It's common knowledge that the universe has three spatial dimensions, and one temporal. But how can we be so sure? Physics tells us that vector forces act independently along different dimensions, so without a source of force along the fourth dimension, or with a component in the fourth dimension, nothing would ever pass out of our little slit of 3D space in a possibly 4D world. Moreover, some materials exhibit different levels of friction, or any other measure of directional "resistance," (using the term as an abstraction for every barrier to motion, physical, fluid and so on) differently along different directions. We could just be stuck on our little 3D slice of 4D universe.  How can we possibly know the world is indeed three-dimensional, especially considering models of two-dimensional motion work so well in our three-dimensional world when motion along the third dimension is heavily restricted?  Note: The question  is somewhat similar in that it asks if spatial dimensions exist at all, but is not a duplicate. I did my research.
Coulomb's Law states that the fall-off of the strength of the electrostatic force is inversely proportional to the distance squared of the charges. Gauss's law implies that the total flux through a surface completely enclosing a charge is proportional to the total amount of charge. If we imagine a two-dimensional world of people who knew Gauss's law, they would imagine a surface completely enclosing a charge as a flat circle around the charge.  Integrating the flux, they would find that the electrostatic force should be inversely proportional to the distance of the charges, if Gauss's law were true in a two-dimensional world. However, if they observed a $\frac{1}{r^2}$ fall-off, this implies a two-dimensional world is not all there is. Is this argument correct? Does the $\frac{1}{r^2}$ fall-off imply that there are only three spatial dimensions we live in? I want to make sure this is right before I tell this to my friends and they laugh at me.
What happens 'internally' on a unstable/overclocked/undervolted CPU? When you undervolt or overlock an CPU, when it passes a certain limit it starts to make the system unstable, reboot, etc. But these are the effects of it.  What I'm curious about is what happens on the CPU. What happens? It gives wrong answers to instructions? Bits on cache flip? queries with db_select don't respect integers When I check "SHOW PROCESSLIST" in phpMyAdmin I can see lots of queries where argument-integers are being inserted as strings (with single quotation marks).  I'm wondering if this is due to the way phpmyadmin reports the queries (if it puts every argument between quotation marks) of if it is db_select that puts every argument between quotation marks.  An example of my code where integers are being used as arguments:  $select-&gt;join('taxonomy_index', 't_'.$key, 't_'.$key.'.nid = n.nid AND t_'.$key.'.tid = :tid_' . $key, array(':tid_'.$key =&gt; (int)$tid)); $select-&gt;condition('t_'.$key.'.tid', (int)$tid);   As you can see I explicitly write (int)$tid but in my query log I can see the following:  ... WHERE (t_0.tid = '10') ...   The big question is: How can I be sure that integers are being treated as integers in my queries? I don't want MySQL having to transform strings to integers. Calculus function create Create an example of a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(f(\mathbb{R}))) = f(f(\mathbb{R})) \neq f(\mathbb{R})$. If G is a group of order n=35, then it is cyclic I've been asked to prove this.   In class we proved this when $n=15$, but our approached seemed unnecessarily complicated to me. We invoked Sylow's theorems, normalizers, etc. I've looked online and found other examples of this approach.    I wonder if it is actually unnecessary, or if there is something wrong with the following proof:   If $|G|=35=5\cdot7$ , then by Cauchy's theorem, there exist $x,y \in G$ such that $o(x)=5$, $o(y)=7$. The order of the product $xy$ is then $\text{lcm}(5,7)=35$. Since we've found an element of $G$ of order 35, we conclude that $G$ is cyclic.   Thanks. Trying to run a Ubuntu server for the first time... But everything is in a terminal. Is there a way to switch to a GUI? Prove $\lim_{x\to0}\sqrt{4-x}=2$ using the precise definition of limits. (Epsilon-Delta) Prove $$\lim_{x\to0}\sqrt{4-x}=2$$ using the precise definition of limits. (Epsilon-Delta)  I am not sure how to link $0&lt;\left |x  \right |&lt;\delta$ with $\left |\sqrt{4-x}-2\right |&lt;\epsilon$ .  EDIT (Trying it out now)  I worked till here, then I basically got stuck. Cannot update/get any packages. gedit not installed my problem is that my sources.list is out of date for some reason. I already found the fix in another port, but it doesn't work. It says to use gedit to edit sources.list and replace everything with the proper sources. Well, for some reason I don't have it installed (Ubuntu Server 14.04) and I can't install it or else I get the main problem again.  