We all have seen and know of Euclid’s proof by contradiction, but I’m wondering if there is such thing as a direct proof? Also, the theorem isn’t posed as an “if-then” statement, so I can not even imagine how an alternative proof would be structured.
This question relates to a discussion on . Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My understanding is that Intuitionists reject such proofs because they rely on the Law of the Excluded Middle, which they don't accept. Does there exist a direct and constructive proof of the infinitude of primes?
Which is Most TimeMachine-like Backup Program for Ubuntu? I know this  but there is no comparison to OSX's TimeMachine.  I would like to find a backup software which is about the same as TimeMachine or better.  I would like to have such a software which can use some format of HDD which is suitable for big files (50 GB) in Linux.  I noticed that   OSX format of HFS+ is not supported by Linux distros, since no maintainer of HFS in Linux since 2006 NTFS format made by Debian 8.1 is not supported by older Linux distros   If Windows support, good.   Try with the Backups (Deja Dup) software  I did start the backup first time but I wanted to go to lunch after backup was running already two hours of just 10 GB files.  So it gave me this after the lunch    which is really insane, reported now , since the software is not designed to have pauses in doing the backups. Why was the DeLorean in Back To The Future not always covered in ice after time travel, and why didn't it fly in 1885? I can't open/ping any antivirus sites I have a problem with a virus in windows 7, I don't know what kind of virus is infected my OS, I installed kasersky antivirus, and clear all detected viruses. After that I can't open any antivirus sites, can't ping any antivirus sites. example:  C:\Windows\system32&gt;ping kaspersky.com Pinging kaspersky.com [38.117.98.231] with 32 bytes of data: General failure. General failure. General failure. General failure.  Ping statistics for 38.117.98.231:     Packets: Sent = 4, Received = 0, Lost = 4 (100% loss),   When I ping to other sites, I get reply normally: example woth pinging to google.com  C:\Windows\system32&gt;ping google.com  Pinging google.com [31.7.160.224] with 32 bytes of data: Reply from 31.7.160.224: bytes=32 time=2ms TTL=61 Reply from 31.7.160.224: bytes=32 time=2ms TTL=61 Reply from 31.7.160.224: bytes=32 time=1ms TTL=61 Reply from 31.7.160.224: bytes=32 time=1ms TTL=61  Ping statistics for 31.7.160.224:     Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:     Minimum = 1ms, Maximum = 2ms, Average = 1ms   I tryed any antispyware/antimalware softwares, but null results.  Anyone else encountered this problem? Notify us when one of our posts is edited Your summary page shows "N Edits to your posts" for the selected time period but you really have to be looking for it.  Could we add edits to the notification feed? You could put the reason for the edit (entered--or not--when the editor changes it) and just a link to the post. How to append something to an array? Parameter Sniffing (or Spoofing) in SQL Server A while ago I had a query that I ran quite a lot for one of my users. It was still being evolved and tweaked but eventually it stablised and ran quite quickly, so we created a stored procedure from it.   So far, so normal.   The stored procedure, though, was dog slow. No material difference between the query and the proc, but the speed change was massive.   [Background, we're running SQL Server 2005.]  A friendly local DBA (who no longer works here) took one look at the stored procedure and said "parameter spoofing!" (Edit: although it seems that it is possibly also known as 'parameter sniffing', which might explain the paucity of Google hits when I tried to search it out.)   We abstracted some of the stored procedure to a second one, wrapped the call to this new inner proc into the pre-existing outer one, called the outer one and, hey presto, it was as quick as the original query.   So, what gives? Can someone explain parameter spoofing?   Bonus credit for    highlighting how to avoid it  suggesting how to recognise possible cause discuss alternative strategies, e.g. stats, indices, keys, for mitigating the situation What is the opposite of "preaching to the choir"? In Euclidean geometry texts, it is always mentioned that point is undefined, it can only be described. Consider the following definition: "A point is a mathematical object with no shape and size." I do not understand what is the problem with this definition. Please give the detailed reasons. Thanks in advance! Is the set of all unitary matrices in $M_2(\mathbb{C})$ is compact? I can show that as determinant map is continuous so unitary matrices are closed but how to show they are bounded?   Please help. If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite.  so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this point, and also $f(x)$ must be continuous at x as well  so that mean that:  $\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h} =  \lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$  and that: $\lim_{x\to x_0^+}=\lim_{x\to x_0^-}=f(x_0)$  I know I should probably use arithmetic limit laws to prove this but I can't see how what I figured out could help me. any help with that?  Thanks! Like many others, I commonly find myself ending a sentence with a preposition.  Yes, it makes me cringe.  I usually rewrite the sentence, but sometimes (in emails) I just live with it.  To, with... you know who you are.  Should I keep fighting myself on this one, or is it okay in some circumstances? How much creative independence are the Show runners accorded in Game of Thrones? Since Season 3, we have been observing that the show was taking a different direction than the books. Many important characters never made it into the show, others who are still alive were killed off.   As we know the show is supposed to be based on the books, not to be direct adaption of the books from print to television media, there's bound to be a certain degree of creative independence accorded to them by the author of the series, George R.R. Martin.   