At school I asked the question and I kept wondering "Can fractions or decimals be odd or even?"
Is the concept of even/odd numbers applicable to decimal numbers? For e.g. - 4.222 is a even number?
Why can't I use Stack Exchange if I'm under 13 years old, or if I'm under 16 years old and from the EU? I was just reading the , and noticed something:  You must be at least 13 years old to access or use the Network or Services, including without limitation to complete a Stack Overflow account registration. By accessing or using the Services or the Network in any manner, you represent and warrant that you are at least 13 years of age. If you are under 13 years old, you may not, under any circumstances or for any reason, access or use the Services or Network in any manner, and may not provide any personal information to or on the Services or Network (including, for example, a name, address, telephone number or email address). If you are located within the European Union, you must be at least 16 years old to access or use the Network or Services, including without limitation to complete a Stack Overflow Account Registration.  I signed up for Stack Overflow when I was around 11 or 12 and I did provide my email address. Are you saying that my use of Stack Overflow was not legal? I'm 16 now (so I'm in the clear), but that just seems a little harsh. Is that possibly related to the age limit for accepting a binding contract/agreement (although I thought that was 18 in the U.S.)? Is there a way for users under the age of 13, or for users from the European Union who are under the age of 16, to use the site legally? When A and B are positively related variables, can they have opposite effect on their outcome variable C? A is positively related to B.  C is the outcome of A and B, but the effect of A on C is negative and the effect of B on C is positive.  Can this happen? Dedicated mouse and keyboard for a virtual machine, is it possible? Covering estate tax on US assets for non-resident alien I am a non-resident alien holding a brokerage account in the USA but residing outside the USA. My country does not have an estate tax treaty with the USA. So my estate is liable for a significant estate tax up to 40%. Is there any way to avoid or to cover this tax? I thought of buying term insurance for myself, but I am not sure whether US companies will sell term insurance to a non-resident. I want to buy insurance from a US company because I want the death benefit to be in USD. Can someone suggest as to what the best way to approach this issue is? Story about building up speed leading to increased quality of life It's a story I read in 2 or 3 parts in a Greek sci-fi magazine called 9 around 2004 (give or take a few years), but it may be a bit older than that. This is what I remember (and I hope it's all accurate):   The protagonist used to travel a lot, mainly using high speed media like aircraft. It was established in the story that as he traveled more frequently and with more speed, he would gradually cross over towards (sort of) higher planes of existence (and that everyone in the setting could do the same, although most people simply didn't bother). These higher planes of existence were apparently still Earth, but the environments and the inhabitants were increasingly more alien and life in them was increasingly luxurious. As far as I recall, the protagonist's goal was to reach those higher levels (either for the luxury or for some other reason I can't recall). If he failed to maintain his "speed" for too long, he would fall back to lower planes of existence, with more familiar characteristics and significantly lower quality of life. His wife had died some time ago and her grave was apparently in the lower planes. He did visit her once near the end of the story, before starting to build up "speed" again.   I don't happen to know the original language of the story because what I read was a version translated into Greek. How bias arises in using maximum likelihood to determine the variance of a Gaussian? While reading , I ran into the following explanation for why there is an error in estimating the variance of Gaussian distribution using Maximum Likelihood. On ,      Illustration of how bias arises in using maximum likelihood to   determine the variance of a Gaussian. The green curve shows the true   Gaussian distribution from which data is generated, and the three red   curves show the Gaussian distributions obtained by fitting to three   data sets, each consisting of two data points shown in blue, using the   maximum likelihood results (1.55) and (1.56). Averaged across the   three data sets, the mean is correct, but the variance is   systematically under-estimated because it is measured relative to the   sample mean and not relative to the true mean.   