In lecture I learned the following definition of valuation: Let $(K,+,\cdot)$ be a field and let $(G,+)$ be a totally ordered group. A map $v: K \longrightarrow G\cup\{\infty\}$ is a valuation if the following properties hold:  $v(ab) = v(a)+v(b)$ $v(a+b) \ge \min\{v(a),v(b)\}$ $v(a) = \infty \iff a = 0$  Then we proved that:  $v(1) = 0$ $v(a^{-1}) = -v(a)$ $v(-a) = v(a)$ $v(a - b) \ge \min\{v(a), v(b)\}$  Later in the lecture we used the &quot;easy fact&quot; that $$ \text{If }v(a) \ne v(b), \text{ then } v(a+b) = \min\{v(a), v(b)\}$$ I do not understand why this is true. I looked on  and there they define a valuation to so that it has this property. But is it possible to derive this from my definition?
Let $F$ be a field and $v:F\rightarrow G\cup\{\infty\}$ be a valuation on $F$ so $G$ is a totally ordered abelian group with $\infty$ having the properties $\infty+\infty=g+\infty=\infty+g=\infty$ and $\infty\leq\infty$ for all $g\in G$, $\infty \notin G$.     Show if $v(a)\neq v(b)$ then $v(a+b)=\min(v(a),v(b)$).   I must be missing some trick here on how to show this is true.  My approach has been to use that $G$ is a totally ordered group so I start by assuming $v(a)\neq v(b)$ and letting my first case be $v(a)&lt;v(b)$ as trichotomy holds in $G$. By $v$ being a valuation on $F$, we then have that $v(a+b)\geq \min(v(a),v(b))=v(a)$. From here what I've tried doing doesn't seem to help much.  I've said lets assuming $v(a+b)&gt;\min(v(a),v(b))=v(a)$ and we also have $$v(a+b)&gt; v(a) \Rightarrow v(a+b)+v(a^{-1}) &gt; 0 \Rightarrow v(1+ba^{-1}) &gt; 0$$ and $$v(a)&lt;v(b) \Rightarrow 0&lt; v(b)+v(a^{-1}) \Rightarrow 0 &lt; v(ba^{-1})$$ also $$v(1+ba^{-1})\geq\min(v(1),v(ba^{-1}))=\min(0,v(ba^{-1}))=0.$$  How does one solve this?
How does JavaScript .prototype work? I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. Is the "live" in "He only lived a few days after the accident" intrasitive or transitive? p:commandbutton action doesn't work inside p:dialog Prove that to each $\epsilon >0$, there exists a $\delta >0$ so that the Lebesgue integral... Suppose $f$ is in $L^1$ space of $\mu$, where $\mu$ is the Lebesgue measure. Prove that to each $\epsilon &gt;0$, there exists a $\delta &gt;0$ so that the Lebesgue integral of the absolute value of $f$ is less than $\epsilon$ (over the set $E$) whenever the measure of $E$ is less than $\delta$.  I know that the integral of $f$ is finite. So for arbitrary $f$ we can find the integral of $f$ to be less than some $\epsilon$, but how would I connect that with the measure of $E$ less than $\delta$. Any help would be welcome. Should I upgrade my body or lens first? I have an entry-level Nikon DSLR, and I cant decide if I should upgrade my camera body or upgrade from a  AF-S DX Nikkor 18-200mm to whatever else might be recommended? Is it possible to make a sequence of renders and give the user the option to cancel the process through the UI at any given time? I'm trying to make a script where I have to render several images of the same scene at the same frame, and write to disk with appropriate names. The problem is that although this works fine with a non-responsive UI, it doesn't work well when the render progress appears in the Blender UI.  I mean,  bpy.ops.render.render()   uses "EXEC_DEFAULT" when called from a script. This leaves a blocked User Interface until the render finishes, meaning that you can't cancel the process, except by killing Blender. The problem is that the render may be too big in resolution, and the user could want to stop the process when realizing that the render could take hours, so the idea is to cancel this process without losing the currently opened project.  If I call,  bpy.ops.render.render("INVOKE_DEFAULT")   I get a nice preview bar in the Info Window with a clickable "X" button. I want the user to be able to press that button to cancel the operation if needed, but for a series of renders (not animation).  The problem here seems to be when trying to loop this command with the INVOKE_DEFAULT context to write several renders. I made a lot of testing, and this way of calling the render operator seems to continue with whatever is next in the script, so it loops very fast and ends before the rendering actually finishes.  