I've seen a proof that shows that rational numbers are not the same infinity as real numbers.   I'm curious if there is a proof that shows $\mathbb{R}^1$ and $\mathbb{R}^2$ are the same infinity. If so, is there a way to map from one to the other for all points?
Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?  Thank you.
Mathematical symbol for "and" I have found some pretty complete lists (I think) of mathematical symbols  and , but I don't see a symbol for the word "and" on either list.  A person could easily just write the word "and" or use an ampersand, but I was wondering if there was an actual mathematical symbol for the word "and".  Also, if anyone knows any lists that are more complete than the ones I have linked to please provide a link. I have been using the Copy Attributes add-on to copy some properties from active object to selected objects. This add-on is not enabled by default and must be explicitly enabled. But its a very useful workflow enhancing tool.    There is a limitation though. Not all properties can be copied and even those that can be copied are copied in groups (Its hard-coded in to the add-on).  My question is if there is a built-in feature or any another other way to copy any property from the active object to the selected objects? Storing datetime values in Shapefiles using QGIS? I have a shapefile with date in a string field. I create a new field with date type and use the calculator to make the conversion. But the result only provide the date part, the time part is removed. Even when the preview show the correct conversion.  Here you can see the original string field time and the result date time2. And the use of the conversion function to_datetime() and the preview result at the end    If you go to the layer properties you can change the edit mode to date/time. Then if go to the data and try to edit time2 you get the current date/time. So is possible save date and time in the same field. But the update tool still doesn't work as expected      As J.R said. Even when you can change QGIS edit to allow Date/Time the save to shapefile will discard the time. Soql query on parents other child object I want to write a soql query , My scenario is as follows. I have three objects objectA,objectB,objectC. objectA is the parent object for both objectB,objectC.   objectA - objectB  (Master-detail relationship) objectA - objectC  (Master-detail relationship)`   ObjectB has two fields:   Start_date__c End_date__c   ObjectC has a field:    Date__c   Now i want to retrieve records with date__c in objectC those were in between start_date__c and end_date__c of objectB. how can i achieve such functionality . Is my expansion of the state $| x \rangle$ correct? Why linear and logistic regression coefficients cannot be estimated using same method? I read in a machine learning book that parameters of linear regression can be estimated (among other methods) by gradient descent, while parameters of logistic regression are usually estimated by maximum likelihood estimation.  Is it possible to explain to a novice (me) why we need different methods for linear/logistic regression. a.k.a why not MLE for linear regression and why not gradient descent for logistic regression? In the Wikipedia article , it says     After Newtonian gravitation was found to be inconsistent with special relativity, . . .   I don't see how Newtonian gravitation itself is inconsistent with special relativity.  After all, Newton's Universal Law of Gravitation is completley analogous to Coulomb's Law, so it would seem that, if there were an analogous "gravitational magnetic field", one could formulate a theory of gravitation in exact analogy with Maxwell's Theory of Electromagnetism, and of course, this would automatically be consistent with Special Relativity.  What about this approach to gravitation does not work?  The only problem I could see with this is the lack of evidence for a "gravitational magnetic field".  That being said, gravity is "weak" as it is, and my guess is that it would be extremely difficult to set up an experiment in which a massive body were moving fast enough for this "magnetic field effect" to be obsevable.  EDIT:  As has been pointed out to me in the answers, Newton itself is inconsistent with SR.  Similarly, however, so is Coulomb's Law, yet, electromagnetism is still consistent with SR.  Thus, how do we know that it is not the case that Newton's Law is the special "static" case of a more general gravitomagnetic theory exactly analogous to Maxwell's Theory:  Let $\mathbf{G}$ be the gravitation field, let $\rho$ be the mass density, and $\mathbf{J}$ be the mass current density, let $\gamma _0$ be defined so that $\frac{1}{4\pi \gamma _0}=G$, the Universal Gravitation Constant, let $\nu _0$ be defined so that $\frac{1}{\sqrt{\gamma _0\nu _0}}=c$, and suppose there exists a field $\mathbf{M}$, the gravitomagnetic field, so that the following equations hold: $$ \vec{\nabla}\cdot \mathbf{G}=-\frac{\rho}{\gamma _0} $$ $$ \vec{\nabla}\cdot \mathbf{M}=0 $$ $$ \vec{\nabla}\times \mathbf{G}=-\frac{\partial \mathbf{M}}{\partial t} $$ $$ \vec{\nabla}\times \mathbf{M}=\nu _0\mathbf{J}+\gamma _0\nu _0\frac{\partial \mathbf{G}}{\partial t} $$ where theses fields would produce on a mass $m$ moving with velocity $\mathbf{v}$ in our inertial frame the Lorentz force $\mathbf{F}=m\left( \mathbf{G}+\mathbf{v}\times \mathbf{M}\right)$.  