I'm pasting figure 1.15 and the required equations (1.55, 1.56) here for convenience.  $\mu_{ML} = \frac{1}{N}\sum_{n=1}^{N}x_n$   $\sigma_{ML}^2 = \frac{1}{N}\sum_{n=1}^{N}(x_n - \mu_{ML})^2$    I can imagine how maximum likelihood estimation would look pictorially. However, I'm not able to understand the figure or the explanation. I would appreciate if someone could label the figure or explain what points are considered in each part of the figure and why the curve could look like that in each case.  I looked up , and I'm able to understand why there is a factor of $\frac{N-1}{N}$ between the true $\sigma^2$ and $\sigma_{MLE}^2$.  ${\rm I\!E}[\sigma_{MLE}^2]= (\frac{N-1}{N})\sigma^2 $ Questions on "All Horse are the Same Color" Proof by Complete Induction I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$  I'm not allowed to use taylor expansion. So my first thought was to rewrite the function in a form where I can apply the basic identity:  $$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n $$  So let's rewrite the original function $f(z) = \frac {1}{1+z^2} $ and use above identity:  $$\frac {1}{1+z^2} = \frac {1}{1-(-z^2)} $$ But observe that I need to center it at $z_0=1$ so I rewrote it to:  $$\frac {1}{1-(-z^2)}=\frac {1}{2z-(-(z-1)^2)} $$  I can factor out $\frac {1}{2z}$ and use the basic identity to get : $$\frac {1}{2z}\frac {1}{1-\frac{(z-1)^2}{2z}} = \frac {1}{2z}\sum_{n=0}^\infty \left(\frac{-(z-1)^2}{2z}\right)^n =\frac {1}{2z}\sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^n} = \sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^{n+1}}$$  The problem is if I ask wolfram alpha: power series$ \frac{1}{1+z^2} $centered at z = 1 it will give me a different answer that doesn't match the sum I found.  Can anyone help me figure this one out? What am I doing wrong? In Star Wars: "The Force Awakens," how could this lightsaber have turned up where it did? In The Force Awakens:      Maz Kanata tells Rey that the lightsaber she just picked up and had visions from belonged to Anakin and Luke and now it's hers.   The lightsaber that Luke got from Obi-Wan Kenobi in A New Hope was lost/destroyed when Luke fought Darth Vader in Star Wars: The Empire Strikes Back.  In Return of the Jedi Darth Vader compliments Luke on building a new lightsaber and tells Luke that his skills are now complete.   How could that possibly be the lightsaber that Anakin used originally, and that was then passed on to Luke, when that lightsaber no longer exists?  Writers notes, director's notes, interviews, cast interviews, etc. are welcome. Thanks. Find the coefficient of the term $x^2$ in $\left(x+\frac 2x\right)^{4}$ Find the coefficient of the term $x^2$ in $$\left(x+\frac 2x\right)^{4}$$   This is what I have done so far.  We know $a$ = $x$ and $b$ = $\frac2x$ and $n$ = 4  So what I have done is $4c2$ x $(x)^2$ x ($\frac2x$)$^2$  As we know that $^2$ + $^2$ = 4, and 4 is the $n$ value.  The answer is one though?  What have I done wrong?   P.S.  was the tutorial that helped me. What counts as an attack? I was looking over the class feature of the 'fiend' patron warlock in 5e. I'm a bit confused on what actions the player needs to take before they can use the 'Hurl Through Hell' ability on an enemy. The PHB states:     Starting at 14th level, when you hit a creature with an attack, you can use this feature to instantly transport the target through the lower planes.   I'm not clear on what an 'attack' means in this context. Does it have to be a physical attack like a sword or crossbow? Do magical attacks like Eldritch Blast that require an attack roll count? What about spells with no attack roll? Birational map from elliptic curve keeps smoothness? Let $E$ be a elliptic curve that is , genus $1$ smooth curve with base point. And suppose that the map $φ$, which sends $E$ to $E'$, is birational. Birational means there are two rational maps on both direction which composites identity. Then, can we say that $E'$ is smooth? Trying to run a Ubuntu server for the first time... But everything is in a terminal. Is there a way to switch to a GUI? There is an alignment issue in the stackexchange.com search page .  The bottom bar is missing in the search input box.  Sample URL:   Screen shot for reference:    Firefox version: 54.0b8 (32-bit) OS Version: Windows 10 Pro  Also happens with Chrome latest version. Show that $f$ is differentiable on $(−1, 1)$. Let $f$ be a function defined on the interval $(−1, 1)$ such that for all $x, y \in (−1, 1)$, $$f(x + y) = \frac{{f(x) + f(y)}}{{1 - f(x)f(y)}}.