According to what I was taught as school, the past tense of 'get' is 'got' and 'gotten' is "an American corruption and, therefore, is not a proper word".  Example:     "Should auld acquaintance be forgot,   And never brought to mind?" &mdash; opening lines for Auld Lang Syne   Note that this is not as follows, which seems to be popular usage:     "Should auld acquaintance be forgotten, And never brought to mind?"
I see these two expressions are used almost identically in different contexts. Is there a difference between I have got and I have gotten?
Is the empty set a group? I know that a set G with a binary operation $*$ is a group, if:   $a*b\in G$, for all $a, b \in G$. $*$ is associative:   $$(a*b)*c=a*(b*c) \\ \text{for all }a, b, c\in G.$$   An identity element $e \in G$ exists, such that   $$a*e = e*a = a\\ \text{for all }a\in G.$$   For all elements $a \in G$, there exists an $a^{-1} \in G$, such that:   $$a*a^{-1} = a^{-1}*a=e.$$  Can I use that to show that the empty set is a group? Print xml with php What is the correct way to completely remove an application? I've searched the net for such information and found different command lines, like these ones:  sudo apt-get remove application sudo apt-get remove application*  sudo apt-get remove --purge application sudo apt-get remove --purge application*  sudo apt-get purge application sudo apt-get purge application*   So, what is the correct way? Is it necessary to use that "*"?  After that, I also found these commands:  sudo updatedb sudo locate application sudo rm -rf (file/folder name) After my online research on the subject, I learnt that, biologically speaking, many scientists believe that there is no such thing as a race. Homo sapiens as a species is only 200,000 years old, which has not allowed for any significant genetic diversification yet, and our DNA is 99.99% similar. I've read statements that there can be more genetic variation inside a racial group than between different racial groups, meaning that, for example, two individuals from the same "race" can have less in common with each other than with an individual from another "race".  :     Scientists consider biological essentialism obsolete, and generally discourage racial explanations for collective differentiation in both physical and behavioral traits     Q1: If Homo sapiens has no races (according to biologists), why are we so different morphologically? (hair/eyes/skin colour and even athletic performance seem to differ between human populations)  Q2: Is it common for other species too, when genetically close populations have very different morphological traits? Are there any other mammal or animal species that exhibit biological diversity comparable to human diversity, and how do taxonomists treat these species? (excluding intentionally bred domestic species to keep the comparison fair)    The question has been paraphrased to emphasize that it is the biological debate that is in question, not the sociopolitical. I.e., why is there no consensus in evidence and opinions of scientists? I need to upload my signature in a PDF document, but I do not know how to edit the background of this image.   I wish to remove the bluish-tint background, and make it pure white.   Can anyone help me with this, please? Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? Is ArcSDE no longer needed at ArcGIS 10.1? elementary ring proof with composition of functions and addition of functions as operations Consider the set $\mathcal{F}=\lbrace f\mid \mathcal{f}:\Bbb{R}\to\Bbb{R}\rbrace$, in which an additive group is defined by addition of functions and a second operation defined as composition of functions. The question asks to verify that the resulting structure does not satisfy the ring properties.     This is the question 24.10 from Modern Algebra, by Durbin, 4th edition.  So what I have so far is that all of the properties for the additive group (with addition of functions as the operation here) hold, associativity of the composition of functions hold, and that the failure must come from the distributive laws.  My proof of my claim, so far;  $$1\,\,\,\,\,[(\mathcal{f}\circ\mathcal{g})+\mathcal{h}](\mathcal{x})=\cdots=\mathcal{f}(\mathcal{g}(\mathcal{x}))+\mathcal{f}(\mathcal{h}(\mathcal{x})).$$ ...but,  $$2\,\,[(\mathcal{f}\circ\mathcal{g})+\mathcal{h}](\mathcal{x})=$$ $$3\,\,(\mathcal{f}\circ\mathcal{g})(\mathcal{x})+\mathcal{h}(\mathcal{x})=$$ $$4\,\,\,\mathcal{f}(\mathcal{g}(\mathcal{x}))+\mathcal{h}(\mathcal{x}).$$ ...which brings me to my contradiction of the properties of rings that;  $$5\,\,\,[\mathcal{f}\circ(\mathcal{g}+\mathcal{h})](\mathcal{x})\neq[(\mathcal{f}\circ\mathcal{g})+\mathcal{h}](\mathcal{x}).$$  So my question is, is this the correct path to showing the distributive laws are not obeyed when composition is taken as an operation on $\mathcal{F}$? particularly my proceeding from 2 to 3 to 4 is what i feel uncomfortable about, like i may be missing a step in there.  