How to prove $\mathbb{R}\times \mathbb{R}$ is equipotent (i.e. there is a 1-1 map between the two) to $\mathbb{R}$? How to prove $\mathbb{R}\times \mathbb{R}$ is equipotent (i.e. there is a 1-1 map between the two) to $\mathbb{R}$? The way to prove $\mathbb{N}\times \mathbb{N}$ is equipotent to is to $\mathbb{N}$ is to construct a function of two independent variables and one dependent variable, e.g. $g(m,n)=(m+n)^2+n$, and prove that it is bijective. How can we construct such a function for $\mathbb{R}\times \mathbb{R}\to\mathbb{R}$, or $\mathbb{C}\to\mathbb{R}$? It seems that we can use fractals to cover a two dimensional plane with a line. But what exactly are such fractals? And how can we write them with functional expression?
Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?  Thank you.
is a light side power that many Jedi, including  are capable of. In Episode I, Qui-Gon uses it twice (once to outrun droids and once on Jar Jar's tongue), Obi-Wan uses it once and Anakin uses high speed reflexes for the entire duration of the pod race with no formal training.  According to  the power has a high drain and requires a longer recovery period than other powers. And then of course there are hilarious drawbacks such as .  But why wasn't this power used in the following scenarios (and many, many others):   The Redshirt Jedi who accompanied Windu to arrest Palpatine, before they were summarily dispatched to the Nether. Qui-Gonn, to avoid a lightsaber through his gut. Yoda, to avoid flying senate seats. By Windu, who is supposedly a notable practitioner. Multiple File Extension Rename with single command I received some important files via mail. There are about 150 files like these. The extensions of the files are named as zip1,zip2,zip3, zip_ etc. I just want to change the extension of these files back to zip using a single command to avoid the manual task of renaming each file. Can I travel to Croatia with single-entry Schengen visa? I am traveling to Zurich with a single-entry Schengen visa.  While coming back can I travel to Croatia for 3 days and then return to India (Country of Origin)? Determining Min - Max within zip code (statistics by area)? I'm new to ArcGIS Desktop. I have created a layer representing zip codes and overlayed it with a layer containing some data points. I want to find the min and max value of the data points in each zip code. Currently, I am manually checking the values if all the data points in each zip code but this is becoming very time-consuming and cumbersome.   Is there a faster way to do this? In case of shell-builtins (eg type itself):  $ type type type is a shell builtin  $ which type &lt;Doesn't return anything since it's a shell builtin, silently exits&gt;   In case of commands (normally) (eg python):  $ type python python is /usr/bin/python  $ which python /usr/bin/python   In case of which (which is a command located at /usr/bin/which)  $ type which which is hashed (/usr/bin/which) $ which which /usr/bin/which   Why does type which say that which is hashed? What is the significance of which being hashed and what does it actually mean? What should a rational person accept as a miracle? How to monitor process/program execution in windows? We are trying to develop a small application that can monitor the programs/processes that are executing in a windows machine.  If the program/process is not supposed to run, it should be blocked. It works similar to an antivirus.  This is the basic idea.  I want to know the ways to hook into the OS to get notified about every single program/process trying to run in the machine. I several days ago somewhere read that category and tag pages should be noindexed. But later on somewhere else, I read that, they only increase rank for search term, but overall traffic decreases- said one from his experience.  So, if I just concern about traffic, not about individual search term that much, should I make those category and tag pages as indexed to google? I am confused which is the best to do.  Thanks in advance.  Regards Rana In the biblatex doc, it says ( Version 1.7, p.11):     thesis      A thesis written for an educational institution to satisfy the   requirements for a degree. Use the type field to specify the type of   thesis.      Required fields: author, title, type, institution, year/date      Optional fields: subtitle, titleaddon, language, note, location,   month, isbn, chapter, pages, pagetotal, addendum, pubstate, doi,   eprint, eprintclass, eprinttype, url, urldate   I would like to know if there is any "canonical" way to add the name of the supervisor(s) in a field of the .bib file (no .bbl manipulation please, I want my bibliography to be usable "as it" in several documents).  