Failed to fetch http:// some ubuntu site Could not resolve hostname "us.archive.ubuntu.com"  Is there any other way to edit sources.list? Or do I need to do a re-install of the OS? Introductory Calculus textbook for physics? Custom function won't refresh as inputs are changed How can I stop the symbolic x coords to be plotted twice? Bullied boy given immense powers by an alien race who destroys the human race Looking to identify a short story I read in a sci-fi anthology as a child, about an angry, bullied somewhat misanthropic boy who is given immense near-omnipotent powers by an alien intelligence. I read this in English, sometime in the late 1990s/early 2000s. He goes on to use this power to destroy the human race, which causes him to age terribly in a very short amount of time. At the end of the story he is a lonely old man stripped of the powers he held briefly, and is alone on an isolated Earth. The aliens then proceed to invade Earth. How can I have one keyboard input layout and multiples languages in Ubuntu 17.10? If I have English language I can't type "'e" and get an "é", as I can do in Portuguese and Spanish.  When I manage to do that on Settings -> Region &amp; Language, I'm able to change languages, but it also changes the keyboard layout.  Is there any way to change language (Portuguese or Spanish for example) and set the keyboard input layout (US International for example)? The binomial expansion:  $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$  The General Leibniz rule (used as a generalization of the product rule for derivatives):  $$(fg)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}$$  Both formulas can be obtained simply by induction; Newton's binomial also has a combinatorial proof . It's striking how these formulas are similar; is there a possible connection between them? I was thinking that maybe the General Leibniz rule could be obtained using a combinatorial argument as well (hence the binomial coefficients)... In the Harry Potter world, there's the term "muggle" for people with no magical abilities, "muggle-born" for those with magical abilities who's parents are muggles and "squib" for those with no magical abilities who's parents are magical.  But is there any collective term for all magical people (as opposed to wizards for males and witches for females)?  I don't recall seeing any word that refers to all men and women and children with magical abilities, and it's the only group left without noun or collective noun to describe them. In our application, we receive text files (.txt, .csv, etc.) from diverse sources. When reading, these files sometimes contain garbage, because the files where created in a different/unknown codepage.  Is there a way to (automatically) detect the codepage of a text file?   The detectEncodingFromByteOrderMarks, on the StreamReader constructor, works for UTF8  and other unicode marked files, but I'm looking for a way to detect code pages, like ibm850, windows1252.     Thanks for your answers, this is what I've done.  The files we receive are from end-users, they do not have a clue about codepages. The receivers are also end-users, by now this is what they know about codepages: Codepages exist, and are annoying.  Solution:    Open the received file in Notepad, look at a garbled piece of text. If somebody is called François or something, with your human intelligence you can guess this. I've created a small app that the user can use to open the file with, and enter a text that user knows it will appear in the file, when the correct codepage is used.  Loop through all codepages, and display the ones that give a solution with the user provided text.  If more as one codepage pops up, ask the user to specify more text. Simulate a uniform distribution on a disc I was attempting to simulate injection of random points within a circle, such that any part of the circle has the same probability of having a defect.  I expected the count per area of the resulting distribution to follow a Poisson distribution if I break up the circle into equal area rectangles.    Since it requires only placing points within a circular area, I injected two uniform random distributions in polar coordinates: $R$ (radius) and $\theta$ (polar angle).  But after doing this injection, I clearly get more points in the center of the circle compared to the edge.     What would be the correct way to perform this injection across the circle such that the points are randomly distributed across the cirlce? How to install software or upgrade from an old unsupported release? Recently I have installed an older version of Ubuntu on my old machine. Whenever I try to install any software, I get an error saying it couldn't be found:  $ sudo apt-get install vlc Reading package lists... Done                Building dependency tree        Reading state information... Done     E: Couldn't find package vlc How do I tell Finder to always show my Library folder on Mavericks? I thought there was a checkbox to suppress the hiding of my user Library folder on Mavericks, but I can't seem to find it. I appreciate hiding Library by default, but  have missed it being easily accessible.  How can I get the library folder to be visible in Finder, the Finder go menus / file save dialogs without needing to hold down shift? What is the difference between lands and mana? I am very confused about how to compute  $$\sum_{n=0}^{\infty}nk^n.$$  Can anybody help me?