I am aware of why the Axiom of Choice is equivalent to the the statement that every surjection splits. However, I don't see why we don't also need AC to show that every injection splits.  In particular, for any injection $f$, we need to pick some element in the domain of $f$ to which elements in the codomain of $f$ are sent in the case that those elements are not in the image of $f$. This shows moreover that left inverses are not in general unique. How can we do this without AC?    EDIT: My real issue here is in the comparison of the proof of "$f: A \to B$ is a surjection implies $f$ has a right inverse" with the proof "$f: A \to B$ is an injection implies $f$ has a left inverse", and the use of the Axiom of Choice therein.   In particular, one apparently need not invoke AC in the injection proof to select in advance a single element $a_0 \in A$ for which to send $b \in B$ in the case that $b \notin f(A)$. But then, in the same way, in the surjection proof surely one may select for each $b \in B$ a single element $a_b \in f^{-1}[\{b\}]$ which then determines a function $g: B \to A$ such that $fg = \text{id}$, all without AC? We know each $f^{-1}[\{b\}]$ is not empty, so pick an element in $f^{-1}[\{b\}]$ (in the same way we did so in the injection proof by picking $a_0 \in A$) and send $b$ to that element.
Suppose $f:X\rightarrow Y$ is a surjection and you want to show that there exists $g:Y\rightarrow X$ s.t. $f\circ g=\mathrm{id}_Y$. You need the AC to show this.  However, suppose $f$ is a injection and you want show there is $g$ s.t. $g\circ f=\mathrm{id}_X$. Then, according to my textbook, you don't need the AC to show this.  This is counterintuitive to me, because it's like you need a special axiom to claim that an infinite product of big sets is nonempty, while you don't need one to claim that an infinite product of singleton sets is nonempty, which seems smaller than the former.  So why don't you need the AC to show the latter?  EDIT: $X$ should be nonempty.  EDIT 2: I realized (after asking this) that my question mostly concerns whether the AC is needed to say that an infinite product of finite sets is nonempty, and why.
Everytime, I read about blackbody, I always get confused at the point where it is written     Under thermal equilibrium conditions , the blackbody radiation depends only on temperature. ..... . At equilibrium , electromagnetic waves bounce around with the walls of the cavity.   What is this equilibrium all about? Why is it important here?? If there is thermal equilibrium, why does the body radiate?? Please help. I was getting the usual 4-8mbps in 12.04 when copying files via usb sticks. However after upgrading to 12.10, I only get 200-300kbps when I try to perform the same operation.   I have tried different usb sticks (which work flawlessly on Windows 7) and still experience the same issue. I experienced the same issue sometime ago on 11.10 but everything was normal when I upgraded to 12.04 so it might be a kernel issue.  Here's the output of dmseg |tail:      mysterio@mysterio-HP-Pavilion-dv6700-Notebook-PC:~$ dmesg |tail   [  871.023093] sdb: sdb1  [  871.026909] sd 5:0:0:0: [sdb] No Caching mode page present  [  871.026920] sd 5:0:0:0: [sdb] Assuming drive cache: write through  [  871.026927] sd 5:0:0:0: [sdb] Attached SCSI removable disk  [ 1036.226206] ISO 9660 Extensions: Microsoft Joliet Level 3  [ 1036.233561] ISO 9660 Extensions: RRIP_1991A  [ 1086.342973] ISO 9660 Extensions: Microsoft Joliet Level 3  [ 1086.343010] ISO 9660 Extensions: RRIP_1991A  [ 1173.971357] ISO 9660 Extensions: Microsoft Joliet Level 3  [ 1173.971441] ISO 9660 Extensions: RRIP_1991A   I have updated my system with the latest updates. What could be wrong? Geometrical representation of Contravariant and covariant vectors After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, which is contravariant to its basis. However, using the metric tensor, we can map this vector to a one form in the dual space, which acts as a vector space for this example. This one form, or covactor, varies similarly to the basis in our original vector space. However, the covectors have a different basis, which is the dual basis, and are expanded in terms of those. Here's where my confusion arises : Many diagrams over the internet give a geometric picture of the scenario, by claiming that contravariant components of an 'arrow' are found by drawing parallel lines along the axes, and checking where they intersect. Co-variant components could be found by sketching the perpendicular lines on these axes. So, these materials are treating covariant and contravariant as different representations of the same arrow, while I'm inclined to believe they are completely different. However, if we let go of rigour, in exchange for a 'more' geometric understanding, I think we are allowed to do this. The diagrams are usually of the form :  However, even if we assume that contravariant and covariant are different representations of the same object, this particular diagram still seems wrong to me. Particularly the locations of $x_1,x_2$. In this diagram, they are located on the span of the original basis. Shouldn't covector components be located on the span of the dual basis ? This diagram suggests to me, that co-variant components have the same basis, as the contravariant ones. Shouldn't the diagram look more like this ?  