In a simple physics experiment, we take the average a few readings to reduce the random errors. I apply  rules to these.  Say we round off at each step:  (8.0+9.0+10.0)/3 = 27.0/3 (addition keeps dp) = 9.00 (division keeps sf)  Say we choose to only round off at the end by taking the least precise of given readings:  (9.0+10.0+11.0)/3 = 30.0/3=10 (2sf because 9.0 is 2 sf  'common sense' tells us that in the first example, the decimal place should be kept at 9.0, and in the second, it should be 10.0, not 10.   Another example: Given the external (19.2) and internal (18.3) diameter of a pipe, calculate its thickness: (19.2-18.3)/2 =0.45 or 0.5 or 0.450?  All these significant figure rules seem to only work out in theory to me. Can someone explain?
I'm a little confused as to when to use  for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.  Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$  Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$  Should I round $125.664~\mathrm{m}/\mathrm{s}$  to  $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two?
How to determine CPU and memory consumption from inside a process? How to justify $\phi(n) \ge \sqrt{n}$ If $\phi(n)$ is the Euler-totient function, how can I show that $\phi(n) \ge \sqrt{n}$? Is $2^\sqrt{2}$ irrational? Is it transcendental? Apply empty style to the entire bibliography $A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected Let $X$ be a topological space. Let $A$ and $B$ be two subsets of $X$ such that: $A$ and $B$ are closed, $A \cup B$ is connected, $A \cap B$ is connected     Prove that $A$ and $B$ are connected.   Thoughts:  Suppose that $A$ is disconnected. Since $A$ is closed, there are two non-empty, disjoint, closed sets $E$ and $F$ in $X$ such that $A = E \cup F$. Then I attempted many things but all were not really useful.  Please help. Thanks. Loop-cut won't go all the way around Alright so I've seen some postings about loop cuts and done some research. I am very new to blender and modeling overall. I am trying to loop cut this concrete barrier. Now I'm following a tutorial and kind of did my own thing in some areas. I cleaned it up following some instructions in another answer here on stack exchange.  I can't for the life of me figure out why I can loop cut around the whole bottom:   ...but not the whole middle:   I see this marked as answered by another question. That question refers to ngons and inside faces. Now once again I'm totally new to this. I looked on the inside. I don't believe I have inside faces or any ngons by the terms I've read online in my research.  Here is a wireframe picture and the inside view:     Maybe I'm missing something the answers on other posts are saying. If so and you know what it is I'd be very appreciative of what it is. I have several input and option elements on my page, each (well almost) have an event attached to update some text on the page once they change. I use jQuery which is really really cool :)   I also use Microsofts  framework, utilizing the UpdatePanel. The reason why I do that is that certain elements are created on the page based on some server-side logic. I don't really want to explain why I use the UpdatePanel - even if it could (it can with quite some effort) be rewritten to use only  I still want that UpdatePanel.  You probably guessed it - once I have a postback on the UpdatePanel the jQuery events stops working. I actually was expecting this, since the "postback" is not really a new postback so my code in document.ready that binds the events won't be fired again. I also confirmed my suspicion by reading up on it in the jQuery help libraries.  Anyway I'm left with the problem of rebinding my controls after the UpdatePanel is done updating the . I preferably need a solution that does not require adding more .js files (jQuery plug-ins) to the page but something as simple as being able to catch the UpdatePanel's 'afterupdating' where I can just call my method to rebind all the form elements. Can a service written in .NET self-terminate? I have a service application written in C# and under certain circumstances, I would like it to terminate itself.  This would happen after the service has been running for a while, so this would not be happening in the OnStart() event.  Everything that I have read so far suggests that the only safe way to terminate a service is through the Service Control Manager.  My service runs as Local Service and does not have the rights to start or stop services, so I can't access the SCM from the service itself.  Is there another way to self-terminate while still playing by the rules of the SCM? How can I find a mentor to critique my work? What is the significance of a mostly blank user card? I want to show that for each $n \in \mathbb N$ it holds that $$\sum_{k = 0}^\infty \frac{n^k}{k!} \vert k - n \vert = 2n \frac{n^n}{n!}.$$ But I have absolutely no idea how to do that. I tried to split the series like this:  $$\sum_{k = 0}^\infty \frac{n^k}{k!} \vert k - n \vert = \sum_{k = 0}^{n - 1} \frac{n^k}{k!}(n - k) + \sum_{k = n + 1}^\infty \frac{n^k}{k!} (k - n)$$ Now I want to show that $\sum_{k = 0}^{n - 1} \frac{n^k}{k!