I'm very new to data science. I'm wondering how is it possible to graph data of n dimension. For example by Clustering high-dimensional data      How do we graph these data points? Do we use a specific data structure? Do we take only the three more relevant dimensions?
I am trying to learn data in higher space into lower space. To have a clue, I'd like to know how to transform the data in the image below into a lower dimension preserving the structure. Hope to hear some explanations and what should I study to learn mapping data to lower dimension?
Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? I think the answer is yes.   Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then $\{(e_{\lambda_1},e_{\lambda_2}):\lambda_1,\lambda_2\in\Lambda\}\subset\mathbb{R}^2$ is a Hamel basis of $\mathbb{R}^2$. Consider set theoretic bijection $i:\Lambda\to\Lambda\times\Lambda$ define map $\varphi:\mathbb{R}\to\mathbb{R}^2$ by its action on element of Hamel basis of $\mathbb{R}$ by equality $\varphi(e_\lambda)=(e_{i(\lambda)_1},e_{i(\lambda)_2})$. This is an isomorphism between $\mathbb{Q}$-vector spaces $\mathbb{R}$ and $\mathbb{R}^2$.  Question Could you tell me is this proof correct, and if it is not, where is the mistake?  EDIT: Thanks to GAJO we found that this proof is wrong. The correct one basis is  $$ \{(e_\lambda,0):\lambda\in \Lambda\}\cup\{(0,e_\lambda):\lambda\in\Lambda\} $$ So we had to look at bijections of the form $i:\Lambda\to\Lambda\coprod\Lambda$ and the desired linear operator is defined by equalities $\varphi(e_\lambda)=(e_{i(\lambda)},0)$ if $i(\lambda)$ lies in the first copy $\Lambda$ and $\varphi(e_\lambda)=(0,e_{i(\lambda)})$ otherwise. How does Ubuntu partition the disk when we choose "Erase disk and install Ubuntu" during installation? How will Ubuntu automatically allocate the disk partition sizes when we select Erase disk and install Ubuntu during installation?    Can the settings for automatic space allocation be changed? When does decision tree perform better than the neural network? The space obtained by identifying the antipodal points of a circle Run a batch file in a completely hidden way I remember a story I have read buy could not remember the author the name or anything apart from the basic plot. The story is about some future where humanity has to fight with some aliens (not 100% sure). The story tells about a guy who was some kind of activist and who was in a prison two or more times and somehow helped humanity by being held in a prison (can't remember details). I think the story starts by saying that there is a statue in his honor that is not very well known but he did more to help humanity than famous politicians.   The story is most likely written by some famous English-speaking author, though I read it in a translation. If light rays obey to the wave equation, why can they be thought as straight lines? I'm a newbie with physics but I'm wondering how a ray of light can essentially be represented. I have always known that a ray of light proceeds in a straight line until it encounters another object (or material) that refracts/reflects it. But a light ray should be part of an electro-magnetic wave, is this correct? If it is so, it should obey to the  and this doesn't seem to me to describe a straight line ray.  I'm having problems visualizing how light is emitted and how it relates with the wave equation. Can someone with a clear understanding of the problem explain it to me in simple terms? I have one script to allow for one user to run. The string calling it from the command line looks like  ./script path/to/folder -a option='value'   When I try to add it to the other commands in Cmnd_Alias line, sudo blames sudoers file for incorrect syntax. How can I set such commands properly? {5,15,25,35} is a group under multiplication mod 40 Show that the set $\{5,15,25,35\}$ is a group under multiplication modulo $40$. What is the identity element of this group. Can you see any relationship between this group and $\mathbb U(8)$?   I am very stuck on this question and I think my knowledge of abstract algebra is my liability at the moment. Right now I'm most confused by the fact that I learned: An element of a group of this nature must be relatively prime with the working mod.  Obviously here this is not the case, so I don't really know where to begin.  Do I need a Cayley Table?  Thanks. Partial Sums of Geometric Series This may be a simple question, but I was slightly confused.  I was looking at the second line $S_n(x)=1-x^{n+1}/(1-x)$.  I was confused how they derived this.  I know the infinite sum of a geometric series is $1/(1-x)$.  I just can't figure out how the partial sums, $S_n(x)$, have $1-x^{n+1}$ on the numerator.  How was this derived?  Thank you.     Example 5.20.   The geometric series   $$     \sum_{n=0}^\infty x^n   = 1 + x + x^2 + x^3 + \dotsb $$   has partial sums   $$     S_n(x)   = \sum_{k=0}^n x^k   = \frac{1 - x^{n+1}}{1 - x} \cdotp $$   Thus, $S_n(x) \to 1/(1-x)$ as $n \to \infty$ if $|x| &lt; 1$ and diverges if $|x| \geq 1$, meaning that   $$     \sum_{n=0}^\infty x^n   = \frac{1}{1-x}   \qquad   \text{pointwise on $(-1,1)$}. $$   (Original image .) How is the popularity of "recent tags" calculated? How are these calculated? I see them going up and down all the time, and there are obviously way more than 48 questions about C# so how are these calculated? Show that quotient rings are not isomorphic I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic.    