I have wrote the following in LaTeX:  \begin{align} &amp; \tan(\Gamma_{4}) =  \begin{cases} \frac{x_{T3} - x_{T2}}{y_{T3} - y_{T2}} &amp; \text{if} \qquad x_{T3}&gt;x_{T2} \\ \\ \frac{x_{T2}-x_{T3}}{y_{T2}-y_{T3}} &amp; \text{if} \qquad x_{T3} &lt; x_{T2} \end{cases} \\ &amp; (T_{2})_{Calc}: ((x_{T2})_{Calc},(y_{T2})_{Calc}) = (x_{T3} - (y_{T3}-y_{L2}) \cdot \tan(\Gamma_{4}),y_{L2}) \end{align}   and I obtain the following results:    The thing is that the text inside the fraction becomes very small compared to the other equations.  Is there a way to make them have the same size as the rest?
what should I do to stop the cases environment compressing the first line of the following? \begin{equation} \alpha_{mn} = \begin{cases} \frac{16\mu^2}{mn\pi^2\left[\pi^2(m^2/a^2 + n^2/b^2) - \mu^2\right]} &amp;\mbox{for } m \mbox{ and } n \mbox{ odd;}  \\ 0 &amp;\mbox{for } m \mbox{ or } n \mbox{ even;} \end{cases} \\[1ex] \end{equation}  I'm not sure to do tex on this site, but if anyone tries it they see the first case is compressed because it's a fraction.
Proof of continuity of Thomae Function at irrationals. How do I make a Lightning Bow in vanilla Minecraft? I'm trying to make a setup with command blocks that summons lightning at the location of a shot arrow. The setup should recognize the arrow that is shot or the bow that is used by its name or another property, so that lightning only strikes at that specific arrow.   Also, the lightning should only strike once when the arrow hits the ground. I know this is somehow possible with scoreboard, but I'm no good with that. Does anyone know how to make a Lightning Bow in vanilla Minecraft? Eigen-Decomposition of Householder operator Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$. Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$.   see also   I figured this out:  $$ P(Z=z)=\cases{\sum_{i=o}^{min(m,n)} Bin(k+i,n,p)*Bin(i,m,p), &amp;if $z\ge0$;\cr \sum_{i=0}^{min(m,n)} Bin(i,n,p) * Bin(i-z, m, p),&amp;otherwise. \cr}$$  I also validated it by Monte Carlo simulation. For $n=30$, $m=20$ and $p=0.5$, I get the following distribution, where the circles are the analytical probabilities and the line connects the MC estimates.    Because that looked to me pretty much like a binomial distribution, I gave it a try and figured out that its actually a binomial, just shifted by m to the left. This can be simply written as $P(Y=y) = Bin(y+m, m+n, p)$. Hence, given equal success probabilities, the sum of two independent binomially distributed random variables is binomial, but also their difference, just shifted to the left.  This question here  is actually the same as mine, but received no answer and only the comment that there would be no simple formula. But the above formula looks pretty simple to me.   Is it correct that for the case of equal success probabilities, the above equations actually describes the distribution of $Z=X-Y$? I read in a book that $Z$ could not be binomial distributed because it had negative support. Is it right to call it a shifted binomial? Symbolizing layer with graduated colors by date/time field? I've got a layer of polygons where each has a date associated with it.  I want to color the polygons with graduated colors, but I want the colors to start with the newest polygons and fade to the oldest polygons.  So far I haven't been able to find a way to do this in ArcMap (&lt;= 9.3.1) other than to create a new field and calculate some integer or float value based on the date.  Is it possible to use a date or time field directly?   The data is natively a date-time value There are far too many records to use any unique-value renderer Converting the date to a numerical field is an answer for a different question. Agreement With Compound Subjects Joined by And I was surfing the internet the other day when I found this phrase: Instead your precious time and attention is wasted. To my ears, it sounds wrong. But I'm not a native English speaker, so I consulted with a number of them, and some of them said that it was "is" because the adjective modifies only the noun "time" and not "attention". Others said it "is" because that's just how it is. But I still believe it should be "are". Any help is appreciated. How common is "thrice"? How can you use optional parameters in C#? Note: This question was asked at a time when C# did not yet support optional parameters (i.e. before C# 4).  We're building a web API that's programmatically generated from a C# class. The class has method GetFooBar(int a, int b) and the API has a method GetFooBar taking query params like &amp;a=foo &amp;b=bar.   The classes needs to support optional parameters, which isn't supported in C# the language. What's the best approach? why arcane legacy steps to boot from USB with GPT and UEFI? To run the live AVG rescue "disk" on USB for Windows 10, how is it made to work with a GPT and UEFI system?  (Possible solution listed below.)  I enabled the boot menu so that pressing F12 will display a list to select from.  Also, I put everything USB at the top of the boot sequence and Windows and the HDD at the bottom.  However, I hadn't realized GPT and UEFI are so different from a BIOS.  Because the USB flash drive isn't compatible with GPT it's not being picked up?  This  was suggested:   Open the notebook and press quickly F2 to enter the BIOS. Under the MAIN menu, enable the F12 Boot Menu option. Under the BOOT menu, select Legacy boot and press F10 Save &amp; Exit. While the notebook is rebooting, press F2 again to enter the BIOS. Under the BOOT menu, select as your first boot option the USB HDD stick (your USB stick). Press F10 Save &amp; Exit.    Perhaps this will work, but what is the problem that prevents UEFI boot from registering the thumb drive?  Because it's using GPT and not an MBR?  I don't want to switch to legacy, but stay with UEFI.  The laptop: How are the degrees of freedom in the emmeans package calculated? - R I think the answer is yes but I know a lot of people disagree. So, I would like to ask these people when exactly does it stop being me.  Let's say I want to upload my brain into a computer using the following procedure:  I have each of my neurons replaced by electronic neurons. By electronic neurons I mean chips with the necessary hardware and software to perform the exact same function that the neuron being replaced was performing. This means that given the same inputs that the biological neuron was given it would produce the same outputs. Then I connect that electronic neuron to all the neurons that former biological neuron was connected to.   