Let X be a compact Riemann surface of genus $g$ corresponding to the equation $w^3 = P(z)$, P is a polynomial without multiple roots and $degP = 8$ It's known that there are $g$ holomorphic differentials on this surface. How can I construct them explicitly?  If my curve was defined as hyperelliptic one, $w^2=P(z)$, then I would use $\frac{z^k \cdot dz}{w}$ for $k = 0, \cdots ,g-1$. $\frac{z^k dz}{w} = \frac{2z^k dw}{P'(z)}$, we would check that the form is holomorphic at $z \to \infty$ and $\omega \to 0$. Mention here that $deg P = 2g+2$. (I don't understand to the full how we do this). Then $\frac{z^k dz}{w}$ is well defined when $z$ is a coordinate. and $\frac{2z^k dw}{P'(z)}$ is well defined when $w$ is a cordinate.  By implicit function theorem $w$ is a coordinate everywhere where $P'(z) \neq 0$ so the second representation works and we get over the case of $w=0$.  $z$ is a coordinate everywhere where $2w \neq 0$. so the first representation works and when $z \to \infty$ then we have a double pole from $dz$ and $g+1$-ble pole from $\frac{1}{w}$  Our curve equation means that $dw = \frac{P'(z)}{3w^2} dz$ so if we try the same way it is done for hyperelliptic curve, then we should seek the answer as $\frac{z^k dz}{w^2}= \frac{3z^k dw}{P'(z)}$.   How can we check the holomorphoucity of forms more rigorously and how can I construct the forms in the case $w^3 = P(z)$
Let us have a surface $X$ which satisfy the equation $\omega^3 = (z-1)(z-2)\cdots (z-8)$ It's stated in the task, that there are exactly $g$ holomorphic differentials on the curve. How can I find $g$ and construct them?  The genus $g$ of this curve (I tried to use Riemann-Hurwitz theorem)  The map $X(z,\omega) \to CP^1(\omega)$ has degree 3. and we have 8 branching points with 3 leaves at each. $\chi(Domain) = DegreeOfMap * \chi(Target) - \sum_{points}(BranchingLeaves - 1) $ $2-2g = 3 *(2) - 8*2 $ hence $ 12 = 2g$ and  $g = 6$ I know that in case of hyperelliptic curve there are $g-2$ (If i'm not mistaken) holomorphic differetials and they are of sort $\frac{z^k dz}{w}$. But the form of the curve equation was heavily used. UPD: there is a branhing point at $\infty$, because if we go around all points $ 1 \cdots 8$ at once, we get that cubic root of the right side is multiplied by $\exp{4 \pi /3}$ $2-2g = 3 *(2) - 9*2 $ hence $ 12 = 2g$ and  $g = 7$ (I hope that this is true)
Formatting a float to ###.## (two decimals) Newbe Ubuntu installer - Ubuntu 14.04.3 LTS 64bit - Windows 10 Having some issues installing Ubuntu 14.04.3 LTS 64bit next to Windows 10.  This is what I did:  1) Created a bootable usb with ubuntu on it using Universal-USB-Installer-1.9.6.3  2) Changed the boot mode  3) Boot from USB - It showed the option to test drive or install - I chose install  4) Choose Language  5) Setup Wireless - I didn't choose anything here  6) Then it showed me the preparing to install  7) After that a message popped up saying:      Unmount partitions that are in use The installer has detected that the following discs have mounted partitions /dev/scd Do you want the   installer to try and unmount these partitions before continuing? If   you leave them mounted, you will not be able to create, delete, resize   partitions on these discs, but you may be able to install on an   existing partition there.   So obviously I said yes.  8) Installation type - I chose something else - This looked nothing like the screen shots I saw on the installation guide.    9) Then it only gave me the option to choose my bootable usb for the installation. With a bunch of options to choose, so I got scared and quit.     Can any one please help?  Keep in my mind I do not know a lot, so please dumb it down. Why can't I choose to run it along side Windows 10? Why doesn't the installation pick up my hard drives?  If you need any other info, please ask. But I think I put everything in here. When I'm on my mod site I have my diamond dropdown:    But on Meta.SE and other sites, I have the StackBurger:    This doesn't show on my mod site, so I can't switch between sites easily. How do I prompt for Yes/No/Cancel input in a Linux shell script? Lets say I have something like:   \[ Let x = number of cats \]   (This is a contrived example, but pretend we are actually inside a long align environment block...) "Let" and "number of cats" need to be written like normal text, while x= needs to be math type text.  I've seen this:  \[ \mbox{Let } x = \mbox{ number of cats} \]   and this:  \[ \textrm{Let } x = \textrm{ number of cats} \]   They seem to produce equivalent output, but I'm not sure which is the "correct" one. Also, they are not ideal in that the extra space needs to be placed inside the text mode block, otherwise it is ignored as math mode and butts the text right up against the math.  Ideally there would be some command which allowed embedding text into an equation, which correctly inserted leading and trailing space around the block as required without adding space inside the text block.  How does one do that "correctly"? Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$ How to show $u \in R^{x} \implies u \in R[X]^{x}$ for integral domain $(R,+, \times)$ Passing data between two controllers using storyboards I recently upgraded a c# windows service to run as a 64 bit .net process. Normally, this would be trivial, but the system makes use of a 32-bit DLL written in C++. It is not an option to convert this DLL to 64 bit, so I wrapped the DLL in a separate 32 bit .net process and exposed a .net interface via remoting.  This is quite a reliable solution, but I would prefer to run the system as a single process. Is there any way I can load my 32 bit DLL into a 64 bit process and access it directly (perhaps through some sort of thunking layer)? How do I detect a click outside an element? I have some HTML menus, which I show completely when a user clicks on the head of these menus. I would like to hide these elements when the user clicks outside the menus' area.  Is something like this possible with jQuery?  $("#menuscontainer").clickOutsideThisElement(function() {     // Hide the menus }); What does "...from nonmagical attacks not made with silvered weapons" mean? I'm sure your first thought is, that's perfectly clear.  Unfortunately my group who aren't normally rules lawyers disagree.  Party encounters a  (MM p.211).  It has:     Damage Immunities: bludgeoning, piercing, and slashing from nonmagical   attacks not made with silvered weapons.   My reading of that is if the weapon has to be magical and silvered to hurt it.  So a +1 silver dagger will hurt it, but a +1 regular dagger or a +0 silver dagger won't.  The group's interpretation was that all three of those weapons would hurt and only a +0 regular dagger would be ineffective.    I would also note my interpretation would mean something like blade barrier would also not harm it since you could argue they're magical blades but nothing indicates that they're silvered.    This also appears to be the same wording for some undead so I'm not looking for a lycanthrope specific interpretation but something general for every time that wording is used.    Thanks in advance. How can I get the MAC and the IP address of a connected client in PHP? In Java, why do people prepend fields with `this`? When referencing class variables, why do people prepend it with this? I'm not talking about the case when this is used to disambiguate from method parameters, but rather when it seems unnecessary.  Example:  public class Person {             private String name;      public String toString() {         return this.name;     } }   In toString, why not just reference name as name?  return name;   What does this.name buy?   a stackoverflow question whose code has this pre-pending. Does the universe have an edge/boundary/barrier? If yes, what is at the edge? My question related kind of to asking what the shape of the universe is. Say a hypothetical alien civilization built an faster-than-light spaceship. If they keep flying would they end up where they began? ie space itself is curved...or would they hit some invisible barrier? Brief Description:  I need my Ethernet card (en0) to see all traffic under 192.168.2.xxx.  However I also need to use WiFi card (en1) for all other traffic.  OS: OS X Lion    I am using a separate network location to deal with this specific use (since this is a for a robotics project that communicates via Ethernet). Ethernet (en0) had to be set up as a static IP (192.168.2.10) per the requirements for the device I am connecting to. All connections on the Ethernet card (en0) should be 192.168.2.xxx as specified above. WiFi (en1) is set up as DHCP to a router since their is no point in assigning it as a static for general purpose traffic. Ethernet (en0) is set up as priority over WiFi (en1).  I would also like to keep this under the network location preferences if at all possible since I do not require this feature to be on all the time.  I would really appreciate your help on this. No one I have talked to knows how to solve this. Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from. How to change horizontal alignment of \frac? Typing \frac{W+W^*}{2} gives a fraction in which the denominator appears to the eye to be in the wrong position. Is there a way to correct this, i.e. place the 2 in under the + (without using arbitrary negative spacing)? Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? Performance difference between dot notation versus method call in Objective-C You can use a standard dot notation or a method call in Objective-C to access a property of an object in Objective-C.  myObject.property = YES;   or  [myObject setProperty:YES];   Is there a difference in performance (in terms of accessing the property)? Is it just a matter of preference in terms of coding style? I have a MacBook Pro with Mini DisplayPort and I want to use target display mode to use a Thunderbolt iMac as an external display. If this works, would I need a Mini DisplayPort cable or a Thunderbolt cable?