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The set of integers $n$ expressible as $n=x^2+xy+y^2$ Let $S$ be the set of integers $n$, such there exist integers $x,y$ with $$n=x^2+xy+y^2$$   Is the implication $$a,b\in S\implies ab\in S$$ true? If yes, how can I prove it?  I worked out $$n\in S\iff 4n\in S$$ and $$n\in S\iff 3n\in S$$ I tried two approaches. The first is to express $$(a^2+ab+b^2)(c^2+cd+d^2)$$ in the form $$f^2+fg+g^2$$ with polynomials $f,g$ with integer coefficients. I however could not find suitable $f$ and $g$. The second approach is based on $$x^2+xy+y^2=\frac{(2x+y)^2+3y^2}{4}$$ If we have $n=x^2+xy+y^2$ , we have $u^2+3v^2=4n$ for some integers $u,v$ with equal parity. The main problem of this approach is to consider the equal parity. Any ideas ? Can someone pgfplots the following picture concerning confidence intervals? I've been trying to use the following code to produce it, but have no idea how to shade in the tails I am interested in. By the way, the line pointing the (100-C/2)% can be an arrow, does not have to be squiggly as depicted in the picture.   \begin{tikzpicture}[ every pin edge/.style={&lt;-}, every pin/.style={fill=yellow!50,rectangle,rounded corners=3pt,font=\small}] \begin{axis}[every axis plot post/.append style={   mark=none,domain=-3:3,samples=50,smooth}, clip=false, axis y line=none, axis x line*=bottom, ymin=0, xtick=\empty, ] \addplot {\gauss{0}{0.5}}; \addplot {\gauss{0}{1}}; \node[pin=70:{$\hat{\theta}_1$}] at (axis cs:0.57,0.5) {}; \node[pin=270:{$E(\hat{\theta} = 0$}] at (axis cs:0,0) {}; \draw[dashed] (axis description cs:0.5,0) -- (axis description cs:0.5,0.92); \end{axis} \end{tikzpicture} I'm writing up my dissertation and I'm really confused where to use "the". Examples:     In this experiment, (the?) heat transfer coefficient was calculated, allowing to estimate (the?)frost thickness.    And:     Firstly, (the?) air properties are defined.   And:     As (the?) crystals grow in radius, the amount of air in (the?) ice is reduced, due to (the?) ice becoming more packed.   None of the nouns mentioned above have been used for the first time, so I rule out the use of "a".   Would appreciate any help. Uniform convergence of $f_n(z)=nz^n$ in the set $|z|<\frac{1}{2}$ passthrough graphics card to virtualbox So I'm trying to get my laptops NVIDIA fx 880m to pass to a virtual box running window seven on a linux mint 17 install.  So far everything seems to be (maybe) heading in the right direction:  the device passed through happily, when I booted the virtual box it installed a bunch of new drivers, but when I try to install the NVIDIA driver on the guest it can't find the card. I looked under the device manager and there's no listing for the nvidia card under the pci bus, my guess would be that I need to disable it in the host so that it can be passed through to the guest (the virtual box manual said they can't be shared) but I'm not sure how to do so. Can anybody help with this? This question does not seem to me to be a duplicate of , because here the given solution was that it was not possible with a windows host, however in this case the host is linux Value of $x=\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of second digit $1$ of $x$ What is the value of the series   $\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}}$   and can one find closed forms for the sums   $\sum\limits_{n=1}^N {\frac{1}{10^n-1}}$?   I found that $\sum_{n=1}^N {\frac{1}{10^n-1}} $  =  $\sum_{n=1}^N (\lim_{m\to\infty  }{\frac{10^{-n}-10^{-nm}}{1-10^{-n}}})$  =  $0.111\ldots+0.0101\ldots+\ldots$ but I can't go farther.     The original question is: "Where does the second '1' appear?" Windows 10 "Enable NTFS long paths policy" option missing MapThread with non-rectangular lists "(" or "\left(" parentheses? What is the difference between ( and \left( in LaTeX? Sometimes, when the content is small, it does not seem to matter which pair I use.  What is the best practice when it comes to which parentheses use? Should laptops remain plugged in when their battery is 100% charged? When your laptop's battery is 100% charged, should you leave it plugged in so any battery power doesn't get used, or will that cause overcharging, overheating, etc.? Should the laptop be unplugged when the battery level is 100%?  I'm asking because my laptop's screen tends to get dim when unplugged, so I don't like to run it on battery. Abstract algebra book recommendations for beginners. I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved problems and exercises? The lecturer said in this semester, he will cover group theory, ring theory and a bit of field theory. Which book contains extensive information in these three fields? So it looks like Stack Exchange now supports HTTPS (to some extent). Which is awesome! But there are a few problems, the main one being that some content is delivered over the CDN, which is plain HTTP. This causes browsers to complain about "unsecured content": How do programmatically create fields for content types, and ADD them to the the content type form Say I have this field, "map_description". I know I would this function to define the field:   $field = array(   'field_name' =&gt; 'map_description',   'cardinality' =&gt; 1,   'type' =&gt; 'text', ); field_create_field($field);   And I have this code which I'm not sure what it does but I'm told that I'll need it:   $instance = array(     'field_name' =&gt; 'map_description',     'label' =&gt; 'The map description.',     'bundle' =&gt; 'my_content_type',     'entity_type' =&gt; 'node',     'widget' =&gt; array(     'type' =&gt; 'text_textfield',  );  field_create_instance($instance)   Those two code bits are both in my install hook and run when I install the module. But while the fields are in fact created, I have to manually assign them to the content type via "manage fields", is there a way to have the fields automatically assigned to the content type? How to access object attribute given string corresponding to name of that attribute Impulsioned jumping There's one thing that has been puzzling me, and that is how to implement a 'faux-impulsed' jump in a platformer. If you don't know what I'm talking about, then think of the jumps of Mario, Kirby, and Quote from Cave Story. What do they have in common? Well, the height of your jump is determined by how long you keep the jump button pressed.  Knowing that these character's 'impulses' are built not before their jump, as in actual physics, but rather while in mid-air - that is, you can very well lift your finger midway of the max height and it will stop, even if with desacceleration between it and the full stop; which is why you can simply tap for a hop and hold it for a long jump -, I am mesmerized by how they keep their trajetories as arcs.  My current implementation works as following:    While the jump button is pressed, gravity is turned off and the avatar's Y coordenate is decremented by the constant value of the gravity. For example, if things fall at Z units per tick, it will rise Z units per tick.  Once the button is released or the limit is reached, the avatar desaccelerates in an amount that would make it cover X units until its speed reaches 0; once it does, it accelerates up until its speed matches gravity - sticking to the example, I could say it accelerates from 0 to Z units/tick while still covering X units.  This implementation, however, makes jumps too diagonal, and unless the avatar's speed is faster than the gravity, which would make it way too fast in my current project (it moves at about 4 pixels per tick and gravity is 10 pixels per tick, at a framerate of 40FPS), it also makes it more vertical than horizontal. Those familiar with platformers would notice that the character's arc'd jump almost always allows them to jump further even if they aren't as fast as the game's gravity, and when it doesn't, if not played right, would prove itself to be very counter-intuitive. I know this because I could attest that my implementation is very annoying.    Has anyone ever attempted at similar mechanics, and maybe even succeeded? I'd like to know what's behind this kind of platformer jumping. If you haven't ever had any experience with this beforehand and want to give it a go, then please, don't try to correct or enhance my explained implementation, unless I was on the right way - try to make up your solution from scratch. I don't care if you use gravity, physics or whatnot, as long as it shows how these pseudo-impulses work, it does the job.  Also, I'd like its presentation to avoid a language-specific coding; like, sharing us a C++ example, or Delphi... As much as I'm using the XNA framework for my project and wouldn't mind C# stuff, I don't have much patience to read other's code, and I'm certain game developers of other languages would be interested in what we achieve here, so don't mind sticking to pseudo-code.  Thank you beforehand. Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$ zsh scripting - How do I NOT match a pattern in filename generation? I have recently been reading about Quantum Electrodynamics which I found very interesting, but even more confusing.  I understand photons mediate the electromagnetic force and interactions between charges can be described by virtual photon exchange. And that in Quantum Mechanics particles paths can be evaluated by sum over all paths...  An electron travelling from one point to another will take all paths, emitting and reabsorbing virtual photons. Does the electrons take all paths including those that involve faster than light velocities?  Is this question null, Does it mean anything to ask about properties of virtual particles?  Do I have the wrong idea about what virtual particles are? Words to describe people with good/poor vocabularies Words to describe people with good/poor vocabularies. I am looking for words which I can use to describe characters and I would also like to describe their vocabularies, and explain how their vocabularies stem from their upbringing Finding the search terms behind 'not provided' We use Google Analytics to track the vistors to our websites, were currently putting together a big audit of our SEO and one of the things we want to see is what search terms are working for us.  About 90% of our organic search traffic comes under 'not provided' which as i understand means a user is signed into a Google account when coming to our site, thus Google doesn't track their search terms.   This makes it really hard to conduct a meaningful study, (i suppose we could get the data of the 10% that we do have search terms for and multiply them it but i dont think thats a particularly good or meaningful approach).  Is there a workaround to see behind the 'not provided' data and see the search terms actually used ?    UPDATE  I read  last night from Hacker news, which states :     [using https will allow you] more complete referrer information in Google Analytics for people   visiting from sites already using HTTPS   and contained a link to this , it seems to say that because Google's users are using https that an https site cannot pass data to an http site, as the http site is not secure, none of the sites i manage have https / sll setup, could some one confirm if Google will pass you more referrer data if your site has https implemented ?