I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$  $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$  $ z_{0}^{d}+z_{1}^{2}+...+z_{n}^{2}=0$ and $\vert z_{0}\vert^{2}+\vert z_{1}\vert^{2}+...+\vert z_{n}\vert^{2}=2\rbrace$  The second cuestion I don't know what it means.
I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold.  How give the charts. For $n=1$ and $d=2$ not distinguish which 1- manifdold is?
How to include PDF pages without a newpage before the first page? Probability question with interarrival times (Why did the chicken cross the road?) A chicken wants to cross a single-lane road where the cars arrive according to a Poisson process with rate $\lambda$. She needs at least $k$ minutes to cross the road safely, so she will have to wait until she sees a gap of at least $k$ between the oncoming cars. If the gap between the car that just arrived and the next one is at least $k$ then she starts crossing the road immediately. Let $T$ denote the random time she needs to wait by the road.  Find the expected time needed to cross the road. Pygame: Is there any easy way to find the letter/number of ANY alphanumeric pressed? I always know that the natural logarithm $\log_e$ is the inverse of the exponential function $e$, but to my surprise when reading elementary functions in complex analysis, I discover that that is not always true. $$\log(e^z)=z+2k\pi i$$  My question here is that, why is the above expression true? Creating mask file from shapefile using ArcGIS for Desktop? I have a shapefile of Lakes over my study area. Each polygon has attribute information of each lake, name, shape etc. Actually I want to eliminate all lakes from analysis. In other word, I want lakes to have 0 value and the area that out of boundary of lake to have their orginal pixel values. when I do subset using this shapefile, the output is not I want because my shapefile deoesn't have any value outer boundary of lake.   How can I build a proper mask file (inner part of lake has 0 value, outer is 1) using this shapefile in ArcGIS Desktop? Today after I installed Ubuntu 16.04 via USB on my pre-installed Windows 8 machine, I was required to restart my computer. After doing so I wasn't offered to select an OS, so after searching the internet, I came across Boot-Repair which I ran using the Try Ubuntu option in my Ubuntu installation USB. Following the boot repair I got a message saying,  If your computer directly reboots into Windows, try to change the boot order in your BIOS. If your BIOS does not allow to change the boot order, change the default boot entry of your Windows bootloader. For example, you can boot into Windows then type the following in an admin command prompt: bcdedit /set {bootmgr} path \EFI\ubuntu\shimx64.efi   After I rebooted my PC, I am receiving the following error message for about 3 seconds, after which it boots automatically into Windows 8, without letting me access Ubuntu. Error received while booting after manufacturer's logo, before grub: Failed to open \EFI\BOOT\grubx64.efi - Not Found   Failed to load image \EFI\BOOT\grubx64.efi: Not Found     start_image() returned Not Found    So I tried the other solution of typing that command in the cmd, to which my surprise, worked, and I am now prompted a window (grub) to choose my OS each time I turn on my PC, but unfortunately, I still receive an error each time, that lasts about 3s, after which I can choose my OS (both are working just fine) and work normally. Do you have any suggestions to get rid of the error message each time I boot my PC? Windows 7 finding location of installed program Using mask as a garbage matte to hide an object I'm trying to use a mask to hide a part of a render layer, so that when the render layer is shown over a movie, some parts in the movie will appear to be in front of the render layer.  For example, assume I want to add a box to this image, so that it would look like it's behind the person:    I created the following simple object:    If I render the object, it looks great - the object is solid, everything else is transparent (I didn't use any 'sky'):     Now I create a mask to describe the standing person (or at least the part that would otherwise be covered by the box):    What I want now is to create a "hole" in the rendered box, so that when I use Alpha-over, the box look as if it's behind the person, but I can't find an easy way to do that. My first guess was to use 'Set alpha' to apply the mask (inverted) as an alpha for the rendered layer, but this doesn't look good:    The mask did create a "hole" where it should, but now all the other pixels are not transparent. How can I "retain" the transparent pixels while setting more pixels to be transparent based on the mask (use it as a matte)?  