Kernel of a map $\mathbb{C}[x,y,z] \to \mathbb{C}[t]$
Kernel of a substitution map
Two homologous but not homotopic loops on a closed surface of genus greater than one Suppose $S$ is a closed surface with genus greater than $1$. Give an example of two loops on $S$ which are not homotopic, but are homologous to each other.   I need help solving this question. I have read the Wikipedia page on it and I realize that it does not contain the necessary knowledge to visualize this problem.  And I was also wondering the relation between the fundamental group $\pi_1(S)$ and the first homology group $H_1(S)$? Some restrictions is that $S$ is a closed surface and the genus is greater than $1$. Mount error: "unknown filesystem type 'exfat'" When trying to mount an exfat filesystem, I get the following error:  Error mounting /dev/sda6 at /media/gkp/Backup: Command-line  `mount -t "exfat" -o "uhelper=udisks2,nodev,nosuid" "/dev/sda6" "/media/gkp/Backup"'  exited with non-zero exit status 32: mount: unknown filesystem type 'exfat'   Exfat is used on some USB sticks and camera sd cards. What can I do to mount this type of filesystem? In Windows 8, when Alt+Tab, the desktop would appear as an option to switch to at the end of the list of open programs (thus Alt-Shift-Tab helpfully took you right to the desktop).  In Windows 10 that desktop option is gone. Is there any way to get it back?  I know about Windows+D but Alt+Tab is just more convenient in my opinion. And  also has no desktop and is uglier. generic function with a "has property X" constraint? How do I give myself an item that can only be placed on a block with a certain damage value? I'm currently in the process of making a custom map and I want to add blocks that can only be placed on a certain colour clay.  This is the command I tried using.  /give @p cobblestone 1 0 {CanPlaceOn:["hardened_stained_clay"],Damage:3}   However, even with this command the cobblestone can be placed on any colour of stained clay. What command can I use for placing only on a certain block? Using adjustbox along with table footnote Shortcut or button for copying posted code from Stack Overflow I might just be lazy, but do you ever find it tiresome to select all the source code from a large post on Stack&nbsp;Overflow to copy it?  I would like a feature, possibly a Firefox extension, but preferably a JavaScript script, for automatically copying source examples to the clipboard. It's no big deal for a small snippet, but a 1000+ line posting is cumbersome to copy.  Do you guys have a better way of copying large source posts? I'm using Firefox 3.5.5 on Windows&nbsp;7. Windows is unable to detect disk partitions I installed Windows 8.1 Preview using the DVD yesterday after wiping off my HDD (I made a backup prior to that). While installing Windows I created only the partitions necessary to install Windows (System Reserved, C: shown in second screenshot). Further I installed Ubuntu 13.04. During Ubuntu installation I created necessary extended partitions (for OS, swap and home directory - all ext4 except swap). Later within Ubuntu I created a NTFS partition at the end (partition 4 in first screenshot). This was a primary partition. The objective behind this was that the partition must be accessible from Windows.  While Ubuntu disk utility reports the partitioning just fine (as below)  Windows disk management is showing all partitions as primary.   Thus within Windows I am able to access only C:! What puzzles me is one can't create so many primary partitions! How can I solve this issue? Where to put my code: plugin or functions.php? How to find hydrogen wave-functions? I have found the hydrogen wave functions and would now like to calculate the function that describes the orbitals so that I can plot this function and see how they look.  I don't know how I can do that and it is crazy how I cannot find anything on the subject on the internet as if there was no relation between the wave functions and the orbitals.  Could you please tell me what I need to do to get a function $r(\theta,\phi)$ out of my wave functions $\psi(r,\theta,\phi)$. I am trying to perform centrality analysis with sDNA with ArcGIS Pro (ver. 2.7.1). However, I have failed to run the Preparation &amp; Integral Analysis in sDNA with the same reason. What is the problem and What should I do? My code:  a = '2.3'   I wanted to display a as a floating point value.  Since a is a string, I tried:  float(a)   The result I got was :  2.2999999999999998   I want a solution for this problem. Please, kindly help me.  