}(n - k) = \sum_{k = n + 1}^\infty \frac{n^k}{k!} (k - n) = n \frac{n^n}{n!}$ holds. But I got no idea how to achieve that. I tried to use induction but things became pretty messy and I don't think it will work like that. On the other hand I got the idea that it might have to do with $\exp$ or binomial coefficients but I don't see where I can facilitate that. I would appreciate some hints :) Cursor is set to move by increments. how to reset to normal? I am currently running a Dungeons and Dragons 4 campaign for a group of 9 players.  Whittling the group down to a more manageable size is not really an option since they are all family.  When we do combat it takes forever even if I have one player help me and allow them to go 2 at a time.  When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs.  I've tried encounters with many foes, encounters with only a few tough foes, dialog only encounters, social puzzle encounters, everything I can think of.  What else can I try to get this group more engaged? Scaling problem with fillbetween in pgfplots From everything I've read about quantum mechanics and quantum entanglement phenomena, it's not obvious to me why quantum entanglement is considered to be an active link. That is, it's stated every time that measurement of one particle affects the other.  In my head, there is a less magic explanation: the entangling measurement affects both particles in a way which makes their states identical, though unknown. In this case measuring one particle will reveal information about state of the other, but without a magical instant modification of remote entangled particle.  Obviously, I'm not the only one who had this idea. What are the problems associated with this view, and why is the magic view preferred? Suppose $\mu(X)&lt;\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q&lt;\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r&lt;q$   In the solution the assistant wrote this:  If $1\le r&lt;q$ then $q/r\ge 1$. Let p be the dual exponent to $q/r$ (i.e. $\frac{1}{q/r}+\frac{1}{p}=1)$  then by Hölder Inequality  $\int_{X}|f|^rd\mu\le\underbrace{(\int_{X}|f|^qd\mu)^{r/q}}_{??????}(\int_X1^pd\mu)^{1/p}&lt;\infty$  for me the middle part doesn't make sense (it is not the hölder inequality) can somebody show it ?  instead of this can't we say (only in this case) that   $\int_{X}|f|^rd\mu\le (\int_{X}|f|d\mu)^r$ then  $(\int_{X}|f|d\mu)^r\le ((\int_{X}|f|^qd\mu)^{1/q}(\int_X1^pd\mu)^{1/p})^r&lt;\infty$  ? What would be the effects of warp collisions? Triggered by a  I was wondering what actually would happen to a ship (e.g. the Enterprise-D NCC-1701) that collides with a stationary object such as a) another ship of a similar configuration or b) a gigantic structure like a moon or even a planet.  We know the deflector is supposed to keep interstellar dust (not to be confused with ordinary dust, which is a different thing) from ripping large holes in the ship.  At first, one thinks "Obvious. ." But does it really? Would the deflector manage to actually deflect both masses? If not, where is the threshold of what a deflector can handle? Would the ship maybe tunnel right through the object, because of wibbly-wobbly space wobblification by the warp field? It does do some weird things to the space right in front and behind the vessel.  Important remark: We cannot approach this with what we know about physics at all. For all intents and purposes, both ships must measure the other ship travelling faster than light, resulting in it having infinite (or rather undefined) mass and by extension momentum. If we start with that reasoning any ship jumping to warp would suck in the entire universe (it takes a moment, but think about it). So, obviously we cannot apply our intuition (even our relativistic intuition) here. Is this noun used as an adjective? I read this recently in :     At the end of the summit, the French and European officials had   claimed a points victory over the Germans by getting them to agree   more firmly to a target date of January 1st next year to entrust the   European Central Bank (ECB) with the ultimate authority to supervise   the euro zone’s 6,000-odd banks.   "Points" is a plural, and I suppose it doesn't function as a noun in this sentence since it's following the article "a", so is it an adjective adorning "victory"? I want to learn more about hopf algebras but I am having trouble finding a down to earth introduction to the subject with lots of motivation and examples. My algebra knowledge ranges from Dummit and Foote, Atiyah-Macdonald commutative algebra, and Hartshorne level algebraic geometry.  What are good places to look at for a hopf algebra beginner? My main motivation is for studying groupoid schemes from an algebraic point of view. Best vehicle for each stage in Hill Climb Racing In Hill Climb Racing we have several vehicles and several stages.  Different vehicles have different performances on the same stage, and I'd like to know what are the best combinations to maximize distance in each stage.    Also, what is the best combination of a vehicle and a stage to get lots of coins?