This is my attempt at a solution:  For $\mathbb{C}[x,y]/(y - x^2)$, we can parametrize in the following way: $x = t$ and $y = t^2$. Then this ring is isomorphic to $\mathbb{C}[t]$.  For $\mathbb{C}[x,y]/(xy-1)$, we can parametrize $x = t$ and $ y = 1/t$. Then this is isomorphic to $\mathbb{C}[t, 1/t]$.   But $\mathbb{C}[t, 1/t]$ is not isomorphic to $\mathbb{C}[t]$.  Am I on the right track? If not, any helpful hints? Background  Recently  was implemented. With this feature the amount of questions with new activity is shown in parenthesis at the beginning of the page title, e.g.     (2) Meta Stack Overflow   means that MSO has two questions with new activity since the user last viewed it.   this feature is unrelated to inbox updates. Here is an example to illustrate this:    Even though there are two inbox messages in this screenshot this is not reflected by the page title. The page title is currently only sensitive to questions with new activity.  Suggestion  I suggest that the page title also reflects inbox updates. For the screenshot above this would mean that the page title shows that there are two inbox updates and two questions with new activity.  I am not sure what format is best for showing this. For chat * in the page title is used to show that there are messages directed at you which is similar in meaning to inbox updates. However, * if is placed after the number in parenthesis to denote inbox updates there might be confusion as there is no connection between the number in parenthesis and * like in chat. Thus, one might prefer another format such as one of the following (all are meant to reflect the status of the screenshot above):      * (2) Meta Stack Overflow     2* (2) Meta Stack Overflow     [2] (2) Meta Stack Overflow     2i 2q  Meta Stack Overflow     (2/2)  Meta Stack Overflow     (2|2)  Meta Stack Overflow    For the alternatives without an explicit hint of which number refers to what it might be best to place the number for questions with new activity first as it will in most cases be greater. This will to a greater degree avoid combinations such as "(0/2)" which may be confused zero of two.  An advantage of the suggested feature is that if the user has a browser tab with a Stack Exchange site then they will be instantly notified of inbox updates. They do not need to focus a tab with a Stack Exchange site to know if there are inbox updates, they merely have to check the relevant page titles among their tabs.  A disadvantage of the suggested feature is that it might be experienced as verbose. I suggest that it is implemented such that it is as unobtrusive as possible while still being informative. Think of a situation where a user has   one Meta Stack Overflow tab with two inbox updates relating to Meta Stack Overflow and two questions with new activity, and one Super User tab with one inbox updates relating to Super User and three questions with new activity.   Then the sum of the inbox notification in the page titles should be local to sites and not global. The page titles for the tabs above should be something like the following pair:      2i 2q  Meta Stack Overflow     1i 3q  Super User Why is the prospective new kilogram standard a sphere? I can understand the choice of material, silicon 28, but why is it a sphere rather than (say) a cube? Article   I would have thought that a sphere would have been the hardest shape to machine accurately. What version of Ubuntu should I pick? I'm just a simple user frustrated with Windows 10 and I want to leave Windows altogether.  However, I don't want to spend three weeks learning how to install the new system and then get all my current programs to work.  I am not setting up a server, I am not a developer and I do not want all my stuff on a cloud. How can I pass a command line argument into a shell script? I know that shell scripts just run commands as if they were executed in at the command prompt.  I'd like to be able to run shell scripts as if they were functions... That is, taking an input value or string into the script.  How do I approach doing this? How do you compute negative numbers to fractional powers? My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a positive. I know that if a negative number is raised to an odd power it is negative, but fractional powers are neither odd or even. Is there a general rule for dealing with these types of problems? How to use a variable for a key in a JavaScript object literal? Automatically convert quotations in the form of "abc" to become ``abc" How do I apply differential cryptanalysis to a block cipher? I have read a lot of summaries of block ciphers particularly with regards to the NIST competitions stating that reduced-round block ciphers are – for example – vulnerable to differential cryptanalysis.  I have a general idea that the application of differential cryptanalysis is to look at the difference between inputs;  makes that fairly clear. However, I could take any two inputs for any given block cipher and I am pretty certain I'd be staring at random differences.    I am aware this is the idea of a well written block cipher; however, assuming a broken or vulnerable cipher (feel free to provide simple examples) how do I go about choosing differences to try?  Are there any clues in algorithm design that would inform a decision on which values to choose?  How does being vulnerable to differential cryptanalysis impact a cipher in the wild?  If all I have are differences between known plain-texts and known keys as my analysis and a captured ciphertext as my data to exploit, what can I actually deduce?