In $F = kma$, why $k$ is taken to be unitless?  If $k$ is unitless and 1, then we have $F=ma$. This means (I guess) the physical quantity Force is product of different (from Force) physical quantities mass and acceleration. But doesn't that mean we are equating entities that are physically different? What is the basis for the choice of $k$ to be unitless.
It seems intuitive that $a\; \propto \frac{F}{m}$, as the greater the force that is applied on an object, the greater its acceleration will be. Inversely, the greater the mass of the object, the slower the acceleration will be.  However, when rewriting proportions as equations, you must introduce a constant of proportionality, and in this case of a direct proportion, if $a \propto \frac{F}{m}$ then when rewriting as an equation you will have $$a = k\cdot\frac{F}{m}$$  In order to get the standard formula $F = ma$ this constant must be $1$. However, how do we know that this is the case? How do we know that the constant isn't $2$ and the formula $F = \frac{1}{2}ma$, for instance?
I am travelling form Delhi (DEL) to San Francisco (SFO), I have to make a connection flight in Vienna that would take me to Frankfurt (FRA). My flight to SFO is from FRA, I have one ticket for entire journey and the flights are operated by Lufthansa or its subsidiaries.  Do I need a transit visa? My flight arrives at Vienna international airport and leaves from the same, and for second leg it arrives at Frankfurt international airport.  I asked this question as an addendum to my  but was advised to make a new post.  I hold Indian passport and valid US visa. Configuring, compiling and installing a custom Linux kernel I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go.  In the truth table, where $A \implies B$ we obtain this result:  A   B   T/F ___________ T | T | T T | F | F F | T | T F | F | T   Now, my confusion here is regarding why $A\implies B$ given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.  Now... then, why is B true despite the fact that A is false? Or rather, why is the statement B given not A, True. The truth of B is implied from the truth of A, but A is a false statement. So it's very contradictory that this may be, and in fact, if you go through a real life example, you would find that the third statement seems to be false.  Say $A$ is the statement that it is dark outside, and $B$ the statement that the sun is not on our side of the earth. If it is not dark outside, than stating that the sun is on the other side of the planet must be wrong (in my example i'm going to claim that the sun is the only light source).  Side note:  I just had a thought, if A is false, does that mean you can conclude... mmm, whatever? I mean, B is implied via the truth of A, if A is false, B is NOT implied, but then B could be EITHER true or false, I suppose. Is this correct?  However, would that not mean that A does not imply anything when false. Would that not set the 3rd and 4th statement to be undefined? how many conditional sentence types are possible? I wonder how many conditional sentence types are possible. Basically, there are four main and two mixed types. By mixing tenses, it could be up to nine of them. Are the rest viable?  I came across the following examples in Full Spate Pre-Profficiency Primer:   After you have just sold alcohol to a young guy. If he is if under eighteen, you should not have sold him the alcohol. If he promised to bring the drinks, he will. If you will take a seat, the doctor will see you shortly. (Likely, the former will demonstrates politeness only.) I would be grateful if you would tell me a little about your friends. (The same politeness, but would instead.) If he will keep eating all those biscuits, he can not expect to stay as slim as he used to be.   I would be grateful if someone could shed more light on the topic. A story where technology is forbidden, but used by priests I am looking for a story I remember reading in the mid-90s. I can't remember if it was a trilogy or just a three part book, but I know it was in paperback book form. I also know it was found in the Young Adult section of the library at that time.  The general premise was that there had been some sort of war in the past and machines were outlawed. People visited priests who used magic. I remember that the main character discovers that the priest aren't using magic but are actually using machines.  I can also remember the main character ends up leaving the nation he is in and ends up in a neighboring nation where machines are allowed. I remember there being a scene where he watches a "movie" which I think was more like an old fashioned moving picture.  For some reason I keep wanting to tie this whole thing back to a sword, or blue sword, or something like that, but I might be mixing stories. Other things I keep linking to it: the character was in training to be a priest, he was forced to leave, he ended up using the neighboring nation to conquer his homeland. Again, still not sure if I am mixing things but they keep popping up.  I do know that it is not The Blue Sword by Robin McKinley, or its prequel, The Hero and the Crown. Leaving the States on Visa Waiver and coming back I'm travelling on the ESTA visa waiver, I'm now in the States but would like to go to Costa Rica for a bit (say three weeks) then come back to the States. All will be within my 90 day limit.  Is that allowed?  (I have my onward ticket home.) What is the difference between "I've been to China" and "I've gone to China"? Both are in the present perfect, but one uses the verb 'to go' and the other 'to be'. Is there a rule for this?  Is there any chance that the differences between "I have been" vs. "I have gone", are differences in English dialect (I don't know if this is the right word).  Edit Could I say that North American English speakers use "I have gone", and maybe U.K speakers say "I've been"? Or vice versa. Proof that all abelian simple groups are cyclic groups of prime order $n$ choose $k$ where $n$ is less than $k$ Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$ I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{\lambda ^2x^2+\lambda x+1}\text{d}x =\frac{1}{\lambda}\int_0^{\infty}\frac{(\ln x-\ln \lambda)^2}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\int_0^{\infty}\frac{1}{x^2+x+1}\text{d}x+\frac{1}{\lambda}\int_0^{\infty}\frac{\ln ^2x}{x^2+x+1}\text{d}x \\ =\frac{\ln ^2\lambda}{\lambda}\frac{2\pi}{3\sqrt{3}}+\frac{2}{\lambda}\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x$$ But I hav no idea on process the remaining integral:( Anyone knows how to solve it?  