How do I load an existing public and private key pair into SSH?
SSH and home directory permissions
Search across Stack Exchange sites in Android app Show that a continuous function has a fixed point Question: Let $a, b \in \mathbb{R}$ with $a &lt; b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.  I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ? Let $B$ be a commutative domain and $A \subseteq B$ a subring. Suppose that the extension $A \to B$ is finite, i.e. $B$ is finitely generated as an $A$-module. If $I \cap A = \lbrace 0 \rbrace$ for some ideal $I$ of $B$, can we conclude that $I = \lbrace 0 \rbrace$?  I know that if $I$ contains a nonzero prime ideal, the statement holds since $A \to B$ is an integral extension, whereby no two comparable prime ideals of $B$ can have the same intersection with $A$. But does the statement still hold if $I$ does not contain a nonzero prime ideal? And what if $A \to B$ is only assumed to be an integral extension? $$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you Lebesgue measure of the set of zeroes of Brownian motion Let $B_t(w)$ be a Brownian associated with a measure $P(dw)$. Suppose that $X$ is a random set of times $t$ where $0 \le t \le 1$ such that $B_t=0$. Show that $X$ has Lebesgue measure zero $P$ almost surely. What is the difference between ethics and morals? I am currently writing an essay on artificial moral agents, and I need to explain the difference between ethics and morals. However, I am finding it really hard to find a good book or research paper which contains a citable definition.   Could someone point me in the right direction? Official StackExchange app says my question is not available? I recently posted a new question on Android Enthusiasts using the official StackExchange app for Android. Right after posting the question, the app said that my question is not available.     The question is not available, it may have been deleted.   Deleted? Already?...  Then I navigated to my user name to see my recent activity. There it was! I could see the question I just posted. But if I tapped on the link to try to open it, the app kept complaining about my question not being available.     The question is not available, it may have been deleted.   But 7 minutes after posting the question, I could see my question. Here are some screenshots that showcase this issue.      This is not the first time this happens. This appears to be more or less normal behavior of the app. I don't use the StackExchange app very often to post questions. But I had exactly the same message come up when I posted a question a few months ago (and some app version numbers earlier).  This is not a major issue, but you should try to address this nevertheless. It's a small bug, but addressing it might imply much bigger changes... which might explain why it's still present... I don't really know, I'm guessing. I just thought I would let you know about it, in case you are not aware of it.  For the record, the phone is Galaxy S4 Plus (GT-I9506) running Android 4.4.2. Version number of the app is 1.0.50. Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous. Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty$. Prove that $f$ is uniformly continuous.   Since $f(x)\to 0$ as $x\to \pm\infty$ we have $|f(x)|&lt;\epsilon $ whenever $x&lt;-K$ and $x&gt;K$ where $K&gt;0$.  Also $f$ is uniformly continuous on $[-K,K]$ as the domain is compact.  Hence $f$ is uniformly continuous in $(-\infty,-K)\cup [-K,K]\cup (K,\infty)$.  But how to conclude that $f$ is uniformly continuous on $\Bbb R$ from above as I may chose $x\in (-\infty,-K);y\in [-K,K]$ ,how to conclude that $|f(x)-f(y)|&lt;\epsilon $ from above for a given $\epsilon$?  Please suggest the edits required. How do I sort a list of dictionaries by a value of the dictionary? I have a list of dictionaries and want each item to be sorted by a specific value. Take into consideration the list: [{'name':'Homer', 'age':39}, {'name':'Bart', 'age':10}]  When sorted by name, it should become: [{'name':'Bart', 'age':10}, {'name':'Homer', 'age':39}] Voldemort couldn't kill Harry as long as he called the Dursleys home. Why didn't he just murder the Dursleys? This would prevent Harry from being able to live there. What is the simplest SQL Query to find the second largest value? Satisfying explanation of Aristotle's Wheel Paradox. The paradox:    We have a circle and there is another circle with smaller radius. They are co-centeric.  If circle make full turn without sliding, both smaller and bigger circle make full turn too. If we assume that the passed road is equal to the circumference of circles. We have got smaller circle's radius is equal to bigger one's.  Unsatisfying Solutions I found:    "Do not assume that smaller circle's circumference is equal to passed road since the surface that contacts to the ground is bigger one. " // Okey but it does not explain the paradox, it explains just what is the wrong assumption (even does not explain why it is a wrong assumption.) It is undeniable that every point on both smaller and bigger circle will contact exactly one and only one point on their path. Therefore we can think that this is a bijective maps and smaller circle is isomorphic to bigger one. (Okey but ....)   Question: What is the true answer? What is wrong with the definition of circumference of a circle and relationship with its taken path. Separate long math text under sum symbol into different lines? I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines? How can light be called electromagnetic if it doesn't appear to be electric nor magnetic?  If I go out to the sunlight, magnets aren't affected (or don't seem to be).  And there is no transfer of electric charge/electrons (as there is in AC/DC current in space).   In particular, the  (which light is supposed to be composed of) have no electric charge (nor do they have ).   I'm looking for an explanation that can be appreciated by the average non-physicist Joe. Headline Language Is there a particular term for the abbreviated language used in headlines (the removal of at least articles and conjunctions)? Could a ship equipped with Alcubierre drive theoretically escape from a black hole? Could a ship equipped with Alcubierre drive theoretically escape from a black hole?  Also, could it reach parts of the universe that are receding faster than the speed of light from us? How to clear current “read-only” state on a USB flash drive I setup with Rufus? How to open files from other machine running ubuntu I tried to take data from my pc to laptop. Both are using ubuntu 11.10 os. I googled and tried to connect both the systems. Now it is showing connected symbol. Dont know how to proceed further. I need to take data from my pc using this lan connection. Please provide me solution for this... Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$.  For example: $32 = 21 + 8+3 = F_8+F_6+F_4$  Research effort:  The base step it's simple:  Let $k=1$ it can be britten as $k=1=F_2$  For the inductive step I considered:  Let $k = k  F_2 =F_2+F_2+\cdots+F_2 = \sum_{i=2}^w a_iF_i, a_i =\{0,1\} $  Then if $k$ suffice I want to see if $k+1$ suffices too... But I'm not realy seeing how to use the inductive hypothesis, so I assume It's wrong.  Any thoughts on which can the inductive step be? I was just wondering whether the following statement is correct.  Let R be a ring and M a noetherian R module. Then M is finitely generated.