At the end of this process is it still me? If not when did it stop being me? When one neuron was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  (assuming it is still me)  Then, I divide these electronic neurons into groups and for each group I perform the following:   Write software to simulate the group of electrical neurons in my computer such that given the same inputs that go into the group of electronic neurons in my brain, it will produce the same outputs. Replace each group of neurons with another chip that communicates with my computer. Each of the chips will communicate to the corresponding software module in order to determine how to transform its inputs into the necessary outputs. Connect all of the inputs and outputs of the group of electronic neurons to that new chip.   At the end of this process is it still me? If not when did it stop being me? When one group was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  (assuming it is still me) Now, what I have is a bunch of these new chips in my head, connected to each other and for each of these chips I have a small neural network software module in my computer. So, now for each pair of chip/software module I do this:   Transfer all connections going into and out of the chip into the computer version (i.e. connect the software module to the other software modules that represent the other chips that this chip is connected to) Disconnect the chip from the other chips and remove it   At the end of this process is it still me? If not when did it stop being me? When one of the new chips was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  If it is still me then I have successfully moved my brain into a computer without ceasing being myself.  So, again the question is if at the end it is not me anymore when exactly does it stop being me and why does it stop being me?  EDIT: This question assumes that our universe and everything in it (including ourselves) follows the (testable) laws of physics. As it stands it also assumes that neuroscience is mostly right, although the question could also be considered without that requirement if instead of neural simulations we were to implement physics simulation of the brain. Of course this is a lot less practical. However, if your view is that our minds have a supernatural component then the question does not make any sense.  It was also pointed out that this is related to the Ship of Theseus, i.e. is something still the same if all its parts have been changed. With regards to that I think it is useful to consider that most of the cells in our body (not including brain cells) are regularly replaced anyway and some people even get entire organs replaced. Yet, we don't say that they stopped being themselves. Note, however, that if you are assuming a supernatural component of the mind that this explanation is not really adequate or relevant. Do muggleborns get their letter on their 11th birthday or the summer before they start Hogwarts? A lot of the fanfiction seems to like to give Hermione her letter in September to coincide with her birthday. That just doesn't seem very logical for the Statute of Secrecy, as you then have 10 out of every 12 10-year-old wizards and witches knowing about magic while finishing the school year (assuming even distribution of birthdays).   It seems to me that this is just asking at least one of them to brag about knowing magic and spill the secret.   Also this would mean that the poor kid born on 30 of August would only have 2 days to adjust to being magical before being shepherded off to a boarding school in the middle of nowhere.  From another point of view, getting such news on your birthday can screw up a lot of celebration plans.  Also, from rereading the memory from Dumbledore where he introduced magic to Tom, there is no mention of snow or even closeness to his birthday from the caretaker.  I believe this view came from the fact that Harry got his letter on his birthday but that piece of fanon seems to forget that letters were being sent up to at least a week or 2 beforehand (leading to the Dursleys hiding on a rock in the sea in the first place). I can't understand where the final keyword is really handy when it is used on method parameters.    If we exclude the usage of anonymous classes, readability and intent declaration then it seems almost worthless to me.   Enforcing that some data remains constant is not as strong as it seems.    If the parameter is a primitive then it will have no effect since the parameter is passed to the method as a value and changing it will have no effect outside the scope.  If we are passing a parameter by reference, then the reference itself is a local variable and if the reference is changed from within the method, that would not have any effect from outside of the method scope.    Consider the simple test example below. This test passes although the method changed the value of the reference given to it, it has no effect.  public void testNullify() {     Collection&lt;Integer&gt; c  = new ArrayList&lt;Integer&gt;();           nullify(c);     assertNotNull(c);            final Collection&lt;Integer&gt; c1 = c;     assertTrue(c1.equals(c));     change(c);     assertTrue(c1.equals(c)); }  private void change(Collection&lt;Integer&gt; c) {     c = new ArrayList&lt;Integer&gt;(); }  public void nullify(Collection&lt;?&gt; t) {     t = null; } How to return multiple objects from a Java method? I want to return two objects from a Java method and was wondering what could be a good way of doing so?  The possible ways I can think of are: return a HashMap (since the two Objects are related) or return an ArrayList of Object objects.  To be more precise, the two objects I want to return are (a) List of objects and (b) comma separated names of the same.  I want to return these two Objects from one method because I dont want to iterate through the list of objects to get the comma separated names (which I can do in the same loop in this method).  Somehow, returning a HashMap does not look a very elegant way of doing so. What is the CVE-2014-6271 bash vulnerability (Shellshock) and how do I fix it? A problem about wave equation Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$ I'm just wondering the reason behind being able to do this:  $$\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$$  Is there an elementary proof to the reasoning behind this? how do i change the color of typeface on photoshop? How do I delete a file whose name begins with "-" (hyphen a.k.a. dash or minus)? How do you remove a file whose filename begins with a dash (hyphen or minus) -?  I'm ssh'd into a remote OSX server and I have this file in my directory:  tohru:~ $ ls -l total 8 -rw-r--r--    1 me  staff  1352 Aug 18 14:33 --help ...   How in the world can I delete --help from a CLI?  This issue is something that I come across in different forms on occasion, these files are easy to create, but hard to get rid of.  I have tried using backslash  rm \-\-help   I have tried quotes  rm "--help"   How do I prevent the minus (dash or hyphen) character to be interpreted as an option?