Prove the inequality for n>2.
Show that if $n>2$, then $(n!)^2>n^n$.
Equivalent definitions of convexity for $f\in\mathcal C^1(\mathbb R^n)$ I noticed many identical questions on this site relating to equivalent definitions of convexity, see for example , , , , , ,  and there might be more. So I wanted to create a &quot;big list&quot; of equivalent definitions of convex functions, I will start with some equivalent definitions and post my proofs in my answer below. Feel free to add any other equivalent definitions. This question can then serve as an abstract duplicate. Claim. Let $n\in\mathbb N$ and $f\in\mathcal C^1(\mathbb R^n)$. Then the following are equivalent:  $f$ is convex, i.e. $f(\lambda x_1 + (1-\lambda x_2))\le\lambda f(x_1)+(1-\lambda)f(x_2)$ for all $\lambda\in[0,1]$ and $x_1,x_2\in\mathbb R^n$. For all $x,y\in\mathbb R^n$, $\langle\nabla f(x)-\nabla f(y),x-y\rangle\geq0$. For every $x,y\in\mathbb R^n$, $f(y)\geq f(x)+\langle y-x,\nabla f(x)\rangle$. In which cases is a word, or a group of words written in italics? Is italics used in specific contexts, or it is quite normal to write words in italics? How long does the Truffle take to move in after building the mushroom biome? I currently find myself in a pickle because even after I've built the mushroom biome and met all other requirements, the Truffle NPC won't move in.   I've defeated not only the Wall of Flesh but Ocram already. Created a surface mushroom biome. Built a house made out of mushrooms with mushroom furniture.   I have no idea how long it takes for the NPC to move in or if I'm wrong in any of the steps. How to do integer & float calculations, in bash or other languages/frameworks? RegEx match open tags except XHTML self-contained tags Extracting usernames from the /etc/passwd file I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ I know that $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{1}{4^{2m+1}}\left(\zeta\left(2m+1,\frac14\right)-\zeta\left(2m+1,\frac34\right)\right)$$ so maybe I could simplify the above more? I have a big pot full of chicken necks and backs on the stove which I need to separate into (a) stock and (b) everything else.  I'm wondering the best way to go about doing this. I've previously laid cheesecloth into a fine-mesh strainer and poured it through that, but the cheesecloth always seems to kind of clog up, making straining 4 litres of stock a very lengthy process. I also tried a coffee filter laid inside a strainer and got more-or-less the same results.   Alternatively, I ditched the cheesecloth and just used the fine-mesh strainer by itself, but, unsurprisingly, this method leaves some bits I'd rather not have.   I'm wondering: is there a better way? Or should I stick with the cheesecloth and work on being more patient? What WordPress API function lists active/inactive plugins? How to use both binary and continuous variables together in clustering? I need to use binary variables (values 0 &amp; 1) in k-means. But k-means only works with continuous variables. I know some people still use these binary variables in k-means ignoring the fact that k-means is only designed for continuous variables. This is unacceptable to me.  Questions:     So what is the statistically / mathematically correct way of using binary variables in k-means / hierarchical clustering? How to implement the solution in SAS / R? How soon should I "vote to close"? Installing Ubuntu on a single partition I’m trying to install Ubuntu 16.04 alongside with Windows 7. My HDD supports only 4 partitions. Three of the partitions are taken up by windows. The fourth one is labelled as free space.   Now, how do I make root, swap and home partitions? Should I format my free space as primary with EXT4 first (if so, what should be the mount point?) and then create logical partitions for root/Home/swap? How do I go about this?  Edit: I tried to use the automatic installation option for dual boot. However I got the error ubi partman exit code 141. Probably because it tried to create two separate partitions for root and swap.   $ sudo parted -l Model: ATA ST1000DM003-1CH1 (scsi) Disk /dev/sda: 1000GB Sector size (logical/physical): 512B/4096B Partition Table: msdos Disk Flags:   Number  Start   End    Size   Type     File system  Flags  1      1049kB  106MB  105MB  primary  ntfs         boot  2      106MB   277GB  277GB  primary  ntfs  3      277GB   903GB  627GB  primary  ntfs How to convert Bedrock texture pack to Java Edition Minecraft I'm trying to use a texture pack designed for the Bedrock Edition of Minecraft, on Java Edition. I looked at the texture names for a Java Edition texture pack and compared those to the names of the Bedrock Edition texture names, and they aren't the same. A lot of the texture names are flipped, such as:   Jungle Log (Java = jungle_log.png) (Bedrock = log_jungle.png)   Most of the texture names are flipped, like the Jungle Log (and all of the other log types) and there are some .json files that are different and/or not there.  Trying to open the Bedrock texture pack would result in some animated blocks not being animated, other blocks play the wrong sound (or no sound) when broken, if OptiFine is in the mix, the skybox could be missing, which I am pretty sure would result in an always black sky, along with no sun and moon textures.  Any help in finding a program or site that can do this would be helpful, because I tried doing it manually, but almost every texture is different.  Any help is appreciated! How to animate A curve segment? How do I drag and drop files into an application? I managed to successfully publish my old bachelor thesis in a good quality national journal in my country. However, I am thinking of retracting the article due to certain circumstances. First, I felt really embarrassed by the low quality research methodology used in my bachelor thesis. I only used simple correlation in my research. After it was published, I realized that the journal editor team forgot to include a reference in that article because they were rushing me throughout the editing process. I informed them about their mistake and it seems that they have done nothing to fix the problem. I became a bit annoyed because the editor team has never made this problem in the past.  I felt that it would be better for me to simply ask for retraction to solve this issue, but I am a bit afraid of negative consequences in the future. Is it okay to do this? I am really interested of entering the academic world in the future. I am trying to solve a variation on the .  In this scenario, someone is selecting coupons at random with replacement from n different possible coupons. However, the person is not selecting coupons one at a time, but instead, in batches.  Here's an example problem formulation: There are 100 distinct coupons. A person makes selections in 10-coupon batches at random (each coupon with replacement). What is the expected number of batches necessary to have selected 80 unique coupons?  I have been able to determine the expected number of selections necessary to have selected k unique coupons when selecting one at a time (much like Henry's ), but I'm a bit stumped as to how to go about solving it with this particular wrinkle.  Any tips/guidance would be greatly appreciated. Finding out the area of a triangle if the coordinates of the three vertices are given What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?   One approach is to find the length of each side from the coordinates given and then apply . Is this the best way possible?   Is it possible to compare the area of triangles with their coordinates provided without actually calculating side lengths? By just typing some commands, how can Ubuntu be uninstalled? I don't want to install Windows or anything, leaving my PC without an OS.  Would this command work?  sudo rm -rf / Do I need swap space if I have more than enough amount of RAM? From what I understand, the purpose of a swap partition in Linux is to free up some "not as frequently accessed" information from RAM and move it to a specific partition on your harddrive (at the cost of making it slower to read from or write to), essentially allowing active applications more of the "high speed memory".  This is great for when you are on a machine with a small amount of RAM and don't want to run into problems if you run out. However, if your system has 16 GB or 32 GB of RAM, and assuming you aren't running a MySQL database for StackExchange or editing a 1080p full length movie in Linux, should a swap partition be used?