The question is, how much independence do they have? Do they still have to take GRRM on board with whatever changes they might intend to make? By "they" I mean D&amp;D and the whole writers crew for the show. Given the number of significant changes they have made, I reckon the answer would be a lot. Has GRRM or HBO ever commented on that? How can I visualize or simulate the effect of different focal lengths? I have seen many camera review sites illustrate the effect of using different focal lengths on the same photograph using frames. There are also similar illustrations to compare different sensors (). I find this kind of visualization to be very useful in comparing the actual effect of the longer focal length\different sensor size. Of course the ability to do a wider focal length visualization would also be equally useful, but that's not possible.  Is there any ready-made software or plugin that enables this? If not, is there any simple technique to visualize the effect of longer focal lengths on an image (one would of course have to use the zoom information in the image EXIF data to accurately represent this)? All I can think of is to use some trigonometry to do the necessary calculations for the frame sizes for cropping. Is it possible to use JSF+Facelets with HTML 4/5? When electric dipole placed in non uniform electric field, what is the approach to calculate torque acting on it? Can it be zero? What is the translation of what's written below?  I think it may be the poem of the One Ring, but I'm not sure. Badass and BAMF are both modern words with approximately the same meaning: "Someone who is awesome to an extreme level, thereby leveraging unquestionable authority." Is there another noun or title that would convey the same level of awe, admiration and toughness that calling someone a badass does, without the vulgarity?  Thanks for any help. :) Here is a DOM-based XSS flaw reported by SalesForce Security Review team in our code:   RootComponent  &lt;aura:component extensible="true" abstract="true" access="global"                 description="Root template component"&gt;     &lt;!-- ... lot of other code ... --&gt;     &lt;aura:attribute name="label"                     type="String"                     access="global"                     description="Label of the form input element" /&gt;     &lt;!-- ... lot of other code ... --&gt;  &lt;/aura:component&gt;   SubComponent  &lt;aura:component access="global" extends="c:RootComponent"                  description="Child extension of RootComponent"&gt;     &lt;!-- ... lot of other code ... --&gt;     &lt;label&gt;{!v.label}&lt;/label&gt;     &lt;!-- ... lot of other code --&gt;  &lt;/aura:component&gt;   According to SFDC, "any app can extend SubComponent component and pass in  as a label and trigger a DOM-based XSS attack." The solution would be to "sanitize" label global attribute to avoid such attack. What is meant by sanitization of global attribute?  I tried to reproduce the so-called security flaw , but as label attribute is of type "String" , when we send it HTML, it is displayed as a string, not as interpreted HTML ... so I don't see where the security flaw could be :/  &lt;aura:application &gt;     &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}" /&gt;      &lt;c:SubComponent aura:id="TestXSS" label="DOM-based XSS attacks" /&gt;  &lt;/aura:application &gt;   JS Controller  ({     doInit: function(component, event, helper) {         var XssAttack = '&lt;img src=x onerror=alert(1) /&gt;';        component.find('TestXSS').set('v.label',XssAttack);     } })   Output    We have global attributes everywhere. Our components are packaged and meant to be extended by client implementation. Do we actually need to sanitize them? Openlayers using Google Maps as base layer: possible to zoom out farther than max extent? See attached picture. This is how it is when the page loads. I'm currently using  map.zoomToMaxExtent();     Everything is working pretty well but I'd like to zoom out more/see the whole world using OSM/Google Map Layers: is this even possible?  Note that the zoom bar on the left is set to the lowest bar when the page loads (if you can't see the map well, just right click the image and paste into a new browser tab).   This  but even when I do MIN_ZOOM_LEVEL that doesn't do much.  Also note I am not setting any bounds, extents, or resolutions. Toying with those did not seem to do the trick either.   Any input is appreciated.   Edit: here's a codeshare if you want to take a look:   UPDATE:  This is not perfected yet (and it makes me think something may be wrong with my OL code itself) but currently I am using this   var extent = new OpenLayers.Bounds(-1.32,81.71,-1.18,-61.80).transform(new OpenLayers.Projection("EPSG:4326"), new OpenLayers.Projection("EPSG:3857"));   I tried   var extent = new OpenLayers.Bounds(-20037508.34279,-10018754.17139,20037508.34279,10018754.17139);   (both with and without transform) but this did not do anything.   UPDATE 2:   wrapDateLine: false works fine for now: can zoom out farther, pan around, etc etc.  Now I would like to cut off everything to the left of the date line on the left side of the map (the 'first' portion), and cut off everything to the right of the date line on the right side of the map (the '3rd' portion) WHILE still keeping the map full screen. I've tried bounds, resolutions, but no go. Is this even feasible?  Something like this but it should take up the full screen: possible to do? See below: Induction proof: n lines in a plane Assume that there are $n$ infinitely long straight lines lying on a plane in such a way that no two lines are parallel, and no three lines intersect at a single point. Prove that these lines divide the plane into $\frac{n^2+n+2}{2}$ regions. (Hint: Any two non-parallel straight lines on a plane must intersect at exactly one point.)  Can someone tell me where to start here? I know to prove the base where $n=0$, but don't know how to proceed.  Any help is greatly appreciated!