknown filesystem  grub rescue&gt;ls (hd0,msdos1) unknown filesystem  grub rescue&gt;ls (hd0) unknown filesystem I have an Ubuntu 14.04 server that has no external monitor connected.  I use NoMachine to remote control the machine.  When I do so, the Unity/Gnome interface doesn't see any monitors connected, so I can only use 800x600 when connecting with NoMachine.  If I plug in a monitor, I can set the resolution to whatever size the NoMachine window is on the remote computer.  Is there a way to create a "fake" monitor device on Ubuntu so I can set a desktop resolution in Unity/Gnome? "It is to be discussed", what is the infinitive doing in this sentence? It is to be discussed.   Is be + infinitive forming the future tense here?      You are to be dressed and ready by 8:00.   I was thinking it's almost commanding (or speaking of a command) but this doesn't seem to be the case, as sentences like this are commanding without this structure:     You will be dressed and ready by 8:00.   or      You are going to be dressed and ready by 8:00.   And there's sentences like this that don't imply commandment in any way:     I am to go home tomorrow!   or     I am to be home tomorrow!   (Slightly different meaning, same general sentiment)  So what's going on with the be + infinitive structure? I am trying to solve the following exercise:     Prove that complex multiplication does not extend to a multiplication   on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division   algebra.   I am aware of Frobenius' theorem that there are only three finite dimensional real associative division algebras. But the exercise would be pointless if the theorem was assumed.   Let $x,y \in \mathbb R^3$. Since $x y$ has to extend the complex multiplication:  $$xy = (x_1y_1-x_2y_2, x_1y_2 + x_2 y_1, ?)$$  I have no idea how I can proceed from here. Could someone tell me how to do this? It should be easy as the other exercises I did so far were also easy.