I feel like I am stuck. How can I interpret the equation with Einstein summation? Here is the exercise:      you should see that r′ can be written as a linear transformation of r. This means we should be able to write r′=Ar for some matrix A.    The exercise is depicted on the picture.   I will be more than thankful if someone explains that to me in minute detail.
I'm trying to compute the shadow of 3D objects on the ground. To do this I'm assuming parallel rays from Sun origin.   Let's assume the Sun direction is given by azimuth $\varphi$ and elevation $\theta$ angles. So the direction unit vector is retrieved as:  $$ \hat{\mathbf{s}} = \mathbf{R}_y\left(\theta\right)\mathbf{R}_z\left(\varphi\right)\hat{\boldsymbol{\imath}}$$  where $\mathbf{R}_{x/y/z}$ are the rotation matrices along the $x, y, z$ axes given by, for example, Euler-Rodrigues equation, and $\hat{\boldsymbol{\imath}}$ is the unit vector along $x$-axis.  A point on a ray-line has the equation:  \begin{equation} \mathbf{r} = \mathbf{r}_p + \lambda\hat{\mathbf{s}}  \tag{1} \end{equation}  being $\mathbf{r}_p$ a point on that line.  A point on a general plane is given by the equation:  \begin{equation} \left(\mathbf{r} - \mathbf{r_0}\right)\cdot \hat{\mathbf{n}} = 0 \tag{2} \end{equation}  being $\mathbf{r}_0$ a point on the plane and $\hat{\mathbf{n}}$ the unit vector normal to that plane.   Ensuring the validity of both equations (1) and (2) allows to compute the $\lambda$ multiplier and retrieve the position vector of the projected point (writing $\tilde{\mathbf{r}} = \mathbf{r_p} - \mathbf{r_o}$)  \begin{gather} \lambda = - \left(\frac{\tilde{\mathbf{r}}\cdot \hat{\mathbf{n}}} {\hat{\mathbf{s}}\cdot\hat{\mathbf{n}}}\right) \\ \mathbf{r}_\text{proj} = \mathbf{r}_p - \left(\frac{\tilde{\mathbf{r}}\cdot \hat{\mathbf{n}}} {\hat{\mathbf{s}}\cdot\hat{\mathbf{n}}}\right)\hat{\mathbf{s}} \tag{3} \end{gather}  Applying eq. (3) to all the vertices of a mesh $\left\{\mathbf{r}_{p_i}\right\}$ .   A problem can be noted: $\hat{\mathbf{s}}\cdot \hat{\mathbf{n}}$ can be zero (e.g. Sun is horizontal, i.e. the ray is ortogonal to the normal of plane).   Are there some conceptual mistakes in my reasoning? Is there a way to rewrite the operation as a linear operator applied to the set of vertices position vectors?   $$ \left[\mathbf{A}\right]\left\{\mathbf{r}_{p_i}\right\} = \mathbf{r}_\text{proj}$$
Experimental data for asymmetric Newton cradle Using a "successive impact model" (as if each ball were seperated from the other ones), I produced the following animations:    You can see any combination of balls with masses of 1 or 2 (left) or 1 and 4 (right).  Unfortunately, I do not have any Newton cradle to make some experiments, and I'd like to compare my results with observations. I contacted the author of , which is the most instructive one I found about asymmetric Newton cradle. In particular, he writes that the "OoO" configuration behaves as the "ooo" one, which is not what I simulate. But the author is not sure whether who is wrong because he did the experiments a long time ago.  I am especially interested in asymmetric Newton cradle with simultaneous impacts: left and right balls collinding at the same time.   Please let me know if you have any idea on where to find such experimental data.  Edit Just of few general remarks to avoid extending the comments.    Energy and momentum are conserved. However, they are sufficient to ensure uniqueness of the post-impact velocities only when there are 2 balls. The successive impact model precisely consists in propagating the impact ball after ball, and therefore leads to a unique solution (which conserves momentum and energy globally). From the waves point of view, changing the mass of a ball can be done by increasing the diameter, or the density (or a combination of both). This simple model cannot account for such subtleties of course, but maybe it does not matter when the balls are "small enough". Even if I think it's quite basic, following WetSavannaAnimalakaRodVance's comment, I am ready to hand out my Mathematica code to anybody who wants it. I'll check for good ways to share it. Submitting form elements with the same name I accidentally killed the orphanage lady before I started the quest. She just pissed me off, sadly it will not acknowledge that I have finished the quest and I haven't been contacted. Is there another way to join the Dark Brotherhood? 