For $Y_1,Y_2$ closed subsets of $\mathbb{A}_k^n$, $\mathcal{I}(Y_1 \cap Y_2)= \text{rad}(\mathcal{I}(Y_1)+\mathcal{I}(Y_2))$
Proving an inclusion related to algebraic sets and interpreting it
Create a virtual file that is actually a command How long was Luke Skywalker's training with Yoda? From the answer to the question  :     We also don't know how much time Luke spent training.   I tried to check some references like  and  related section on their Wookieepedia articles, but there is not even a clue of the amount of time he spent there.  Is there any canon information on the subject? Or any extrapolation of how long Luke's journey on Dagobah was? Was it a day? Several weeks? Some months? How to convert milliseconds into human readable form? I need to convert an arbitrary amount of milliseconds into Days, Hours, Minutes Second.  For example: 10 Days, 5 hours, 13 minutes, 1 second. The bug displayed in the attached screenshot occurs consistently for me while running the developer beta of iOS 10 using v1.5.2 of the Stack Exchange app on an iPhone 6.  I was able to recreate this by simply tapping on any question or answer in my feed with a long title. This doesn't appear to be a bug with iOS 10 itself since I haven't seen it in other apps, but I could be wrong. The validation tab in your script tool properties allows you to put parameters in categories. The :     Put parameters in different categories. Categories are expandable   window shades with a collection of parameters.     I am using the categories to organize my parameters, so I don't care about the expandable/collapsible feature;  it just adds a lot of time to the user.  This is all within a normal toolbox, not a script toolbox.  v10.1    Does anyone know how to have these categories expanded by default?  Here is my only validation code.  I am only utilizing categories.  The other parts are the default.  def initializeParameters(self):     """Refine the properties of a tool's parameters.  This method is     called when the tool is opened."""     self.params[0].category = "General"     self.params[1].category = "General"     self.params[2].category = "Inudaciones"     self.params[3].category = "Cienegas"     self.params[4].category = "DrenajesPrincipal"     self.params[5].category = "DrenajesSensillos"     self.params[6].category = "Inudaciones"     self.params[7].category = "Cienegas"     self.params[8].category = "DrenajesPrincipal"     self.params[9].category = "DrenajesSensillos"     return In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in this case, we should make sure that $\tau$ is imaginary, right? However, I often see places where people seem to handle $\tau$ as a real number, for example setting $\tau$ to $\pm\infty$. Moreover, I remember in one of my quantum mechanics classes, the professor said the Euclidean time $\tau$ was real. I am so confused and need some good explanations. I've recently started playing Minecraft, and I only have a little experience with enemies (I've been playing with the peaceful setting so I can figure out what the heck I'm doing).  When I did play some survival mode, I happened to be at the top of a pretty large cliff when night hit, and this seemed to protect me - the baddies just piled up at the bottom of the cliff.  However, it seems like this probably only worked due to pathing flukes.  The terrain behind me was pretty easy going, so I suspect if enemies spawned in the right areas, they could still come up and visit.  That said, how can I set up a quick protective shelter for the first night or two?  The days don't seem to last very long, so I don't expect I would be able to generate a lot of high-end materials, tools, etc. to work with.  Also, the zombies caught fire nicely when the sun came up, but there were still some spiders and some other things (creepers? - they were explosive) that didn't go away at daybreak.  Do I need multiple exits for my shelter or some other way to avoid running into all the enemies outside the front door that have accumulated overnight? First off I'll let you know I'm a total noob at blender I've used 3ds Max up until this point, but the company I'm working for at the moment needs me to use blender so I'm trying to make the transition over.  Literally all I'm making at the moment is a simple book and cannot understand how the bevel tool isn't working similar to how the chamfer tool would in 3ds Max.  I want a nice curved edge for the spine of the book however when I press control + B and try to bevel it bevels the edge unevenly with the new bevel pushing forward.    