I think that I need some way to stop the script and make it wait until render finishes to continue looping with the next ones. I tried the "render_complete" and "render_cancel" handlers, but I can't use them to start a new render or even make the rest of the script wait until the loop of renders is complete. I tried threading and events without success either. Maybe I'm doing something wrong.  import bpy  render = bpy.context.scene.render path = render.filepath = "/tmp/" shots = ["one", "two", "three"]  for x in shots:     render.filepath = path + x     bpy.ops.render.render("INVOKE_DEFAULT", write_still=True) # this doesn't work      # but if I use:     # bpy.ops.render.render(write_still=True)     # without the INVOKE thing, images are written as expected,     # but losing the possibility to see the progress and cancel rendering Limit of $L^p$ norm when $p\to0$ Let ($\Omega$, $\cal{F}$, $\mu$) be a probability space and $f\in L^1(\Omega)$. Prove that  $$\displaystyle\lim_{p\to 0} \left[ \int_{\Omega}|f|^pd\mu \right]^{\frac{1}{p}}=\exp \left[ \int_{\Omega}\log|f| d\mu  \right],$$  where $\exp[-\infty]=0$. To simplify the problem, we may assume $\log|f|\in L^1(\Omega).$ Spotlight overrides first character What's wrong: I hit CMD+SPACE to open up spotlight, typically type in the name of the application I want to start, hit enter and either the wrong app or no app at all will start, because the app name is missing the first character. "Learn to type" I hear you say but wait: as I enter the first character in the search bar, the cursor jumps to the beginning of the line and my second character input overrides the first one. So entering for example "ever", in order to start Evernote, will result in "ver". "php" to start PHPStorm will result in "hp" and start my damn HP printer app and so on.   This cursor jump to the beginning of the line will not happen in any other application and even if I accept the fact that my cursor jumps from time to time, I don't get why the second character overrides the first one. I'm not even sure if you can configure your Mac to always override the character at current position, but I surely haven't.  So this keeps on happening quite regularly and I cannot reproduce it. It just happens and when I try it again (close Spotlight, CMD+SHIFT again), it won't happen again, even if I give it like 5 tries or so.  It won't kill me but I am a software developer myself and this kind of bugs just leave me frustrated plus I can not imagine that this is the desired behavior. I have seen some talk about this issue on the internet but I did not come across a solution..  I have 3 MacBooks, different ages, all with German keyboard layouts and German set as system language and they all do this. Just mentioning this, because after that fact, I can barely imagine this isn't affecting at least a lot, if not all, MacBook (Pro?) users.   13" Pro 2010 or so 15" Pro Retina 2013 13" Pro Retina 2015   They all run El Capitan but I already encountered this before. I don't know the OS X version I had installed when I first stumbled across it. DIY.Blogoverflow.com is redirecting to stackexchange.com/blogs I seem to be the unlucky recipient of a 302 redirect from the blogoverflow site, and was asked by Grace Note to upload the issue here. While trying to access the DIY blog site, I'm getting bounced over to the generic SE blog page. I've tried resolving this by changing DNS servers, clearing my browser cache, and trying a different browser. Others going to the same site aren't experiencing the issue, but I'm at a loss for what's different on my system that would cause this. Here's the wget output of the redirect:  --2014-02-18 14:59:28--  http://diy.blogoverflow.com/ Resolving diy.blogoverflow.com (diy.blogoverflow.com)... 67.228.172.87 Connecting to diy.blogoverflow.com (diy.blogoverflow.com)|67.228.172.87|:80... c onnected. HTTP request sent, awaiting response...    HTTP/1.1 302 Found   Date: Tue, 18 Feb 2014 19:59:28 GMT   Server: Apache   X-Powered-By: PHP/5.4.22   Location: http://stackexchange.com/blogs   Content-Length: 1001   Connection: close   Content-Type: text/html; charset=UTF-8 Location: http://stackexchange.com/blogs [following] --2014-02-18 14:59:28--  http://stackexchange.com/blogs Resolving stackexchange.com (stackexchange.com)... 198.252.206.140 Connecting to stackexchange.com (stackexchange.com)|198.252.206.140|:80... conne cted. HTTP request sent, awaiting response...    HTTP/1.1 200 OK   Cache-Control: private   Content-Type: text/html; charset=utf-8   X-Frame-Options: SAMEORIGIN   Date: Tue, 18 Feb 2014 19:59:28 GMT   Content-Length: 16775 Length: 16775 (16K) [text/html] Saving to: ‘index.html.1’       0K .......... ......                                     