THX guys! I came with another method might work for this: Recall the handy GF $$\frac{1}{x^2+x+1}=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ Then we have :  $$\int_0^1\frac{\ln ^2x}{x^2+x+1}\text{d}x=\frac{2}{\sqrt{3}}\sum_{k=0}^{\infty}\frac{2}{\left(k+1\right)^3}\sin \left(\frac{2\pi}{3}\left(k+1\right)\right)x^k$$ With $$\sum_{k=1}^{\infty}\frac{\sin (kx)}{k^3}=\frac{\pi ^2}{6}x-\frac{\pi}{4}x^2+\frac{x^3}{12}$$ What can we arrived? Term for words like Snowmageddon, Nipplegate and even cheeseburger? $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible. The problem is from  Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful.      Let $A,B$ be $m \times n$ and $n \times m$ matrices. Prove that $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.    Please provide with only a hint. Missing new file in Ubuntu 13.04 The New file operation is missing in Ubuntu 13.04. How can I fix it?  I have only &quot;New Folder&quot;. How can I add the New File option again? Maybe a stupid question but how can I tell if I have a static IP address or a dynamic IP address? Running Fedora-13 How can I find hard links on Windows? I've created some hard links on my Windows 7 file system using mklink. It was some time ago and I can't remember for sure where, or which files. When I use Explorer, all files look the same. When I use the command line and type "dir", they all look the same.  How do I find hard links? Or how do I determine whether a specific file IS a hard link? Is there an idiom for people who boast too much? I am looking for idioms or informal/slang/colloquial expression for some people that make you think that they are able of building a skyscraper, constructing a spaceship, playing the piano better than Mozart or something very fancy that requires remarkable skills, but when it comes to practice they prove to be completely inadequate. Difference between static class and singleton pattern? What real (i.e. practical) difference exists between a static class and a singleton pattern?  Both can be invoked without instantiation, both provide only one "Instance" and neither of them is thread-safe. Is there any other difference? In Bash, two integers can be compared using conditional expression     arg1 OP arg2      OP is one of -eq, -ne, -lt, -le, -gt, or -ge. These   arithmetic binary operators return true if arg1 is equal to, not equal   to, less than, less than or equal to, greater than, or greater than or   equal to arg2, respectively. Arg1 and arg2 may be positive or negative   integers.   or arithmetic expression:     &lt;= &gt;= &lt; &gt; comparison      == != equality and inequality   Why do we have two different ways for comparing two integers? When to use which?  For example, [[ 3 -lt 2 ]] uses conditional expression, and (( 3 &lt; 2 )) uses arithmetic expression. Both returns 0 when the comparison is true  When comparing two integers, can these two methods always be used interchangeably? If yes, why does Bash have two methods rather than one? Using trickle to limit the bandwidth of a VirtualBox VM I'm trying to limit the bandwidth of a virtual machine running via VirtualBox.   trickle -d 5 virtualbox doesn't work, I believe because of an issue with child processes (trickle -d 5 sudo apt-get upgrade doesn't work either)  How can I limit the download/upload bandwidth of VirtualBox's VMs? I am trying to sniff on a custom SPI-like bus protocol, which uses a unified bidirectional data line instead of separate MISO/MOSI. To make sense of the communication, I need to split it up into unidirectional lines before feeding them to my logic analyzer. Otherwise I wouldn't be able to see which device is responsible for which part of the signal. Since my devices won't talk to each other unless the protocol is carried out correctly, it's obvious that the bus splitting device has to be transparent to the actual agents communicating.  I tried to construct a bus splitting device, but failed as described . Since this question was maybe too specific and has not been answered, I want to ask more generally: Is there a bidirectional bus splitter that is able to split  into four wire SPI? Where could I buy such a device? (This seems similar to  question, only that I'm not talking about two really different protocols, so the answer does not apply). In principle only getting one direction would also suffice, since I could just subtract it from the full signal in order to retrieve the other half. And I'm also only interested in the master side of the communication.    The  seems to be a good candidate. Although it's advertised for use with I2C, it seems to be agnostic to the actual protocol on the lines. (As it does not even differentiate between DATA and CLOCK line for I2C.)     Splits I2C-bus signal into pairs of forward/reverse Tx/Rx, Ty/Ry signals for interface with opto-electrical isolators and similar devices that need unidirectional input and output signal paths.   As my devices operate on SPI mode 3 (polarity 1, idling on HIGH), the implementation of the receive pin should work:        sense the voltage state of the I2C-bus pin Sx (or Sy) and transmit this state to the pinTx (Ty respectively), and   sense the state of the pin Rx (Ry) and pull the I2C-bus pin LOW whenever Rx (Ry) is LOW.      But I am still puzzled on how this would actually work in sniffing this bidirectional bus. I could connect one device with the I2C line and grab its MOSI at Tx and MISO at Rx, but I would eventually have to connect both lines before feeding it to the other device, therefore making it impossible to separate the signals at my logic analyzer again. How can I connect the Tx/Rx for the slave while still being able to read them separately with a third device?  Edit: I now think this is actually not helping at all, since it leaves me with the same splitting issue as with out the device. I already have separate master and slave lines, I just don't know how to join them while still retrieving the separate directions.    I also stumbled upon the . It seems like this is a device designed for sniffing on buses, but I don't think it can help me in my case. Is there a mode where I can use it as a bridge on a bidirectional line and grab unidirectional signals off of it simultaneously?    Edit: I found this  which seems related to the issue. This does supposedly split the bidirectional line to feed it to two unidirectional isolators. The question is: Does this work for my setup? It describes the I2C lines being open-drain, whereas my lines are open-collector and also I'm not entirely sure if I could just leave out the isolator part and grab the signals from the lines or if these would have to be also hooked up to some transistor or be pulled high.