thanks I recently received an invitation to a conference and I'm trying to determine how reputable it is. Is there a good way to go about this? E: Package 'guile-1.8-dev' has no installation candidate this is an error i got I have apt-get 0.7.23.1 on host machine. I do not have root, thus not able to update it, or install other packages.  I want to download .deb packages with it, but neither  apt-get -d bash   (Invalid operation bash)  nor  apt-get -d install bash   (do not have root)  work.  On my home machine with a newer version of apt-get I can  apt-get download bash   and it does exactly what I want it to.  How can I perform the same on the host machine? What causes the 'Connection Refused' message? This is a  about Connection Refused    We see a lot of questions to the effect      When I try to connect to a system I get a message      Connection refused      Why is this ? How does a spinning object "know" that it is spinning? I am constructing a thought experiment about a spinning object that is floating in intergalactic space.  I assume that this object is about the size of a planet so that it will have enough gravity so that a Foucault pendulum will work, although I'm not sure that this is necessary for the thought experiment.    I can easily determine that this object is spinning if I stand on the object and observe the galaxies around me rise and set.  Similarly, I can determine the axis of rotation.    For simplicity, I go to one of the poles of my object, and I set up my Foucault pendulum.  What will I see?  And why will I see it?  I assume that in intergalactic space, the gravity is very small (galaxies are very far away, and the mass of my object is very small compared to the mass of a galaxy) so that there will be little coupling between the gravitational field of my object and the gravitational field of the galaxies around me. In the beginning of the series, it's mentioned that Muggle-born wizards/witches are identified due to inadvertent spell-casting. However, it seems like later in the book, disarmed wizards/witches are completely helpless.   What is the explanation for why some spells can be cast without a wand and is there an in-universe explanation for why this isn't done? Also, are there spells that can be cast non-verbally (with/without a wand)? I am planning an hour and a half ride with my 1 month old baby girl. I heard that being in the car seat for a long time is not good for newborns.  Is an hour and a half too long? Should we do a stop in the middle of the trip? Possible Duplicate:          On  it's a stated fact that if $I$ and $J$ are coprime ideals of a ring $A$, then $I^k$ and $J^\ell$ are also coprime for any integers $k,\ell$.   There's no proof given. Why is this fact true? Predicting the number of decimal digits needed to express a rational number The number $1/6$ can be expressed with only two digits (and a repeat sign denoted as $^\overline{}$), $$ \frac{1}{6} = \,.1\overline{6}$$  Meanwhile, it takes 49 digits to express the number $1/221$, since a string of 49 digits repeats: $$\frac{1}{221} = .\overline{004524886877828054298642533936651583710407239819}$$ Yet for $1/223$, 222 digits repeat, giving a total of 224 digits needed to express the number.  If $f:\mathbb{Q}\rightarrow\mathbb{N}$ is a function that gives the smallest number of digits needed to express a rational number in decimal notation, what can we say about $f$?  For example, if we do not consider the negative sign to be a digit, then $f$ is an odd function.  Other than that, is there any pattern to it at all? In the DC Comics Universe, What Happens to Earthlings Under a Red Sun? I know in the earliest versions of Superman, his powers weren't from the yellow sun, but that was changed and, in general (at least as I understand it), Superman's powers are because he was native to a world with a red sun and now he's under a yellow sun.  What happens to an Earthling or other native to a yellow sun planet when near a red sun?  And what if the sun is blue?  Or a brown dwarf?  Please note: I'm asking about a red sun, not about living on Krypton.  Okay, that's heavy gravity, but it's not the only planet under a red sun.  (But, still the point about whether yellow sun natives would have super powers would come into play there, wouldn't it?) Make Blender usable on high resolution screen vim-clang-format: clang-format is not found In all the Star Wars movies, R2D2 just boops and beeps his way around. And everyone understands him perfectly, though he is obviously not speaking Basic.  How do people understand him - is there like an astromech language? If so, then would one have to learn it, or is it comprised of feelings and emotions (expressed in beeps), essentially making a tonal language?  If I may be permitted to use an example from the prequels without too many lynch mobs forming, Anakin Skywalker converses frequently with Artoo while in the Naboo fighter, but he couldn't have had much schooling at the age of eleven, especially because he was a slave.  So how would people, Anakin especially, be able to understand astromechs?