Why there is the requirement for derivatives no higher than second order in free quantum fields equations?   We can get the equations for the free fields of an arbitrary spin by using the requirements of linearity of the equations, derivatives no higher than second order. But why there is the second requirement? Is it connect with initial conditions $A_{\mu}(0), \partial^{\mu}A_{\nu}(0)$ for the field? For example, Schrodinger equation is first-order time-derivative equation, because we have the postulate that if we know the $\psi (\mathbf r , t)$ in the present, we know it in the future.
What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?    If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\phi\Delta\phi$.  I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem.  Furthermore you could come up with quantities in the spirit of $F\wedge F$ and $F \wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too.   Do we know the limit for making up these objects? What is the fundamental justification for order two?
Why is the gravitational force always attractive? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time?   PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely? Do 'learn by heart' & 'learn by rote' mean the same? Here in India, both the phrases learning by heart and learning by rote are taken to have the same meaning, i.e., blind memorisation without true understanding.    However, some sources say that to learn something by heart implies that one knows something so well that it has been thoroughly internalised and imbibed.    So is it wrong to conflate the two? Why does this not work with an imaginary exponent? Could someone please tell me why this exponentiation rule does not work when the exponent is imaginary?  The following seems to be true but is only true if $x$ is an integer.  $$e^{2\pi ix} = \left( e^{2\pi i} \right)^x = 1^x = 1$$ Accuracy Vs. Precision  What I would like to know is whether I should use System.currentTimeMillis() or System.nanoTime() when updating my object's positions in my game?  Their change in movement is directly proportional to the elapsed time since the last call and I want to be as precise as possible.  I've read that there are some serious time-resolution issues between different operating systems (namely that Mac / Linux have an almost 1 ms resolution while Windows has a  50ms resolution??).  I'm primarly running my apps on windows and 50ms resolution seems pretty inaccurate.  Are there better options than the two I listed?  Any suggestions / comments? Problems with pie charts There seems to be in increasing discussion about pie charts.   The main arguments against it seem to be:   Area is perceived with less power than length. Pie charts have very low data-point-to-pixel ratio   However, I think they can be somehow useful when portraying proportions. I agree to use a table in most cases but when you are writing a business report and you've just included hundreds of tables why not having a pie chart?  I'm curious about what the community thinks about this topic. Further references are welcome.  I include a couple of links:         In order to conclude this question I decided to build an example of pie-chart vs waffle-chart. Remove .php extension with .htaccess What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$? Integral of sinc function multiplied by Gaussian I am wondering whether the following integral $$\int_{-\infty}^{\infty}  \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ exists in closed form. I would like to use it for numerical calculation and find an efficient way to evaluate it. If analytical form does not exist, I really appreciate any alternative means for evaluating the integral. One method would be numerical quadrature including Gaussian quadrature, but it may be inefficient when the parameters $a$ and $b$ are very different in scale.  EDIT : In view of  discussion, we have decided to add OP's self-answer to the end of the question, for it does not qualify as an answer yet contains vital details. The copying is unabridged. Thanks very much for your comments, and the following result was obtained including the case for $x_0 \ne 0$: $$ \int_{-\infty}^{\infty} dx \exp[-a(x-x_0)^2] \frac{ \sin(bx) }{ x } = \pi \exp(-a x_0^2) \mathrm{Re}\left(\mathrm{erf}\left[\frac{b+2iax_0}{2\sqrt{a}}\right] - \mathrm{erf}\left[\frac{2iax_0}{2\sqrt{a}}\right]\right) $$ where $a\gt0, b, x_0$ are assumed to be all real. (note: coefficients etc may be still wrong...) This integral appears in a type of electronic structure calculation based on a grid representation (sinc-function basis). I believe the above result should be definitely useful. Thanks much!! --jaian i want to compile several pgfplots pictures with lualatex, since it dynamically allocates memory for compiling (in contrast to pdflatex).  