I wonder what is the usefulness of k-means clustering in high dimensional spaces, and why it can be better (or not) than other clustering methods when dealing with high dimensional spaces.
How would you know if your (high dimensional) data exhibits enough clustering so that results from kmeans or other clustering algorithm is actually meaningful?  For k-means algorithm in particular, how much of a reduction in within-cluster variance should there be for the actual clustering results to be meaningful (and not spurious)?  Should clustering be apparent when a dimensionally-reduced form of the data is plotted, and are the results from kmeans (or other methods) meaningless if the clustering cannot be visualized?
proving how many permutations exist. Show that the number of permutations of n numbers, for n ≥ 2, with two cycles is at most (n−1)!logn. Unsure as to were to start any help would be appreciated. The usual Schrodinger Lagrangian is $$ \tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi, $$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ vanishing. This Lagrangian density is not real but differs from a real Lagrangian density  $$ \tag 2 \frac{i}{2}(\psi^{*}\partial_{t}\psi -\psi \partial_{t}\psi^{*} ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi $$ by a total derivative.  My trouble is that these two lagrangian densities lead to different conjugate momenta and hence when setting equal time commutation relations, I am getting different results (a factor of 2 is causing the problem). I can rescale the fields but then my Hamiltonian also changes. Then applying Heisenberg equation of motion, I don't get the operator Schrodinger equation.   Is it possible to work with the real Lagrangian density and somehow get the correct commutation relations? I would have expected two Lagrangians differing by total derivative terms to give identical commutation relations (since canonical transformations preserve them). But perhaps I am making some very simple error. Unless all conjugate momenta are equivalent for two Lagrangians differing by total derivatives, how does one choose the correct one?   I guess the same thing happens for other first order systems like Dirac Lagrangian also. Prevent back button after logout Regularization using methods such as Ridge, Lasso, ElasticNet is quite common for linear regression. I wanted to know the following: Are these methods applicable for logistic regression? If so, are there any differences in the way they need to be used for logistic regression? If these methods are not applicable, how does one regularize a logistic regression? Time limited and band limited signal What does a leading zero do for a php int? When should I use the Visitor Design Pattern? How can I add a bookmark in my nautilus folder explorer In Ubuntu 11 I just dragged the folder there.  This seems to be not possible on my new Samsung ultrabook with Ubuntu 12.04   I've tried Ctrl+D and using the top menu but nothing happens and no error messages, just nothing.  How can I add a bookmark? How to combine measurement error with statistic error BSOD: Driver State Power Failure For the past two weeks, I've been getting this BSOD error on my Lenovo laptop. I have even reset my laptop ("rest this PC"), but it's still happening.  My laptop details:     I have the latest Nvidia driver for my Nvidia Geforce MX150: 431.60 released on 23 Jul 2019.  I have the latest BIOS for my laptop: 8TCN53WW15  released on 15 Jul 2019.     I have run WhoCrashed, it says it says the problem was caused by some driver which could not be identified. If I get to know the problematic driver I will update/uninstall it.  How can I analyze my dump files, so that I can find the problematic driver?  Edit 1: If you get "Oops there was a problem with the network" on google drive page while accessing dump files, just click on download. They will be downloaded. I don't know what's causing this. Evaluate forecasting ability of GARCH models with RMSE and MAE She put forward two proposals, .......... of which didn’t make any sense to the participants in the summit.  The answer for this question in grammar exercises is ‘either’. I am doubtful whether to use either or neither. Can any one clarify this with reason. Programming language books explain that value types are created on the stack, and reference types are created on the heap, without explaining what these two things are. I haven't read a clear explanation of this.  I understand what a stack is. But,    Where and what are they (physically in a real computer's memory)? To what extent are they controlled by the OS or language run-time? What is their scope? What determines the size of each of them? What makes one faster? How do I work recursively through a directory tree and execute a specific command on each file, and output the path, filename, extension, filesize and some other specific text to a single file in bash. As per title, I wish to know if I can, after a successful 32bit setup and consequent software installations, update to a 64bit version.  I know how to partition (actually one of the solutions is to set /, /etc, /home, /var/www, and /opt as separate partitions) and I know that a clean install is way better than a dirty one, yet I would like to know if/how it's possible to do that. How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$. Prove that there is no continuous functions $f:[0,1]\to R$, such that   $$ f(x)+f(x^2)=x. $$   My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\in [0,1]$, \begin{align*} f(x)&amp;=x-f(x^2)=x-\big(x^2-f(x^4)\big)=x-x^2+\big(x^4-f(x^8)\big)=\cdots\\ &amp;=x-x^2+x^4-\cdots+(-1)^n\left(x^{2^n}-f\big(x^{2^{n+1}}\big)\right) \end{align*}  since $f(0)=0$ and $\displaystyle\lim_{n\to \infty}x^{2^{n+1}}=0$ for any $x\in(0,1)$,it follows by the continuity of $f$ that $\displaystyle\lim_{n\to \infty}f\big(x^{2^{n+1}}\big)=0$, hence $$f(x)=x-x^2+x^4-x^8+\cdots+(-1)^nx^{2^n}+\cdots$$ for any $x\in (0,1)$  Why do I have prove that there is exists such a continuous functions $f$? Maybe  my example is wrong? Why?  if my   method is wrong,then How prove this problem ? Thank you. To flag or not to flag question that don't show own effort? I am a little confused about what do with questions that are just asking      "I want this and that (...) and I want it to look exactly like that   (...) please help"   without showning ANY own effort?  According to  those kind of answers are not welcome  also the  says those questions should not be asked:     Don't ask about...   Questions you haven't tried to find an answer for (show your work!)   But does this mean that I should flag those kind of "helpdesk" questions or not? And if yes as what? low quality ? flag for moderator attention ? Or is just adding a kind comment asking for the asker's own attempts enough? What does 'exec {fd}</dev/watchdog' do in Bash That is literal, {fd} isn't a placeholder. I have a script that does this, and does not source in anything, nor does it reference {fd} anywhere else.  Is this valid bash?  exec {fd}&lt;/dev/watchdog Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational? Suppose we wish to prove that the square root of two is irrational. We begin by assuming that it is rational.  Namely, where both $a$ and $b$ are integers $$\frac{a}{b} = \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GH9aqpdaGcaaqaaiaaikdaaSqabaaaaa!31D3! $$  Why is it so important that both $a$ and $b$ be coprime and the fraction be irreducible?  That is, if $a$ and $b$ are not coprime and the fraction is indeed reducible, then$$\frac{a}{b} \ne \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GHGjsUdaGcaaqaaiaaikdaaSqabaaaaa!3294! $$ Create veracrypt traveler disk in Ubuntu According to , it is possible to create a portable "Traveler Disk" in the application:      Tools -> Traveler Disk Setup      You can use this facility to prepare a   special traveler disk and launch VeraCrypt from there. Note that   VeraCrypt ‘traveler disk’ is not a VeraCrypt volume but an unencrypted   volume. A ‘traveler disk’ contains VeraCrypt executable files and   optionally the ‘autorun.inf’ script (see the section AutoRun   Configuration below). After you select Tools -> Traveler Disk Setup,   the Traveler Disk Setup dialog box should appear. Some of the   parameters that can be set within the dialog deserve further   explanation:   My problem is I can't seem to find Tools → Traveler Disk Setup.    Am I missing something? I want to be able to share files on this disk with people running MS Windows.