$$ Suppose that $f$ is differentiable at $x = 0$, show that $f$ is differentiable on $(−1, 1)$.  Need your kind guidance and help. Thanks in advance! Cannot add PPA: 'ppa:~darklin20/ubuntu/bomi'.     ERROR: '~darklin20' user or team does not exist.   Showing this message when suing this command:    sudo add-apt-repository ppa:darklin20/bomi   please help me. Why does unlocking the bootloader wipe phone? How To Change The Fonts In Windows 8 In Windows 7, there is an option to change the fonts for things such as the active title bar, menu, etc. It was found by opening the Window Color and Appearance settings in the Control Panel. I just installed Windows 8 and would like to change the fonts, but I can't find anything to change the actual fonts (there is however a limited drop-down list to change a few font sizes in the appearance settings). Does anyone how to change the fonts used in Windows 8? Why are all snaps being mounted and listed as block devices or partitions for Ubuntu 18.04? As of Ubuntu 18.04 running lsblk shows 16 snap loops (2-3 times for each snap). The question is, why are they being listed as results for lsblk, fdisf-l, and blkid?    It creates a lot of clutter from the actual disks drive partitions I need to see, namely /dev/ partitions. I know a purported duplicate of this question exists, but it only asks why three loops are being listed per snap. I want to know why these snaps are being listed in the first place, and the purported duplicate does not answer this (perhaps those marking this as duplicate could help me by explaining why it is a duplicate). Technically, they qualify as file systems (which I neither created nor asked for), but they are getting in the way of the information output for the /dev/ partitions I am interested in. This becomes a problem when fdisk -l outputs a three page+ list filled mainly with snaps.  The output of a recent (1 week old) Ubuntu install and I have not installed any snaps:    $ lsblk   NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT   loop0    7:0    0  14.5M  1 loop /snap/gnome-logs/37   loop1    7:1    0   2.3M  1 loop /snap/gnome-calculator/170   loop2    7:2    0  86.6M  1 loop /snap/core/4486   loop3    7:3    0  86.6M  1 loop /snap/core/4650   loop4    7:4    0   1.6M  1 loop /snap/gnome-calculator/154   loop5    7:5    0  14.5M  1 loop /snap/gnome-logs/34   loop6    7:6    0   3.3M  1 loop /snap/gnome-system-monitor/36   loop7    7:7    0   2.3M  1 loop /snap/gnome-calculator/178   loop8    7:8    0    13M  1 loop /snap/gnome-characters/101   loop9    7:9    0   3.7M  1 loop /snap/gnome-system-monitor/45   loop10   7:10   0 139.5M  1 loop /snap/gnome-3-26-1604/64   loop11   7:11   0   140M  1 loop /snap/gnome-3-26-1604/59    loop12   7:12   0   3.7M  1 loop /snap/gnome-system-monitor/41   loop13   7:13   0    21M  1 loop /snap/gnome-logs/25   loop14   7:14   0  12.2M  1 loop /snap/gnome-characters/69   loop15   7:15   0    13M  1 loop /snap/gnome-characters/96   sda      8:0    0 298.1G  0 disk   ├─sda1   8:1    0   512M  0 part /boot/efi   └─sda2   8:2    0 297.6G  0 part /   sr0     11:0    1  1024M  0 rom     (supplemental screen capture of above text):     My snap list shows 6 results:       core   gnome-3-26-1604   gnome-calculator   gnome-characters   gnome-logs   gnome-system-monitor     Meanwhile, gnome-disk-utility shows nothing at all for snaps, only showing my HDD and optical drive.    It won't be very efficient if every installed snap gets listed as a block device (2-3 times each to add). Should I expect future updates to deal with this?  Edit: fdisk-l also dumps out a very long list with 16 instances of these "disk loops" (Disk /dev/loop0, Disk /dev/loop1, etc., each with details which I won't show here because it's too long). This can't be intended behaviour, can it? blkid also lists 16 loops, as TYPE="squashfs". At least parted -l works as expected, only outing my actual disk partitions.    I just tested this, and installing more snaps does add more to the lsblk output. Therefore, fdisk, lsblk, blkid could have potentially huge output lists, according to the number of snaps available, and installed. I'm a bit confused in describing 1.5hrs in words.   Is writing one and a half an hour correct or should it be one and half hour?  E.g:     I'll see you there in one and a half an hour.     OR       I'll see you there in one and half hour.   Or is there any other correct way of writing this?