If $U\subset \mathbb{R^n}$ is open then $\dim U=n?$
Dimension of open subsets of $R^n$
How do I change the full background color of the console window in C#? Number of suggested edits seems to include own edits How can I randomise the output of a redstone circuit? I have 5 buttons and only one of these can be correct, all others must be false. These 5 outputs will go into 2 outputs, in between there needs to be a circuit which changes which single button out of the 5 will go down the &quot;correct&quot; output of the final 2. The 4 remaining buttons must go to the second of the final 2 outputs. Both final outputs must give out a true signal for a brief period the same as a button on its own would produce. There should also be a way of activating the circuit. Not sure if that is explained well enough so I have made a basic diagram of what I need.  The diagram shows how I need the signal to be split. However in between the buttons and the final two wires there needs to be a circuit (marked by the grey box). This circuit changes the single button connected to the left wire. The right input should start the circuit and randomise the 5 buttons. There are no limits to size, amount of materials used or method used. This question is similar to the one asking about , but I am specifically working with Redstone circuitry - the commands solutions do not apply here Question about the Riemann-Lebesgue Lemma proof Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$  I get all the steps of the proof, except the one which says that if $f$ is an arbitrary integrable function, it may be approximated in the $L^1$ norm by a compactly supported smooth function $g\in L^1$. My question is: Why? How do you know that for all $\epsilon&gt;0$, and for all arbitrary function $f\in L^1$ exists a compactly supported smooth function $g$ so that $||f-g||_{L^1}&lt;\epsilon$? The "proof" can be found in many places, but this step can be found in Wikipedia What is the name for the grammatical device of putting "not" after a verb to negate it? MacBook Pro never goes to sleep I got a brand new mid-2015 MacBook Pro 15", OSX 10.11.3.  Display never turns off even it is configured to turn off after 10 mins in power adapter AND in battery mode. Screen saver (aerial) stays active forever. I tried disconnecting all USB peripherals connected - it did not help.  Any ideas? What IDE to use for Python? What IDEs ("GUIs/editors") do others use for Python coding? I asked a question almost as soon as I got my account. I got a few decent answers, but none of them helped solve my problem. The question has long since become buried by newer questions, and there's still no accepted answer. Unfortunately, because there are quite a few answers with votes, to a casual observer it probably looks like the question was answered and I just neglected to mark an answer as accepted, and the system does not consider it "unanswered".  Or, I asked a question that was a bit more difficult to answer. After waiting for some time, I received no answers, little or no comments, and only a few views. The question does show up in the "unanswered" list, but no one has answered my question.  In either case, what's the protocol for me to try to get this question answered?  I see several options:   Post the question a second time, possibly linking to the original, to get attention for the original question. This is encouraged on sites like Reddit.  However, it's probably not appropriate here, since we're encouraged to check for duplicates and not submit a question that's already been asked by someone else, so this probably falls under that heading. Wait for some kind of wiki-like talk page to be implemented, then post a plea on the talk page for input and hope that activity on the talk page bumps up the question. Edit the question, but do nothing further on the Stack Exchange site. If I edit my post to say that I haven't accepted any of the current answers but am looking for more, then the question might get answered eventually by people browsing through old questions to find something they can answer. Post the question on other sites, but link to my original question on Stack Exchange. People will either answer here, or they'll answer on the other site and I can post the accepted answer here myself.  This might be considered spamming, so it might be better to do this without the link.      For more information, see the "" page in the . $\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$ Let $a,b,c,x,y,z&gt;0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq  a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}&gt; 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq  \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ First solution Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz)$$ for all non-negative $x,y,z$ such that $x+y+z=1$. How about another solution? I really hope to see that. Thanks! Prove recursively defined sequence converges I would like some advice on how to solve problems like the following: Let $(x_n)$ be a sequence defined by $x_1= 3$ and $x_{n+1} = \frac{1}{4-x_n}$. Prove that the sequence converges.  My strategy is to use the Monotone Convergence Theorem, but I am having trouble showing that the sequence is decreasing and bounded below. Here's my work so far:  Decreasing: The first 4 values are $3,1,1/3,3/11$, so let's assume $x_n \leq x_{n-1} \leq \ldots \leq x_1$. Want to show $x_{n+1} \leq x_n$. We have $x_{n+1} = \frac{1}{4-x_n}$. I want to have an upper bound for the RHS, but can't find one and don't really know where to go from here.  Bounded below: I wanted to show that all values are positive, but if we assume $x_n &gt; 0$, that doesn't rule out $x_{n+1}$ from being negative. Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum_{k=1}^n\frac1{\sqrt k}&lt;2\sqrt n$$ Please help! How can i show with quite simple methods that a group of order 15 is cyclic? real memory usage How do I install the Tor Browser Bundle in Ubuntu? In Ubuntu 13.04 I used to use the ppa:upubuntu-com/tor64  PPA to install the 64-bit Tor Browser Bundle. However, in Ubuntu 13.10 it doesn't work anymore. How do I install the Tor Browser Bundle in Ubuntu? How can I run an external program from C and parse its output? I've got a utility that outputs a list of files required by a game.  How can I run that utility within a C program and grab its output so I can act on it within the same program? UPDATE: Good call on the lack of information.  The utility spits out a series of strings, and this is supposed to be portable across Mac/Windows/Linux.  Please note, I'm looking for a programmatic way to execute the utility and retain its output (which goes to stdout). What’s the best way to reload / refresh an iframe? I would like to reload an &lt;iframe&gt; using JavaScript. The best way I found until now was set the iframe’s src attribute to itself, but this isn’t very clean. Any ideas? Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.  But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.  This brings me to my question (provided there are any)     Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?   Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help! How to cancel a credit card and verify it was closed Proof that no term (of a language of predicate logic) is a (non-empty) initial segment of another Field between the plates of a parallel plate capacitor using Gauss's Law