Natural Logarithm contradiction?
What's wrong with this?
Can we know the fundamental nature of space and time? Boggle: What is the dice configuration for Boggle in various languages? This is the dice configuration for the English Boggle:  Other variants and discussions concerning letter distributions for English can be found here:  , , ,  (Wikipedia).  I am interested in the 16 dice configurations for also other languages (Spanish in particular) and I am unable to find it anywhere online.  Versions I have managed to find so far:  ,  English 25 dice version:   Spanish 25 dice version:   Italian 25 dice version: Putting a piece of code in between two html tags in Solaris So I had asked a question prior to this one about recurrence relations, but apparently it was a bad one to ask. So I'm trying again to understand how to solve these babies... Here it is:  $$ 3a_{n+1}-4a_n=0, n&gt;=0, a_1=5 $$  What is the general process for solving a relation like this? List questions: Community Wiki? swap x and y axis on data generated plot How often did Gandalf return to Hobbiton after the events of The Hobbit? It has been a while since I read the Lord of the Rings novels (don't have a copy at the moment) but I seem to recall he went to Hobbiton at least once, as the young hobbits were always excited to see him.   Did he travel there often? Within my web application directory, how can I change all my folder permissions to rwxr-xr-x (755)? And how can I change all file permissions to rw-r--r-- (644)? I want to change all folders and files including sub-sub files and folders.   I know the following command would change all files and folders but I want to discrimate between a file and a folder.  chmod -R 755 ./my_web_app_project Show all of my question/answers to me even if they are deleted I would really like to see all of my questions and answers on the profile page, even if some of them were deleted, and I don't have enough rep to see them on the site. (Note that some questions are  after 30 days or 1 year, and the author might be oblivious about what happened.)   , the &quot;deleted recent [questions/answers]&quot; links in one's own user profile now list all respective posts that were deleted in the past 60 days, rather than just those that were originally posted in the last 60 days. , users with the &quot;access to moderator tools&quot; privilege (10k+ on designed sites, 2k+ on beta and non-designed sites) can use the search operator deleted:1 to see all their own deleted posts. , deleted questions and answers that were posted in the past 60 days, can be seen using the &quot;deleted recent questions&quot; and &quot;deleted recent answers&quot; links on the questions and answers tabs (at the bottom of the tab) in your user profile. For older posts: as of 2013-04-23 you can  if you already have a link to them, but they still aren't linked from the user profiles (not even just for you). Nor do you see inbox notifications for comments on them if you come back a few hours later. You have to have thought to bookmark your question, or you have to go digging for it in your browser history. I would like to add a new cable outlet to an outside wall and wanted to find out what kind of drill bit is needed to drill through and feed the coax cable as well as anything else to be mindful of. Comcast wanted $70 for a new outlet installation ($30 install, $40 tech visit), so I figured I could do it myself. The part that I'm most unfamiliar with is how to properly weather seal it. The outside just has vinyl siding and there's no electrical outlets nearby.  So far, I think I need the following   Coax Cable Coax Cable Wall Plate Long drill bit Add a 3-way Cable Splitter to cable box Weather sealer? Here is the top output which I gathered:    I noticed that top is showing vlc's cpu usage is > 100%.  