I seem to recall that there is a relatively easy method for determining the associativity of an operation by using its Cayley table. What is it?
Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot &amp; a &amp; b &amp; c &amp; d &amp; e\\\hline a     &amp; e &amp; d &amp; b &amp; a &amp; c\\ b     &amp; d &amp; c &amp; e &amp; b &amp; a\\ c     &amp; b &amp; e &amp; a &amp; c &amp; d\\ d     &amp; a &amp; b &amp; c &amp; d &amp; e\\ e     &amp; c &amp; a &amp; d &amp; e &amp; b \end{array}$$ it is easy to check that it is closed (no elements occur in the table which don't occur as row or column index), commutative (the table is symmetric), has a neutral element (the row and column of $d$ are copies of the index row/column), and has an inverse element for each element (there's a $d$ in each row and column). In other words, almost all important properties can immediately be seen. The only part missing is associativity.  Therefore my question: Is there a simple way to see directly from the operation's table (i.e. without doing explicitly all the calculations) if an operation is associative?
Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$  Hint: First show that $f(z)$ is a polynomial.  Clearly one can not use Cauchy Estimates to prove that $f(z)$ is a polynomial, the other way is to prove that $f(z)$ has a pole at infinity which I am not sure how to prove that, are there any other ways to prove that a given entire function is a polynomial ?  Thank you ! Two submit buttons in one form I have two submit buttons in a form. How do I determine which one was hit serverside? Which databases use real multidimensional indices? Is oracle ever using several indices to get data from tables, or will it always take the one that seems to have the highest selectivity? How about other dbms? Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Is there a built-in function to print all the current properties and values of an object? So what I'm looking for here is something like PHP's  function.  This is so I can debug my scripts by seeing what's the state of the object in question. How to determine which program is making a certain sound in windows 7? I'm experiencing an annoying "female robotic" sound effect saying "loading complete" at uncertain interval in windows 7.  I want to shut down that noise for good. To do that, I have to determine which program is making this sound. I tried to open the "sound mixer" window, only to find to program last made any sound, one is official GTalk client, one is system sound. I tried to close GTalk and the sound still existed.  I started experiencing this issue not long after I installed a "driver" program for Rapoo E9198 wireless keyboard. Does the sign of $\Delta x$ matter in this time dilation calculation? How to find the real user home directory using python? More than a decade ago, I read a short story about a small group of people at the end of the world. The story was premised on some sort of mass-produced device which took ambient heat and turned it into power. The devices had been overused, and now the world was freezing.   I think the people were essentially bandits; the main character may have been a woman who lured men to her house and killed them for their body heat? There was also a strong implication that one person she had done this to was an alien who had come to warn that the devices were destroying the world.  I think the story was in a hardbound collection, and I think the collection was from the late 70's or very early 80's.  (Calling it a "short story" is just meant to distinguish it from a full novel — it may have actually been a novella or similar.) Let $({\Bbb R},{\mathcal A},\mu)$ be the measure space where ${\mu}$ is the Lebesgue measure.  Assume that $\int_{\Bbb R}|f|^pd\mu&lt;\infty$ ($p\geq1$). There is an exercise for proving that       $$\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s).$$   If one formally let $s=t^p$, then $ds=pt^{p-1}dt$ and one would immediately get the equality above. However, how can I justify it rigorously in the sense of Lebesgue integration?    [Edited] What puzzles me is that I have the "change of variable" theorem for Riemann integration, but all the integrations here are in the Lebesgue sense. Unless one has shown that the integrands are also Riemann integrable, how can I directly use "change of variable"? Using BibTeX/biblatex to make a list of references without having citations in the body of the document? Is there any way to have BibTeX or biblatex insert a bibliography without having citations in the body of the document itself? Maybe a way to hide the in-text citations, so that BibTeX still sees them and includes them in the bibliography, but does not print them in the body? What's the easiest way to set up a LAMP stack? I set up a new VPS instance of Ubuntu and am wondering what the easiest way is to get up and running with a basic LAMP stack (i.e. which packages are required, which configuration options need to be tweaked, if any, etc.). How to get the ASCII value of a character If anonymity is so important, why is it not universal? A frequent argument brought up in support of voting anonymity is that it can lead to potential grudges. Certainly the review system is of the same nature.  