I see countless of people using &quot;can&quot;, &quot;could&quot;, &quot;will&quot; and &quot;would&quot;, simply using them, mixing them up and more. I am not sure and want to be clear of the meaning, usage and difference of these words. These are a few examples: Can and could a) I can check your order and see how is the status. b) I could check your oder and see how is the status. a2) Can you give me a lift back home? b2) Could you give me a lift back home? Will and would a) I will help you with your homework tomorrow. b) I would help you with your homework tomorrow. a2) Will you refund my money? b2) Would you refund my money?
I can work. I could work. I will work. I would work. I shall work. I should work.
Linux - Help, I'm running out of inodes! Why do clocks measure arc-length? Apologies in advance for the long question.  My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$   t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$.    Why is this so?     Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$.  Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$.  If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$.  If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers.  I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$   T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$.  Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$   I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align}   I_{ba}  &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\ &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\ &amp;= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result.  How do others feel about this argument?  I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$.  I imagine that in GR a similar argument could be made by invoking local flatness of the metric. What happens if you use a drone to build a building while he's carrying minerals? Kind of a basic question, but what happens? Do you lose it? Does it count as harvested?  In a similar fashion, what happens when a drone/scv/probe goes harvesting gas when he's holding a mineral? Why can't Python's raw string literals end with a single backslash? General Information about Eigenvalues for an 3x3 symmetric matrix How do we find general information about the eigenvalues of an arbitrary 3x3 symmetric matrix without resorting to explicitly computing the solutions to the cubic characteristic equation?:  \begin{bmatrix}     a   &amp; b &amp; c  \\     b    &amp; d &amp; e \\     c &amp; e &amp;f \end{bmatrix}  (In this case the sum and product of the eigenvalues, and the sum of the reciprocals of the eigenvalues.) Since diagonalization preserves the trace and determinant of matrices, would it be best to just find the corresponding diagonal matrix and read off the sum and product of the eigenvalues from there? The problem is I can't think of a way to diagonalize a matrix without explicitly computing the eigenvalues. I recently updated my operating system from Ubuntu 13.04 to Ubuntu 13.10. Since then, I am facing difficulties compiling my TeX code in which the french babel package is called. Please note that all my sources were compiling very fine before the system upgrade. However, now, given the following header:   \usepackage[francais]{babel}  I receive the following error:  ! Package babel Error: Unknown option `francais'. Either you misspelled it   What should I do? Please note that if I remove this part of code (\usepackage[francais]{babel}), everything is compiling well. I also tried it with french, and I receive the same result. Thank you in advance for your help.  MINIMUM WORKING EXAMPLE : \documentclass[11pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[francais]{babel} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}  \begin{document} This is a MINIMUM WORKING EXAMPLE. \end{document}   ! Package babel Error: Unknow option `francais'. Either you misspelled it How to circumvent using an out parameter in an anonymous method block? The following method does not compile. Visual Studio warns "An out parameter may not be used within an anonymous method". The WithReaderLock(Proc action) method takes a delegate void Proc().  public Boolean TryGetValue(TKey key, out TValue value) {     Boolean got = false;     WithReaderLock(delegate     {         got = dictionary.TryGetValue(key, out value);     });     return got; }   What's the best way to get this behavior? (Please refrain from providing advice on threadsafe dictionaries, this question is intended to solve the out parameter problem in general). Find $\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$ I am trying to find  $$\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$$  My first guess is to look at the reciprocal and isolate factors:  $${2 \over 1}{5 \over 4}{8 \over 7} \dots {3n+2 \over 3n+1}= {\left(1+1\right)}\left(1+{1 \over 4}\right)\left(1+{1 \over 7}\right) \dots \left(1+{1 \over 3n+1}\right)$$ Now we take the natural log: $$\ln{ + \ln\left(1+1\right)} + \ln\left(1+{1 \over 4}\right) +\ln\left(1+{1 \over 7}\right) \dots +\ln\left(1+{1 \over 3n+1}\right)$$ Using $\ln(1+x) \le x$, we get: $$\ln{ + \ln\left(1+1\right)} + \ln\left(1+{1 \over 4}\right) +\ln\left(1+{1 \over 7}\right) \dots +\ln\left(1+{1 \over 3n+1}\right) \le 1 + {1 \over 4} + {1 \over 7} + \dots + {1 \over 3n+1}$$ Now, I'm stuck. I suppose I might use the fact that the RHS is similar to the harmonic series and show that it converges to some log, but I'm not sure how to do that. Do you have any clues? As the title suggests, I’m simply looking for a method to compute the number of Eulerian Paths in the complete graph $K_5$, where all paths start from the same node. Since there are 5 nodes, all Eulerian paths must be Eulerian circuits as well.  Thanks. How can I determine the name of the Bash script file inside the script itself?  Like if my script is in file runme.sh, then how would I make it to display "You are running runme.sh" message without hardcoding that? Rules for Profiles Are there any rules concerning what I can put in my profile? Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.  So far, I have not found any precise definition in literature. lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn. Where is the default folder for Apps? I am relatively new to the linux world. I use OSX at home and Windows at work. In those worlds, there is a default folder for applications (C:\Program Files, /Applications). Is there any such directory in Ubuntu?   I understand that this is not very important and any location in the file system should be ok, but was just wondering whether there are such standard locations (I think there would be one). For instance where is Firefox or Libre Office (that are installed by default)?  Got this question when I was trying to install eclipse. Downloaded the zip, and was wondering... How to validate an email address using a regular expression? Tracking Object Position - Firing on a Trajectory How can I "grep" patterns across multiple lines? It seems I am misusing grep/egrep.   I was trying to search for strings in multiple line and could not find a match while I know that what I'm looking for should match. Originally I thought that my regexes were wrong but I eventually read that these tools operate per line (also my regexes were so trivial it could not be the issue).  So which tool would one use to search patterns across multiple lines? The "I Get It" Reputation Problem Pass an extra argument to a callback function What is the meaning of p values and t values in statistical tests? After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value?