Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each.  $H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively.  What will the area of the square formed in the middle be?  I know that this problem can be solved by trigonometry by using Area of triangle ($\frac{1}{2}ab\sin{c}$) but,  is there another method or visual proof?
If the lines intersect the vertices of the square. The area of the square is $1$ and the lines also intersect the midpoints of the square lines. How to find the area of shaded region?
Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$. So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity element, other than 0. Because if 0 is the identity element, then this group won't have inverses.  The set explicitly excludes -1, which I found to be its identity element, which makes going about proving that this is a group mighty difficult. Writing USB Joystick for PIC18F using MikroC Continuous weighted random distribution, biased toward one end I am currently contributing to a particle system for our game and developing some emitter shapes.  My uniform random distribution along a line or along a rectangular area works fine - no problem.  But now I would like to have something like a 1 dimensional gradient in this distribution. This would mean for example lower values are more common than higher values.  I don't know what would be appropriate mathematical terms for this problem, so my search skills are rather useless with this one. I need something that is computationally simple, as the particle system needs to be efficient. Why do we need to dummy code categorical variables Showing $\gcd(2^m-1,2^n+1)=1$ A student of mine has been self-studying some elementary number theory. She came by my office today and asked if I had any hints on how to prove the statement      If $m$ is odd then $\gcd(2^m-1,2^n+1)=1$.    It's been a while since I took number theory and I'm not sure what to do. She said she is learning about congruences, primitive roots, and power residues.  She has not taken any group theory. At some point in (I think) the 90s, I saw an episode of something like Outer Limits or Twilight Zone where nobody was allowed to have more than average intelligence. Smarter children had to wear headsets which inhibited them from being too intelligent, administering minor electric shocks if intellectual thought was detected.  I can remember a final (or near final) scene of somebody watching his suicide and being upset but not remembering why because of the headset.  Can anybody identify the show and episode? Network-wide Help Center bug How can you combine all lines that end with a backslash character? Using a common command line tool like sed or awk, is it possible to join all lines that end with a given character, like a backslash?  For example, given the file:  foo bar \ bash \ baz dude \ happy   I would like to get this output:  foo bar bash baz dude happy Student t as mixture of gaussian Using the student t-distribution with $k &gt; 0$ degrees of freedom, location parameter $l$ and scale parameter $s$ having density  $$\frac{\Gamma \left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\sqrt{k \pi s^2}\right)} \left\{ 1 + k^{-1}\left( \frac{x-l}{s}\right)\right\}^{-(k+1)/2},$$  how to show that the Student $t$-distribution can be written as a mixture of Gaussian distributions by letting $X\sim N(\mu,\sigma^2)$, $\tau = 1/\sigma^2\sim\Gamma(\alpha,\beta)$, and integrating the joint density $f(x,\tau|\mu)$ to get the marginal density $f(x|\mu)$? What are the parameters of the resulting $t$-distribution, as functions of $\mu,\alpha,\beta$?  I got lost in calculus by integrating the joint conditional density with the Gamma distribution. What are the regulations for transporting medication abroad? This is in reference to medications, both prescription and non-prescription, not illicit drugs. Since the legality may vary on country, let's say a round trip from the US to Singapore, as an example.  When travelling with prescription medication, I am aware that you are supposed to bring your prescription with you. What about over-the-counter medication, such as Tylenol/aspirin? How about medication which is restricted or prescription-only in one country, yet available OTC in another (think Viagra)? Is denying "Paradoxical Partitioning" equivalent to accepting the Axiom of Choice? In  (to ) it has been noted:     One more point to make about the paradoxical decomposition to more parts than elements [...]   currently we do not know of any model of +$\lnot$ where such decomposition does not exist.    Namely, as far as we know, in all models where choice fails there is some set which can be partitioned into more parts than elements.   (Just for clarification: From the context of the above references the comparison "more parts than" is obviously meant in terms of the  of the suitable  being strictly larger than the cardinality of the suitable initial set itself.)  Now, I find the negation or denial of such "paradoxial partitioning" interesting; i.e. the statement (proposition, "$\lnot$PP"):  "Each partition of each given set has cardinality less than, or at most equal to, the cardinality of the given set."  And I like to further explore how this suggested "proposition $\lnot$PP" (being considered along with ZF, of course) relates to "the standard set theory including Axiom of Choice", . Therefore  My questions:      Are there any models of ZFC known (or could there be any such models, in principle) in which there is some set which can be partitioned into more parts than elements ?  And:  Are there any models of ZF known (or could there be any such models, in principle) in which there is no set which can be partitioned into more parts than elements, and which is not also a model of ZFC ?  (And just for reference: Is there a conventional or concise way of expressing the suggested "proposition $\lnot$PP" in terms of standard notation, such as used in the sources linked above? Has it perhaps been discussed already, by some other name? ...) Why isn't my String comparison working in iOS 6? Configure Postfix to use external MX servers for delivery of local mail if user is unknown Allow more than one @name notification per comment Now this is probably a surprise to many since I always see thousands of comments like :  @Sarfraz, @balpha just told you they did in the first comment he wrote. Also, allowing only one Stack Overflow user to be logged in per IP address would be problematic for networks with shared IPs.  What's surprising is that apparently the notification will trigger only for Sarfraz and not for balpha. As you can see from the :  Only one person can be replied to at a time in a comment. The first one in the string wins.  I propose to increase the limit to let's say 3 or 5 people; it won't hurt (never seen that feature being abused) and especially many people believe it already works that way. Why is there no absolute maximum temperature? Is Stack Exchange / Stack Overflow available for private or internal use? How to know if something is a primitive concept, a law, a definition or a theorem Some basic Physics books are often misguiding in the sense that they don't make clear whether something is a primitive concept, a law, a definition or a theorem. This is often a little confusing. I've asked here some moments ago if there's a definition of force, and I've been redirected to a question about whether Newton's Laws are laws or definitions of mass and force.  This is just an example, but there are many others, like for instance, the conservation of energy, which some books present as a theorem, and some places says that this is a law of Physics. Time on the other hand, is something that according to Feynman's lectures on Physics, cannot be defined, so we just say how to measure it and leave it as a primitive concept.  In this context: how in general can we identify whether something is a primitive concept, a fundamental law of Physics, a definition or a theorem? Why is King Arthur female? Why is Iskander tall? I've watched both the Fate/Zero and Fate/Stay Night anime (including the Unlimited Blade Works movie), but none of the visual novels, so no spoilers from that unless they are light spoilers. One thing that bothered me was that despite the fact that heroic spirits are defined by the public opinion/view of them, some heroes had qualities that were in direct opposition of what the public thought.  We know that heroic spirits, being sealed on their Throne of Heroes, are meant to be completely unchanged. This is the entire point of the Throne of Heroes. But this seems to be in opposition of the previous fact that heroic spirits are defined by what others think they are. Then there are Counter Guardians who are completely different, I believe, in that they are not sealed on their Throne of Heroes.  One particularly important Counter Guardian of the series is Arturia, or Saber. Arturia is female, despite the fact that she was treated by even her contemporaries as male, and has gone done in history as male. In fact, there isn't even any small hint as to Arturia's gender being anything other than male in our history, or the history of the Nasuverse (I believe). This being true, shouldn't Arturia be male?  This not being the case shouldn't Arturia be erased, or else create instability in the World? Same with Iskander. In history he is known for being pretty short, even for a Macedonian. That he is very tall means there is an incongruity with human knowledge and true fact.  Furthermore, in the Unlimited Blade Works movie Archer is revealed to be Heroic Spirit Emiya. If Heroic Spirit Emiya is summoned before he is even a hero, or acquired his gift from the World, etc., then how is it that Heroic Spirit Emiya is even strong at all? The power of a Servant being based on their popularity at the time of their summoning, Emiya should be weaker than pretty much every other Servant, and then some.  Maybe I'm just over-analyzing things, or I have some misconception. In any case, answers are appreciated. Using special auto start servlet to initialize on startup and share application data How to install Tor?