In some plots, like my MWE.tex given below, i want to use a contour plot. This is typically done with gnuplot. To compile a document like this i need to use --shell-escape, such that gnuplot command can be executed while compiling the document.  \documentclass[a4paper]{article}  \usepackage[utf8]{luainputenc} \usepackage{pgfplots} \pgfplotsset{compat=newest} \begin{document} \begin{tikzpicture}   \begin{axis}     \addplot3[contour gnuplot]      {x+y};       \end{axis}   \end{tikzpicture} \end{document}   If i use  pdflatex --shell-escape MWE.tex   everything works fine, and the log says something along the lines  This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016) (preloaded format=pdflatex 2016.6.12)  13 JUN 2016 10:46 entering extended mode  \write18 enabled. .... Package pgfplots notification 'compat/show suggested version=true': document has been generated with the most recent feature set    (\pgfplotsset{compat=1.13}).  \openout3 = `MWE_contourtmp0.dat'.  \openout3 = `MWE_contourtmp0.script'. runsystem(gnuplot "MWE_contourtmp0.script")...executed.   If i use     lualatex --shell-escape MWE.tex i get the following error:  This is LuaTeX, Version 0.95.0 (TeX Live 2016)  (format=lualatex 2016.6.12)  13 JUN 2016 10:52  system commands enabled. ... Package pgfplots notification 'compat/show suggested version=true': document has  been generated with the most recent feature set (\pgfplotsset{compat=1.13}).  Package pgfplots info on input line 10: Using 'lua backend=false' for plot 0 (type 'contour external'): plot handler does not support LUA.  \openout3 = MWE_contourtmp0.dat  \openout3 = MWE_contourtmp0.script gnuplot "MWE_contourtmp0.script"  ! Package pgfplots Error: sorry, plot file{MWE_contourtmp0.table} could not be opened.   The .table file exists and looks identical to the one created by pdflatex, but it says that it cannot be executed.  (I just installed texlive 2016 yesterday, so everything should be up to date.) Clusering based on categorical variables? I am working on a project and currently experimenting cluster analysis. The dataset is mainly categorical variables and discrete numbers. Please pardon my poor programming skills as I am not very familiar with MathJax, but I will try to summarize the data set in words in the following three examples.   imagine column 1 is participants names of course, from column2 - 5, each column's value range from 1 (least important) - 5 (most important). so in this case, column 2-8 only have discrete data.  column 6 for example, since this is a multiple choice question. row 1 chose "nice" as an answer, however, row 2 chose "poor". In this case, we have one column that contains multiple categorical answers. for column 7-9, this is another type of multiple choice questions. However this time, each column represents only one answer. column 7 only allows string values "true", column 8 only allows string "somewhat". So in this case, we have multiple columns that represent multiple answers of the same question.    Any ideas how to solve this? Thankful to any input!   A2       A3       A4       A5       A6       A7         A8         A9 1        4        5        4        nice     true       somewhate  false 2        4        3        1        poor     true                  false 1        5        2        1        nice                somewhate     3        2        1        5        nice     true                  false .fbx export why there are no materials or textures? I posted an answer (pointing to another question already asked) but when I hit submit, I saw my answer appear as a comment to the question instead.   I tried it a couple of times and still the same result. The answer gets eaten up and is transformed into a comment.   Is this a bug? Reputation misaligned in new topbar Has the disputed flag message changed? I know how disputed flags work. Today, I reviewed a "not an answer" flag and marked it as invalid.   I've later seen a disputed message with this reason:     invalid flag         disputed - a moderator reviewed your flag, but found no evidence to   support it   Given that disputed flags are in the ,  Is this message applicable only to my flag or is this message also for the original flagger?  Or put otherwise, are all disputed flags now shown this message (or similar decline reason)?  I've run out of flags for today to test this further. Delete from uitableview + write to plist not working How to setup a restricted SFTP server on Ubuntu? I would like to know how to set up root, sudo, sftp-only user accounts which won't be required public key authentication at log in. I would also like to know how to set up sftp-only users' home directories where they can't access upper level other directories. ODBC connection to Access database in 64-bit Windows My daughter asked me to cosign on a car loan.  Somehow I ended up as the borrower and she a the cosigner.  Is that legal? Travelling as an unmarried Western couple in Indonesia Regarding running Watir scripts for UI Automation from Jenkins CI - How to hook up Jenkins and watir-automations