Show that $d_1=\min(d(x,y),2)$ is a metric space if it is given that $d(x,y)$ is a metric space.   I am stuck at the triangle inequality part, to show that   $d_1(x,z)\leqslant d_1(x,y)+d_1(y,z)$ i.e to show that:   $$\min(d(x,z),2)\leqslant\min(d(y,z),2)+\min(d(x,y),2)$$  It is not a duplicate of  "brute force" question, maybe it is a duplicate of , can someone please post another answers before you are colsing as duplicate, I promise to vote.
Is the following just a matter of showing the 3 properties that make up a metric??  Define d on $\Bbb R\times\Bbb R$ by $d(x,y)=\min \{1,|x-y|\}$. Show that $d$ is a metric on $\Bbb R$     $d(x,y)=0$ if $x=y$   $d(x,y)=d(y,x)$ for every $x,y \in X$   $d(x,y)\le d(x,z)+d(z,y)$ for every $x,y,z \in X$
How do I recover the default version of some configuration file? By accident, I messed up a configuration file. How do I recover the default version without reinstalling or affecting other things? Inverse elements in the absence of identities/associativity. when upgrading to a 200 amp service panel with service drop coming from bottom underground do I need southern California Edison to come disconnect and then reconnect or can I just do it on my own. I do have a permit. How to remove film from stainless steel pan I have an unsightly film around my stainless pan as shown here:    What is causing this to happen?  And how can I remove the film? I have soaked it in soapy water, scraped it with a plastic scraper, and run it in the dishwasher several times, but I can't seem to get rid of the film.   UPDATE: the pan is an All-Clad stainless, fully-clad frypan: I am asking this question because I've always had the impression that manuscripts submitted to a journal in my field (theoretical linguistics) take an insanely long time to get published, and sometimes I get the impression that this is because, most of the time, nobody other than the author cares about getting things done within a reasonable amount of time. Some examples from personal experience follow.   For the last paper I submitted before getting my PhD, I had to wait nine months from the submission date to get reviews back from the referees. Last week I got a review request from a journal I had never reviewed for previously. The review deadline is March 15 (almost four months). In early October, I submitted a manuscript to a journal that I know asks reviewers to return reviews within 6 weeks. I checked the status of the manuscript online and it still says "with editor". Earlier this year, I was about to start writing a review when my wife went into labor. By the time we came back from the hospital, I had forgotten about the review, so I missed the deadline. The editor didn't contact me to ask about the review until two months after the deadline. Late in 2013, I was asked to write a survey chapter for a handbook. The expected publication date of the handbook is summer 2016.   All of this is not me being unlucky with editors and referees. In private conversations through the years, many colleagues have confirmed that it is virtually unheard of to get reviews back within 3 months of submission; 5 or 6 months seems to be a good average, but in some cases it can take longer (see my first point above). In general, everybody seems to have accepted that the submission-to-printed-journal process is going to take a bare minimum of one year, and that's assuming that the manuscript is accepted without revisions. If revisions are required, two years is not out of the question. Also, it's not like our papers are ridiculously long. I've reviewed about 100 papers so far, and the longest one was about 50 single-space pages; 30-35 single-spaced pages is closer to average.  I remember that, when I was a postdoc, my astrophysics housemate mentioned that in his field the entire submission-to-printed-journal process took only three or four months. I really can't see a good reason why theoretical linguistics shouldn't operate on similar timeframes, other than the fact that everybody seems to have grown used to things taking way longer than they really need to. What are the wait times in other fields? How do you manage to get things done quickly? Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$ Let $n$ be a positive integer. Prove that   \begin{align} \sum_{k=0}^n \left(-1\right)^k \binom{n}{k} = 0 . \end{align}   I tried to solve it using induction, but that got me nowhere. I think the easiest way to prove it is to think of a finite set of $n$ elements, but I can't find the solution. How should I structure my URLs for both SEO and localization? When I set up a site in multiple languages, how should I set up my URLs for search engines and usability?  Let's say my site is www.example.com, and I'm translating into French and Spanish. What is best for usability and SEO?  Directory option:  http://www.example.com/sample.html http://www.example.com/fr/sample.html http://www.example.com/es/sample.html   Subdomain option:  http://www.example.com/sample.html http://fr.example.com/sample.html http://es.example.com/sample.html   Filename option:  http://www.example.com/sample.html http://www.example.com/sample.fr.html http://www.example.com/sample.es.html   Accept-Language header:  Or should I simply parse the  header and generate content server-side to suit that header?  Is there another way to do this? If the different language versions don't have different urls, what do I do about the search engines?    