How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.
If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at  $$ \frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0. $$ Now mathematically this is clearly a divergent series, but using zeta function regularization here we are taking  $$ \sum_{n=1}^\infty n = \zeta(-1) = -\frac{1}{12}. $$ And obtain $ D = 26 $ where $\zeta $ is the analytic continuation of the zeta function we know. But it makes no sense in putting $ s = -1 $ in the formulae $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}. $$ As the above is only valid for $ Re(s) &gt; 1 $. So what is going on in here? Can anyone give me a reasonable explanation about obtaining $ -1/12 $?
DMed for the first time yesterday and one of the things I feel I need to sort out, among other comments, improvements, and similar, is how combat itself is actually initiated. The main example I want to use involves  attempting to generate surprise conditions by players. Here the characters have entered a ruined city and generally been noisy about the whole thing - using a spell to make their voice louder and literally ask if there is anyone about. Any enemies around probably know what something is about to kick off but arguably don't know precisely what. After resting the PCs head up a long flight of steps to a temple on the edge of the city. At the top only one crests and finds 3 bandits lounging around, chatting, and doing whatever bandits do in their spare time. After a brief exchange of pleasantries a combat starts:  DM &quot;The bandit spits in your direction 'we're not done looting the temple, so you can clear off', and starts raising his crossbow as his mates emerge from their cover behind the pillars... PC &quot;I cast Firebolt at Bandit Keith&quot; DM &quot;Hang on, I hadn't finished, everyone rolls for initiative...  Which sparked a debate about whether or not the PC could preemptively cast/attack before the combat starts by &quot;instant actioning&quot; (as above) as I was describing the start of the combat. As far as the bandits are concerned a lone half-elf who is pretty heavily armed has popped up in front of them, not even tried to be diplomatic, and ended up saying something rather suggestive about their mothers. The PC then asked about it being &quot;surprise&quot; because the bandits wouldn't necessarily expect an attack, but my view of &quot;surprise&quot; is closer to an ambush where one party is unaware of the other's existence. (This precise example is close to how  works, however this specific PC was always the first of the party to go into rooms (only one with darkvision) and consistently felt they should get free rounds at the start of combat based on declaring attacks before I could &quot;start&quot; the combat.) The second example the party came to more bandits, who were not alert immediately, in an underground room. They used magic to yell something obscene at the bandits and shut off all the torches so the bandits were effectively blinded. At this point they wanted to burst into the room casting spells, swinging maces, and playing some &quot;meaty chords&quot; (the bard's words, not mine). Again they wanted surprise arguing that, despite the fact that the bandits were yelling and cursing in confusion, they couldn't be sure they were actually being attacked as such. In the end a character with darkvision wanted to sneak in to a corner and again wanted surprise based on passing stealth rolls, so they would start the &quot;combat&quot; by firing a crossbow. At that point I think I relented because it was clear that they would do anything to try and win combats in a single round. In a nutshell, my question is: &quot;how liberally can the concept of surprise be applied to combat - is it only when one party is literally unaware of a threat (as I read the rules), or just when they are caught slightly off guard by the exact situation?&quot; NB I already plan to address the attempts at interrupting descriptions with actions next session, because it happened at other times and actually threw me off quite a bit. When using Alt+Tab, I can't seem to get back to Skyrim. When clicking its icon or Alt+Tab to get back, I just see an empty window. Windowed mode doesn't allow me to use it in a Windowed (Fullscreen) manner as seen with other games, so that doesn't seem to be an option either. Is there any way to fix this behavior? I used a  for Photoshop, and was able to create some boxes with it. These boxes were shaped in the form of Software Boxes, while I require Customized Dimensions Boxes, which currently I'm trying to create with .   This software creates a box with no Reflection, and I disabled shadows in it, cause it doesn't allow transparency while saving to png. It allows custom dimensions, rotation, etc settings, but doesn't save transparency or reflections.  Here's the difference in output from both the above mentioned apps.. I'm trying to get the reflection + shadow in the 3D box Generator to the Protomissume generated image.   My question: How can I create similar Reflection &amp; Shadow in Photoshop?   Images:  3D Box Generator   Protomissume Box Shot   Thanks for any help.  Here are the Box Shot images (Photoshop, Protomissume) :  [  ] Can't access pre populated SQLite database using PhoneGap/Cordova in Android Violation of linearity assumption in Logistic Regression I built a logistic model with 7 continuous independent variable and 8 categorical (dummy) variables in R. I tested for the linearity assumption using the Box Tidwell test and all my variables show high significance.  I am wondering now, if the assumption is violated what it would mean for the coefficient estimates and if I still can run the model with caution.   