100% 1.04M=0.02s I often see statements in discussions about web displayed image quality that go something like this: "I only upload pictures at 72 dpi and 1200 pixels on the long edge, so if someone copies and prints the image it won't look as good as if it were 300 dpi."  What? Have I missed something here?  Does an image edited and saved/exported with 1200x800 pixels at 300 ppi look any different online than the same image saved/exported with 1200x800 pixels at 72 ppi? Other than the metadata regarding ppi, is there any difference in the two images at all? If so, what is the difference? If I tell my printer to print the images at 4R (4X6 inches) will there be any difference at all in how the printer creates a print from the 1200x800 @ 300 dpi image versus the 1200x800 @ 72 dpi image? I was going through fluid dynamics and I came upon this section on the shape of a liquid surface in a rotating cylindrical vessel. The working in the book is something like this:   It considers a mass $m$ of water on the surface of the rotating liquid at a distance $x$ from its axis of rotation .It takes into account a pseudo force ($m \cdot x \cdot \omega^2$) radially outwards and its weight force ($mg$).It then proceeds to calculate the net force and the angle it makes with the vertical.From there through basic integration it gets the equation of the surface of liquid  i.e. $y= \dfrac{(x^2 \cdot \omega^2)}{2g}$ .  I get this working.  But if I were to view the same rotation through an inertial frame,how would I proceed? I mean, there wouldn't be any pseudo force, rather a centripetal acceleration of same magnitude radially inwards.This is where I am facing certain doubts.  The forces on the liquid would be  the centripetal force inwards and the weight force downwards.The net force would therefore be in a different direction,and consequently the slope of the surface at that point would be in a different direction.How is it possible for the same surface to have two different shapes in two different frames?   I would be obliged if someone could help me out by pointing out the flaw in my understanding. Is there a difference between "think of something" and "think about something"? I've also met "have heard of/about something". How to see time stamps in bash history? Is there any way I can see at what time the commands were executed from the bash history? We can see the order but is there any way I can get the time also?  Bottom-Line: Execution time in the Bash history Okay. 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Is it possible in Ruby to get a reference to methods of an object ( I would like to know if this can be done without procs/lambdas ) , for example , consider the following code :   class X   def initialize     @map = {}     setup_map   end    private   def setup_map     # @map["a"] = get reference to a method     # @map["b"] = get reference to b method     # @map["c"] = get referebce to c method   end    public   def call(a)     @map["a"](a) if a > 10     @map["b"](a) if a > 20     @map["c"](a) if a > 30   end    def a(arg)      puts "a was called with #{arg}"   end    def b(arg)      puts "b was called with #{arg}"   end    def c(arg)     puts "c was called with #{arg}"   end end   Is it possible to do such thing ? I would like to avoid procs/lambdas because I want to be able to change the behaviour of A,B,C by subclassing . Is there a correct resistance value for I2C pull-up resistors? Formula Assistance I am creating a formula variable in a Flow but keep getting this error. Crop net Acres is a Number field. IF(     ISBLANK( [Opportunity].Crop_Net_Acres__c),     0,      ( [Opportunity].Crop_Net_Acres__c) )  Error Message:  netAcresFormula (Formula) - The formula expression is invalid: Syntax error. Missing '='  Any thoughts here? Why is a T distribution used for hypothesis testing a linear regression coefficient? In practice, using a standard T-test to check the significance of a linear regression coefficient is common practice. The mechanics of the calculation make sense to me.  Why is it that the T-distribution can be used to model the standard test statistic used in linear regression hypothesis testing? Standard test statistic I am referring to here:   $$ T_{0} = \frac{\widehat{\beta} - \beta_{0}}{SE(\widehat{\beta})} $$