This second diagram seems to fit the concept better according to me. This is because the dual vector must be 'contravariant' with respect to the 'dual basis'. This means, their components must be found out by sketching parallel lines along the span of the dual basis. These lines intersect the original axis at a right angle, which is to be expected, as dual bases are orthogonal to the original bases. Moreover, this second picture can also show in a much better way, how scaling up of the original basis, scales down the 'contravariant components' and the 'dual basis', which in turn scales up the 'covariant' components.This shows that the components of the dual vector are covariant to the original basis. This is something that is not readily visible in the first diagram. So, am I correct in assuming this second 'geometrical' representation is correct ? I know this doesn't make much sense, because as the mathematicians said, that vectors are completely different from one forms, and one forms should be represented using the number of hyperplanes intercepted by our arrow. However, I've seen most course material refer to it in this way, and frankly it is easier to visualize. However, most of this material used the first picture. Can anyone point out my mistakes, if any, and tell me if the first picture is correct, or the second one ? Or in this case 'less-flawed'. Can a Raspberry Pi run Ubuntu? I was thinking about buying a , and running Ubuntu on it. I have looked around the web, and a lot of stuff says that it's difficult for it to work on ARM processors. Has anyone figured out how to get a Raspberry pi to run Ubuntu? SQlite: select into? Opening PDF generated by Python AddIn using Report (*.rlf) file automatcally? As a test I have:   Authored a test.mxd with one layer called Localities in a single data frame called Layers Authored a test.rlf using View | Reports > Create Report of ArcGIS 10.2.2 for Desktop Created a Python AddIn which creates the expected PDF file from the Report (*.rlf).   This is the code which all works to create the PDF report from the Localities features within the current extent:  import arcpy  import pythonaddins  class ReportForExtent(object):     def __init__(self):         self.enabled = True         self.checked = False      def onClick(self):         mxd = arcpy.mapping.MapDocument("CURRENT")         df = arcpy.mapping.ListDataFrames(mxd,"Layers")[0]         lyr = arcpy.mapping.ListLayers(mxd,"Localities",df)[0]         arcpy.mapping.ExportReport(lyr,r"C:\temp\test.rlf",r"C:\temp\test.pdf","EXTENT",extent=df.extent)         del mxd   Is there a way to have the Python AddIn not just create the PDF file but also open it in Adobe Acrobat Reader? Am I going to have trouble flying without my full name on the ticket? How do I use an NBT editor to modify item properties? In Minecraft, the NBT (Named Binary Tag) format is used to store data of our Minecraft worlds, including block, entity, and item properties. In the Java Edition of Minecraft, you can modify this data by including the data in the command, like so: /give @s diamond_sword{Enchantments:[{id:&quot;minecraft:knockback&quot;,lvl:10s}]}  In the Bedrock Edition, on the other hand, NBT is inaccessible from commands. In order to modify NBT, we need to use an external editor. What are some useful item properties that I can modify with an NBT editor, and how can I do so?  Related: Work done changes between reference frames? (This is not homework; a friend shared with me this puzzler and neither of us can figure it out.) Suppose you are in a plane traveling at velocity $v_1$ relative to the ground. The flight attendent pushes a cart of mass, say, $m$, accelerating it from rest to $v_2$, relative to the plane (so relative to the ground, its velocity goes from $v_1$ to $v_1+v_2$). From your perspective, the work done is then $$W = \Delta E = \dfrac{1}{2}mv_2^2$$ but from the perspective of the ground, the work done is $$W = \Delta E = \dfrac{1}{2}m(v_1+v_2)^2 - \dfrac{1}{2}mv_1^2 = \dfrac{1}{2}mv_2^2 + mv_1v_2.$$ Why are these quantities unequal?  