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? Deep learning model (LSTM) with temporal and non temporal attributes How to self study Linear Algebra I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is indeed the wrong forum for the type of question I'm about to ask, do please say so).  I am a 16 year old guy who is passionate about physics and as a result wants to increase his knowledge in mathematics, the language of physics. I've read and heard a lot about Linear Algebra and how crucial it is to physics and I am deeply motivated to self study this intriguing part of mathematics. However, I have limited knowledge of maths. I (for example) know basic algebra, trig, diff/int calc and some analytical geometry, but I wouldn't say I master these subjects past the high school curriculum. Now my question is: Would you kind people say I am able to self study Linear algebra or is it too tough and/or does it require too much of a math background? And are there any good books out there for BEGINNERS in LA? I found this (free) e-book called: Elementary Linear Algebra by Kenneth Kuttler and another one called 'Linear Algebra: Theory and its applications' also by Kuttler? Are these any good? Or would you recommend other books? If you guys have any tips regarding books for LA but also tips in general, please, I'd appreciate them! A complete ring with respect to an ideal I would like to know what does it mean to say " a ring $R$ is complete with respect to some ideal $I \subset R$.  Is it like, we define a metric by saying two elements in $R$ is "close" if the difference lies in a higher power of $I^n$ and $R$ is a complete metric space? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation. How can I use JUnit4 idiomatically to test that some code throws an exception?  While I can certainly do something like this:  @Test public void testFooThrowsIndexOutOfBoundsException() {   boolean thrown = false;    try {     foo.doStuff();   } catch (IndexOutOfBoundsException e) {     thrown = true;   }    assertTrue(thrown); }   I recall that there is an annotation or an Assert.xyz or something that is far less kludgy and far more in-the-spirit of JUnit for these sorts of situations. In the article class, it seems that using the abstract environment has an effect to the rest of the document. More precisely, it seems that it redefines the \linewidth, which equals \textwidth without abstract, is smaller with the abstract. My feeling is that \linewidth is redefined for the abstract environment, but the value is not reseted at the end of the abstract.  Here is a minimal example:  \documentclass{article} \begin{document} \abstract{This is an abstract. This is an abstract.}\bigskip  This is text. This is text. This is text. This is text. This is text. This is text.\\  \rule{\textwidth}{3em} \rule{\linewidth}{3em} \end{document}   The output is not the same if you comment the \abstract line or not.  Do you have the same problem as I do when you compile (if not, I will upload some PDF files to show)? Is this a (known) bug in LaTeX2e? How can we circumvent the problem?  Note: I compile with the pdflatex command, and I use pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian). Is an irrational number times a rational number always irrational?  If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number? What is behind the definitions of work and energy? I read a short story (I think, but it might have been a novel), where most races discover a very easy way to travel faster than light at a certain point in their technological development.   What I remember:   The guys in wooden ships bounced around subjugating the people without FTL travel. Earth had never figured out the FTL, and so had poured its effort in a different path, achieving present day technology. The aliens were freaking out about handing the humans FTL travel. The story was told from the point of view of the aliens. I read it in the last 10 years, but there is no telling how old it actually was, although I do remember them mentioning F-18s or F-14s. What is the intuition behind beta distribution? Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial for other people here. So I would be very thankful if answers included less specific terms and more explanation. Imagine that you are talking to your grandma :)  I'm trying to grasp the nature of beta distribution – what it should be used for and how to interpret it in each case. If we were talking about, say, normal distribution, one could describe it as arrival time of a train: most frequently it arrives just in time, a bit less frequently it is 1 minute earlier or 1 minute late and very rarely it arrives with difference of 20 minutes from the mean. Uniform distribution describes, in particular, chance of each ticket in lottery. Binomial distribution may be described with coin flips and so on. But is there such intuitive explanation of beta distribution?    Let's say, $\alpha=.99$ and $\beta=.5$. Beta distribution $B(\alpha, \beta)$ in this case looks like this (generated in R):     But what does it actually mean? Y-axis is obviously a probability density, but what is on the X-axis?   I would highly appreciate any explanation, either with this example or any other. Stop all active ajax requests in jQuery If my data is overdispersed when binomial but not when Poisson does that alone mean I should use Poisson GLM? Top is Poisson, bottom is Binomial Source of variation is different people observed over a week long period. 118 people to be exact.  want to make predictions about the proportion of days/week one exercises if they wake up at a given time.  &gt; summary(FMR7)  Call: glm(formula = No.Days.Ex ~ WaketimeStand, family = poisson, data = PSD)  Deviance Residuals:       Min        1Q    Median        3Q       Max   -2.99043  -0.64677  -0.03492   0.67393   1.51943    Coefficients:               Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)    1.36991    0.04689  29.218   &lt;2e-16 *** WaketimeStand -0.08309    0.04806  -1.729   0.0839 .   --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for poisson family taken to be 1)      Null deviance: 124.55  on 115  degrees of freedom Residual deviance: 121.51  on 114  degrees of freedom AIC: 479.71  Number of Fisher Scoring iterations: 4  &gt; summary(FMR6)  Call: glm(formula = cbind(No.Days.Ex, No.Days.NotEx) ~ WaketimeStand,      family = quasibinomial(link = logit), data = PSD)  Deviance Residuals:      Min       1Q   Median       3Q      Max   -3.7483  -0.9585  -0.0625   1.0428   2.9739    Coefficients:               Estimate Std. Error t value Pr(&gt;|t|)   (Intercept)     0.2595     0.1009   2.571   0.0114 * WaketimeStand  -0.1869     0.1020  -1.832   0.0695 . --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for quasibinomial family taken to be 2.015388)      Null deviance: 288.42  on 115  degrees of freedom Residual deviance: 281.57  on 114  degrees of freedom AIC: NA  Number of Fisher Scoring iterations: 4  Quite clear that there is much more overdispersion on the quasibinomial than the poisson. The response variable is no.days exercise per week and so my original thought was to analyse with binomial GLM as it is the proportion of days in a week someone exercises. However, when modelled with a Poisson GLM it is much less overdispersed. Is this a sign I am using the wrong distribution. How can I prove the Carmichael theorem I am trying to prove that these two definitions of Carmichael function are equivalent.  I am using this definition of Carmichael function:     $\lambda(n)$ is the smallest integer  such that $k^{\lambda(n)}\equiv1 \pmod n\,$ $\forall  k$ such that $\gcd(k,n)=1$.   to prove this one:     $\lambda(n) = \begin{cases}\phi(p^a) &amp; (p=2\text{ and }a\le 2)\text{ or }(p\ge 3)\\\dfrac 1 2 \phi{(2^a)}  &amp; a\ge 3\\ \text{lcm}\left((\lambda(p_1^{a_1}),\cdots,\lambda(p_k^{a_k})\right)&amp; \text{for }n=\prod_{i=1}^k p_i^{a_i} \end{cases}$   Since $a^{\lambda(n)}\equiv 1 \pmod n$,  and $a^{\lambda(m)}\equiv1 \pmod m$ with $\gcd(m,n)=1 \implies a^{\text{lcm}({\lambda(n)},\lambda(m)}\equiv1 \pmod {nm} $   Hence $\lambda (mn)=\text{lcm}(\lambda(n), \lambda(n))$ for $m$ and $n$ coprime to each other.  How can I prove the other relations? Writing a git post-receive hook to deal with a specific branch How to install updates via command line? I am being driven crazy by the fact that my media keys aren't working for iTunes. Such a pain! How can I fix this? I've tried the whole Chrome Extensions thing, that being I deleted all my extensions. Still, nothing has fixed this. Thanks to all who reply!