Why does the chair and the grandpa look bigger and inflated?
Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens?
How to "cat" or "less" a libreoffice writer .odt file? How to fix task manager memory column if it shows lies I have a problem in showing memory in task manager with memory and paging columns. Why it shows so small size usage? And maybe anyone knows how to fix it?  Here is a screenshot(working set, memory, page pool) sorted by most consuming: Unitarily evolving a separable state into an entangled one Suppose, I want to refer multiple equations (1), (2), (3), (4), (5) and (7) so that they appear as (1-5, 7):  \documentclass[fleqn,preprint,10pt]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \begin{document} \begin{align} \label{1}\psi([e_1,e_2]) = \psi(e_2) = e_2 = (e_2,2e_3) = (-2e_3,e_2) = (\psi(e_1), \psi(e_2))\\ \label{2}\psi([e_1,e_3]) = \psi(2e_3) = 2e_1 = (e_1,2e_3) = (-2e_3,e_1) = (\psi(e_1),\psi(e_3))\\ \label{3}\psi([e_2,e_3]) = 0 = (e_2,e_1) = (\psi(e_2),\psi(e_3)),\\ \label{4}\left\{e_4, e_1\right\},\,\,\,\left\{e_4, e_2\right\},\,\,\,\left\{e_4, e_3\right\},\,\,\,\left\{e_4, e_1+\epsilon\,e_2\right\},\\ \label{5}\left\{e_1, e_2\right\},\,\,\,\left\{e_3, e_1\right\},\,\,\,\left\{e_3, e_2\right\},\\ \label{6}\left\{e_4, e_1\right\},\,\,\,\left\{e_4, e_2\right\},\,\,\,\left\{e_4, e_3\right\},\,\,\,\left\{e_4, e_1+\epsilon\,e_2\right\},\\ \label{7}\text{Nor}_{L_4\oplus L_3}\left(\{e_2\}\right)/\{e_2\}=\,\{e_1, e_3, e_4\} \end{align} \end{document} What is the sum of $$\sum_{n=0}^{\infty}\frac{1}{2^n(n+1)}$$    I've spent an insane amount of time on this problem. I checked on Wolfram and it gives $ln(4)$, which I assume you get from $2\ln(2)$.   I know that the integral of $\frac{1}{n+1}$ is $\ln(n+1)$. What am I missing?  Thank you! DRAM Frequency showing 800 instead of 1600 I am running a Core i7 CPU originally at 3Ghz overclocked to 4Ghz using a multiplier of x20 and a Bus Speed of 200 Mhz  My memory is 1600 Mhz OCZ.  I installed CPU-Z to see what is going on - but it shows me (under the "Memory" Tab):  DRAM Frequency 800 Mhz  Channel Triple - NB Frequency 3800.2 Mhz  So what is the actual frequency its running on then? I am slightly confused. Labeling all sides of the matrix with kbordermatrix I have a matrix with labels on the left and top. Now I want to have a label on all sides of the matrix. How can I change my code?  \documentclass[10 pt, a4paper, leqno, oneside] {report} \usepackage{kbordermatrix} \begin{document}  \renewcommand{\kbldelim}{(}% Left delimiter \renewcommand{\kbrdelim}{)}% Right delimiter \[   \tilde{\textbf{m}} = \kbordermatrix{     \mbox{} &amp; c_1 &amp; c_2 &amp; c_3 &amp; c_4 &amp; c_5 \\     % &amp; c_1 &amp; c_2 &amp; c_3 &amp; c_4 &amp; c_5 \\     r_1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 \\     r_2 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\     r_3 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1 \\     r_4 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 \\     r_5 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1    } \]  \end{document} Why do some websites show the company name next to the URL? Some websites show the company name next to the URL (in Chrome) and some don't.  How do I set this up for my website? Is there a reason why or why I wouldn't want to do this? How can I resolve the error "cannot execute binary file"? When I login using SSH, all I can see is this...  -bash: /usr/bin/id: cannot execute binary file -bash: [: : integer expression expected   I couldn't do anything in here. Commands such as halt, poweroff, reboot will return command not found.  How can I fix this? I am using Debian Squeeze Linux Slower download speed compared to Windows on Ubuntu 20.04 I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD.   I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are all irreducible. Is there someway to conclude that $\mathbb{Z}[\sqrt{-n}]$ is not a UFD based on this? Thanks. It is quite easy to show that (by using residue theorem) $$\int_0^{\infty} \frac{1}{1+x^n}dx = \frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)} $$ for $$n \ge 2$$  Is there any possibility to simplify $$\frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)}$$ or it is best result?  Thanks in advance! Video editing software options? How do I get a consistent byte representation of strings in C# without manually specifying an encoding? Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$ The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$  But I don't know how to explain it. I understand that the derivative of the equation is a truth clause, but why would the following be true, intuitively?  $$\int^{x}_{C1}\frac{1\cdot dx}{\sqrt{1-x^{2}}} + \int^{x}_{C2}\frac{-1 \cdot dx}{\sqrt{1-x^{2}}} =\\  \arcsin(x) - \arcsin(C1) + \arccos(x) - \arccos(C2) = 0 \\  \text{while } \arcsin(C1) + \arccos(C2) = \frac{\pi}{2}$$  I can't find the right words to explain why this is true?    