The expect result could be something like  :     Meand Myself. “A brilliant result, explained”. PhD thesis**, under the   joint supervision of Pr. Toto and Pr. Tata**. Université Paris 24,   Nov. 2013.   The "under the (joint) supervision" could be a string that would be declined in four ways:   under the supervision of  under the joint supervision of and the abridged forms Lightning Cancel Enqueued Server Side Action Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result? I am trying to remember the title of a children's SF book I once read, maybe 20 to 25 odd years ago.   I don't have a great number of details (and some of what I recall may be completely wrong), but the details (such as they are):   A war between two alien groups ended, and some technology that should have been destroyed was instead imprinted into a human baby. A few years later, both groups find this out and attempt to retrieve it. The main characters are the child, and an alien that I believe was called "Yacob" (or similar) who was sent to protect the child/retrieve the technology, and the antagonists were the other group of aliens that had previously lost the war and were trying to get the technology to start it up again (again, I don't remember the exact details, but I think it was a fairly simple black &amp; white/good vs. bad scenario, being a children's book and all). There was a scene in which Yacob/the alien gives the child blue food, who complains that it's blue, and the alien states "close your eyes and it's just food". The alien had a space shuttle/some other piece of tech disguised as a van. I vaguely recall a scene in which it is under attack and it flies or something. Cyclic Group Generators of Order $n$ How many generators does a cyclic group of order $n$ have? I know that a cyclic group can be generated by just one element while using the operation of the group.  I am having trouble coming up with the generators of a group of order $n$.  Any help would be great!  Thanks! A question about long exact sequence of finite dim vector spaces Q) Prove that if $0\rightarrow V_1\rightarrow V_2\rightarrow...\rightarrow V_k\rightarrow 0$ is an exact sequence of finite dimensional vector spaces, then $\sum_{i=1}^{k}(-1)^idim (V_i) = 0$.  For $k=3$, I see that the statement is true. For $k&gt;3$, am I correct in understanding that $V_2 = V_1\oplus V_3 (\implies dim(V_2)=dim(V_1)+dim(V_3))$ and $V_3 = V_2\oplus V_4(\implies dim(V_3)=dim(V_2)+dim(V_4))$ and so on?   If it is true, then for $k=4$, $\sum_{i=1}^{4}(-1)^idim (V_i) = -m_1+m_1+m_3-m_2-m_4+m_4$ and am not sure why $m_2=m_3$ for the sum to be $0$? where $m_i = dim(V_i)$. Thanks. Unable to 'apt-get dist-upgrade' due to appstreamcli database error I have created a USB stick with a persistent Ubuntu 16.04 (release version). I tried a couple of ways, using Startup Disk Creator and MultiSystem, but I get the same end result.  When I boot into the USB and do 'apt-get update', all is well.  When I try 'apt-get dist-upgrade' I get:     ** (appstreamcli:2761): CRITICAL **: Error while moving old database out of the way. AppStream cache update failed.   The only reference to this error message I can find is  but would appear to be fixed, at least fixed by the time the release version of Ubuntu 16.04.  Aside: I want to test Unity8, which includes installing software.  I don't have a spare computer, so I thought a persistent USB would do the trick.  Any takers? How to disable creation of desktop workspaces on macbook?   Somehow they get created when I don't need them. [OS: Mojave] How do I generate the upper triangular indices from a list? What is the preferred way to find focused control in WinForms app? Strange black shading. Cannot fully light mesh. (Cycles) Here is the viewport in material view.  The node setup is just a PNG texture into a diffuse shader.    And this is in rendered view.  I've tried using environmental lighting as well as a directional light, and while the lit parts become brighter/washed out, the darker parts stay dark.  Normals are correct and there is no z-fighting.  Ambient Occlusion is not enabled.  Changing to Flat shading makes it worse.    Update:  Any modifier that alters the mesh (bevel, subdivide, etc) make the problem go away instantly. I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:     Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for all $x \in \mathbb{R}$.   I am also wondering, can we find $f$ so that is continuous?  I was thinking of letting $f$ be a periodic function, and adding half the period to x each time. I had no success with this, and am now thinking that such a function does not exist.  Source: