I am struggling to find an explicit bijection $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ when $\mathbb{N}$, by definition, includes $0$. If $\mathbb{N}$ were to exclude $0$, I could note that any positive integer can be written uniquely in the form $2^{a-1} (2b - 1)$ and then map $(a,b) \to n$. Including $0$, I have to change $a - 1$ to $a$ and $2b - 1$ to $2b + 1$, but I still run into the issue of what to map to $0$. I can't map $(0,0)$ to $0$, for example, because $2^{0} (2(0) - 1) = 1$ is the unique element of the form $2^{a-1} (2b - 1)$ mapping to $1$. Does anyone have any tips for how to repair this bijection?
Towards the end of , how do I get  $$ \frac{\sqrt{8z + 1} - 1}{2} &lt; w + 1 $$  Here $w = x + y \geq 0, t = (w^2 + w)/2 , z = t + w, x,y \in \mathbb N_{\geq 0}$.  Thanks.
Getting Joker from the existing groups in Rummikub I have a group of tiles made up of 3 fours and a joker:     4 4 4 Joker   Now, if a player removes the joker, do they have to replace it with another 4? Trying to Remember a Book that explored evolution of Dinosaurs to intelligence and then encountering humans Many, many years ago, I read a book where dinosaurs were assumed to have suvived the K-T event (or it never happened), and they evolved to intelligent species that used not only language (which combined sound, posture, and colour changes on their skin) but also used technology.  In the series, a human is captured or somehow gets integrated with one of the evolved dinosaur groups.  Again, from a very faint memory, the dinosaurs had a level of technology that would be about middle ages, and the humans were much more analogous to American Indian.  That's really all I can recall about the book, and I am trying to find it for my daughter to read.  Can anyone help me out? While accepting an offer to grad school, one is basically entering into a lasting relationship with one's adviser - most likely, someone whom the applicant has never met before, and the only exposure has been through the potential adviser's website/publications. It is in the interest of both parties to ensure (to the greatest possible extent) that there are no personal/professional traits of either that hamper the formation of a pleasant working relationship - no-one would want to go through the ordeal of having to change advisers midway!  While the faculty has a chance to have a good look at the applicant's profile as well as his motivations (through his grades and SOP), the applicant doesn't have a similar opportunity. So, I'm interested to know what parameters can be used to gauge a potential fit. I've thought of the following:   Past students:  Did anyone ever drop out/change advisers midway, and if so, for what reasons? These would be a bit hard to find though, as I don't expect the faculty concerned would list them on their website. It would be great if anyone could let me know how to find out the list of incoming students to a department for any year. Publication rate, taking into account the venues where they were accepted. Time taken to graduate - though I accept this is more dependent on the student, a median figure should be telling ... What they did post Ph.D. - did anyone get tenure if they went into academia, or is almost everyone unable to break out of being a post-doc?  Tenure status: I'm a bit unsure about this, so wanted the community's opinions about it. Just so that I'm clear, I'm only trying to calibrate the applicant's expectations about the working style of his potential adviser - and hence need to know to what extent are the following "typical" assumptions valid.  Tenured professor  A full professor is more likely to get grant funding, hence less time spent on TAship - but could also mean less time/effort spent on interactions with students (either being busy with other projects/talks, or due to more commitments to family at that age).  Tenure-track faculty (Assistant profs)  More likely to be young and energetic, and could translate to more time spent on one-to-one discussions with grad students - but funding may prove to be an issue, and may have to be on TA for a longer period.   What other factors would be relevant in this matter, and to what extent am I correct/incorrect in either the factors considered, or for undertaking this exercise at all? Sorry for the title, its hard to put the problem into proper words.  Basically, in Firefox or Chrome, I try to navigate to stackoverflow.com.  The request fails and the URL I see in the navigation bar is:  http://d2e24t2jgcnor2.webhostoid.com/Secure/Error?URL=http%3A%2F%2Fstackoverflow.com%2F&amp;ResponseCode=504&amp;user_id=5b5033d3-3543-4b7c-9844-055afde93841&amp;uc=20140915&amp;subid=20140915&amp;source=browsersafeguard-rockettab-spigot&amp;version=1.0.5370.12627&amp;implementation_id=browsersafeguard-rockettab-spigot%20&amp;block_host=False&amp;reg=False&amp;redirectms=True   This looks shady and my internet seems slower lately. I have a few questions:   Why is my request going through this strange URL? Do the query parameters reveal my identity? How can I stop this?   I use my cell phone’s 3G network and tether it to my laptop. When do I clear US customs and immigration if connecting at a US airport to my final destination elsewhere in the US? I'm a British citizen travelling from London Heathrow (LHR) to the Hartford, Connecticut airport, and have a two hour stopover in Newark, New Jersey (EWR) on my itinerary. At what stage would I clear immigration and customs on arriving in the USA? Apply subsurf modifier to only part of an object How can I apply a modifier only to a part of an object? I want to model a goblet with a pedestal, and if I add a subsurf, the pedestal, of course, gets an unwanted round shape. How can I avoid that? I have tried Googling it, but, to my surprise, found nothing. parsing XML file gets UnicodeEncodeError (ElementTree) / ValueError (lxml) How do you get the current time in milliseconds (long), in VB.Net? To an an external observer it appears that time has stopped for photon. But this relation is reflexive, so for an observer travelling with the photon it appears the universe has stopped everywhere.  Is this right?  Space also gets distorted parallel to the direction of motion, but not perpendicular to it.   Does this mean that for an observer travelling with a photon sees spacetime as a flat plane?  Note 1: I'm using language vividly not literally when I say a photon experiences space and time. Not that I'm against idealist or panpyschist interpretations of matter or energy come to that.  Note 2: Since it's been pointed out that an inertial frame can't be boosted into the frame of a photon and moreover there is no limiting behaviour here - no matter what frame you are boosted into, light will always appear to travel at c. Nevertheless, when an external observer watches a train chasing a photon by gradually increasing its speed, he will note that the train speed is approaching the speed of the photon (but will never match it) and when he looks at the clock inside the train he will not too that it is gradually slowing down (but never actually stops). Using this picture, we can attempt the thought experiment - if not one that can be carried out - of what a world looks like to a photon. Do $T$-invariant subspaces necessarily have a $T$-invariant complement? How do I install Ubuntu from an USB drive / create a live USB-stick in Windows? How do I see my name in the panel in Gnome Shell? I have Ubuntu GNOME 14.04 LTS and from installation, my name has not appeared in the panel. I have enabled it via dconf in org.gnome.desktop.privacy, but it has yet to appear even after a re-login and restart.  Can I receive some assistance into the cause of this and a solution?  This is where I would like it to appear: What was with the two different colored prison uniforms in Daredevil season 2? In Daredevil S02, we see two kinds of prison uniforms, full orange and full white. We see in the flashback that Wilson Fisk gets a white prison uniform but his current one was orange.  The Punisher was also given a white prison uniform but most of the prisoners had orange uniforms. What was with the two different colored uniforms? Does it have some significance or is it just random? Does it happen similar in real life US prisons too? "contribute to investigating" or "contribute to investigate" I'm trying to find, which of the following sentences is correct?  A) We try to highlight how they contribute to investigate their importance. B) We try to highlight how they contribute to investigating their importance.  I found that both cases were used frequently in formal writing. Edited: Sources for each case: For A: &quot;Their multifaceted nature means that there are very many routes to investigating their importance for understanding social change&quot; in &quot;A lineage for contemporary inter-city studies&quot; book. B: &quot;More factors are analysed to investigate their importance in scoring from shots and shot probabilities&quot; in &quot;Applications of logistic regression to shots at goal at association football&quot; book. Is it true that $\lfloor\sqrt{n}+\sqrt{n+1}+\sqrt{n+2} \rfloor=\lfloor\sqrt{9n+7}\rfloor$? I know it's true that  $$\lfloor\sqrt{n}+\sqrt{n+1} \rfloor=\lfloor\sqrt{4n+1}\rfloor,\forall n\in \mathbb N. \tag 1$$ Is it true that  $$\left[\sqrt{n}+\sqrt{n+1}+\sqrt{n+2} \right ]=\left[\sqrt{9n+7}\right], \tag 2$$ $$\left[\sqrt{n} +\sqrt{n+1}+\sqrt{n+2}+\sqrt{n+3}\right ]=\left[\sqrt{16n+20}\right], \tag 3$$ $$\left[\sqrt{n} +\sqrt{n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\right ]=\left[\sqrt{25n+49}\right], \tag 4$$ for all $n\in \mathbb N$? I have checked $(2),(3),(4)$ through $10^6.$   (I think $(4)$ maybe has counter-examples, but I cannot find it.) How to export SVG without lines, polylines and polygons? How can I prove  from first principles that $0!$ is equal to $1$? Proving that if $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$, then $A$ is diagonalizable. A question about the higher-order Weyl variation for the geodesic distance I have a question in deriving Eqs. (3.6.15b) and (3.6.15c) in Polchinski's string theory vol I p. 105.   Given      $$\Delta (\sigma,\sigma') = \frac{ \alpha'}{2} \ln d^2 (\sigma, \sigma') \tag{3.6.6}$$   where $d(\sigma,\sigma')$ is the geodesic distance between points $\sigma$ and $\sigma'$.   It is said for the Weyl variation $\delta_W$,     $$\partial_a \delta_W \Delta(\sigma,\sigma')|_{\sigma'=\sigma}= \frac{ 1}{2} \alpha' \partial_a \delta \omega(\sigma) \tag{3.6.15a} $$   $$ \partial_a \partial_b' \delta_W \Delta(\sigma,\sigma')|_{\sigma'=\sigma}= \frac{ 1+ \gamma}{2} \alpha' \nabla_a \partial_b \delta \omega(\sigma) \tag{3.6.15b} $$   $$ \nabla_a \partial_b \delta_W \Delta(\sigma,\sigma')|_{\sigma'=\sigma}= - \frac{  \gamma}{2} \alpha' \nabla_a \partial_b \delta \omega(\sigma) \tag{3.6.15c} $$   Here $\gamma=-\frac{2}{3}$.   I know the reason for the factor of $1/2$ in the RHS of Eq. (3.6.15a) and I can derive it. But, putting $\nabla$ on LHS and RHS of Eq. (3.6.15a) will not lead to (3.6.15c). I guess it is because there is some higher-order terms in the geodesic distance where the treatment of (3.6.9)-(3.6.11) omitted. But I don't know how to get it. I tried $ds^2= g_{ab} dx^a dx^b= (\eta_{ab} + h_{ab})dx^a dx^b $ to separate the deviation of Minkowski metric, but I cannot get the factor $\gamma$   My question is, how to derive Eqs. (3.6.15b) and (3.6.15c) (especially there is a prime, $'$, namely $\partial'_b$ in the LHS of (3.6.15b) but not in the RHS) Legacy Debian versions and Bash Shellshock We are running Debian Etch, Lenny and Squeeze because upgrades have never been done in this shop; we have over 150 systems running various Debian versions.   In light of the "shell shock" of this week, I assume I need to upgrade bash.  I do not know Debian so I am concerned.   Can I merely execute apt-get install bash on all of my Debian systems and get the correct Bash package while my repository is pointed at a Squeeze entry. If not, what other course of action do I have?