Make greek numeration
Greek numbering
This question has an inverse:     What would be the most efficient way to translate a combination of $k$-tuple into its positions within the $\left(\!\!\binom{n}{k}\!\!\right)$ combinations? I need this to be fast for combinations of $\left(\!\!\binom{70}{7}\!\!\right)$ order of magnitude - very large, but not exceeding 2 billion (fits into int32 maximum value).  Below is an example of $\left(\!\!\binom{6}{3}\!\!\right)$, where the aim is to quickly translate (a, d, f) tuple to value 9, while in the real problem $k$ ranges between 5 and 8.  $$\begin{array} {cccccc|c} a&amp;b&amp;c&amp;d&amp;e&amp;f&amp;^{combination}/_{sort\_order}&amp; \\\hline x&amp;x&amp;x&amp; &amp; &amp; &amp;1\\ x&amp;x&amp; &amp;x&amp; &amp; &amp;2\\ x&amp;x&amp; &amp; &amp;x&amp; &amp;3\\ x&amp;x&amp; &amp; &amp; &amp;x&amp;4\\ x&amp; &amp;x&amp;x&amp; &amp; &amp;5\\ x&amp; &amp;x&amp; &amp;x&amp; &amp;6\\ x&amp; &amp;x&amp; &amp; &amp;x&amp;7\\ x&amp; &amp; &amp;x&amp;x&amp; &amp;8\\ x&amp; &amp; &amp;x&amp; &amp;x&amp;9\\ x&amp; &amp; &amp; &amp;x&amp;x&amp;10\\ .&amp;.&amp;.&amp;.&amp;.&amp;.&amp;.\\  &amp; &amp; &amp;x&amp;x&amp;x&amp;20\\ \end{array}$$  I know that I could pre-calculate all the combinations and reverse the lookup dictionary. However, such dictionary would not be efficient in terms of memory usage. Therefore I am looking for either calculation-based approach, or a more efficient data structure to perform this mapping. I want to make a plane for my supply on my PCB and a plane for my MCU to isolate the MCU from the power supply noise.  I found this article in Altium designer :  But it isn't clear how the split plane tool work. Can someone explain it in a different manner?  [EDIT]  This is a picture how I see the planes: Flow On Schedule I"ve created a flow that I'd like to run monthly on the 7th day of each month. How can I do this with scheduled apex. I have no experience writing apex from scratch. Any suggestions or starting off points appreciated.  Thanks,  Kris When to use an object pronoun or a possessive adjective before a gerund Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist? Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area. From this  on Bayesian Least Mean Square estimation:  $$\mathbf E\left[(\Theta -\hat\theta)^2\right] = \rm{var}(\Theta -\hat\theta) + \left(\mathbf E[\Theta -\hat\theta]\right)^2 = \rm{var}(\Theta) + \left(\mathbf E[\Theta -\hat\theta]\right)^2$$  which is to say that $\rm{var}(\Theta -\hat\theta) = \rm{var}(\Theta)$.  I would like to ask how that comes about?  Thanks.  $\Theta$ is the random variable being estimated, and $\hat\theta$ is a constant estimate. Unable to load styles and script from static resource in LWC after summer'19 release How can I detect mouse events on sprites in a canvas? I'm making an HTML 5 game. I want mouse clicks on sprites drawn in a canvas element cause events that my code can react to. At the moment, I'm doing it by checking through all of my sprites in a for-loop on each click.  Is there a way to do that by adding an event listener to a sprite so it would respond just like that?  Are Javascript's custom events useful in this situation? I know it's possible with KinectJS, but I don't have any experience with it and really I'd like to learn how to do this myself without the help of a library. How to fix Terminal error DYLD_ environment variables being ignored because main executable (/usr/bin/login) is setuid or setgid? Is there a backquote on the iPhone keyboard? Is there a backquote (backtick `) on the iPhone keyboard, and if so, where is it?  When I'm writing a post on a site in the SE family (especially SO), I want to use the backquote for inline code blocks. Under the single quote key, I've found two curly/angled quotes, but those (’ and ‘) don't seem to do the trick... I am trying to automate various tasks in ArcGIS Desktop (using ArcMap generally) with Python, and I keep needing a way to add a shapefile to the current map. (And then do stuff to it, but that's another story).  The best I can do so far is to add a layer file to the current map, using the following ("addLayer" is a layer file object):  def AddLayerFromLayerFile(addLayer):  import arcpy  mxd = arcpy.mapping.MapDocument("CURRENT")  df = arcpy.mapping.ListDataFrames(mxd, "Layers")[0]  arcpy.mapping.AddLayer(df, addLayer, "AUTO_ARRANGE")  arcpy.RefreshActiveView()  arcpy.RefreshTOC()  del mxd, df, addLayer   However, my raw data is always going be shapefiles, so I need to be able to open them. (Equivalently: convert a shapefile to a layer file without opening it, but I'd prefer not to do that). To make Latin-letter variables bold I can use e.g. \mathbf{a}, but while putting Greek letters or symbols such as \nabla inside \mathbf doesn't cause any errors or warnings, it also doesn't do anything else.  What is the best way to make bold math symbols, in particular Greek letters and \nabla? Applications of complex numbers to solve non-complex problems Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating that for a question which only concerns integers, and whose answers can only be integers, such an elegant solution comes from the seemingly unrelated complex numbers - looking only at the question and solution one would never suspect that complex numbers were lurking behind the curtain!  Can anyone give some more examples where a problem which seems to deal entirely with real numbers can be solved using complex numbers behind the scenes? One other example which springs to mind for me is solving a homogeneous second order differential equation whose coefficients form a quadratic with complex roots, which in some cases gives real solutions for real coefficients but requires complex arithmetic to calculate.  (If anyone is interested, the original question I asked can be found here: )  EDIT:  I just wanted to thank everyone for all the great answers! I'm working my way through all of them, although some are beyond me for now! Best resources to learn bash scripting? Why is angular velocity of any point about any other point of a rigid body always the same? I have the following question regarding an ideal rigid body.  Firstly, is it always true without any exceptions that the angular velocity of any point about any other point on a rigid body is always the same?  If so, what happens in case of precession, in which two different angular velocities are predefined? understanding Debian dpkg package version numbers How can I show that the sum   $$ \sum_{k = 0}^n 2^k \binom{n}{k}$$  is equal to $3^n$? What is the purpose of LongLength for .Net Arrays? What is the purpose of the LongLength property for arrays in .Net.  Using a standard integer for length, you could accommodate up to 2 billion indices.  Are there really people using .Net to maintain a single array with more the 2 billion elements.  Even if each element was a single byte, that would still be 2 GB of data.   Is it feasible to use such a large array in .Net? There are two vectors called $\vec{a}$ and $\vec{b}$. Vector $\vec{c}$ is the bisector and it can be given as  $$\vec{c} = |b|\vec{a} + |a|\vec{b}$$  How to prove that? I have used the dot product method. But there I can't find the angle between them. Then I tried to do this by using unit vectors. Then I got an answer like in the given picture. . Can you help me to get the correct answer