Can anybody please justify me why it is showing like this, whether this is a bug in top application or something else. Is doing two PhDs a good path? Is Feyerabend confusing discovery and justification when he criticizes the scientific method? I am reading Feyerabend's "Against Method", where he uses Copernicus's (and Galileo's confirmation) discovery of the fact that the Earth orbits around the Sun and other examples to show that irrational approaches can lead to legitimate scientific progress. He then claims that no unified epistemic principle or principles can be used to define science, and that science is best characterized by anarchy.   Is he confusing the way scientific truth is discovered with the way scientific truth is epistemically justified?   Even if it is true that Copernicus, Galileo and others used irrational methods to arrive at valid scientific discoveries, why does that affect science's special epistemic status compared to other fields of inquiry? Question about Angle-Preserving Operators This an exercise out of Spivak's "Calculus on Manifolds".  Edit:  There was a typo in the exercise as is noted below in the answers.  The statement has been edited to reflect this.  Given $x,y\in\mathbb{R}^{n}$, the angle between $x$ and $y$ is defined by  $$\angle(x,y) = \arccos\left(\frac{\langle x,y \rangle}{|x|\cdot |y|}\right),$$ where $\langle x,y \rangle$ denotes the standard Euclidean inner product.  A linear operator $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is said to be angle-preserving if $\angle(T(x),T(y)) = \angle(x,y)$ for every $x,y\in\mathbb{R}^{n}$.  The exercise as stated:  Let $\{x_{1},\dots, x_{n}\}$ be a basis for $\mathbb{R}^{n}$.  Then suppose that $\lambda_{1}, \dots, \lambda_{n}\in \mathbb{R}$ are such that $Tx_{j} = \lambda_{j}x_{j}$ for each $j = 1,\dots, n$.  Then $T$ is angle-preserving only if (not if and only if!)$|\lambda_{i}| = |\lambda_{j}|$ for every $1\leq i\leq j\leq n$.  I'm having problems with the $(\Rightarrow)$ direction.  My best attempt (which seems to lead nowhere) is to suppose that $|\lambda_{j}|\neq |\lambda_{k}|$.  Then by assumption, \begin{align*} \angle(Tx_{j},Tx_{k}) &amp; = \arccos\left(\frac{\langle Tx_{j},Tx_{k} \rangle}{|Tx_{j}|\cdot |Tx_{k}|}\right)\\ &amp; = \arccos\left(\frac{\langle \lambda_{j}{x_{j}},\lambda_{k}{x_{k}} \rangle}{|\lambda_{j}{x_{j}}|\cdot |\lambda_{k}{x_{k}}|}\right)\\ &amp; = \arccos\left(\frac{\lambda_{j}\lambda_{k}\langle {x_{j}},{x_{k}} \rangle}{|\lambda_{j}|\cdot|\lambda_{k}|\cdot|{x_{j}}|\cdot |{x_{k}}|}\right)\\ &amp; = \arccos\left(\text{sign}(\lambda_{j})\text{sign}(\lambda_{k})\frac{\langle {x_{j}},{x_{k}} \rangle }{|{x_{j}}|\cdot |{x_{k}}|}\right)\\ \end{align*} may also be calculated as \begin{align*} \angle(Tx_{j},Tx_{k}) &amp; =  \angle(x_{j},x_{k})\\ &amp; = \arccos\left(\frac{\langle x_{j},x_{k} \rangle}{|x_{j}|\cdot |x_{k}|}\right). \end{align*}  Then since $\arccos$ is injective, I believe I can make the jump that $\text{sign}(\lambda_{j})\text{sign}(\lambda_{k}) = 1$, which does not resemble the conclusion that I should arrive at.  Note:  I wasn't sure what tag to put this under, so anyone who knows better please feel free to adjust.  Thanks for any help you can give. Sorry if this is very basic but here's a question.   Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$.      Why do we have   $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|,  $$   where the left-hand side can be thought of as the $2$-norm $\|\mathbf{v}\|_2$ on $L^2(k^n)$?   $\mathbf{General \; case}$: If this is true for $p=2$-norm, I am guessing that this is true for all $p\geq 1$:  $$ \left( \sum_{i=1}^n |v_i|^p\right)^{1/p}\leq \sum_{i=1}^n |v_i|.  $$     In fact, does this inequality hold when $p$ is a rational number? Override a method at instance level "Arnold raced out of the door": grammatical or not? Is there a PPA with the latest version of LibreOffice? There was recently another release of , but it is  a pain to have to manually go to their website and download the packages every time there is a new release. Is there a PPA that always has the latest version of LibreOffice? Is it possible to be admitted to a masters in Germany with a 3-year bachelors from another country? Why haven't I received the "Curious" badge?