UPDATE 2011-12-06  Google has new recommendations for meta tags for explicitly pointing to other language content: .  UPDATE 2012-05-25  Related but not precisely:   UPDATE 2013-06-12  includes discussion of several URL schemes directly relevant to the question. What did Ken Thompson mean when he said, "I'd spell creat with an 'e'."? What is Ken referring to? Is there a "creat" command? Decomposition of semisimple Lie algebra via its roots Workflow for UV orientation How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 Simple Hadamard Circuit gives incorrect results? extending convergence radius using one boundary point Show that if $f(z)=\sum\limits_{n\geq0}a_nz^n$ is analytic in $\{z\in\mathbb{C}:|z|&lt;1\}\cup\{1\}$ and $\forall n\geq0:a_n\geq0$ then the radius of convergence of the power series is strictly larger than $1$.  My approach:   Let $D=\{z\in\mathbb{C}:|z|\leq1\}$.  We have that $f(1)$ converges, so $\sum\limits_{n\geq 0} a_n&lt;\infty$. Moreover since the $a_n$'s are all positive we have that $|a_n|=a_n$, therefore $\sum\limits_{n\geq 0} |a_n|=\sum\limits_{n\geq 0} a_n&lt;\infty$.  Let $f_n(z)=a_nz^n$, then $\forall z\in D, n\geq 0$ we have:  $$|f_n(z)|=|a_nz^n|=|a_n|\cdot|z^n|=a_n\cdot |z|^n\leq a_n\cdot 1^n\leq a_n$$  Then, applying Weierstrass M-test we get that the power series is convergent in $D$.  Now, I need to "find" an open ball $B$ such that $D\subset B$ and the power series is convergent in it.  Edit  $f$ is analytic in $1$, so there is an open set $U$ that contains $1$ such that   $$f(z)=\sum\limits_{n\geq 0} b_n(z-1)^n$$  Does it mean that the series is convergent in $U$?   I can see that we have that the two series coincide in $D\cap U$. How to sum data.frame column values? Positioning of decorations on parallel lines I'm displayed three lines on a pgfplot. I want to show a marker for the slope of each line and the difference in height between lines.     I'm showing the slopes by adding triangles under the lines  (at a manual position) and placing an annotation inside the triangle (at a manual position). I'm showing the difference between the height of the lines by adding a brace manually as well.   How can I place the braces on the plot and anchor them at a certain point on each line? The triangles can be added at the end of each line declaration, but doing it like this is ok. The braces are put in place manually, which is not sustainable as I create more plots.   I want to avoid using xshift, yshiftand manual adjustment to put the braces,traingles and annotations in the right place. How can I do this?      \documentclass{article} \usepackage{pgfplots}  \pgfplotsset{compat=1.9}  \begin{document}   \begin{tikzpicture}[scale=0.45,trim axis left, trim axis right]     \begin{axis}[ylabel={$\hat{S(4)}$},xlabel={$S{^*}_{breakeven}$},ylabel style={rotate=-90},domain=1:25, legend style={legend pos=north west,font=\scriptsize}, legend cell align=left]  \addplot+[samples=200,mark=none]                                    {6.26 + 0.28*x} coordinate[pos=0.5](x) coordinate[pos=0.9] (y) (x)-| (y) node [midway,xshift=-0.6cm,yshift=0.3cm] {\footnotesize $Slope X$};;  \addplot+[samples=200,mark=none, dotted, thick]             {2.66 + 0.28*x}  coordinate[pos=0.3](a) coordinate[pos=0.6] (b) (a)-| (b) node [midway,xshift=-0.5cm,yshift=0.2cm] {\footnotesize $Slope 1$};  \draw [decorate,decoration={brace,amplitude=5pt,raise=4pt},xshift=4pt]  (225,107) -- (225,73)  node [midway,xshift=0.6cm] {\footnotesize $\beta_1$};  \end{axis} \end{tikzpicture} \end{document} Can I calculate the size of a real object by just looking at the picture taken by a Camera? Is it possible to write one CSS rule for several prefixed selectors? How do I use the latest GCC on Ubuntu? I want to compile my program with the latest version of gcc.  Ubuntu 14.04 comes with gcc 4.8.2, however there's 4.9.0 available, moreover, I see that it is available as a package: . I tried to install it  sudo apt-get install gcc-4.9   but it says  Reading package lists... Done Building dependency tree        Reading state information... Done Note, selecting 'gcc-4.9-base' for regex 'gcc-4.9' gcc-4.9-base is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   Looks like it is already installed, just not as the default one? How do I utilize it to build my program? Do the following negations mean the same thing? I don't think you understood me. / I think you misunderstood me.    Do these senteces mean the same thing? If not, what's the difference?  Edit: I just realized that I asked something different from what I originally wanted. So I am also interested in the following comparison:     I don't think you understand me. / I think you don't understand me. Activate Field Service Lightning feature in a Salesforce DX scratch org? I am struggling to figure out how to -- or if it's even possible to -- create a DX scratch org with the "Field Service Lightning" (FSL) feature installed. Based on the feature list  I'm guessing the answer is no and I'll just have to wait. But I'm not sure I'm interpreting that source correctly or missing something entirely since I'm just wrapping my head around DX.   Background: we have a production org with FSL installed and active, along with it's accompanying managed packages. I can create sandboxes from it and do development using standard Metadata API tools, but I was hoping to start a new project using DX. But I'm stuck configuring a scratch org with the proper features. Are Love Potions legal?