Showing that an entire function is a polynomial based on its values on the unit circle
Characterizing non-constant entire functions with modulus $1$ on the unit circle
Requires installation of untrusted packages? What's the Best Way to Shuffle an NSMutableArray? If you have an NSMutableArray, how do you shuffle the elements randomly?  (I have my own answer for this, which is posted below, but I'm new to Cocoa and I'm interested to know if there is a better way.)    Update: As noted by @Mukesh, as of iOS 10+ and macOS 10.12+, there is an -[NSMutableArray shuffledArray] method that can be used to shuffle. See  for details. (But note that this creates a new array, rather than shuffling the elements in place.) Up until now, I implemented all-against-all collision detection in my games. It worked fine.  But I reached a point where implementing this kind of collision detection, makes the game run poorly, because there are two many objects to iterate through and check for collision.  I have searched this site and found something about 'uniform grids'. But I didn't completely understand what it is and how to implement it. Tried to Google it but nothing easy enough to understand came up.  I am a beginner to 2D-game programming. Do you mind trying to explain to uniform-grids, or link me to some good tutorial?  Thanks a lot How to make files protected? How to make some file that contains crucial data protected? For instance, suppose I have some files that are important, is there any method or strategy to make these kind of file more protected on Ubuntu? I even want to make some files ask for a password whenever someone tries to open it. Does it affect the version of Ubuntu in terms of security policy? May I use any encryption algorithm externally? Regards. How to prove that if A, B, C are submodules of M module then $$ A \cap (B+C) = (A\cap B)+C \iff C \subseteq A $$  Any help would be appreciated! What rules say that summoning undead is evil, and what rules say why? How do you delete untracked local files from your current working tree? Is there Ubuntu's analogue of snapshot.debian.org? What are the pros and cons of having a separate home partition? I installed a mainline kernel for testing purposes. I would like to set grub to boot from the older kernel by default.  I know I can set the GRUB_DEFAULT=0 setting for the first page of grub but how do I set it to boot by default from one kernel in the second page (Advanced page)?  I would prefer doing this without installing other software (ex. grub-customizer).     Related Questions: For any triangle with sides $a$, $b$, $c$, prove the inequality $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0 .$$   This is IMO 1983 problem 6.  I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to nothing (please don't mind the joke). Using $a=2R\sin A$, $b=2R\sin B$, $c=2R\sin C$ also didn't lead to anything for me. Could you give me a hint for finding the proper substitution? On Latin.SE, I am getting a 404 error when clicking any of the three "history" links on the .  That is, this:        Gives this:        All three editable sections have a revision history with two revisions or more, and all three give an error.  The actual links are , , and . Why doesn't \makeatletter work inside \newcommand? After hours of banging my brain against my skull, I narrowed down an issue I'm having to the fact that \makeatletter doesn't work inside \newcommand. Commands that I call after \makeatletter are registered as undefined, as if \makeatletter didn't do anything.  Why is this the case?  In the example below, I define two commands. Calling one of them from the document body causes the undefined error, while the other doesn't.  \documentclass[11pt]{book} \usepackage{kvsetkeys}  \newcommand{\commandThatFails}{     \makeatletter \comma@parse{}{} \makeatother }  \makeatletter     \newcommand{\commandThatWorks}{         \comma@parse{}{}     } \makeatother  \begin{document}  Hi there.  \commandThatFails \commandThatWorks  \end{document} TikZ or cryptocode package for cryptographic protocol diagrams Why does \noindent have such an effect on \rlap and \llap.? If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$ This is a question from the book Methods of Real Analysis by R. R. Goldberg.     If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$.   I don't have any idea how to start working on this problem. Please help. Thanks. Does this polynomial evaluate to prime number whenever $x$ is a natural number? 3D software like GeoGebra Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on. $f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors Let $f(x,y)=4x^3y^2$  How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most rapidly from the point $(2,1)$? And, what is the rate of change of $f$ in the direction given in the second question?  My Work (What I have done so far):  For the first question: $\nabla f(x,y)=12x^2y^2$i$+8x^3y$j$\nabla f(2,1)=48$i$+64$j$u=(3/5)$i$-(4/5)$j $\therefore D_u f=\nabla f\bullet u=0$  I'm not entirely sure if this is right? Can someone please verify?  For the second question: $u=(3/5)$i$-(4/5)$j $\therefore -||\nabla f(2,1)||=$? How do I find this? And am I doing this right?  I have no idea what to do for the third question. Switching between windows with scroll wheel on Ubuntu Dock