This theory would automatically be consistent with SR and would reduce to Newton's Law of Gravitation for the case of gravitostatics:  $\frac{\partial \mathbf{G}}{\partial t}=\frac{\partial \mathbf{M}}{\partial t}=\mathbf{J}=0$.  (To be clear, you can't just set the time derivatives equal to $\mathbf{0}$ as it seems I have done.  Upon doing so, you obtain the corresponding static theory, which is technically incorrect (as you can easily see because it won't be relativistically invariant), but is nevertheless often a useful approximation.)  My question can thus be phrased as:  without appealing directly to GR, what is wrong with this theory? I've never really got how chmod worked up until today. I followed a tutorial that explained a big deal to me.   For example, I've read that you've got three different permission groups:   owner (u) group (g) everyone (o)   Based on these three groups, I now know that:   If the file is owned by the user, the user permissions determine the access. If the group of the file is the same as the user's group, the group permission determine the access. If the user is not the file owner, and is not in the group, then the other permission is used.   I've also learned that you've got the following permissions:   read (r) write (w) execute (x)   I created a directory to test my newly acquired knowledge:  mkdir test   Then I did some tests:  chmod u+rwx test/ # drwx------ chmod g+rx test/ # drwxr-x--- chmod u-x test/ # drw-r-x---   After fooling around for some time I think I finally got the hang of chmod and the way you set permission using this command.    But...  I still have a few questions:   What does the d at the start stand for? What's the name and use of the containing slot and what other values can it hold? How can I set and unset it? What is the value for this d? (As you only have 7=4+2+1 7=4+2+1 7=4+2+1) Why do people sometimes use 0777 instead of 777 to set their permissions?   But as I shouldn't be asking multiple questions, I'll try to ask it in one question.  In UNIX based system such as all Linux distributions, concerning the permissions, what does the first part (d) stand for and what's the use for this part of the permissions? How to configure extra buttons in Logitech Mouse Comparing complex numbers If $a+ib$, $c+id$, $e+if$ are three complex numbers, than can we tell which one is greater or smaller between them? If yes, then how and if no then why not?   Can somebody give explanation on this.... I will be grateful to him. Summoning an item in 1.9 with motion When I put   /summon Item ~ ~2 ~ {Item:{id:"minecraft:diamond",Count:1b,Motion:[0.0,0.8,0.0]}}    in a command block, it summons the diamond with Count:1b but it doesn't follow the Motion Tag. Does anyone know how to get the Motion Tag working? What triggers a full garbage collection in Java? The curious case of Dmitri Valtane $\mathbb S_n$ as semidirect product In  note, I've read that $\mathbb S_n$ is a semidirect product of the alternating group $A_n$ by $\mathbb Z_2$. So I am trying to define a morphism $\rho: \mathbb Z_2 \to Aut(A_n)$ to show that $\mathbb S_n \cong A_n \rtimes Z_2$. I would appreciate suggestions on how could I define the morphism. Thanks in advance. I would say $\infty - \infty=0$ because even though $\infty$ is an undetermined number, $\infty = \infty$. So $\infty-\infty=0$. What is the fate of the Asgardians at the start of Avengers: Infinity War? Thor: Ragnarok ends with the remaining people of Asgard aboard a refugee ship that meets Thanos' ship. Avengers: Infinity War opens with      Thanos and his minions having commandeered the ship and several bodies can be seen on the floor.   This begs the question, what happened to these characters:   Korg Miek Asguardian extra #3 Valkyrie how to prove the chain rule? I have just learnt about the chain rule but my book doesn't mention a proof on it. I tried to write a proof myself but can't write it. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus. How do I change the hostname without a restart? I want to change the OS hostname but I do not want to restart.  I have edited /etc/hostname but it requires a restart to get implemented. How to avoid this? Most of the time, I use good and well interchangeably. However, on many occasions I would find well or good a misfit. Please suggest the proper usage. Just installed Ubuntu for the first time... Is there any simple way to browse installed programs?  On Mac there is the Applications top-level folder... here I see Desktop, Documents, Downloads etc but no Apps? That'd be a good idea, no? (is there some way to create this myself?)  On Windows there is the Start menu...  Here it seems like I have to click the 'Dash Home' and then I can see 'Recent Apps' ... I can search for them (I'd rather not)  ...or I can click the ruler/pencil icon at the bottom and see 'Recent Apps' and click to expand 'Installed Apps' ...ok that's all of them - is it possible to get a straight list view instead of the hard-to-scan wrapped rows?  Also, is there any way to get rid of the pointless random selection of 'Apps available for download' at the bottom?