Note: I can do this with (I'm ashamed to say) some math work on the alpha channel, but it just doesn't seem right:    Thanks, and sorry for the [oh so] long question. Counting number of subsets which contain a specific element There is a set $x = \{A,B,C,D,E,F,G\}$ and the question ask for counting the number of subsets which contain $D$. And the answer is $2^6$. why would it be $2^6$? I know the total number of subsets is $2^{|x|}$ is it realated? Incredible frequency of careless mistakes Limit of a particular variety of infinite product/series I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn. So the vending machine tore my $5 in pieces. What now? The vending machine swallowed half of my $5 bill, and the other half remained in my hand. Can I get my $5 back, or is it all lost forever? In the  a line is delivered at around 1:41 where Han Solo says "a 190 years old!? You look great" and I was wondering if there is a defined concept of a year in the Star Wars universe or would Han Solo just be familiar with the duration of a  solar orbit. Send data from Lightning Web Component (LWC) to a Visualforce page I am adding an LWC component in visualforce and need to add a handler for an event fired in the LWC component on the visualforce page. When googling all I can find is examples of how to do this with aura component using application events. LWC controller: handleSearchClick() {              const searchinfo = {          shipper: this.shipper,         system: this.system,         product: this.product,         status: this.template.querySelector('c-lwc-multi-select').value(),         period: this.period       };     this.dispatchEvent(new CustomEvent(         'dosearch',          {             detail: { data:  searchinfo},             bubbles: true,             composed: true,         }     ));     console.log({ 'lwcNominationSearch searchinfo': searchinfo }); }  AURA app: &lt;aura:application extends=&quot;ltng:outApp&quot; access=&quot;GLOBAL&quot;&gt; &lt;aura:dependency resource=&quot;c:lwcNominationSearch&quot; /&gt; &lt;aura:dependency resource=&quot;markup://force:*&quot; type=&quot;EVENT&quot;/&gt; &lt;/aura:application&gt;  Visualforce: &lt;div id=&quot;rcontainer&quot;&gt; &lt;/div&gt;  &lt;script&gt;     $Lightning.use(&quot;c:NotificationSearch_ltngoutApp&quot;, function()     {         $Lightning.createComponent(         &quot;c:lwcNominationSearch&quot;,         {},         &quot;rcontainer&quot;,         function(cmp)         {             console.log('component created');              //$A.eventService.addHandler({ &quot;event&quot;: &quot;c:myEvent&quot;, &quot;handler&quot; : visualForceFunction})               $A.eventService.addHandler({                 event: 'dosearch',                 handler: function (event) {                          console.log(&quot;vf event success&quot;);                 }             });          });     }); &lt;/script&gt; Check if convergent in the end points of the convergence interval I have to find the radius of convergence for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. I think it is $r^{-1}=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{ \frac{n+1}{n+2} }{ \frac{n}{n+1}}=1$ so r=1. Then I have to find out if $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$ is convergent in  the end points of the convergence interval. I think I have to check for $|r|$. But we can't use ratio test (L=1) or integral test(not decreasing) for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}$. How can I then check if it's convergent in the end points of the convergence interval? How to change the name of document elements like "Figure", "Contents", "Bibliography", "Appendix", etc.? How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$? Problem was to find $$\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}.$$ My partial progress i tried to motivate such that upper term in binomial terms gets constant rather than variable , so i thought of changing it to form $\binom{n}{n-r}\binom{n-r}{n-2r}$. Now it's of the form $\binom{n}{m}\binom{m}{k}$. We know it is equivalent to $\binom{n}{k}\binom{n-k}{m-k}$. Doing this we get $\binom{n}{2r}\binom{2r}{r}2^{n-2r}$. Now what I need to do as next step as such still nothing the upper or lower terms r any constant values(sum to ) , all are variables still. Now by one answer by Robpratt i got by snake oil , is there any other way to solve this problem ? I recently made my pen-drive bootable with Ubuntu 14.04. But I need to unboot my pen-drive now. I formatted my pen-drive once,But the size of the pen-drive still shows 2.25 MB. Whereas,this is an 8GB pen-drive. How can I get rid of this problem. I'm still using Windows 7. Can someone please explain this weird Pygame importing convention?