I read about the approach of using regression splines but I am confused how I would determine those and interpret the model then. Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us:   Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because that would mean it's not a function as it fails the vertical line test). This function also has a vertical tangent at $x=0$.   My question is: how can we have a function that also has a vertical tangent? To get a vertical tangent we need 2 vertical points, which means that we are not working with a "proper" function as it has multiple y values mapping to a single $x$. How is it possible for a "proper" function to have a vertical tangent?  As I understand, in the graph I pasted we cannot take the derivative of x=0 because the slope is vertical, hence we cannot see the instantaneous rate of change of x to y as the y value is not a value (or many values, which ever way you want to look at it). How is it possible to have a perfectly vertical slope on a function? In this case I can imagine a very steep curve at 0.... but vertical?!? I can't wrap my mind around it. How can we get a vertical slope on a non vertical function? Behave as if it was or it were You can place the script in the head or body as you like. The   script will behave as if it was located exactly where you put the   script tag in the document.   The sentences above are taken from  (at the very bottom). I think they made a grammatical mistake here: it should be behave as if it were rather than it was. I'm not 100% sure, so want to confirm it. unable to use synaptic via launcher When I open synaptic via launcher on gnome-panel, it opens and asks me for the password but if I enter the password, it says its wrong even though the password is correct and it used to work OK few days ago. If I open synaptic via terminal using sudo synaptic then it opens OK without any problem. if p is prime then 1920 divides $p^4-10p^2+9$ How can I formalize that the function $$f:\mathcal{P}(A)\to(A\to \{0,1\})$$ is bijective? It seems clear to me but I don't know how to write a proof. Thanks in advance. How can I use xindy on a system with libncurses.so.6? self referential struct definition? Tail sum for expectation In , the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$:  $$E(X) = \sum_{i=0}^\infty P(X &gt; i).$$   I wonder if there is a similar formula for nonnegative continuous random variable $X$:  $$E(X) = \int_0^\infty P(X &gt; x) dx?$$  If no, are there some conditions for it to hold? And how can it be proved?  Here is my thought:  If the cdf $F$ of $X$ is bijective, then $X=F^{-1}(U)$ for some random variable $U$ uniformly distributed over $[0,1)$. So $$E(X) =     \int_0^1 F^{-1}(u) du.$$  To prove the tail sum formula, it suffices to prove $$\int_0^1     F^{-1}(u) du = \int_0^\infty P(X &gt; x) dx.$$ But I am stuck here.   What's more, is the condition that the cdf $F$ of $X$ is bijective really necessary for tail sum formula to hold? Can tail sum formula be generalized to a random variable that is not necessarily nonnegative?   Thanks! What is "it" in the following sentence: It is clear that Bob likes doughnuts Finding a distribution from the sum of two random variables in a joint trinomial distribution Let X and Y have a trinomial distribution with join probability function:  $ f(x,y) = \frac{n!}{x!y!(n-x-y)!}p^x q^y (1-p-q)^{n-x-y}; x=0,1,...n, y=0,1,...n$  Let $T=X+Y$  I can't seem to figure out where to go exactly. Here's where I am at:  $\sum_{x=0}^{n} \frac{n!}{x!y!(n-x-y)!}p^x q^y (1-p-q)^{n-x-y}\mbox{ substitute in Y=T-X}$ $\sum_{x=0}^{n} \frac{n!}{x!(t-x)!(n-t)!}p^x q^{t-x} (1-p-q)^{n-t}$   I think I could I figure out the algebra.. if I could just see what to do with the summation. I know I have to involve a t somewhere, but I'm not sure. Maybe somebody could just show me how to set up my summation correctly with the substitution? Can you use another material as input to material nodes in cycles? Dynamically set value of a file input A standard document (e.g. scrreprt) uses roman font family for text and Sansserif font family for titles etc. It is the other way round in beamer. How can I switch the two font families? Or why do beamer define the font families in this way?  For the background: I think this is my problem when using formulas, because they use the Sansserif font instead of the roman version. Formulas using the euler package look really poor.  P.S.: I compile with LuaLaTeX using TeXlive 2013. In my resume, I'm describing several projects I've worked on in the past. Most of them were done in teamwork. What is the best way to say this?  Examples:  Together with XY, I built ... OR  In collaboration with XY, I built ... OR  XY and I built ...  Are the first two examples grammatically correct? I'm unsure since first I state that something has been done in collaboration with a second person, but then I use "I built ...". Relationship between balloon size and pressure I see numerous websites talking about how a small balloon is at higher pressure than a big balloon. This is a fun counter-intuitive factoid... unless it isn't a fact at all.  Young-Laplace gives  $$T = \Delta p \frac R2 $$  Which means that for a given tension, increasing the radius does decrease the pressure. But why would the tension be constant? A latex balloon is like a 2D spring, right? So the tension in a inflated spherical balloon should be proportional to the surface area, correct?  $$T \propto R^2 $$  But this gives  $$\Delta p \propto R$$  Which means the bigger the balloon, the higher the internal pressure. Which, again, would be the normal intuition - but it would burst the bubble of some simplified explanations. (har har)