Is there a way I can get it to bevel evenly like it does in Max? Is there a tool that combines zcat and cat transparently? When handling log files, some end up as gzipped files thanks to logrotate and others not. So when you try something like this:  $ zcat *   you end up with a command line like zcat xyz.log xyz.log.1 xyz.log.2.gz xyz.log.3.gz and then with:  gzip: xyz.log: not in gzip format   Is there a tool that will take the magic bytes, similar to how file works, and use zcat or cat depending on the outcome so that I can pipe the output to grep for example?  NB: I know I can script it, but I am asking whether there is a tool out there already. The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra.     Recall that an infinite cyclic group is isomorphic to $\mathbb{Z}$. We wish to show that we do not have an isomorphism between $\mathbb{ZxZ\;and\;Z}$. Note that $\mathbb{ZxZ}$ is an infinite group (under addition of course). Now, in order for there to even be potential for an isomorphism, two spaces must have equal dimension. Since the $\mathbb{dim(\mathbb{ZxZ})}=2&gt;\mathbb{dim(\mathbb{Z})}=1$, we know that $\nexists$ an isomorphism between our spaces. Hence, $\mathbb{ZxZ}$ is not a cyclic group.   My question (besides a validity check): was there a better way to prove this? I just found this to be the easiest way.  EDIT Totally wrong with this one. Back to the cutting board. How to evaluate the integral $$\int_{25\pi / 4}^{53\pi / 4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx\ ?$$ How to check if list item already exist when adding it with Disable white background of pasted text in PowerPoint When I paste text from another source with "Keep source formatting option" into PowerPoint 2010 I receive a white background, that I don't want.    How can I make this text background transparent, so there would be just red text on purple? Declare variable to hold job name and if job running kill it I’m looking for a single file that does these things in order.   Declare a variable to hold a job name  Run a query to check whether that job declared is currently running Set up an if loop If the job is running, execute the stored procedure that kills it.   My question is to write a query that declare a variable to hold a job name and gives us result the job is still running or not if the job is running execute the stored proc to kill it.  My manager was asking me to listing running jobs and suspended jobs and declare a variable to hold job name and setup an IF loop like if job is running run proc to kill it. Difference between "git add -A" and "git add ." The command git add [--all|-A] appears to be identical to git add .. Is this correct? If not, how do they differ? The text for  says:     {1}{G}, {T} : Target creature you control fights another target creature. (Each deals damage equal to its power to the other.)   Does this have anything to do with combat? Or is it just direct damage between two creatures? For example, can one of the creatures have defender? Or what if one of them is tapped?   I've been assuming it is more like the damage dealt by a .  Also: are the two creatures then “tapped” until their next untap step? ie: could you use Ulenvald Tracker to make two creatures fight in your pre-combat step, then enter the combat phase with a creature who just “fought” untapped? Could you make two creatures fight who are both tapped? Calculate battery life Op Amps: how do they know whether the supply is split? Most op amps can operate in single and split supply configurations. However, I cannot see how this is possible.  If I connect voltages \$V_\text{SY}^+\$ and \$V_\text{SY}^-\$ to the positive and negative supply pins of a op amp, and \$V_+\$ and \$V_-\$ to the non-inverting and inverting inputs, then the output is (if \$A_\text{OL}\$ is the open loop gain of the op amp): $$V_\text{out}=A_\text{OL}(V_+-V_-)+V_\text{SY}^-\tag{Single-supply}~\text{ }~~\text{ }~~\text{ }~~\text{ }~~\text{ }~~\text{ }$$ The above equation is from personal experience using op amps in single supply (i.e. \$V_\text{SY}^- = \text{GND}, V_\text{SY}^+ &gt; \text{GND}\$).  In split supply, we can use the following diagram from MIT's 6.002 introduction to electronics lecture video:  $$V_\text{out}=A_\text{OL}(V_+-V_-)+(V_\text{SY}^+-V_\text{SY}^-)/2\tag{Split-supply}$$  These two equations, however, are mutually inconsistent: If I take an op amp and hook up a single voltage source between its supply pins, how does the op amp know that the resulting voltage is not split between two sources?   It must know - but how does it know? How do I open the user folder of a hard drive that I transfered? My Windows 10 notebook died but fortunately I was able to rescue the hard drive and put it into another PC.   Unfortunately, the other PC can't open the C/Users/Joe folder. What do I have to do to open the folder? Uploading an image from the web can leave paste broken in editor