I've seen some similar problems, like a body colliding with a wall in different reference frames, but I don't have the physics experience to tell if that question is analogous to this one. So, my apologies if this is a duplicate. :) How do I print an IFrame from javascript in Safari/Chrome Can someone please help me out with printing the contents of an IFrame via a javascript call in Safari/Chrome.  This works in firefox:  $('#' + id)[0].focus(); $('#' + id)[0].contentWindow.print();   this works in IE:  window.frames[id].focus(); window.frames[id].print();   But I can't get anything to work in Safari/Chrome.  Thanks  Andrew Iterating each character in a string using Python How can I change xfce bottom panel to the top? I am using Xubuntu 12.10 and would like to change the panel which is currently on the bootm to the top.  This is the opposite of what most people using Unity want to do.    I have done this in the past but have forgotten how and could not find the info easily this time. Cheers, Is it possible to build a .dmg file (for distributing apps) from a non-Mac platform? And if yes, how? Is there a term for this word play where a song intentionally avoids completing a rhyme? Take this for example:  In this , at 1:04 to 1:10, the person goes on rapping with a rhyming word at the end of each line, but he pauses before the end of the last word and, being humans, we predict the next word (due to the hint from the rhyme) that he is going to say but he avoids it intentionally. Is there a word or phrase for this word play?     ...and the beat still knocks when I sort my socks   I'm five foot eleven of sex   from the tip of my head to my gorgeous...   knees    Not the best example I could find but I think it sums up my point quite well. These days, the standard way to present differential calculus is by introducing the Cauchy-Weierstrass definition of the limit. One then defines the derivative as a limit, proves results like the Leibniz and chain rules, and uses this machinery to differentiate some simple functions such as polynomials. The purpose of my question is to see what creative alternatives people can describe to this approach. The nature of the question is that there is not going to be a single best answer. I have several methods that I've collected which I'll put in as answers to my own question.  It's not reasonable to expect answers to include an entire introductory textbook treatment of differentiation, nor would anyone want to read answers that were that lengthy. A sketch is fine. Lack of rigor is fine. Well known notation and terminology can be assumed. It would be nice to develop things to the point where one can differentiate a polynomial, since that would help to illustrate how your method works and demonstrate that it's usable. For this purpose, it suffices to prove that if $n&gt;0$ is an integer, the derivative of $x^n$ equals $0$ at $0$ and equals $n$ at $1$; the result at other nonzero values of $x$ follows by scaling. Doing this for $n=2$ is fine if the generalization to $n&gt;2$ is obvious. How to extend my root (/) partition? Till today I was using ubuntu 13.04 in my windows7/ubuntu dual boot. Just now I have upgraded to ubuntu 14.04. My software updater showed an updates available message. I proceeded with  Install Now . Soon I got the following window.    Then I tried sudo apt-get clean as recommended. Nothing happened! Not even a window opened. Teach me deleting the temporary packages.   Deleting temporary files may not be highly helpful in case of further updates and installs for I don't have too much space left in my /. I wanna extend my / partition with the help of the unallocated space in my HD. The following is my Gparted window.       I got an answer . But it wasn't useful for me. I had two choices. But the answer wasn't working for both my chioces.  Choice 1: I may use the 4.25 GiB unallocated space to extend my /dev/sda10. To do so, I must need to move my linux-swap partition. But my Resize/Move option is disabled for the linux-swap partition. Choice 1 failed!   Choice 2: I may use the 14.35 GiB unallocated space instead. In this case I must move more than one partition. /dev/sda4  partition is the first on the list. Even that partition has the same problem. Resize/Move option is disabled. Choice 2 failed!  I am out of choices now.  What shall I do? Help me to extend my / with Gparted. zsh I/O redirection: redirecting and then piping; what is going on? How to setup Squid Proxy (for localhost services and Tor and I2P) How does one manually restart graphics drivers? I'm suffering from the dreaded . The problem is intermittent and can be "solved" by putting the computer to sleep or restarting it. I hope simply restarting the graphics drivers will remove the symptoms of the problem.  would seem to indicate that restarting the graphics drivers is possible in Windows 7. My question is: how? If $-1&lt;x&lt;1$ show that $\lim\limits_{n \to \infty}{n\,x^n} = 0$. I don't have idea. I only prove that $n\,x^n$ is decreasing.