I was following . What are the limitations of a typical kit lens as a general purpose lens? I have a Canon EOS 550d with the 18-55mm IS kit lens. I have done  (portraits, nature, night shots etc), and find the image quality to be pretty good. In fact most of the reviews have also been positive (at least in terms of image quality):   Quote:     indeed Canon's main concern may   ultimately become whether users have   as much incentive to upgrade to more   expensive optics as they did before       So, what are the key limitations of this lens (or any other similar kit lens) that would compel one to upgrade to a more expensive general purpose zoom lens (apart from the wanting to upgrade the zoom range)? Forgive the possibly naive question, but I'm returning to a basic undergrad-level study of chaotic dynamical systems and can't quite seem to reconcile the notion that in order for a dynamical system to be considered chaotic, it must have positive global Lyapunov exponents. Such exponents, of course, drive the divergence of trajectories. However, topological transitivity implies that trajectories can get arbitrarily close as well over some finite time. It seems that positive Lyapunov exponents would disallow this mixing. Is there a point at which these positive exponents no longer describe the system, and the mixing behavior begins? The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ What is this element on the far right of schematic described as "270 OHM"? Is this some kind of inductor? What is its purpose in this device (this is a reference design of dc-dc power supply of sim900 taken from its datasheet) and what would be an exemplary part number? What's the best way to extend the User model (bundled with Django's authentication app) with custom fields? I would also possibly like to use the email as the username (for authentication purposes).  I've already seen a   to do it, but can't decide on which one is the best. Python gives us the ability to create 'private' methods and variables within a class by prepending double underscores to the name, like this: __myPrivateMethod(). How, then, can one explain this  >>> class MyClass: ...     def myPublicMethod(self): ...             print 'public method' ...     def __myPrivateMethod(self): ...             print 'this is private!!' ...  >>> obj = MyClass() >>> obj.myPublicMethod() public method >>> obj.__myPrivateMethod() Traceback (most recent call last):   File "", line 1, in  AttributeError: MyClass instance has no attribute '__myPrivateMethod' >>> dir(obj) ['_MyClass__myPrivateMethod', '__doc__', '__module__', 'myPublicMethod'] >>> obj._MyClass__myPrivateMethod() this is private!!   What's the deal?!  I'll explain this a little for those who didn't quite get that.  >>> class MyClass: ...     def myPublicMethod(self): ...             print 'public method' ...     def __myPrivateMethod(self): ...             print 'this is private!!' ...  >>> obj = MyClass()   What I did there is create a class with a public method and a private method and instantiate it.  Next, I call its public method.  >>> obj.myPublicMethod() public method   Next, I try and call its private method.  >>> obj.__myPrivateMethod() Traceback (most recent call last):   File "", line 1, in  AttributeError: MyClass instance has no attribute '__myPrivateMethod'   Everything looks good here; we're unable to call it. It is, in fact, 'private'. Well, actually it isn't. Running dir() on the object reveals a new magical method that python creates magically for all of your 'private' methods.  >>> dir(obj) ['_MyClass__myPrivateMethod', '__doc__', '__module__', 'myPublicMethod']   This new method's name is always an underscore, followed by the class name, followed by the method name.  >>> obj._MyClass__myPrivateMethod() this is private!!   So much for encapsulation, eh?  In any case, I'd always heard Python doesn't support encapsulation, so why even try? What gives? A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix Numlock turned on on boot-up I have a desktop running ubuntu 10.04  I want numlock to be turned on initially.  What do I have to change to accomplish this?  ~~edit~~  I checked in my BIOS.  It was already set up to have the numlock turned on initially.  Yet it isn't being turned on initially.  I have a USB keyboard: would that break this functionality?   Or, should I ask elsewhere (superuser, perhaps) about how to get that BOIS setting working properly?