How long does it take an investment to double if continuously compounded at a 6% rate? Is this right?  \begin{align*} e^{0.06 \cdot t} &amp; = 2\\ 0.06t &amp; = \ln(2)\\ t &amp; = \frac{\ln 2}{0.06}\\   &amp; \approx 11.55 \end{align*}  Why does $e$ equal the $\lim_{n \to \infty} (1+\frac{1}{m})^m$?  I know that if $r$ is the rate and $n$ is the number of compounding periods in $t$ years, then the value of an investment is:  $$A_0\left(1+\frac{r}{n}\right)^{nt}$$  But how do we get from here to the definition with $e$ in continuous compounding?
I was wondering if it is possible to get a link to a rigorous proof that  $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
Variance of sample variance? What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.  I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$  I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as far as I got. Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$ top command in cygwin? I am starting to use Cygwin(64) on my Windows 8 laptop, however, the top command is not recognized. I have read online that I need to get the procps package, but it is not available on the 'Cygwin Setup' package selections. How should I go about installing the procps package so that I can use top?  ps. for whatever reason ps and kill both work so I would think top would be included in whatever package allowed those guys to work, but there is no top command. What is this Windows 8 icon for as rectangle icon in the title bar? Every Window in Windows 8 got this icon if it gots the focus. When clicking on it, nothing seems to happen.   How does this icon work? Show $1/(1+ x^2)$ is uniformly continuous on $\Bbb R$. How can an int be cast to an enum in C#? I have to do some manual reordering or rows in Excel and cut/paste is too hard to use. I'm looking for something that would enable me to use drag'n'drop or to add some buttons to move-up/down move-top/bottom. Indefinite article and people's names Sometimes, on the internet, particularly in online games, I see people using the  indefinite article before someone's name: "I see a Joey" or "I hug a Polly". I know some of these people and I'm positive that they are from English-speaking countries, which means it is not a mistake, they do it intentionally.  I was always taught that articles are not used in front of people's names. Maybe it is some sort of slang? What does it mean, when the indefinite article is put in front of someone's name?  Edit: There is a group of people standing somewhere. And somebody, named Joey, appears. They all know Joey. And someone in this group says "I see a Joey". So it is used when someone sees his or her friend. But I don't understand why they use the indefinite article here. I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two?  Linearly interpolating between the two almost works, if they are similarly distributed and sized.   provides a forumla at the bottom for combination but it doesn't seem right; two identically distributed distributions should have the same covariance, but the formula at the bottom of the page doubles the covariance.  Is there a way to combine two matrices? For realism should I use the maximum number of Subsurf levels? I was wondering if I always need to use the maximum value of the options from the elements of "Add Modifier". For example, do I need to use the view 6 in Subdivision surface? I know it depends of what we are trying to achieve, but we all want the same. Make something look the most believable. So do you always use the maximum value Blender offers? If you activated a free upgrade to Windows 10 or bought and activated   Windows 10 from the Microsoft Store, you have a digital license for   your device. This means you can reinstall the same edition of Windows   10 that your device has a digital license for without entering a   product key. During reinstallation, if you’re asked to enter a product   key, you can skip this step. Windows 10 will automatically activate   online after the installation is complete.   source:   Does MS digital license 'fingerprint' hardware information  during the free upgrade / activation and store it (at Microsoft) , so that the next re-install queries MS for the fingerprint and activates Windows 10?  Thought the disk drive may change, I expect that none of the other fingerprinted hardware will change. Unexpected behavior of rule matching a pattern For thematic reasons, I am wanting to make a Zatoichi-style character based on using  and Decisive Strike monk alternate feature from the PHB2. The only catch is that Decisive Strike says:     . . . make one attack with an unarmed strike or a special monk weapon . . .   What are all of the official ways to add a weapon to the list of ? I have found only one in the Shou Disciple prestige class (Unapproachable East, p.32), but doing so would mean the earliest the character could use decisive strike with a katana would be level 10. I would like to be doing it no later than level 5 if such an option exists.  