Find the number of triangles formed by n lines.
Number of triangles formed by $m$ lines
Cancellation of addition on convex sets I  found a question about a property of the . However the question was not properly answered (it used a projection argument which might not be true in a general Banach space).  I was wondering whether the following (weaker) statement holds:  Let $X$ be a Banach space and suppose $A,B,C_0\subset X$ are bounded, closed, convex and non-empty subset. Do we then have $$A+C_0=B+C_0\implies A=B?$$ $ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$ Could someone please help me with this proof?     Suppose that $a, b \in N$, and $d = \gcd(a, b)$. Since $d$ divides $a$, we have $a = de$ for some integer $e,$ and similarly $b = df$ for some integer $f$. Prove that $\gcd(e, f) = 1$.   I understand why it works. Since d is all the common factors of $a$ and $b, e$ and $f$ had no common factors, therefore the $\gcd(e,f) = 1$. But how do I prove this?  Thanks in advance. HD bad blocks: is there a way to repair them? I want to write $ k = \{k1,k2,k3,...,kn\} $. How could I do this? How to move and overwrite subdirectories (and files) to parent directory? Product of square of distances from vertices of a polygon of radius a I want to find out the following product.  $\prod_0^{n-1}  (r^2 + a^2 -2ra\cos(2k\pi/n - \theta))$ I have been trying to use complex numbers but did not work out. The book I am reading the result is directly stated as $r^{2n} + a^{2n} -2r^na^n\cos(\theta)$. Please help. why don't we define vector multiplication component-wise? I was just wondering why we don't ever define multiplication of vectors as individual component multiplication.  That is, why doesn't anybody ever define $\langle a_1,b_1 \rangle \cdot \langle a_2,b_2 \rangle$ to be $\langle a_1a_2, b_1b_2 \rangle$?  Is the resulting vector just not geometrically interesting? I am trying to run a buffer tool as part of script that I will both use as a standalone script and import into ArcToolbox. I'm just wondering, if I use GetParameterAsText for the buffer distance, how do I set the linear units? E.g. I want it preset so they only have to type 300 and the script takes that to be meters? Here is what I have so far:  # Buffer the Thames feature class to create the Flood Risk Area in_features = "Thames" out_feature_class = "Flood_R_A" # Run the Buffer Tool arcpy.Buffer_analysis (in_features, out_feature_class, arcpy.GetParameterAsText(5),    "FULL", "ROUND", "ALL", "") Supporting Multiple Screen Sizes in Storyboards (3.5 and 4 inch screens) Alternative Method to Acquire El Capitan Upgrade How do you mark a struct template as friend? An excited electron loses energy in the form of radiation. The radiation constitutes photons which move at a speed $c$. But is the process of conversion of the energy of the electron into the kinetic energy of the photon instantaneous? Is there a a simple way to visualize this process rather than math? If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ Let $f\colon\mathbb R\to\mathbb R$ be twice differentiable with $f(x)\to 0$ as $x\to\infty$ and $f''$ bounded. Show that $f'(x)\to0$ as $x\to\infty$. (This is inspired by  to a different question) javascript document.getElementsByClassName compatibility with IE What is the optimum button size of touch screen applications? There was a not so recent blog post about the ideal button size of touch screen control sizes (sorry there is no link, the website is now a spam trap) that was based on the  form the MIT touch lab that gives some measures of the size and sensitivity of the finger. If found that averages for finger pads are between 10–14mm and fingertips are 8–10mm. This would make 10mm x 10mm a good minimum touch target size.  As the author of the blog pointed out, this ideal contact size is much larger than the suggested size for Windows and Apple devices that is in their design guides. I am wondering why there is such a difference, and whether it makes more sense to increase the size of controls on touch screen applications, or if this is not really a usability issue at all? Also, it seems that many mobile websites and apps seem to completely ignore these principles. I am sure the screen size poses a limitation to how big the interface controls can be, so is this simply a trade-off between being able to fit in as much information versus how easy it is to interact with the control? It feels as if the stylus is not something that is in much use. A similar question has been raised on the minimum/smallest size that you can get away with, but I would consider this to be a different question to what the optimum size should be, because the smallest size accommodates the minimum usability requirement, whereas the optimum size caters for the best usability requirement. Archived link of the blog (original domain shows spam)  And here is another reference it to (just to show that it is not the problem with the link but the website () UPDATE #1: another new reference that has a very precise figure of a minimum of 44 x 44 points, not sure how it is derived exactly but interesting (). Read Section 2. Provide Large Clickable Elementss. UPDATE #2: another reference, this time from  citing touch target size to 9 mm square or greater (48x48 pixels on a 135 PPI display at a 1.0x scaling plateau). They also suggest avoid using touch targets that are less than 7 mm square. UPDATE #3: something from the  that I referenced recently citing 44 x 44 points. UPDATE #4: NN/g has published an article on  and the information/conclusion remains unchanged. UPDATE #5 A really nice write-up on the need to enhance the  of common UI elements, citing the  that the &quot;minimum target size for touch or mouse should be 44×44 pixel&quot; (2.5.5 - Target Size - Level AAA). Pointer to class data member "::*" I have Ubuntu 13.1 installed with the default window manager and DE.   I am trying to enable single click so I can open files and folders with only one click. I tried googling and all results said to open files, then go to preferences from top bar but my top bar has no options.  picture of desktop: If a SHA256 hash with high entropy is then hashed with one made from low entropy, is the resulting hash higher/same/lower entropy? If one were to create a SHA256 hash using 256 coin flips converted to hex (which, as I understand it, is as "entro-phized" as one can capture in SHA256) then take that hash as a string, combine that with a hash of the word "cat" and hash the resulting 512 character string, would the resulting hash from that exercise be any less random than our "perfect" hash we began with?  My thinking is the final string would capture the same amount as the original randomness as what we started with because the "extra" entropy can not be captured and is lost. Is this the case and am I thinking about this correctly? Would hashing something with high entropy with something simpler ever cause a lower amount of randomness or does it only go up?  Thanks to anyone who can help explain this a little better. For a book cover, I want to write the title vertically (like in Japanese, but using English text):           T T        I H        T I  I  M  L S  S  Y  E   How do I do this without jumping through a host of \vbox and \hboxes? Why is Apache executing .php.html files as PHP? I have an odd problem...I'm using a documentation generator which generates a lot of output like docs/foo.php.html. It's XHTML, and thus contains &lt;?xml...&gt; tags at the beginning of file. The problem is, Apache has somehow decided to run it through the PHP interpreter, even though ".php" appears in the middle of the filename, and not at the end. This, in turn, triggers a PHP error, because it sees "&lt;?" as the command to start executing PHP code, and immediately gets confused by the "xml..." which follows it.  How do I configure Apache to ONLY execute .php files and not .php.html files? The string "php.html" does not appear explicitly anywhere in my Apache config files. There is a line "AddHandler php5-script .php", but I don't see how that would also include ".php.html" files.