Edit #1 (25 Jan, 20:10 UTC): The following is a truth clause: $$ \begin{array}{ll} \frac{d}{dx}(\arcsin(x) + \arccos(x)) = \frac{d}{dx}\frac{\pi}{2} \\ \\ \frac{1}{\sqrt{1-x^{2}}} + \frac{-1}{\sqrt{1-x^{2}}} = 0 \end{array} $$  By integrating the last equation, using the limits $k$ (a constant) and $x$ (variable), I get the following:  $$ \begin{array}{ll} \int^x_k\frac{1}{\sqrt{1-x^{2}}}dx + \int^x_k\frac{-1}{\sqrt{1-x^{2}}}dx = \int^x_k0 \\ \\ \arcsin(x) - \arcsin(k) + \arccos(x) - \arccos(k) = m  \text{ (m is a constant)}\\ \\ \arcsin(x) + \arccos(x) = m + \arcsin(k) + \arccos(k) \\ \\ \text{Assuming that } A = m + \arcsin(k) + \arccos(k) = \frac{\pi}{2} \text{ ,for } x \in [-1,1] \end{array} $$ Using Calculus, why is that true for every $x \in [-1,1]$?    Edit #2:  A big mistake of mine was to think that $\int^x_k0 = m \text{ (m is const.)}$, but that isn't true for definite integrals.  Thus the equations from "Edit #1" should be as follows: $$ \begin{array}{ll} \int^x_k\frac{1}{\sqrt{1-x^{2}}}dx + \int^x_k\frac{-1}{\sqrt{1-x^{2}}}dx = \int^x_k0 \\ \\ \arcsin(x) - \arcsin(k) + \arccos(x) - \arccos(k) = 0\\ \\ \arcsin(x) + \arccos(x) = \arcsin(k) + \arccos(k) \\ \\ A = \arcsin(k) + \arccos(k) = \frac{\pi}{2} \text{ ,for } x \in [-1,1] \end{array} $$ First time player in a group figuring out things as we all go.  I just hit level four as a Ranger, and I get to improve one ability score of my choice by 2, or I can increase two ability scores of my choice by 1.  My first thought was Strength, as I've been rocking a score of 11 with a mod of +0 so far. But I read that Rangers use (wisdom mod + 10) for their spell casting modifier. Currently my wisdom is 13, with a mod of +1. If I increase either STR or WIS, will the modifier increase as well? How and why?  A Monk in our party went from 9 CON to 11 during their ability score improvement, and by doing so their modifier went from −1 to 0.  We're all pretty new, so between eight people we're currently sharing one book, and all trying to locate our own copies for later campaigns. That's making it… inefficient for us to learn these things quickly. Why didn't the Ministry use owls to find Sirius? Harry, before the Triwizard Tournament, was able to send owls to Sirius when he was in hiding (not at Grimmauld Place) just by telling them "just find him, will you?", and the owl would come back a few days later with an answer. This doesn't happen with Hedwig only, because, if I recall correctly, he uses school owls as well.  If finding Sirius was as easy as telling an owl "find him", and a 13-year-old wizard could do it, how come the Ministry had no clue as to where he was? They could've sent an owl with a summons and followed it from a distance.  Two theories, neither mentioned in the book, and both a bit unlikely:   He was living as a dog and -hopefully- the Ministry would not be able to recognize him. Unlikely because the owl would have got there anyway, so they would have guessed the dog was Sirius in disguise, not necessarily as an animagus, but maybe he could perform human-animal transfiguration (remember he was brilliant at school).. Kingsley Shacklebolt was already in charge of the hunt and Dumbledore told him Sirius was innocent and all that. Then, Kingsley would have prevented the Ministry from using any method that could've actually worked. Unlikely because they could've used owls before Dumbledore's tip, when Kingsley didn't know Sirius was innocent, and also because Kingsley was quite powerful and had the support of Crouch and others at the Ministry, so if he was in charge, they would probably have succeeded.   Your thoughts? What would be the optimum design/structure for product data database, where each product have many variations and many attributes? Attribute sets vary between products, and some attributes have values common for all variations of a given product. A typical example of product variations &amp; attributes table is in the image below:    (The 'ATTRIBUTE X' is here to give you and idea of a different attribute type than for 'Product 1').   Would it be practical to try to 'group' some of the attributes (to save time and minimise errors when creating new product entries)? If so, how complicated would it need to be?  If it's relevant - I'm thinking of using mySQL engine, but I'd be happy to see any applicable example from other engines.  EDIT : More information: at the moment we have over 400 variations in a spreadsheet, each one about 15-20 attributes, so that would give about 6000-8000 'data points'. It would be great to be able to list all possible values for a given attribute (to create a dropdown menu when adding new entries), and also to have a choice of possible attributes for a given product. It would be great to be able to specify which attributes are applicable for product variations of each product. I am trying to host some static content in an Amazon S3 bucket and to have that content available through my own domain.  The bucket in question is called "e.tl-dr.org" and I have "index.html" set as the index document. I can visit  and I see the contents of index.html (just "test"). This works fine.  I also have added the CNAME entry for "e.tl-dr.org" pointing to "s3.amazonaws.com" from within my registrar's site. (I am not using the AWS Route 53 DNS services, which is what most tutorials about this seemed to be geared at.) So now "" correctly shows the file, but just the bare domain only gives me S3's standard 403 message.  Is there anything else I need to set up to get this working? chkrootkit shows "tcpd" as INFECTED. Is it a false positive?