I cannot wrap my mind around the fact that voting is anonymous and reviewing is not. Either both should be public--and this would be fine since public reviewing has been successful--or both should be anonymous--and I assume this would be the desired option since this community holds anonymity so dear.  One could say that voting rates the posts, the actual meat of the site, while reviewing is based on clarifying the purpose of the site and cleaning it up.   answer makes the point that the community can trust that votes are a good measure of quality for the very reason that they do not know who cast them; the mentality that a high-ranking member may have supported something could blur the line between "science" and "politics/religion."  Reviewing, then, can be seen as a judgment of specific members rather than quality of edit/questions, which seems to fall precisely under the "politics/religion" umbrella--the review system is much more vulnerable to grudges due to this.  So let's discuss the differences and the implications of these disjointed anonymity policies. How can I reflect a transparent background? I'd like to make a glossy surface which reflects objects above it, but that also reflects a transparent background in the final render.  So in effect, the final render would contain only the reflections, not the surface itself.  Is this possible please? After seeing a question regarding the acquisition of Data's cat Spot, the point was brought up that Data's  midway through the series.  My question is, how? I have thought up two possibilities (see below to my own answer) but I'm wondering if there aren't possibly others that might fit better. Exhibit A- Spot the Younger  Exhibit B- Spot the Older I want to use  to display my CPU-Core temperatures (haswell i7). However the setting:  order += "cpu_temperature 1" #... cpu_temperature 1{         format = "T: %degree °C" } #   doesn't display the correct core temperature. The numbers it shows seem to correspond to the value xsensors shows for temp1, if I change the 1 to 2 above it corresponds to xsensors temp2. Trying 3 or 4 doesn't have any effect. However I want to get the true core temperatures of all 4 cores with i3 status. How can I do this? What is an "AppImage"? How do I install it? I just read . It talks about downloading an AppImage.     Yes, Linux; the Linux packages is distributed as an .appimage for 32-bit and 64-bit distributions, and should run across all major Linux distributions without any issues. The team currently have no plan to provide a native .deb (or .rpm) installer.   What are AppImages? How do they differ from snaps? Inserting newlines in Rectangle Text elements via ArcPy causes overlap? I ran across an issue the other day when I tried to use ArcPy's mapping module to edit rectangle text elements with newlines (\n) in an ArcMap document.  Here's what the output looked like:    Here's the code I used to generate that output.  The first column are rectangle text elements Text1, Text2, Text3 going down; the second column are "plain" text elements Text4, Text5, and Text6 going down.  import os import arcpy  HomeDir = r"C:\Desktop" arcpy.env.workspace = HomeDir  CurrentMXD = arcpy.mapping.MapDocument(r"C:\Desktop\TextTest.mxd") OutputFilename = r"C:\Desktop\TextTest.pdf" if os.path.exists(OutputFilename):     os.remove(OutputFilename)  for TextElement in arcpy.mapping.ListLayoutElements(CurrentMXD, "TEXT_ELEMENT"):     TextElementName = TextElement.name      String1 = "The quick brown fox jumped over the lazy dog.\nShe sells sea shells by the sea shore."     String2 = "The quick brown fox \njumped over the lazy dog.\nShe sells sea shells by the sea shore."     String3 = "The quick brown fox jumped \nover the lazy dog.\nShe sells sea shells by the sea shore."      if TextElementName == "Text1":         TextElement.text = String1     if TextElementName == "Text2":         TextElement.text = String2      if TextElementName == "Text3":         TextElement.text = String3     if TextElementName == "Text4":         TextElement.text = String1     if TextElementName == "Text5":         TextElement.text = String2     if TextElementName == "Text6":         TextElement.text = String3  arcpy.mapping.ExportToPDF(CurrentMXD, OutputFilename)   So far, it looks like the presence of the messed up text depends on whether the line is longer enough to wrap, and whether the line before the newline is longer than the line after the newline.  Any ideas about what could be going wrong?  Is there a workaround?  I could use plain text elements and worry about wrapping lines using Python, but I'm hoping I can figure something out. 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