Note that, for the purposes of this question, anything that works for "Flurry of Blows" can be interchanged with "Decisive Strike" (e.g. the "Martial Flurry" ability of a Shou Disciple gains ability to use other weapons with "Flurry of Blows," but that is fine). I've installed VM VirtualBox v. 4.3.8 on my Ubuntu 12.04 LTS (Lenovo G500S with Intel 1005M processor) and I can't configure any 64-bit guest (nor Windows, neither other OS). Inslalation of Win 8.1 64-bit under 32-bit profile obviously fails... My processor supports VT-x technology, but in my BIOS there is no enable/disable option for VT-x. Is there any way to force VirtualBox to support 64-bit guest ? I have a CSV file which has postcode data in a field, but it is non-spatial e.g. no Easting and Northing.   I have a shapefile with postcode in a field and some additional data.  I want to join the CSV with the shapefile based on attributes (postcode fields). I know I can do this in ArcMap, but how can I achieve the same thing in QGIS? L293, L298 and SN754410 H-bridge drivers on low voltage power supply This question is about the following three integrated H-bridge drivers: -    or L293D (D = protection diodes added)  (protection diodes included)  (no protection diodes)   Time after time the same question keeps coming up - someone is using one of these devices (on a low voltage, usually around 6V or less) and they are just not performing adequately. The reasons are listed further below but my question is this: -  What H-bridge drivers are preferred when controlling a low-voltage motor?   Information  The L293 and the SN754410 are nearly identical and crucially, if you try and control a 1 amp load, you are faced with dismal performance: -    The tables tell you (typical conditions) that the upper transistor drops (loses) about 1.4 volts when driving a 1A load and, the lower transistor drops (loses) about 1.2 volts when driving a 1A load. The upshot is that if you have a 6V, 1A motor and 6V battery, don't expect to see more than 3.4 volts across the motor: -  \$V_{OUT} = 6V - (1.4V + 1.2V) = 3.4V\$  Worst case scenario is you might only see 2.4 volts across it.  What about the L298? It's got a nice big heat-sink whereas the L293 and SN754410 are regular-looking chips. Here's what the volt drop (losses) look like: -    It's the same story - for a 1A load, you can expect to lose up to 3.2 volts and, what you thought might be 6V across your motor, is at best 4.2 volts and at worst only 2.8 volts.  Clearly none of the devices listed are suitable for low voltage applications where the motor might be expected to draw in excess of 0.5 amps. Does magic missile hit silent image's illusory creature? I need to build a probabilistic cellular automaton for simulating land-use changes and am wondering how to implement this in a GIS. I understand that Markov chains are a way of doing this, and IDRISI has some sort of module for Markov chains, but I have no access to IDRISI.  that, with raster data, this can, in theory, be done with any GIS that can do raster algebra. Presumably this means that it can even be done in ArcGIS, and I think I've figured out how to build cellular automata in ModelBuilder (once I know with certainty how to make iteration work there).  Still, I can find next to no practical advice on this. Once I have two datasets, one showing the land use at a certain point in time (time A) and the other at a later time (time B), I imagine doing the following (correct me if I'm wrong):   If there are (for example) 3 types of land use, then there are 9 possible things that can happen to a cell from time A to time B; thus, reclassify each dataset such that each land use is represented by a different number (for example, 9, 11, and 14 at time A and 3, 4, and 5 at time B); Divide one raster by the other, obtaining a raster with 9 possible cell values. Determine the percentage of cells in the new raster that has each of the 9 values (I forgot, how does one do this in ArcGIS?); those will be the probabilities of each type of change (or lack thereof) in land use.   What I cannot figure out, though, is how to apply these probabilities during the next iteration. Finally, I wonder if there is a way of making an external factor (say, proximity to roads) influence the spatial distribution (not the overall likelihood) of change in my model.  Can anyone recommend where to find information on how to do this in ArcGIS? After Doomsday was defeated, a spec-ops team sent to the Kryptionian ship in Metropolis found Luthor submerged in water, in front of a... thing, which "held" three metal cubes. The thing looked up, roars, and dissipates into thin air.    What the hell was that all about? I am really confused. What was that creature and what was Luthor doing with it? I am trying to do a new environment or command (i tried both) where i can put in the name, description and code of a Method in c++ and it gives me a new section with the MethodName, the code listed and the description below. But i am not able to use the listings environment in a new command or new environment.. Consider my MWE   \documentclass[a4paper,11pt]{article} \usepackage{listings}  \newcommand{\method}[3] { \section{#1}\label{#1} \begin{lstlisting} #2 \end{lstlisting} #3 }   \begin{document}  \method{MethodName}{Method Void MethodName()}{Description of the Method blablabla}  \end{document}   How can I fix this?   I would prefer to do it using a macro since I don't need an ending.